On coherent structures in shallow flows (18-point font size)
M.S.Ghidaoui (14-point font size)
Hong Kong University of Science and Technology, Hong Kong (12-point font size, italic)
A.A.Kolyshkin
Riga Technical University, Riga, Latvia
ABSTRACT (12-point font size): The presence of two-dimensional coherent structures is evident from experimental and theoretical analyses of turbulent shallow flows. This paper discusses the connection between
shallow water equations and the two dimensional coherent structures (2DCS) and illustrates the ability of the
shallow water equations in modeling 2DCS. In addition, the onset and development of 2DCS in shallow waters is analyzed by means of the linear and weakly nonlinear analysis. The Ginzburg-Landau amplitude evolution equation is derived from the rigid-lid shallow water equations. The applications of the weakly nonlinear
analysis are discussed.
1 INTRODUCTION (12-point font size)
Turbulent shallow flows such as shallow mixing
layers, shallow wakes and shallow jets are widely
observed in practice. Examples include flows in
compound and/or composite channels, flows on the
leeward side of islands and headlands, waste and
thermal discharges into shallow waters.
Turbulent shallow flows are studied experimentally in Chu & Babarutsi (1988), Chen & Jirka
(1995), Lloyd & Stansby (1997), (Balachandar et. al.
2000), Uijttewaal & Booij (2000), Balachandar &
Tachie (2001), Tachie & Balachandar (2001). Chen
& Jirka (1995) classified experimentally observed
patterns in shallow wakes behind bluff bodies as
vortex shedding, unsteady bubble and steady bubble
using a shallow wake stability parameter,
S  c f D / H introduced in the paper by Ingram &
Chu (1987), where c f is the friction coefficient, D
is the diameter of the obstacle and H is water depth.
Experimental results indicate that the development
of shallow wakes is different from the development
of wakes in deep water due to the following reasons:
first, limited water depth prevents the development
of three-dimensional instabilities and second, bottom
friction tends to suppress the transverse growth of
perturbations.
2. SHALLOW WATER EQUATIONS & 2DCS
Turbulent shallow flows are characterized by the
presence of turbulent structures whose transversal
horizontal length scale L is considerably larger than
the water depth H (i.e., L/H>>1). These flows contain structures with broad range of scales ranging
from the Kolmogrov scale to L. When the flow is
stable (i.e., when the bed friction number S is
large), all scales of motion are disordered and no coherent structures are present (Uijttewaal & Booij
2000).
To test the ability of shallow water equations in
modeling 2DCS, consider the following depth averaged mass and momentum equations in surface water
u
u
u p c f
u
v 
 u u2  v2  0
t
x
y x h
(1)
v
v
v p c f
u v 
 v u2  v2  0
t
x
y y h
(2)
h hu hv


0
t
x
y
(3)
where u and v are the depth-averaged velocity
components, p is the pressure, h is water depth and
c f is the friction coefficient.
System (1), (2) and (3) is solved numerically by a
finite volume scheme where the fluxes are determined from the Boltzmann Bhatnagar-Gross-Krook
(BGK) model. Details of this model can be found in
0.4
0.2
Cl
Ghidaoui et al. (2001). The BGK model is applied to
the wake flow data of Chen and Jirka (1995) and
Lloyd and Stansby (1997) and to the shallow mixing
layer of Uijttewaal & Booij (2000). Given the space
limitation, only a small sample of the results is presented in this paper.
Figures 1, 2 and 3 clearly show the 2DCS obtained from the shallow water equations. Comparison between the shallow water model and the data of
Chen and Jirka (1995), Lloyd and Stansby (1997)
and Uijttewaal & Booij (2000) show good agreement in terms of shedding frequency and overall
wake structure. However, the size of the 2DCS obtained from the BGK shallow water model is found
to be 20 to 40% smaller than that observed in the
experiments. One possible reason for this discrepancy maybe due to the fact while experimentalists
reported the size of the 2DCS as observed at the free
surface, the BGK model produces the average size
of 2DCS over the water depth. It is expected that the
2DCS at the water surface is larger than the depthaverage of the 2DCS. Another possible reason is that
the shallow water equations ignore part of the interactions between the unresolved and the resolved
scales since the various averaging coefficients are
set to 1 and no turbulence model has been used.
0
-0.2
1000
2000
3000
Time
Figure 5: ``Lift Coefficient (CI) ” versus time for a convectively unstable wake flow for S about 46% of S cr .
3. ACKNOWLDGEMENT
This work is supported by project No:
DAG01/02.EG24. The authors would like to thank
Dr Qibing Li for performing the BGK experiments
and producing the figures in the paper.
REFERENCES
-3.0 -2.4 -1.8 -1.2 -0.6 0.1 0.7 1.3 1.9 2.5
y
1
0
-1
0
1
2
3
x
Figure 1. Near-wake details of unsteady bubble wake with
S  0.51. This example corresponds to the wake flow of
Chen and Jirka (1995).
It is interesting to investigate the applicability of
the GL equations when S b is far from S cr . The BGK
model is applied to wake flow for the case with
S b  0.42 and S cr  0.91 and the corresponding CI
versus time curve is given Fig. 5. Note the contrast
between Fig. 4 and Fig. 5. The disordered type motion in Fig. 5 which appears to indicate that away
from the critical condition, either the wake is susceptible to further bifurcations or the scale of disorganized (incoherent) structures exceeds the water
depth and is thus resolved by the shallow water
equations. Careful 3-D numerical modeling and/or
experimental data would be required to understand
the apparent disorder in Fig. 5.
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