FEA by nitin gokhale

- Some fatigue analysis softwares supports only time domaln analyùs and then it become necessary to perform dynamicas wellas fatigue analysisin time domain.- Some times input providedby client / experimentai departmentIdata acquisitionagency is of theform gZl Hz.lt should be convertedto (m1sec')'lHz by multiplyingail the terms by (9.8lP. files, MT5 RPC III files, Test time histoh files. ~preadsheetext files, Binaty files etc. While importing the data default setting and scale factors shouMbe adjusted properly. importedReferences1) R.D.Cook, D.S. Malkus M. E. Pierha.Concept &applicationsof FmiteElement Analysis,Third EditionJohn Wiley& sons.2) W.T.Thornron :Theory of vibrations,Third Edition, CBSPublishers& Distributors3) J.S.Rao,K.Gupta :TheThwry &Practiceof MechanicaiVibrati~isF, ihh edition, Wile Eastern Lirnited,4) KewalPulara:Vibration& NoireFor EnQineers Fourthedition-Dhanuat Rai &Sons51 [rwin ~r&szig:~dvanccdïng:nee<in;~athematicr, hfth edition. G l c ~artern~ m i t e d-61 V. Adam,&A. Arken~r:iBuildirq Belter produrtr with FiniteElrment anaiysir. OiiwardPress71 SrinivaranP : NonlinearVibration Analvsis. New Aae International Publishers8) Nastran, Famfet, 1-deaoOptistruct help documentation

Thermal maiysis m15.1 IntroductionHeat transfer is defined as energy in transit. Analysis of a system using the laws of heat transferis named asThermalAnalysis. Heattransferisa branchofthermodynamicswhich deals with rateof heat transfer between two or more equilibrium states of a system.Thermal analysis is investigation of the part or a system, to calculate heat transfer rate andtemperature distribution. Different approaches for thermal analysis are, 1)Theoretical(Analytical) 2) Experimental 3) Numerical 4) GraphicalWhy thermal analysis?To understand the physical phenomenon either natural or man made & most important tocontrol theinTheoreticalapproach of thermal analysis is applicableto simplegeometries(exactsolution). Experimental approach is limited by the the, cost and uncertainty in measuring theproperty data ( e.g. pressure, temperature, velocity) due to complexity of geometry, inducederrorsc Instrumental, human) and environmental disturbance. Numerical approach is probablythe best approach i.e. using the present high end hardware &software computational support& validated computer codes.The forth approach. using graphical techniques is a history & nowoutdated.In order to understand the importanceof thermal analysisin Our daily life, let us ask somequestions:-* Why a sugarcube dissolves faster in warm water than in cold? Why soiar cookers workefficientlya t high altitudeslike in Himalaya? Why ventilators & exhaust fan are mounted on top position on wall? Why wefeel cooleras we gain altitude? Why a deep freeze compartrnent is always in top positionin refrigerator? Why we see condensation on wings of air plane while landing? Why a golf bal1is having dimples on itsface?9 Why in the design of a compact heat exchanger turbulent flow is preferredover laminar? Why tornado's rotatesanticlockwisein southernhemisphere & clockwise in Northern?Do we ever thinkof answering questions like these.Principlesof heat transfer &Fluid mechanicsare useful in answering questions like above and many more...To solveany problern,first step is to understandthephysicsof theproblem. Severalsoftwaresare 237

Thermal Analysis available to performThermal& Fluid analysis. Cornmonlyused Thermal / Fluid analysis software's : Ansys Sinda Flowtherm/Radtherm MSC Nastran Abaqus IDEASNx (TMG, ESC) lcepack Airpack Fluent ( CFD solver) CFX ( CFD solver) STAR-CD ( CFD solver) Pure thermal analysis is rare as most of the industrialproblemsare coupled(Flow+ HeatTransfer analysis) calledas Conjugate HeatTransfer Analysis.The aim of suchanalysis is either to find the heat transfer coefficient or temperature distribution(when heat transfer coefficient is known).ln order to performthermal analysis of a system, wefirst need to distinguish theheat transfermode & then type of flow if it's a conjugate heat transfer analysis. Conduction Convection Radiation /\ /\ /\ Natural ForcedSteady Unrteady Specular Dlffused Compressible Incompressible J\"\"\ /\ Vixous Invistid/ lnvisc~d (Doe~n'iexirt) /\Rotational lrrotatlonal Rotatmal Irrotationai RotationalLaminar Laminar LarninarTurbulent Turbulent Turbulent

15.2 ConductionHeat Tranrfer Thrs is a hemdral FE model of O pole assemblyofAir CircuitBreokertoconducf Thermal Elecrric Analysls. The main objectiveofthe application ofslmuhtlon methadsinSwitchoearddesonistocutthe corrr for experimenlul inve~tigalionfor tliir thegeomelryofonecompletepoied a Fr1 circuil brmker l6WAmo har been modeied uriny ho-E. ~ ~ ~ & , 4 Neor h uwd to generole FE model. ANSIS ir the solver and port-proca5sing is done usrng HypetWew. Alr Circull BreokerThermal- ElectricAnaiysis (ImageSource: AltairCalendar2006,Courte5y:LorsonandToubro Lld,, Switchgear division)The transfer of heat is normally from a high temperature object to a lower temperature object.Conduction heat transfer is observedin al1typesof phases i.e. solid, liquid & gas. In solidsit is dueto lattice vibration (translational& rotational) & mainly due to free electrons.Hence pure metalsIikeCopper, Aluminiumare goodconductorsofelectricityandalsogoodthermalconductors.But,the surprising thing is, sometimes a bad conductor of electricity can be a very good conductorof heat (e.g. Diamond - used in special cutting tool). In liquids, conduction is due to molecularmovement & in gases due to molecular collisions.Let's first define temperature and internal energy. Temperature i s a measure of the averagetranslational kinetic energy associated with the disordered microscoplc motion of atoms andmolecules. Interna1energy is defined as the energy associated with the random, disorderedmotion of molecules. Temperature i s not directly proportional to internal energy sincetemperature measures only the kinetic energy part of the internal energy, so, two objects withthe same temperature do not in general have the same internal energy. Heat transfer changesinternal energy of both the systems involvedaccording to the First LawofThermodynamics.Fourier's law governsconductionheat transfer. Fourier provedby experiments, that heat transferrate is proportional to area, temperature differenceand inverselyproportional to thickness.WhereQ = Heat transfer rate (W)A= Area (m2)dT=Temperature differential (\"ClW OR)dX =Thickness (m)k=Thermal conductivity (Wlm K)The bracketed term is called as temperature gradient, its been assigned with negative sign, as

Thermal Analysis the temperature gradient vector is in oppositedirectionof heat flow as shown below. nie mi hwt îranskr k tn the direction of negatiw afthe temperatcmgradlent.The mathematicalgradient of a function is a directional derivative, which points in the directionof the maximum rate of change of the function.The direction of heat transfer will be oppositeto temperature gradient, since net energy transfer will be from high temperature to low. Thisdirection of maximum heat transfer will be perpendicular to the equal-temperature surfacessurroundinga source of heat.Conduction heat transfer is significant in solids. Any heat transfer problem obeys the first law ofthermodynamics (Energy i s conserved) & to satisfy this, we have to apply boundary conditions.(e.g. knowntemperature, heat flux, heat sourcelgeneration,convective like known heat transfercoefficient or radiation).In general there are three types of boundary conditions to solve conductive heat transferanalysis. 1. Dirichlet Boundary condition (Boundarycondition of lstkind) Wall temperatures (TlVT2,T3.T4)are known &constant 2. Neuman Boundary condition (Boundary condition of 2d kind) When this gradient of temperature is fixed alsocalledasvariabletemperatureboundary condition d$ldx= c,d$ldy= c,d$ldz= c 3. Robin Boundary condition (Mixed) (Boundaryconditionof 3\" kind) +A$ B d@/dx= 0q = constant

Exercise 1:Pin fin S e d y state conduction usingFEA softwareInput:Geometry (diameter & length of fin, materialproperties k -Thermal conductivity)Meshing: 1Dlinear beam elementsBoundary conditions : Fin base temperature, convective BCs data [heat transfer coefficient,ambient temperature)Output: Temperaturedistribution (spatial), heat fluxProcedure for analysis:Step1 I Dmesh is used as the problem can be considered as one dimensional.Cross Sectionof beam is circular (dia. 0.02m). Fin is represented via 5 beam elements.Step 2 Apply thermal boundary conditions ApplyThermal BC1 on lumped mass element & speclfyBC type as const.Temp. = 200°C Apply Thermal BC 2 on Beam element free faces & specify BC type convectivecoefficient. = 22.11 W/m2 OC & environmentaltemperature= 30°C+Step 3 Model manager study setupCheck al1Boundaryconditions & solve.Step 4 Solver control Select steady state option for solution. Set the heat balance residual (itis to check whether problem obeys first law of thermodynamics). Solve & check for convergence ( Influx = efflux) Post-processthe results & compare with analyticalresults.

