25

Reynolds Stresses In and Out

[upon losing the use of his right eye] Now I will have less distraction. (Leon-
hard Euler)

25.1 Introducing Reynolds Stresses

The traditional approach to mathematical modeling of turbulence is to seek
modified NS equations satisfied by some mean value (¯

u, ¯

p) of the true velocity-

pressure (u, p). In Reynolds Averaged Navier–Stokes equations RANS the
mean value is an ensemble mean, or a time average taken over long time,
while in Large Eddy Simulation LES the mean value is more local in space-
time. The modified equations for the mean-values are sought by taking mean
values of the NS equations to get the Averaged (or Filtered) NS equations:

∂ ¯

u

i

∂t

+

j

(¯

u

j

¯

u

i

)

,j

− ν∆¯u

i

+ ¯

p

,i

+

j

τ

ji,j

= ¯

f

i

,

∇ · ¯u = 0,

(25.1)

where

τ

ji

= u

j

u

i

− ¯u

j

¯

u

i

are the so-called Reynolds stresses. The idea is then to seek to model the
Reynolds stresses in terms of the mean-values (¯

u, ¯

p) in a turbulence model (or

subgrid model) to get a set of modified NS equations for the mean value (¯

u, ¯

p).

Many turbulence models have been proposed in the literature, see e.g. [98],
but all models only seem to cover the set of test problems they were designed
for, and thus lack the generality required to be able to model new problems
and make predictions.

So, designing turbulence models of the Reynolds stresses seems to be a

very difficult if not an unsurmountable problem. But do we really need to
model the Reynolds stresses?

202

25 Reynolds Stresses In and Out

25.2 Removing Reynolds Stresses

Suppose that we are interested in some output M (u, p) which itself is a mean
value. Using a turbulence model we would then obtain the output M (¯

u, ¯

p)

involving two mean value operations, to be compared with M (u, p) with only
one.

Now, averaging twice seems to be one too much, but what would be the

evidence that we could live without Reynolds stresses? This would be possible
if the effect of the Reynolds stresses on the output M (u, p) would turn out to
be small. Below we shall give computational evidence that this is true in many
cases. More precisely, the computational model we use contains a stabilizing
term, which may be viewed as a simple turbulence model, and we shall give
evidence that the exact nature of this model has little effect on mean-value
outputs. The net result is that very crude modeling of the Reynolds stresses
seems to be sufficient in many cases of practical importance. We expand on
this aspect in the next chapter. This means that we do not have to introduce
any Reynolds stresses at all, nor model them, even if the flow is turbulent! The
stabilizing term in the computational model will handle all that automatically!
In particular, we settle directly for computing the mean value M (u, p) and
avoid introducing the double mean value M (¯

u, ¯

p).

Obviously, avoiding Reynolds stresses greatly simplifies computational tur-

bulence modeling, since any chosen known turbulence model could be ques-
tioned on very good grounds.