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AN INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS. Shuisheng He School of Engineering The Robert Gordon University. OBJECTIVES. The lecture aims to convey the following information/ message to the students: What is CFD The main issues involved in CFD, including those of Numerical methods

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AN INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS

Shuisheng He

School of Engineering

The Robert Gordon University

Introduction to CFD (Pisa, 30/09/2005)

OBJECTIVES

The lecture aims to convey the following information/ message to the students:

• What is CFD

• The main issues involved in CFD, including those of

• Numerical methods

• Turbulence modelling

• The limitations of CFD and the important role of validation and expertise in CFD

Introduction to CFD (Pisa, 30/09/2005)

OUTLINE OF LECTURE

• Introduction

• What is CFD

• What can & cannot CFD do

• What does CFD involve …

• Issues on numerical methods

• Mesh generation

• Discretization of equation

• Solution of discretized equations

• Turbulence modelling

• Why are turbulence models needed?

• What are available?

• What model should I use?

• Demonstration

• Use of Fluent

Introduction to CFD (Pisa, 30/09/2005)

1. INTRODUCTION

Introduction to CFD (Pisa, 30/09/2005)

What is CFD?

• Computational fluid dynamics (CFD):

• CFD is the analysis, by means of computer-based simulations, of systems involving fluid flow, heat transfer and associated phenomena such as chemical reactions.

• CFD involves ...

Introduction to CFD (Pisa, 30/09/2005)

What does CFD involve?

• Specification of the problem

• Development of the physical model

• Development of the mathematical model

• Governing equations

• Boundary conditions

• Turbulence modelling

• Mesh generation

• Discretization of the governing equations

• Solution of discretized equations

• Post processing

• Interpretation of the results

Introduction to CFD (Pisa, 30/09/2005)

Depth of sea: 500m ~ 1000m

• Tidal current: 10 to 20m/s

• Waves (unsteady): -5m/s to +5m/s

• Diameters: 150~200mm

• Gap above sea bed: 10mm

An example

• Initiation of the problem

• DP Offshore Ltd is keen to know what (forces ) caused the damage they recently experienced with their offshore pipelines.

• Development of the physical model

• After a few meetings with the company, we have finally agreed a specification of the problem (For me, it defines the physical model of the problem to be solved):

Introduction to CFD (Pisa, 30/09/2005)

Symmetry

Inlet:

Flat inlet profiles

V=25m/s

Turbulence=5%

10D

Outlet:

fully developed

Flow

Smooth wall

20D

10D

An example (cont.)

• Development of the mathematical model

• Governing equations

• Equations: momentum, thermal (x), multiphase (x), …

• The flow is turbulent!

• Boundary conditions

• Decide the computational domain

• Specify boundary conditions

Introduction to CFD (Pisa, 30/09/2005)

An example (cont.)

• Development of the mathematical model (cont.)

• Turbulence model

• Initially, a standard 2-eq k-ε turbulence model is chosen for use.

• Later, to improve simulation of the transition, separation & stagnation region, I would like to consider using a RNG or a low-Re model

• Mesh generation

• Finer mesh near the wall but not too close to wall

• Finer mesh behind the pipe

Introduction to CFD (Pisa, 30/09/2005)

An example (cont.)

• Discretization of the equations

• Consider to use QUICK for velocities, later.

• There is no reason for not using the default SIMPLER for pressure.

• Solver

• Use Uncoupled rather than coupledmethod

• Use default setup on under-relaxation, but very likely, this will need to be changed later

• Convergence criterion: choose 10-5 initially: check if this is ok by checking if 10-6 makes any difference.

Iteration

• Start iteration

Failed

• Plot velocity or other variable to assist identifying the reason(s)

• Potential changes in: relaxation factors, mesh, initial guess, numerical schemes, etc.

Converged solution

• Eventually, solution converged.

Introduction to CFD (Pisa, 30/09/2005)

An example (cont.)

• Post processing

• Interpretation of results

Force vector: (1 0 0)

pressure viscous total pressure viscous total

zone name force force force coefficient coefficient coefficient

n n n

------------------------- -------------- -------------- -------------- -------------- -------------- --------------

pipe 8.098238 0.12247093 8.2207089 13.221613 0.1999 13.421566

------------------------- -------------- -------------- -------------- -------------- -------------- --------------

net 8.098238 0.12247093 8.2207089 13.221613 0.199 13.421566

Introduction to CFD (Pisa, 30/09/2005)

Specify the problem

Select turbulence model

Generate Mesh

Discretize equations

Solve discretized equations

Post processing

Pre-processor

Solver

Post-processor

Introduction to CFD (Pisa, 30/09/2005)

Why CFD?