Thermal Analysis 15.3 Steady State Conduction Steady state conduction problems are of elliptic in nature (partial differential equations (PDE) are of elliptic type). For elliptic PDE, al1 walls are required boundary conditions of any type as discussed earlier. It is also called as equilibrium type problems, like problems of structural mechanics satisfyingequilibrium conditions. Governing PDE for steady state conduction problems is Laplace's Equation. Numerical approach is used for solving such non-linear PDE. Most preferred method to solve Steady state conductionproblemsis Finitedifferencemethod. Finitedifferencemethod: lt is a numerical methodin which computationaldomainis subdivided in cells, either of equal spacing or varying & evaluate properties like temperature, pressure, velocitiesat nodal points usingTaylor'sseries, we assumethat properties are continuousin space & time. Thus using Taylor's series, we convert second order PDE to Linear algebraic equations. Using Taylor's series, we can convertequation 2 as (2Dcase):- T,, = Ax2Ay2/f2*(Ax2+Ay1)*l { (T,, +TL,)/(Ax2) *(T,+, +TI.,)/( Ay2)) ----------------------.-- (3) Using any of the availablecomputer language (e.g. FORTRAN, C, C++) equation3could be solved for unknown temperature a t different nodal points. Steps for solving equation 3 DefineAx&Ay lnitializethe completedomain with a guess temperature Apply boundary conditionsat boundaries Solve equation 3 recursivelyusing DoIforloop&updatetemperatureatal1nodalpoints after each iteration. Keep track of number of iterations. Applyconvergence criteria, asvariationin temperatureat any nodein domainbetween consecutive iteration, to be less than O.Ol°C (temperature residual). For convective

boundary conditions, also apply heat baianceresidual.Aftertheresidualsfortemperature&heatbalanceareachievedprint nodaltemperaturedata tofiie or via contour plots.Post processinq is a very important process as like understandingthe physics, as in this step, weare correlating-the output data with physics again. Temperaturecontour plot 1 1with goodquality tria1 1 1elements using FEMTemperaturecontour plot Temperaturecontour plot with highiy skewed tria with highly skewed tria elements using FEM elements usingFDMAbove comparisonshowseiernent quaiity&element typedo not haveanyeffecton temperaturecontours for pure conduction problems using FEM but FDM (Finite Difference Method) rnakesdifference in temperaturecontour plots.Using the temperature distributions, we can calculate thermal strains, which could be further+used to calculatecombinedstresses (thermal structural)using iinear static analysis.+-Thermal straina - Coefficient of linear thermalexpansionAT -Temperature differenceReal life steady state ronduction examples:- Connectina rod bras bush Dress fit simulation : Connecting rod at small end has a bras bush with an oil hoie.The assembly of bush is done by pressfitTo simulatepress fit effect temperaturegradientisappiiedoninner &outersurfaceofbush (toaccountforcompression loading inside small end of connecting rod). Temperaturedifferentiai causes thermal strain & create effect of pressfit. Expansionof railwav tracks:We know, why there isagapbetweenrail trackjoints.In summer soiar heating of tracks give riseto expansion in longitudinal direction, which if arrested can cause deflection of track & hence a serious issue. Gap at rail joints allow thermal expansion of rails. But solution for one could be a problem for other, as this gap is a source of noise generation due to impact of wheel &rail nearjoints.Nowa day's special materialsare used as filier in gap, which wiil allow thermal expansion but also reduce noise levels.

Thermal Analysis15.4 Un~tcrdyState GmductionUnsteady means, temperature is a function of time. As such al1problems of heat transfer areunsteady in nature. To reduce computational complications & solution time, we assumeproblems as steady. As per thermodynamics every processfinally reaches t o steady state, as rimeapproaches to infinity.Problems like design of electric heater, water heater, solar cooker, satellite heat exchangers andwelding simulation are some of the typical examplesof unsteady conduction analysis. Unsteady HeatTransfer Analytical Method 1(Lumped HeatCapacity) Numerical MethodThe governing equation for 1D unsteady heat transfer is stated asWe canuse analyticalmethod with the condition t o satisfy Biot Number, Bi < 0.1.Physical significance of Biot Number is it's a ratio of conductive resistance to convectiveresistance.Hence, analytical method canbe used in cases, where thermal conductivity is higher&convection heat transfer is insignificant. (e.g.Thermocouple bead, superconductors).Lumped Heat Capacity :This method is governed by equationWherem= Mass flux= p *Vc = Specific heat (Kllkg K)T = Ambient temperaturet =TirneEquation 6 can be re written asSolving equation 7 we get-(T-T )I(T,- T _ ) = e ( - h N p V d tWhere~ = T i m econstant = hAlpVcT,= initial temperature

-Practical FiniteElernent AnalysisT = Unknown temperatureh =Heat transfer coefficient (W/rn2)Exercise 2:Pin fin unsteady state conduction using FEA software- - -Rod Length = 10 cm p . ,. . -Rod Diameter= 2 cm 2O 3 22.1 1°k=12W/m C 750 20.00 1006 19.50c = 480 Jlkg OC 1250 19.30p = 7800 kg/m3 79.27 19.00h =As per table shownFin initial temperature=20O0CFin base temp (TJ = 200 OCBasic steps forThermal Analysis: Simplify the geometry (De-featuresmall fillets. notches, pin holes) Decideif to simplify as 1D, 2D or 3D Meshing:. 9 Create finer mesh in high temperature gradient zones, heat source or a t interfaces in composite > Use iinear elements to reduce computationalt h e . 9 Use couplingapproachbetweentwo parts, when temperaturedistribution i s not required. (Assignconstant thermal conductivity to coupling part) 9 Lookfor symmetry, periodicity in the model Neglect lessdominant physicai phenomenonlike radiation, natural convection.1 9 5 Convection Wat TransferConvection heat transfer is due to molecular movement of fluid such as air or water, when thefluid is caused to move away from the source of heat, carrying energy with it.Many industrial thermal probiems are convective in nature. Aim of convective heat transfer 245

analysis is to calculate heat transfer coefficient.Basic governingequationis Newton's Law of CoolingWhereQ - Heat transfer rateh -Heat transfer coefficientA - Surface areaT - Surface temperatureTl- Fluid temperatureConvection heat transfer requires fluid media (e.g. air, water etc.). Convective heat transfer isclassified as follows EanrccMwt Ha*frsnsfk INatural (Çree) ForcedFurther fluid flow canbe compared in following way Incompressible(If Mach number < 0.3) /Compressible (IfMach number > 0.3) lnvicid (if p=0) /Viscous (if p t O) Irrotational 1Rotational Steady 1Unsteady LaminarITurbulent1Mixed lD/2D/3DFrom thermal analysis point of view, it's important to know, if it is Laminar orTurbulent, as heattransfer coefficient is a function of this.LaminarVs.Turbulent Flow:Heat transfer is mainly due to conductlon nea Wall Heat transfer 1smalnly dueto mokcular movement,& then convcction, as molecuks do not cross fluid as thcre 1s cross movement withln fluid laminas.laminasForexternalflow Re < 5 x 10\" For external flow Re> 5 x 10<For Interna1flow Re <2300Shearstress- T = pduldy For Cnternalflow Re > 2300 1 - --Advantageousfor smooth surface geometries (e.g. 1 r reuion of aerofoil Shearstressr= p du/dy+ (Reynolds's stresses] ! 1 Advantageousfor rough surface (e.g.Turbulent flow overa gdfball providesrequlreddlrectlonalstablHty in fllghtf

