'j
Numerical Investigation of Turbulent Coupling
Boundary Layer of Air-Water Interaction Flow
by
Song Liu
Submitted to the Center for Ocean Engineering
Department of Mechanical Engineering
in partial fulfillment of the requirements for the degrees of
Master of Science in Ocean Engineering
and
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2005
o Massachusetts
Institute of Technology 2005. All rights reserved.
A u th or ...................................................
Center for Ocean Pngineering
Department of Mechanical Engineering
/
-August
29, 2005
C ertified by .........................
'L/Dick K. P. Yue
Professor, Center for Ocean Engineering
Department of Mechanical Engineering
Thesis Supervisor
A ccepted by ............................
Professor Lallit Anand
Chairman, Departmental Committee on Graduate Student
Department of Mechanical Engineering
MASSACHUSETTS INS
OF TECHNOLOGY
NO
E
7 2005
LIBRARIES
_
2
Numerical Investigation of Turbulent Coupling Boundary
Layer of Air-Water Interaction Flow
by
Song Liu
Submitted to the Center for Ocean Engineering
Department of Mechanical Engineering
on August 29, 2005, in partial fulfillment of the
requirements for the degrees of
Master of Science in Ocean Engineering
and
Master of Science in Mechanical Engineering
Abstract
Air-water interaction flow between two parallel flat plates, known as Couette flow, is
simulated by direct numerical simulation. The two flowing fluids are coupled through
continuity of velocity and shear stress condition across the interface. Pseudo-spectral
method is used in each flow subdomain with Fourier expansion in streamwise and
spanwise directions and finite difference in vertical direction.
Statistically quasi-steady flow properties, such as mean velocity profiles, turbulent intensities, Reynolds stress and turbulent kinetic energy (TKE) budget terms
show significant differences between air-water interface turbulence near the water
side (IntT-w) and wall-bounded turbulence(WT) while there are some similarities
between Int T-w and free surface turbulence (FST). Due to the velocity fluctuation
at the interface, water side near interface turbulence flow (IntT-w) is characterized
with a thinner viscous sub-layer and decreased intercept parameter B in log-law layer,
strengthened Reynolds stress and eddy viscosity, together with a stronger production
term, decreasing-then-increasing dissipation term and negative turbulent diffusion
term in TKE budget. Abundant physical phenomena exist on the water side turbulent flow with four major types of three-dimensional vortex structures identified near
the interface by variable-interval spacing averaging (VISA) techniques. Each type of
vortex structures is found to play an essential role in the turbulent energy balance
and passive scalar transport.
Thesis Supervisor: Dick K. P. Yue
Title: Professor, Center for Ocean Engineering
Department of Mechanical Engineering
3
4
Acknowledgments
First and foremost I would like to thank my supervisor Professor Dick K. P. Yue for
being the best advisor who is always trying to give me great help in my research. His
broad knowledge of this field has greatly benefited my research.
Special thanks to Professor Lian Shen for co-supervising me and giving me hand-on
instructions. Ideas and support have been provided by him throughout the course of
this thesis.
Much of my graduate student's time was spent in my office and interacting with fellow
students. I would like to acknowledge a few in particular: Dr. Yuming Liu, for giving
me very important inspiration and help in life; Dr. Kelli Hendrickson, Dr. George
Papaioanou, Areti Kiara and Dr. Guangyu Wu, for helping me a lot in my research.
Every discussion with them inspired my thoughts in many ways.
None of this would have been possible without the support and encouragement of
my parents and my wife, Min Jiang. I am so grateful for their love, support and
understanding.
Finally, I would like to thank the Office of Naval Research for providing funding for
this project.
5
6
Contents
23
1 Introduction
1.1
Introduction of air-water interaction flow . . . . . . . . . . . . . . . .
23
1.2
Wall and free surface turbulence flow . . . . . . . . . . . . . . . . . .
25
1.2.1
Wall turbulence flow . . . . . . . . . . . . . . . . . . . . . . .
27
1.2.2
Free surface turbulence flow . . . . . . . . . . . . . . . . . . .
28
Research on air-water interaction flow . . . . . . . . . . . . . . . . . .
29
1.3.1
Previous research review . . . . . . . . . . . . . . . . . . . . .
29
1.3.2
Development of simulation tools . . . . . . . . . . . . . . . . .
30
1.3.3
Our research objectives . . . . . . . . . . . . . . . . . . . . . .
31
1.3.4
Outlines of this thesis . . . . . . . . . . . . . . . . . . . . . . .
31
1.3
2
Mathematical formulation and numerical method
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
. . . . . . . . . . . . . . . . . . . . . . . .
34
. . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.3.1
Numerical schemes . . . . . . . . . . . . . . . . . . . . . . . .
36
2.3.2
Computational grid . . . . . . . . . . . . . . . . . . . . . . . .
39
2.3.3
Initial condition . . . . . . . . . . . . . . . . . . . . . . . . . .
40
Parallel computation . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
. . . . . . . . . . . . . . . . .
44
2.1
Problem statement
2.2
Mathematical formulation
2.3
Numerical algorithm
2.4
33
2.4.1
Spectral method parallelization
2.4.2
Parallel performance
. . . . . . . . . . . . . . . . . . . . . . .
45
2.4.3
Parallel environment . . . . . . . . . . . . . . . . . . . . . . .
47
7
3
Computational result of Direct Numerical Simulation:
results
51
3.1
Overview of numerical simulation and flow characteristics . . . . . . .
51
3.1.1
Initial flow condition and time evolution
. . . . . . . . . . . .
51
3.1.2
Quasi-steady state testification
. . . . . . . . . . . . . . . . .
55
3.2
3.3
3.4
3.5
4
Statistical
Flow property profiles and near boundary behavior
. . . . . . . . . .
57
. . . . . . . . .
57
. . . . . . . . . . . . . . . . . . . . . .
60
3.2.1
Velocity fluctuations and turbulence intensity
3.2.2
Mean velocity profiles
3.2.3
Reynolds shear stress and eddy viscosity
. . . . . . . . . . . .
62
3.2.4
Near interface behavior of vorticity fluctuation . . . . . . . . .
65
3.2.5
Passive scalar transfer
67
. . . . . . . . . . . . . . . . . . . . . .
Distributions of turbulent fluctuations
. . . . . . . . . . . . . . . . .
75
3.3.1
Skewness and Flatness . . . . . . . . . . . . . . . . . . . . . .
75
3.3.2
Probability density function of turbulent fluctuations
. . . . .
76
3.3.3
Joint probability density functions
. . . . . . . . . . . . . . .
84
3.3.4
Weighted function
. . . . . . . . . . . . . . . . . . . . . . . .
84
3.3.5
Correlation coefficients between turbulent fluctuations . . . . .
84
Two-point correlation and integral scales . . . . . . . . . . . . . . . .
92
3.4.1
Horizontal two-point correlation . . . . . . . . . . . . . . . . .
92
3.4.2
Three-dimensional two-point correlation
94
3.4.3
Spectrum and Co-spectrum of turbulent fluctuations
94
3.4.4
Integral scales . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
Turbulence transport budget . . . . . . . . . . . . . . . . . . . . . . .
103
3.5.1
Turbulence kinematic energy (TKE) budget
. . . . . . . . . .
103
3.5.2
Reynolds stress budget . . . . . . . . . . . . . . . . . . . . . .
107
3.5.3
Enstrophy dynamics
110
3.5.4
Budget for scalar transfer
. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
112
Computational result of DNS: Coherent Structures
115
4.1
115
Two-dimensional streaky structures . . . . . . . . . . . . . . . . . .
8
4.2
5
4.1.1
Low-speed and high-speed streaks . . . . .
. . . .
115
4.1.2
Distribution of streaky structures . . . . .
. . . .
118
. . . . . .
. . . .
120
4.2.1
Definition of vortex core in shear flow . . .
. . . .
122
4.2.2
Instantaneous coherent structures . . . . .
. . . .
123
4.2.3
Conditional averaging coherent structures
. . . .
129
4.2.4
Dynamic scalar transfer with structures .
. . . .
132
4.2.5
Air-water interaction near the interface .
. . . .
138
Three-dimensional coherent structures
155
Conclusions
9
10
List of Figures
. . . . . . . . . . . . . . . .
24
. . . . . . . . . . . . . . . . . . . .
24
1-3
Air-sea interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
1-4
Turbulence boundary layer profiles. . . . . . . . . . . . . . . . . . . .
26
1-5
Identification of coherent structures that are responsible for turbulence
1-1
Atmospheric water vapor concentration.
1-2
Scalar transport at the interface.
production near the wall. . . . . . . . . . . . . . . . . . . . . . . . . .
27
1-6
Hairpin vortex in the wall turbulent boundary layer . . . . . . . . . .
28
2-1
Definition sketch of air-water interaction Couette flow . . . . . . . . .
34
2-2
Subdomain-to-subdomain alternation . . . . . . . . . . . . . . . . . .
38
2-3
Computational domain and meshes. . . . . . . . . . . . . . . . . . . .
40
2-4
Two-point correlations:
streamwise separations.
(a) R,',';(b)R,'v,;
(c)R.'2'. Correlations are calculated at two different vertical positions
on the air side, corresponding to z = 0.007 and z = 0.993 respectively.
2-5
41
Two-point correlations: spanwise separations. (a) Re'a';(b)Rv'o'; (c)RWIWI.
Correlations are calculated at two different vertical positions on the air
side, corresponding to z = 0.007 and z = 0.993 respectively . . . . . .
2-6
42
Grid resolution validation: (a) mean velocity; (b) turbulent intensity.
--
, 64x 64x 96 x 2; -
, 128 x
128 x 128 x 2; o, 256 x 256 x 256 x 2.
All velocity components are normalized by shear velocity. . . . . . . .
43
2-7
Sketch of transpose method. .... ......
. . .
46
2-8
Parallel efficiency (comparing with original in-serial source. . . . . . .
48
2-9
High performance cluster in VFRL. . . . . . . . . . . . . . . . . . . .
48
11
. ...
. . . . ......
2-10 HPC cluster structure diagram. . . . . . . . . . . . . . . . . . . . . .
3-1
449
Time evolution of mean velocity ii profile on the (a) air side and (b)
water side; turbulence intensity q2 profile on the (c) air side and (d)
w ater side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-2
Quasi-steady state validation: (a) time-evolution of different flow properties; (b) time-evolution turbulent intensity in different flow regions.
3-3
54
56
Quasi-steady velocity fluctuation profiles: (a) velocity fluctuation profiles before normalization; (b) near boundary behavior of u'ms; (c)near
boundary behavior of
v'ms; (d)near boundary behavior of w'ms. Here
h+ refers to the normalized distance from each boundary. . . . . . . .
3-4
58
Quasi-steady velocity fluctuation profiles comparing with experimental
result at Re = 180 [27] and DNS results at Re = 194 [19].
. . . . . .
3-5
Mean velocity profiles: (a) Linear law region (b) Log-law region
3-6
Quasi-steady flow properties:
. . .
59
61
(a) overview of normalized turbulent
shear, mean shear rate and eddy viscosity; (b) near boundary behavior of mean velocity gradient; (c) near-boundary behavior of turbulent
shear stress; (d) near-boundary behavior of turbulent shear stress in
log-scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-7
Quasi-steady vorticity fluctuation profiles:
profiles; (b) near boundary behavior of w'
(a) vorticity fluctuation
n;
(c) near boundary be-
havior of woms; (d)near boundary behavior of w'rm8.
All vorticity
components are normalized by v/u*2 . . . . . . . . . . . . . . . . . . .
3-8
63
66
Time evolution of mean scalar c on the (a) water and (b) air side; Time
evolution of scalar fluctuation c'c' on the (c) water and (d) air side.
12
.
69
3-9
Quasi-steady scalar properties profiles: (a) mean scalar; (b) root-meansquare value of scalar fluctuation; (c) turbulent transport term; (d)
molecular and turbulent diffusivity. Comparison with molecular diffusivity: for Sc = 1.0, the molecular diffusivity is 1.14 x 10- 6m 2 S- 1 and
1.45 x 10- 5 m 2 S-
1
for water and air side respectively; for Sc = 4.0, the
value is about 2.85 x 10- 7m 2 s 1 and 3.63 x 10- 6m 2 S- 1 correspondingly. 71
3-10 Near boundaries behavior of quasi-steady scalar properties : (a)-(b)
scalar fluctuation profile on the (a) water and (b) air sides; (c)-(d)
turbulent scalar transport on the (c) water and (d) air sides. . . . . .
72
3-11 Near boundaries behavior of quasi-steady scalar properties log-scale:
(a)-(b) scalar fluctuation profile on the (a) water and (b) air sides;
(c)-(d) turbulent scalar transport on the (c) water and (d) air sides. .
73
3-12 Near boundaries behavior of turbulent scalar diffusivity in the water
and air side: (a)-(b) plotted in linear scale; (c)-(d) plotted in log-scale.
74
3-13 Skewness profiles of three velocity components on the (a) water and
(b ) air side.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3-14 Flatness profiles of three velocity components on the (a) water and (b)
air sid e.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3-15 Skewness profiles for the scalar fluctuations on the (a) water and (b)
air sid e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3-16 Flatness profiles for the scalar fluctuations on the (a) water and (b) air
sid e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-17 Skewness profiles for u'V, c'w' and L'
3-18 Flatness profiles for u'w', c'w' and
o'
78
on the (a) water and (b) air side. 78
on the (a) water and (b) air side.
79
3-19 PDF of (a) u'; (b)c'; (c)w' and (d)u'w'; (e) c'V'; (f)c'u' on the water
side at different horizontal planes. . . . . . . . . . . . . . . . . . . . .
80
3-20 PDF of (a) n'; (b)c'; (c)w' and (d)u'w'; (e) c'w'; (f)c'u' on the air side
at different horizontal planes.
. . . . . . . . . . . . . . . . . . . . . .
13
81
3-21 Conditional PDF of (a) u'; (b)c'; (c)w' and (d)u'w'; (e) c'w'; (f)c' '
on the water side near the interface at h+ = 10. 1-(u' > 0, w' > 0),
2-(u' < 0, w' > 0), 3-(u' < 0, w' < 0), 4-(u' > 0, w' < 0).
. . . . . . .
82
3-22 Conditional PDF of (a) u'; (b)c'; (c)w' and (d)u'w'; (e) c'w'; (f)c''u'
on the air side near the interface at h+ = 5.5.
1-(u' > 0, w' > 0),
2-(u' < 0, w' > 0), 3-(u' < 0, w' < 0), 4-(u' > 0, w' < 0).
. . . . . . .
83
3-23 Joint PDF of u' and w' at different horizontal planes on each fluid
side: (a) z=-0.001; (b)z=-0.019; (c) z=-0.27; (d) z=0.001; (e)z=0.019;
(f)z= 0.27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
3-24 Joint PDF of c' and w' at different horizontal planes on each fluid side:
(a) z=-0.001; (b)z=-0.019; (c) z=-0.27; (d) z=0.001; (e)z=0.019; (f)
z= 0.27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-25 Weight function of
86
V'W' on the (a) water and (b) air side near the
interface at different horizontal planes (z=-0.00112 and z=0.00112 for
water and air side respectively). . . . . . . . . . . . . . . . . . . . . .
3-26 Weight function of
1i''
87
on the (a) water and (b) air side near the
interface at different horizontal planes (z=-0.27 and z=0.27 for water
and air side respectively).
. . . . . . . . . . . . . . . . . . . . . . . .
88
3-27 Weight function of c'w' on the (a) water and (b) air side near the
interface at different horizontal planes (z=-0.00112 and z=0.00112 for
water and air side respectively). . . . . . . . . . . . . . . . . . . . . .
89
3-28 Weight function of c'W' on the (a) water and (b) air side near the
interface at different horizontal planes (z=-0.27 and z=0.27 for water
and air side respectively).
. . . . . . . . . . . . . . . . . . . . . . . .
90
3-29 Correlation coefficients of turbulent variables on the (a)-(c) water and
(d )-(f) air side.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-30 Two-point correlation of ', v',
91
w', c' in the x- direction with (a) z=-
0.001; (b) z=-0.28 on the water side and (c) z=0.001; (d) z=0.28 on
the air side.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
92
3-31 Two-point correlation of u', v', w', c' in the y-
direction with (a) z=-
0.001; (b) z=-0.28 on the water side and (c) z=0.001; (d) z=0.28 on
the air side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3-32 Specified 3D two-point correlation of c'(Sc = 1) and w': correlation
coefficient contours in horizontal planes with (a) z=-0.0034 and (b)
z=0.22; (c) correlation coefficient contours in the vertical Ax - z plane
with Ay = 0; (d) correlation coefficient contours in the vertical Ay - z
plane with Ax = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
3-33 Spectrum and co-spectrum in the x-direction near the interface on the
water side (z=-0.022) and air side (z=0.007).
. . . . . . . . . . . . .
96
3-34 Spectrum and co-spectrum in the x-direction on the water side (z=0.28) and air side (z=0.28).
. . . . . . . . . . . . . . . . . . . . . . .
97
3-35 Spectrum and co-spectrum in the y-direction near the interface on the
water side (z=-0.022) and air side (zz=0.007). . . . . . . . . . . . . . .
98
3-36 Spectrum and co-spectrum in the y-direction on the water side (z=0.28) and air side (z=0.28).
. . . . . . . . . . . . . . . . . . . . . . .
99
3-37 Taylor lengthscale profiles in the interaction flow. (a)-(b): lengthscale
in the x direction; (c)-(d): lengthscale in the y direction for air (a, c)
and water (b, d) sides respectively.
. . . . . . . . . . . . . . . . . . .
101
3-38 Macro-lengthscale profiles in the interaction flow. (a)-(b): lengthscale
in the x direction; (c)-(d): lengthscale in the y direction for air (a, c)
and water (b, d) sides respectively.
. . . . . . . . . . . . . . . . . . .
102
3-39 Turbulent kinetic energy (TKE) budget terms: (a) overview of TKE
budget; (b) near-interface amplification. All terms are normalized by
v/ l * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 4
3-40 Turbulent kinetic energy (TKE) budget terms: (a) Production term;
(b) Dissipation term; (c) Turbulent transport term; (d) Viscous diffusion term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
3-41 u'u' budget on the (a) water and (b) air side . . . . . . . . . . . . . .
108
3-42 v'v' budget on the (a) water and (b) air side. . . . . . . . . . . . . . .
108
15
3-43
'w' budget on the (a) water and (b) air side. . . . . . . . . . . . . .
3-44 u'w' budget on the (a) water and (b) air side.
. . . . . . . . . . . . .
109
109
3-45 Enstrophy dynamics of c'.: (a) near the interface behavior; (b) near
the wall boundaries.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
111
3-46 Enstrophy dynamics of w': (a) near the interface behavior; (b) near
the wall boundaries.
. . . . . . . . . . . . . . . . . . . . . . . . . . .111
3-47 Enstrophy dynamics of 2': (a) near the interface behavior; (b) near
the wall boundaries.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
112
3-48 c'c' budget on the (a) water and (b) air side. . . . . . . . . . . . . . .
113
3-49 c'w' budget on the (a) water and (b) air side . . . . . . . . . . . . . .
114
4-1
Time series of streaky structures at the air-water interface: (a) t=4000;
(b) t=4050; (c) t=4100.
4-2
. . . . . . . . . . . . . . . . . . . . . . . . .
Macro-lengthscale and Taylor lengthscale profiles in the interaction
flow: (a) Macro-lengthscale; (b) Taylor lengthscale.
4-3
116
. . . . . . . . . .
117
Distribution of streaky structures on air and water sides: (a) z = 0,
h+ = 0, at the interface; (b)h+ = 1.2 (z = 0.005 for the air side and
z = -0.01
for the water side); (c) h+ = 11 (z = 0.04 for the air side
and z = -0.1 for the water side. . . . . . . . . . . . . . . . . . . . . .
119
4-4
Indicators of streaky structures: (a) water side; (b) air side.
121
4-5
Vortex definition for numerical result of flow past wavy wall([59]): (a)
. . . . .
isosurface of vorticity; (b) isosurface of A2 . . . . . . . . . . . . . . . .
4-6
Isosurface of A2 on the water and air side: air-water interaction flow
with Re* = 120 and Re* = 271. . . . . . . . . . . . . . . . . . . . . .
4-7
124
Isosurface of A2 on the water and air side: Gas-liquid count flow with
R e' = Re* = 180( [36]).
4-8
124
. . . . . . . . . . . . . . . . . . . . . . . . .
125
Vortex structures on the air and water sides near the interface, and
scalar concentration on the vertical cross-section cutting through the
head portion of a hairpin-shaped vortex.
16
. . . . . . . . . . . . . . . .
126
4-9
Vortex inclination angles diagram:
(a) 0,z; (b)Oyz defined by O.,z
tan- 1 (OX/wz) and Oyz = tan 1 (ky/wz).
wz will be defined by Oz
tan-' (w'/w) based on vorticity fluctuation values.
. . . . . . . . . .
4-10 Histograms of vortex inclination angles, Oyz in the (y, z)Oxz in the (x, z)-
=
126
plane and
plane, at various distances from the interface.
. . .
127
4-11 Time-evolution of hairpin vortex near the interface on the water side
(vortex illustrated by iso-surface of A2 =-0.0008).
. . . . . . . . . .
130
4-12 Hairpin vortices in the conditional averaged VISA flow field of L':
(a)iso-surface of A2 =-0.003; (b) vortexlines.
. . . . . . . . . . . . .
133
4-13 Single and paired interface-attached ("U"-shape) vortices in the conditional averaged VISA flow field of wz. . . . . . . . . . . . . . . . . .
134
4-14 Quasi-streamwise vortices in the the conditional averaged VISA flow
field of w,: streamlines and iso-surface of A2 =-0.00034. . . . . . . .
134
4-15 Coherent hairpin vortex structures in the conditional averaged VISA
flow field of w': (a)iso-surface of A2 =-0.003 with w' at the interface;
(b) streamlines and turbulence production contours on the vertical
cross-section cutting through the hairpin vortex. . . . . . . . . . . . .
135
4-16 Coherent interface-attached vortex structures in the conditional averaged VISA flow field of wz: (a)interface-attached single and interfaceattached paired ("U"-shape) vortices with wz at the interface;
(b)
turbulence production contours on the vertical cross-section cutting
through U-shape attached vortices.
