John C. Tannehill

Upwind Algorithm for the Parabolized Navier-Stokes Equations

A new upwind algorithm based on Roes scheme has been developed to solve the two-dimensional parabolized Navier-Stokes equations. This method does not require the addition of user-specified smoothing terms for the capture of discontinuities such as shock waves. Thus, the method is easy to use and can be applied without modification to a wide variety of supersonic flowfields. The advantages and disadvantages of this adaptation are discussed in relation to those of the conventional Beam-Warming (1978) scheme in terms of accuracy, stability, computer time and storage requirements, and programming effort. The new algorithm has been validated by applying it to three laminar test cases, including flat-plate boundary-layer flow, hypersonic flow past a 15-deg compression corner, and hypersonic flow into a converging inlet. The computed results compare well with experiment and show a dramatic improvement in the resolution of flowfield details when compared with results obtained using the conventional Beam-Warming algorithm.

Application of an upwind algorithm to the three-dimensional parabolized Navier-Stokes equations

A new computer code for the solution of the three-dimensional parabolized Navier-Stokes equations has been developed. The code employs a state-of-the-art upwind algorithm to capture strong shock waves. The algorithm is implicit, uses finite volumes, and is second-order accurate in the crossflow directions. The new code is validated through application to laminar hypersonic flows past two simple body shapes: a circular cone of 10 deg half-angle, and a generic all-body hypersonic vehicle. Cone flow solutions were computed at angles of attack of 12, 20, and 24 deg and results are in agreement with experimental data. Results are also presented for the flow past the all-body vehicle at angles of incidence of 0 and 10 deg.

A new PNS code for chemical nonequilibrium flows

A new parabolized Navier-Stokes (PNS) code has been developed to compute the hypersonic laminar flow of a multicomponent, chemically reacting mixture of thermally perfect gases over two-dimensional and axisymmetric bodies. The new PNS code solves the gas dynamic and species conservation equations in a coupled manner using a noniterative, implicit, space-marching finite-difference method. The conditions for well-posedness of the space-marching method have been derived from an eigenvalue analysis of the governing equations. The code has been used to compute hypersonic laminar flow of chemically reacting air over wedges and cones. The results of these computations are in good agreement with the results of reacting boundary-layer calculations.

Numerical solution of Space Shuttle Orbiter flowfield including real-gas effects

The hypersonic, laminar flow around the Space Shuttle Orbiter has been computed for bath an ideal gas (y = 1.2) and equilibrium air using a real-gas, parabolized Navier-Stokes code. This code employs a generalized coordinate transformation; hence, it places no restrictions on the orientation of the solution surfaces. The initial solution in the nose region was computed using a 3-D, real-gas, time-dependent Navier-Stokes code. The thermo- dynamic and transport properties of equilibrium air were obtained from either approximate curve fits or a table look-up procedure. Numerical results are presented for flight conditions corres- ponding to the STS-3 trajectory. The computed surface pressures and convective heating rates are compared with data from the STS-3 flight.

Upwind parabolized Navier-Stokes code for chemically reacting flows

A new upwind, parabolized Navier-Stokes (PNS) code has been developed to compute the hypersonic, viscous, chemically reacting flow around two-dimensional or axisymmetric bodies. The new code is an extension of the upwind (perfect gas) PNS code of Lawrence, Tannehill, and Chaussee. The upwind algorithm is based on Roes flux-difference splitting scheme, which has been modified to account for real gas effects. The algorithm solves the gasdynamic and species continuity equations in a loosely coupled manner. The new code has been validated by computing the Moo = 25 laminar flow of chemically reacting air over a wedge and a cone. The results of these computations are compared with the results from a centrally differenced, fully coupled, nonequilibrium PNS code. The agreement is excellent, except in the vicinity of the shock wave, where the present code exhibits superior shock-capturing capabilities.