MosttyrequlredIn many altuatlonr tiy physics. 1caser.Expermentalfindingsawiequkedto perlorm numerlcal analysIr. I Analflical ireatmcnt 15possible. Velocltyvectorsfromwall - 1/7* power prdie -. &Wall 1 -- [ & ~ c i t ~vectors from wdl -Ilneai profilesl2u15.6 Forced Convection (Internai Flow)Flow through pipes, ducts, water cooling jacket of ICengine and air flow in intake system are fewexamplesof internal flow. it i s carried out t o find out HeatTransferCoefficient (HTC) also knownas film coefficient & pressure drop.For internal flow, there are lot of analytical correlations, especially for simple geometriesof different sections like circular (used in shell & tube heat exchanger), elliptical (automotiveradiators), square/ rectangular (used in air conditioning ducts).Before starting with convection heat transfer, it is necessary to know few non-dimensionalnumbers &their physical significance.Reynolds's Number:It is a ratio of inertia forces t o viscousforces. Afunction of velocity, density,viscosity & characteristic length. Mud flow or slurry flow has very low Re number. ln general, ifRe is dominant then, we have t o consider convection heat transfer, it could be an internal orexternal flow.For low Re number (Re < 2300), i.e. Laminar flow, in pipe there are exact resultsof HTCNu =3.657 ---------------------------Constantwall temperatureFor Turbulent flowNu =0.023 Re08Prn --- Dittus- Boelter equation for fully developed flow. n = 0.4 Fluid being heated n = 0.3 Fluid being cooledNusselt number: Itis ratio of convectivet o conductive heat transfer.Where h - Heat transfer coefficient (HTC)

ThermalAnolysisL - Characteristic length (m)k-Thermal conductivity of fluid ( W/m K)Biot number (hUk) of Unsteady Conduction looks very similar to Nusselt number but both aredifferent.In Nusselt number, kis thermalconductivity offluid&inBiot itsofsolid.TheBiotnumberis a measure ofthetemperature drop in thesolid materialand the temperature drop between thesolid and the fluid. The Nusselt number is a dimensionless version of the temperature gradientat the surface between the fluid and the solid, and thus it providesa measure of the convectionoccurring from the surface.Validation case samole1Dexamole (Fuliv develo~edflow):1 Twail= 8O0C ShellMesh ThermalProperties for water at 60°C k =0.654 W/ m OC p =4.71 e-4 kg1m.s I1 p = 983.3 kg/m3 BeamMesh 1 Cp=4179J/kgC T , , = 6OoC 222 4.195 9702 \"\" -0.67'Step1 Create mesh of -.:::!i?ments, cirr-.,. -,-ta 0.0254 m.

Step 2 Define material as waterStep 3 Boundary Conditions Create velocity inlet BC at inlet as the flow is incompressibleVw=2 mlsec Thermal BC to shell elements 80°C (T) Set ambient conditions t o 60°C (TF) Create forced coupling between shell & fluid elements using analytical correlationStep 4 SolverControl Select steady state option for solution After solution recover results(select al1data for recovery). Postprocess the resuihfor viewing velocity, temperature, Reynolds nurnber etc. at nodes & elements, record the outlet water temperature Boundary conditions &set run directory(a)\"\"Step 5 Result VerificationwithTheoryNu =0.027 * ReoB* Pr'\" ( Sieder &Tate) - For Fully DevelopedTurbulent FlowAll properties except y evaluatedat bulk fluid temperature.T water,out (Theory ) = 69.5\"CT water,out ( software with constant properties ) = 67.9 OCT water,out (withvariable properties ) = 68.6OCp= Fluid viscosity evaluatedat TmTm(Bulkfluid temperature) = (T,+TJ 1 2Comrnon practical examples o f interna1flow are:-* Waterjacket flow analysis ( I.C. engines) Heat exchanger analysis* Flow simulation in intake&exhaust manifoldsFor suchcases, there is noanalytical correlationsavailable, hence wehavetogoforcomputationalanalysis commonly called as \"Computational Fluid Dynamics (CFD).\" From CFD, we can acquireproperty data like pressure, temperature, velocity, turbulence quantities, density & viscosity atany point in fluid domain.This helps in optimization of the geometry t o reduce pressure drop &enhancethe heat transfer coefficient.The main aim of external flow analysis is to calculate drag & lift on body due to pressure &viscous forces acting on body. Empirical correlations are available for external flow over simplegeometries. This subject is used in advanced applications like aerospace, in order t o calculateliftldrag on aerofoil and in automobile to optimize the external shape of car bodies.Normally external flow is turbulent in nature, except near wall (region of flow near solid body isknown as boundary layer region).Boundary layer is a thin layer of flow in which viscousforcesaredominant & which are responsibleforfrictional drag on body.Flow around a half hemispherical body (Incompressible, Viscous, Rotational &Turbulent 249

ThermolAnaiysis External flow)Quad~!tateraml esh arnundhalfhemirphere VelocityvectorsaroundhalfhemisphefeContours Ofstaticpre~sure[Pa) .- Contours of totaipressure(PolThe flow around a half hemisphericalbody, is numerically simulated using computational fluiddynamics. Numerical analysis shows gradients of different flow variables (pressure, velocity) &the stream line contours. In external aerodynamic flows, area near wall plays important role, asviscous shear stresses are dominant.We can clearly see the separation point in velocity & total pressure contours plots. Bluerecirculation zone is visible in total pressure plot, here flow is rotational. Also static pressure islowest, where velocity is highest.In the present case, adverse pressure gradient is the cause of flow reversal& it activates at pointof separation.It's a general rule, that for long slender bodies (e.g. aerofoil) laminar flow is advantageous asviscous drag is dominant & hence, we can avoid separation by limiting drag in boundary layeronly. That's why boundary layer is retained on the aerofoils to avoid flow separation usingtechniques like suction & hence lessdrag.

In case offlow around bodies, which are blunt shapes, the profile or pressure drag is significant &in order to reduce this, normally such bodies prefer turbulent flow to reduce the wake region.Sirnplest validation case forexternal flow isflow overa flat plate, was usedby Prandtltoformulatetheory popularly called asT'BoundaryLayerTheory\"in 1950. Typicai velocity profile in the laminar regirne & RexThe above figure shows various thermal boundary layers in presence of heat transfer.Analytical correlationsare available for laminar & fully turbulent zoneLarninar regirne Nu = 0.332 PrTt3Re'\"Turbulent regime Nu =0.0292 PrIt3Reo8Both correlationsare for constant wall temperature case.i, As Re nurnber increases, pressure drop increasesi, As diameter increases. pressure drop increasesk Profile /Pressure drag > Viscous dragImportant terms in Fluid Mechanics : Absolute pressure: - Pressure rneasured from absolute zero Gauge pressure: - Pressure rneasured above or below atmospheric pressure Static pressure: - Pressure a t a point away from dynamic effects = pgh Dynamic pressure: - Pressure caused due to velocity of fluid = 112pv2 +Total pressure:- Static pressure Dynarnic pressure Profile drag :- Drag force due to wake formulation behind a submerged body in a flow domain

rhermal Analysis -Viscous drag: Drag force due to friction between fluid & solid surface at fluid-solid interface. +Total drag:- Profile drag Viscous drag 15.8 Meshingfor ThermalAnalysis: For simulatingboundarylayer and interna1or external flows,biasingaf mesh is reeommended. -In general following 3 types of biasing are usedLinearBiasing Bell curveBiasing Exponential Biasing (Mostly usedfor externalMostlyin structural analysis) (Mostly usedin pipe flow) awoàynamics)Mesh biasing, is coagulationof mesh cells in the high property gradient zones (e.g. in boundarylayer, around heat source, at interface between two solids of different materials.)Unstructuredmeshlngaround 2Dcar body:Number of elements - 10184Calculationis expensivedue to higher number of elementsTime requiredto generate grid is Jess, as it is fully automatic

