A non-local model is presented for approximating the pressure diffusion in calculations of turbulent free shear and boundary layer flows. It is based on the solution of an elliptic relaxation equation which enables local diffusion sources to be distributed over lengths of the order of the integral scale. The pressure diffusion model was implemented in a boundary layer code within the framework of turbulence models based on both the kappa-epsilon-(bar)upsilon(exp 2) system of equations and the full Reynolds stress equations. Model computations were performed for mixing layers and boundary layer flows. In each case, the pressure diffusion model enabled the well-known free-stream edge singularity problem to be eliminated. There was little effect on near-wall properties. Computed results agreed very well with experimental and DNS data for the mean flow velocity, the turbulent kinetic energy, and the skin-friction coefficient

Demuren, A. O.

Durbin, P.

Lele, S. K.

English

NASA Technical Reports Server

Center for _rbulence Research
Proceedings of the Summer Program 199J1
Role of pressure
non-homogeneous
N95- 21053
diffusion in
shear flows
313
By A. O. Demuren, 1 S. K. Lele 2 AND P. Durbin 3
A non-local model is presented for approximating the pressure diffusion in calcula-
tions of turbulent free shear and boundary layer flows. It is based on the solution of
an elliptic relaxation equation which enables local diffusion sources to be distributed
over lengths of the order of the integral scale. The pressure diffusion model was
implemented in a boundarylayer code within the framework of turbulence models
based on both the k - e - v2 system of equations and the full Reynolds stress equa-
tions. Model computations were performed for mixing layers and boundary layer
flows. In each case, the pressure diffusion model enabled the well-known free-stream
edge singularity problem to be eliminated. There was little effect on near-wall prop-
erties. Computed results agreed very well with experimental and DNS data for the
mean flow velocity, the turbulent kinetic energy, and the skin-friction coefficient.
1. Introduction
In higher-order turbulence models 'pressure diffusion' is usually neglected, or at
best added to 'turbulent diffusion' (Launder 1984) and the two modeled in aggre-
gate. Pressure diffusion refers to the term Oi-_jp in the Reynolds stress budget;
turbulent diffusion refers to C3k_,ujuk. The latter represents the ensemble averaged
effect of turbulence convection and can often be modeled as a diffusion process;
the former, however, is harder to explain as diffusion. Turbulent diffusion is usu-
ally considered to be the dominant diffusion mechanism and pressure diffusion is
considered to be negligible. However, Lumley (1975) showed that, for homoge-
neous turbulence, the application of symmetry constraints to the exact equation
for the "slow" or non-linear part of the pressure diffusion led to the result that its
magnitude is 20% of that of the triple velocity correlation. In the present study,
DNS databases for several shear flows were examined, namely: the mixing layer
simulation of Rogers and Moser (1994); the wake simulation of Rogers (private
communication); the boundary layer simulation of Spalart (1988); and the back-
ward facing step simulation of Le & Moin (1994). These confirm that, in the main
shear regions, pressure diffusion is roughly 20-30% of turbulent diffusion. However,
it appears to be mostly counter-gradient transport, so that it merely reduces the
effect of turbulent diffusion, which is mostly gradient transport. Thus, the current
practice of absorbing pressure diffusion and turbulent diffusion into a single model
1 Old Dominion University
2 Stanford University
3 Center for Turbulence Research
_,k-'_ PAGE. B_#,HK NOT FIL_,_P
"o{5 . ,.,_......
https://ntrs.nasa.gov/search.jsp?R=19950014636 2020-06-16T08:28:01+00:00Z
314 A. O. Derauren, S. K. Lele _ P. Durbin
FIGURE 1.
80
6O
4o i
2oi
o!
.7.5
/" \..
,'l'''''''l'''''''''l'''''''''l'''''''''l ......... I''" ......
-5.0 -2.5 0 2.5 5.0 7.5
(y - yc)/6. 
Computed eddy viscosity distribution in a two-stream mixing layer.
term appears reasonable, as far as the main shear regions are concerned. But the
DNS data show that near the edges of the shear layers, turbulent diffusion decreases
rapidly to zero, while pressure diffusion decreases only very gradually, so the latter
then becomes dominant. Thus, the budgets show that near the free-stream edge
the balance is between pressure transport and mean convection, or temporal drift,
rather than between turbulent transport and the latter. When applied to one or
two-equation models (Coles 1968, Cazalbou et al. 1994), an assumed balance be-
tween evolution and turbulent diffusion leads to an unsteady non-linear diffusion
problem whose solution has a propagating front at the edge of which the eddy vis-
cosity vt, the turbulent kinetic energy k, and its dissipation rate _ all go sharply
to zero. Fig. 1 illustrates how the eddy viscosity drops abruptly at the edge of the
shear layer, even though a non-zero value was imposed in the free stream. This sin-
