## Chapter 7Reynolds-Averaged Turbulence Modelling

“I get the news I need on the weather report. Oh, I can gather all the news I need on the weather report.”

Simon & Garfunkel, The Only Living Boy in New York (1970).

For CFD to be useful, it must calculate the ﬂow, forces, heat transfer, etc. with suﬃcient accuracy. The accuracy of any calculation is always compromised by various approximations.

First, there are approximations from the numerical methods that are employed. Discretisation schemes, described in Chapter 3 , aﬀect accuracy which can be improved by reducing the size of cells and time step, at the cost of increased solution times.

The algorithms and solvers, described in Chapter 5 , are a further source of numerical approximation. Linear solvers are generally iterative with tolerances that balance computational cost against accuracy of the solution. Algorithms may include loops with a ﬁxed number of iterations that limits the overall solution convergence.

Boundary conditions, described in Chapter 4 , are applied as part of the conﬁguration of the CFD simulation. Invariably there is some level of approximation in the way the conditions represent the problem of interest. The approximation is physical, not numerical, and includes everyday examples such as:

• setting a ﬁxed value condition with a uniform value, e.g. on velocity at an inlet;
• applying a zero gradient condition, e.g. to pressure at a wall boundary;
• imposing the direction of inﬂow, e.g. with the inlet-outlet-velocity condition, Sec. 4.15 .

Beyond the numerical methods and boundary conditions, are the models that describe ﬂuid dynamics. The governing equations and basic models in Chapter 2 are accurate for laminar ﬂows with modest speeds and heat transfer. However, as the ﬂow physics becomes more complex, the models become increasingly approximate.

It is clear from Chapter 6 that turbulence is highly complex, so turbulence modelling is often a major source of error in a CFD solution. It exempliﬁes the observation of statistician George Box that all models are wrong.1 He advises that we should seek “an economical description of natural phenomena” and “simple but evocative models”.

It is with that advice we present some of the Reynolds-averaged turbulence models commonly used in CFD. The models take the form of transport equations for turbulence ﬁelds, typically turbulent kinetic energy and dissipation rate.

The models are accompanied by advice on speciﬁcation of the inlet and initial conditions for the ﬁelds. Particular attention is given to the modelling of the ﬂow within turbulent boundary layers at wall boundaries.

1George Box, Science and statistics, 1976.

Notes on CFD: General Principles - Chapter 7 Reynolds-Averaged Turbulence Modelling