Boundary layer regionis very well captured, using finer grids in these regions Convergence is slower. Mesh quality - poorStructured meshing around 2D car body : Notch back vehiciewifh baundaryfiffedgriduring trianguior&quadriiateraIceils. Numberof elements-6516 Calculation are 33% cheaper, as less numberof elements than unstructured Time required t o generateis more. (Blocking techniquesare used) Boundary layer regionisvery well captured, using exponential biasing Convergenceis more faster. Mesh quality - goodObservations: Near wall mesh density is more. Exponentiaibiasing used to capturevelocity gradient. Aspect ratio away from body of vehiclecould be as high as 10-15. Aspect ratio near wall region is approximately 1. Mesh isfiner in the rear portion of the vehicle to capturevortices produced by the body. In the viscous region near wall, skewness of cells is maintained close t o ideal elements skewness. As shear stress calculationsare carried out in near wall region.15.9 FreelNaturalConvectionHeat transfer is a natural phenomenon, taking place due to density variation, which in turn i s afunction of temperature. In natural convection, momentum & energy equations are solved incoupled manner, hence computation is more expensive.Velocity is function of density, density is function of temperature hence velocity is function oftemperature.

To solve natural convection problems, boundary conditions at lnlet & outlet of flow domain are set to atmospheric pressure. For faster convergence, predicted Reynolds's number can be imposed at inlet by using velocity inlet boundary condition. For analysis definition of gravity direction is very important. Fluid properties are normally evaluated at mean temperature & wall temperature is fixed (constant). For simple geometries like vertical plate correlationsare available. As per Churchill and Chu average Nusselt number is Another important number is Grashof's number. It is defined as follows (Buoyancy force) X (Inertia force) (Viscous force) Natural convection plays major role in natural phenomenons like TornadoICyciones Movement of clouds Underwater currents All above phenomenons are function of density and in turn pressure variations in environment. The practical use of Natural convection in modern age i s in laptops. Here the space is limited to add forced convection (Cooling fan). Hence the cooling of PCB components is carried using natural convection. Natural convection answers the question why the coolest portion in refrigerator is arranged at top most position.254

15.10 Radiatian HeatlransferRadiationheat transferis consideredto beadvanced subjectin theheat transfer studies, detailedtreatment is beyond the scope of this book and in following pages we will have overview ofradiation heat transfer fundamentals.Thismode of heat transferdo not require medium, as like in conduction &convection. Radiationis the transfer of energyvia electromagnetic waves.So the next questionsarises, what is wave &how to find out the heat transfer rateby mode of radiation.For this, we first see what is wave, the following figure explainsthat. Itis form of sinusoidalshapewith vertical pick distance defined as amplitude&pickto pick distance is the wavelength.Basedon variation in the wavelength, we can differentiate wave types. Following figure shows typesof waves.All object that have temperaturemore than O K emit heat by radiation.Now, we will answer thenext question,as how tofind outrateof heat transfer by radiation. Forthis, we will introduceonemore definition as black body.

T i i e r i ~ i oAi ~iniysis It is an obiect, which is perfect absorber as well emitter. For the obiect to behave like black body does not hean it to be black coloured. Before starting the study of radiation heat transfer, let us discuss an important finding by Max Planck, popularly known as8'ThePlanck hypothesis': Max Planck proposed that hot body radiate the energy only in discrete quanta (also termed as photons), which were proportional to the frequency. This became the foundation of modern quantum theory,\"Quantum mechanics': Where E =Quantum energy of photon. h =Planck constant - 6.626 x IO3' ).sec The radiation formula derived by Planck says that, the average energy per\"niode\"or\"quantum\" is the energy of the quantum times the probability of its occupation. The probability function could be derived by Bose-Einstein Distribution. Where Q - Average energy per\"mode\"or\"quantum\" k - Boltzmann constant - 1.3806505 xlO-*l JoulelKelvin. The Boltzmann constant'k'is to remember contribution by Austrian physicist Ludwig Boltzmann. It is a bridge between macroscopic and microscopic physics. Macroscopically, absolute temperature i s proportional to the product of the pressure P and the volumevof an ideal gas at that temperature. An experimental finding of product IkT] at room temperature is 4.14~10-\"J, by which we can conclude, why Kelvin temperature scale i s used in radiation heat transfer. -T (Room temperature - 27°C) = (4.14~1 (1.3806505~1O-23=) 299.85 300 T - Absolute temperature of a radiant body (K) N - Number of molecules of gas The curly bracketed term of equation (9) is the probability function. Total power per unit area from a black body radiator can be obtained by integrating the Planck's radiation formula over al1wavelengths. After integration one can arrive at the same result, which was proved experimentally by Stefan &Boltzmann working independently. The outcome of the integration is as follows

PtacticaiFiniteEiementAnalysis m Where c- Speed of light - 299,792,458 mlsec.Thebracketed term is nothingbut the Stefan-Boltzmannconstant. Stefan-Boltzmannlaw i s usedfor calculating radiationemitted by an object to its temperature.WhereE =Total amount of radiation emitted by an object per square meter(Wattsm ')o= Stefan-Boltzmannconstant = 5.669 X 1Oa W/ m2 K4E = Emissivityof the bodyTl &T, is the temperatures of the objects in KWien's dispiacement law states that, the hotter an object is the shorter the wavelength a t whichit wiil emit most of its radiationand the frequency for maximal or peak radiationpower is foundby dividing Wien's constant by the temperature of the radiant body in Kelvin. Maximum useof Wien's iaw is in astronomy & thermal management with flexibility from designing infraredsensing equipments to study of big bang. By using Wien's law & measuringthe wavelengthfromradiant source, one canfind out the temperature of the body from a remotelocation.Whereb - Wien's displacement constant - 2.8977685(51) x 10\" m K A hm) Wien's distribution curvesAs such practical use of radiation heat transfer i s limited to mostly orbital applications likedesigning satellite, heat exchangers, solar thermal analysis (Diurnai solar heating) and soiarequipment design, radioactive heating in nuclear power plants, combustion process in ICengines, in short at al1high temperature applications. For general applications radiation heattransfer is normally neglecteddue to lower surface temperature.

Thermal Analysis Itis also usedin environmentalsciences, study of green houseeffectandrelatedsubject. As such al1the available thermal analysissoftware's have capabilityto analyse the radiation heat transfer. But al1the software do not support both the forms of radiations i.e. specular & diffuse. Following figure shows the meaning of the specular & diffuse radiation. 'bIncidentray///////O / //////ffSpecularradiation is mostlyin mirror likesurface, combmed specular&diffuse is obsewedin therealsurfacesandpure diffuse radiationisobservedin the atmasphereassolarradiation onearthisassumedas diffise.Th1s approximationwork wellfor manyengineeringproblemsand hencemostthermal analysis softwareare basedon a$sumptionof diffuse radiation.Anaîyiical approachto radiationheat transfer is limifed &bas& on calculationshadowmg viewfactor between the objects using calculationmethodslikeoppenheimand Gebhardt.15.11 PractlcalApplicationof Thermal Analysis1) Design of cooling system inTractor1CarlTruckCooling system design is an important area in engine design. Cooling system is heat sink forengine and a proper design of cooling system enhance output power and efficiencyof I.C.engine.Coolingsystemdesign includesfollowing steps:1. Determination of heat rejection:The starting point of cooling system design is to get engine heat release curve either fromthermodynamiccalculationsor by conducting experiments.It is an important task, to find maximum heat rejection point. Heat release ratelQ'is highest atrated engine rpm, but the fan & water pump speed is low at maximum torque point. Hence theheat rejectionat maximumtorquepoint is consideredas more severe in some caseS.Thisisvalidonly for fans coupled directly to engine crankshaft.

