The results of numerical simulations of turbulent channel flows were used to examine the validity of the local approximation of the pressure-strain term in the Reynolds stress transport equation. Outside of the viscous sublayer the local approximation compares very well with the exact pressure strain. This agreement is due, at least in part, to the high correlation between the rapid pressure and its Laplacian, which suggests that only the near parts of the flow contribute to the rapid pressure at a point. In the viscous sublayer the distance over which the mean shear can be considered constant is comparable to the length scale in the normal direction of the correlations of velocity gradients, leading to failure of the local approximation

Bradshaw, P.

Mansour, N. N.

Piomelli, U.

English

NASA Technical Reports Server

Center for Turbulence Research
Proceedings of the Summer Program 1987
159
N 88 - 23 1 0
On Local Approximations of the
Pressure-Strain Term in Turbulence Models
By P. BRADSHAW 1, N. N. MANSOUR 2 and U. PIOMELLI 3
The reason for the success of the approximation that uses the mean velocity
gradient OU/Oy outside the integral solution of the Poisson equation for pressure,
and sets the gradient equal to its value at the point where the pressure is being
calculated, has been explored. This approximation is implicit in most existing
turbulence models, where the pressure-strain terms are assumed to be functions of
local variables rather than of the proper spatial integrals. Direct simulation data
for the channel were used to evaluate spatial correlations of pressure and velocity
gradients. The results show that a correlation coefficient of about -0.5 between
the rapid pressure and its Laplacian (proportional to Ov'/Oz); this is in favor of
the local assumption. Analysis of the solution to the Poisson equation indicates
that the assumption will be valid when the mean velocity gradient varies slowly as
compared to the correlation length of the fluctuating velocity gradients.
1. Introduction
The pressure-strain "redistribution" terms in the Reynolds-stress transport equa-
tions, which are mean products of the pressure fluctuation and various components
of the fluctuating rate of strain, have received much attention from turbulence mod-
elers for two reasons: first, the pressure-strain terms in the shear stress equations are
the largest of the terms that need to be modelled in those all-important equations;
secondly, little experimental data is available to properly assess the models.
In an experimental setup, pressure fluctuations within a boundary layer cannot be
measured with any assurance of accuracy, because the velocity fluctuations induce
pressure fluctuations on any solid probe inserted in the flow, and these spurious
fluctuations are usually larger than those in the undisturbed flow. Using the in-
stantaneous velocity field in the entire domain, turbulence simulations (numerical
solutions of the full time-dependent Navier-Stokes equations without any modeling
approximations) can be used to deduce statistics of measurable and of "unmeasur-
able" quantities such as those we will discuss in the next section. We shall discuss
in this paper an approximation to the pressure-strain term which is commonly used
in modeling the Reynolds-stress transport equations.
1 Imperial College
2 NASA Ames Research Center
3 Stanford University
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160 P. Bradshaw, N. N. Mansour and U. Piomelli
2. Local approximation to the pressure
For an incompressible flow the fluctuating pressure can be obtained from the
velocity field by solving the Poisson equation
02P - 2 0Ui Ou_ I Ou_ Ou_ Ou_ Ou_ _ (la)
O_jO_,j _ OX i _ OZj OX i OXj OZ i ]
with the boundary condition at the walls
Op = l___O_v ' (lb)
Oy Re Oy 2"
Equation (la) shows that any correlations of the pressure with other fluctuating
quantities will have a part proportional to the mean velocity gradients OUi/Oxj
and a part depending only on velocity fluctuations. These are called the "rapid"
and "slow" parts respectively, because only the former will respond at once to a
change in the mean velocity field. An inhomogeneous boundary condition makes
the split less obvious. However, for the case of the fully developed channel the
volume integrals of each of the terms on the right hand side of (la) will integrate
to zero and pressure in this case can be split in three parts: the rapid pressure pl,
which satisfies
02p I _ 2 0Ui Ou_ (2a)
OxjOxj Oxj Ozi
with the boundary condition at the walls
pl
-0; (2b)
0y
the slow pressure p2, which is the solution to:
] (3a)
with the boundary condition at the walls
aPl O; (3b)
oy
and, finally, the Stokes pressure pS, which is the solution to Laplace's equation with
the boundary condition (lb) at the walls.
The pressure-strain terms that appear in the Reynolds stress equations are linear
in p, so that the total term will be equal to the sum of the rapid pressure-strain, slow
On Local Approzimations of the Pressure.Strain term 161
I? o. /;...../-_.......................
o 50 lOO 15o
y+
FIGURE 1. Pressure-strain term in the -_-_ transport equation.
pressure-strain and Stokes pressure-strain. The solution of (2a) with the boundary
condition (2b) at the walls is
p] _ 1 Iv OUi Ou_4rr 2 0zj Oxi
Gdl; (4)
in which G is Green's function with homogeneous Neumann boundary conditions at
the walls. Note that most modelers neglect the surface integral terms that should be
added to (4) if non-homogeneous Neumann conditions are used for the pressure. The
use of the homogeneous boundary condition (2b) at the walls for the rapid pressure
is consistent with (4) and the approximation used by the modelers. Although, by
(4), the rapid pressure is an integral of the weighted mean velocity gradients over
the entire domain, in modeling the assumption is often made that pl depends only
on the local mean velocity gradient, which allows one to take the mean velocity