gular solution is not in agreement with experimental data, which show that all these
properties asymptote gradually to free-stream values. A consequence of the singu-
larity is that computations with these turbulence models are unable to properly
account for free-stream turbulence effects since the free stream and the main shear
regions are decoupled, except where insufficient grid resolution produces smearing
via numerical diffusion. This result would also apply to second moment closures
which use gradient diffusion models for turbulent transport.
The free-stream edge singularity can be removed by introducing a model for the
pressure diffusion which does not vanish abruptly at the edge of the main shear flow.
Since pressure transport should be non-local, a new model for pressure diffusion is
introduced based on the elliptic relaxation concept introduced by Durbin (1991,
1993) for modeling the pressure-strain correlation in non-homogeneous turbulent
flows. Computations with the new model are compared to experimental data for
the plane mixing layer and the boundary layer flows.
Role of preJsure diffuJion in shear flows 315
2. Mathematical model
In the present study only simply shear flows (mixing layers, wakes, boundary
layers) are considered, so that boundary layer forms of the governing equations are
applicable. For simplicity, we shall only present equations for the k - _ - v 2 model
(Durbin 1991, 1994). Though the _-_ model was also utilized, the results are
virtually the same for the simple shear flows considered here.
The mean flow equations are:
D,U = O_[(v+ v,)OvUl (i)
v. u = o (2)
for incompressible thin shear layers. (D, represents the total derivative and O_ the
cross-stream partial derivative.)
_.I k - _ - v2 model
The usual k- _- v-'_ model (Durbln 1994_.)_ismodified by the addition of a pressure
diffusion term to the k-equation and the v2 equation. Thus, we have:
D,k = Pk - _ + Ol,[(v + v, lat)cgyk] - cgs,(-pT) (3)
r + Ov[(v + vt/a,)£gy_] (4)
-- --C
D,v 2 = _f22 -- IJ2"_ -_- Oy[(V "_- Vl/Crk)ay_ -_] -- 20.(_--_)
where the rate of turbulent energy production is
Pk = v,(OyU)2
(5)
(6)
and the eddy viscosity is given by
v,= C_-_T (7)
The term kf2"-'_ is the source of v 2 via redistribution from the streamise com-
ponent u-'_. This is the pressure strain term in homogeneous turbulent flow. In
non-homogeneous turbulent flows, non-local effect are introduced through the ellip-
tic relaxation equation
v2/k2 2 -- (8)L o_f22-/22 = (1 - c,) 2/3 c2P_
T k
The lengthand time scaleswhich appear in the equationsare givenby
L = eL max [--7' C,,
(9)
316 A. O. Demuren, S. K. Lele _ P. Durbin
T = max , CT (10)
In free shear flows, equations (9) and (10) are modified by removing the Kol-
mogorov limits which are strictly only relevant to wall bounded flow. The bound-
ary conditions follow Durbin (1994) and are not repeated. Their main effect is to
produce the proper behavior of k, ¢, and v2 near no-slip boundaries.
The empirical coefficients, are CL = 0.3, C_, = 0.19, C_ = 1.3 and C_ 2 =
1.90, CT = 6.0, C_ = 70.0, C1 = 1.4, C2 = 0.3, ak = 1.0, a, = 1.3.
2.2 Pressure diffusion model
The pressure transport is modeled as:
-cgy_ = ayf (11)
Non-local effects are introduced via the elliptic relaxation equation
_ / = _/L (12)
where fL is a local gradient diffusion model
fL : c _tcgyk (13)
Uk
It is assumed that the same length scale which governs the non-locality in the
pressure redistribution would also govern the non-locality in the pressure trans-
port. C° is a free parameter which is determined by optimization. The boundary
conditions are
f = 0 ; free stream
Oyf = 0 ; no-slip wall
(14)
3. Results and discussion
Model computations were performed for a two-stream mixing layer with the ratio
of low to high free-stream velocities of 0.6 chosen to coincide with the experimental
study of Bell and Mehta (1990). A parabolic forward marching code was utilized.
Starting from hyperbolic tangent profiles, the solution was marched until self-similar
results were obtained. Computations were performed with 3 values of Cs, namely
0.0, 0.5, and 1.0. C, = 0.0 represents the case with no pressure diffusion. Com-
puted results of vt/v and k are compared in Figs. 2(a) and 2(b), respectively. It is
clearly seen that, at the higher values of Co, the edge singularity present with Co
= 0 has been removed, k profiles now go gradually to zero exhibiting long tails.
Also vt profiles gradually approach the small values set in the free stream, though
Role of pressure diffusion in shear flows 317
./:_ . o_
o.2O" 0.1-]
OJ_l_nl TM ...... g ......... i ......... , ......... i .........
°.7_......:_'_......:5._...... :8........_'_,......._o.......;rs .7.s .5.o a.5 o 2.s 5.0 7.5
FIGURE 2. Effect of diffusion coefficient C, on computations of the two-stream
mixing layer: ........ , C, = 0.0;--, C, = 0.5; .... , C, = 1.0. (a) eddy viscosity
(b) turbulent kinetic energy.
<1
I
0.50. -" •
0.25.
0
-0.25 i
-0.50- . ...... • .., ......... L
-7.5 -5,0 -2.5 0 2.5 5.0 7.5
-
FIGURE 3. Comparison of velocity profiles in the two-stream mixing layer to
• experimental data of Bell & Mehta (1990). (See figure 2 for legends.)