Critical heat rejection:The maximum ambient temperature is an important parameter to be considered. Criticalcoolingcondition, i s the condition, where, the requirements by the engine on the cooling system resultsin Maximum EntranceTemperatureDifference (ETD).ETD = lnlet coolant temperature - lnlet air temperatureDecisionof critical cooling condition is based on following assumptions:i l Fan speed = VA(Inlet airflow rate)2) Air flow rate (at one ETD) .;Radiator heat release rate (Q)3) Fan speed (n), = Engine speed (n,\",&4) From above three assumption, we can conclude- Q/nEngn=nCeonstant.The ETD will be maximum when Q/nEnqimisemaximum. Normally it happens a t maximum torquepoint.Typical heat release curve of Engine: Typiral Engine Heat Releare Curve2. Design of Radiator (Heat exchanger) :Design constraints from vehicle side Compact Low pressure drop Low weight Higher heat rejection. Low costActual Radiator design requires > Large frontal area (preferablysquare) > Optimal heat rejectionDesign engineer has to make a fine balance between above two constraints. Normally fortractor or truck large frontal area radiator is required but this requirement is constraint by the

Tliermal Analysis space available.In order to match heat rejection requirement, variouspatternsof radiator fins as discussed below could be tried out to increase air side heat transfer coefficient. -- CORRUGATEDFIN DESIGN CORRUGATEDFIN WlTH TURBULENCE STRIPSSTRAIGHTFIN DESIGNFor al1types of radiator design, water tube cross section i s normally elliptical. 2-3rows of tubesare recommended, to reduce the pressure drop.Manufactuiing cost c04yRiskof Clogglng Less Higheriieatrejection forequal frontal Moderate 1Higherarea J -Pressure dmpSpacerequlied for a dehned ( &&ztors, I n d u t r i ' Truck, b u jeep & heat rejectionrate \"1power plants ppllcation militarytankFor design finalization, following input graphs (characteristicof a heat exchanger)are requiredby coolingsystem designer.These are usuallyprovided by radiator manufacturer.

Radiofor pieirure drap curve Air flow rateva trn3/hr1 Radiator hearreleaie w r v eAfter finalization of Radiator design the next step is to design the fan,3. Design of Fan :Thisis the most challengingstep,in thedesign ofcooling system.Differenttypesoffan mountingsare as follows.a) Engine driven fans normally mounted on water pump pulley (Beltdriven)b) Motor driven fansc)Viscouscoupling fansEngine driven fan design is more difficukThis type of fan also extract power from engine, whenactually it is not required from cooling point ofview. Reason to adopt this types offans is easy tomanufacture (mostly metallic, plastic) and low cost.So, how to tune a\"FAN\"to Our cooling requirement?Following parameters affect performance of cooling system 1) Fan speed 2) Fan diameter 3) Air flow rate 4) Number of blades 5 ) Cliord length 6) Fan hub dianieter 7) Fan to cowl tip clearance 8) Distance between fan & engine block 9) Fan blade and cowl ring overlap 10)Cowl ring shapes1)Fan speed: Fans speed = Crankshaft speed * (Crank pulley diameterl Water pump pulley diameter)This formula is for fans mounted on water pump pulley, where water pump i s belt driven by 261

TtiernlnlAnalysis crankshaft. Present days in cars, electric motor driven fans are used. These are controlled by a temperature sensor placed in the fan hub. Motor driven fan are costlier, but well sulted for fast rnoving vehicles, as they take the advantage of ram air at high speeds. 2) Fan diameter: Fan diarneter is decidedby air flow rate requirement.Normallyfan diameter is limited to the size of radiator frontal area as shown below r Fan diarneter r Rachator Fmnal Area Case 1, optimum Case 2, Radiatornot Case3,fan ir under desiun. hence even evenk cooledflow rhroughradiator3) Air flow rate : it is function of fan diameter, number of blades and chord length. The initialguess value is taken from radiator heat release curve & then iterations are carried out till we getoptimized radiator outlet water temperature. This iterative process i s normally carried out bynumerous experiments, but present days CFD techniques are well adopted to reduce the t h e& cost of these experiments. CFD requires only one validation test & further iterations could becarried out nurnerically.4) Number of blades: Air flow rate is directly proportional to number of blades. In general itis lirnited by flow overlap of blade to blade & noise constraints. Mostly 6, 8 or 12 blade fan ispreferred.5 )Chord length :Air flow rate is proportional to chord length.6) Fan hub diameter :Reduction in hub diameter results in more fan blade area & thus the airflow rate. But there is a constraint from structural strength point of view i.e. it weakens the fanblade at high speeds.'-I\1IFanHubFan Made

rn Practical Finite Element Analysis7) Fan tip clearance: It i s the most important parameter as it account for re-circulation of flowa t fan tips & hence the reduction in fan efficiency. Low values (3-5 mm) of tip clearances arepreferred. Normally fan is mounted on engine & cowl on radiator. Hence, relative movement ispossible which in turn force the tip clearancevalues in the range 8-15 mm. - ,CowlRing8) Distancebetween fan & engine block:More the gap better will be the results(as it reducesthe recirculation). Distance from blockage H9) Overlap of blade t o cowl ring : Following figure explains the effect of variation of thisparameter. Normally overlapof 45% is recommended.

10) 5hape of outlet of cowl ring:Curved cowl ring increasesefficiency of coolingsystem. Consideration of ail above parameters will lead to design finalization of radiator, fan &cowi for çooling system. A typical performance mapping of cooling system is carried out & plotted as follows 10 15 Volume h w rale Above figure shows the operating point, if direction\"a\"is followed then we are over designing the fan & if its'lb\" then the fan is running in stall condiion, in which the pressure drop due to system is more than the pressure rise by fan. Itis a situation which must be avoided in order to increase life of fan &adverse effects on enginefrom overheatingdue to improper cooling. +System resistance = Radiator pressure drop pressure drop due to frontal obstructions, parts +kept in front of radiator (e.g. battery, air cleaner, horns etc.) + pressure drop due to front grill 10% of radiator pressuredrop (In order to consider fouling lossesof radiator). We normaily over design the cooling system. But reduction in performance of cooling system will result in higher water temperature & indirectly oil temperature. Sometime the overheating c m leadto engine seizure.264

We can further optimizethe cooling systemby parameterizationof above discussed parametersand by using the softwares (suchas KULI) along with statisticai techniques iikeTaguchimethod.Present days, CFD combined experimental method is very common. it gives fast & accuratesolutions of flow variables.II) I CEngineblock thermal analysis:iCenginecooling analysis can be categorised asa) Water cooled enginesb) Air cooled enginesWater is a beîter heat conductor than air and hence water cooled engines are more efficientcomparedto air cooled engines, especiailyas the engine compression ratio or rpm goes up.Theseadvantages of water cooling motivatemost engine designersto use water as heat transfermedia. Significantdifferencesare as followsFor high compressionratio &RPM enginer For low power engines (Iwo wheeler)Thermodynamicallymore efficient Lesefficient ICompler design Simpledesignlncreasesthe total weight of engine Negligible rirein weightLeads to warmnty problems like llner Wariantyproblems are negligible .cavltations, corrosion, overheating due to ~heat exchanger clqging etc. ,'.lncreasesnumberof partstovehicle (mdiator, Noadditionaiparts areaddedthermostat, surgetank,fanetc) I =\"Thermal analysis of water cooled engine:Following steps are followed, afterfinaiizingengine specification.1) Design of water core & head core2) Design of liner (Basedon bore, stoke &thickness)3) Design of water pump4) Design of cooiing system components (radiator, fan, oil cooler etc.)Few important aspects of thermal analysis of engine block:1) Cylinder head valvebridge water velocities(design of cross-sectionsin head water core)2) Piston &vaive (exhaust & intake)cooling3) Liner cavitation4) Cyiinder head gasket designThermal analysis of water cooledengineblock starts withainput of engineblock CAD geometry.+Thermal analysis is carried out in coupled rnanneri.e. CFD +Thermal Structural,also known as\"ConjugateHeat Transfer Anaiysis:

