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NASA/TM–2020–5000766VULCAN-CFD Theory Manual: Ver. 7.1.0Robert A. Baurle, Jeffery A. White, Tomasz G. Drozda, and Andrew T. NorrisNASA Langley Research Center, Hampton, VirginiaApril 2020

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NASA/TM–2020–5000766VULCAN-CFD Theory Manual: Ver. 7.1.0Robert A. Baurle, Jeffery A. White, Tomasz G. Drozda, and Andrew T. NorrisNASA Langley Research Center, Hampton, VirginiaNational Aeronautics andSpace AdministrationLangley Research CenterHampton, Virginia 23681-2199April 2020

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not constitute anoffical endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics andSpace Administration.Available from:NASA STI Program / Mail Stop 148NASA Langley Research CenterHampton, VA 23681-2199Fax: 757-864-6500

Contents1Introduction12Governing Equations2.1 Thermodynamic Nonequilibrium . . . . . . . . . . . . . . . . . . . . . . .133Reynolds-Averaged Equations64Spatially-Filtered Equations115Hybrid RAS/LES Closures146Boundary Layer Bleed Models167Performance Extraction Methods18A Thermodynamic Properties26B Transport Properties28C Finite Rate Chemistry Extensions30D Reynolds-Averaged Turbulence Closure ModelsD.1 Spalart Allmaras One-Equation Model . . . .D.2 Menter k-ω Models . . . . . . . . . . . . . .D.3 Wilcox 1998 k-ω Model . . . . . . . . . . .D.4 Wilcox 2006 k-ω Model . . . . . . . . . . .D.5 Gatski and Rumsey 2003 k-ω EAS Model . .D.6 Wall Functions . . . . . . . . . . . . . . . .D.7 Compressibility Corrections . . . . . . . . .E Spatially-Filtered Turbulence Closure ModelsE.1 Smagorinsky Model . . . . . . . . . . . . .E.2 Vreman Model . . . . . . . . . . . . . . .E.3 Yoshizawa Model . . . . . . . . . . . . . .E.4 Dynamic SGS Closures . . . . . . . . . . .3232333536384043.4444444546F Blended RAS/LES Formulation48G Improved Delayed Detached Eddy Simulation49H Bleed ModelsH.1 Doerffer/Bohning Model . . . . . . . . . . . . . . . . . . . . . . . . . . .H.2 Slater Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .515153

IConserved Flux Method Data ReductionI.1 CMME Method Data Reduction Details . . . . . . . . . . . . . . . . . . .I.2 CMES Method Data Reduction Details . . . . . . . . . . . . . . . . . . . .565658

1IntroductionVULCAN-CFD offers a comprehensive set of capabilities to enable the simulation of continuum flowfields from subsonic to hypersonic conditions. The governing equations thatare employed include allowances for both chemical and thermal nonequilibrium processes,coupled with a wide variety of turbulence models for both Reynolds-averaged and largeeddy simulations. A description of the physical and numerical models available in thesoftware are presented in this document. However, it is emphasized that the descriptionsprovided are not intended to fully document every aspect of the models employed. Instead,the governing equations, and the models required to simulate them numerically, are presented with the goal of providing a sufficient level of detail to understand their strengthsand limitations. The reader is encouraged to access the references provided throughout thisdocument for a more complete explanation of the formulations presented.2Governing EquationsThe fundamental conservation equations employed by VULCAN-CFD to describe thermallyequilibrated chemically-reacting flows are the Navier-Stokes equations coupled with masscontinuity equations for the chemical species. These partial differential equations can bewritten as ρui t x j ρE t x j ρYm t x j ρ ρu j t x j ρui u j δ i j P τi j x j ρHu j q j τi j u i x j ρYm u j ρYm V (m)j x j 0 0 0 ẇm(1)where ρ is the density, ui is the velocity, P is the pressure, E is the total energy, H is thetotal enthalpy, τi j is the stress tensor, q j is the heat flux vector, and Ym , V (m)j , and ẇm are themass fraction, diffusion velocity, and production rate, respectively, of species “m”.The total energy is the sum of the internal and kinetic energy, i.e.,E e 1ui ui2(2)and the total enthalpy is given by the expression shown below.H E P1 h ui uiρ2(3)Here, the mixture enthalpy, h, is defined through the relationshiph nsXm 11Ym hm(4)

