J. of Thermal Science Vol.7, No.1
J l ~ ¢~
C o m p u t a t i o n a l S t u d i e s of L o b e d F o r c e d M i x e r F l o w s
H. H u
S.S. W u
Jet Propulsion Laboratory, Beijing University of Aeronautics, Beijing, China
S.C.M. Yu
Thermal and Fluids Engineering Division School of Mechanical and Production Engineering,
Nanyang Technological University, Singapore
Full Navier Stokes Analyses have been conducted for the flows behind the trailing edge of a lobed forced
mixer. The governing equations are derived from the time-dependent compressible Navier-Stokes equations and discretized in the finite-difference form. A simple two-layer eddy viscosity model has also
been used to account for the turbulence. Computed results are compared with some of the velocity
measurements using a laser-Doppler anemometer (Yu and Yip (1997)). In general, good agreement can
be obtained in the streamwise mean velocity distribution but the decay of the streamwise circulation
is underpredicted. Some suggestions to the discrepancy are proposed.
Keywords: lobed forced mixer flows, eddy viscosity model, velocity distribution.
INTRODUCTION
Splitter plates with a convoluted trailing edge (commonly referred to as lobed forced mixers) are passive,
fluid mechanical devices which generate in the coflowing streams a three-dimensional shear layer with
strong secondary flow. As shown in Fig.l, the geometry of a lobed forced mixer is characterized by a
periodically alternating, lobed trailing edge surface.
The lobe causes large scale streamwise vortices to be
shed at the trailing edge so that the downstream flow
field is embedded with an array of streamwise vortices of alternating sign. The enhanced mixing of the
lobed mixers is believed to be directly attributable to
the large mixing scales generated by the streamwise
vortices. Application of lobed mixers in the turbofan engine exhausts has been studied by Presz, Blinn
and Morin [1J and Presz, Gousy and Morin [2], where
the rapid mixing of the core and the bypass flows can
achieve noise reduction and thrust enhancement.
Received August, 1997.
Cold 11111flou
Inlel
ixert{o~laust
Lobed mixer
Fig.1 Typical lobed (forced) mixer exhaust geometry
Paterson [3] measured velocity and turbulence characteristics downstream of a lobed mixer using laserDoppler a n e m o m e t e r and concluded t h a t b o t h the
lobe shape and lobe penetration were the important
p a r a m e t e r s to determine the effectiveness of mixer
performances. Paterson also found that the generation of streamwise vortices was due mainly to the secondary flow shed by the lobe and, therefore, concluded
t h a t the streamwise vortices ,were inviscidly generated.
The subsequent investigation by Barber, Paterson and
Skebe [4] confirmed the inviscid nature of the secondary
flow generated by a lobe and the strength of the secondary flow was also found to be higher when the
H. Hu et al.
Computational Studies of Lobed Forced Mixer Flows
lobe penetration consisted of straight parallel sidewalls. They also found that higher strength would
result in a faster mixing rate and hence would facilitate the condition of achieving spatial uniformity
downstream of the mixer.
Based on flow visualization tests in a water tunnel,
Werle, Paterson and Presz [5] suggested that the flow
structure of the wake region behind the lobed mixer
follows a three-step process by which the streamwise
vortex cells form, intensify, and then break down.
Most intense mixing seemed to occur in the third region. By varying the velocity ratio across the lobe,
the location of these three regions could be shifted.
Velocity measurements by McCormick and
Bennett [G] and Yu and Yip [7] concluded that intense
small-scale turbulence and mixing occurred at about
two to four wavelengths downstream of the mixer trailing edge is mainly due to the deformation of the normal vortex into a pinched-off structure by the streamwise vorticity.
The complicated three-dimensional features of the
flows behind the mixer trailing edge provide challenges
to use CFD methods to predict them. Some early
computation works were focused mainly on the region
downstream of the lobe exit whereby the accuracy of
the lobe exit velocity information was crucial to the
success of the predictions. The methods have been
improved later on by modelling the lobe itself (Refs.9).
K o u t m o s and McGuirk Is], Tsui and Wu [9] solved
fully elliptic equations using the finite volume method
coupled with a k - e turbulence model to predict the
flow behind the lobed forced mixer trailing edge. In
general, good agreement can also be obtained against
measured results. Although these calculations were
promising, confidence in the predictions requires further rigorous testing against suitable measurements.
The work presented here has the objective in achieving this goal.
T h e following section describes briefly the mathematical model. It will be followed by presentation and
discussion of the results. The paper ends with a summ a r y of important findings.
