3. TURBULENCE MODELS IN PHOENICS

Contents

1. Classification
2. Recommendations
3. Models using the Effective Viscosity Hypothesis (EVH)
   1. Sub-group in which no differential equations are used
      1. The prescribed effective-viscosity model
      2. The LVEL Turbulence Model
      3. The Prandtl mixing-length model
      4. The Van-Driest low-Reynolds-number mixing-length model
   2. Sub-group in which one differential equation is used
      1. Prandtl one-equation k-L_m model
      2. Spalart-Allmaras model
   3. Sub-group in which 1 or 2 differential equations are used
      • Two-Layer k-ε model
   4. Sub-group in which two differential equations are used
      1. Standard k-ε model
      2. Chen-Kim k-ε model
      3. RNG k-ε model
      4. Lam-Bremhorst low-Re k-ε model
      5. MMK and Kato-Launder k-ε models
      6. Realisable k-ε model
      7. Saffman-Spalding k-Ω model
      8. Wilcox (1988) k-ω model
      9. Wilcox (2008) revised k-ω model
      10. Menter (1992) k-ω model
      11. k-ω SST model
   5. Sub-group in which four differential equations are used
      • Two-scale k-ε model
4. Models not using the EVH
   • Reynolds-stress turbulence model
5. Models which may make some inessential use of the EVH
   1. Large-Eddy Simulation: the Smagorinsky subgrid-scale model
   2. Two-fluid turbulence model
   3. Multi-fluid models of turbulence
6. Models for chemical reaction
   1. The special difficulty about turbulent chemical reaction
   2. The "presumed-pdf" approach
   3. The "eddy-break-up" model
   4. The 4-fluid model
   5. The multi-fluid model
   6. The probabilistic (Monte-Carlo) approach
7. Models for multi-phase flow
8. Wall functions
9. Low-Reynolds-number models

1. Why turbulence models are needed
2. List of models
3. Full descriptions of turbulence models

1. Why turbulence models are needed

The small-eddy problem

The time-scale of these fluctuations is usually orders of magnitude smaller than that of the phenomena af practical interest. Therefore, if PHOENICS were required to simulate them in detail, the computer time would be orders of magnitude greater than would otherwise be needed.

Asociated with the variations with time are variations with position in space. Therefore, if PHOENICS is to simulate them, its computational grid must consist of cells which are exceedingly small in comparison with the dimensions of the whole geometric domain. This requirement further increases the computer time.

The variations in space and time can be thought of as caused by a population of eddies of a wide range of sizes, orientations and speeds of rotation, which are in ceaseless and random motion.

The Reynolds-Number criterion

These eddies come into being spontaneously as a consequence of the inherent instability of even a steady flow when the velocity of the fluid is much greater than its kinematic viscosity divided by the distance to the nearest solid object.

For flow in a pipe for example, 'much greater' is found to signify 'more than one thousand times greater'; which happens when the so-called Reynolds Number (= velocity times diameter divided by kinematic viscosity) exceeds 2000.

For other geometries, the critical value of the Relnolds number will be different in value; but, from the practical point of view it can be said that it scarcely matters what the criterion is, because it is vastly exceeded: most flows which PHOENICS is called upon to simulate are turbulent.

Why the eddies cannot be ignored.

These effects include:

1. Increased macro-mixing, an example of which is that the heat transfer from a hot turbulent fluid to the colder wall of the pipe through which it is flowing is much greater than it would be for a laminar flow.

2. Increased micro-mixing, an example of which is the greatly increased rate of chemical reaction between co-flowing streams of fuel and oxidant gases, when the velocity is raised to the critical level.

Micro-mixing is handled in PHOENICS by the Multi-Fluid Model.

The three approaches to simulating turbulence

1. Turbulence models of the macro-mixing kind were invented several decades before digital computers; but it was only the arrival of such computers which enabled them to be used for engineering purposes. It is these which receive the greatest attention in the 'Turbulence Models in PHOENICS' Encyclopaedia articles.

2. In the last two decades, as digital computers have become more and more powerful, increasingly successful attempts have been made to use fine-enough grids and small-enough time steps to enable the behaviour of even the smallest eddies to be computed numerically. This practice has become known as Direct Numerical Simulation or DNS.

   DNS is still practicable only for very simple flows and for modest Reynolds numbers; and its expense is far too great to be afforded in engineering practice. Nevertheless such results as have been published are beginning to serve the needs of turbulence-model developers who require empirical data for the calibration of their models.

   Hitherto 'empirical' has been synonymous with 'experimental'; but now its meaning can be extended so as to include 'DNS-generated' as well.

   DNS is not discussed further in the present Encyclopaedia articles.

3. A third approach to the problem of predicting the behaviour of turbulence phenomena can be regarded as combining some aspects of both DNS and macroscopic modelling. It is called 'Large-Eddy Simulation', commonly abbreviated to LES. Its nature is this:

   • The CFD code is run in stepping-through-time mode, even when, turbulence fluctuations apart, the flow is steady.
   • A macrocopic turbulence model is employed, but with modifications intended to represent only the eddies which are small compared with the cells of the computational grid which is in use.

   Although still much more expensive in respect of computer time than wholly macroscopic modelling, LES is beginning to be used in engineering practice.

   PHOENICS is equipped with one version of LES, the nature of which is explained here.

   A useful source of recent information concerning both DNS and LES (and its recent variants) is:

   K Hanjalic, Y Nagano and S Jakirlic (Eds)
   Turbulence, Heat and Mass Transfer 6
   Begell House, 2009.

2. Summary list of models

The keywords in () are the keywords used in the TURMOD command to activate the model, and also displayed in the VR-Editor, Main menu, Models, Turbulence models dialogs.