gradients outside of the integral in (4) to yield the following approximation for the
pressure:
pB 1 _ OUi Ou_ 'd
Direct experimental measurement of either p] or pB is impossible, but the use
of direct simulation results allows us to test the approximation (5) directly by
computing the integrals in (4) and (5) and comparing the approximate value pB with
p]. The purpose of the present paper is to examine the results of this comparison
for channel flow at low Reynolds number.
3. Results
The results of the direct numerical simulation of fully developed channel flow at
Re,. = 180 (Kim, Moin and Moser, 1987) were used to compute all the statistics
presented in this work.
162 P. Bradshaw, N. N. Mansour and U. Piomelli
X
I!
II
t'v
FIGURE 2. Two point correlation of the rapid-pressure term with Ov/Oz.
Figure 1 shows the contribution of the rapid pressure-strain term in the -u'v'
transport equation. All results are normalized by pub v: on this scale the maximum
rate of production of turbulent energy, which occurs where r = r_/2, i.e. at about
y+ = 12, is 0.25. Note that the plotted terms are negative if -u'v' is being reduced
and positive if it is being increased by the pressure-strain redistribution.
The accuracy of the local approximation to the pressure-strain terms is sur-
prisingly good: differences between the exact and approximate values are gener-
ally within the scatter of the statistical averaging, except in the viscous sublayer
(y+ < 30). The rapid pressure-strain contribution to the other Reynolds stress
transport equations shows similar trends.
To further investigate the reasons for the success of the approximation outside of
the sublayer, as well as to ascertain the causes of its failure inside it, we examined
the correlation coefficient between the rapid pressure pl and various components of
the fluctuating strain-rate tensor for varying distances from the wall.
The most noticeable feature of the simulation results is the high correlation be-
tween pressure and velocity gradients. In particular, the coefficient of correlation
between pl and Ov'/Oz (Figure 2) is numerically as large as 0.5. This implies a
correlation coefficient of -0.5 between the pressure and its Laplacian. The rela-
tively high value for pl is noteworthy. It is not high enough to legitimize the pB
On Local Approzimations of the Pressure-Strain term 163
I 1
I.................... ................. .......i -.n.i-t:--.V.........: ....y.:...=...Zs.,6.1.......-.I
i ! i - _il_ \ i y,=46.63 i
d ...................] .................i............. ii..............i y+=18o.oo ....
x : : I _ _. : i i
n,"i ...............---.ii..................._7i........ _'_q-:':f_-_ '_,i'.......... i ...................i_...................I
! . / i.k" ' io-L ............! ........../.Z_ .....i....._i_.....k ........, I
I-_. ....2-" " .: "_':__Z_i
I..................., ........., ........., .........; .........;. ...!
d _ : : : :,I, ; ; • , /
-o.3 -o.2 -o.1 o.o _,.1 d._ d.3
x/L
FIGURE 3. Two point auto-correlation of the rapid-pressure.
approximation directly, but it suggests that the rapid pressure is not so dependent
on contributions from distant parts of the flow as its governing Poisson equation
nominally implies. A demonstration of this dependency is the rapid increase of
the length scale of pl near the channel centerline, which can be inferred from the
auto-correlation of pZ Figure 3. Near the centerline the source term in the Poisson
equation for pl, being proportional to the mean shear, vanishes; contributions to p_
on the centerline, therefore, must necessarily come from regions off the centerline,
and have their high wavenumber parts attenuated by distance.
For homogeneous flows in the z and z directions, the pressure-strain term can be
written as follows:
Substitution of (4) into (6) gives
' f?£ " "7
OUt . t
-_x (z - z',y',z - z') COu_(z,y,z)G(x',y,y',z')dxdzdx'dz'dy '
Oxj (7)
164 P. Bradshaw, N. N. Mansour and U. Piomelli
whereas the local approximation (5) yields
o.,_
1 O0
Ov' ,, ,)
-_z (z - x ,y ,z - z -_zj(:c,y,z)G(x',y,y',z')dz'dz'dxdzdy'. (8)
The two inner integrals in (7) and (8) represent the two-point correlation between
Ov(y')/Ox and Ouj(y)/Ozi, weighted by Green's function G, as a function of y and
y'. This correlation can be expected to vanish as y - y' increases, since Green's
function decreases as y - y' increases and the velocity gradients are not correlated
over large distances. As long as the length scale of this correlation is small with
respect to the length over which the mean shear can be considered constant, the
local approximation will hold. In the logarithmic region the mean shear varies
slowly (02U/Oy _ ..- 1/y 2 ) and the local approximation holds• At the sublayer edge,
where the velocity gradient changes significantly, this is not true and the local
approximation fails.
4. Conclusions
The results of numerical simulations of turbulent channel flows have been used to
exanfine the validity of the local approximation of the pressure-strain term in the
Reynolds stress transport equation.
Outside of the viscous sublayer the local approximation compares very well with
the exact pressure strain. This agreement is due, at least in part, to the high
correlation between the rapid pressure and its Laplacian, which suggests that only
the nearer parts of the flow contribute to the rapid pressure at a point.
In the viscous sublayer the distance over which the mean shear can be considered
constant is comparable to the length scale in the normal direction of the correlations
of velocity gradients, leading to failure of the local approximation.
REFERENCES
KIM, J., MOIN, P. & MOSER, R. D. 1987 Turbulence statistics in fully developed
channel flow at low Reynolds number. J. Fluid Mech. 177, 133-166.


On local approximations of the pressure-strain term in turbulence models

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