CFD for Hyper Sonic Applications

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COMPUTATIONAL FLUID DYNAMICS TECHNOLOGY FOR HYPERSONIC APPLICATIONS Peter A. Gnoffo NASA Langley Research Center Hampton, VA 23681-0001 Abstract Several current challenges in computational fluid dynamics and aerothermodynamics for hypersonic vehicle applications are discussed. Example simulations are presented from code validation and code benchmarking efforts to illustrate capabilities and limitations. Opportunities to advance the state-of-art in algorithms, grid genera
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    COMPUTATIONAL FLUID DYNAMICS TECHNOLOGYFOR HYPERSONIC APPLICATIONS   Peter A. Gnoffo   Peter.A.Gnoffo@nasa.govNASA Langley Research CenterHampton, VA 23681-0001 Abstract  Several current challenges in computational fluid dynamics and aerothermodynamics for hypersonicvehicle applications are discussed. Example simulations are presented from code validation and code benchmarkingefforts to illustrate capabilities and limitations. Opportunities to advance the state-of-art in algorithms, gridgeneration and adaptation, and code validation are identified. Highlights of diverse efforts to address thesechallenges are then discussed. One such effort to re-engineer and synthesize the existing analysis capability inLAURA, VULCAN, and FUN3D will provide context for these discussions. The critical (and evolving) role of agilesoftware engineering practice in the capability enhancement process is also noted. Introduction Computational fluid dynamics (CFD) is thenumerical simulation of flowfields through theapproximate solution of the governing partialdifferential equations for mass, momentum, andenergy conservation coupled with the appropriaterelations for thermodynamic and transport properties.Aerothermodynamics is the branch of fluid dynamicsthat focuses on the effects of thermodynamic andtransport models on aerodynamics and heating. It isespecially focused on conditions of hypersonicvelocities where the energy content and exchangebetween kinetic, internal, and chemical modes in theflow precludes the otherwise common use of calor-ically perfect gas assumptions. Computational aero-thermodynamics is therefore defined in exactly thesame manner as CFD, with the added emphasis thathigh temperature gas effects on pressure, skinfriction, and heat transfer are included in thenumerical simulation. The fundamental role of computational aerothermodynamics is the simulationof aerodynamic forces and heating for external andinternal high speed flows. Reference 1 presents areview of recent applications for access to space andplanetary missions.The approximate solution of the governingequations requires that the domain be subdivided intomany small control volumes. The accuracy of thesolution depends upon a variety of factors; the mostcritical factors are the size of each control volume,the orientation of its boundaries relative to a varietyof flow features, and the order of accuracy of thediscretization. At the risk of belaboring the obvious,note that hypersonic flows routinely involve extremesof pressure, density, and temperature separated byshocks, expansions, shear layers, and boundarylayers. These extremes in conditions and topology of flow structure complicate computationalaerothermodynamic simulation. The ability to orienta grid with evolving flow structures is particularlychallenging. For example, accurate simulation of heat transfer requires adequate resolution of theboundary layer (or merged layer) and accuraterepresentation of conditions at the boundary-layeredge. Conditions at the boundary-layer edge in turn,particularly in the stagnation region of hypersonicflows, are dependent on entropy carried alongstreamlines from the shock. Any irregularities in thecaptured shock create associated irregularities inentropy that feed the rest of the domain. Furthercomplicating the simulation is the fact that the naturalstability properties of upwind schemes so often usedin hypersonic applications are in turn most poorlyrealized in the broad, stagnation region of bluntbodies where system eigenvalues are small.Nevertheless, computational aerothermo-dynamic tools that generally do a good job inhandling the difficulties of steady, laminar,hypersonic, blunt body and attached flow simulationhave evolved over the past 15 years. The greatestuncertainty in such applications is the validation of physical models for a host of non-equilibriumprocesses 2 . Greater challenges are encountered in the  simulation of separated flows, complex shock-shock and shock – boundary-layer interactions, predictionof transition, time-dependent flows, and plasmaflows. The application of unstructured grid methodsfor such three-dimensional simulations (even therelatively simple blunt body problem) has proven tobe especially challenging when heat transfer isrequired as will be discussed herein.The goal of this paper is to reflect on theneeds of computational aerothermodynamics