• Continuity and Navier-Stokes equations for incompressible fluids:

Introduction to CFD (Pisa, 30/09/2005)

• Flow in a pipe

• For laminar flow:

?

• For turbulent flow:

Or

Why CFD? (cont.)

• Analytical solutions are available for only very few problems.

• Experiment combined with empirical correlations have traditionally been the main tool - an expensive one.

• CFD potentially provides an unlimited power for solving any flow problems

Important conclusion: There is no analytical solution even for a very simple application, such as, a turbulent flow in a pipe.

Introduction to CFD (Pisa, 30/09/2005)

CFD applications

• Aerospace

• Automobile industry

• Engine design and performance

• The energy sector

• Oil and gas

• Biofluids

• Many other sectors

Introduction to CFD (Pisa, 30/09/2005)

CFD applications (cont.)

• As a design tool, CFD can be used to perform quick evaluation of design plans and carry out parametric investigation of these designs.

• As a research tool, CFD can provide detailed information about the flow and thermal field and turbulence, far beyond these provided by experiments.

Introduction to CFD (Pisa, 30/09/2005)

What can CFD do?

• Flows problems in complex geometries

• Heat transfer

• Combustions

• Chemical reactions

• Multiphase flows

• Non-Newtonian fluid flow

• Shock waves

Introduction to CFD (Pisa, 30/09/2005)

What can’t CFD do?

• CFD is still struggling to predict even the simplest flows reliably, for example,

• A jet impinging on a wall

• Heat transfer in a vertical pipe

• Flow over a pipe

• Combustion in an engine

• Important conclusions:

• Validation is of vital importance to CFD.

• Use of CFD requires more expertise than many other areas

• CFD solutions beyond validation are often sought and expertise plays an important role here.

Introduction to CFD (Pisa, 30/09/2005)

Validation of CFD modelling

Errors involved in CFD results

• Discretization errors

• Depending on ‘schemes’ used. Use of higher order schemes will normally help to reduce such errors

• Also depending on mesh size – reducing mesh size will normally help to reduce such errors.

• Iteration errors

• For converged solutions, such errors are relatively small.

• Turbulence modelling

• Some turbulence models are proved to produce good results for certain flows

• Some models are better than others under certain conditions

• But no turbulence model can claim to work well for all flows

• Physical problem vs mathematical model

• Approximation in boundary conditions

• Use of a 2D model to simplify calculation

• Simplification in the treatment of properties

Introduction to CFD (Pisa, 30/09/2005)

Validation of CFD modelling (cont.)

• CFD results always need validation. They can be

• Compared with experiments

• Compared with analytical solutions

• Checked by intuition/common sense

• Compared with other codes (only for coding validation!)

Introduction to CFD (Pisa, 30/09/2005)

Commercial CFD packages

• Phoenix

• Fluent

• Star-CD

• CFX (FLOW3D)

• Many others

• Computer design tools – integrating CFD into a design package

Introduction to CFD (Pisa, 30/09/2005)

Specify the problem

Generate Mesh

Select equations to solve

Select turbulence models

Define boundary conditions

Select numerical methods

Iterate – solve equations

Fail – calculation does not converge or converges too slowly

Improve model:

Physical model

Mesh

Better initial guess

Numerical methods (e.g., solver, convection scheme)

Under-relaxations

Post processing

Interpretation of results – Always question the results

How to use a CFD package?

Introduction to CFD (Pisa, 30/09/2005)

How to use a CFD package? (cont.)

• Important issues involved in using CFD:

• Mesh independence check

• Selection of an appropriate turbulence model

• Validation of the solution based on a simplified problem (which contains the important features similar to your problem)

• Careful interpretation of your results

Introduction to CFD (Pisa, 30/09/2005)

How to use a CFD package? (cont.)

• The commercial packages are so user friendly and robust, why do we still need CFD experts?

Because they can provide:

• Appropriate interpretation of the results and knowledge on the limitations of CFD

• More accurate results (by choosing the right turbulence model & numerical methods)

• Ability to obtain results (at all) for complex problems

• Speed: both in terms of the time used to generate the model and the computing time

Introduction to CFD (Pisa, 30/09/2005)

Basic CFD strategies

• Finite difference (FD)

• Starting from the differential form of the equations

• The computational domain is covered by a grid

• At each grid point, the differential equations (partial derivatives) are approximated using nodal values

• Only used in structured grids and normally straightforward

• Disadvantage: conservation is not always guaranteed

• Disadvantage: Restricted to simple geometries.