23
transformed to a b o d y conforming curvilinear coordinate system (~,77,(). Then, in the absence of b o d y
forces, the equations for three-dimensional turbulent
flow in non-dimensional form can be written as:
-OQ*
--+
Ot
Jr
O{F* - F ; } +
O~
O{H* - H*}
o(
0{C*-C;}
0~
--0
(1)
where:
Q, = j-1Q
F* = J - l ( ~ F
+ ~yG + ~zH)
G* = J-I(~?~F + ~?yG + ~?~H)
(la)
H* = J - Z ( ( ~ F + ( y G + (~H)
G* = J-l(77xF. + ,TyG. + UzH,,)
S:, = J-I((~F~ + (~C. + (~U~)
j _ a(~,77,() _ 1/
o(x,y,z)
Q =
x~
xv
x(
Y~
Yv
Y(
z¢
zv
z(
(~b)
p
pu
pu
pu 2 + P
puv
F =
pv
pwl
paw
(E + P)u
0
Txx
F~=~
1
Txy
rxz
M A T H E M A T I C A L MODEL
Urxx T vrxy T w1"xz
Governing Flow Equations
T h e governing equations are derived from the
time-dependent, compressible Navier-Stokes equations (strong conservation law form) cast in terms of
mass-averaged variables and Cartesian coordinates.
In order to facilitate the implementation of a finitedifference solution procedure and the treatment of
b o u n d a r y conditions, the Navier-Stokes equations are
pv
puv
G --
pv u + P
pvw
(E + Ply
,
"r#
O(P/p)
( ~ - 1)Pr
Ox
24
Journal of Thermal Science, Vol. 7, No.l, 1998
is evaluated with Sutherland's law, and the turbulent
viscosity is computed with a turbulence model, which
will be discussed later. The molecular and turbulent
Prandtl numbers are assumed to be constant at 0.72
and 0.9 respectively.
0
~y
1
G
Tyy
= -ffe
ry,
7#
a(P/p)
(~/- 1)Pr
Oy
Numerical
Algorithm
In integral form, the governing Eqs.(1) can be written as
p12~
puw
H =
pvw + P
,
0
pw 2
(E + P)w
+ff
ds=0
(6)
for a fixed region V with boundary S. Here Q represents the conserved quantity and E is the corresponding flux terms, where
"rxz
1
= ( F - F,,)~--) + ( a - G , ) 7
I4,, = -~ee
+ (H - H,)-~
(7)
Tzz
ur~z + Vry, + w%~ -
7~
O(P/p)
Dz
(lc)
Flow stagnation conditions are used as reference quantities, the density p, the velocity components u, v, w,
and the internal energy E are expressed by
p'
=--, PP
Po
n--~-----n= z + n y j
(8)
+nz
( 7 - 1)Pr
p'
p'
, # =--,
poa~
#o
u', if, w I
E'
u,v,w=--,
E=-T=
ao
poa~'
T'
a2~!
where 'indicates a dimensional quantity and a0 is the
stagnation speed of sound. Thus, the Reynolds number Re is based on stagnation condition. The temperature is given by
E = p [ c , T + ~(u2 + v2 + w2)]
The governing Eqs.(6) are discretised by first dividing the. computational domain into hexahedral cells
(Fig.2) and then approximating Eq.(6) for mass, momentum, and energy conservation in each cell. We
assume that the dependent variables Q are known at
the point (i, j, k) where each such point is the center of
one of the cells. If we apply a semi-discretisation (i.e.
a discrete approximation to the spatial terms only),
we obtain the following discrete analog of Eq.(6).
(9)
d(Q~a,k) + E~,~-k = 0
where
(Io)
(2)
and the pressure can be determined with the ideal gas
equation of state:
P = pRT
(3)
The viscosity, p and the ratio p / P r , where P r is the
Prandtl number, are defined as
,j,k+ ~
= ~.,.
+ ~,u~b
i,j,k-
(4)
--4
~-~lan
~£urb
- + - (5)
Pr
Brian
Prturb
where the subscripts "lam" and "turb" refer to laminar and turbulent, respectively. The laminar viscosity
T h e vector S i + ½,j,k denotes the cell face between the
points (i, j, k) and (i + 1, j, k). The value of E at the
cell face (i + 1 / 2 , j , k ) is taken as the average of E at
the points (i, j, k) and (i + 1, j, k).
