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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 l.jpg

AN INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS

Shuisheng He

School of Engineering

The Robert Gordon University

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


Objectives l.jpg

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 l.jpg

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 l.jpg

1. INTRODUCTION

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


What is cfd l.jpg

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 l.jpg

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)


An example l.jpg

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)


An example cont l.jpg

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 cont9 l.jpg

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 cont10 l.jpg

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 cont11 l.jpg

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)


Cfd road map l.jpg

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 l.jpg

Why CFD?

  • Continuity and Navier-Stokes equations for incompressible fluids:

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


Why cfd cont l.jpg

  • 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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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)


How to use a cfd package l.jpg

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 l.jpg

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 cont24 l.jpg

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 l.jpg

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 l.jpg

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 cont27 l.jpg

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 l.jpg

2. ISSUES IN NUMERICAL METHODS

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


Mesh generation l.jpg

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 l.jpg

- 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 l.jpg

- 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 mesh32 l.jpg

- 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 l.jpg

- MESH GENERATION -Multi-Block and Overset Mesh

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


Mesh generation body fitted mesh transformation l.jpg

- MESH GENERATION -Body fitted mesh - transformation

Regular mesh

Body fitted mesh

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


Mesh generation adaptive mesh generation l.jpg

- 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)


Equation discretization l.jpg

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)


Equation discretization staggered collocated variable arrangement l.jpg

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)


Equation discretization staggered collocated variable arrangement38 l.jpg

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 l.jpg

- 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 l.jpg

- 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 cont41 l.jpg

- 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 l.jpg

- 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 l.jpg

- 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 l.jpg

- 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 l.jpg

- 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)


Solution of discretized equations l.jpg

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 l.jpg

- SOLUTION OF DISCRETIZED EQUATIONS -Solvers

  • Discretized Equations – large linearized sparse matrix

=

*

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


Solution of discretized equations solvers cont l.jpg

- 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)


Solution of discretized equations solvers cont49 l.jpg

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 l.jpg

- 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 l.jpg

- 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 l.jpg

- 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 l.jpg

- 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 l.jpg

3. Turbulence modelling

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


Turbulence modelling l.jpg

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 l.jpg

The governing equations

  • Continuity and Navier-Stokes equations for incompressible fluids:

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


The reynolds averaged navier stokes equation l.jpg

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 l.jpg

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 l.jpg

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 models60 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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)


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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)


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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 l.jpg

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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