• Finite Volume (FV)

• Finite element (FE)

Introduction to CFD (Pisa, 30/09/2005)

Basic CFD strategies (cont.)

• Finite difference (FD)

• Finite Volume (FV)

• Starting from the integral form of the governing equations

• The solution domain is covered by control volumes (CV)

• The conservation equations are applied to each CV

• The FV can accommodate any type of grid and suitable for complex geometries

• The method is conservative (as long as surface integrals are the same for CVs sharing the boundary)

• Most widely used method in CFD

• Disadvantage: more difficult to implement higher than 2nd order methods in 3D.

• Finite element (FE)

Introduction to CFD (Pisa, 30/09/2005)

Basic CFD strategies (cont.)

• Finite difference (FD)

• Finite Volume (FV)

• Finite element (FE)

• The domain is broken into a set of discrete volumes: finite elements

• The equations are multiplied by a weight function before they are integrated over the entire domain.

• The solution is to search a set of non-linear algebraic equations for the computational domain.

• Advantage: FE can easily deal with complex geometries.

• Disadvantage: since unstructured in nature, the resultant matrices of linearized equations are difficult to find efficient solution methods.

• Not often used in CFD

Introduction to CFD (Pisa, 30/09/2005)

2. ISSUES IN NUMERICAL METHODS

Introduction to CFD (Pisa, 30/09/2005)

Specify the problem

Select turbulence model

Generate Mesh

Discretize equations

Solve discretized equations

Post processing

Mesh generation

Why do we care?

• 50% time spent on mesh generation

• Convergence depends on mesh

• Accuracy depends on mesh

Main topics

• Structured/unstructured mesh

• Multi-block

• body fitted

Introduction to CFD (Pisa, 30/09/2005)

- MESH GENERATION -Computational domain and mesh structure

• Carefully select your computational domain

• The mesh needs

• to be able to resolve the boundary layer

• to be appropriate for the Reynolds number

• to suit the turbulence models selected

• to be able to model the complex geometry

Introduction to CFD (Pisa, 30/09/2005)

- MESH GENERATION -Structure/unstructured mesh

• Structured grid

• A structured grid means that the volume elements (quadrilateral in 2D) are well ordered and a simple scheme (e.g., i-j-k indices) can be used to label elements and identify neighbours.

• Unstructured grid

• In unstructured grids, volume elements (triangular or quadrilateral in 2D) can be joined in any manner, and special lists must be kept to identify neighbouring elements

Introduction to CFD (Pisa, 30/09/2005)

- MESH GENERATION -Structure/unstructured mesh

• Structured grid

• Economical in terms of both memory & computing time

• Easy to code/more efficient solvers available

• The user has full control in grid generation

• Easy in post processing

• Difficult to deal with complex geometries

• Unstructured grid

Introduction to CFD (Pisa, 30/09/2005)

- MESH GENERATION -Multi-Block and Overset Mesh

Introduction to CFD (Pisa, 30/09/2005)

- MESH GENERATION -Body fitted mesh - transformation

Regular mesh

Body fitted mesh

Introduction to CFD (Pisa, 30/09/2005)

- MESH GENERATION -Adaptive mesh generation

• The mesh is modified according to the solution of the flow

• Two types of adaptive methods

• Local mesh refinement

• Mesh re-distribution

• Mesh refinement/redistribution are automatically carried out during iterations

• Demonstration – flow past a cylinder

Introduction to CFD (Pisa, 30/09/2005)

Specify problem

Select turbulence model

Generate Mesh

Discretize equations

Solve discretized equations

Post processing

Equation discretization

Relevant issues

• Convergence strongly depends on numerical methods used.