. . . . . . . . . . . . . . . . . . .
136
4-17 Coherent quasi-streamwise vortex structures in the conditional averaged VISA flow field of w,: (a)quasi-streamwise vortices (A2 =-0.00034)
with passive scalar transport rate Dc/Dz at the interface; (b) turbulence
diffusion associated with quasi-streamwise vortices.
. . . . . . . . . .
137
4-18 Vortexlines near the interface on both sides (Conditional average is
made by vertical vorticity at the interface). . . . . . . . . . . . . . . .
17
139
4-19 Vortexlines near the interface on both sides with the hairpin vortex
near the interface on the water side (Conditional average is made by
C'
4-20
near the interface on the water side).
. . . . . . . . . . . . . . . .
139
' in the horizontal planes at different vertical position with hairpin
vortex near the interface on the water side (Conditional average is
made by 2' near the interface on the water side). k refers to the grid
number away from the interface on each flow sides.
. . . . . . . . . .
140
4-21 w, in the horizontal planes at different vertical position with hairpin
vortex near the interface on the water side (Conditional average is
made by w' near the interface on the water side). k refers to the grid
number away from the interface on each flow sides.
. . . . . . . . . .
141
4-22 u' in the horizontal planes at different vertical position with hairpin
vortex near the interface on the water side (Conditional average is
made by w' near the interface on the water side). k refers to the grid
number away from the interface on each flow sides.
. . . . . . . . . .
142
4-23 v' in the horizontal planes at different vertical position with hairpin
vortex near the interface on the water side (Conditional average is
made by w' near the interface on the water side). k refers to the grid
number away from the interface on each flow sides.
. . . . . . . . . .
143
4-24 Dw'/Dz in the horizontal planes at different vertical position with hairpin vortex near the interface on the water side (Conditional average is
made by w' near the interface on the water side). k refers to the grid
number away from the interface on each flow sides.
. . . . . . . . . .
144
4-25 D2 1'/Dz 2 in the horizontal planes at different vertical position with
hairpin vortex near the interface on the water side (Conditional average
is made by w' near the interface on the water side). k refers to the
grid number away from the interface on each flow sides. . . . . . . . .
145
4-26 u' in the x - z vertical plane with hairpin vortex near the interface on
the water side (Conditional average is made by w' near the interface
on the water side).
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
146
4-27 w' in the x - z vertical plane with hairpin vortex near the interface on
the water side (Conditional average is made by w' near the interface
on the water side).
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
147
4-28 Ow'/Dz in the x - z vertical plane with hairpin vortex near the interface
on the water side (Conditional average is made by w' near the interface
on the water side).
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
4-29 Dw'/&z in the horizontal plane at different vertical position (Conditional average is made by velocity divergence at the interface, &w'/Dz).
k refers to the grid number away from the interface on each flow sides. 150
4-30 aw'/&z in the x - z vertical plane (Conditional average is made by
velocity divergence at the interface, Dw'/&z). . . . . . . . . . . . . . .
151
4-31 u' in the x - z vertical plane (Conditional average is made by velocity
divergence at the interface,
Ow'/&z).
. . . . . . . . . . . . . . . . . .
152
4-32 w' in the x - z vertical plane (Conditional average is made by velocity
divergence at the interface, Ow'/Dz).
. . . . . . . . . . . . . . . . . .
153
4-33 Isosurface of A2 near the interface on each side, showing splat effect.
(Conditional average is made by velocity divergence at the interface,
0 w '/ z). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
154
20
List of Tables
2.1
Difference of fluid properties between air and water. All fluid properties
([26]).
. . . . . . . . .
36
. . . . . . . . . . . . . . . . . . . . . . .
36
valued are given at 150C and 1.0 atm pressure
2.2
Computational parameters.
2.3
Parallel efficiency. Test case: air-water interaction flow with a mesh of
N, x N, x N, = 64 x 64 x 96 with FFT parallelized along y direction.
Time T is the computational time per 50 time steps.
3.1
. . . . . . . . .
47
Scalar transfer parameters in our computation comparing with physical
problem s.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
68
22
Chapter 1
Introduction
1.1
Introduction of air-water interaction flow
As a very common occurrence, air-water interaction flow is a problem of great importance in many industrial applications, such as in chemical engineering, the fluid flow
and heat transfer in gas-liquid contactor or evaporators. There are many environmental problems related to gas-liquid interaction flow ranging from pollution diffusion to
global ocean circulation (see figure 1-1). Research on sea-atmospheric coupled flow
is especially important in meteorology and marine engineering where more precise
climate and sea-state prediction is needed (figure 1-2-1-3).
Air-water interaction flow is also a problem of fundamental interest with very rich
physics. The turbulent air and water boundary layers, coupling through the exchange
of mass, momentum and heat at the interface, is still an open problem without clear
understanding. Most of the ongoing research projects in air-water interaction flow
are focused on,
" Turbulence transport in both air and water fluid domains, especially in the
air-water coupled boundary layer
" momentum flux, heat transfer, mass transfer such as the exchange of gases and
aerosols near and across the air-water interface
" Chemical and biological processes at and near the air-water interface
23
Figure 1-1: Atmospheric water vapor concentration.
Figure 1-2: Scalar transport at the interface.
24
. Wave dynamics and wave-turbulence interaction
* Experimental research such as sensing of surface winds, waves and temperatures, technology development for high resolution wave and flux measurements,
calibration of meteorological and oceanographic sensors
The coupled boundary layer of air-sea transfer is of particular interest in marine science and technology. In the oceanic and atmospheric wave boundary layers,
momentum, mass and heat transfer processes are the key factors affecting the local
weather and sea state. To understand the influence of ocean surface waves and the
three-dimensional fluid structures in these boundary layers, there is a nation-wide research program, the Coupled Boundary Layers Air-Sea Transfer (CBLAST) project
sponsored by Office of Naval Research (ONR). The research work within this problem
ranges from experimental study of remote sensing of fluid structures in the boundary
layer and ocean surface to direct numerical simulation of small-scale air-sea turbulent
flow, for low or high speed winds.
As part of the CBLAST program, our main research objective is to develop numerical simulation mthod of small-scale air-sea interaction flow and then, from the
numerical simulation results, to investigate the physical mechanism by combining the
turbulent kinetic energy budget, the momentum and mass transfer budgets with the
coherent structures in the coupled boundary layers.
1.2
Wall and free surface turbulence flow
Two problems are closely related with air-water turbulent interaction flow and provide
a solid base for our research. One is the turbulent flow near the wall boundary (WT),
the other is the free surface turbulence in open channel flow (FST). We will discuss
the flow phenomena in WT and FST in the following sections.
25
Figure 1-3: Air-sea interaction.
1.0
35
H
U
U
30
-
--
-
- ---
-
K- --
Imer regn
-
-
-
- -
Outer rtV an
U
Pate - Pi - (aro)
25
0.8
Lawo f
te WqIl
575I51 gi0
-
15
.+S5 5
Po'o law
-
116
10
-
0.4
-
-
vu
S
-
v
)
Blendingregon
5
0
S Laranarib-ayer
u77--y
2
Turbulmt Boundary Layer Pro alesl
4
3
10
Figure 1-4: Turbulence boundary layer profiles.
26
.Streaks
Z"
2. Burst
3. Swieep
Figure 1-5: Identification of coherent structures that are responsible for turbulence
production near the wall.
1.2.1
Wall turbulence flow
The wall bounded turbulence has been an active research area for about one-hundred
years. Comparing with air-water interaction turbulence flow(AWIT),wall turbulence
(WT) is relatively clearer as well known as turbulent boundary layer theory near
the plate or tube, burst-sweep mechanism of turbulence generation in the near-wall
region and the development of hairpin vortex ([37], [38], [18]).
In turbulent plate boundary layer, the inner layer can be divided into three layers:
viscous sub-layer, log-law layer and the inter layer which is a blending region between viscous sub-layer and log-law layer. From the Reynolds averaged N-S equation
(RANS) in the turbulent boundary layer, we can get the linear relationship between
velocity and vertical distance from the wall in viscous sub-layer, in log-law layer
where turbulence is the strongest, logarithmic relation is satisfied. See the turbulent
boundary layer profiles in figure 1-4.
Very organized coherent Structures can be identified which are responsible for
turbulence production near the wall. Contours of vorticity magnitude (see in figure 15) near the walls show the characteristic streamwise streaks of a turbulent boundary
layer. These streaks highlight streamwise-coherent structures that occur in the viscous
sub-layer very near a wall. The streaks are characterized by lower streamwise velocity
than the mean flow in the sub-layer (figure 1-5). As the steaks develop downstream,
27
Hairpin vortex near the wall
Figure 1-6: Hairpin vortex in the wall turbulent boundary layer.
they are lifted from the wall do to self-induction and the mean shear in a process
called an ejection. During the ejection phase, the streaks begin to oscillate eventually
leading to a rapid breakdown of the coherent structure with an increase in small-
scale, chaotic motion.
Immediately following a burst, high-speed fluid away from
the wall moves towards the wall to "sweep" away the fluid from the previous burst
event. Turbulence is burst through such an ejection-sweep mechanism.
Hair vortex
is developed from the initial structures (see in figure 1-6).
For high Reynolds number, there is strong shear existing in the near-wall region,
which is a similar problem with that strong wind shear blowing across the air-sea
interface. The similarities and differences between AWIT and WT will be one of our
main interests.
1.2.2
Free surface turbulence flow
FST is a new emerging problem comparing to WT study. Water-side free surface
turbulence investigation has been carried out by many studies in recent years, such as
open channel flow investigation with direct numerical simulation approach ([32], [25],
[6],
[48]), and the interaction between a turbulent shear flow and a free surface through
large eddy simulation ([55], [56]). These direct numerical simulation (DNS) and large
eddy simulation (LES) results provide detailed information on the statistical and
structural properties of the free-surface turbulence (FST) and have led big progress
28
towards its understanding and provide an appreciable base for air-water coupled flow,
although the calculation is limited to water side only and no wind shear is imposed
on the free surface.
Flow with a low shear rate boundary is characterized by "patchy" flow structures
near the boundary, caused by impinging eddies that flatten into a pancake shape as
they approach the boundary ([12]). For free-surface turbulence flow, as a limitation of
low shear rate flow, numerical simulation result shows that a dominant characteristic
is the presence of hairpin vortex inclined against the mean flow with head portion
near the free-surface and the two legs extending into the bulk region.
During the
free-surface attachment of hairpin vortex, decaying, stretching and merging of the
hairpin vortices or legs could be found ([55]).
1.3
1.3.1
Research on air-water interaction flow
Previous research review
Although air-water interface turbulence has much more abundant physical phenomena (turbulent kinetic energy transfer across the interface, structures in the coupled
boundary layer and wave turbulence interaction for a deformable interface, and so
on) than wall turbulence and free-surface turbulence, it has not received enough attention until very recent years. Considering the huge difference of length and time
scales between these two flow regions (as a result of the big fluid properties difference), the air-water interaction problem need more computational resources and is
more challenging for measurement techniques. Direct numerical simulation investigation is initially performed where the two fluids are decoupled from each other with a
mean-shear boundary condition imposed on the liquid side
vestigation
([49])
([32]). Experimental in-
and numerical result ([24]) of superimposed low wind stress on the
interface indicate that turbulence at interface is somewhat similar to wall turbulence
in some aspects which means that turbulence intensity and flow structures appear to
be dominated by the shear rate at the interface, rather than the coupling between air
29
and water structures.
To simulate the real interaction flow with two fluid domains coupled through the
interface, interaction between count-cross gas-liquid flow without the surface deformation is studied with direct numerical simulation approach ([36]). This investigation
provides a systematical view of the flow characteristics which includes the mean statistical turbulence properties near the interface region in both domains and the physical
mechanisms related to the coupling fluxes between two fluid phases. In a recent paper
([11]), the gas-liquid problem is calculated with a deformable interface for capillary
waves where Fr number is very small. The Reynolds numbers in these two studies
are quite low as general and an assumption of unrealistic flow density and viscous
coefficient ratios are made to keep Reynolds number the same on both sides. The
real air-water coupling flow remains open, especially for higher shear rates near the
interface. Also, the influence of density and viscous ratio remains to be a problems
for gas-liquid interaction flow.
1.3.2
Development of simulation tools
Most of the numerical results mentioned above are gained through DNS for low
Reynolds numbers, with shear based Reynolds numbers in the range of 100 ~ 200,
when the flow characteristics are relatively simple. In the application level, Reynolds
numbers are much higher when direct numerical simulation is too expensive to be
possible.
Then large eddy simulation (LES) becomes a promising approach. The
review on LES could be found in the papers of Moin ([39])and more recently in the
book of Sagaut
([53]). The development of LES for free-surface turbulence has been
limited until recently ([54],[58]).
From these studies it is clear that the effectiveness
of LES for free-surface turbulence flow would be enhanced if sub-grid scale (SGS)
models could capture the dynamic features of the flow.
30
1.3.3
Our research objectives
We will investigate the real air-water interaction flow with high shear stress on the
interface in this study. Our objective is to understand the coupled boundary layer and
turbulence or scalar transfer mechanism through DNS, by combining the statistical
fluid properties and coherent structure characteristics.
This will provide a better
understanding of the problem and benefit our future work to develop more effective
LES models for high Reynolds number air-water interaction flow.
Our objective
also includes developing numerical method for air-water coupling flow simulation and
computation code parallelization.
1.3.4
Outlines of this thesis
This thesis will be structured in the following way: problem statement, mathematical
formulation and numerical algorithm for DNS are described in chapter 2. In chapter 3,
we present the DNS statistically results, which include the mean velocity velocity,
turbulent kinetic energy, Reynolds stress and vorticity fluctuations, turbulent kinetic
energy budget, as well as the statistical vertical profiles of passive scalars.
Two-
dimensional streaky structures are investigated as well as three-dimensional vortex
structures in chapter 4. The coherent vortex structures are also investigated through
a conditional averaging technique with three kinds of vortex structures categorized
in the flow region near the interface. Finally, conclusions are drawn in chapter 5.
31
32
Chapter 2
Mathematical formulation and
numerical method
2.1
Problem statement
The physical problem to be simulated numerically is the air-water coupling flow between two infinite large parallel plate as that in Couette flow. A definition sketch is
shown in figure 2-1. The computational domain is split into two subdomains, the airside subdomain and water-side subdomain respectively. The coordinate axes x, y, z
point to the streamwise, spanwise and vertical direction with the origin located at
the center of the interface.
The flow is driven solely by the shear stress imposed by the movement of the top
plate sliding at a fixed streamwise velocity U, without any body forces or external
pressure gradient along the streamwise direction. Air-water two phase flow couples
across the interface which is generally deformable. As a first step in our research,
here the interface is kept flat as a physically realizable situation hence our research on
coupled shear flow is isolated from wave-turbulence interaction. The Froude number,
Fr, defined by Fr = U//g-h, is assumed to be zero which is reasonable in the limit
of small free surface deformation.
33
Z
Interface
Figure 2-1: Definition sketch of air-water interaction Couette flow
2.2
Mathematical formulation
The continuity equation and the Navier-Stokes equations are solved for each fluid.
Both fluids are incompressible at current fluid condition. Normalized governing equation for each fluid is in the same form as follows,
- 0.
(2.1)
and
1
a(ujuj)
aP
aui
+ RP
1
=uu
au,+
+1 Ox,
Ox2
a2U
U
Re OxOx'
= 1, 2, 3
(2.2)
For each flow side, all variables are normalized by the half width between the two
plates (depth of each fluid) and the constant velocity of the top plate U. As a result,
the Reynolds number is defined by
Re
Uh
V
(2.3)
where v is the kinematic viscosity of each fluid.
Along the edges in streamwise and spanwise direction, period boundary conditions are used in both direction. The simulated turbulent flow is fully-developed and
homogeneous in streamwise and spanwise direction. No-slip boundary conditions are
34
imposed at the top and at the bottom. At the interface, continuity of the velocity
and continuity of shear stress are required, written as, at z =
19a&
(2.4)
ay
(2.5)
"aD
PW
av",
az -P
IPW
0
=_w Ua
(2.6)
V=
(2.7)
a
(2.8)
= Wa
W =0
W
where p is the dynamic viscosity. In the statistical analysis of our computational
result, shear velocity is employed, named also as friction velocity in wall boundary
turbulent flow, u* and shear unit
l* to normalize the velocity and length-scale,defined
by
U* =
*
where
r/ p
T
(2.9)
(2.10)
v/U*,
O is the initial shear stress at the interface, same as the shear stress at the top
or bottom wall due to the equilibrium along the streamwise direction. Remembering
that there are big differences of fluid density and viscosity between two flow phases,
shear based Reynolds numbers Re*, defined by Re* =
u*hl/v, are quite different in the
two fluid subdomains. For real air-water interaction fluid flow, the shear Reynolds
number on the air side is more than twice of that on the water side,
Re*
-- _-
e .Va
-
Re*
Pa v,,
1
2.23
(2.11)
The shear Reynolds numbers are assumed to be the same on both flow sides
in some numerical researches ([36],
[11]).
Equation (2.12) need to be satisfied for
such an assumption, which is approximately right for oil-gas coupling flow while not
appropriate for air-water interaction flow. How the fluid properties such as density
35
and viscosity influence the flow characteristics is an open problem in stratified flow.
The subscripts
L ,G
in the equation above refer to liquid side and gas side respectively.
[tG
PL
(2.12)
1
/IL PG
Densityp(kgm- 3 )
0.999 x 103
1.227
814
Water
Air
Ratio(w/a)
v(m 2
1.14 x
1.46 x
7.81 x
s')
10-6
10-5
10-2
,(kgm- 1 s 1 )
1.14 x 10-3
1.79 x 10-5
63.7
Table 2.1: Difference of fluid properties between air and water. All fluid properties
valued are given at 15'C and 1.0 atm pressure ([26]).
Table 2.2 gives a list of the computational parameters based on the shear Reynolds
number on the water side. Direct numerical simulation is carried out for Re*
120,
responding to shear Reynolds number on the air side at Re* = 268.
Re*,
120
DNS
Re*
268
Re,
3657
Rea
9378
Grid size
128 x 128 x 128 x 2
At
0.01
Table 2.2: Computational parameters.
2.3
Numerical algorithm
2.3.1
Numerical schemes
To solve the Navier-Stokes equations and the continuity equation, fractional-step
method with approximate factorization technique ([17],
[34]) is used in order to solve
the implicit coupling between the continuity equation and pressure in the momentum
equation. Following equations need to be solved with different Reynolds number and
fluid density in air and water sides,
_
_-
_
1
_I
(3H
- H
(
+
+R
2
2
+S2
36
+
+
2
2
(
u)
(2.13)
2on+1
____+_
+
a2pn+-
Q2g/n+i
a21
-
At
1i
at
t,
(2.14)
(2.15)
x,
where fi is a temporary velocity, superscripts n
-
1, n and n + 1 refer to the
previous, current computational time step and the next time step. The Poisson's
equation of
# is solved
based on the residue of compressible "velocity" ft and then the
velocity of the new time step is corrected by equation (2.15) to satisfy the continuity
equation.
Pseudo-spectral method is used in the flow field in each subdomain to solve the
Poisson's equations of the intermit velocity iii and Pn+l.
Fourier expansions are
employed in the homogeneous plane, i.e., in streamwise (x) and spanwise (y) directions
while in the vertical (z) direction a second-order finite difference method is applied.
Numerical solution for Poisson's equation of fi by spectral method is given as an
example.
To solve the Poisson's equation of t(x, y, z, t)
92 it
(92
+
02&
Dy
2
+
02 i
(z
2 -
2Re~
At
= 0-
(2.16)
in the physical domain (x, y, z), we first transfer the Poisson's equation into the
spectral domain, (i, j, k, t) through Fourier expansion of ft and
N, /2
a =
(i, j, k, t)
Ny/2
Z(
'2
o
with
i27 r~2L J ."'
, m, k, t)e eN
N9.
(2.17)
=-Nx/2 m=-Ny/2
Equation (2.16) in spectral domain will be
dz2 (l, m, k, t) -
(27r N, -Ax) +
I = -N,/2
(2-rNy
-Ax )1
~ N,1/2 - 1, m = -Ny /2
i (, m, k, t),
~ Ny1/2 - 1.
(2.18)
The velocity U in spectral domain will be obtained by solving the tridiagonal system
along the vertical direction. To get the physical velocity ft the inverse Fourier trans-
37
DNS solving
(air side)
DNS solving
-MMMO (air side)
O
Air
Domain :r,
Dirichiet B C
Interface: z o
Neumann
s.C.
Water
Domain
DNS solving
(water side)
DNS solving
(water side)
t=t"
t = I'"
Figure 2-2: Subdomain-to-subdomain alternation
fer is needed. The period boundary conditions at the edges in the streamwise and
spanwise direction are easy to satisfied with this spectral method.
For air-water coupled problem, how to satisfy the the continuities of velocity and
shear stress at the interface is a big concern in the numerical calculation. The direct
way is to perform iterative procedure in each time step in order to satisfy both the
velocity and the shear stress boundary conditions ([24]). This is very time-consuming
and expensive because the complete velocity field in each subdomain need to be calculated for up to many times in each time step. To limit CPU time requirement, one
kind of time-split strategies is suggested that two subdomains are calculated separately during one time step ([36]). During each time step, the velocity continuity and
shear stress continuity are not satisfied simultaneously which will introduce a small
error depending on time steps. This subdomain-to-subdomain alternative calculation
strategy is also employed here in our research. Figure 2-2 gives the sketch of this
method. To make the numerical calculation more stable, for the water side where
the vertical velocity gradient is much smaller, Neumann type boundary condition is
applied at the interface with velocity gradient prescribed by latest air side result. In
air side, we use Dirichlet boundary condition with velocities at the interface given by
water side motion. By this kind of alternation, the boundary conditions are approximately satisfied simultaneously with a small error of At. This error will not lead
to instability which is proved by our result that interface shear stresses reach fixed
38
values with the decreasing of computational time step.