H. Hu et al.
Computational Studies of Lobed Forced Mixer Flows
/
.//~ i+1/2,/,
25
where Qn and Qn+l are the values of Q after the nth
and ( n + l ) t h time steps respectively and
k
1
a1=~,~2=~,
3".,i.k
~
1
1
~3=~,~4=1
In order to increase the Courant number, the residual averaging is used implicitly in the product form:
~ i
(1 - e¢6~)(1 - ev6nv)(1 - e~g~i)Ri,j,k = Ri,j,k
Fig.2 Hexahedral cell around point (i, j, k)
In order to eliminate the numerical odd-even decoupling and capture shocks without any preshock oscillation, artificial viscosity Di,j,k is employed in the
equations (Ref.10), so the governing Eqs.(9) can be
written as
d (Q~j,k)) + Eij,k - Di,j,k = 0
(12)
where
(15)
where Ri,j,k is the residual at each cell, 5~,m5~¢ are
the three second order differential operator and the
~ ~, e¢ are three coefficients correspondingly.
Turbulence
Modeling
Because algebraic models are less CPU intensive
than other models, previous works have shown that
the turbulence model variations had little impact on
the flow-field solution of the mixer flow, so the twolayer eddy viscosity model of Baldwin-Lomax (Ref. 11)
is employed for wall boundary layers and mixing region. The turbulent viscosity is calculated as follows:
D,j,k = (D~ + D . + D¢)Qij,k
D~Qi,£k = di+t/2,j,k -- di-I/2,j,k
hi + l /2,j,k
di+l/2"j'k =
At
(4)
X [e(2+)l/2j,kAxQi,j,t ~ -- £ i+l/2,j,k
h 3 ~
x(~i-l,J, k]]
(13)
AQi,j,k = Qi+l,j,k - Qi,j,k
i+l/2,j,k = k (2) m a x ( v i + l , j , k ,
E (2)
(16)
The internal formulation of the turbulent viscosity
is based on a mixing length l, function of the local
aerodynamic quantities of the flow and of the normal
distance from the wall.
vi,j,k)
(#t)i,*n~r = Pl21al
e(i4.l)l/2,j,k ~- m a x ( 0 , k (4) - e (2)
i+l/2,j,k)
In the outer region of the boundary layer,
is expressed as below:
Pi+l,j,k -- 2Pi,/,k + Pi-xj,k
vij,,~ = I ~ ; ~
+ P~-lj,k I
K (2) = 0.5,
{ (ut)i ........... y < y~
(m)o., ........ y --- y~
u, =
(Pt)o,~t~r : 0.618ccvpFwFk
(18)
F k ( y ) = [ 1 - 5 5 ( cky ~°1-1
(19)
K (4) = 0.01
The governing Eqs.(12) can be solved in a class of
four stage R u n g e - K u t t a schemes, so it can be written
as
where
"
\Ymax/
J
U2
[ 1 Cwk
-F~ma
-DtFx )'~
F., = y m a x F m a x min ~,
Q(O) = Q .
ccp = 1.6;
Q ( I ) __ Q;:j,k - °tl R(Q(°))i,J,k
Q(2) = Q .i,j,k _ a2R(Q(1))ij,k
(14)
Q n + l ____Q(4)
(20)
c,~k = 0.25; ck = 0.3
The quantities Ymax and Fm~x are determined from
the maximum value of the function
Q(3) = Qni,j,k _ a3R(Q(2))ij,t~
Q(4) __ Q~,j,k - ot4R(Q(a))i,J, k
(17)
(•t)outer
{ yll~l[1-exp(-y+/26)]
F(y) =
yll21for boundary flow
for
wake
(21)
26
Journal of Thermal Science, Vol. 7, No.l, 1998
Geometry Modeling
The schematic of the studied lobed mixer is shown
in Fig.3. Because of the symmetric character of the
2-D mixer, only one half of lobe of the geometry is
modeled. According to the different boundary conditions, three computed domains are separated for the
flow field of lobed mixer, namely the upper stream,
the lower stream and the wake.
Synmactric
f-
_ ..~"_
"
,~,- - - - ~
;.1..~...4.
:
--" d,
.e *
.....
-
,
r ......
,
.
.- = 1 2 :" m / s
"
' Domain
z*
,'egio0.
v .I.._ _ 2"~I... _-_--.
phme
.."2-~_~ . . . .
D,,m,,,n 2
i.=66mm
!,~"=~2
" "
F~Co,nptLted
Syntmetri,~
plane
~
).-=-6~_~ ram"
3
....................
