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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

zero gradient

Flow

Smooth wall

20D

10D

An example (cont.)

  • Development of the mathematical model

    • Governing equations

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

      • Phase 1: 2D, steady; Phase 2: unsteady, …,

      • 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

    • Start with 1st order upwind, for easy convergence

    • 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

CFD road map

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

  • Unsteady flows

  • 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)


CFD Road Map

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

  • Adaptive mesh generation

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

    Advantages:

    • 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

      Disadvantages

    • Difficult to deal with complex geometries

  • Unstructured grid

    • Advantages/disadvantages: opposite to above points!

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

  • 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

  • Dynamic adaptive method

    • Mesh refinement/redistribution are automatically carried out during iterations

  • Demonstration – flow past a cylinder

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


CFD Road Map

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

  • Pressure-velocity link

  • 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)


- EQUATION DISCRETIZATION -Pressure-velocity link

  • 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

  • Unsteady flow solvers

CFD Road Map

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)

    • ADI-TDMA

    • Strongly implicit procedure (SIP)

    • Conjugate Gradient Methods (CGM)

    • 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)


CFD Road Map

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.

    • Advantages: simple

    • 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)


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