in orderto meet these challenges. Some of the approachesdiscussed here are under development within theHigh Energy Flow Solver Synthesis 3 (HEFSS)project at NASA Langley Research Center. Thisproject combines the physical models and simulationcapabilities in the structured codes LAURA 4 (focuson external, hypersonic flow simulations) andVULCAN 5 (focus on internal, scramjet flowsimulations) into the unstructured flow solverFUN3D 6 (perfect gas flow simulations with adjoint-based design and optimization capabilities). Theultimate goal of this project is to develop robustschemes with quantified uncertainties requiringminimal user intervention. In this endeavor to re-factor existing sources we modularize the code tosimplify future coupling with other physics. Grid Requirements   Criteria: The HEFSS team is investing inunstructured grid technology, including the use of mixed elements - tetrahedron (tets), prisms,pyramids, and hexahedron (hexes) - as the underlyingsupport for future hypersonic flow simulation. Wedefer any investment in patched or overset multiblock structured grid systems at this time. The rationale forthis approach is based on our assessment of currentand anticipated future capabilities for grid generationand grid adaptation. This assessment considers:(1) efficiency in the initial generation of a“reasonable” grid over complex configurations;(2) ability to adapt the grid to flow structures (steadyand time-dependent) and changes in parameterizedstructures on the vehicles; and (3) solution qualityand efficiency on a “good” grid using best availablealgorithms. A “reasonable” grid is one that willaccommodate a converged solution with at leastapproximate resolution of flow structures. It providesan initial condition for future adaptation andrefinement. A “good grid is one that will produce agrid converged solution such that an additionalrefinement will result in dependent variable changeswithin user specified tolerances. Initial Generation: Unstructured gridgeneration on complex configurations is already judged superior to structured grid capabilities forinitial grid generation. As an example, consider thegrid shown in Fig. 1 and Fig. 2 which is discussed inReference 7. A grid around mated ascent vehicles,including a fine grid near all solid surfaces to resolveboundary layers, was generated in approximately fourhours (including all user interactions) from a cleanCAD file. A subjective judgment of the time requiredto create an equivalent, multiblock structured gridusing continuous or patched interfaces is on the orderof days to weeks. The time to generate an overset,structured grid is estimated to be approximately oneday. The extra time in these cases derives from thetopological decisions required by the user so that thegrid will be sufficiently adaptable to accommodatesolution quality.Fig. 1: GridEx application of unstructured grid forsimulation of viscous, hypersonic flow over matedreusable launch vehicle system.  Ability to Adapt: The ability to adaptstructured grids versus unstructured grids forhypersonic applications is considered to be a draw atpresent, with unstructured capability advancingrapidly. The external hypersonic flow, structured gridsolver LAURA 4 , for example, routinely appliesquasi-one-dimensional adaptation as part of thesolution procedure to align a structured grid with thecaptured bow shock and control near-wall grid sizeand grid stretching factors to obtain grid-convergedheating rates. However, a lot of time goes into the  grid generation process in order to enable suchadaptive capability for complex configurations, andsome topologies (e.g. elevon gaps as seen in inset of Fig. 2) are not amenable to such simple adaptationstrategies. Many other solution-adaptive tools areavailable to adjust grid spacing based on userspecified, weighted functions of gradients (featurebased adaptation) but defining the weight functions isoften more art than science in order that all flowstructures are adequately resolved.Fig. 2: Details of unstructured grid in elevon gap forRLV application. Scalloped edges are artifacts of presentation of tetrahedral facets.   Feature based adaptation is routinely appliedto unstructured grids as well. But unstructured gridsprovide additional degrees of freedom that enablelocal enrichment, coarsening, and re-orientation(edge-swapping) without propagating these effectsthroughout the domain. However, strategies foranisotropic adaptation are required across shocks andshear layers to prevent excessive numbers of nodesfrom being drawn into the high gradient regions. Asan example, the method of Ref. 8 has been applied toMach 3 flow (inviscid) through a scramjet inlet andMach 2 flow (viscous, Re 10000) over a NACA 0012airfoil showing very crisp looking captured shocksacross tightly clustered, high aspect ratio triangles(two-dimensional test cases). The algorithmexamines the eigenvalues and eigenvectors associatedwith the Hessian of each dependent variable andderives a transformation