2.3.2
Computational grid
Computational domain and grid information is also list in table 2.2. For DNS computational with Re* = 120, The streamwise and spanwise computational domains
are chosen to be 27rh and 7rh (here h is the cross length of half domain) which is
about 754 and 377 in wall or interface shear unit for water subdomain, 1703 and 852
air side shear units. Mesh number is 128 x 128 x 128 in x, y, z directions respectively
for each subdomain. The grid spacings in the streamwise and spanwise directions are
AX+ = 5.9, Ay+ = 3.0 for water side and Ax+ = 13.2, Ay+ = 6.6 for air side in shear
units. Here superscript + values refer to the length scales normalized by shear units
1*.
Adaptive non-uniform meshes are used in the vertical direction with high grid
resolution near the wall and near the interface where the shear rate is very high, see
figure 2-3. Smallest grid size is satisfied the condition that the first grid point should
be kept within the wall turbulence viscous sub-layer. Grid convergence is validated
with three different grid densities. Mesh resolution is Az+ = 0.045 for water and
Az+ = 0.10 for air near each boundary. The maximum spacing, locating at the
center line of each sub-domain, are Az+ = 1.80 for water and Az+ = 4.05 for air all
in shear units.
The computational domain size is validated by the two-point correlations in figure 2-4 and figure 2-5 in streamwise and spanwise respectively (with low computational grid resolution of 64 x 64 x 96 x 2).
The computational grid resolution is
validated by the results in figure 2-6 and the spectrum of velocity fluctuation twopoint correlation in each horizontal direction.
Three different sets of grids are carried out with different resolutions as 64 x 64 x
96 x 2, 128 x 128 x 128 x 2 and 256 x 256 x 256 x 2. Comparison among the different
resolutions shows that the difference between the results on 128 x 128 x 128 x 2 and
256 x 256 x 256 x 2 mesh resolutions is small while the difference is much bigger
in coarser grid (see figure 2-6 for shear rate at the interface varying with the time).
39
0-
Figure 2-3: Computational domain and meshes.
Small time step At = 0.005 (At = 0.01 for mesh resolution 64 x 64 x 96 x 2) satisfies
the Cf number requirement for computation stability and ensures the dynamically
significant time scales are resolved.
2.3.3
Initial condition
Initial condition is given by velocity fields including mean streamwise velocity and
random velocity fluctuation. The initial averaged velocity profile in the z-direction
near the wall and near the interface is given by the wall turbulence boundary layer
theory with linear law in viscous sub-layer and log-law at above.
The calculation is carried out forward in time until the flow reaches statistically
steady state (quasi-steady state). The steady state can be identified by profiles of
flow properties such as the total shear stress and statistically steady turbulent kinetic
energy. Once the velocity field reaches the statistically steady state, the results are
integrated further in time to obtain a time average of various statistical variables. The
statistical samples are further increased by averaging over horizontal plan (homogeneous direction). In this thesis, we use - or <> (in figures) to indicate systematical
average over (x, y, t), and prime ' to indicate fluctuations from the average value.
40
(a)
1
- - - - - - Near interface
-Near wall boundary
0.5
0
'.',
-0.5
0.5
1
. '
. . - 1 - .
15
2
2.5
3
2
2.5
3
2
2.5
3
x
(b)
1
-t
0.5
A
0.5
0.5
1
1.5
x
(c)
1
0.5
0
-0.5
0
0.5
1
1.5
x
Figure 2-4: Two-point correlations: streamwise separations.
(a) Ro,,';(b)R,',';
(c)R'.-,. Correlations are calculated at two different vertical positions on the air
side, corresponding to z = 0.007 and z = 0.993 respectively.
41
(a) 1
- - - - - 0.5
Near interface
Near wall boundary
-a
-a
a
-
0
' ' ' '
-0.5
0.5
1.5
1
Y
(b)
1
0.5
0
-a
-
-
-1
-
-
0.5
1.5
1
Y
(C)
1.
0.5
0
-3
-
-
'
-0.51
'
'
0.5
15
Y
Figure 2-5: Two-point correlations: spanwise separations. (a) Re'a';(b)Ro'c,; (c)Rw'l.
Correlations are calculated at two different vertical positions on the air side, corresponding to z = 0.007 and z = 0.993 respectively.
42
(a)
(b) 1
0.5 -
0.5-
-0.5-
-0.5 -
-1
0
20
10
-1
0
30
U
1
2
Q
4
5
U u u
Figure 2-6: Grid resolution validation: (a) mean velocity; (b) turbulent intensity.
, 128 x 128 x 128 x 2; o, 256 x 256 x 256 x 2. All
-,64
x 96 x 2; x 64
velocity components are normalized by shear velocity.
43
2.4
Parallel computation
DNS need a large number of of computational grid and time step. Comparing with
RANS and LES, DNS is much more time-consuming and has higher requirement
on computer capacity. For three-dimensional DNS, the computational grid number
(N) required by DNS increases with Reynolds number by an order of 9/4, that is,
N
~' Re 9 / 4 .
In our calculation, more time is needed to get a quasi-steady state on the water
sides. The shear velocity has a big difference between two flow sides due to the same
shear stress at the interface and huge difference of fluid densities. The water shear
velocity is about 1/30 of the shear velocity on the air side. While shear velocity is the
changing speed of the flow property in the whole fluid domain, larger turnover time
is needed in the water subdomain with a lot of computational time "wasted" on the
air side. This is the main reason why we need much more time to get a quasi-steady
water side turbulent flow. Parallel computation on high performance cluster (HPC)
becomes our choice in our research.
2.4.1
Spectral method parallelization
There are two different ways to parallelize our numerical method. One is the parallel
solution of tridiagonal systems in vertical (z) direction where finite difference method
is employed. Cyclic reduction algorithm, usually being recursive, is practical to solve
single or block tridiagonal systems. Triadiagonal system or some banded system are
usually amenable to efficient parallel solution by iterative method. The second way is
to parallelize Fourier transform in spectral method which is used in our computational
code. We will discussion the parallel solution of Fast Fourier Transform (FFT) in
detail.
In applications such as the pseudo-spectral methods for solving partial differential
equations (PDE's), a number of multidimensional FFTs are computed per time step.
The speed of the FFT computation is therefore very critical to any large application
using the spectral method. Since such very large computations are feasible mostly
44
only on parallel machines, there is a need for fast multidimensional FFT algorithms
for parallel machines. There are two approaches that are possible, binary exchange
algorithm and transpose method.
The transpose method is used in the parallelization of our computational codes.
In this method, data are divided by planes among nodes. For example, in the three
dimensional transform, each node has a number of planes on which it computes two
dimensional FFTs. Next, a distributed transpose rearranges the data in such a way
that the FFT along the third dimension can be computed locally. The parallel aspect
of this approach is limited to the distributed transpose, which is equivalent to a
standard exchange problem. Here, each node sends data to and receives data from
all other nodes during distributed transpose.
This method is fairly easy and has
been implemented for a number of applications. Another approach, binary exchange
algorithm, is to design a distributed FFT algorithm which operates without collecting
planes.
In our 2-D FFT parallel program, the whole domain is split into slides along y
direction and each slide is solved by one processor.
The basic idea is, first, FFT
along x-direction are computed locally and then data need to be transposed in order
to carry out one-dimensional FFT in another direction (y-direction).
Since data
communication in the transpose process is very time-consuming, overall performance
of transpose algorithm mostly depends on particular implementation of all-to all
collective communication.
method.
Sketch in figure 2-7 shows the basic idea of transpose
During the transpose, we set Aij as the data need to be sent from
7th
process to J"h process. After transpose, FFT is continued in another direction.
2.4.2
Parallel performance
Parallel performance can be evaluated by the parallel efficiency or speed up. The
parallel efficiency is defined by e =
S/P, where S is speedup and P is the number of
processors. Theoretically speedup is defined by the express in equation (2.19), where
a c [0, 1] is the proportion of the parallelizable operation. Speedup S is less than
P due to a lot of reasons such as non-parallelization part of the problem, parallel
45
SendBuff A,
A3
A31
A21A22
I
A1J
I
ith
A32
o
A31
A2ranspose
All
712
A1
AO,
A02
Ao
Process
1DFT
403
A13
A23
A33
02
A12
A22
A32
A
A21
A
P
Aj7o AoA30
ith Process
1-D FFT
Figure 2-7: Sketch of transpose method.
overhead, higher numerical complexity, synchronization etc.
S =
1
1
(I - a) + a/P
(2.19)
Considering the influence of non-parallelization part only, speedup can be expressed as a linear function of P. Given total computation time T which includes parallel computational time Tp
=
fT and sequential computational time T, = (1 - f)T,
experimentally speed up can be calculated by
Spf = fP + (1 -
f),
(2.20)
where Spf is a linear function of P.
Parallel performance in table 2.3 shows a parallelizable operation proportion of
about 100% and speed-up of 4.032 with 16 processors (efficiency about 25.2%).
Note that T is much larger than in-serial calculation time due to the new parallel FFT source code is based on complex number comparing with Cosine Fourier
transform in original in-serial calculation code (real number). The result could be
46
Table 2.3: Parallel efficiency. Test case: air-water interaction flow with a mesh of
N, x N, x N, = 64 x 64 x 96 with FFT parallelized along y direction. Time T is the
computational time per 50 time steps.
np
np=1 (Serial)
np=1 (Parallel)
np=2
np=4
np=8
np=16
Memory (512M:100%)
40.8%
41.2%
22.0%
14.0%
8.0%
4.4%
T
1101s
2365s
1191s
707s
452s
274s
Speedup S
0.466
0.985
1.540
2.424
4.032
Efficiency e
-
46.6%
49.2%
38.5%
30.3%
25.2%
improved by parallel FFT for real numbers only. Besides FFT parallel, the precision
order for the
1 s'
and
2 nd
derivatives in y-
direction has great influence on parallel
performance. 6"' higher order is used in our calculation.
2.4.3
Parallel environment
The parallel computer cluster (High Performance Cluster) at vortical flow research
lab (VFRL) has 32 nodes with each node having two processors and 512M memory.
Data in each processing have communication or reduce through switches. Two kinds
of switch are used for data communication in our cluster, one is HP Ethernet switch
for data transfer between outside and cluster master node, the other one is Myrinet
fiber-optic switch with much faster data communication speed which is used for data
communication among computational nodes. The cluster system is shown in figure 2-9
with a diagram shown in figure 2-10.
For the parallel programming environment, MPI (Message Passing Interface) is
provided in vortical flow research lab.
47
80
8
6
0.
4
-A3-
-
-
-i----
Memory (512Mb=100%)
Speed-up
Parallel efficiency
--
-
-
60
^
40
a)
0
E
w
20 2
2
'
0
0
2
'-'-''
4
'
6
8
p
10
12
14
16
18
0
Figure 2-8: Parallel efficiency (comparing with original in-serial source.
Figure 2-9: High performance cluster in VFRL.
48
C
5'
.1'
Parallel Computer Cluster
Figure 2-10: HPC cluster structure diagram.
49
50
Chapter 3
Computational result of Direct
Numerical Simulation: Statistical
results
3.1
Overview of numerical simulation and flow characteristics
3.1.1
Initial flow condition and time evolution
The direct numerical simulation starts with the initial mean velocity profile given
by the wall turbulence boundary layer theory with linear law in the viscous sublayer and log-law at above in order to shorten the turbulence developing time for the
flow to reach statistically steady states. The Nikuradze logarithmic law, given by
equation(3.1) with parameters
K=
2.5 and B = 5.5, is employed near the interface
in each subdomain as well as near the wall boundary. Details are given as follows.
1
= -Inz+
l+
+ B,
K
f or
z+ > 30
(3.1)
Single-phase Couette flow on the water side with interface velocity of Ui1 t is first
considered. To be general, we assume that shear stresses (thus the shear velocities)
51
at the interface and in the bottom wall are different with each other, given as
Trt,
and
rbt respectively. Shear velocities and shear-based Reynolds numbers are given by,
Tt0 P =
P(u*op)'
=
bot2
Tbot
(3.2)
and
Rebot (3.3)
Re*
topv =
According to wall turbulence boundary layer theory, we can get an approximate
velocity profile as
_
N
1-
- mn*(bzo)
n
*
hRe*ot, 2.51n (h
bo)
+
.h
0
Re* , 2.51n (r-zRe* ) + 5.01
-
zot
-
Re*O+
>
forzh
Re+*0 .
(3.4)
In this problem, body force
f = (uot2
-
Utop2) /h exists in the flow to balance the
difference between shear stresses.
Following this way on both air and water sides and being aware that velocity need
to be continuous at the interface, we will get the profiles for air-water interaction
two-phase problem. For velocity it
U*, -in
z
on the water side, there is,
Re* 2.51n (
hRc* ) + 5.0]
U,,t - u* -min (- Re*,, 2.51n (
f or - 1 <
+ 5.0]
+Re*)
-
h
2
(3.5)
for - 1 < z < 0
For air side, the velocity profile is given as,
Ut
u*,
,t+
U* Min z Re*, 2.51n ( Re*) + 5.0]
f or0 < z < 1
U -Wu
- mnin h-z Re*, 2.51n h-zRe* +.0
for
a
)+50
fo'r!
<h-'<
1
(3.6)
u* and Uint in the two equations above can be expressed by U and Re*, Re*
through the continuity of shear stresses at the interface (see equation 3.7) and mass
conservation in each fluid subdomain (see equation 3.8 and 3.9). Equations (3.8) and
52
(3.9) are deducted assuming that log-law is satisfied in all flow regions above each
boundary considering that the linear viscous sublayer is much thinner than log-law
layer and could be neglected.
(3.7)
Pa
~2 2.5ln
(
U - Uine
~ 2 2.51n
L
Ua
) + 5.0
(3.8)
R
") + 5.0
2
(3.9)
Hereby, we have set the initial mean velocity profile based on the physical parameters of the air-water flow, flow property ratios pw/Pa,
ivw/Va
and Re* (or Re*). As
we have already mentioned in chapter 2, velocity components in the calculation are
normalized by the top plate speed U.
Small amplitude divergence-free velocity fluctuations are imposed upon the mean
velocity, serving as seeds for turbulence development.
Initial turbulence intensity
will have influence on the intermitted computational time before the turbulent flow
reaches full developed state while the final numerical results should be irrelevant,
which is only depending on the flow parameters.
Figures 3-1 (a)-(d) show the time evolution of the mean streamwise velocity and
turbulence intensity profiles on the air and water sides respectively with t = 0 corresponding to the initial condition. In view of horizontal homogeneity, these flow
properties are spatially averaged over the horizontal plane. As expected, energy is
extracted from the mean shear for turbulence production.
Turbulence intensity is
indicated by turbulence kinetic energy as,
q2 =
2
+v 2 +
2
.
(3.10)
Comparison between two fluid domains shows that long time taken on the water side
in order to reach quasi-steady state.
53
(a) 1
(b) 1
,,
Wa -
0
--------------------
Water
'--
i i
0
Water
1 0.0000
--
-- -- -- 4.0000
.40.0000
100.0000
200.0000
500.0000
-0.5|1
0
,
Air
Air
z
, ,
0.0000
40.0000
100.0000
2000.0000
3000.0000
4000.0000
0. 5
0.5 -
,
-1I
0.2
0.6
0.4
I
,
-0.5
I -
0.8
1
0.01
0
0.02
0.03
0.0
Mean Velocity
Mean velocity
(d)
(c)
0.5 -
0.5Air
r
-- - ~
z
z
Water
Water
0.0000
- -
--
-
-
-
-
-C
).5
- -
-
0.02
100.0000
0.06
2
2
200.0000
-0. 5
200.0000
0.04
4.0000
- 40.0000
500.0000
0
0.00
------
4.0000
40.0000
100.0000
2
0.018
500.0000
0
2E-05
4E-05
8E-05
6E-05
2
0.0001
2
Turbulent intensity: <u, +v, +w,2>
Turbulent intensity: <u, +v, +w, >
Figure 3-1: Time evolution of mean velocity ii profile on the (a) air side and (b) water
side; turbulence intensity q2 profile on the (c) air side and (d) water side.
54
3.1.2
Quasi-steady state testification
To get DNS converged statistical results, flow properties are first averaged over horizontal plane (in homogeneous directions). Then statistical samples are increased by
averaging in time range after the flow reaches quasi-steady state.
Statistically steady state is testified by mean velocity and fluctuating variables.
From figure 3-2, we can find that it takes up to 4000 non-dimensional time for the flow
to reach quasi-steady state. There are mainly two reasons for being time-consuming.
The first reason is that higher-order velocity fluctuation correlations, such as turbulence kinetic energy and eddy viscosity, need longer time to reach steady state than
lower order correlation, mean velocity for example, which is illustrated through flow
properties at the interface in figure 3-2(a). Here eddy viscosity Vt is defined by,
v
-U'W'
1-=m
(3.11)
Figure 3-2(b) compares the evolution of turbulence intensity at three different vertical
positions, z = -0.34 on the water side, z = 0 at the interface and z = 0.34 on the air
side. The turn-over time for turbulence development has big difference between two
fluid domains as analyzed in chapter 2, thus much longer time is needed on the water
side for the full development of turbulence. This is also shown by the time evolution
results in figure 3-1. In current DNS results, the statistics is taken from t = 4500 to
t = 7500 with 600 samples in this time range.
55
-(a)
<U>
<u' 2+v,2+w' 2>: z=-0.34
- Vt
- - - -. .
2000
1000
0
3000
5000
4000
6000
t
0.03
4E-05
()
(b)
C.)
co
3E-05
0.02
2E-05
0.0
1
C)
N
-1
0
E-05
0
a)
-
-0.01 !
0
.a
-0.02la
z=-0.34: water side
- -- -- - z=0: interface
-.-.-.-z=+0.34:airside
C..
C
a)
'5-1 E-05
40
I_2E-O5c
i
iI
i
1000
i
i
I
2000
i
I
3000
I i i
4000
i
i
i
i
5000
'
'
'
60
t
Figure 3-2: Quasi-steady state validation: (a) time-evolution of different flow properties; (b) time-evolution turbulent intensity in different flow regions.
56
3.2
Flow property profiles and near boundary behavior
3.2.1
Velocity fluctuations and turbulence intensity
Our interest is the statistically steady flow properties near the interface. Near the
wall boundary, the root-mean-square value of velocity fluctuation could be write as
equation (3.12) since velocity fluctuations are not permitted at the wall boundary.
The z-order behavior of the root-mean-square (rms) value of horizontal velocity
fluctuation and the z2-order behavior of the rms value of vertical velocity fluctuation
are expected from the no-slip wall boundary condition and the mass conservation at
the wall.
U'rms ~a, z + a 2z 2 + O(z 3 )
V'ms ~ biz + b2 z2 + O(z 3)
Wrms '-~
(3.12)
c2 z 2 + O(z 3 )
Near the free surface or interface, however, the scales of velocity fluctuations are
quite different from the wall interface, see equation(3.13).
r'__'
U rms
'
wrM,,
+ 11
~rmsz=O
ms
z+
1 -2 + 0(z 3 )
22 +
o+miz+n 2z 2 + 0(z
+ niz + 7n2Z2
~a
3
(3.13)
)
msz=O
For indeformable free-surface or interface (Fr = 0), there is wjms ~
O(z).
Figure 3-3(a) gives the velocity fluctuation profiles in each subdomain.
Near
boundary results, normalized by the shear velocities, are shown in figures 3-3(b) and
(c) with comparison between the four boundaries -
near interface and near the
wall boundary in each flow field. Computational result near the wall verifies the zbehavior of horizontal velocity fluctuation and z2 behavior of the vertical velocity
fluctuation.
The normalized horizontal velocity fluctuation on the air side near the interface
is very small although not zero (shown in figure 3-3 b).
57
On the water side, there
0.06
0.04
0.02
1
. '0.1
0.08
(b) 30
-v---
Wall (water, btmf)
Wall (air, top) 6
Interface (water),
Interface (air)
0.5
20
Air
----------- ~-------
C
z 0
P0?
Water
-0.5
I
-
I
-
-1
10
0.001
)
0.003
0.002
0.004
.
. 0.005
00
' '
(c)
I
...
I
U'
(d)
-
Wall (water, btmY
Wall (air, top)
Interface (water>
Interface (air)
P
-
4
3
2
1
velocity fluctuation
Wall (water, btm
Wall (air, top) Interface (water)
Interface (air)
20
20
h'
10
10
0
0.
' 1
0
15
5
05
2
w '/u
V'r,JU
Figure 3-3: Quasi-steady velocity fluctuation profiles: (a) velocity fluctuation profiles
before normalization; (b) near boundary behavior of ',ms; (c)near boundary behavior
of v'ms; (d)near boundary behavior of wms. Here h+ refers to the normalized distance
from each boundary.
58
-
4 --
.
--
A
o
V
a-
--
3
3
-A------ v--
U.
03
W'./U
Kreplin et al. Um,
Kreplin et al. v',
Kreplin et al. w'Kim etal.u
Kimetal.v'
Kim et al. wm
-
2
0
1
0
A
20
80
60
40
100
120
h+
Figure 3-4: Quasi-steady velocity fluctuation profiles comparing with experimental
result at Re = 180 [27] and DNS results at Re = 194 [19].
is a large value of horizontal velocity fluctuation (UrmsVrms) at the interface while
wrms is linear against z near the interface, which indicates a stronger vertical velocity
2
fluctuation than that near the wall boundary where w'ms - O(z ).
For near-interface behavior, the difference between water and air side is due to
dynamic boundary condition at the interface (see equation 2.4) and velocity normalization based on different shear velocities.
The shear velocity on each side has a
difference as big as 28.8 times due to big density ratio between air and water (see
equation 3.14). The interface seems like a wall boundary for air side while it is more
like a free surface for water side.
=
U*
"
Pw
1
28.8
.(
4
(3.14)
Our results are also validated by comparison of the water side (Re* = 120) velocity
fluctuation profiles with the experimental and numerical results ([273, [19]), shown in
figure 3-4. The Reynolds numbers locate in the same range as Re* = 100 ~ 200.
59
3.2.2
Mean velocity profiles
In the inertial part of turbulent boundary layers, referring to both wall boundary
layer and coupled interface layer in our case, the following dynamic equilibrium
equation(3.15) is satisfied on both air and water sides,
1j
-
=w'
(3.15)
pipu*2.
The viscous sub-layer exists in a very thin layer near the wall where
P
>> -p'u'w'.
