Notes on Computational Fluid Dynamics: General Principles

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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 $\text{[math]}$ for multiple realisations of the same ﬂow.

Each ﬁeld, e.g. $\text{[math]}$ , is decomposed into the averaged ﬁeld $\text{[math]}$ and ﬁeld of random ﬂuctuations $\text{[math]}$ , according to

-- 0 (t) = + (t): \relax \special {t4ht=

(6.11)

We can apply this decomposition to ﬁelds in the momentum conservation Eq. (2.27 ) with stress $\text{[math]}$ split into $\text{[math]}$ and $\text{[math]}$ as follows:

@--u-+ r ( uu) = r ☐☐☐ rp + b: @t \relax \special {t4ht=

(6.12)

Let us assume constant $\text{[math]}$ and that the body force $\text{[math]}$ is not subject to turbulent ﬂuctuations. Splitting the remaining ﬁelds, i.e. $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ , 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=

(6.13)

The averaged Eq. (6.13 ) is derived using the following relations for general ﬁelds $\text{[math]}$ and $\text{[math]}$ : $\text{[math]}$ ; $\text{[math]}$ ; $\text{[math]}$ ; $\text{[math]}$ . Relations for averaged derivatives are: $\text{[math]}$ ; $\text{[math]}$ ; $\text{[math]}$ .

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 $\text{[math]}$ term containing ﬂuctuations $\text{[math]}$ .

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

|------------| ☐☐☐t = u0u0 | ------------- \relax \special {t4ht=

(6.14)

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 ( $\text{[math]}$ ) to give

@--u- t @t + r uu = r ☐☐☐ + ☐☐☐ rp + b: \relax \special {t4ht=

(6.15)

This ensemble-averaged equation is the same form as Eq. (6.13 ) but with the addition of $\text{[math]}$ . Solving this Reynolds-averaged equation is the key to low cost CFD with turbulence — but it requires a model for the additional unknown $\text{[math]}$ .

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