The simulation of high-speed flow involves unique challenges such as the treatment of strong flow gradients associated with shock waves and expansion fans, as well as the modeling, at hypersonic speeds, of air chemistry effects. However, supersonic flow also produces the effect of limited zones of influence, which allows for single-sweep flow solution processes. The parabolized Navier-Stokes (PNS) equations are a modification of the equations of fluid flow that extend this inherently supersonic property into the subsonic portion of the boundary layer in viscous flows. Solution methods for systems that are hyperbolic/parabolic in a spatial direction can be obtained using a space-marching approach that can be an order of magnitude faster than using a time-marching method on the time-dependent Navier-Stokes equations.

The original PNS solvers used central-differencing with shock-fitting to represent the strong bow shock waves present in high-speed flows. However, the central-differencing methods were weak in capturing embedded shocks within the outer layer, and the shock-fitting methods tended to limit the complexity of the geometries that could be considered because of the requirement that the grid generation be done concurrently with the flow solution. Shock-capturing is the method whereby the shock waves are resolved naturally as part of the solution, allowing the grid to be generated independent of the flow solution. Some PNS codes did use shock-capturing exclusively, but the central-differencing algorithms required the addition of a copious amount of user-specified artificial dissipation to maintain stability in the presence of strong shocks.

At the time of the development of this software, upwind algorithms had been developed and applied to the time-dependent Navier-Stokes equations. In upwind algorithms, an inherent recognition of local wave propagation speeds and direction is built into the algorithm, enabling the algorithm to capture strong shock waves much more accurately than can central-differencing schemes.

The purpose of this software is to provide a single-sweep, shock-capturing solution process for supersonic and hypersonic flows. Single-sweep methods are much less computationally expensive in terms of CPU time and memory requirements, and shock-capturing methods enable the treatment of more geometrically complex configurations. The approach taken was to apply an upwind algorithm to the parabolized Navier-Stokes equations. This involved the development of an approximate Riemann solver especially for the steady form of the Euler equations. This solver is the basis for the inviscid flow treatment, and central-differencing is used for the viscous terms.

Additionally, the differencing is based on a finite-volume formulation rather than finite-differences, as had been used in previous PNS solvers. This is believed to provide better accuracy, especially in regions where the grid is not ideal, as well as more physical application of boundary conditions. Also, traditional approximate-factorization is applied to make the algorithm implicit, allowing much larger marching steps to be taken than for explicit methods.

Convection-diffusion solvers for turbulence (Spalart-Allmaras) and finite-rate air chemistry (Park 5-species model) are loosely coupled to the fluids solver. In addition, the Baldwin-Lomax algebraic turbulence model is included, as are the Srinivasan-Tannehill curve fits for equilibrium air chemistry. These models enable the code to be used for high-Reynolds-number and high-enthalpy flows.

This code is unique in its application of an upwind, finite-volume algorithm to the parabolized Navier-Stokes (PNS) equations. The implicit and non-iterative nature of the algorithm enables very rapid flowfield solutions for non-separated supersonic flow. Use of a PNS code can reduce the CPU time per solution by one to two orders of magnitude over that required by time-dependent Navier-Stokes solvers. This speedup is achieved not only because of the use of the PNS equations, but because of the implicit nature of the algorithm and also, importantly, the non-iterative nature of the algorithm. The method avoids the necessity for iteration by taking small spatial steps downstream, performing interpolation between input grid planes to obtain discretization.

The use of a shock-capturing PNS solver enables simulation of configurations with wings, tails, and fins that produce complex outer shock surfaces. Because the shock surface provides the outer grid surface in shock-fitting methods, the grid generation is doubly challenged with complex inner and outer boundaries. In addition, the shock evolves as part of the solution so the grid generation must be performed along with the fluids integration, i.e., it must be completely automated. The present code employs an upwind algorithm that allows for capture of very strong shock waves of complex shape with little or no user input.

This work was done by Scott Lawrence of Ames Research Center and John Tannehill of Iowa State University. This software is available for use. To request a copy, please visit here .

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This article first appeared in the October, 2017 issue of Tech Briefs Magazine.

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