(3.16)
In the viscous sub-layer, there is a linear relationship between mean streamwise velocity and wall distance, i.e.,
u+ = z+ (for z+ <~ 5 which is generally accepted).
The logarithmic law is satisfied above z+ > 40 in the outer layer.
In the log-law
equation 3.1, the parameter B is gained analytically by an assumption that there is
no transition region between the viscous sub-layer and log-law layer.
Figure 3-5 shows the mean velocity profiles near each boundaries. Again our computational verifies that for air side flow properties near the interface and near the wall
have the same characteristics. The linear layer exists for h+ < 5 and the parameters
in log-law are 1/K = 2.5 and B = 4.5 respectively. For water side boundary layer near
the interface, our results show a much thinner viscous sub-layer within h'
= 1 ~ 2
away from the interface. Figure 3-5(b) shows a modified log-law relationship near the
interface on the water side with the same sloop 1/K, however parameter B need to
be adjusted greatly from 5.0 to 2.5. This could be explained quantitatively through
the characteristics of Reynolds shear stress -u'w'
in the boundary layer. It has been
shown by Taylor expansion and by numerical results that normalized Reynolds shear
stress -uwf/u*
2
has a z- behavior near the interface on the water side which is much
stronger than that on the air side or near the wall. Thus it is not surprising a thinner
viscous sub-layer and a smaller parameter B will happen near the interface on the
water side.
60
(a)
10
...
....
- D-8
-
Walfwl
-Wall'a
-Interface(w
y'
Interface
--------
-
-----U+=h
-
-
6
4 -
01
10
8
6
4
2
h+
(b) 20
Walfw)
Wal(A)
15
0
------ -- ----
-
- -
Interfacei")
Interface*"
u+=2.5inh'+5.5
u=2.51nh'+3.0
U~ 10
5
0
20
40
60 80
h+
Figure 3-5: Mean velocity profiles: (a) Linear law region (b) Log-law region
61
From the viewpoint of shear stress scale, for non-normalized Reynolds shear stress,
there is
(-
-7.
~ (-Ti'').
vw (Oii/Oz)w << va (&U/&z)a
can be given from
dynamic boundary condition at the interface (equation 2.4), thus width of the viscous
sub-layer on the water side can be expected to be much smaller within which equation
(3.15) is satisfied.
Experimentally, increasing the roughness of the wall boundary will induce a smaller
B in log-law relation. Here it seems that the horizontal velocity fluctuations at the
interface have the similar effect as the wall roughness.
For a deformable interface
where vertical velocity fluctuation is also permitted as well as that in the horizontal
plane, we can expect a much smaller log-law parameter B and a diminishing viscous
sub-layer.
3.2.3
Reynolds shear stress and eddy viscosity
From Taylor expansion, the Reynolds shear stress -u'w' can be expressed as follows
near each boundary (for Taylor expansion analysis, z refers to the vertical distance
from each boundary),
z'w'
=
'w'1o +
au'W
OZ
0
Z+ 1
2!
Z2 + 1
(z 2
Z
3
z 3 + 0(z
4z 3
!
4
)
(3.17)
0
where
aju'w'
__/
a2
C9Z3
Uz
=
/
,&9 w'+
aZ
U~g
aZ3+
u'
- 2__'
+_
+3
,(9U
w-5 33
/
3
+3
(3.18)
+.
gu/ a3
Z3/
323
3a
y
( 9X+
e
With mass continuity equation (2.1), it's easy to find that
a''
-3
3!
=3W
z3 ±O(z4 )
(3.19)
az az3 0
at the solid wall boundary (for our case near the up and bottom plate).
At the
interface or free-surface, for Fr = 0 that -u'w' diminishes at the interface, the
62
(a) I
(b) 30
S- - (a)-<U'W'>/U2
-
-- d<u>*/dz+'*
1v)
D -
Wall (water, btm)
Wall (air, top)
Interface (water)
Interface (air)
0.5
-
20
Air
z 0
-- -----------------------
h
Water
10-
-0.5 r
00
1
0
0
2
1
4
3
05
0
5
1.5
1
d<u>'/dz*
(C)
(d)
30
--
- -
- - - - --
Wall (wate , bt
)
Wall (air,
Interface ( ate
r)
10
Interface(
20
------
Wall (water, btm)
Wall (air, top)
Interface (water)
Interface (air)
C
-
10
10
10
0
1
0.5
10,
15
-<u'w'>/u-2
10
-<U'w'>/U*2
101
Figure 3-6: Quasi-steady flow properties: (a) overview of normalized turbulent shear,
mean shear rate and eddy viscosity; (b) near boundary behavior of mean velocity
gradient; (c) near-boundary behavior of turbulent shear stress; (d) near-boundary
behavior of turbulent shear stress in log-scale.
63
Reynolds shear stress has a z-
-U'w'
behavior as
=
-
&w'
&u'
, 9 + WI'azI z
az
i9z 0
+ O(z 2 ).
(3.20)
Figures 3-6(b) and (c) verify the Taylor expansion analysis of near-boundary behavior
of Reynolds shear stress with a z- behavior near the interface on the water side and
z 3 behavior near the wall and near interface region on the air side. It is not surprising
that the interface acts like a wall for air side considering the air density is much
smaller than water density.
The near interface behavior of eddy viscosity, which is defined by the ratio of
Reynolds shear stress and mean velocity gradient, is given by equation (3.22) for
water side and (3.23) for air side. The same result of z 3 behavior can be gained near
the wall boundaries. Here the eddy viscosity is normalized by the molecular viscosity
on each side.
-''W
-"W'
Vt
- =
=
((3.21)
V
VaalaZ
U.2
_
-7)iw
-
(3.22)
0 (z)
(3.23)
((_3a
From the eddy viscosity profile in figure 3-6(a), we can see that in the air domain,
the eddy viscosity has a symmetric profile, while in water domain eddy viscosity decreases faster to zero near the interface than near the wall, which indicates a stronger
eddy viscosity near the interface on the water side. Away from the boundaries,
is supposed to be a linear function of z in the log-law region withVT
=
nU*Z.
VT
Peak
value of vT/v exposes to be about 20 in water domain and double to 40 in air domain.
The only reason for the difference shall be the different of shear Reynolds number
between water and air sides.
64
3.2.4
Near interface behavior of vorticity fluctuation
Kim et al. gave a systematic analysis of vorticity fluctuation near the wall boundary
for channel flow at Re* = 180 ([19]). For streamwise vorticity fluctuation
W'rms,
they
located a local maximum value of 0.13u* 2 /v at about z+ = 20 corresponding to the
average location of the center of the streamwise vortices. A local minimum value of
0.09u* 2 /v was shown at about z+ = 5 due to the opposite-sign streamwise vorticity
being created at the boundary of non-slip wall.
In our computational results, on the water side (see figure3-7(a),there are also
a peak value and a valley value near the interface existing at z+ = 5 and z+
1
respectively, which are both much closer to the boundary than the DNS results for
Re* = 180 ([19], z+ = 20 and z+ = 5 respectively). These results also indicate that
the streamwise vortice center becomes closer to the interface boundary. There could
be two possible reasons for the difference, strong shear and velocity fluctuation at the
interface.
For zero-shear free surface with small deformation, spanwise vorticity fluctuation
W'rms
decreases to zero at the free-surface ([55]). For wall boundary and high shear
air-water interface in our
case,
cbrms
reaches peak value at the boundary. Figure 3-
7(b) also shows a similar varying tendency of j',,s
with the streamwise vorticity
fluctuation, with a local maximum value and a local minimum value. This might be
attributed to the high-shear interface boundary properties. Dynamically, this boundary has a high shear rate with peak value reached at the boundary; kinematically,
fluctuation is permitted at the interface, which decreases the vorticity fluctuation to
some extent, just like what happens in the free-surface turbulence. For vertical vor([55]) shows that
ticity fluctuation w'rms, free-surface
Z turbulence research
2
wIus
= 0
az
at the free-surface which indicates a surface layer induced by the dynamic boundary
condition.
65
(b)
(a) 1
-
/
-
Wall (water, btm)
Wall (air, top)
Interface (water)
Interface (air)
-
0.5
20
--
Air
N
-
z
-
9
-
0
Water
-0.5 -
-1
10
p
0.1
0.3
0.2
0.4
0
0 .5
(C)
0.3
0.2
0.1
vorticity fluctuation
0.4
0.5
0)
(d)0
3
Wall (water, btm)
Wall (air, top)
Interface (water)
Interface (air)
c
C
-_---
20
20
10
10
Wall (water, btm)
Wall (air, top)
Interface (water)
Interface (air)
9
cC
0
0.1
0.2
0.3
0.4
~O
0.5
(I..-
0.1
0.3
0.2
0.4
C
OUz)
Figure 3-7: Quasi-steady vorticity fluctuation profiles: (a) vorticity fluctuation profiles; (b) near boundary behavior of w'rms; (c) near boundary behavior of wbrms;
(d)near boundary behavior of W' m. All vorticity components are normalized by
2
66
3.2.5
Passive scalar transfer
Our scalar transfer simulation chooses a boundary value problem in two-phase couette
flow.
Scalar concentration on the top moving plate is given by a fixed value (as
sources) and scalar is transported through the turbulent shear flow. The scalar in our
investigation could be any passive scalars.
The governing equation of scalar transfer in each fluid domain is given by equation(3.25),
where c is the concentration of passive scalar and D is the diffusivity of the scalar in
the specific fluid. The problem is comparable to heat transfer if energy dissipation is
neglected.
&c
at
+
D(u
c)
axj
C)
D
&2c
(3.24)
axjaxj
For each flow side, all variables are normalized by the half width between the two
plates, the constant velocity of the top plate U and the fixed scalar concentration on
the top plate Co. As a result, normalized scalar governing equation for each fluid is
in the same form as follows,
Dc
9t
&(ugc)
Oxj
&2 c
1
ReSc 9x, x.
1
Pe
9 2c
(x3.x2
where Re and Pe are the Reynolds number and Peclet number, defined by equations (2.3) and (3.26).
Schmidt number, defined by (3.27) is the ratio of viscous
diffusivity and scalar transfer diffusivity (It is called Prandtl number for heat transfer, defined by ratio of viscous diffusivity and heat diffusivity.).
Pe
Uh
D
Scm=
D
(3.26)
(3.27)
Two sets of scalar computational parameters, with Sc = 1.0 and Sc = 4.0 respectively, are chosen in our numerical investigation. We also make assumption that
Schmidt number is identified on both air and water sides. In physical reality, the
67
Schmidt number usually has big difference between the air and water side. Parameters of our simulation cases are compared with those in two physical problems, heat
transfer and vapor transport, in table 3.1.
D,(m 2 s 1 )
Casel: Sc = 1.0
Case2: Sc = 4.0
Heat transfer:
Vapor transport:
1.14 x 10-6
2.85 x 10-7
1.43 x 10-7
2.4 x 10 5
Da(m 2 s 1 )
1.45 x 10-5
3.625 x 10-6
2.2 x 10-5
1.32 x 10-5
Sc"
1.0
4.0
8.0
0.05
SCa
Scw/Sca
1.0
4.0
0.66
1.0
1.0
1.0
12
0.05
Table 3.1: Scalar transfer parameters in our computation comparing with physical
problems.
Figures 3-8 (a)-(d) show the time evolution of mean scalar and velocity fluctuation
profiles on the air and water sides with t = 0 corresponding to the initial conditions.
Initial scalar profiles are given by a similar way of velocity fluctuation in each side. For
mean scalar profile, it is given by linear relationship with vertical coordinate in each
subdomain, with boundary condition at the interface satisfied through continuity of
scalar and scalar transfer. Scalar statistics are taken during a time range after quasisteady state is reached which is also similar with velocity statistical procedure. Again,
similar with velocity field, the time evolution results show that it takes longer time
for scalar field on the water side to reach quasi-steady state than air side scalar field.
To short the developing time of scalar field, the scalar calculation is started from a
quasi-steady velocity field (quasi-steady reaches around t=1500).
Figure 3-9 (a)-(d) show the profiles of quasi-steady scalar properties which include
the mean scalar concentration, scalar fluctuation rms value, turbulent scalar transport (-c'w') and turbulent diffusivity of scalars. The turbulent diffusivity for scalars,
similar with eddy viscosity defined by turbulent shear stress and mean velocity gradient, is given by
Dt =
.1
(3.28)
The profiles for Sc = 1.0 and Sc = 4.0 are plotted together for comparison.
For both sides, higher Schmidt number (Sc = 4) is corresponding to a deeper scalar
profile near the boundaries (near the wall boundaries and near the interface). Again
68
(a)
(b)
'i
Air
Water
0.81
1
0.98
0.6
A
0.96
U
V
V
t=00
0.4
-
-.
-.. -.-
-
.-
. . .
. .,
-
0.94
-0.8
-0.6
z
(c
t=1000
t=2000
t=2500
.,
-0.2
-0.4
-t=500
-----------
I,
-1
-
---------
t=2000
t=2500
-------
0.2 -
t=O
-
8=500
=1000
-
0.2
0
(d)
0.01
Water
,..
I ,,I
0.4
z
0.6
0.008
--------
1
Air
8=0
0.15
0.8
t=500
t=1000
t=2000
t=2500
-
0.006
-A
0.1
V
-
/
---
0.05 J
-
-1
-- ---
--
-0.8
-0.4
-0.6
z
V
0.004
8=0
-
-
t=506
t=100
-t=200Qt
0.002
t--2500Q
-0.2
0
0
0.2
0.4
z
0.6
0.8
1
Figure 3-8: Time evolution of mean scalar 2 on the (a) water and (b) air side; Time
evolution of scalar fluctuation c'c' on the (c) water and (d) air side.
69
on the air side, mean scalar and scalar fluctuation show symmetric properties near the
wall and near the interface, while on the water side, the results are quite different with
a flatter mean scalar profile and a much stronger scalar fluctuation near the interface.
Turbulent scalar transfer profiles in figure 3-9 (c) show that higher Peclet number
(i.e., bigger Schmidt number and smaller molecular diffusivity) is corresponding to
a compressed scalar transfer in the vertical direction. Turbulent scalar diffusivity is
not depending on molecular diffusivity (Schmidt number), as shown in figure 3-9 (d),
which has been verified by turbulent analysis and other numerical results.
Figures 3-10 (a)-(d) show the near boundary behavior of the scalar fluctuation and
turbulent scalar transport on both sides. The vertical coordinates are plotted up to
h+ = 50 and h+ = 120 to cover all the boundary behavior. Same results in log-scale
are shown in figure 3-11. Figure 3-12 (a)-(d) show the near boundary behavior of the
turbulent scalar diffusivity, in both linear scale and in log scale.
With a similar Taylor expansion analysis as velocity fluctuation (see equation 3.29
for near wall region and equation 3.30 for near the interface region), we could get the
near-boundary behavior (z" orders, here z refers to the the vertical distance from
each boundary) of each statistical scalar property.
Crm
ez + e 2 z + O(z)
c'ns ~c'rmsIz=o-+fiz+f~z2 + O(z 3 )
(3.29)
(3.30)
All these near boundary behaviors are verified by our numerical simulation results in
figure 3-11 and figure 3-12(c)-(d). For scalar fluctuation, c'c', there is a relationship
of z 2 for near wall region. For near interface region, it is approximately kept as a
constant (c'c'I2=o).
The turbulent scalar transfer, -c'w', has an order of Z3 in near
wall region (also near the interface on the air side) and an order of about z 2 near the
interface on the water side. For turbulent diffusivity, the behavior is quite the same
as c'c'.
70
(a)q
0.99
0.98
0.97
0.96
15
-------- | Sc=1.0
-----Sc=4.0
---------
Sc=1.0
--------
Sc=4.0
0.02
-
-5
-
0.015
0.01
0.005
1
0.5
0.5
Air
Air
\
z0
z0
Water
'Water
-0-5
-
-0.5
-V
I
-1
0.6
0.8
6E-05
8E-05
0.4
0.2
I
2E-05
4E-05
0. 0001
(d.
0.004
0.002
----
---------- SC=1.0
-.-
---
-
.- Sc=4.0
-
I
-
0.15
0.1
C'
0.05
0
<C>
0.006
0.008
0.2
0.
Turbulent diffusivity: Sc=1
Turbulent diffusivity: Sc=4
diffusivity
S--- 'lecular
0.5 -
0.5
Air
Air
Water
Water
zo
~
.-
-0.5 -
-0.5
-'
2E0
E0
0
2E-05
4E-05
6E-05
E-09
00001
0
0.0002
0.0004
0.0006
0.0008
0.001
diffusivity
<C'M
Figure 3-9: Quasi-steady scalar properties profiles: (a) mean scalar; (b) root-meansquare value of scalar fluctuation; (c) turbulent transport term; (d) molecular and
turbulent diffusivity. Comparison with molecular diffusivity: for Sc = 1.0, the mole5 2
6 2 1
side
cular diffusivity is 1.14 x 10- m 8- and 1.45 x 10- m S-' for water and air
7 2 1 and 3.63 x 10- 6m 2 S -1
Sm
10x
2.85
respectively; for Sc = 4.0, the value is about
correspondingly.
71
(h
(a)0
'I
20
- ---- -
110
Wall (water, btm)
Interface (water)
40
-
Wall (air, top) 7
Interfaice (air)
100
-
90
,
80
30
-
-
0
70
h'
0
20
40
30
-
''..
10
50
20
'I-
10
o
0.015
0.01
0.005
0.02
0
2E-05
4E-05
6E-05
8E-05
0.00 01
<C'C>
<CVC>
(d).
(C)
Wall (w ter, btm)
Interf&1 (water)
-a---
-
Wall (air, top)
Interface (air)
110
100
40
0
80
30
-
1
-
70
G
h+>
0-
-
20
20
40
-
-
30
i , , , , , , ,
0
10
20
10
0
2E-05
4E-05
6E-05
BE-05
'II
0.0001
2E-05
4E-05
SE-05
BE-05
00001
<C'W'>
Figure 3-10: Near boundaries behavior of quasi-steady scalar properties : (a)-(b)
scalar fluctuation profile on the (a) water and (b) air sides; (c)-(d) turbulent scalar
transport on the (c) water and (d) air sides.
72
(b)
(a)
--
o-
- -
-
-
10,
Wall (air, top)
nterface (air)
ater, btm)
Inte ce (water)
Wall
10'
P
h+
10*
10
10,
101
10"
10,
1
04
(C)
10,
10,*
10,
<CVC>
<C'C>
(d)
- --
- -
Wall (air,
Wall (water, blm)
Interface (water)
--
a---
Interface(
t)
)
10,
10'
*i
-
10
10'
,0
-
100
10"
10*'
0w
'
10'
-7
-
10,
r
10'
I ti l sii rl
101
a i i i i ll0
10,
<C'w'>
Figure 3-11: Near boundaries behavior of quasi-steady scalar properties log-scale:
(a)-(b) scalar fluctuation profile on the (a) water and (b) air sides; (c)-(d) turbulent
scalar transport on the (c) water and (d) air sides.
73
Wall (air, top)
-
050
-- -o--
-
Wall (water, btm)
Interface (water)
110
-
(air, top)
Interface (air)
- -Wall
-4-
100
40
90
0
80
p
30
70
h')
-
20
40
30
10
20
10
0
5E-05
0.0001
0.002
0.0 002
0.00015
(C) -
'
---
C
--
-
Wall (water, btm)
Interface (water) ' .
10,
0.004
diffusivity
diffusivity
I ~I~
1)0
_+---
Wall (air, top)
Interface (air)
o
-1
0
0
10'
h*
h1'
i i i il lli i 111,
i i e1 11 7
10,
10,
10
10iu
t
d
diffusivity
C)
10-
10
10"
10
diffusivity
Figure 3-12: Near boundaries behavior of turbulent scalar diffusivity in the water and
air side: (a)-(b) plotted in linear scale; (c)-(d) plotted in log-scale.
74
3.3
3.3.1
Distributions of turbulent fluctuations
Skewness and Flatness
Skewness is an indicator of asymmetry in a distribution while flatness is the relatively
flat appearance for a probability distribution. In our numerical simulation, for each
horizontal plane, their definitions are given as follows with velocity components "u"
as an example.
/=
F,1 =
U it/3(3.31)
U'm
1(3.32)
Skewness of a symmetric distribution is zero and flatness of the Gaussian distribution is exactly 3. Figure 3-13 and figure 3-14 show the skewness and flatness profiles
of three velocity components in each fluid side.
On the air side near the boundary, our numerical results show agreement with the
results in wall turbulence (experimental results, [43]; numerical results,
[461, [13]).
Streamwise velocity fluctuation is positively skewed with a highest skewness factors
of about 1 near the interface while u' is highly negatively skewed near the upper wall
where the air flow is stretched by the viscous shear stress. Vertical (wall normal)
velocity fluctuation w' is moderately skewed near the boundaries which relates to
the ejection process. Spanwise velocity fluctuation is slightly skewed on both air and
water side. For flatness, bulk flow on the air side showing a Gaussian distribution
indicates that the turbulent flow is more isotropic. Affected by the interface, flatness
factors at the interface are slightly smaller than those near the upper wall boundary
(see figure 3-14).
Near the interface on the water side,
W' is less flatted than the wall boundary and
horizontal velocity fluctuation flatness factors are trivial, similar with free-surface
characteristics ([46]). The huge positive skewness near the bottom wall on the water
side can't be well understood. Since large turbulence events such as sweeping, ejection
and splat directly affect the vertical velocity fluctuation, skewness and flatness of w'
near the interface would be very important in the investigation of turbulent transport
75
(b)
(a) 0
0.9
su
-- SV
su
-
- - - Sv
sw
------
0.8
SW
-0.2 -
0.7
0.6:-
-0.4-
0.5
0.4-
-
-0.6-
I
0.3
-0.8-
0.2 -
-
It
.
0.1
-2
-1
0
Skewness
1
-2
2
-1
0
Skewness
1
2
Figure 3-13: Skewness profiles of three velocity components on the (a) water and (b)
air side.
and scalar transfer near the interface.
Figure 3-15 and figure 3-16 show the skewness and flatness profiles of two passive
scalars at different Schmidt number. Numerical results ([9]) shows that the concentration field of scalar directly affected by the largest event which also affects the vertical
velocity fluctuation. In our numerical results, however, this is not very clear.
