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

2
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

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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

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1. INTRODUCTION
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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 ...

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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

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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)

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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

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An example (cont.)

• Development of the mathematical model (cont.)
• Turbulence model
• Initially, a standard 2-eq k-e 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

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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 coupled method
• 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.

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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
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CFD road map
Pre-processor
Solver
Post-processor
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Why CFD?

• Continuity and Navier-Stokes equations for
  incompressible fluids

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Why CFD? (cont.)
Important conclusion There is no analytical
solution even for a very simple application, such
as, a turbulent flow in a pipe.

• 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

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CFD applications

• Aerospace
• Automobile industry
• Engine design and performance
• The energy sector
• Oil and gas
• Biofluids
• Many other sectors

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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.

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What can CFD do?

• Flows problems in complex geometries
• Heat transfer
• Combustions
• Chemical reactions
• Multiphase flows
• Non-Newtonian fluid flow
• Unsteady flows
• Shock waves

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What cant 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.

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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

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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!)

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Commercial CFD packages

• Phoenix
• Fluent
• Star-CD
• CFX (FLOW3D)
• Many others
• Computer design tools integrating CFD into a
  design package

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How to use a CFD package?

• 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

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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

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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

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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)

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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)

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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

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2. ISSUES IN NUMERICAL METHODS
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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

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- 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

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- 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

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- 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!

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- MESH GENERATION - Multi-Block and Overset Mesh
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- MESH GENERATION - Body fitted mesh -
transformation
Regular mesh
Body fitted mesh
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- 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

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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

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- 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
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- EQUATION DISCRETIZATION - Staggered/collocated
variable arrangement

• 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 60s and early
  80s
• 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 80s and such methods are
  often being used since.
• Used for non-orthogonal, unstructured grids, or,
  for multigrid solution methods

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- 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.

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- 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

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- 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

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- 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.

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- 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

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- 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

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- 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

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Solution of discretized equations
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- SOLUTION OF DISCRETIZED EQUATIONS - Solvers

• Discretized Equations large linearized sparse
  matrix

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- 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

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- 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

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- 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

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- 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 (a)
• Different variables often require different
  levels of under-relaxation
• Iteration diverged? Relaxation is the first thing
  to look at

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- 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

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- SOLUTION OF DISCRETIZED EQUATIONS - Unsteady
flow solvers

• Explicit method
• use only the values of the variable F from last
  time step.
• Conditionally stable, first order
• Implicit method
• Mainly use the values of the variable F from the
  current time step
• Unconditionally stable, first order
• Crank-Nicolson method
• Use a mixture of values of the variable F at the
  last and current steps
• Unconditionally stable, second order
• Predictor-Corrector method
• Predictor Explicit method
• Corrector (Pseudo-) Crank-Nicolson method

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3. Turbulence modelling
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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

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The governing equations

• Continuity and Navier-Stokes equations for
  incompressible fluids

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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

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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 70s
• Almost all commercial CFD packages are RANS based.

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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

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Classification of turbulence models

• Eddy viscosity turbulence models
• The key issue is to model the eddy viscosity ?t
• 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-e, k-?, k-t models

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An example of the two-equation model
Jones and Launder (1972) k-e two equation model
Eddy viscosity
Turbulence kinetic energy
Dissipation rate
Closure coefficients
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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)
63
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 y50.
• 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

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What model should I use?

• Algebraic models
• Main models used until early 70s, and still in
  use.
• Advantages simple
• Disadvantages lack of generality, ?t vanishes
  when du/dy0, etc.
• Two-equation models (especially k-e 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 ?t vanishes
  when du/dy0.
• 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

65
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.

66
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