"~"
Sy nuuetric
(~
"'"
plane
d
d,,mai,12
Fig.3 Schematic of the lobed mixer and
studied three domain
Grid G e n e r a t i o n
The grid is generated in the three separate regions
defined in Figs.4-6. The 3-D finite difference grid is
constructed as a sequence of 2-D grid using T T M
method. Mesh points in each cross-sectional plane
in the physical space are related to those in a corresponding uniform computational space (A~ = At/ =
A~ = 1) by the transformation derivatives and transformation Jacobian in Eq.(lb). The results presented
in the following, obtained using a 28 x 20 (on a crosssectional plane)x 30 (in the streamwise direction).
.v/2-0.25
(b)
,v/2-1,0
(c)
x/A-2.0
(d)
x / 2 - 3.0
(e)
.v/2- 5,0
(f)
F i g . 6 Grid in the studied domain 3
Boundary Conditions
Z
X
a , 3 - d ~ t d , , f , , m d i e d d,~mam ]
Ibl
.'¢ - - S2 rren
,el
x=-41
q,lJ
mm
x = ii
F i g . 4 Grid in the studied domain 1
The boundary condition required for Navier-Stokes
calculation are the total pressures (PT,p and PT,~)
at the primary stream and secondary stream in flow
b o u n d a r y of the computational domain, and static
pressure (Pa) at the downstream b o u n d a r y of the
computed domain 3. The wall boundary conditions
are used at the top of the computational domain 1 and
the b o t t o m of the computational domain 2. Symmetrical b o u n d a r y conditions are employed at the other
sides of the computational domain.
RESULTS AND DISCUSSION
( a '~ 3--d ~ t d o f s t u d i e d d o m a i n 2
x - - 82 m m
x = - 41 m m
x = cl
S t r e a m w i s e M e a n Velocity C o n t o u r s
F i g . 5 Grid in the studied domain 2
T h e axial velocity contours at selected downstream
H. Hu et al.
locations are presented in Figs.Ta-e. In the wake region the streamwise vortex convected the high speed
stream into the lower speed region, and by the same
token, the low speed flow also p e n e t r a t e d into the high
speed region. Subsequently, turbulence was being generated and diffusion enable the s m o o t h i n g out of the
velocity gradients. Effective mixing can be achieved in
the wake between the two streams. T h e velocity distribution becomes more uniform at the further downstream station. In general, the c o m p u t e d results capture the m e a n flow field reasonably well when comparing with measurements.
.--..
I
f,
•',C
ii
• (. . . .
'g"
.,
v/2 = o.25
(a)
.v/2 = 1.0
(b)
x/2 = 20
(e)
v/2=5 o
(e)
.v/2 =3,o
(d)
Fig.7 Contour of the normalized streamwise mean
velocity (U/U,.) at domain 3 (U~=IO.O m/s)
Secondary
Flow
Velocity
Vectors
T h e secondary velocity vector plots are shown in
Figs.8a-e.
As expected, a streamwise vortex was
formed immediately at the trailing edge with centre of
the vortex core located at a r o u n d the inflexion point of
the lobe (Fig.5a). T h e s t r e n g t h of the secondary m e a n
velocity maintains nearly the same s t r e n g t h from the
trailing edge to a b o u t x / A = 2 . 0 (see Figs.5b-c). At
region after x/A -- 3.0 (Fig.5d) and to the end of
the m e a s u r e m e n t range, the secondary flow decreases
rapidly. C o m p a r i n g the corresponding vector s t r e n g t h
with those measured, it was found t h a t the decay for
the secondary m e a n velocity was actually slower t h a n
t h a t found in the experiment.
..,..
.
......
'
.,,•.
I
(a)
F = ~ Y . dS
x/;~= I.o
(b)
x/,;, = 2.0
(¢)
x/2=3.0
(d)
:5:-- I
x/),=5.0
(e)
P i g . 8 Velocity vector field in domain 3
(22)
where the circuit C is the outer p a t h of the c o m p u t a tional domain, as d e n o t e d by the s h a p e d area given
in Fig.3. T h e result of which are shown in Fig.9.