Figure 3-17 and figure 3-18 show the skewness and flatness profiles of Reynolds
stress u'w', vertical scalar turbulent flux c'w' for Sc = 1.
u'w' and c'w' are both
negatively skewed in bulk flow, especially near the wall boundary. Again the results
show significant difference between the interface and the wall boundary on the water
side.
3.3.2
Probability density function of turbulent fluctuations
Probability density function show another way to look at things like fractional contributions and sknewness shown above can be well illustrated by the probability density
function of each turbulent fluctuation ([43], [47]).
Figure 3-19-3-20 show the probability density function of a variety of turbulent
fluctuations each normalized by its rms value. Shown in the same picture is the
76
(a)
(b),
I||
- -------
-0.2
FU
FV
FW
FU
-- - FV
fr------- FW
0.8
-
-
IiI
-0.4
-
z
I
0.6
z
-
0.4
-0.6
I..-
- I
0.2
-0.8
5
~'C
10
Flatness
15
0
0
10
5
15
20
Flatness
Figure 3-14: Flatness profiles of three velocity components on the (a) water and (b)
air side.
(b)
(a)
----- ---0.2
Sc=1
Sc=4
-
-
0.6
-0.4
z
-1
4
-
-
-
z
-/
-
Sc=
-
0.8
0.4
-0.6
.
I-
-
-
-0.8
-2
-1
0
0.2
I
-1
1
-
0
1
2
Skewness
Skewness
Figure 3-15: Skewness profiles for the scalar fluctuations on the (a) water and (b) air
side.
77
(b)
(a)
SC=4
-------
-----
-Sc=1
-
-
Sc=1
Sc=4
0.8
-0.2
-
*
i-
0.6
-0.4 .
z
z
0.4
-0.6 .
-uw
0.2
-0.8-
-1
-
2
4
;
0. .. ...
1
8
6
t ;
2
Flatness
3
Flatness
5
4
Figure 3-16: Flatness profiles for the scalar fluctuations on the (a) water and (b) air
side.
(b)
---- ---
-------
SC1W
suw
SC1w
0.8
-0.2
0.6
-0.4
z
z
w|
.- ai
-0.6
0.4
0.2
-0.8
-31'5
-10
-5
0
5
915
-10
-5
-5
Skewness
Skewness
Figure 3-17: Skewness profiles for u'w', c'w' and 2
78
0
0
5
on the (a) water and (b) air side.
(a) 0
(b)
---------
-
FUW
FC1W
---------
FUW
FC1W
0.8-
-0.2-
0.6-
-0.4z
z
-0.6-
0.4-
-0.8-
0.2-
0
10
20
30
40
0
50
Flatness
10
20
30
40
50
Flatness
Figure 3-18: Flatness profiles for u'w', c'w' and w on the (a) water and (b) air side.
Gaussian normalized distribution. Results at different vertical planes are drawn together to show the pdf variation along the normal direction. Shapes of p.d.f.s of u',
c' and w' change greatly from near interface region out to the bulk flow region and
become more and more similar with Gaussian normalized distribution. P.d.f.s shapes
of U'w', c'w' and c'u', however, change little within a broad region near the interface
with a long tail in the negative side for u'w', c'w' and a long positive tail for c'u',
which marks the momentum transfer and scalar transfer in the flow region near the
interface.
Besides probability density function on several horizontal planes with different
distances from the interface, we also show the conditional pdf results from different
to one
(u', w') quadrants (see figure 3-21-3-22), with each quadrants corresponding
kind of flow event (coherent structure). Quadrant u' > 0, w' > 0 corresponds to
outward interaction (splat) motion, u' < 0, w' > 0 ejection, u' < 0, w' < 0 towardwall fluid motion and u' < 0, w' < 0 for sweep motion (see [43]).
79
(d)
(a)
0.8
0.8
- ----- *11
0.6
2
-
j --i
- -
2
- -
-
-
1
- 0.6
-
-
e0.4
L L-
- - - - -- a - ----- ---- - " "
0.4
---- ----
-
-
- -
-- ---
r--
--
--
-
--
r
I
SI
N
- -
- ---
-
- --
-
-.
0021
0.2
0.2
CI
-2
-4
-
*--4
4
2
2
-2
(e)
(b
Gaunman
-I--T
--------- --------
0.00112
i"- - -- -- -0.002610
0.019450
S
.--
0.8
0.8
- --
r---r--------
--------
0.6
0.6
-+-+-
0.4
0.4
- ----- ---
0.2
0.2
0
-- -- - -----. - -
0.271620
1 ,
-----------
--
-
--
-
7 1 - --
- -,r - -
-
--- --- -- - - -
p
-24
-2
2
2
4
(f)
(c)
1
i
---------
-
--
GTmaa
-
-
0.8
- - --
-
-7-
--
--
-
--
-
- -- .- -
-T - -
- -iT
0.8
42142
.
--
-
-
.00112o
-0.002610
- - --
0.27162 0
G
--..
..
0.6
--
J....J
--
-
-
- -
-
0.6
-
1
0.4
- -- -
-
--
.0
12
0.4 r
----------
0.2/
--
-T
r-------
---
-
--
r--r--r--
--
r--
0.2
0
-4
2
p
2
"'4
4
-2
P
2
4
Figure 3-19: PDF of (a) u'; (b)c'; (c)w' and (d)u'w'; (e) c'w'; (f)c'u' on the water side
at different horizontal planes.
80
(b)
(a)
-
--
0.020
---------- I--
0.8
0.8
-- - --- --' -
+
0.6
-+ --+ -+
-
---
- ---
0.00B1r
0.4
0.2
0.2
i
p
-2
-4
2
-
- - - --- - - -
0.6
0.4
fl
-
-4 ---- --4-- ------
/7
T
4
2
-2
-4
4
i i 0 '/
i
(a
(a)
- - -- - -
-
0.8
08
-
- - - --.- 6-
-r -- --
G------ ---
- ---------- --- - - -- -
-"
0
0.6
:
0
-----
-T
TT
0.4
-:.,1:J
------
4
2
0.2
0.2
4k
$
2
-2
4
04
4
4
2
-2
(a
(a)
0.8
0.8
0.6
0.6
Ti~
-L
--
e
0.4
- 1-
Iy|- -
S
~
L
J-
I
I |
k--
0.4
0.2
0.2
-4
I
_
-2
..
Q
"-4
2
-2
R
2
4
Figure 3-20: PDF of (a) u'; (b)c'; (c)w' and (d)u'w'; (e) c'w'; (f)c'u' on the air side
at different horizontal planes.
81
(a)
(d)
1.
-7
-
0.8-8
-.
--.
4101003
0.8
~
-0
0. 6-
-
-
01
0.6
-.
0. 40.2 -
0.101302
G
150
0.010
0.4
0.2
-.
//-
.
I
1 91
2
-2
' -4
2
-2
(e)
(b)
0.8
0.8
0.6
0.6
0.4
010
-
-
0.4
0.2
0.2
-4
-0110
-.
s
-
-7
-2
2
-
-4
4
(c)
-2
2
(f)
1r1
0.8
0.8
0.6-
0.6
0.4-
0.4
0.2-
0.2
0I
-4
-2
0
2
0-4
4
..~....4.lO1000
-
-2
2
4
Figure 3-21: Conditional PDF of (a) u'; (b)c'; (c)w' and (d)u'w'; (e) c'w'; (f)c'u' on
the water side near the interface at h+ = 10. 1-(u' > 0, w' > 0), 2-(u' < 0, w' > 0),
3-(u' < 0, w' < 0), 4-(u' > 0, w' < 0).
82
(d)
(a)
0.8
0.019433
-
0.8
-
0.6
0.6
0.4-
0.4
0.2-
0.2
1
*
.4.- '---
-:4
p
-2
I~
*
0-4
2
~1~
-2
*40'
. "i
r
4
2
(e)
(b)
0.8
0. 8
0.6
0. 6-
0.4
0.4
01~5*
-.
f
0.2
0 .2-
44
.---....,-
-4
2
-2
'4
0-4
-2
2
-
(C)
(f)
1
0.8
0.8
0.6
0.6
0.4
0.4
- *N
0.2
I)
-''4
I;
0.2
44
I'
- 07-
Q4
0'-4
2
-2
R
2
4
Figure 3-22: Conditional PDF of (a) u'; (b)c'; (c)w' and (d)u'w'; (e) c'w'; (f)c'u' on
the air side near the interface at h+ = 5.5. 1-(u' > 0, w' > 0), 2-(u' < 0, w' > 0),
3-(u' < 0, w' < 0), 4-(u' > 0, w' < 0).
83
3.3.3
Joint probability density functions
Figure 3-23 and figure 3-24 show the joint probability density function of (u', w'), (c',
w') at different horizontal planes on the water and air side. Our numerical results show
significant different in the near interface region (z = -0.019)
([43]).
and theoretical prediction results near the wall
In the (u', w') joint p.d.f.
with the experimental
results in figure Figure 3-23 (a)-(b), contours are
concentrated in two regions, corresponding to a positive and negative u' respectively.
This indicates the effect of spat which including the ejection part and sweeping part.
Considering an imposed flat interface boundary condition with no vertical velocity
fluctuation permit at the interface, the contours are only concentrated on zero vertical
fluctuations (w').
3.3.4
Weighted function
Figure 3-25 and figure 3-26 show the distributions of weighted function for Reynolds
stress u'w' at two horizontal planes. Figure 3-27 and figure 3-28 show the distributions
of weighted function for scalar flux c'V'. Like the conditional p.d.f.s discussed above,
these weighted function distributions are made by statistics upon velocity fluctuation
U' and w'.
3.3.5
Correlation coefficients between turbulent fluctuations
The correlation coefficients (as a function of vertical coordinate z-) of two turbulent
fluctuations a' and b' are defined as,
ra/
a' b'
amb.
(3.33)
Figure 3-29show the profiles of the correlation coefficients between a variety of
turbulent fluctuations.
of a' - w', w'
-
Among them, (a) and (d) show the correlation coefficient
' and w'
-
correlated with i', a', w','w' -
+ ay , (b) and (e) the correlation coefficients of c'
,
u + '),
84
(c) and (f) the correlation coefficients
(d)
(a)
2
~>
~~rs-
(
1
1
//
y
y
-.1
-10
9/
-9
2
p
-1
-
\
1
2
1
2
(e
(b)
fK$
2
u* 0
K
~
LL 0
-1 -
-1
~
N
-2
(c)
----
-1
-
/
1
2
2
-1
-1
H
1
1
2
(f)
2
2
u0
1
L-
\
2~2
0
1
-1
-1
0-
-1-
1
.J
1
2
1
2
Figure 3-23: Joint PDF of u' and w' at different horizontal planes on each fluid side:
(a) z=-0.001; (b)z=-0.019; (c) z=-0.27; (d) z=0.001; (e)z=0.019; (f)z=0.27.
85
l;
(d)
(a)
2
1
1
V
'I
L 0
-1
/
1'
N
-1
V.
'I,
/
N~
2
-1
-
1
2
2
(e)
(b)
2
1
1
LI 0
2
)
K
-
*/)
-1
-
-
i2 0
I
-
-1
U'
,
- -
/~
-
-2'-2
-1
(f)
(c) 2
U-
-1
0
F
1
2
1
2
2
0
1 -
1
-2
2
-2-1
Figure 3-24: Joint PDF of c' and w' at different horizontal planes on each fluid side:
(a) z=-0.001; (b)z=-0.019; (c) z=-0.27; (d) z=0.001; (e)z=0.019; (f) z=0.27.
86
(a)
(b)
WFUW
WFUW
j
2
--
-
.
-.
-- -- - -
-
------.--.
-
1
0
--
0.036
0.020
-
- --------------
-----
-
0.004
-0012
-0.028
-0.044
-0.060
- - --------
-1
-1
---
--- - ------------------------
- --
- --
-2
-3
---------
2
- -- --- ----
-- -, -
0.052
3
0.052
0.036
0.020
0.004
-0.012
-0.028
-0.044
-0.060
------ ---
---- -- --
-2
2
0
-2
-2
U
0
2
U
006
8
0
-340
+3
-
-2
-
:
-
.2
0
2
00
2
2
*
on the (a) water and (b) air side near the
Figure 3-25: Weight function of u'
interface at different horizontal planes (z=-0.00112 and z=0.00112 for water and air
side respectively).
87
(b)
(a)
WFUW
WFUW
-
-
0.087
-0.060
0 60
00
2 ---------2-00007
1
-0.087
2
-----
--
- ---
0.033
- --- --- -- -----
-0020
-O.G47
-0073
-
0.007
-0.020
-0.047
-0073
-0100
-0.100
-- - -
-- - -- -
-3
-2
-2
0
-2
2
0
2
U
U
Z2
0,.
.0
-005
-005
Figure 3-26: Weight function of u'w' on the (a) water and (b) air side near the
interface at different horizontal planes (z=-0.27 and z=0.27 for water and air side
respectively).
88
(b)
(a)
WFUW
WFC1W
- ------------------------------
2
1
3
0.027
3
--.---- ---------
--- - ----------
0.027
0.000
-0.027
0.000
2 ------ I ----------------------- - -
0.053
0.080
------
-0.053
-0.080
-0.107
-0.133
-0.160
-0.107
-0.133
-0.160
0
-1
0 ---
---------
-
-
-
-1
-2
- -- ------------- ----------- --- --
0
-2
2
0
-2
- -0
----- ---
U
-
------
2
005
~2
2
2
4
-3
-2
-2
0
2
1
22
22
Figure 3-27: Weight function of c'V' on the (a) water and (b) air side near the
interface at different horizontal planes (z= =-0.00112 and z=0.00112 for water and air
side respectively).
19
(b)
(a)
WFUW
WFC1W
0.027
0.027
0.000
2-
1
-----
-
-- -------------- -------
- -- - - - -
-
-- - -- - -----
------
2
-0.053
-0,080
-0.107
--0133
.1-0.160
2 --
0.000
---------
--------------
-0 027
-0.053
-0,080
--------------
-0.107
-0.133
-0.160
- --- ---- --- - ---------1
-2
-- - -
------------- ---
-----
------ -- --- --- - --- --- ---2
0
- -- -------- -- ---- - - --- - -
-2 -
0
-2
2
U
2
U
8
-2
-2
Figure 3-28: Weight function of c'w' on the (a) water and (b) air side near the interface
at different horizontal planes (z=-0.27 and z=0.27 for water and air side respectively).
90
Air
Wate r
-CEUW
......
.--
(a)
(d)
CEWOZ.
CEWDIV
0.5
0.5
0
0
......
CEUW
CEWOZ
..-.-.
CEWDIV
-.
-
-...........
..........
-0.5
-0.5
-1.
-
-...
...----
(b)
0.5
0.4
0.6
-0.8
1
0.2
0
-02
-
CECIW
.............
o............
(e)
E
.....
CEC1OZ
-
CECIW
CECIU
......................
- - CECIDIV
E OI
-
0
-.
,-
5
-0.5
-0.51
-1
-8.5
.
(C)
1
08
.5
.d.........
%
0.6
0.4
--.........
-0.4
-0.6
-0.2
0
0
CEC1ZW
CECIZU
CECIZOZ
.CEC1ZDIV
05
0.2
04
0.6
0.8
0.6
U.6a
1
CEC1ZW
"'"--...
CEC1 ZU
...... CEC1 ZOZ
- - - - CEC1 ZDIV
0
0
-0.5
-0.8
-0.4
-06
-0.2
*0
0
0.2
Z
0.4
Z
Figure 3-29: Correlation coefficients of turbulent variables on the (a)-(c) water and
(d)-(f) air side.
of a
correlated with w', U',
', w -W ', I
+
i-).
Here
+- 2)
is the horizontal
deviation of velocity fluctuation, which is highly correlated with vertical velocity
component.
Beside the high negative correlation of c' and w', it is shown in figure (b) and
(d) that a high correlation also exists between scalar fluctuation c' and streamwise
velocity fluctuation u'. For scalar transfer near the interface on the water side, 2rrz
are extraordinarily correlated with u' and w'.
91
(b)
(a)
Water
u-u
v-v
0.8
0,6 -
-102
-IC
0.6
0.4
-
0.4
-
0.2
0.2 --
0
-.2 0
-0.2
0.
1.0
1
Air
x
2
2.5
05
3
3
2.5
2
1.
1
x
(d)
(c)
----
0.8
I
u-u
_-_
---
0.8
W-c1-c1l
u-u
-- v
-- -w-
0-6
0.6
0.4
F
Ii ~.-iVi.
0.2
02
I--
-0.2
-04,
00
1
0
-0.4
3
25
x
05
1
1s
21
Figure 3-30: Two-point correlation of u', v', w', c' in the x- direction with (a) z=0.001; (b) z=-0.28 on the water side and (c) z=0.001; (d) z=0.28 on the air side.
3.4
3.4.1
Two-point correlation and integral scales
Horizontal two-point correlation
Horizontal two-point correlation coefficient of two turbulent fluctuations, a' and b', is
defined by
StA~
IR''LlL (
A,
a'b'a
,A
_
a'(x,
a' y, z)b'(x + b'Ax, y + Ay, z)
b'l
.9
(3 314)
Two-point correlations have been shown in many results to illustrate the adequacy of
the computational domains and scales of the coherent structures ([19], [171, [20], [8]).
Figure 3-30 show the two-point self-correlation of u', v', w', c'(Sc = 1) in the
streamwise direction (Ay = 0) and Figure 3-31 show the two-point self-correlation of
U', v', w', c'(Sc = 1) in the spanwise direction (Ax = 0). Results near the interface
are shown at two different horizontal planes for each direction.
92
(a)
Water
(b)
ae8
0,8
0.6
0.4
0.2
0
v-v
0.6
OA4
-
0.2
0
-
0611
-C
-
-0.2
-0.2 ,
-0.4
-0.4
-0.6
-0.8
-0.5
-0.S
1
Air
-IC
0.0
11
5
5
-1 0
u-u
v-v
--w
-- c
-___
1
(d)
(C)
0.2
04
0.2
-0.4
-0.2
-0.6
-0.8
-0.4
........
0.2
04
0.6
0.8
1
1.2
--6
0.8
0.6
0.4
-u-u
v-v
w-w
cl-cl
-10
1.0
----
w-
-0.
02
1A4
Y
0.4
0.6
0.8
y
1
1.2
1.4
Figure 3-31: Two-point correlation of U' , v', w', c' in the y- direction with (a) z=0.001; (b) z=-0.28 on the water side and (c) z=0.001; (d) z=0.28 on the air side.
93
3.4.2
Three-dimensional two-point correlation
To reveal a quantitative correlation between the interface field and near-interface field,
we need to computer the three-dimensional two-point correlation coefficient([46],[13].
[21]). The two point correlation function relating the interface field fluctuation with
velocity, vorticity and scalar fluctuation in the subsurface near the interface is defined
by
Rab/(AXAYZ)
=a'(x, y, z = 0)b'(x + Ax, y + Ay, z)
a(msb(ms
(335)
Figures 3-32(a)-(d) show the three-dimensional two-point correlation results of
c'(Sc = 1) and w' in different planes.
Figures 3-32(a) and (b) show correlation
coefficient contours in the horizontal planes near the interface on the air and water
side respectively. Correlation coefficient contours in the vertical planes with Ay =
0
or Ax = 0 are shown in Figures 3-32(c)-(d).
3.4.3
Spectrum and Co-spectrum of turbulent fluctuations
To see how energy density is distributed in the x or y direction, spectra of a variety
of turbulent fluctuation and co-spectrum of two different turbulent fluctuations are
shown in figure 3-33-figure 3-34 and figure 3-35-figure 3-36, corresponding to different
horizontal planes.
For spanwise cospectrum of co-spectrum of Reynolds stress and turbulent scalar
flux, our results show a similar with former numerical study ([46],
[8]) where the
co-spectrum reaches a minimum at low wavenumber. For the streamwise direction,
however, these results show big difference. Other spectrum and co-spectrum results
could be found in the references ([411, [42]).
3.4.4
Integral scales
Quantitatively, lengthscale is calculated through the correlation of fluctuations ([13],[46],[7]).
As an example, the macro-lengthscale and Taylor lengthscales of velocity fluctuations
94
(bl
(a)
0
0.
0
70
4 02
050
0.
0
0
2S
(C)
0.4
0.09
0.2
>-
0.03
-002
-0.07
.12
0
ia
A
-0.23
-028
-Om3a
-0.39
-0.4
(d)
1.9 0.0
003
.0 02
.07
.12
:'.:2"
0
2"
0.3
d .0.39
-0.,
Figure 3-32: Specified 3D two-point correlation of c'(Sc = 1) and w': correlation
coefficient contours in horizontal planes with (a) z=-0.0034 and (b) z=0.22; (c) correlation coefficient contours in the vertical Ax - z plane with Ay = 0; (d) correlation
coefficient contours in the vertical Ay - z plane with Ax = 0.
95
Water
0
Air
U
10
-U
10-1
-V
10
V
-
W
UW
1 0-2
10-2
10'
10-
10*4
10-4
. ~ . .I ~
10'
-U-W
....
. ..
U -W .
10-5
106
10-7
10"
10-8
-
10
10'
101
1
1-
I
''',,,,'
)0
10'
102
I,
100
10
C1-W
Cl-U
101
Cl-DIV
10-2
Cl-w
Cl-U
----
C1-Oz
-.-.- Cl-DIV
10-2
10-1
10-3
10-4
10-4
"A'
10
100
10
'' ' '
108
108
109
101
10'
-
10,
10
102
k.
k-.
10,91
10
C1z-W
C1z-U
C1z-Oz
C1Z-DIV
102
10-1
Clz-w
I
~
Ciz-oz
ClZ DIV
-
10-
10-3
I',w
10-4
'Ai
10-4
10-1
10"
10~
10'
10-1
106
10.
10~-9
D
10
10'
k
a
ii102
10"
10'
10'
Figure 3-33: Spectrum and co-spectrum in the x-direction near the interface on the
water side (z=-0.022) and air side (z=0.007).
96
Water
Air
100
U
10'
101
r
W
-
..
..
....
K..
Fl
10-
10-6
10-a
107
10"
10'1--
. . . ...
- .