Three-dimensional upwind parabolized Navier-Stokes code for real gasflows

A real gas, upwind, parabolized Navier-Stokes (PNS) code has been developed to compute the three-dimensional hypersonic flow of equilibrium air around various body shapes. The new code is an extension of the upwind (perfect gas) PNS code of Lawrence et al. (1986). The upwind algorithm is based on Roes (1981) flux-difference splitting scheme which has been modified to account for real gas effects using the nearly exact approach of Vinokur and Liu (1988). Simplified curve fits are employed to obtain the thermodynamic and transport properties of equilibrium air. The new code has been validated by computing the M-infinity = 25 laminar flow of air over cones at various angles of attack. The results of these computations are compared with the results from a conventional centrally-differenced, real gas PNS code and the previous axisymmetric, upwind, real gas code. The agreement is excellent in all cases.

Comparison of a two-dimensional shock impingement computation with experiment

Results of computations of two-dimensional viscous blunt-body flowfields with an impinging shock wave, with a time-dependent finite-difference method employed to solve the complete set of Navier-Stokes equations, are compared with experimental results. The experimental results were obtained in a 20-inch hypersonic tunnel with a planar shock impinging on the cylindrical leading edge of a fin, hence with the shock parallel to the centerline of the leading edge, so that type III and type IV interference patterns were generated. Close agreement is found. The overall effects of smoothing and grid size on the calculations are determined. A 31 x 51 mesh is adequate for wall pressure values (except in peaked regions).

Development of an iterative PNS code for separated flows

A new iterative parabolized Navier-Stokes (PNS) algorithm is being developed to efficiently compute supersonic viscous flowfields with embedded separated regions. In the vicinity of these embedded regions, the PNS equations are solved iteratively in order to duplicate the results that would be obtained with the complete Navier-Stokes equations. The algorithmsplits the streamwise flux vector using either the Vigneron or the Steger-Warming methods. Once a separated flow region is computed, the algorithm returns to the usual PNS space-marching mode until, the next embedded region is encountered. The algorithm has been successfully incorporated into NASA’s upwind PNS (UPS) code. The new algorithm has been validated by applying it to three separated flow test cases consisting of flow over a compression ramp, a shock impingement flow and flow over a cone-flare geometry. The present numerical re&lts are in excellent agreement with other NavierStokes computations and experimentaldata.

Numerical computation of the hypersonic leading edge problem using the Burnett equations

The hypersonic rarefied flow near the sharp leading edge of a flat plate is computed for the first time using a finite‐difference solution of the complete, unsteady Burnett equations. The computation is advanced in time from the initial conditions until the steady‐state solution is reached. The computational region extends from the leading edge to the strong‐interaction regime. Schamberg’s second‐order wall slip and temperature jump boundary conditions are employed at the wall. The numerical results are compared with experimental data, a Monte Carlo simulation, and a finite‐difference solution of the complete Navier–Stokes equations. Based on these comparisons, it is evident that the Burnett equations in conjunction with Schamberg’s second‐order boundary conditions give a much less accurate description of the rarefied flow field near the leading edge than do the Navier–Stokes equations.

Three-dimensional upwind parabolized Navier-Stokes code for supersonic combustion flowfields

A new upwind, parabolized Navier-Stokes (PNS) code has been developed to compute the three-dimensional chemically reacting flow in scramjet (supersonic combustion ramjet) engines. The code is a modification of the three-dimensional upwind PNS (UPS) airflow code which has been extended in the present study to permit internal flow calculations with hydrogen-air chemistry. With these additions, the new code has the capability of computing both aerodynamic and propulsive flowfields. The algorithm solves the PNS equations using a finite-volume, upwind TVD method based on Roes approximate Riemann solver that has been modified to account for nonequilibrium effects. The fluid medium is assumed to be a chemically reacting mixture of thermally perfect (but calorically imperfect) gases in thermal equilibrium. The new code has been applied to two test cases. These include the Burrows-Kurkov supersonic combustion experiment and a three-dimensional shockinduced combustion flowfleld. The computed results compare favorably with the available experimental data.