• Accuracy – discretization errors

Main topics

• Staggered/collocated variable arrangement

• Convection schemes

• Accuracy

• Artificial diffusion

• Boundedness

• Choice of many schemes

• Linearization of source terms

• Boundary conditions

Introduction to CFD (Pisa, 30/09/2005)

V

U,V,P,T

U

P,T

- EQUATION DISCRETIZATION -Staggered/collocated variable arrangement

• Collocated variable arrangement

• All variables are defined at nodes

• Staggered variable arrangement

• Velocities are defined at the faces and scalars are defined as the nodes

Collocated Arrangement

Staggered Arrangement

Introduction to CFD (Pisa, 30/09/2005)

The problem:

Unless special measures are taken, the collocated arrangement often results in oscillations

The reason is the weak coupling between velocity & pressure

Staggered variable arrangement

Almost always been used between 60’s and early 80’s

Still most often used method for Cartesian grids

Disadvantage: difficult to treat complex geometry

Collocated variable arrangement

Methods have been developed to over-come oscillations in the 80’s and such methods are often being used since.

Used for non-orthogonal, unstructured grids, or, for multigrid solution methods

- EQUATION DISCRETIZATION -Staggered/collocated variable arrangement

Introduction to CFD (Pisa, 30/09/2005)

- EQUATION DISCRETIZATION -Convection schemes

The problem

• To discretize the equations, convections on CV faces need to be calculated from variables stored on nodal locations

• When the 2nd order-accurate linear interpolation is used to calculate the convection on the CV faces, undesirable oscillation may occur.

• Development/use of appropriate convection schemes have been a very important issue in CFD

• There are no best schemes. A choice of schemes is normally available in commercial CFD packages to be chosen by the user.

Introduction to CFD (Pisa, 30/09/2005)

- EQUATION DISCRETIZATION -Convection schemes (cont.)

The requirements for convection schemes:

• Accuracy: Schemes can be 1st, 2nd, 3rd...-order accurate.

• Conservativeness: Schemes should preserve conservativeness on the CV faces

• Boundedness: Schemes should not produce over-/under-shootings

• Transportiveness: Schemes should recognize the flow direction

Introduction to CFD (Pisa, 30/09/2005)

- EQUATION DISCRETIZATION -Convection schemes (cont.)

Examples of convection schemes

• 1st order schemes:

• Upwind scheme (UW): most often used scheme!

• Power law scheme

• Skewed upwind

• Higher order schemes

• Central differencing scheme (CDS) – 2nd order

• Quadratic Upwind Interpolation for Convective Kinematics (QUICK) – 3rd order and very often used scheme

• Bounded higher-order schemes

• Total Variation Diminishing (TVD) – a group of schemes

• SMART

Introduction to CFD (Pisa, 30/09/2005)

• The problem

• The pressure appears in the momentum equation as the driving force for the flow. But for incompressible flows, there is no transport equation for the pressure.

• In stead, the continuity equation will be satisfied if the appropriate pressure field is used in the momentum equations

• The non-linear nature of and the coupling between, the various equations also pose problems that need care.

• The remedy

• Iterative guess-and-correct methods have been proposed – see next slide.

Introduction to CFD (Pisa, 30/09/2005)

- EQUATION DISCRETIZATION -Pressure-velocity link (cont.)

Most widely used methods

• SIMPLE (Semi-implicit method for pressure-linked equations)

• A basic guess-and-correct procedure

• SIMPLER (SIMPLE-Revised): used as default in many commercial codes

• Solve an extra equation for pressure correction (30% more effort than SIMPLE). This is normally better than SIMPLE.

• SIMPLEC (SIMPLE-Consistent): Generally better than SIMPLE.

• PISO (Pressure Implicit with Splitting of Operators)

• Initially developed for unsteady flow

• Involves two correction stages

Introduction to CFD (Pisa, 30/09/2005)

- EQUATION DISCRETIZATION -Linearization of source terms

• This slide is only relevant to those who develops CFD codes.

• The treatment of source terms requires skills which can significantly increase the stability and convergence speed of the iteration.

• The basic rule is that the source term should be linearizated and the linear part can the be solved directly.

• An important rule is that only those of linearization which result in a negative gradient can be solved directly

Introduction to CFD (Pisa, 30/09/2005)

- EQUATION DISCRETIZATION -Boundary conditions

• Specification of boundary conditions (BC) is a very important part of CFD modelling

• In most cases, this is straightforward but, in some cases, it can be very difficult ...,

• Typical boundary conditions:

• Inlet boundary conditions

• Outlet boundary conditions

• Wall boundary conditions

• Symmetry boundary conditions

• Periodic boundary conditions

Introduction to CFD (Pisa, 30/09/2005)

Relevant issues

• Cost/speed

• Stability/Convergence

Main topics

• Solver – solution of the discretized equation system

• Convergence criteria

• Under-relaxation

• Solution of coupled equations

Specify problem

Select turbulence model

Generate Mesh

Discretize equations

Solve discretized equations

Post processing

Solution of discretized equations

Introduction to CFD (Pisa, 30/09/2005)

- SOLUTION OF DISCRETIZED EQUATIONS -Solvers

• Discretized Equations – large linearized sparse matrix

=

*

Introduction to CFD (Pisa, 30/09/2005)

- SOLUTION OF DISCRETIZED EQUATIONS -Solvers (cont.)

