50 YEARS OF 50 YEARS OF 50 YEARS OF 50 YEARS OF CFDCFDCFDCFD

Dr. Fakhir HasaniDr. Fakhir HasaniProfessor & ChairmanProfessor & Chairman

Department of Mechanical EngineeringDepartment of Mechanical EngineeringNED University of Engineering & TechnologyNED University of Engineering & Technology

What is CFD?What is CFD?What is CFD?What is CFD?It is the analysis of systems involving fluid It is the analysis of systems involving fluid flow, heat transfer and associated flow, heat transfer and associated phenomena such as chemical reactions by phenomena such as chemical reactions by means of computer based simulations. means of computer based simulations.

Ho as CFD e ol ed?Ho as CFD e ol ed?How was CFD evolved?How was CFD evolved?FLUID

DYNAMICS

COMPUTERSCIENCE

MATHEMATICS

ApplicationsApplicationsApplicationsApplications

Aerodynamics of aircraft and vehicles: lift and Aerodynamics of aircraft and vehicles: lift and dragdragH d d i f hiH d d i f hi Hydrodynamics of shipsHydrodynamics of ships

Power plants: combustion in diesel/petrol Power plants: combustion in diesel/petrol engines and gas turbinesengines and gas turbinesengines and gas turbinesengines and gas turbines

Turbomachinery: flows inside rotating passages, Turbomachinery: flows inside rotating passages, diffusers etc.diffusers etc.diffusers etc.diffusers etc.

Electrical and electronic engineering: cooling of Electrical and electronic engineering: cooling of equipment including microequipment including micro--circuits circuits

Applications (cont )Applications (cont )Applications (cont)Applications (cont)

Chemical process engineering: mixing and Chemical process engineering: mixing and separation, polymer moldingseparation, polymer molding

External and internal environment of buildings: External and internal environment of buildings: wind loading, heating and ventilationwind loading, heating and ventilation

l d ffl d ff hh Marine engineering: loads on offMarine engineering: loads on off--shore shore structuresstructuresE i t l i i di t ib ti fE i t l i i di t ib ti f Environmental engineering: distribution of Environmental engineering: distribution of effluents and pollutants effluents and pollutants

Applications (cont )Applications (cont )Applications (cont)Applications (cont)

Hydrology and oceanography: Flow in Hydrology and oceanography: Flow in rivers, estuaries and oceansrivers, estuaries and oceans

Meteorology: Weather predictionMeteorology: Weather prediction Manufacturing engineering: Flow of Manufacturing engineering: Flow of g g gg g g

material and temperature distribution in material and temperature distribution in moulds used in sand and die castingmoulds used in sand and die casting

Biomedical engineering: blood flow Biomedical engineering: blood flow through arteries and veinsthrough arteries and veins

Governing EquationsGoverning EquationsGoverning EquationsGoverning Equations

Conservation of MassConservation of MassCo se at o o assCo se at o o ass

0)( =+ Vdiv

t

Conservation of MomentumConservation of Momentum

pgV ij += '.

Conservation of EnergyConservation of Energy

pgt ij

+ .

j

iij x

uTkdivDtDp

DtDh

++= ')(

wherewhere )()(' divVxu

xu

iji

j

j

iij +

+=

Mathematical character of Mathematical character of governing equationsgoverning equations

22ndnd Order PDEsOrder PDEs Coupled in pressure and velocityCoupled in pressure and velocity Solution is very complexSolution is very complex Solution is very complexSolution is very complex

C l t A l iC l t A l i 3 t3 tComplete Analysis Complete Analysis 3 step process3 step process

PrePre--processingprocessing Processing (Solution)Processing (Solution) Processing (Solution)Processing (Solution) PostPost--processing processing

P eP e p ocessingp ocessingPrePre--processingprocessing

Defining the geometry: The computational Defining the geometry: The computational domaindomain

Grid generation Grid generation the subdivision of the the subdivision of the domain into a number of smaller, nondomain into a number of smaller, non--overlapping suboverlapping sub--domains: a grid (or mesh) domains: a grid (or mesh) of cells (or control volumes or elements)of cells (or control volumes or elements)

Selection of the physical and chemical Selection of the physical and chemical phenomena that need to be modeledphenomena that need to be modeled

P eP e p ocessing (cont )p ocessing (cont )PrePre--processing (cont)processing (cont)