I
10'
11
U
V
W
W
10-4
10*
10.6
7
10
.. I
2
10
102
10,
L,
100
--- --- C1-W
101
Cl U
10"
.-
-
102
10-2
-
--
C1-U
Cl-OZ
N
Cl DIV
102
-
10-
k
102
-
Cl-DIV
kt
10-3
10-1
10-1
10-6
10-5
10-
6
~'I
10'
7
10-
10-'
10-8
,,I
1
10,
10"
102
10'
K
10'
1I
C1z-W
10-2
-
-
C1z-U
C1z-Oz
ClZ-DIV
1
Clz-W
C1z-U
2
10
1
V1
C1z-Oz
,ClZ-DIV
10-3
10
10i
10-
10-1
10-6
10"
10.
r
10'a
10-9 3'1
'' '--
V
10"
i i i -iis 10'
k
''I
10
100
101
k
102
Figure 3-34: Spectrum and co-spectrum in the x-direction on the water side (z=-0.28)
and air side (z=0.28).
97
Water
Air
100
Uw
10-,
U
10"
_ _W
-
--
---
-U-W
10 -
W
U-WN
-
1.'
1072
10-2
VV
w4.
10-4
10 4
10-
10*5
h
10*6
10-
10'
10 7
10-1
-8
10
.I
109
1'
10'
10"
- - --
102
10 1
102
101
C1-w
- -
- ............
101
10-2
-.......
10'
Cl-DIV
C 1-W
Cl-U
Cl-OZ
Cl-DIV
-Cl U
V1
- .
10-3
10'
10'
10-5
10'
10-6
10*
10-7
10-7
105
10-1
1-9
101
10,
0
k
101
k.
C1Z-W
10-2
ClZ-U
C1Z-OZ
C1Z-DIV
-7
10-1
-
ClZ-W
C1Z-U
-C1Z-OZ
C1Z-DIV
10.3
10-3
10
1 02
10.
1
1i
102
-'
"
10.4
10
10,
1010
N'!
-
10.
10-"
10~,9
10
10
0
-
'I
~''' ~~~
10,
k
'
102
100
I
101
10
I
10'
10,
Figure 3-35: Spectrum and co-spectrum in the y-direction near the interface on the
water side (z=-0.022) and air side (z=0.007).
98
Water
Air
10"
-_
U
10-1
V
101
W
AJ---U-W
10.2
102
a
10
10-
103
u3w
10~1
10-
105
1010 -6
14
108
106
''''
108
10,
11
L,
100
'
10'
Cl-U
C 1-Oz
1
-
101
C1-W
101
10'
10"
-.......
-C1-U- .... C 1-W
I
10-
Cl-DIV
2
Cl-u
Cl-DIV
pN
1010-3
10'
Zj
V
4
10-
106
106
10-7
10-8
108
10-1
10'9
11 00
' '10''' -
10
K
101
I
I
1 )0
10'
10
k.
C1z-W
-
-
C1Z-U
10'
Y. 1..'
C1Z-DIV
2
_Clz-W
C1Z-U
C1Z-Oz
10-2
/~*~(
C1Z-DIV
10*
10
F
10
101k
10
106
10-'
10'
10,..
10-
10
10
1
10-1
101
k
10'
10"
i
I. ..
10
1-'0 '
102
Figure 3-36: Spectrum and co-spectrum in the y-direction on the water side (z=-0.28)
and air side (z=0.28).
99
in the streamwise and spanwise directions (xi, i = 1, 2), are given as follows through
the velocity correlations Rjj,
Ai (Z) =
SLi
/2 Rj (xi) dxi
i = 1, 2; j
1, 2, 3
(3.36)
and
2
Ax (z) =
lim
zo2y(xi)Tdxi
i(= 1, 2; j
Note that summation notation here is not implied for
1, 2, 3.
j
(3.37)
1, 2, 3. As an example,
All refers to the streamwise macro lengthscale of the streamwise velocity while A21
the spanwise macro lengthscale of the streamwise velocity. All and A21 refer to the
streamwise and spanwise Taylor lengthscale of the streamwise velocity respectively.
Figures in figure 3-37 and figure 3-38 show the micro-lengthscales and macrolengthscales of u', v', w' and c'(Sc = 1) in the x- and y-
100
direction respectively.
(a)
(b)
-U-X
-0.2
-------
-
-U-X
0.8
V-X
------- I V-X
-----W-X
- -W-X
-- x-
-x
-0.4
0.6
z
-
-1
-0.6
0.4
-0.8
0.2
0
0.2
0.4
0.8
0.6
1
0
(C)
-0.2
0.4
0.2
0
-
-
-
0.8
0.6
1
(d)
0.8 -
-
-U-Y
S----------
-0.4
-
V-Y
- W-Y
.--- C1-Y
-------
U-Y
V-Y
- -- -C1-Y
-
.6
--
-
w
-l-
0.6
--
-
-
z
-0.6 --
-0.8
0.4-
0.2-
Iv
6
0.2
0.4
0.6
0.
0
1
0
0.2 '
O.4
0.6
0.8
Figure 3-37: Taylor lengthscale profiles in the interaction flow. (a)-(b): lengthscale
in the x direction; (c)-(d): lengthscale in the y direction for air (a, c) and water (b,
d) sides respectively.
101
-W-Y
(a)
(b)
-
-
-0.2
U-X
--------
1
U-X
------V-X
-------. W-X
0.8
V-X
W-X
C1-x
-0.4
0.6
-
u-x
-
-
-
-,
z
z
-0.6
0.4
a
-
-0.8
0.2
-1'0
'/ '
0.5
2.5
2
1.5
1
0-0
3
0.5
2
1.5
1
2.5
(d)
(C)
-0.2 -UY
-------
U-Y
0.8
V-Y
-------------.-----
W-Y
-..--.
--
C1-Y
-0.4
-
V-Y
W-Y
C1-Y
-
0.6 --
-
zv-
z
-0.6
0.4
-0.8
0.2
0
0.2
0.4
0.6
0.8
00
.
-
0.2
0.4
0.6
0.8
Figure 3-38: Macro-lengthscale profiles in the interaction flow. (a)-(b): lengthscale
in the x direction; (c)-(d): lengthscale in the y direction for air (a, c) and water (b,
d) sides respectively.
102
3.5
Turbulence transport budget
3.5.1
Turbulence kinematic energy (TKE) budget
For the turbulence kinetic energy, k = u'u'/2, its budget equation is given by:
Dk
Dt
1
,Du
= P
+
Dxi
D2 Uul
2 Re axjDxj
I
1
au, au,
Re Dxj Dxj
II
III
1 DUUa
_
Dxj
2
,
-- al
IV
,DW
D'4
(.8
(3.38)
Dx.
OX2
V
VI
The transport rate of turbulent kinetic energy is governed by all six terms above.
I is the pressure strain term. For incompressible fluid flow, it is identically zero from
continuity. V refers to the turbulence energy production, which is positive and III is
the turbulent energy dissipation (always negative). There are three diffusion terms
such as II the viscous diffusion term, IV the transport due to velocity fluctuations
(turbulent diffusion) and VI the transport due to pressure fluctuations (pressure diffusion).
For the problem of air-water interaction channel flow, turbulent kinetic energy
budget equation is given by:
Dk
Dt
1
Re
2k
=-
(z
2
+
1 DU' DU'
Re
Oxj Dxj
II
+
Dv' &v'
Ox
+
Ow' Dw'
Oxj
Dx
III
x
+ --
Dkw'
IV
+ -'w'-+
V
DTi
DZ
--
Dpv'
I+
V
(3.39)
The transport rate of turbulent kinetic energy is governed by all six terms above. V
refers to the turbulence energy production, which is positive and III is the turbulent
energy dissipation (always negative).
There are three diffusion terms such as VI
the transport due to pressure fluctuations (pressure diffusion), IV the transport due
to velocity fluctuations (turbulent diffusion) and II the viscous diffusion term. I is
the pressure strain term.
For incompressible fluid flow, it is identically zero from
continuity.
Notice that the time varying rate of turbulent kinetic energy, the left-side of
equation (3.38), is identically zero, since the flow under consideration is statistically
steady and homogeneous in the streamwise and spanwise directions.
103
(b)
(a)
I
0.2
-
0.5 z
o
------
-
-
'I
- z-0.10- - --.Air
--
:. .-
--
---
Water
I
x
-
p
i-0.1
-0.5-I
I
-04 -2
0
-0.2
2-04
0.2
-0.2
0
0.2
0.4
0.4
Figure 3-39: Turbulent kinetic energy (TKE) budget terms: (a) overview of TKE
budget; (b) near-interface amplification. All terms are normalized by V/u* .
Figure 3-39(a) shows profiles of each term in the TKE equation along the vertical
direction for both subdomains.
As observed for the mean velocity and turbulent
intensity, all significant differences in the profiles are localized in the near-interface
region up to IzI = 0.2 (see figure 3-39 b).
Figure 3-40 shows the profiles of the
production, dissipation and turbulent transport terms up to h+ = 30. All terms in
figures 3-40(a,b,c) are normalized with U4/v in order to compare the results between
water and air flow field. Profiles near the interface and near the wall are shown in
the same figure.
Energy production term reaches the largest value at locations close to the air-water
interface, but reduces rapidly as the interface is approached. There is no energy production at the indeformable interface although strong shear stress exists. For the air
side, the profile is almost symmetric with tiny difference between the maximum values near the bottom wall and near the interface. The global maximum value (V)max,
reaches about 0.27 at h+ = 9. For the water side, the peak value near the interface,
(V)max
0.28 at h+ = 7, is 20% bigger than that in the wall boundary region with
(V)max
0.22 at h+ = 11, which means that stronger turbulence generated near the
104
(b
(b)
-
(a) (a-
~ ~
0
-
--
=30
Wi(wtrbn)
-Wall
(water, btm)Wall (air, top)
Interface (water) -
--
Interface (air)
-
Wall (water, tn)
Wall (air, to
Interface (w er)
0
-
Interface (ai
-
20
20
-
--
10
10
-0
-
-or
h-(O.5, 3)
0
-0.5
0.5
0.4
0.3
0.2
0.1
(C)
30 '
-0.2
-0.3
-0.4
Pk
-0.1
0
Ek
' ' '
' 'T
-
-
(d)
30
--
Wall (water, btm) Wall (air, top)
Interface (water)
Interface (air)
-- -
-
20 -
20 -
10
10
Wall (water, btm).
.
(air, top)
Interface (water) Interface (air)
toJ
-0.2
-0.1
0
0.1
-0.4
0.2
-0.2
0
0.2
0.4
Dk
Figure 3-40: Turbulent kinetic energy (TKE) budget terms: (a) Production term; (b)
Dissipation term; (c) Turbulent transport term; (d) Viscous diffusion term.
105
.Wall
interface.
For dissipation term III, on the air side, dissipation increases towards the interface
and reaches a maximum at the interface as that near the wall boundary. Near the
boundaries, III is mainly balanced by the viscous transport term II. The result for
water side is more interesting with dissipation decreasing first and then increasing
as the interface is approached. III is abnormally small within h+ = (0.5 ~ 3) (see
figure 3-40 b). It can be expected that net energy is transferred to neighbor region
in this low dissipation region where more energy is produced than dissipation. Considering the interface is fluctuation permitted, a lower dissipation near the interface
is reasonable.
Turbulent kinetic energy is transported mainly through turbulent diffusion (IV)
and viscous diffusion (II). Corresponding to negative turbulent diffusion, the turbulent
kinetic energy is transported into that area while positive turbulent diffusion means
that energy is taken away. It is very clear that near the interface for water side, shown
in region I in figure 3-40 (c), energy is taken away (IV < 0). As a comparison, for air
side, energy is transported into (IV > 0), shown in region II in figure 3-40 c). The
energy transport process is indicated as follows. On the air side, turbulent velocity
fluctuations transport turbulent kinetic energy from the bulk region to the nearsurface region, while on the waterside, a portion of the energy within the near-surface
region is transported to the bulk flow by turbulent velocity fluctuations.
Viscous diffusion II is only significant very close to the interface or wall boundary.
Near the interface, viscous dominating region on the water side is much thinner than
that on the air side (see figure 3-40 d). The negative viscous diffusion region (see
where Dk < 0 in figure 3-40 d), which reflects the energy transfer direction through
viscous force, disappears near the interface in waterside.
This illustrates the same
energy transport process from air side to water side as that of turbulent transport.
In our case where the interface is indeformable and associated with strong-shear, the
viscous transport is more important than the turbulent diffusion effect.
The pressure transport term, VI, is found to be much smaller than other processes
(see in figure 3-39). Here, the profiles of VI is much flatter near the interface on the
106
water side. We could expect a much more important pressure diffusion effect if the
interface is deformable.
As a summary, interface strengthens the turbulence in the
IntT-w region, which
has been indicated by a stronger turbulent production V, a weak dissipation and a
(
thinner viscous dominating layer comparing with the wall turbulence results
[38];
[19]). Again, we can see that for the air side, the profile of each term near the interface
and near the wall is always the same.
3.5.2
Reynolds stress budget
For air-water interaction flow with a two-dimensional mean shear, the equations for
the primary components of the Reynolds stresses
OU'2
2p'
Dt
a,
DOI
I D2 U/ 2
2 Di' Du'
RD'
Dz
+
Re OZ2
Re
Dt
DvI
= 2p'
Dy
2
2p'
Dt
Di'
Dy
+
1 D2 ? 2
+
Re Dz
2
1
-
Re
at
Dw
+
x
Dz
u'
P'
+
D
-2'u'w'
Oz
D
2
w'
2
Dz 2
2 Dv' Dv'
D
Re xk xk
Oz
2 Dw' Dw'
Re xk xk
1 D2 v'w'
Re Dz 2
Rexkxxk
D 3
w'
Oz
V'2W'
(3.41)
-
D
2-p'w'.
Dz
(3.42)
IV
D
D
U'w'w'
Dz UOz
IV
IIIII
(3.40)
Dxk
az'
IV
III
2 Du' Dw'
au
V
II
II
Du'w'
v' 2 and W' 2 are
IV
I
I
D'
Ox
III
II
DV' 2
L'2,
Dp'v'
DwzwD
az
V
VI
(3.43)
Here I are the pressure-strain correlation terms, II the viscous diffusion terms, III
the dissipation terms, IV the transport terms and V the shear flow production terms,
VI the pressure diffusion term.
107
(a)
(b)
-0.2
-
PROD
TURD
PRES
VISD
DISS
-----
I
-0.4
-
0.8
-
-------...-.
PROD
TURD
PRES
------
VISO
-
DISS
-
-
0.6
a--
z
1
z
-
ji
-
-0.6
0.4
-
-
0.2
-0.8
4f.1
-1
1
0.5
0
-0.5
0.5
0
-0.5
-1
Figure 3-41: u'u' budget on the (a) water and (b) air side.
(a)
(b)
.2----PRES
a
-0.4
0.8 -
.----- TURD
-0
-
-
-
-DISS
0.6--
z
I!
z
a-
-0.6-
0.4 -
0.2
-
-
-1
TURD
--------- PRES
-- - - - - VISD
-
-
-0.8
-------
VISD
DISS
I
-
\I
-0.2
0
0.2
-0.2
--
i--0.2
Figure 3-42: v'v' budget on the (a) water and (b) air side.
108
(a)
(b)
------
.-
-0.2
TURD .
--
------- TURD
--- -- -- PRES
-----VISD
- - - DISS
0.8
PRES
------. VISD
DISS
I I
-0.4
0.6
z
z
-0.6
0.4
-1 1
-
/
IA
0.2
-0.8
-1
0.2
0
-0.2
0.2
0
-0.2
Figure 3-43: w'w' budget on the (a) water and (b) air side.
(a) 0-
(b)
-
PROD
PRED
-0.2-TURD
-----0.4
0.8 -
PROD
PRED
-------- TURD
PRES
-----VISD
- - DISS
PRES
VISD
DISS
0.6-
-
z
z
-0.6-
0.4
-
-t
-0.8
-
GY
-
F-
-0.2
I
K
-1
-0. 4
-_I_
_
-0.2
' 0I
-__-_I__II'__
0
0.2
0.4
-0.4
-0.2
..
~
-
0
0.2
Figure 3-44: u'w' budget on the (a) water and (b) air side.
109
0.4
3.5.3
Enstrophy dynamics
As for free surface turbulence, the surface layer is manifest primarily in the horizontal vorticity components rather than in the vertical vorticity component. Thus the
surface layer has disparate effects on the dynamics of the horizontal versus vertical
enstrophy components.
The equation for the balance of the enstrophy components is given by
2
1DW
X_-
Dw2 w'
4-2
2
W/
al'
-
+ ~W ,
II
au'
ai
9z
IV
9u'
1
- +_ -
W'
(9y
at
= --2-
(92 i
/
Yjjj9az Z2
I
w- w'
-'
+2
(2
2W'
Re jz
7
Ov,
W'w'
+
xO
'2
av' 1 a2
+
+'
az Yay Re az 2
V
at
II
Re xk
X
(3.44)
X
Dxk
(9V'
(y
'
+ W'W O'
YzZ
IIIII
a9
w) 2w'
z
X
VII
+2-W'
aw) 2
' w'
2
2
VI
V
S
±2w1 wL
+
III
+2
aw'2
auKz
+4W/4L7
2
Y,
292
-
VII
aw'
2
OX +L
y
(3.45)
y
Re axkx xk'
VI
Ow'
9w'
y
au aw'
aw'
<
az
z &y
III
V
+
I az
ReOZ2
a2
VI
2
aw'9a'
' 2 R al
zRe
Oxk
Ox
.
(3.46)
VII
Here the terms are: I, gradient production; II, transport by velocity fluctuations;
III, production due to the gradients of velocity fluctuation; IV, production due to
mean shear; V, mixed production; VI, viscous diffusion; and VII, dissipation.
The vertical variation of the above terms is plotted in figure 3-45110
3-47.
(a)
(b)
II
~..---.1
---.IV
II
-0.2
-------
-0.4
V
V
-------
VII
- - -
VII
-
0.8
-
--------
-0.6
~
mI
-
IV
V
VI
.
-
-
0.6
-
:
i'
0.4
'
-0.8
.
-I I
-.-
-
0.2
.
-0.02
-6.04
0.04
0.02
0
-0.04
-0.02
0
0.02
0.04
Figure 3-45: Enstrophy dynamics of 2': (a) near the interface behavior; (b) near the
wall boundaries.
(b)
(a)
0
II
-
-0.2
I
-------
-
0.8
y
-------
VI
-0.4 -
-
V
VI
VI/
-
0.6
N
N
-0.6
-
I
-
I-
0.4
0.2
-0.8
-
04
-0.02
0
0.02
0.04
-0.04
-0.02
0
0.02
0.04
Figure 3-46: Enstrophy dynamics of 2': (a) near the interface behavior; (b) near the
wall boundaries.
111
(a)
(b)
\
*
-
-----S ------- - -
-0.2-
III
0.8-
V
VI
VI-
------
-0.4 -
VI
VII
0.6
-0.6-
0.4-
-
-0. 8-
0.2
-0
6.02
-0.01I
0.01
0
0.02
-8.02
-0.01
0
0.01
0.02
Figure 3-47: Enstrophy dynamics of W/: (a) near the interface behavior; (b) near the
wall boundaries.
3.5.4
Budget for scalar transfer
For passive scalar transfer through the air-water inteface, the equilibrium equations
for the scalar concentration Z, scalar turbulent fluctuation c'd and turbulent scalar
flux c'w' are given in equation (3.47)-(3.49). Here, I are the pressure-strain correlation
terms, II the viscous diffusion terms, III the dissipation terms, IV the transport terms
and V the shear flow production terms, VI the pressure diffusion term.
DE
-- = - &c'w'
'
+
at
z
-
IV
ac'c'
Ot
I
a2C'C'
1
ReScz
II
2
1
a2g
91
ReScaz 2
.
(3.47)
II
ac' o8c'
a
6C_
c --d
c'c'w' -2c'w'
ReSc &Xk DXk
9Z
1Z
2
-2~2
-
III
112
IV
V
(3.48)
(a)
(b)
-
-0.2
- ------------
PROD
TURD
VISD
DISS
PROD
0.8 -
-0.4
-------
TURD
------
VISD
-- - - -
DISS
-
-
0.6-
:
z
z
-0.6
0.4-0
-
0.2 -
-0.8
-1 1 ,
-0.0002
-0.0001
0
1E-04
(9c'
I'
at
az
I
ReSc
Re
III
-1E-05
-2S-OS
0
1 E-05
budget on the (a) water and (b) air side.
Figure 3-48: c'
ac'w'
0.0002
+-
1 a 2C'w'
Re
Dz
2
+
II
a
Ow' ac'
0
aXk
z
(ReSc
1
Re]
1
aw
8z
ap'c'
az
aXk
9z
7
,ae
IV
V
-
VI
Profiles of each term in the c'c' and c'w' budget are shown in figure 3-48-
113
(3.49)
3-49.
(a)
-~
~-'--
-0.2-
(b)
-
PROD
TURD
VISD
---- - - -
'
~
0.8
DISS
-0.4
------
PRO D
TUR D
VISD
- - - ---
DISS
0.6-
-
z
-0.6-
0.4
-0.8-
0.2-
-0.00
02
-0.0001
0
1E-04
0.0002
-2
A-0
5
-1E-05
0
1E-05
Figure 3-49: c'w' budget on the (a) water and (b) air side.
114
Chapter 4
Computational result of DNS:
Coherent Structures
4.1
4.1.1
Two-dimensional streaky structures
Low-speed and high-speed streaks
Streaky structures are hallmarks of wall-bounded turbulence flows. The primary determinant of streaky structures is the shear rate near the boundary. In wall-bounded
turbulence at high shear rates the structures form low-speed streaks with alternating high- and low-speed regions. The low-speed streaks are periodically disrupted
by spectacular instabilities ("ejections"), which are confined to relatively low shear
stresses.
Streaky structures are shown in figure 4-1 through the streamwise velocity fluctuation at the interface which verifies that a solid boundary is not necessary for streaks
while high shear rate condition is sufficient ([35]). Figures 4-1(a-c) show a time series
of streaks near the interface. It is very clear that there are alternating high-speed
and low-speed regions. The streaks are characterized by higher or lower streamwise
velocities than the mean flow velocity at the interface. As we know for the burst
mechanism of wall-bounded turbulence, these streaks highlight the streamwise coherent structures that occur in the viscous sub-layer very near a wall. When evolving
115
U'
0.006
0.004
0.003
0.002
0.000
-0.001
-0.003
-0.004
-0.006
-0.007
4W
- -
U'
-
-
0.006
0.004
0.003
0.002
0.000
-0.001
-0.003
-0.004
-0.006
-0.007
(b)
U'
..