based on this geometricinformation that guides directional refinement. Italso enforces a surface boundary orthogonalitycondition to improve solution quality of skin friction.This issue is likely important for the simulation of heating in hypersonic applications as well.Fig. 3: Mach number contours and feature-based,isotropic adapted grid from Venditti 9 for Mach 3,inviscid flow over airfoils.Adjoint-based grid adaptation is a recentdevelopment that refines mesh as a function of userdefined error tolerances. The adjoint equations arisefrom the linearization of the effects of residualchanges at every node on an output function (lift,drag, heating). A user can specify the desired  tolerance on the output function and the adaptationprocess continues until the tolerance is met or systemresource limits are exceeded. In this sense, adjointadaptation is an objective error minimizationtechnique whereas feature adaptation is a subjectiveerror reduction technique.An example of adjoint-based adaptation andfeature-based adaptation applied to supersonic flowover a pair of airfoils is shown in Fig. 3 and Fig. 4taken from Ref. 9 and 10. The objective function inthis example is the drag on the lower airfoil. Theadjoint adaptation yields the same drag coefficient asthe Hessian based adaptation to about 0.013% usingnearly a factor of ten fewer nodes. Note that aniso-tropic adaptation has not been included in thisexample but the capability is currently being testedwithin HEFSS.Fig. 4: Isotropic, adjoint-based grid adaptation toobtain grid converged drag on lower airfoil fromVenditti 9 . Only parts of flow required to get gridconverged drag on lower airfoil are resolved.It is worth noting that the adjoint equationset provides capability for configuration design andoptimization 11 . This capability is currently applied toperfect-gas, low speed to transonic applications inHEFSS. It has not been applied to hypersonic appli-cations because of uncertainty in the choice of optimum algorithm (reconstruction, limiting) re-quired for flow simulations including strong shocks.Still, it is anticipated that this capability will be of fundamental value and was one of the major reasonsfor choosing the FUN3D code as the baseline forextensions to the hypersonic domain. Solution quality and efficiency: Structured grid solvers are currently superior to un-structured grid solvers as regards quality andefficiency for viscous, hypersonic flow applicationsincluding heat transfer. This assertion, based onpersonal observations, ignores the time required togenerate an initial grid and assumes that the numberof nodes in both approaches is comparable. Un-structured grid solvers carry an overhead associatedwith connectivity of nodes, edges, and volumes –particularly in the case of mixed elements. The con-vergence rate of unstructured solvers, especially inthe presence of strong shocks is slower thanstructured grid algorithms. This behavior is thoughtto be a consequence of greater problems with“ringing” associated with the application of limitersin the flux reconstruction algorithms and with lessefficient implicit relaxation algorithms but moreresearch is required to confirm such speculation.The solution quality concern stems fromobservations of blunt body stagnation region heatingin 3D using HEFSS on simple test problemsinvolving cylinders and spheres. (This issue will bediscussed in more detail in the Algorithms section.)Heating contours in these tests display fair to poorsymmetry in the stagnation region, depending on thereconstruction algorithm, whereas symmetries arevery good away from the stagnation point. It isbelieved that unstructured-grid-induced irregularitiesin the captured shock promote entropy gradients thatfeed directly to the boundary layer edge in thestagnation region. These irregularities tend to bedissipated downstream as the entropy layer enters theboundary layer. The tested grid in the cylinder casewas entirely composed of tets derived from anadapted, structured grid. The diagonals inserted intothe structured grid to create the tets were all orientedidentically; creating a highly biased grid that providesan extreme test case. (A literature search for similartests focused on 3D heating was unsuccessful.) Caveats: (1) The problem of relative motion of tworigid bodies where boundary layer resolution isrequired on each body is probably better suited tooverset grid algorithms. Both structured 12 andunstructured 13 grids may be used as the baselinediscretization.(2) The “unstructured” Cartesian gridmethod CART3D 14 is a highly efficient alternative totraditional unstructured grid approaches in terms of time to generate and refine a grid and in terms of timeto generate a solution on the grid. This approachemploys a structured, rectangular grid systemthroughout the domain (Fig. 5) but enables localrefinements to better resolve regions of highcurvature in the body and high gradient in the flow.
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