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Gray, during the preparation of this editionLIST OF SYMBOLSJacobian of convective Tuxes4acobian of viscous fluxesconstant depth of control volume in two dimensionsCeed of soundspecific heat coefficient at constant pressurespecific heat coefficient at constant volumevector of characteristic variablesmolar concentration of species m(=pYm/wm)Smagorinsky constantow av au aw av ducurl vcurl of vdzdz dx dx dydistancediagonal part of implicit operatoraal dispatioeffective binary diffusivity of species mdu av dugence of v+—yInternal energy per unit massEtotal energy per unit massFourier symbol of the time-stepping operatorf evector of external volume forcesflux vectoAux tensoramplification factorrelocityenthal△hlocal grid(cell)sizeHotal(stagnation) enthaljHHessian matrix (matrix of second derivativesimaginary unitList of Symbolsidentity matrixunit tensor2hinterpolation operator2hrestriction operator1JJprolongation operatorsystem matrix (implicit operatorinverse of determinant of coordinate transformation Jacobianthermal conductivity coefficientKturbulent kinetic energyKf, Kh forward and backward reaction rate constantsturbulent length scalestrictly lower part of implicit operatorcomponents of leonard stress tensorMMach numberMmass matrixunit normal vector(outward pointing) of control volume facex components of the unit normal vector in x-, y-,2-directionnumber of grid points, cells, or control volumesnumber of adjacent control volumesFnumber of control volume facesstatic pressureproduction term of kinetic turbulent energytransformation matrix from primitive to conservative variablesleft and right preconditioning matrix(Krylov-subspace methods)Prandtl numberqhheat flux due to radiation chemical reactions. etcsource termposition vector(Cartesian coordinates); residual (GMRES)vector from point i to point 1Rspecific gas constantRuniversal gas constant(=8314.34/kg mol k)Rresidual, right-hand sideRsmoothed residualrotation matrixList of SymbReeynolds numberrate of change of species m due to chemical reactionsface vector(=△Scomponents of strain-rate tensorSx, Sy, s Cartesian components of the face vectorsurface element△Slength/area of a face of a control volumetimeturbulent time scaletime stepstatic temperaturematrix of right eigenvectorsT-1matrix of left eigenvectorsCartesian velocity componentsskin friction velocity(=tulp)general (scalar) flow variablestrictly upper part of implicit operatorvector of general How variablesvelocity vector with the components u, v, and icontravariant velocityIIwcontravariant velocity relative to grid motioncontravariant velocity of a face of the control volumemolecular weight of species mvector of conservative variables(= lp, pu, pv, pw,pEIvector of primitive variables([p, u,v,w, TI,y,之Cartesian coordinate system△cell size in x-directionnondimensional wall coordinate( p yur/uw)mass fraction of species mFourier symbol of the spatial operatorangle of attack, inlet anglecoefficient of the Runge-Kutta scheme (in stage m)parameter to control time accuracy of an implicit schemeblending coefficient(in stage m of the Runge-Kutta schemeList of Symbolsratio of specific heat coefficients at constant pressure and volumecirculationpreconditioning matrix (low Mach-number flowKronecker symbolrate of turbulent energy dissipationc∈cλA-Asmoothing coefficient(implicit residual smoothing); parameterthermal diffusivity coefficientsecond viscosity coefficienteigenvalue of convective flux Jacobiandiagonal matrix of eigenvalues of convective fux Jacobian^A^Aspectral radius of convective Aux Jacobianspectral radius of viscous fiux Jacobiandynamic viscosity coefficientkinematic viscosity coefficient(= u/p)5,n,scurvilinear coordinate systemdensityCourant-Friedrichs-Lewy(CFL)numberCFL number due to residual smoothingviscous stresswall shear stressviscous stress tensor (normal and shear stressescomponents of viscous stress tensorcomponents of Favre-averaged reynolds stress tensorRcomponents of Reynolds stress tensorcomponents of subgrid-scale stress tensorSEcomponents of Favre-filtered subgrid-scale stress tensorSRcomponents of subgrid-scale reynolds stress tensorrate of dissipation per unit turbulent kinetic energy (=E/kpressure sensorcontrol volumecomponents of rotation-rate tensor02boundary of a control volumemiter functionList of SymbolsXVau du dUVUgradient of scalar Uy02a-u du 0UVULaplace of scalar U2-norm of vector uU. USUBSCRIPTSconvective partrelated to convectiondiffuusive partnodal point indexL,,K index of a control volumelaminar: leftindex of control volume face; speciesRrightTturbulentViscous partrelated to volumewallx,y,z components in the x-,y-, 2-directionat infinity(far-field)SUPERSCRIPTSI,, K direction in computational spaceprevious time leve1+1new time levetransposeFavre averaged mean value; Favre-filtered value Lesfluctuating part of Favre decomposition; subgrid scale (lesReynolds averaged mean value; filtered value (LES)Fluctuating part of Reynolds decomposition; subgrid scale(LESABBREVIATIONSAGARDAdvisory group for aerospace research and Development(NATO)AIAAAmerican Institute of aeronautics and astronauticsARcAeronautical Research Council. UKASMEThe American Society of Mechanical EngineersCERCACentre de recherche en Calcul Applique(centre forResearch on Computation and its Applications), MontrealCanadaCERFACs Centre Europeen de recherche et de Formation avancee enCalcul Scientifique(European Centre for Research and AdvancedTraining in Scientific Computation), FranceDFVLR(now DlR) Deutsche Forschungs-und Versuchsanstalt fur Luftund Raumfahrt(German Aerospace Research EstablishmentDLRDeutsches zentrum fur Luft- und raumfahrt(German Aerospace Center)ERCOFTAC European Research Community on Flow, Turbulence andCombustionESAEuropean Space agencyFFAFlygtekniska Forsoksanstalten(The Aeronautical ResearchInstitute of Sweden)GAMMGesellschaft fur Angewandte Mathematik und mechanik(German Society of Applied Mathematics and Mechanics)ICASEInstitute for Computer Applications in Science and Engineering,NASA Langley Research Center, Hampton, VA, USAINRIAInstitut National de recherche en Informatique et en automatique(The French National Ie for Research in Computer sId ControlISABEInternational Society for Air Breathing EnginesMAEDepartment of Mechanical and Aerospace Engineering,Princeton University, Princeton, NY,USA