A slower decay is found for the calculation b u t the
t r e n d is very similar. Based on the findings of Yu and
Yip [7], the streamwise vortex development u n d e r w e n t
a t h r e e - s t e p process by which it was formed, intensified and quickly dissipated towards the end of the
near field. T h e spatial spacing between two rows of
streamwise vortices of alternate signs within a lobe
1.l
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
O.1
[~
i
i
~
,
~
i
i
:k I e a M L r e n ! e l l D , ]
]
'
i
'
I
'
i
,
t
0.0 10 2.0 3.0 -ko 5.0 6.0 7.0 8.(1 9.0 100
.v/),
Fig.9 Variation of normalized streamwise circulation
with downstream distance
Nonnal vortex line
__
at~et~ a,:,,:e 1o t!l~
:2""
x/;,=o.~5
Decay of the Streamwise Circulation
T h e streamwise vortex plays an i m p o r t a n t role in
the mixing process. In the results presented next, the
s t r e n g t h of this vortex is characterized by circulation
defined by
Initial pinched-off
"7-:1
7
2~
Computational Studies of Lobed Forced Mixer Flows
]Peak Trough
streanm lse v o r ~ e t ~
.kdditional pinched
- off effects
0
D
due to tile
,~ "1'~'~4.,(~",
nonnal vorticity. ~ . . ~ k . ~
( b ) End view
,/~ \component rides
~, r ~ o n
the stream- ~ ~...
vortex line to
l n'oducetile additional
i l ~inched- off effects
( c ) Tc ) vie~
Fig.lO Schematic of the vortical structure
28
Journal of Thermal Science, Vol. 7, No.l, 1998
reduced with downstream distance causing the mutual annihilation of the vortices. Analyzing this movement of vortices, based on inviscid vortex dynamics
(as shown in Fig.10), suggested t h a t the normal vorticities shed at the trailing edge had actually provided
a "pinched-off' effect to two adjacent streamwise vortex lines within a lobe forcing the streamwise vortices
to interact with each other. However, the vector plots
in Fig.8 did not show the expected movement of the
vortex. It is advisable that further computational domain should encompass one full lobe rather than a half
lobe as most of the calculations were being conducted.
CONCLUDING
REMARKS
C o m p u t a t i o n a l studies has been conducted on the
lobed mixer flows. In general, good agreement can
be obtained for the streamwise mean velocity distribution at successive downstream station. However,
the decay of the streamwise circulation is somehow
underpredicted. The difference m a y be due to the
fact that normal vorticities shed at the trailing edge
actually provided a "pinched-off' effect to two adjacent streamwise vortex lines within a lobe forcing the
streamwise vortices to interact with each other. This
m a y provide a faster decay to the streamwise circulation. Future computational work should include one
full lobe instead of a half lobe.
REFERENCES
[1] Presz, W.M., Blinn, R.F. and Morin, B. "Short Efficient Ejector Systems," AIAA paper No.87-1837, Jan.,
(1987).
[2] Presz, W.M., Gousy, R. and Morin, B. "Forced Mixer
Lobes in Ejector Design," AIAA Journal of Propulsion
and Power, 4, No.4, p.350, July (1988).
[3] Paterson, R.W., "Turbofan Mixer Nozzle Flowfielda Benchmark Experimental Study," ASME Journal of
Engineering for Gas Turbines and Powers," 106, 692,
July, (1984).
[4] Barber, T., Paterson, R.W. and Skebe, S.A., "Turbofan
Forced Mixer Lobe Flow Modeling Vol.l: Experimental
and Analytical Assessment," NASA CR4147, (1988).
[5] Werle, M.J., Paterson, R.W. and Presz, W.M. Jr.,
"Flow Structure in a Periodic Axial Vortex Array,"
AIAA Paper No.87-0610.
[6] McCormick, D.C. and Bennett, Jr. J.C., "Vortical and
Turbulent Structure of a Lobed Forced Mixer FreeShear Layer," AIAA Journal, 32, No.9, pp.1852-1859,
(1994).
[7] Yu, S.C.M. and Yip, T.H., "Measurements of Velocities in the Near Field of a Lobed Forced Mixer Trailing
Edge," The Aeronautical Journal of the Royal Aeronautical Society, 101, pp.121-129, (1997).
[8] Koutomos, P., and McGuirk, J.J.,"Velocity and Turbulence Characteristics of Isothermal Lobed Mixer Flows,"
ASME Journal of Fluids Engng., 117, pp.633-638,
(1995).
[9] Tsui, Y.Y. and Wu, P.W., "Investigation of the Mixing
flow Structure in Multilobe Mixers," AIAA Journal, 34,
No.7, pp.1386-1391, (1996).
[10] Jameson A. and Baker T.J., "Solution of the Euler
Equation for Complex Configuration," AIAA Paper
No.83-1929, (1983).
[11] Baldwin B.S. and Lomax, H., "Thin-Layer Approximation and Algebraic Model for Separate Turbulent
Flows," AIAA Paper No.78-0257, (1978).