Thermal Analysis The mesh generated for engine block & head is same for thermal & structural analysis if it is using hex elements, but if it is tetra then normally lineartetras are used in CFD, thermal analysis &second order tetras for structural analysis.Complete analysis can be broken downin following headings: A. CFD analysis of water jacket & generate Heat Transfer Coefficient (HTC) data on different walls. B. Use HTC data, from CFD & map on inner wails of engine block & head. Perform thermal anaiysis. (conduction+convection) C. Map temperature data calculated from thermal analysis to perform the structural analysis (toevaluate thermal stresses). Now, we wiil brlefly study each. A. CF0 analysis of water jacket: Crankcase&headwaterjacketaremeshedusinglineartetrahedronelementsastheCFDcodesare based on control volume techniques.(ICEMCFD / ANSA 1HYPERMESH). Meshing is a important step in CFD. Crankcase water core is coarsely meshed compared to head as the head core is more complex. Important features, like valvebridge areas, thermostatand sharp curvatures are fine meshed. Approximatetotal mesh size for 6 cylinder inline engine water jacket is 2-3 million cells. One of the CFD solverslikeFluent/ Star-CD/CFXis used to solve the probiem by applyingproper boundary conditions. Normally water is considered as incompressible, it's a fair assumption to reduce computationalefforts. U Crankangk (Qq]Cmnkangle[Deg)ifire redeckregion)lntake Port 120-170

HTC data is exported from CFD soivers & mapped on FE model of engine block for thermalanalysis.The oUtcome of CFD analysis, Mean HTC O 50 1W 150 200 Cyllnder block & head wall temperature(\"C) WC lW/mz k) HTC&GarTempvariaVanvs. piston strokeRThermal analysis :HTC data is imported & patched on t o FE model.The calculatedHTC data in CFD is stored at cell(elernent) centre is mapped in FE model at nodal points using process calledinterpolation.Appropriate thermal boundary conditions & material properties are applied to evaluate thetemperature gradientsin engine block.

C. Structural analysis:The third part of engine water jacket analysis is to perform linear static analysis to evaluatethermal strain & stresses.+=asATO,= ET * EzT-ThermalstrainO,- Thermal stressa - Coefficient of linear thermal expansionAT -Temperature differenceReferences:1) Frank P. lncropera & david P. Delvitt : Fundamentals of heat and Mass Transfer, Fifth edition,John Wiley & Sons2) J.P. Holman : HeatTransfer,8th edition, Mc Graw Hill.19973) Schlichting H. :Boundary LayerTheory, 6th Edition Mac Graw Hill, New york, 19684) Fox R.W. and A.T.McDonald : Introduction To Fluid Mechanics, 3rd edition, Wiley, New York,19855) Prof. K. N. Sitharamu, Prof. Ajit Kumar Kolar, Prof. T. Sunderrajan: Notes from M-Tech coursetaken from IITMadras, Heat Transfer and Thernial Power Engg. Dept.

Computational Fluid Dynamics -116.1 What is CFDCFD Stands for Computational Fluid Dynamics. It is a numericaltool to solve the equations ofFluid Dynamics by suitablemethods which can capturethe essential physics of the fluid.The numerical schemes that are used for discretization of the equilibrium of equations for fluid,i.e. the Navier - Stokes equations can be one of the following : a. FiniteDifferenceMethod b. Finite Volume Method c. FiniteElement MethodCFD is now reckoned as a matured major discipline owing to the hand in hand developmentoverthepast30 years of numericalalgorithmsforcomplex flowcomputations,and of computerswith enormousmemoryandspeed,forchurning out vast amount of numericaldata tosimulatethese complex flow situations.Whilethepossibility to resolve smallscale phenomenon (turbulence) has drivencomputationalFluld Dynamics in the direction of phenomenological studies, such as Direct NumericalSimulation of Turbulence, the possibility to obtain speedy solutions has endeavoured CFDto practising engineers and designers, who view the fast turn around time as an essentialingredient reducing design cycle times and cut in project costs.The recent trend of using too much of CFD in industrialapplications and the consequentimproved performance ofthe designed productshas leadsome ofthe enthusiasts to announcethat the days of the wind tunnels may be limited. While suchanextreme view is unnecessary,a synergetic use of CFD and experimental Fluid Dynamics or wind tunnels should be the keyfor a successful design.The three dimensionsof Fluid Dynamics :As described earlier every problem in CAE canbe described in terms of the following: 1. Level of physics 2. Level of geometric complexity 3. Computing power required. tGeometv

The same i s also true for Coniputational Fluid Dynamics and the classificationof problems can be as follows: 1. Simple Geometry SimplePhysics : Several textbook examples can belong to this category. Following are some examples : a. Quasi one dimensional nozzleflow with various flow conditions. b. Laminar boundary layer flow past a flat plate. c. ShockTube problem for compressible flows. Such kind of problems can be solved on PCs today 2. Simple Geometry ComplexPhysics :A typical exampie can be development of vortices in the following what is calledas driven cavity problem. Here although the geometry is simple, the physics can be very complexand the ultimate model can be Direct Numerical Simulation of Turbulence . 3.ComplexGeometry SimplePhysics: Anexampleofsuchaflowcan besay solution of potential flow (inviscid irrotational) over a full aeroplane or an automobile. Such problems are solved by Source and panel methods in aerodynamics but in principle these are similar to Boundary Element Methods used in Structural Mechanics . 4. Complex Geometry Complex Physics : An example can be solution of Reynolds Stress averaged Navier Stokes (RANS) Equations over a full aircraft. Such problems are solved on parallel computers. Various bevels of Thereare manylevels ofapproximation in Fluid Dynamics and each has gota differentphysical meaning thereby it emphasizes the need for a proper scheme of numerical discretization.That is a numerical method whicli can perform best for a specific flow application may be totally unsuitable for another flow range or type. lt is said that\"Numerica1Fluid shouldfollow the actual fluid\". Thisis similarto the kinds of various analysis tliatwe perform in structural analysis. That is linear static, geometric nonlinearity, material nonlinearity, contacts, a combination of nonlinearities, dynamics in frequency and time domain. One can select a suitable model which can simulate the actual physics to the maximum extent . Similarly a hierarchy of fluid-flow models is obtained by simplifications to the Navier - Stokes equations resulting progressively through the neglect of viscosity, rotationality, t h e dependence or less - dominant terms or through linearization or through simplification of boundary conditions. The following levels of approximations are used in CFD in the decreasing order of complexity: 1. Reynolds - Averaged Navier- Stokes equations with suitable turbulence modeling . 2. Parabolized Navier - Stokes Equations (Streamwise viscous terms are neglected) Which are similar to Boundary Layer philosophy.270