Definition of fluid propertiesDefinition of fluid properties Specification of appropriate boundarySpecification of appropriate boundarySpecification of appropriate boundary Specification of appropriate boundary

conditions at cells which coincide with conditions at cells which coincide with or touch the domain boundaryor touch the domain boundaryo touc t e do a bou da yo touc t e do a bou da y

Mesh t pesMesh t pes StructuredStructured

Mesh typesMesh types

Uniform

StructuredStructured

Non-uniform

dd UnstructuredUnstructuredMeshes can be generated in 2d and 3d. In the first Meshes can be generated in 2d and 3d. In the first d d h i bili li i dd d h i bili li i ddecade mesh generation capability was limited to decade mesh generation capability was limited to 2d structured grids only 2d structured grids only

N me ical Sol tion Techniq esN me ical Sol tion Techniq esNumerical Solution TechniquesNumerical Solution Techniques

Finite Difference MethodsFinite Difference Methods Finite Elements MethodsFinite Elements Methods Finite Elements MethodsFinite Elements Methods Spectral MethodsSpectral Methods

Fi it V l M th dFi it V l M th d Finite Volume MethodsFinite Volume Methods

Finite Diffe ence Methods (FDM)Finite Diffe ence Methods (FDM)Finite Difference Methods (FDM)Finite Difference Methods (FDM)

Finite difference methods describe the Finite difference methods describe the unknown flow variables by means of point unknown flow variables by means of point y py psamples at the grid points. Truncated samples at the grid points. Truncated Taylor series expansions are used to Taylor series expansions are used to y py pgenerate the finite difference generate the finite difference approximations of derivatives in terms of approximations of derivatives in terms of pppppoint samples at each grid point and its point samples at each grid point and its immediate neighbors immediate neighbors gg

Finite Element Methods (FEM)Finite Element Methods (FEM)Finite Element Methods (FEM)Finite Element Methods (FEM)

FEM use simple piecewise linear or FEM use simple piecewise linear or quadratic functions valid on elements to quadratic functions valid on elements to describe the local variations of unknown describe the local variations of unknown flow variables. The governing equation is flow variables. The governing equation is p ecisel satisfied b the e act sol tion Ifp ecisel satisfied b the e act sol tion Ifprecisely satisfied by the exact solution. If precisely satisfied by the exact solution. If the piecewise approximating functions for the piecewise approximating functions for unknown variables are substituted into theunknown variables are substituted into theunknown variables are substituted into the unknown variables are substituted into the equation, it will not hold exactly and a equation, it will not hold exactly and a residual is defined to measure the errors.residual is defined to measure the errors.residual is defined to measure the errors.residual is defined to measure the errors.

FEM (cont )FEM (cont )FEM (cont)FEM (cont)

Next the residuals (and hence the errors) are Next the residuals (and hence the errors) are minimized by multiplying with a set of weighting minimized by multiplying with a set of weighting functions and integrating. As a result we obtain functions and integrating. As a result we obtain a set of algebraic functions for the unknown a set of algebraic functions for the unknown coefficients of the approximating functionscoefficients of the approximating functionscoefficients of the approximating functions.coefficients of the approximating functions.

The theory of FEM was initially developed for The theory of FEM was initially developed for stress analysis and was applied to CFD problemsstress analysis and was applied to CFD problemsstress analysis and was applied to CFD problems stress analysis and was applied to CFD problems only in the last two decades only in the last two decades

Spect al MethodsSpect al MethodsSpectral MethodsSpectral Methods

Spectral methods approximate the unknowns by Spectral methods approximate the unknowns by means of truncated Fourier series or series of means of truncated Fourier series or series of Chebychev polynomials Unlike the FDM or FEMChebychev polynomials Unlike the FDM or FEMChebychev polynomials. Unlike the FDM or FEM Chebychev polynomials. Unlike the FDM or FEM approach the approximations are not local but approach the approximations are not local but valid throughout the computational domain. We valid throughout the computational domain. We g pg preplace the unknowns in the governing equation replace the unknowns in the governing equation by the truncated series. The constraint that lead by the truncated series. The constraint that lead to the algebraic equations for the coefficients ofto the algebraic equations for the coefficients ofto the algebraic equations for the coefficients of to the algebraic equations for the coefficients of the Fourier or Chebychev series is provided by the Fourier or Chebychev series is provided by

Spect al Methods (cont )Spect al Methods (cont )Spectral Methods (cont)Spectral Methods (cont)

the weighted residuals concept similar to FEM or the weighted residuals concept similar to FEM or by making the approximate function coincide by making the approximate function coincide with exact solution at a number of grid pointswith exact solution at a number of grid points