• The discretized governing equations are always sparse,non-linear but linearizated, algebraic equation systems

• The ‘matrix’ from structured mesh is regular and easier to solve.

• A non-structured mesh results in an irregular matrix.

• Number of equations = number of nodes

• Number of molecules in each line:

• Upwind, CDS for 1D results in a tridiagonal matrix

• QUICK for 1D results in a penta-diagonal matrix

• 2D problems involves 5 & more molecules

• 3D problems involves 7 & more molecules

Introduction to CFD (Pisa, 30/09/2005)

Very expensive!

Very effective method used for tridiagonal matrix

Simple and probably most often used method

Used for more ‘complex’ problems

Effective method for more ‘complex’ problems

- SOLUTION OF DISCRETIZED EQUATIONS -Solvers (cont.)

• Direct methods

• Gauss elimination:

• Tridiagonal Matrix Algorithm (TDMA):

• Indirect methods

• Basic methods:

• Jacobi

• Gauss-Seidel

• Successive over-relaxation (SOR)

• Strongly implicit procedure (SIP)

• Multigrid Methods

Introduction to CFD (Pisa, 30/09/2005)

- SOLUTION OF DISCRETIZED EQUATIONS -Convergence criteria

• Two basic methods:

• Changes between any two iterations are less than a given level

• Residuals in the transport equations are less than a given value

• Criteria can be specified using absolute or relative values

Introduction to CFD (Pisa, 30/09/2005)

- SOLUTION OF DISCRETIZED EQUATIONS -Under-relaxation

• Under almost all circumstances, iterations will not converge unless under-relaxation is used, because

• The governing equations are very non-linear

• And the equations are closely coupled

• Under-relaxation (α):

• Different variables often require different levels of under-relaxation

• Iteration diverged? Relaxation is the first thing to look at

Introduction to CFD (Pisa, 30/09/2005)

- SOLUTION OF DISCRETIZED EQUATIONS -Solution of coupled equations

• Governing equations for flow/heat transfer are almost always coupled

• The primary variable of one equation also appear in equations for other variables

• Simultaneous solution – Method 1

• Used when equations are linear and tightly coupled

• Can be very expensive

• Sequential solution – Method 2

• Solve equations one by one - temporarily treat other variables as known

• Iterations include

• Inner cycles: Solve each equation

• Outer cycles: cycle between equations

Introduction to CFD (Pisa, 30/09/2005)

- SOLUTION OF DISCRETIZED EQUATIONS -Unsteady flow solvers

• Explicit method

• use only the values of the variable Φ from last time step.

• Conditionally stable, first order

• Implicit method

• Mainly use the values of the variable Φ from the current time step

• Unconditionally stable, first order

• Crank-Nicolson method

• Use a mixture of values of the variable Φ at the last and current steps

• Unconditionally stable, second order

• Predictor-Corrector method

• Predictor: Explicit method

• Corrector: (Pseudo-) Crank-Nicolson method

Introduction to CFD (Pisa, 30/09/2005)

3. Turbulence modelling

Introduction to CFD (Pisa, 30/09/2005)

Specify the problem

Select turbulence model

Generate Mesh

Discretize equations

Solve discretized equations

Post processing

Turbulence modelling

Turbulence models

• These are semi-empirical mathematical models introduced to CFD to describe the turbulence in the flow

Main topics

• Three levels of CFD simulations

• Classification of turbulence models

• Examples of popular models

• Special considerations

• General remarks about turbulence modelling

Introduction to CFD (Pisa, 30/09/2005)

The governing equations

• Continuity and Navier-Stokes equations for incompressible fluids:

Introduction to CFD (Pisa, 30/09/2005)

The Reynolds averaged Navier-Stokes Equation

The Reynolds averaged Navier-Stokes equations (RANS):

• NOTES:

• The extra terms, Reynolds (turbulent) shear stresses, have

• the effect of mixing, similar to molecular mixing (diffusion)

• These terms need to be modelled

Introduction to CFD (Pisa, 30/09/2005)

The three level simulations

• Direct Numerical Simulations (DNS)

• DNS directly solves the NS equations

• There is no ‘modelling’ in it, so the solution can be considered as the true representation of the flow.