with the enthalpy of each species evaluated from a polynomial curve fit as described inAppendix A.VULCAN-CFD is currently limited to Newtonian fluids, where the stress tensor is related to the strain rate through the following constitutive relation,!! u j1 ui2S ij (5)τi j µ 2S i j δi j S kk ,32 x j xiand the heat flux vector (which contains contributions from heat conduction and the energyflux due to interdiffusion processes) is given by the expression provided below.nsq j λX T ρhm Ym V (m)j x jm 1(6)The diffusion velocity of species “m” is provided by Fick’s law of diffusion, i.e.,V (m) jD YmYm x j(7)and the evaluation of the mixture transport properties (µ, λ, and D) is described in Appendix B.The species production rates are governed by the law of mass action. If a generalreaction is denoted byns 1X0νmlCmm 1ns 1X00νmlCml 1, nr(8)m 1000where νml is the stoichiometric coefficient of reactant “m” in reaction “l”, νml is the stoichiometric coefficient of product “m” in reaction “l”, and Cm is the symbol for constituent “m”(the ns 1 constituent represents the third body species), then the production rate ofspecies “m”can be written in the form shown below.ẇm Wmnr Xl 1 000 ns 1Y ρn !νnlY ρn !νnl ns 1 ,νml νml k fl kbl WWnnn 1n 1000m 1, ns(9)Here, k fl and kbl are the forward and backward reaction rate coefficients of reaction “l”, ρn isthe product of density and the mass fraction of species “n”, and Wm is the molecular weightof species “m”. The molar concentration of the third body constituent, M, is defined bynsXρns 1ρm tbemWns 1Wmm 1(10)where tbem is the third body efficiency given by the kinetic model.The forward reaction rate coefficient is evaluated using the Arrhenius expression,! T alk fl Al T bl exp(11)T2

which mimics the form of the reaction rate derived from kinetic theory. The product Al T blis designed to take into account the frequency of collisions between molecules, and thesteric factor associated with the orientation of the colliding molecules. The exponentialterm is the Boltzmann factor, which represents the fraction of collisions that have an energygreater than the activation energy (defined as the product of the universal gas constant andthe activation temperature, T al ).The backward reaction rate coefficient can be written in a similar fashion, or obtainedfrom the equilibrium constant through the relationshipKcl k flkbl(12)and the equilibrium constant (in terms of molar concentrations) is given by! 0 nl GlexpK cl R u TRu T(13)0where Rhu is the universalgas constant [J/(molK)], Ru is the universal gas constant withil·atmunits of Kg·mol· K , nl is the change in moles from products to reactants, and G l is theGibbs energy of reaction defined by the following expression. Gl ns X000 νml νml (Wm gm )(14)m 1The sensible Gibbs free energy of each species is evaluated from a polynomial curve fit(see Appendix A). VULCAN-CFD can also handle more complex reaction rate representations such as pressure-dependent and arbitrary reaction order kinetics. These extensionsare described in Appendix C.To close the system, an equation of state is needed to relate the thermodynamic properties. At conditions representative of combustion applications (i.e., relatively high temperatures and low pressures), the fluid can be treated as a mixture of perfect gases. Hence, thepressure can be evaluated using the following state equationP ρRT(15)where R is the mixture gas constant given by the expression below.R Ru2.1nsXYmWmm 1(16)Thermodynamic NonequilibriumFlows that are not thermally equilibrated (i.e., the distribution of energy between translational, rotational, vibrational, and electronic modes cannot be described by a single temperature), require the introduction of additional transport equations for those energy modesthat are not equilibrated with the translational energy. The cost of solving these additionalequations, together with the limited availability of physical constants that describe the finite3