0.006
0.004
0.003
0.002
0.000
-0.001
-0.003
-0.004
-0.006
-0.007
(C)
Figure 4-1: Time series of streaky structures at the air-water interface: (a) t=4000;
(b) t=4050; (c) t=4100.
116
(a),,
(b) 0.2
I. I . .
1
1
1
1
.
. . . . . . . .
..
As
-6----..
0.
A2
.
- -
z
z
0.1 -
-0.1 --
-0.2
-)2
-1 X21
A2 2,.
1
'
-0.2
. .
Taylor-Lengthscale
Macro-Lengthscale
Figure 4-2: Macro-lengthscale and Taylor lengthscale profiles in the interaction flow:
(a) Macro-lengthscale; (b) Taylor lengthscale.
downstream, the streaky structures are lifted from the wall due to self-induction and
the mean shear. This lifting process is called an ejection. How the streaks play a role
in air-water coupled boundary layer is still not clear.
Quantitatively, lengthscale of streaks is calculated by correlation of velocity components. The calculation of the macro-lengthscale and Taylor lengthscale are given
as follows by the velocity fluctuation,
- m
)A,1=
Aim = u' x)u'x
1, 2, 3; m= 1,2
(4.1)
and
AIm(z) =u'
2
)2
1/
= 1, 2, 3; m = 1, 2.
(4.2)
where em, is the unit vector in the m-direction and r the vector connecting the two
points. Note that summation notation here is not implied for 1 = 1, 2, 3.
As an
example, All refers to the streamwise macro lengthscale of the streamwise velocity
while A21 the spanwise macro lengthscale of the streamwise velocity. All and
A21
refer to the streamwise and spanwise Taylor lengthscale of the streamwise velocity
117
respectively.
Figure 4-2 plots the profiles of macro-lengthscale and Taylor micro-lengthscale.
Across the interface from water side to air side, both macro-lengthscale and Taylor
microscale drop sharply to a very small value (see figures 4-2). The huge difference
between the lengthscales on the water and air sides will diminish if we normalize the
lengthscales by shear unit 1* in each side. Specifically, the streaks spacing A1 2 , with
a meaning of separation between high-speed and low-speed flow, is about 48 in shear
unit, which is well in coincidence with other numerical result where A1 2 ~ 50 near the
wall
([19]). For wall-bounded turbulence, an increasing of streaks spacing is shown
from A ~ 50 at z+ ~ 10 to A ~ 75 at z+ ~ 30, all in shear unit([19]). In our result
for water side, however, the streaks spacing reaches the maximum value of A12 = 48
at the interface. It appears the lengthscale of the streaky structure at the air-water
interface is mainly determined by the water motion.
4.1.2
Distribution of streaky structures
Research on turbulence structures at high shear rate shows that wall-layer streaks
exist for 2 < z+ < 30 ~ 35 where the energy-partition and streaks elongation parameters are used to indicator where streaky structures exist
([35]). Very near the wall
region, there is no experimental data available. There is numerical research showing
that no-streaky region exists within z+ < 5 where the non-dimensional shear rate,
defined by
-u'w'd6idz
(4.3)
is below a critical value of unit 1 ([36]).
Figures 4-3 (a-c) show the distribution of streaky structures near the interface in
each fluid subdomain. In the bulk flow which is not shown in the figures, turbulence
is more isotropic without dominant streaky structures in the flow. On the air side,
the lengthscale of stready structures decreases abruptly from the interface to near
interface region (see results at h+ = 0, h+ = 1.2 and h+ = 11 respectively), corresponding to the quantitative results in figure 4-2. Velocity fluctuation contours in
118
fI=o
(a)
Wale(
Air
"Ilk
-V
Y 0
r
1.2
Cb I
Y
0
Y
3
1
2
0
0
3
2
1
0
2
1
rc) h411
3
5io
"j2
I
Y 0
Y 0
.
.2
.1
0
1
2
3
3
*2
.1
0
1
2
3
Figure 4-3: Distribution of streaky structures on air and water sides: (a) z = 0,
h+ = 0, at the interface; (b)h+ = 1.2 (z = 0.005 for the air side and z = -0.01 for
0.04 for the air side and z = -0.1 for the water
the water side); (c) h+ = 11 (z
side.
119
figure 4-3 (b) at h+ = 1.2(z ~ 0.005) show a high flow disorder and doesn't give a
clear illustration of streaky structures as those in figures 4-3 (a) or (c). This result is
quite similar with the no-streaks characteristic in a thin layer near the wall boundary.
On the water side, however, the change of streaks lengthscale is quite slow and there
are always streaky structures near the interface up to about 40 shear-based units.
We try to explain the different characteristics of streaks distribution between
air and water subdomains using the non-dimensional shear rate. Definition of nondimensional shear rate S and 9', together with anisotropic indicator K* are given by
equations (4.3-4.5) where k and E are the turbulent kinetic energy and dissipation
respectively. Profiles of these indicators are shown in figure 4-4.
S'zzk
dz
16
(4.4)
20'
K* = 22 +(4.5)
/'
K* and
5'
/2
are good indicators of the presence and distribution of streaky struc-
tures. Streaky structures appear when these indicators are above critical values. On
the water side interface region, with a strengthened turbulence production and a week
dissipation, the non-dimensional shear rate is unanimously above the critical value
for streaks to exist. On the air side, small values correspond to no streaks occur.
Again, streaks distribution shows that interface acts like a wall for the air side,
while different for the water side due to velocity fluctuation on the interface. The
biggest difference between wall boundary and interface is, interface is a high shear
rate layer with fluctuation of shear rate permitted.
4.2
Three-dimensional coherent structures
Three-dimensional coherent structures are always very useful tools for research on turbulence flow. From earlier studies of near wall turbulence, it is shown that turbulence
is burst through an ejection-sweep mechanism very near the wall and streaks forms
in sub-layer at high shear rate region and then hairpin vortex is developed from the
120
(a) 80
60
-t
-
40 1
20
0-
(b)
-0.4
-0.6
-0.8
1
-0.2
)
z
40
----
S
so
5s
------- K*
30
I-
-
20
F11
--
10
/
0'
0.2
0.4
0.6
0.8
1
z
Figure 4-4: Indicators of streaky structures: (a) water side; (b) air side.
121
initial structure. Here "ejections" are confined to relatively low shear stresses while
sweeps are corresponding to high shear stress regions. For low shear rate flows with
free-surface turbulence as a limitation (zero shear rate at the free-surface), the nearboundary structures are "patchy", being caused by impinging eddies that flatten into
a pancake shape as they approach the boundary ([12]). Numerical simulation result
of FST ([55]) shows that free-surface turbulence is characterized by the presence of
hairpin vortex inclined against the mean flow with head portion near the free-surface
and the two legs extending into the bulk region. During the attaching process of
hairpin vortex onto the free surface, decaying, stretching and merging together of the
vortices could be found.
Recent numerical investigation on gas-liquid counter flow shows that sweep events
on the gas side are corresponding to ejections on the liquid side ([36]). It is also shown
that quasi-streamwise vortices are formed in the areas between high and low shear
stresses on both side of the interface while about one-fifth of them appear to be
coupled across the interface. Sweeps usually occur on the high shear stress side of
these vortices and ejections on the low shear stress side. Significant coupling exists
across the interface with over 60% of the Reynolds stresses in the near interface region
being associated with coupled event ([36]). For the gas-liquid counter flow problem, it
is also shown by statistical results that the main coupling coming through gas ejectionliquid ejection events over low shear stress regions, with a lesser but significant number
of gas sweep-liquid sweep events over high shear stress regions.
Three-dimensional, instantaneous flow fields obtained from the present simulation
provide detailed turbulence structures in the air-water coupled flow. Characterized
by the instantaneous or conditional average result, various structures in the nearinterface region of the flow are identified.
The influence of coherent structures on
scalar transport across the interface is also investigated.
4.2.1
Definition of vortex core in shear flow
Before identifying the three dimensional structures in the fluid flow, there is a problem of vortex core identification. A good indicator for vortices in three-dimensional
122
domain should exclude unsteady straining (swirling for example) and some viscous effects like that in the high shear layer, satisfy the Galilean invariant condition and have
a net circulation. Vorticity is not appropriate to identify vortex structures especially
in flow regions with high shear rate. There are many researches made contribution to
the development of vortex indicators. A new vortex indicator was induced by Jeong
& Hussain ([15]) and widely used. Details are given as following.
The vortex core indicator, A2 , is defined by imaginary part of the second largest
eigenvalue of a symmetric tensor
S2
+
Q2,
where S and Q are the symmetric and
anti-symmetric parts of velocity gradient Vu. S and Q are defined by equation (4.6)
and (4.7). A negative A2 has the physical meaning of a local pressure minimum, which
is in general corresponding to a vortex core.
Sr = -
-
+
-
-
2
axj
Qi = 2
axj
2u)
axi
(4.6)
-u
(4.7)
Oxj
A comparison was made between isosurfaces of A2 and vorticity for vortex core
indication in figure 4-5. The physical problem simulated is a strong shear flow near a
wavy wall. It is very clear that with vorticity indicator, vortex structures are totally
in the shallow of the high shear rate contours near the wall, while A2 can grasp the
vortex structures much better.
4.2.2
Instantaneous coherent structures
Our numerical results show that, near the interface, three dimensional quasi-streamline
vortices exists on both sides, which is originally observed in wall-bounded turbulence.
It is called quasi-streamwise vortices because vortex structures are mainly
extending in the streamline direction. Figure 4-6 shows a typical example of quasistreamwise vortex structures in each subdomain through the iso-surface of negative
A2 with A2 =-0.003 on the water side and A2 =-0.20 on the air side. On the air
side, most of the vortices are in streamwise orientation, resembling what have been
123
(a)
§
(b)
Figure 4-5: Vortex definition for numerical result of flow past wavy wall([59]): (a)
isosurface of vorticity; (b) isosurface of A2
lei4
Figure 4-6: Isosurface of A2 on the water and air side: air-water interaction flow with
Re* = 120 and Re* = 271.
124
GAS
Yt .i
0
x
1080
-
*
x
I)
Figure 4-7: Isosurface of A2 on the water and air side: Gas-liquid count flow with
Re* = Re*
= 180( [36]).
125
-Interface
scalar
concentration
Water
Hairpin Vortex
Ouas I-st eamw ise vortex
Interface-attached vortex
Figure 4-8: Vortex structures on the air and water sides near the interface, and
scalar concentration on the vertical cross-section cutting through the head portion of
a hairpin-shaped vortex.
Z
L,>
j+(w
k
3600 00
3600 0
(I
j
+.
k
0
x
y
Figure 4-9: Vortex inclination angles diagram: (a) O6z; (b)0yz defined by O62 =
and 6Oy = tan-1 (w/wz). w'z will be defined by 0, = tan 1(w/wz)
tan-1 (W.,/W
based on vor ticity fluctuation values.
126
Water
Air
(a)
04
z=-0.05
Z=0.01
0.3 .
0.3
0.2-
0.2-
0.1
0.1 -
45
C
90
135
180
0
225 270 315 360
45
90
135 180 225 270 315 360
0.
(tb)
0.05
0.05
z=-0.05
z=0.01
0.04
0.04-
0.03
-
0.03
0.02
-
0.02
0.01
0
45
90
135
0.01 0 . 45
180 225 270 315 360
90
135
(C)
0.05
0.05
0.04-
0.03-
0.03
0.02-
0.022
0.01
0.01
45
90
135
315 360
z=-0.50
z=0.50
0.04-
01'
0
180 225 270
0.
0, Z
180 225 270 315 360
5
45
90
135
180
225 270 315 360
oxz
Figure 4-10: Histograms of vortex inclination angles, Q,, in the (y, z)in the (x, z) - plane, at various distances from the interface.
127
plane and O,
observed in boundary layers at a solid wall as part of hairpin vortices.
It is very
clear that there are great difference in the lengthscales of vortex structures in air
and water subdomains. on the water side, lengthscale of vortex structures is much
larger than the that on the air side, which has been highlighted by the lengthscales
difference of 2-D streaky structures between two fluid sides in the former section. For
gas-liquid interaction problem with the same shear based Reynolds number for both
sides ([36],Re = 170 for both sides is assumed), numerical results have shown the
same size of quasi-streamwise vortices in gas and liquid subdomains (see figure 4-7).
Apparently, lengthscale of three-dimensional vortex structures is directly related to
shear based Reynolds numbers.
On the water side, the dominant vortices are hairpin vortices including the head
portion and two legs before attaching to the interface (see a instantaneous flow sample
in figure 4-8), which is quite similar for hairpin vortex near the free-surface ([55]). The
difference is that in our computational case, the hairpin vortex is highly stretched by
strong shear at the interface (notice that mean shear has been disregarded with the
definition of the vortex indicator A2 ). Scalar concentration is also shown in figure 48 at the cross-section cutting through hairpin vortex with analysis shown in the
following section 4.2.4.
The existence of these vortical structures can be also confirmed by the histogram
statistics of vortex inclination angles.
We employ the same approach which has
been used in the statistics of wall-bounded turbulence and free-surface turbulence
([38],[55]).
Two-dimensional vortex inclination angles are defined in figure 4-9. For
being different with
wYZ, W/ is defined by spanwise vorticity fluctuation values as
Oyz = tan-'(w/wz).
Note that for streamwise and vertical vorticity components,
mean vorticity values are zero and there are no different between vorticity values and
vorticity fluctuations. Figure 4-10(a) shows that near the interface, OYZ is concentrated
around 900 (indicates that
lj >> Iwzj), signifying the presence of strong shear at
the interface or the existence of hairpin heads.
O63 (see in figures 4-10b), being
concentrated around 90' and 2700, signifies the presence of hairpin heads only.
on the air side close to the interface, 0a> is concentrated around 900 and 270',
128
corresponding to streamwise vortices.
peaks around 00/360' and 180'.
vortices (where
|wj > IwI).
In the water subdomain, however, 0.,
has
This indicates the presence of surface-connected
Far away from the interface, OXZ is centered around 450
and 2250, showing the inclination of vortices with the bulk shear flow. For water side
bulk flow (z = -0.50),
the concentration of Oxz scatters around 450, indicating the
stretch effect by mean shear at the interface.
Figures 4-11(a-d) show the time evolution of a hairpin vortex attached to the
interface on the water side. The dominant vortices are hairpin vortices with the head
portion located near the interface in figure 4-11(a) and surface-connected vortices in
figures 4-11 (c-d). The two types of vortices are similar to the ones discovered in shearfree free-surface flows ([55]). Again we can see that vortices in the flow are stretched
significantly in the streamwise direction due to the strong shear at the interface. The
dissipation is very small near the interface from observation, similar with numerical
simulation result at free-surface. Elevations shown at the interface come from the
linear approximation relation between elevation and pressure fluctuation, as
p'/(Fr)2 ([6]).
4.2.3
Conditional averaging coherent structures
To analyze coherent structures in the turbulent flow, conditional averaging methods
are employed.
Following the former research on free-surface turbulence ([58]),
we
use a variable-interval spacing-averaging (VISA) technique, which is based on the
well-developed variable-interval time-averaging (VITA) method ([4]).
The VISA procedure is given as follows with conditional averaging of hairpin
vortex head portion as an example. In order to capture the hairpin vortex events near
the interface, conditional averaging method is applied upon the spanwise vorticity
which is dominated near the interface (big w is corresponding to head portion of the
hairpin vortices). The variable-interval space averaging is defined as,
Y(X, y, z, t, WX, WY) --
(2Wx)(2Wy) I-w
129
y-w
w y ((,z,
t) d~ds,
(4.8)
OW
Ali
t
4L
Figure 4-11: Time-evolution of hairpin vortex near the interface on the water side
(vortex illustrated by iso-surface of A2 = -0.0008).
130
where W2, W. are the half-widths of the averaging window along the streamwise and
spanwise directions respectively, which are set to be about one streaks macro lengthscales in this study. To identify strong wy events, a localized variance is introduced
with
w (x, y, z, t) - L (X, y, z, t, WX, Wy).
W.. (X, y, Z, t, W7, WY)
(4.9)
Strong hairpin head events are detected using the following criterion,
1ifm" > c(W rms)2,
F(x, y, z,
)=
0
Here the detection function
f
>
otherwise
(4.10)
E(x, y, z, 1) = 1 if the hairpin head portion exists. Wr,,
is the root-mean-square variation of wy at the horizontal plane and c is the threshold
level which has the value of 10 ~ 15 in the present study. Conditional averaging
is carried out upon the instantaneous flow data from t = 4500 to
t = 7500 which
includes 600 DNS samples. The three dimensional flow field associated with each
event is then ensemble averaged to yield the VISA field. It is necessary to notice
that before averaging, for each event, the coordinates are transformed horizontally so
that all the events are centered at (0, 0) in horizontal plane. According to vertical
position
(Zd)
of the conditional averaging, in this study, we have investigated hairpin
vortex head events within horizontal planes at different vertical positions, ranging
from
Zd
=-0.025 to
zd =
-0.005.
Increasing Zd is corresponding to hairpin head
portion which is closer to the interface and the later phase of the vortex attachment
process. With special mentions, the conditional average results of hairpin vortex in
this thesis is associated with Zd
-0.005
which is very near the air-water interface.
Through conditional average technique four major types of vortices on the water
side near the interface are identified as, hairpin vortices, quasi-streamwise vortices,
interface-attached single and interface-attached paired ("U"-shape) vortices.
The
first three kinds of vortices can also be seen from the spontaneous vortex structures
in figure 4-11 while the "U"-shape interface attached vortices are only found in the
conditional average result. Two of these four kinds of vortices, the hairpin vortices
131
and interface-attached single vortices, appears also in free-surface turbulence with zero
mean shear at the surface and no air fluid imposed from above. For wall-bounded
turbulence, hairpin vortices and quasi-streamwise vortices also exist near the wall
boundary.
Figures 4-15 to 4-17 show examples of the resulting VISA flow field. Conditional
averaging result of u'
centered at (0, 0, -0.005)
in figure 4-15 clearly illustrates hairpin vortices with head
through iso-surface of A2 and plots of vortexlines. The head
portion and the two legs are seen inclined with the mean shear flow.
Interface-attached single and paired vortices are shown in figure 4-16 through the
vortexlines. The flow fields are conditionally averaged upon large positive 4, at the
interface. Similar vortex structures with opposite vortex orientation can be shown
by conditional average result of negative wz.
Double-attached ("U" shape) vortex
rings appear only on one side of the single-attached vortices (right-side from the
current viewpoint in the figure). Figure 4-17 shows the streamwise vortices through
conditionally averaging upon large positive w. near the interface at
the interface.
zd
= -0.015 below
The quasi-streamwise vortex tubes can be either clockwise or anti-
clockwise direction corresponding to positive or negative w..
Four each conditional
average flow result shown above, the average is made through over 3000 events.
4.2.4
Dynamic scalar transfer with structures
The vortical structures discussed above are found to play an essential role in the
interfacial scalar transport. In figure 4-8, on the vertical cross-section cutting through
the hairpin vortex, the scalar concentration is plotted. It can be seen that upstream
the hairpin vortex, the scalar boundary layer is thinned. This is caused by the upward
convection of the scalar by the upwelling motions induced by the hairpin vortex. As
a result, scalar transfer is enhanced there. For the same reason, the scalar boundary
layer is thickened downstream the vortex and scalar transfer rate is reduced.
These vortices also play a significant role in near-surface turbulent kinetic energy
balance. Hairpin vortices are associated with key features in near surface vertical
velocity and TKE production, shown in the vertical (x, z)section at the center (y = 0)
132
(a)
(b)
0
1.07x10 '
9.35X1 0-02
7.97X1 0-02
6.60x1 002
5.22x1 0-02
02
3.85x1 0.
2.47X1 0-02
1.1Ox10-02
-2.78x1 0-3
02
-1.65x10-
Figure 4-12: Hairpin vortices in the conditional averaged VISA flow field of W': (a)isosurface of A2 =-0.003; (b) vortexlines.
133
02
-3.3x10-
3
-5.8x1Q-o
02
2.lxlO
4.8x10-02
Figure 4-13: Single and paired interface-attached ("U"-shape) vortices in the conditional averaged VISA flow field of u,.
-1.5x10-020 Ox10** 1.5x10-2 3.Ox1002
Figure 4-14: Quasi-streamwise vortices in the the conditional averaged VISA flow
field of wx: streamlines and iso-surface of A2 =-0.00034.
134
(a)
2.9X1U0*
-1.0x0-"4 -6.8x10- -3.6x1e0-3.4x10
K
I
6.1 x10
z
9.4x105
_
-
-
-
-i
-
0
-
-0.1
0.2
-0.2
3
"-
-0.5
-0.3
4
..
-0.-
(b)
z
6x
Pk
S6.0X1
I
_
_
0
-
0-"'
5.2x1 0-m
4.3x10V
3.5x10"
2.6x101 8x10-M
3
9.3x10,9
0
-7.7x10"
7.9x10
-0.2
0.5
-1.6x10*'
Figure 4-15: Coherent hairpin vortex structures in the conditional averaged VISA flow
field of 4: (a)iso-surface of A2 = -0.003 with w' at the interface; (b) streamlines
and turbulence production contours on the vertical cross-section cutting through the
hairpin vortex.
135
(a) (z
-0.02 -0.01 0.00 0.01
0.02 0.03 0.04 0.05 0.06 _Q0Q
z
Tk
1. 5x10,01
9.3x10
07
6.3x10'
1
0.11
-5.7x10-8.7x10
7
-1.2x10*
-0.2
2
-U.b
-0.4 -04-0.3
03
0.4
3.3x10 *0
3.Ox10-w
-2 7x1 0-"
0.5
-.
(b)
z
x
Yl
Pk
1.4x10*
1.2x10*
7
1.0x10-0
8.5x10-0.1
-0.2
0.2
0.4
0.5
6.7x1 04
4.9x10O-**
3.Oxl0.
