AN INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS

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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

• 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

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)

• 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)