Most work on these methods was carried out in Most work on these methods was carried out in h d d dh d d dthe second decadethe second decade

Finite Vol me Method (FVM)Finite Vol me Method (FVM)Finite Volume Method (FVM)Finite Volume Method (FVM)

The FVM was originally developed as a The FVM was originally developed as a special FDM formulationspecial FDM formulationpp

The CV integration distinguishes the FVM The CV integration distinguishes the FVM from all other CFD techniquesfrom all other CFD techniquesfrom all other CFD techniquesfrom all other CFD techniques

The resulting statements express the The resulting statements express the conservation of relevant properties forconservation of relevant properties forconservation of relevant properties for conservation of relevant properties for each finite size celleach finite size cell

FVM (cont )FVM (cont )FVM (cont)FVM (cont)

The clear relationship between the numerical The clear relationship between the numerical algorithm and the underlying physical algorithm and the underlying physical conservation principle forms one of the main conservation principle forms one of the main attractions of the FVM and makes its concepts attractions of the FVM and makes its concepts much more simple to understand by engineersmuch more simple to understand by engineersmuch more simple to understand by engineers much more simple to understand by engineers than the FEM or the spectral methodsthan the FEM or the spectral methods

The FVM use has dominated the CFD communityThe FVM use has dominated the CFD community The FVM use has dominated the CFD community The FVM use has dominated the CFD community for the last three decades and most commercial for the last three decades and most commercial CFD codes are based on this methodCFD codes are based on this methodCFD codes are based on this methodCFD codes are based on this method

P ocessingP ocessingProcessingProcessing

Approximation of unknown flow variables Approximation of unknown flow variables by means of simple functionsby means of simple functionsy py p

Discretisation by substitution of the Discretisation by substitution of the approximations into the governing flowapproximations into the governing flowapproximations into the governing flow approximations into the governing flow equations and subsequent mathematical equations and subsequent mathematical manipulationsmanipulationsmanipulationsmanipulations

Solution of the algebraic equationsSolution of the algebraic equations

Disc etisationDisc etisationDiscretisationDiscretisation

It is the process of approximating the partial It is the process of approximating the partial derivatives in the governing equations using derivatives in the governing equations using g g q gg g q gTaylor series and converting them to Taylor series and converting them to algebraic equationsalgebraic equationsg qg q

E ample of Disc etisationE ample of Disc etisationExample of DiscretisationExample of Discretisation

0=+

xTu

tT

0)( 11

1

=+ ++ TTuTT njnjnjnj 0

2=+ xt

)(2 11

1 nj

nj

nj

nj TTx

tuTT ++ =

Famo s disc etisation schemesFamo s disc etisation schemesFamous discretisation schemesFamous discretisation schemes

FTCS FTCS LaxLax--Wendroff Wendroff

Crank NicolsonCrank Nicolson Crank NicolsonCrank Nicolson RichardsonRichardson Leap FrogLeap Frogp gp g Alternate direction implicit (ADI)Alternate direction implicit (ADI) Alternate direction explicit (ADE)Alternate direction explicit (ADE)

11 tt O d U i diO d U i di 11stst Order UpwindingOrder Upwinding 22ndnd and higher order upwindingand higher order upwinding QUICKQUICK QUICKQUICK

(Most of these schemes were developed during the first (Most of these schemes were developed during the first two decades)two decades)

Nat e of disc etised schemesNat e of disc etised schemesNature of discretised schemesNature of discretised schemes

Explicit schemes: calculate the value of the flow Explicit schemes: calculate the value of the flow variable at the new time interval using values at variable at the new time interval using values at the old time interval for the neighborhood nodesthe old time interval for the neighborhood nodes

Implicit schemes: calculate the value of the flow Implicit schemes: calculate the value of the flow i bl h i i h ii bl h i i h ivariable at the new time using the new time variable at the new time using the new time

interval values for neighborhood nodesinterval values for neighborhood nodes

E o s in n me ical sol tionsE o s in n me ical sol tionsErrors in numerical solutionsErrors in numerical solutions

Truncation errors due to truncation of series. Truncation errors due to truncation of series. Higher order accuracy reduces the truncation Higher order accuracy reduces the truncation errorerrorerrorerror

Machine roundMachine round--off errors result due to chopping off errors result due to chopping of decimal digits in the machine language. Useof decimal digits in the machine language. Useof decimal digits in the machine language. Use of decimal digits in the machine language. Use of double precision machines has almost of double precision machines has almost eliminated machine roundeliminated machine round--off errorsoff errors

Aliasing error is the inability to resolve small Aliasing error is the inability to resolve small components of the solution due to grid size components of the solution due to grid size limitation Larger RAMS reduce aliasing errorslimitation Larger RAMS reduce aliasing errorslimitation. Larger RAMS reduce aliasing errors. limitation. Larger RAMS reduce aliasing errors.

