6.9 Reynolds-averaged simulation

The computational cost of DNS and, to a lesser extent, LES is too great for most practical CFD, as discussed in Sec. 6.8 . Instead, a Reynolds-averaged simulation (RAS) provides a much more affordable method to calculate turbulence.

It solves equations for “averaged” field variables to avoid resolving small fluctuations. Rather than consider an average over a time interval, we imagine the same flow repeated several times under nominally the same initial conditions (2 examples below).

PICT\relax \special {t4ht=

Solutions vary due of differences in initial conditions and the chaotic nature of turbulence. The ensemble average calculates the mean solution eqn for multiple realisations of the same flow.

Each field, e.g. eqn, is decomposed into the averaged field eqn and field of random fluctuations eqn, according to

 -- 0 (t) = + (t): \relax \special {t4ht=
We can apply this decomposition to fields in the momentum conservation Eq. (2.27 ) with stress eqn split into eqn and eqn as follows:
@--u-+ r ( uu) = r ☐☐☐ rp + b: @t \relax \special {t4ht=
Let us assume constant eqn and that the body force eqn is not subject to turbulent fluctuations. Splitting the remaining fields, i.e. eqn, eqn and eqn, into instantaneous and fluctuating components according to Eq. (6.11 ), and taking the ensemble average of Eq. (6.12 ), yields the following:
@ u- --- ---- - -- ----+ r ( uu) + r ( u0u0) = r ☐☐☐ r p+ b @t \relax \special {t4ht=
The averaged Eq. (6.13 ) is derived using the following relations for general fields eqn and eqn: eqn; eqn; eqn; eqn. Relations for averaged derivatives are: eqn; eqn; eqn.

Reynolds stress

The terms for mean quantities in Eq. (6.13 ) are the same as in Eq. (6.12 ). The difference is that Eq. (6.13 ) includes the additional eqn term containing fluctuations eqn.

The argument of this divergence derivative is a tensor known as the Reynolds stress11

|------------| ☐☐☐t = u0u0 | ------------- \relax \special {t4ht=
Substituting Eq. (6.14 ) in Eq. (6.13) eliminates fluctuation terms. The remaining equation is in terms of averaged properties only, so we can dispose of the average notation (eqn) to give
@--u- t @t + r uu = r ☐☐☐ + ☐☐☐ rp + b: \relax \special {t4ht=
This ensemble-averaged equation is the same form as Eq. (6.13 ) but with the addition of eqn. Solving this Reynolds-averaged equation is the key to low cost CFD with turbulence — but it requires a model for the additional unknown eqn.
11Osborne Reynolds, On the dynamical theory of incompressible viscous fluids and the determination of the criterion, 1895.

Notes on CFD: General Principles - 6.9 Reynolds-averaged simulation