1.2x10'
-6.1 x10
-2.4x10"
-0.5
Figure 4-16: Coherent interface-attached vortex structures in the conditional averaged
VISA flow field of w: (a)interface-attached single and interface-attached paired ("U"shape) vortices with w, at the interface; (b) turbulence production contours on the
vertical cross-section cutting through U-shape attached vortices.
136
(a)
)az
EM RK
0.43 3.06 5.70
8.34 10.97 13_-
z
C
0.038
0.027
0.016
-0.005
0.2
00
-0.01
2
-0.5
0.3
0.4.
-0.1
-0.016
-0.027
-0.2
-0.038
-0.049
-0.059
.
(b)
z
Tk
0
-0.1
-
-A
0.2
0.4
0.2
-0.2
-0 .2
-0.5
-0.4
.-
.1.x1
-0.2
0.5
6.3x104.3x10
202x10
0
-3.9x1 0-6 Dx1-8.0x10c*
-0.1
1.2x10
-0.3
Figure 4-17: Coherent quasi-streamwise vortex structures in the conditional averaged
VISA flow field of w,: (a)quasi-streamwise vortices (A 2 =-0.00034) with passive
scalar transport rate ac/az at the interface; (b) turbulence diffusion associated with
quasi-streamwise vortices.
137
of the VISA hairpin vortex in figure 4-15. It is very clear that regions of high TKE
production locate at the near interface region. Distinct near-interface vertical velocity
variations elucidate the hairpin vortex head portion and two legs near the interface.
Interface-attached vortices (see figure 4-16) are also associated with large TKE
production. As shown in the former section, The interface-attached vortices can be
singly or "U"-shape connected with the interface. The "U"-shape connected vortices
significantly enhance TKE transport Tk and turbulence production Pk. The physical
mechanism is that "U"-shape vortices are more associated with distinct counterrotating surface-normal vorticity.
For quasi-steamwise vortices, from conditional average results we can see that
quasi-streamwise vortices (see figure 4-17) play a key role in TKE transport with
strong upward and downward transport on opposite sides.
Quasi-streamwise vor-
tices are also responsible for passive scalar transport, as well as for the turbulent
energy transfer. Corresponding to the strong upward or downward energy transport
regions on opposite sides of the quasi-streamwise vortices, scalar transfer velocities
get increased or decreased.
As a result, there is also big difference between the
boundary-layer thickness on opposite sides, shown by thicker boundary layer on the
side of downward energy transport and scalar transfer.
4.2.5
Air-water interaction near the interface
Our conditional average results also show that the vortex structures on the water and
air sides are coupling through the interface. Figure 4-18 shows the interface attached
vortexlines on both sides by making conditional average of vertical vorticity at the
interface. Figure 4-19 shows the vortexlines on the air side when there is a hairpin
vortex near the interface from water side. Due to the nondeformable interface, the
influence of water side hairpin vortices on the air side is not shown clearly in this
figure. Figures 4-20-4-28 show different flow properties in the horizontal planes or
vertical plane when the hairpin vortex on the water side are near the interface.
By a recent result of conditional average result of interfacial divergence (see figures 4-29-4-33), the splat effect of hairpin vortices on the water side against the
138
Figure 4-18: Vortexlines near the interface on both sides (Conditional average is made
by vertical vorticity at the interface).
Figure 4-19: Vortexlines near the interface on both sides with the hairpin vortex near
the interface on the water side (Conditional average is made by W, near the interface
on the water side).
139
WY
Air
-4.3E-02
6.2E-02
1.7E-01
2.7E-01
3.
k=10
Interface
k=0
Water
-1.OE-02 1.3E-02
3.6E-02
5.9E-
k=0O: int
k=7
k= 1
k=15
k=20
Figure 4-20: w' in the horizontal planes at different vertical position with hairpin
vortex near the interface on the water side (Conditional average is made by w' near
the interface on the water side). k refers to the grid number away from the interface
on each flow sides.
140
Z'
Air
-3.OE-02 -1.7E-02
Water
-3.OE-02 -1.7E-02
-4.7E-03
-47 -3
7.9E-03
2.
E
k=7
k= 1
k=15-Ht
n head,-
k=-20
Figure 4-21: w, in the horizontal planes at different vertical position with hairpin
vortex near the interface on the water side (Conditional average is made by U' near
the interface on the water side). k refers to the grid number away from the interface
on each flow sides.
141
U'
Air
-4.OE-03 -2.3E-03 -6.3E-04
1.1 E-03
2.
Figure 4-22: u' in the horizontal planes at different vertical position with hairpin
vortex near the interface on the water side (Conditional average is made by 2 near
the interface on the water side). k refers to the grid number away from the interface
on each flow sides.
142
Air
-1.OE-03 -5.8E-04 -1.6E-04
2.6E-04
6.
Figure 4-23: v' in the horizontal planes at different vertical position with hairpin
vortex near the interface on the water side (Conditional average is made by Wt near
the interface on the water side). k refers to the grid number away from the interface
on each flow sides.
143
aw/az
Air
-1.6E-02 -1.1 E-02 -5.1 E-03
4.2E-04
.
Figure 4-24: Ow'/&z in the horizontal planes at different vertical position with hairpin
vortex near the interface on the water side (Conditional average is made by W' near
the interface on the water side). k refers to the grid number away from the interface
on each flow sides.
144
d2 u/az
2
Air
-1.OE+01
-5.8E+00 -1.6E+00
2.6E+00
6.8
l
Figure 4-25: 0 2 u'/0z 2 in the horizontal planes at different vertical position with
hairpin vortex near the interface on the water side (Conditional average is made by
U near the interface on the water side). k refers to the grid number away from the
interface on each flow sides.
145
Figure 4-26: u' in the x - z vertical plane with hairpin vortex near the interface on
the water side (Conditional average is made by w, near the interface on the water
side).
146
Figure 4-27: w' in the x - z vertical plane with hairpin vortex near the interface on
the water side (Conditional average is made by L' near the interface on the water
side).
147
Figure 4-28: Dw'/z in the x - z vertical plane with hairpin vortex near the interface
on the water side (Conditional average is made by (2 near the interface on the water
side).
148
interface can be seen. The flow structure characteristics are very abundant and more
profound research is needed. After the physical mechanism of vortex structures interaction is investigated, turbulent transport and scalar transfer within the coupling
air-water boundary layer will be better understood.
149
wlaz
Air
-1.2E-01
-5.3E-02
1.5E-02
8.2E-02
1. '
Figure 4-29: 9w'/&z in the horizontal plane at different vertical position (Conditional
average is made by velocity divergence at the interface, aw'/&z). k refers to the grid
number away from the interface on each flow sides.
150
Figure 4-30: &w'/&z in the x - z vertical plane (Conditional average is made by
velocity divergence at the interface, Ow'/&z).
151
Figure 4-31: u' in the x - z vertical plane (Conditional average is made by velocity
divergence at the interface, aw'/&z).
152
Figure 4-32: w' in the x - z vertical plane (Conditional average is made by velocity
divergence at the interface, aw'/&z).
153
Figure 4-33: Isosurface of A2 near the interface on each side, showing splat effect.
(Conditional average is made by velocity divergence at the interface, &w'/&z).
154
Chapter 5
Conclusions
In this study we investigate the air-water interaction flow by direct numerical simulation. The canonical problem here is a two phase Couette flow where the air and
water flowing fluids are coupled through the continuity of velocity and shear stress
across the interface. As a first step, we investigate a flat interface which is reasonable
in the limit of small free surface deformation (Fr assumed to be zero).
In our numerical simulation, pseudo-spectral method with Fourier expansion in
horizontal plane (streamwise and spanwise directions) and finite difference in vertical
direction are used in each flow subdomain. To simulate the coupling air-water flow,
subdomain-to-subdomain alternation strategy is applied where boundary conditions
at the interface, continuity of velocity and shear stress, are satisfied through a Dirichlet boundary condition on the fluid side with lower dynamic viscosity and a Neumann
type condition on the other side. Other gas-liquid interaction flows, depending on
different density ratio and viscosity ratio, can also be solved by this method. Parallel
computation is carried out to fulfill the high demanding on computation resources of
this problem.
All of the statistically quasi-steady flow properties, such as mean velocity profiles, turbulent intensities, Reynolds stresses and turbulent kinetic energy budget
terms, show significant differences between turbulence near the interface on the water
side (Int T-w) and wall-bounded turbulence (WT) while there are some similarities
between IntT-w and free-surface turbulence (FST). For the air side, turbulence char155
acteristics near the interface (IntT-a) go all the way with WT which indicates that
air-water interface with high shear is quite similar to a wall boundary for the air side.
In bulk flow area, the differences between air and water sides are mainly due to the
big difference of shear-based Reynolds numbers (Re* /Re* = 1/2.26), as a result of
huge density and viscosity differences.
The 0(1) behavior of the horizontal velocity fluctuations and the O(z) behavior of
the vertical velocity fluctuation can be deduced from the interface boundary condition
analytically and verified by our calculation results. The Reynolds shear stress and
eddy viscosity in Int T-w are strengthened comparing with WT or IntT-a. Due to
this Reynolds stress distribution characteristics, water side interface turbulence flow
is characterized with a thinner viscous sub-layer and adjusted intercept parameter
B in log-law layer (B ~ 3.0 for Int T-w comparing with B ~ 5.5 in WT at current
Reynolds number condition).
Regarding the turbulent kinetic energy budget terms, the quasi-steady results on
the water side near-interface flow show a stronger production term, a decreasing then
increasing dissipation term within h+ ~ (0.5 ~ 3) and a negative turbulent diffusion
term in turbulent kinetic energy (TKE) budget.
More abundant physical phenomena exist on the water side turbulent flow with
four major types of three-dimensional vortex structures identified near the interface
by variable-interval spacing averaging (VISA) techniques, which include hairpin vortices, quasi-streamwise vortices, interface-attached single and interface-attached pair
vortices ("U"-shape).
Each type of vortex structures is shown to play an essential
role in the turbulent kinetic energy balance and interfacial passive scalar transport.
The scalar boundary layer is thinned upstream the hairpin vortex, caused by the
upward convection of the scalar due to the upwelling motions induced by the hairpin
vortex. Scalar transfer is enhanced by the hairpin vortex. Hairpin vortices are also
associated with enhanced TKE production near the interface. Near-interface vertical
velocity variations elucidate the hairpin vortex head portion and two legs near the
interface.
For interface-attached vortices, the "U"-shape double attached vortices,
characterized with distinct counter-rotating surface-normal vorticity, also contribute
156
to the large TKE production in IntT-w. Quasi-streamwise vortices play a key role
in scalar transfer near the interface with strong upward and downward transport on
opposite sides.
The conditional average results also show the interaction between vortex structures
on the water and air sides. Physics of coupling vortex structures are very rich and
profound investigations are needed in further research steps.
Through the vortex
structures interaction, understanding on turbulent transport and scalar transfer near
the interface will be greatly improved.
157
158
Bibliography
[11 ANTONIA, R. A. & KiM, J. 1991 Turbulent Prandtl number in the near-wall
region of a turbulent channel flow. Int. J. Heat Mass Transfer 34, 1905-1908.
[2] BANERJEE, S. 1991 Upwellings, downdrafts, and whirlpools: dominant structures
in free surface turbulence. Appl. Mech. Rev.47,
[3]
S166.
BANERJEE, S., LAKEHAL, D., & FULGOSI, M. 2004 Surface divergence models
for scalar exchange between turbulent streams. Int. J. Multiphase Flow 30, 963977.
[4] BLACKWELDER,
R. F. & KAPLAN, R. E. 1976 On the wall structure of the
turbulent boundary layer. J. Fluid Mech. 76, 89.
[5]
BOGUCKI, D., DOMARADZKI, J. A. & YEUNG, P. K. 1997 Direct numerical
simulations of passive scalars with Pr > 1 advected by turbulent flow. J. Fluid
Mech. 343, 111-130.
[6] BORUE, V., ORSZAG, S. A. & STAROSELSKY,
I. 1995 Interaction of surface
waves with turbulence: direct numerical simulations of turbulent open-channel
flow. J. Fluid Mech. 286, 1.
[7] BRUMLEY & JIRKA 1988 . PCH PhysicoChemical Hydrodynamics 10, 295-319.
[8] CALMET, I. & MAGNAUDET,
J. 1997 Large-eddy simulation of high-Schmidt
number mass transfer in a turbulent channel flow. Phys. Fluids 9, 438-455.
[9]
CALMET, I. & MAGNAUDET,
J. 1998 High-Schmidt number mass transfer
through turbulent gas-liquid interfaces. Int. J. Heat Fluid Flow 19, 522-532.
159
[10] PRESS, W. H., TEUKOLSKY, S. A., VETTERLING, W. T. & FLANNERY, B.
P. 1986-1992 Numerical recipes in FORTRAN 77: the Art of Scientific Computing
(Second Edition). Cambridge University Press.
[11] FULGOSI, M., LAKEHAL, D., BANERJEE, S. & ANGELIS, V. D. 2003 Direct
numerical simulation of turbulence in a sheared air-water flow with a deformable
interface. J. Fluid Mech. 482, 319.
[12] HANDLER, R. A., SWEAN, T. F. JR., LEIGHTON, R. I. & SWEARINGEN, J.
D. 1993 Length scales and the energy balance for turbulence near a free surface.
AIAA J. 31, 1998.
[13] HANDLER, R. A., SAYLOR, J. R., LEIGHTON, R. I. & ROVELSTAD, A. L.
1999 Transport of a passive scalar at a shear-free boundary in fully developed
turbulent open channel flow. Phys. Fluids 11, 2607.
[14] HORIUTI, K. 1992 Assessment of two-equation models of turublent passive-scalar
diffusion in channel flow. J. Fluid Mech. 238, 405-433.
[15] JEONG, J. & HUSSAIN, F. 1995 On the identification of a vortex. J. Fluid Mech.
285, 69.
[16] KERR 1990 . J. Fluid Mech. 221, 309-332.
[17] KiM, J. & MOIN, P. 1985 Application of a fractional-step method to incompressible Navier-Stokes equation. J. Comp. Phys. 59, 308.
[18] KIM, J. & MOIN, P. 1986 The structure of the vorticity field in turbulent
channel flow. Part 2. Study of ensemble-averaged fields. J. Fluid Mech. 162, 339.
[19] KIM, J., MOIN, P. & MOSER, R. 1987 Turbulence statistics in fully developed
channel flow at low Reynolds number J. Fluid Mech. 177, 133.
[20] KIM, J. 1988 Transport Phenomena in Turbulent Flows.
[21] KoMORI, UEDA, OGINO & MURAKAMI
521.
160
1982.
J Heat Mass Transfer 25, 513-
[221 KOMORI,
S., MURAKAMI, Y. & UEDA,
H. 1989 The relationship between
surface-renewal and bursting motions in an open-channel flow. J. Fluid Mech. 203,
103-123.
[23] KOMORI, S., NAGAOSA,
R. & MURAKAMI, Y. 1990 . AIChE Journal 36,
957-960.
[24]
KOMoRI,
S., NAGAOSA,
R. & MURAKAMI,
Y. 1993 Turbulence structure
and mass transfer across a sheared air-water interface in wind-driven turublence.
J. Fluid Mech. 249, 161-183.
[25] KOMORI, S., NAGAOSA, R., MURAKAMI, Y., CHIBA, S., ISHII, K. & KUwAHARA, K. 1993 Direct numerical simulation of three-dimensional open-channel flow
with zero-shear gas-liquid interface. Phys. Fluids A 5, 115.
[26] KREITH, F. & GoswAMI, D. Y. 2005 The CRC handbook of mechanical engineering. CRC Press.
[27] KREPLIN, H. & ECKELMANN, H. 1979 Behavior of the three fluctuating velocity
components in the wall region of a turbulent channel flow Phys. Fluids A 22, 1233.
[28] LAKEHAL, D., FULGOSI, M., YADIGAROGLU, G. & BANERJEE, S. 2003 Direct
numerical simulation of turbulent heat transfer across a mobile, sheared gas-liquid
interface. J Heat Transfer 125, 1129-1139.
[29] LESIURE, M. & ROGALLO, R. 1989 Large-eddy simulation of passive scalar
diffusion in isotropic turbulence. Phys. Fluids A 1, 718-722.
[30] Lu, D. M. & HETSRONI, G. 1995 Direct numerical simulation of a turbulent
open channel flow with passive heat transfer.
Int. J. Heat Mass Transfer 38, 3241-
3251.
[31] LYONS, S. L. & HARANTTY, T. J. 1991 Direct numerical simulation of passive
heat transfer in a turbulent channcel flow.
1161.
161
Int. J. Heat Mass Transfer 34, 1149-
[32] LAM, K. & BANERJEE, S. 1988 Investigation of turbulent flow bounded by a
wall and a free surface. In Aundamentals of Gas-Liquid Flows (ed. E. Michaelides
& M. P. Sharma), pp.29-38. ASME.
[33] LAM, K. & BANERJEE, S. 1992 On the condition of streak formation in a
bounded turbulent flow. Phys. Fluids A 4, 306.
[34] LE, H. & MOIN, P. 1990 An improvement of fractional step methods for the
incompressible Navier-Stokes equations. J. Comp. Phys. 92, 369.
[35] LEE, M.J., KiM, J. & MOIN, P. 1990 Structure of turbulence at high shear
rate. J. Fluid Mech. 216, 561.
[36] LOMBARDI, P., ANGELIS, V. D. & BANERJEE, S. 1996 Direct numerical simulation of near-interface turbulence in coupled gas-liquid flow. Phys. Fluids 8, 1643.
[37] MOIN, P. & KIM, J. 1982 Numerical investigation of turbulent channel flow. J.
Fluid Mech. 118, 341.
[38] MOIN, P. & KIM, J. 1985 The structure of the vorticity field in turbulent
channel flow. Part 1. Analysis simulation of three-dimensional open-channel flow
with zero-shear gas-liquid interface. Phys. Fluids A 5, 115.
[39] MOIN, P., SQUIRES, K., CABOT, W. & LEE, S. 1991 A dynamic subgrid-scale
model for compressible turbulence and scalar transport. Phys. Fluids A 3, 2746.
[40] MOIN, P. 2001 Fundamentals of Engineering Numerical Analysis. Cambridge
University Press.
[41] NA, Y., PAPAVASSILIOU, D. V. & HANRATTY, T. J. 1999 Use of direct numerical simulation to study the effect of Prandtl number on temperature fields.
Int. J. Heat Fluid Flow 20, 187-195.
[42] NA, Y. & HANRATTY, T. J. 2000 Limiting behavior of turbulent scalar transport close to a wall. Int. J Heat Mass Transfer 43, 1749-1758.
162
[43] NAGANO, Y. & TAGAWA, M. 1989 Statistical characteristics of wall turbulence
with a passive scalar. J. Fluid Mech. 196, 157-185.
[44] NAGAOSA, R. & SAITO, T. 1997 Turbulence structure and scalar transfer in
stratified free-surface flows. AIChE Journal 43, 2393-2404.
[45] NAGAOSA,
R. 1999 Direct numerical simulation of vortex structures and tur-
bulent scalar transfer across a free surface in a fully developed turbulence. Phys.
Fluids 11, 1581-1595.
[46] NAGAOSA,
R. & HANDLER, R.
2003 Statistical analysis of coherent vortices
near a free surface in a fully developed turbulence. Phys. Fluids 15, 375-394.
[47] OVERHOLT, M. R. & POPE, S. B.
1996 Direct numerical simulation of a
passive scalar with imposed mean gradient in isotropic turbulence. Phys. Fluids 8,
3128-3148.
[48] PAN, Y. & BANERJEE, S. 1995 A numerical study of free-surface turbulence in
channel flow. Phys. Fluids 7, 1649.
[49] RASHIDI, M. & BANERJEE, S. 1988 Turbulence structure in free-surface channel
flows. Phys. Fluids 31, 2491.
[50] RASHIDI, M., HETSRONI, G. & BANERJEE, S. 1991 Mechanisms of heat and
mass transport at gas-liquid interfaces. Int. J. Heat Mass Transfer 34, 1799-1810.
[51] RASHIDI, M. 1997 Burst-interface interactions in free surface turbulent flows.
Phys. Fluids 9, 3485.
[52] ROGERS, M. M., MANSOUR N. N. & REYNOLDS, W. C. 1989 An algebraic
model for the turbulent flux of a passive scalar. J. Fluid Mech. 203, 77-101.
[53] SAGAUT, P. 2004 Large eddy simulation for incompressible flows. Second edition.
Springer.
163
[54] SALVETTI,
M. V. & BANERJEE, S. 1995 A priori tests of a new dynamic
subgrid-scale model for finite-difference large-eddy simulations. Phys. Fluids 7,
2831.
[55] SHEN, L., ZHANG, X., YUE, D. K. P. & TRIANTAFYLLOU, G. S. 1999 The
surface layer for free-surface turbulent flows. J. Fluid Mech. 386, 162.
[56] SHEN, L., TRIANTAFYLLOU, G. S. & YUE, D. K. P. 2000 Turbulent diffusion
near a free surface. J. Fluid Mech. 407, 145.
[57] SHEN, L. 2000 Structures, statistics and mechanisms of low Froude number freesurface turbulence:
a simulation-based study. PhD thesis, Department of Ocean
Engineering, MIT.
[58] SHEN, L., & YUE, D. K. P. 2001 Large-eddy simulation of free-surface turbulence. J. Fluid Mech. 440, 75.
[59] SHEN,
L.,
ZHANG,
X., YUE,
D. K. P.
& TRIANTAFYLLOU, G. S. 2003
Turbulent flow over a flexible wall undergoing a, streamwise traveling wave motion.
J. Fluid Mech. 484, 197.
[60] WANG, L. P., CHEN, S. Y. & BRASSEUR, J. G. 1999 Examination of hypotheses in the Kolmogorov refined turbulence theory through high-resolution simula-
tions. Part 2. Passive scalar field. J. Fluid Mech. 400, 163-197.
[61] WANG, L., DONG, Y. H. & Liu, X. Y. 2005 An investigation of turbulent
open channel flow with heat transfer by large eddy simulation. Computers & Fluids
34, 23-47.
[62] ZHU, J. P. 1999 Solving Partial Differential Equations on Parallel Computers.
World Scientific Publishing Co. Pte. Ltd.
164