C diti f i l l ithC diti f i l l ithConditions for numerical algorithmsConditions for numerical algorithms

How good a numerical algorithm is ?How good a numerical algorithm is ?It is determined by:It is determined by:It is determined by:It is determined by: ConvergenceConvergence

C i tC i t ConsistencyConsistency StabilityStability

The FDM algorithms developed during the The FDM algorithms developed during the first two decades used these criteriafirst two decades used these criteria

Con e genceCon e genceConvergenceConvergence

It is the property of a numerical method It is the property of a numerical method to produce a solution which approaches to produce a solution which approaches p ppp ppthe exact solution as the grid spacing or the exact solution as the grid spacing or control volume size reduces to zero.control volume size reduces to zero.(theoretically difficult to prove)(theoretically difficult to prove)

ConsistencConsistencConsistencyConsistency

Consistency of numerical schemes Consistency of numerical schemes produce systems of algebraic equations produce systems of algebraic equations p y g qp y g qwhich can be demonstrated to be which can be demonstrated to be equivalent to the original governing equivalent to the original governing q g g gq g g gequation as the grid size spacing tends to equation as the grid size spacing tends to zerozero

StabilitStabilitStabilityStability

Stability is associated with damping of Stability is associated with damping of errors as the numerical methods proceeds. errors as the numerical methods proceeds. ppAn unstable scheme can cause wide An unstable scheme can cause wide oscillations and divergence of the solutionoscillations and divergence of the solutiongg

Additional conditions fo FVMAdditional conditions fo FVMAdditional conditions for FVMAdditional conditions for FVM

Patankar (1980) formulated rules that yield Patankar (1980) formulated rules that yield robust finite volume calculationsrobust finite volume calculationsrobust finite volume calculationsrobust finite volume calculations ConservativenessConservativeness

B d dB d d BoundednessBoundedness TranspotivenessTranspotiveness

Conse ati enessConse ati enessConservativenessConservativeness

Conservativeness property ensure global Conservativeness property ensure global conservation of the entire domain. This is conservation of the entire domain. This is physically achieved by means of consistent physically achieved by means of consistent expressions for fluxes through the cell expressions for fluxes through the cell p gp gfaces of adjacent control volumes.faces of adjacent control volumes.

Bo ndednessBo ndednessBoundednessBoundedness

Property is akin to stability and requires Property is akin to stability and requires that in a linear problem without sources that in a linear problem without sources ppthe solution is bounded by the minimum the solution is bounded by the minimum and maximum boundary values of the flow and maximum boundary values of the flow yyvariables.variables.

T anspo ti enessT anspo ti enessTransportiveness Transportiveness

All flow processes involve the phenomena of All flow processes involve the phenomena of convection and diffusion. Convective phenomena convection and diffusion. Convective phenomena involve influencing exclusively in the flow involve influencing exclusively in the flow direction so that a point only experiences effects direction so that a point only experiences effects due to changes at upstream locations Finitedue to changes at upstream locations Finitedue to changes at upstream locations. Finite due to changes at upstream locations. Finite Volume schemes with the tranportivemness Volume schemes with the tranportivemness property must account for the directionality ofproperty must account for the directionality ofproperty must account for the directionality of property must account for the directionality of influencing in terms of the relative strength of influencing in terms of the relative strength of diffusion to convection.diffusion to convection.

Sol tion of disc eti ed eq ationsSol tion of disc eti ed eq ationsSolution of discretized equationsSolution of discretized equations

The discretized equations result in a The discretized equations result in a system of linear algebraic equations which system of linear algebraic equations which y g qy g qcan be represented in matrix form:can be represented in matrix form:

[A] [x]=[C][A] [x]=[C][A] [x] [C][A] [x] [C]where where [A]=Coefficient Matrix[A]=Coefficient Matrix

[x]=Flow variable column vector[x]=Flow variable column vector[x] Flow variable column vector[x] Flow variable column vector[C]= Constant Matrix (Collection of [C]= Constant Matrix (Collection of

sources terms)sources terms)sources terms)sources terms)

Sol tion MethodsSol tion MethodsSolution MethodsSolution Methods

1.1. Direct Methods: provide the solution in a Direct Methods: provide the solution in a finite and predetermined number of finite and predetermined number of ppoperations using an algorithm that is operations using an algorithm that is often relatively complicated.often relatively complicated.y py p

2.2. Iterative Methods: consists of repeated Iterative Methods: consists of repeated application of an algorithm that is usuallyapplication of an algorithm that is usuallyapplication of an algorithm that is usually application of an algorithm that is usually relatively simple.relatively simple.

