Encyclopaedia Index

## PHOENICS Encyclopaedia

### 1. Classification The effective-viscosity concept (EVH)

The simplest view of the behaviour of a turbulent fluid is that it is just like that of a laminar one but with an increased viscosity, which however may vary from place to place.

This concept is usually attributed to Boussinesq (1877) and is called the "EFFECTIVE-VISCOSITY HYPOTHESIS".

One important implication is that turbulent shear stresses are proportional to velocity gradients.

The hypothesis has been known for many years not to be valid in all circumstances; but it is often so close to the truth that it is very widely used.

So central is it to turbulence modelling that models can be usefully classified according to their use or avoidance of it.

### Classification principles

Turbulence models in PHOENICS can thus be classified as belonging to one or other of three groups, namely:-

1. those which employ the Effective Viscosity Hypothesis;
2. those which specifically AVOID the EVH;
3. those which may make some INESSENTIAL use of the EVH.

Group-1 models can then be further classified in accordance with the methods which they adopt for CALCULATING THE MAGNITUDE of the effective viscosity.

Further distinction between models can be made by reference to their handling (or non-handling) of:

• heat and mass transfer,
• chemical reaction, and
• multi-phase effects.

### Group (1), those which employ the EVH

Sub-group 1.1, in which no differential equations are used

The practices for computing the effective viscosity (EV for short) are:
 name practice prescribed EV EV is given a uniform value LVEL EV is computed from the velocity, the laminar viscosity and the distance from nearby walls Prandtl mixing-length EV is computed from the velocity gradient length and a prescribed length scale Van-Driest as for Prandtl mixing length, but with low-Reynolds-number modification

### Sub-group 1.2, in which one differential equation is used

 name practice Prandtl energy EV is prescribed-length * SQRT(KE) where KE is energy of turbulence computed from a differential transport equation

### Sub-group 1.3, in which 1 or 2 differential equations are used

 name practice TWO-LAYER KE-EP as for KE-EP (see below), except that only the KE equation is solved near the wall, where the length scale is treated as known

### Sub-group 1.4, in which two differential equations are used

 name practice k-epsilon (KE-EP) EV is proportional to KE**2/EP, where KE and EP (dissipation rate of KE) are computed from differential transport equations CHEN-KIM KE-EP as for KE-EP, but with a "dual-time-scale concept" making the formulae depend upon the energy-production rate P RNG-derived KE-EP as for KE-EP, but with a "re-normalization- group concept" making the formulae depend upon the energy-production rate P LAM-BREMHORST as for KE-EP, but with low-Reynolds number extension requiring knowledge of the distance from the nearest wall. Saffman-Spalding KE-VO EV is proportional to KE/W**0.5, where KE and W (RMS vorticity fluctuations) are computed from differential transport equations Kolmogorov-Wilcox KE-OMEGA EV is proportional to KE/OMEGA, where KE and OMEGA ("turbulence frequency") are computed from differential transport equations

### Sub-group 1.5, in which four differential equations are used

 name practice TWO-SCALE KE-EP EV is computed in a similar manner to that of KE-EP model; but there are two turbulence- energy variables, KP and KT, and two dissipation-rate variables, EP and ET; so it is necessary to solve four differential transport equations.

### Group 2, not employing the EVH

 name practice REYNOLDS-stress EV is not used. Instead, the shear stresses are themselves the dependent variables of differential transport equations, usually six in number.

### Group 3, which may or may not employ the EVH

 name practice Smagorinsky EV is used only to resolve the small-scale subgrid-scale motion, the main transfers of momentum being computed by performing three-dimensional time-dependent solutions of the Navier-Stokes equations with the finest affordable space and time sub-divisions. Two-fluid EV is either not used at all, or is deduced from the local velocity differences between the two intermingling fluids which are used to describe the turbulent fluid mixture. Multi-fluid as for TWO-FLUID, except that, there being many fluids present, EV can be derived from their various velocities in a wider variety of ways.