• It always solves the unsteady form

• It can only be used for very simple flows at the moment due to its huge requirement on computer power.

• Large Eddy Simulations (LES)

• LES directly solves the NS flow for ‘large eddies’ but uses models to simulate the ‘small scale’ flows

• The solution is again always in unsteady form

• LES can only be used for relatively simple flows

• Reynolds Averaged Navier-Stokes approach (RANS)

• Turbulence models are used to simulate the effect of turbulence

• RANS has been widely used in designs and research since the 70’s

• Almost all commercial CFD packages are RANS based.

Introduction to CFD (Pisa, 30/09/2005)

Classification of turbulence models

• Eddy viscosity turbulence models

• Model Reynolds stresses as a product of velocity gradient and an eddy viscosity

• Solve 0 to 2 transport equations for turbulence

• Reynolds stress turbulence models

• Solve the transport equations of the Reynolds stresses

• Solve 7 transport equations for turbulence

Introduction to CFD (Pisa, 30/09/2005)

Classification of turbulence models

• Eddy viscosity turbulence models

• The key issue is to model the eddy viscosity νt

• Three types of eddy viscosity models

• Algebraic models (e.g., mixing length model)

• One-equation models: solve one transport equation (normally one for turbulence kinetic energy, k)

• Two equation models: solve two transport equations

• K-ε, k-ω, k-τ models

Introduction to CFD (Pisa, 30/09/2005)

An example of the two-equation model

Jones and Launder (1972) k-ε two equation model

Eddy viscosity

Turbulence kinetic energy

Dissipation rate

Closure coefficients

Introduction to CFD (Pisa, 30/09/2005)

An example of the Reynolds stress model

The Launder-Reece-Rodi (1975) Reynolds stress model

Reynolds-stress tensor (six independent equations)

Dissipation rate

Pressure-strain correlation

Auxiliary relations

Closure coefficients [Launder (1992)]

Introduction to CFD (Pisa, 30/09/2005)

Special turbulence models

• ‘Standard’ models and wall functions

• Standard turbulence models are designed only for the core region. Wall Functions are used to bridge the near-wall region for a wall shear flow.

• Standard models are used beyond roughly y+=50.

• Low-Reynolds number (LRN) turbulence model

• LRN models are designed to be used in the near-wall region as well as the core region.

• LRN models are much more expensive – they require much finer grid than used for standard models

• Two-layer models

• In some cases, separate models are used for the wall and core regions

• The wall region model can be a ‘simpler’ model, such as, one-equation model

• This practice can be more economical than using LRN models.

• Other special models

• Realizable models

• Non-linear eddy viscosity models

• Renormalized Group (RNG) models

Introduction to CFD (Pisa, 30/09/2005)

What model should I use?

• Algebraic models

• Main models used until early 70’s, and still in use.

• Disadvantages: lack of generality, νt vanishes when du/dy=0, etc.

• Two-equation models (especially k-ε models)

• Most widely used models, standard model in commercial packages

• Advantages: best compromise between cost and capability

• Disadvantages: no account of individual components of turbulent stresses; νt vanishes when du/dy=0.

• Reynolds shear stress models

• Only recently been included in commercial CFD codes; and still not widely used yet.

• Advantages: provide the potential of modelling more complex flows

• Disadvantages: have to solve up to 7 more differential equations

Introduction to CFD (Pisa, 30/09/2005)

General remarks on turbulence models

• There are no generically best models.

• Near wall treatment is generally a very important issue.

• A good mesh is important to get good accurate results.

• Different models may have different requirement on the mesh.

• Expertise/validation are of great importance to CFD.

Introduction to CFD (Pisa, 30/09/2005)

References

• Numerical Heat Transfer and Fluid Flow

• S.V. Patankar, 1980, Hemisphere Publishing Corporation, Taylor & Francis Group, New York.

• An Introduction to Computational Fluid Dynamics

• H.K. Versteeg & W. Malalasekera, 1995, Longman group Limited, London

• Computational Methods for Fluid Dynamics

• J.H. Ferziger & M. Peric, 1996, Springer-Verlag, Berlin.

• Computational Fluid Dynamics

• J.D. Anderson, Jr, 1995, McGraw-Hill, Singapore

Introduction to CFD (Pisa, 30/09/2005)