3. Euler equations for inviscid but rotational flows.4. Potential Equations for inviscid and irrotational flows.5 . Small- perturbation potential equation with simplified boundary conditions.6. Linearized Potential Equations which are the simplest Laplace Equations.Theequilibriumequationssolvedin CFDare mass,monientumand theenergybalance equationswhich under the assumptions of linear relation between the stress tensor and the strain rate arethe well known Navier - Stokes Equations .By using the standardnotation of Cartesian tensors these are :Mass Conservation IContinuity Equation : - +ap alpu,) axj = O ..................... (1) atMomentum Conservation :alpu) + ~(Pu,u~) JP h', at +-- - a ~ ,axj --- ..................... ( 2 ) 3x1Enerav Conservation :~pe) aipy(pe t P)I a(q,i ~T,,U,I + axl +-- - ..................... (3) at ax, a5In the above equations : P = fluid density x,, x, x, = Cartesian coordinates = fluid velocity along x, P = pressure e = specific total energy per unit mas8 t = time 4, = heat flux vector 7.. = viscous stress

au a (F(- F ~ ) a (G!-G~J a (H,- HJ ,+ +t -- .........(1O)azat ax ;tvWhere we have split the flux vector into insidepartFi, G i ,H & viscous part Fy, G y ,HyIn conventionalnotations ofx, y, zdirectionsand u, v, w being the velocitiesin these directions, theseare givenbyThe above forms of equations of Fluid Dynamics which are nothing but the DIFFERENTIALform is the basis of numerical schemes such as FiniteDifference and FiniteElement schemes. onservation Law5It is also possible to recast the equations in integral form as they are valid for any arbitraryvolume.This forms the basis of finite volume method used extensively in CFD.Consider the inviscid equations ( Euler Equations ) an3 au SF{ 3~~For any volume R, bounded by surface dQ, we canintegrate (12)over volume R, and then writeas

Where we have used Green's divergence theorem which replaces the volume integral by surfaceintegrals.Writing equation (13) separately for each consideration,we obtain,Mass conservationMomentum conservationEnergy conservationWe can regard equations (14)-(16)as more fundamental than the differential form by treatingthem as balance equations.Note that differential form contains partial derivatives of unknowns as against the integral fromwhich contains only integralS.The conditions of integrability are less severe than the conditionsof differentiability. The integral form i s much more meaningful from a physical point of view asit clearly depicts that rate of change of a conserved quantity in volume is due to flux across itsinterface.We have already written the Navier-Stokes equations governing the equilibrium of fluid flow.They are as follows.

Mass Conservation IContinuity Equation: +JP a ( p u i i --- ---- = O at ax,Momentum Conservation : + ~ ( P u , u , ) -- - ap +--a-5- 3puJ axj axj a~, atEneryy Conservation :After understandiny the fact that the left hand side of the NS Equations (above equations)is a nonlinear convection and the riyht hand side is a diffusion phenomena, most of thedevelopment of numerical schemes for CFD was based on usiny schemes for linear convectionequation and the diffusion equation.The linear convection equation in one dimension is :This is known as wave equation and the familiar wave equation is structural dynamics can bederived from (17) by just change of variables in differentiation.This equation has followiny exact solutionU(t)= f (x-ct) --------- (19)The physical meaniny of above is that the wave travels intime with its shape beiny retained inspace as shown in followiny figure Solution oflineor wave eariation.This wave equation mathematically is an hyperbolic equation.

The standard equation for modeling diffusion problem is au = cr- in 1.D .........(20)One can immediately notice the similarities of above equation with heat conduction equationwhich i saT K a2T .........at pcP ax2This is paraboiic equation and cris called the diffusivity of the problem. [The units of diffusivityare in m2/sec1The physical meaning of above equation i s that heat diffuses through the entire structure &ultimately a steady state is reached.Thisi s illustrated in figure below Temperature profile A at later instants ofTypical Solution afa diffusion equation

16.8 NumericalSchemes for a ModelConvection EquationThere are number of numerical schemes for convection equation & to mention a few, followingare some of them 1) Upwind differencing method 2) Lax wendroff central difference method 3) Maccormack method 4) Warming & beam'supwind schemes 5) Euler implicit method 6) Leap frog method 7) Trapezoidaldifferencing method 8) Warming kutter-bmxa methodComparisonsof Some standardmethods & their stability limits along with order of accuracy arepresented in the accompanyingtable.Stability of a numerical scheme is governed by a non dimensional number called the'courant'number & i s given byThe same number also governs the stability of t h e integration schemes (central differencescheme) used for structural crashworthiness analysis.n-Time levelj- spacial discretization index in x-direction Table: Cornparison of numerical scheme for linear convection equation in 1-D

Practical Finile ilenient Analysis1The following methods are to solve the standard diffusion equation are 1 ) Simple Explicit method 2) Richardson's Method 3) Simple lmplicit ILaasonerMethod 4) Crank- NicolsonMethod 5 ) Dafort -Franked MethodThe stability of the standard diffusion equation is governed by a non -dimensional diffusionnumber r=-aAt (Ax)'Following table gives a coniparison of some common methods.

Table: Comparison of scheme for Diffusion Equation in 1-Dk i t and lmplicit NumericalExplicit Scheme : We Say that a scheme is explicit when the information a t time ievel \" n+l\"depends on previous time levels i.e \" n , n-1 etc:' e.g. the upwind difference scheme forthe convection equation is anexplicit scheme.lmplicit scheme : In these schemes the information at time level\"n+l\"is dependent notonly on previous time levels but also on the current time level e.g. an implicit numericalscheme for a convection equation can be developed as follows. U,+CUx=OU,\"\" - Uj\" Cat + Ui+,\"+' - Ui~,\"\" = O .....................0..0..0..0..0..0(23)We can write above equation as

vVWherea= - , d = l , b = - & c = u \" 2 21Written in matrix form7 .................. ..............................-....7........ -. U,\"\" U,\"\" O -O . . . . O b, d, d, JWhere N is the number of grid points in x-direction. erenl Types of alculalionsThe end result of CFD is in the form of a set of computer codes, which can be of two categories:1. Research codes : These are invariably being developed at research Iacademic institutions,typically running on high end Computer configurations to super computers.2. Industrial codes : These are commercial softwares to be run in a production mode, aregeneral purpose codes which tackle themost frequently encountered fluid problems suchas interna1 and external flows involving compressibleand incompressiblefluids and laminarand turbulent flows.A number of commercial softwares are available today in the market. A list of these canbe found on the internet. Most commonly used industry codes are FLUENT, STAR CD andCFX.Schemes used i n Practical CFD Software :We have seen various numerical schemesfor standard Model equations. It must be rememberedthat the practical equations to be solved in fluid mechanics are a\"System of conservation law\"which arennonlinear\"in character.After the development of numerical algorithm for standard Model equation, a straight forwardextension of the scheme to system of conservation laws presented lot of difficulties as fluiddynamic phenomenon such as shocks present a severe challenge to numerics. This renderedonly some of the numerical schemes to be practically useful.

Computotional Fluid Dynamirs One difference that must be kept in mind that for incompressible flow, there is no energy equation to be solved and \"pressure\" has meaning as \"mechanical/ hydrostatic\" rather than \"thermodynamic\" one used in compressibie flows. The foilowing is the iist of most common schemes used in standard CFD softwares available today. FiniteVolume Based Softwares : A) lncornpressible Flow 1. SIMPLE algorithm due to Patankar and Spalding . (Semi lmpiicit Method for Pressure Linked Equations ) 2. Versions of 1. which are SIMPLER and SIMPLEC 3. PIS0 (Pressure lmpiicit Split Operator ) scheme. B) Compressible Flow 01) Upwind Deference category scheme FluxVector Splitting Schemes 1.Schemes at Continuum level given by 1.1 van Leer 1.2 Liou and Stefan's AUSM ( Advection Upstream Splitting Method) 2. Schemes based on KineticTheory of Gases /Boltzmann Equation. In this category several schemes have been developed by researchers in indian lnstitute of Science and the most popular scheme finding application in Defence organizations in Our country i s KFVS (Kinetic Flux Vector Splitting) Method developed by S. M. Deshpande and Mandal. This scheme has been wideiy used and tested for complex geometry aerospace configurations and is aiso used in CFD expert, a software developed by IiT,Mumbai. Flux Difference Splitting Schemes 1. Roe Scheme 2. Osher Scheme 62) Central Difference Schemes 1. Jameson- Schmidt-Turkel Scheme 2. MacCormack Scheme Jameson scheme uses Runge Kutta time discretization and central differencing in space and MacCormack scheme uses a two step predictor corrector approach as discussed for a linear convection equation. 60th ofthese schemes requireanadditionai artificial viscosity for stabilization in presence of shocks and this has been a major drawback of such schenies. FiniteElement Method Based Softwares : The finite eiement method presents lot of difficulties to compressibie fluid flow dynamics

as the schemes ultimately result into central differencetype. The following algorithms havebeen the most popular schemes used in softwares based on FE Technology.1. SUPG ( StrearnlineUpwind Petrov-Galerkin) Method developedby Hughes and researchersin USA. Used in NASTRAN for thermal problems associated with forced convection. Thisscheme has also been used in BOEINGand Dassault for practical configurations and it is alsoused in severalCFD softwaressuchas ACUSIM.2.TaylorGalerkin Method developedby Swansea UK. NlSA software developedby EMRC usesa FEM approach for Fluidflow based on this rnethod.Muchunificationisnowobserved onajointtreatment of FiniteVolumeMeth~d&FiniteElementMethod and research is very active on this front.The reader is referred to the references given at the end of this chapter for a detailedtreatmentof the above methods.16.12 DifferentTypes of Grids Used for CFDPeople use a variety of grids for CFD and the main classification can be structured grids vs.unstructuredgrids. Some examples of these are shown in the following figures.

