High resolution schemes for hyperbolic conservation laws

The conservation equations for simulating hypersonic flows in thermal and chemical nonequilibrium and details of the associated physical models are presented These details include the curve fits used for defining thermodynamic properties of the 11 species air model, curve fits for collision cross sections, expressions for transport properties, the chemical kinetics models, and the vibrational and electronic energy relaxation models The expressions are formulated in the context of either a two or three temperature model Greater emphasis is placed on the two temperature model in which it is assumed that the translational and rotational energy models are in equilibrium at the translational temperature, T, and the vibrational, electronic, and electron translational energy modes are in equilibrium at the vibrational temperature, T sub v The eigenvalues and eigenvectors associated with the Jacobian of the flux vector are also presented in order to accommodate the upwind based numerical solutions of the complete equation set

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Conservation equations and physical models for hypersonic air flows in thermal and chemical nonequilibrium

The development of shock-capturing finite difference methods for hyperbolic conservation laws has been a rapidly growing area for the last decade. Many of the fundamental concepts, state-of-the-art developments and applications to fluid dynamics problems can only be found in meeting proceedings, scientific journals and internal reports. This paper attempts to give a unified and generalized formulation of a class of high-resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock waves, perfect gases, equilibrium real gases and nonequilibrium flow computations. These numerical methods are formulated for the purpose of ease and efficient implementation into a practical computer code. The various constructions of high-resolution shock-capturing methods fall nicely into the present framework and a computer code can be implemented with the various methods as separate modules. Included is a systematic overview of the basic design principle of the various related numerical methods. Special emphasis will be on the construction of the basic nonlinear, spatially second and third-order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. Some perbolic conservation laws to problems containing stiff source terms and terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems. The use of the Lax-Friedrichs numerical flux to obtain high-resolution shock-capturing schemes is generalized. This method can be extended to nonlinear systems of equations without the use of Riemann solvers or flux-vector splitting approaches and thus provides a large savings for multidimensional, equilibrium real gases and nonequilibrium flow computations.

https://ntrs.nasa.gov/api/citations/19890016281/downloads/19890016281.pdf

A class of high resolution explicit and implicit shock-capturing methods

An eddy-viscosity turbulence model employing three additional transport equations is presented and applied to a number of transitional flow test cases. The model is based on the k- framework and represents a substantial refinement to a transition-sensitive model that has been previously documented in the open literature. The third transport equation is included to predict the magnitude of low-frequency velocity fluctuations in the pretransitional boundary layer that have been identified as the precursors to transition. The closure of model terms is based on a phenomenological (i.e., physics-based) rather than a purely empirical approach and the rationale for the forms of these terms is discussed. The model has been implemented into a commercial computational fluid dynamics code and applied to a number of relevant test cases, including flat plate boundary layers with and without applied pressure gradients, as well as a variety of airfoil test cases with different geometries, Reynolds numbers, freestream turbulence conditions, and angles of attack. The test cases demonstrate the ability of the model to successfully reproduce transitional flow behavior with a reasonable degree of accuracy, particularly in comparison with commonly used models that exhibit no capability of predicting laminar-toturbulent boundary layer development. While it is impossible to resolve all of the complex features of transitional and turbulent flows with a relatively simple Reynolds-averaged modeling approach, the results shown here demonstrate that the new model can provide a useful and practical tool for engineers addressing the simulation and prediction of transitional flow behavior in fluid systems. DOI: 10.1115/1.2979230

A Three-Equation Eddy-Viscosity Model for Reynolds-Averaged Navier-Stokes Simulations of Transitional Flow

An upwind-biased, point-implicit relaxation algorithm for obtaining the numerical solution to the governing equations for three-dimensional, viscous, compressible, perfect-gas flows is described. The algorithm is derived using a finite-volume formulation in which the inviscid components of flux across cell walls are described with Roe's averaging and Harten's entropy fix with second-order corrections based on Yee's Symmetric Total Variation Diminishing scheme. Viscous terms are discretized using central differences. The relaxation strategy is well suited for computers employing either vector or parallel architectures. It is also well suited to the numerical solution of the governing equations on unstructured grids. Because of the point-implicit relaxation strategy, the algorithm remains stable at large Courant numbers without the necessity of solving large, block tri-diagonal systems. Convergence rates and grid refinement studies are conducted for Mach 5 flow through an inlet with a 10 deg compression ramp and Mach 14 flow over a 15 deg ramp. Predictions for pressure distributions, surface heating, and aerodynamics coefficients compare well with experiment data for Mach 10 flow over a blunt body.

https://ntrs.nasa.gov/api/citations/19900007726/downloads/19900007726.pdf

An upwind-biased, point-implicit relaxation algorithm for viscous, compressible perfect-gas flows

An approximate method for calculating heating rates on three-dimensional vehicles at angle of attack is presented. The method is based on the axisymmetric analog for three-dimensional boundary layers and uses a generalized body-fitted coordinate system. Edge conditions for the boundary-layer solution are obtained from an inviscid flowfield solution, and because of the coordinate system used, the method is applicable to any blunt body geometry for which an inviscid flowfield solution can be obtained. The method is validated by comparing with experimental heating data and with thin-layer Navier-Stokes calculations on the shuttle orbiter at both wind-tunnel and flight conditions and with thin-layer Navier-Stokes calculations on the HL-20 at wind-tunnel conditions.