E amples of di ect methodsE amples of di ect methodsExamples of direct methodsExamples of direct methods

Cramers RuleCramers RuleGaussian EliminationGaussian Elimination Gaussian EliminationGaussian Elimination

TriTri--diagonal matrix algorithmdiagonal matrix algorithm LU DecompositionLU Decomposition MatrixMatrix--InversionInversionMatrixMatrix InversionInversion

E amples of Ite ati e MethodsE amples of Ite ati e MethodsExamples of Iterative MethodsExamples of Iterative Methods

GaussGauss--SeidedSeided Successive Overrelaxation.Successive Overrelaxation.

PressurePressure Velocity Coupling Algorithms:Velocity Coupling Algorithms:PressurePressure--Velocity Coupling Algorithms:Velocity Coupling Algorithms:

SIMPLE: SemiSIMPLE: Semi--implicit method for implicit method for pressure linked equationspressure linked equationsp qp q

SIMPLER: SIMPLE Revised.SIMPLER: SIMPLE Revised.SIMPLEC: SIMPLE ConsistentSIMPLEC: SIMPLE Consistent SIMPLEC: SIMPLE ConsistentSIMPLEC: SIMPLE Consistent

PISO: Pressure Implicit with Splitting PISO: Pressure Implicit with Splitting f O tf O tof Operatorsof Operators

PostPost P ocessingP ocessingPostPost--ProcessingProcessing

Domain geometry and grid displayDomain geometry and grid display Vector plotsVector plots

Li d h d d t l tLi d h d d t l t Line and shaded contour plotsLine and shaded contour plots 2d and 3d surface plots2d and 3d surface plots Particle trackingParticle tracking Particle trackingParticle tracking View manipulation (translation, rotation, scaling View manipulation (translation, rotation, scaling

etc.)etc.) Real time animationReal time animation Color postscript outputColor postscript output

E pe imental Flo Vis ali ationE pe imental Flo Vis ali ationExperimental Flow VisualizationExperimental Flow Visualization

CFD in the beginningCFD in the beginningCFD in the beginning CFD in the beginning

urs of pressure distribution the blade surfaces in the presence of bubbles

Bubble field simulation around a propeller using an actual nuclei size distribution as measured in the ocean shows where the

b bbl b l ( d th i ibl tbubbles become large (and thus visible to human eyes as cavitation)

erimental setup of the vessel with mixer (left) showing the free surface and vortex ath it; numerical simulation (right) also shows the free surface and vortex as wellath it; numerical simulation (right) also shows the free surface and vortex, as well

as a plane where a relatively steady velocity field can be observed.

Whe e is CFD heading in f t e?Whe e is CFD heading in f t e?Where is CFD heading in future?Where is CFD heading in future?

ODELINGODELINGFluidFluid--solid interactions (FSI)solid interactions (FSI)( )( )MultiMulti--phase flow modelingphase flow modelingModeling of gas mixturesModeling of gas mixturesode g o gas tu esode g o gas tu esTurbulent jet impingementsTurbulent jet impingementsCombustion modelingCombustion modelingCombustion modelingCombustion modelingNonNon--Newtonian flow modelingNewtonian flow modelingWeather prediction modelingWeather prediction modelingWeather prediction modelingWeather prediction modeling

F t e (cont )F t e (cont )Future (cont)Future (cont)

MESH GENERATIONMESH GENERATIONEfforts are underway to develop mesh Efforts are underway to develop mesh y py pgenerators with a selfgenerators with a self--adaptive meshing adaptive meshing capability. Ultimately such programs will capability. Ultimately such programs will refine the grid in areas of rapid variationsrefine the grid in areas of rapid variationsCOMPUTATIONAL SIDECOMPUTATIONAL SIDE

Cluster computingCluster computingGrid computingGrid computingp gp g

Th k Th k Thank youThank you