H Grid Topoiogy There are some solvers which are based on just using a Cartesian grid and in these the point on the mesh need not be a point on the body.Theseare calledas Non-Body fitted grids against theones shown here where each point on the body (airfoil here) isalso a mesh point. In structural mechanics, the mesh continuously deforms with the deformation whereas in fluid dynamics mesh is always fixed, fluid particles enter and leave the control volume. This is why mesh quality criteria are somewhat relaxed in fluid mechanicsas compared to structural mechanics. If you take a look on solving Say a structural mechanics problem of finding out stress concentration in plate due i o applied loads then one can clearly see the deformed mesh pattern as the structure develops stresses. in fluid mechanicr Mesh is aiwoys fixed with respect to rpace. in StrucfuraiMechanicr the mesh deformr continuously as the structure gefr loaded onddevelopr stresser. Theabovedistinction comes into pictureas in solid mechanics,wefollow alagrangian approach and in fluid mechanics, we use Eulerian or field approach.284

Explain thetypical mesh Quality parameters used in CFD and mention how y o u w i l lcheck a mesh quality of a CFD mesh generated b y a software or a vendor:The typical parameters used in CFD are quite similar to structural mechanics except for somedifferences in terminology. For a standard software like ICEM HEXA ,the criteria and theirvalues are shown in the accompanying table. The element is called as cell in CFD , and thedeterminant is nothing but the jacobian of the element. Dihedral angle is the angle betweent w o planes. A standard table to check the mesh created by you or submitted by a vendor isgiven here..Turbulence modeling for CFLI ti s said that \" laminar flows \"exist in textbooks and turbulent flows exist in practice. Laminarflow is characterized by streamlines running in well ordered manner with adjacent fluidlayers sliding relatively t o each other with no motion or change normal t o streamlines. As it iscurrently impossible tosolve full Navier -StokesEquations oruse Direct numerical simulationof turbulence, wealways use atime averagedform calledas the RANS (Reynolds Averged NavierStokes) equations. These introduce a closure problem as the number of unkowns are greaterthan the number of equations and one must introduce some ways t o suitably model theReynolds Stresses. The Reynolds stresses are now new flow parameters apart from the fluid'sown constitution in terms of stresses to kinematic relationships. This is the crux ofTurbulencemodeling right from 1960s and the following models are commonly used.

Coiiipi~iotionoFl luid Dy~iamics1 . Two equation k - E model using turbulent kinetic energy and the dissipation rate.2. Zero Equation or Prandtl's mixing length model which calculates the turbulent viscosity without using the transport equations.3. One Equation model using a transport equation for the turbulent kinetic energy and an algebraic expression for the dissipation rate.4. Reynolds Stress model which rnodels the transport of turbulent shear stresses in each direction rather than using an isotropic turbulence.5. Large Eddy Simulation which uses a spatial filtering removing small scales in turbulence but capturing larger scale fluctuations thereby more accurately representing the true flow condition.16.14 Strengths and Weaknesses of CF erimental FluiDynamics or Wind Tunnel TestingMuch issaid about useof CFD as a \" Numerical WindTunnel \", some enthusiasts mentioningthat CFDcanreplace wind tunnels in thefuture but it isvery important to realize that the role ofCFD is or should be synergetic and the one that compliments experimental Fluid dynamics.We can classify the development of CFD algorithms into the following stages :1. Fundamental research phase : There has been a lot of fundamental research work into the mathematical modeling and numerical simulation of relevant physical phenomena such as turbulence, boundary layers, shock waves.2. Development of basic CFD tools /codes : This involves creating useroriented numerical methods for solving physically approximate functions, applicable to a large range of \" Boundary Value Problems\" of engineering interest sometimes requiring the incorporation of algorithms / modules of Research Phase.These require a VALIDATION, involving the assessrnent of the accuracy withwhich the numerical method solves the boundary value problem posed. Thisis a task undertaken completely by the method developers, involving theexamination and reduction of the numerical errors to an acceptable level froman engineering point ofview.3. Development of Evaluated CFD tools: This involves the comparison of Basic CFD methods with experiment, the aim being t o produce reliable, efficient programs that give quantitativesimulations(ofacceptableengineeringaccuracy) of actual physical flow situations. Ingeneral, the evaluation process will involve both the development of empirical correction procedures and the modification of Basic CFD tools to incorporate the correction procedures. The evaluation process must involve both the method developers and specially trained users.Using the above terminology, Our concern is to let the designer use EVALUATED CFDtools which reduce the design cycle time, cost and uncertainty without compromising designstandards, performance and safety.

In view of the above the strengths and weaknesses of CFD softwares can be convenientlyexamined in the two following contexts : 1. Assessing the quality of the basic CFD tools relative to both experiment and more realistic numerical solutions; such assessments should lead eventually to evaluated CFD Tools. 2. (Most Important) in the context of meeting the goals of evaluated CFD tools described previously.Thus in examining the strengths and weaknesses of CFD tools we distinguish between theevaluation process and the application capabilitiesof evaluated CFD Tools .it i s worth emphasizing that it is the coordinated and complimentary use of validated basicCFD Tools together with Experiment thus giving evaluated CFD Tool that can dramaticallyimprovethe effectiveness of the designer. Thus the strengthsofboth theory andexperimentare coordinated in such a way as to compensate for their individual weaknesses leading toimproved and more credible techniques from the point ofview of project managers. CFD offersa great promise against experimental fluid mechanics / wind tunnel simulation in the corningyears as the numerical algorithms and computers become more powerful.The main strengths of today's CFD tools are as follows :1. Complete domain data and better flow visualization of results : CFD solutions giveyou values of pressure, density etc. at al1 the locations inside the domain as opposed to windtunnels which can give you surface data values. Certainly wind tunnel simulation gives you avery clear picture of flow characteristicsbut fails to provide the information of flow parameterswhich can be very easily obtained as a by-product of a CFD simulation. It is very important torealize that a typical wind tunnel set up can consume enormous amount oftime as comparedto a quickCFD result which can generate data within a much reduced time and at a muchlessercost. The post processing of CFD results can give a moreclearer picture rather than Wind tunnelphotographs.2. Convenient ways t o alter design : The geometry modifications through CAD data cantake place very quickly and remodeling can be done immediately. A practical example is Saya design modification in the side mirror of a car. Physical models require much more time andeffort for adjustments.CFD is thus much cheaper as compared to Wind tunnel simulations asthey require a huge expensive set up (andmaintenance also!). Wind tunnels are found only atthe large companies, universitiesand government laboratories whereas CFD tools can be usedby small companies having sufficient expertise on flow physics and the software use.3. Measurement of certain quantlties:Monitoring and Measuring wind direction, pollutantconcentration, radiation, chemical reactions, species concentration: These are extremelydifficult tasks in wind tunnel simulation whereas the information from a CFD result is muchflexible accounting for each of these unique aspects.