not monotonically. The vt profile obtained with C° = 0.5 is quite similar to that
given by the DNS data of Rogers and Moser (1994) with equation (7) as definition.
The peak value computed with C, = 0.5 was closest to that measured by Bell and
Mehta (1990); hence C_ = 0.5 was chosen for all subsequent computations. Fig. 3
shows a comparison of computed velocity profiles with Cs = 0.0, 0.5 versus experi-
mental data of Bell and Mehta (1990). Both model computations give the correct
spread rate but the computation with the pressure diffusion model shows smoother
profiles near the free-stream edges in agreement with experimental data. Similarly,
k profiles are compared in Fig. 4. In this case, the differences between model com-
putations with and without pressure diffusion are more dramatic. The non-local
role of pressure diffusion in transporting turbulent kinetic energy from the center
of the layer to the edges of the free stream is apparent.
318 A. O. Demuren, S. K. Lele _ P. Durbin
v-'-I
0.4'
0.3'
0.2'
0,1'
O'
-7.5
-5.0 -2.5 0 2.5 5.0 7.5
(u -
FIGURE 4. Comparison of k profiles in the two-stream mixing layer to • experi-
mental data of Bell & Mehta (1990). (See figure 2 for legends.)
With the pressure diffusion model thus calibrated, the question is whether the
improved agreement with data near free-stream edges would also be reproduced
in a wall boundary layer flow without producing an adverse effect on near-wall
agreement, where the role of pressure diffusion should be minimal. Computations
were performed for a developing wall boundary layer using the DNS data of $palart
(1988) as inlet conditions. The solution was then marched until self-similar results
were established. Profiles of t/t/t/across the boundary layer are compared for C, =
0.0, 0.5 and 1.0 in figure 5. The main effect of pressure diffusion is near the free-
stream edge, with little or no effect on near wall values. Skin friction coefficients are
compared to experimental data of Coles and Hirst (1968) in figure 6. This confirms
that the pressure diffusion produces negligible effect on near-wall properties. Finally
k profiles at R0 = 7,500, normalized with the local friction velocity, are compared
to experimental data of Klebanoff (1955) in figure 7. Again, pressure diffusion has
negligible effect in the near-wall region, but produces more gradual decay near the
free-stream edge.
Thus, the sharp free-stream edge problem can be cured by introducing a non-
local model for the pressure diffusion. In the present study a gradient diffusion
model was utilized for fz. Although it enables the edge singularity problem to be
overcome, it appears too simple to reproduce all features of the pressure transport
through the layer and near the free stream. It can be argued that we are only
modeling the deviation from the usual model treatment, i.e., the part which is not
directly proportional to turbulent diffusion. Nevertheless, it is desirable to explore
more general forms of .fL. Consideration is being given to such models. To aid in
this study, we hope to obtain a splitting of the DNS mixing layer data of Rogers
and Moser (1994) for the pressure diffusion into slow and rapid parts. We are
also studying the form of the pressure diffusion term in the shearless mixing layer
simulation of Brlggs et al. (1994). Initial computations of the shearless flows studied
Role of pressure diffusion in shear flows 319
150'
100
5O
......... i | ...... _" " "
0.5 1.0 1.5
tt/a99
FIGURE 5. Effect of diffusion coefficient C, on eddy viscosity distribution in
boundary layer. (See figure 2 for legends.)
7=
r.I
4
0 e
• e_•
...... d • •
, • w • w U • m i @ u u e w w | u g u i w • e | • | • • w u • | w • i | • w I • • w w • •
1000 2000 3000 4000 5000
Ro
FIGURE 6. Comparison of skin friction coefficient in a boundary layer to • exper-
imental data of Coles & Hirst (1968). (See figure 2 for legends.)
320 A. O. Demuren, S. K. Lele _¢ P. Durbin
_ 4"-'.
CD
2" • • .,.
• °"',,..
0 , , o o , , • ,
0 0.5 1.0 1.5
_'/_99
FIGURE 7. Comparison of k profiles in a boundary layer to • experimental data
of Klebanoff (1955). (See figure 2 for legends.)
experimentally by Veeravalli and Warhaft (1989) show that most of the current
turbulent diffusion models with or without the present diffusion model significantly
underpredict the shear layer spread rate.
4. Conclusions
A non-local model for pressure diffusion has been developed based on the solu-
tion of an elliptic relaxation equation. This enabled the free-stream edge singular-
ity problem, which most current turbulence models suffer from, to be eliminated.
The pressure diffusion model did not produce any undesirable effects on near-wall
properties. Current turbulent diffusion models with and without the new pressure
diffusion model are unable to predict observed growth rates in shearless turbulence
mixing layer.
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Role of pressure diffusion in shear flows 321
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Role of pressure diffusion in non-homogeneous shear flows

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