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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

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

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)

- 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)

- 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)

- 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

- 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

- 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

- Governing equations

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

- 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

- Turbulence 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)

- 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)

- 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)

- 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

- 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)

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

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

- 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)

- 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)

- 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)

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)

- 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)

- 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

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

- 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)

- 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)

- 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)

- 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)

- 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

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)

- 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)

- 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)

- 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)

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

Regular mesh

Body fitted mesh

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

- 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

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

- 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)

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)

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)

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)

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)

- 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)

- 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

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

- Discretized Equations – large linearized sparse matrix

=

*

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

- 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

- 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

- Basic methods:

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

- 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)

- 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)

- 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 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)

- Continuity and Navier-Stokes equations for incompressible fluids:

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

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)

- 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)

- 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)

- 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)

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)

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)

- ‘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)

- 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)

- 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)

- 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)