Approximate method for calculating heating rates on three-dimensional vehicles

An analysis of the longitudinal aerodynamics of the shuttle orbiter in the hypersonic flight regime is made through the use of computational fluid dynamics. Particular attention is given to establishing the cause of the 'pitching moment anomaly,' which occurred on the orbiter's first flight, and to computing the aerodynamics of a complete orbiter configuration at flight conditions. Data from ground-based facilities as well as orbiter flight data are used to validate the computed results. Analysis shows that the pitching moment anomaly is a real-gas chemistry effect that was not simulated in ground-based facilities, which used air as a test gas. Computed flight aerodynamics for the orbiter are within 5% of the measured flight values and trim bodyflap deflections are predicted to within 10%.

Navier-Stokes simulations of Orbiter aerodynamic characteristics including pitch trim and bodyflap

A predictive tool for estimating the onset of boundary layer transition resulting from damage to and/or repair of the thermal protection system was developed in support of Shuttle Return to Flight. The boundary layer transition tool is part of a suite of tools that analyze the aerothermodynamic environment to the local thermal protection system to allow informed disposition of damage for making recommendations to fly as is or to repair. Using mission specific trajectory information and details of each damage site or repair, the expected time (and thus Mach number) at transition onset is predicted to help define the aerothermodynamic environment to use in the subsequent thermal and stress analysis of the local thermal protection system and structure. The boundary layer transition criteria utilized for the tool was developed from ground-based measurements to account for the effect of both protuberances and cavities and has been calibrated against select flight data. Computed local boundary layer edge conditions were used to correlate the results, specifically the momentum thickness Reynolds number over the edge Mach number and the boundary layer thickness. For the initial Return to Flight mission, STS-114, empirical curve coefficients of 27, 100, and 900 were selected to predict transition onset for protuberances based on height, and cavities based on depth and length, respectively.

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Overview of Boundary Layer Transition Research in Support of Orbiter Return To Flight

Direct simulation Monte Carlo DSMC simulations are performed for the Apollo capsule in the hypersonic low density transitional flow regime. The focus is on ow conditions similar to that experienced by the Apollo Command Module during the high altitude portion of its reentry Results for aerodynamic forces and moments are presented that demonstrate their sensitivity to rarefaction that is for free molecular to continuum conditions. Also aerodynamic data are presented that shows their sensitivity to a range of reentry velocity encompasing conditions that include reentry from low Earth orbit lunar return and Mars return velocities to km/s. The rarefied results are anchored in the continuum regime with data from Navier Stokes simulations

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DSMC Simulations of Apollo Capsule Aerodynamics for Hypersonic Rarefied Conditions

A new tool was developed to predict the boundary layer quantities required by several physics-based predictive/analytic methods that assess damaged Orbiter tile. This new tool, the Boundary Layer Property Prediction (BLPROP) tool, supplies boundary layer values used in correlations that determine boundary layer transition onset and surface heating-rate augmentation/attenuation factors inside tile gouges (i.e. cavities). BLPROP interpolates through a database of computed solutions and provides boundary layer and wall data (delta, theta, Re(sub theta)/M(sub e), Re(sub theta)/M(sub e), Re(sub theta), P(sub w), and q(sub w)) based on user input surface location and free stream conditions. Surface locations are limited to the Orbiter s windward surface. Constructed using predictions from an inviscid w/boundary-layer method and benchmark viscous CFD, the computed database covers the hypersonic continuum flight regime based on two reference flight trajectories. First-order one-dimensional Lagrange interpolation accounts for Mach number and angle-of-attack variations, whereas non-dimensional normalization accounts for differences between the reference and input Reynolds number. Employing the same computational methods used to construct the database, solutions at other trajectory points taken from previous STS flights were computed: these results validate the BLPROP algorithm. Percentage differences between interpolated and computed values are presented and are used to establish the level of uncertainty of the new tool.

/pdf/development-of-a-boundary-layer-property-interpolation-tool-4jsmq86psw.pdf

Francis A. Greene

Papers

Approximate method for calculating heating rates on three-dimensional vehicles

Navier-Stokes simulations of Orbiter aerodynamic characteristics including pitch trim and bodyflap

Overview of Boundary Layer Transition Research in Support of Orbiter Return To Flight

DSMC Simulations of Apollo Capsule Aerodynamics for Hypersonic Rarefied Conditions

Development of a Boundary Layer Property Interpolation Tool in Support of Orbiter Return To Flight