**FPNS Numerical Solution Scheme:**

The governing flowfield equations [Refs. 1 and 2] used in the FPNS code are based on a unique, scaled form of the Navier-Stokes equations which eliminates the geometric singularity along the stagnation stream line [Refs. 3 and 4]. The overall governing equations consist of the FNS (Full Navier-Stokes) equations scheme for the blunt forebody region [Ref. 4], and the PNS (Parabolized Navier-Stokes) equations for the axially-attached afterbody region [Ref. 3].A high-order, real-gas FVS (Flux Vector-Splitting) scheme [Refs. 4 and 5] is used to approximate flux derivatives, and a fourth-order pressure dissipation model [Ref. 4] is used to suppress the growth of potential numerical oscillations without degrading near-wall flowfield gradients. Bow shock is captured, and a grid-shock adaptation algorithm [Ref. 4] is used to enhance the crispness of the shock and improve the near-wall grid resolution. A unique implicit Predictor-Corrector solution scheme [Refs. 4 and 6] is used to solve the resulting differenced equations in the blunt-body region. This Predictor-Corrector solution scheme is significantly more stable and converges about an order of magnitude faster than a typical Approximate-Factorization algorithm.

Equilibrium-air flowfield predictions use a fast and efficient table-look-up of equilibrium-air properties, which results in computing speeds which are only 10-15% slower than an equivalent perfect-gas flowfield prediction [Ref. 5]. Nonequilibrium-air flowfield predictions use a seven-species finite-rate chemically reacting air model [Refs. 6 through 9] to model the surrounding air as a reacting gaseous mixture of N, N2, O, O2, NO, NO+ and e-, which are assumed to be in thermal equilibrium (i.e., represented by a single gas temperature). This nonequilibrium-air solution scheme is based on a unique Two-Step Algorithm [Refs. 4 and 7] which is considerably more stable than a diagonalized chemistry algorithm [Ref. 7], while being an order of magnitude faster than a fully-coupled chemistry solution [Ref. 7].

**References:**

(1) Peyret, R. and Viviand, H., “Computations of Viscous Compressible Flows Based on the Navier-Stokes Equations,” AGARD-AG-212, 1975.

(2) Viviand, H., “Conservative Forms of Gas Dynamics Equations,” La Recherche Aerospatiale, No. 1, Jan.-Feb. 1974, pp. 65-68.

(3) Bhutta, B.A., and Lewis, C.H., “PNS Predictions of Axisymmetric Blunt-Body and Afterbody Flowfields,” AIAA Paper 93-2725, July 1993.

(4) Bhutta, B.A., Daywitt, J.E., Rahaim, J.J., and Brant, D.N., “A New Technique for the Computation of Severe Reentry Environments,” AIAA Paper 96-1861, June 1996.

(5) Bhutta, B.A., and Lewis, C.H., “Supersonic/ Hypersonic Flowfield Predictions Over Typical Finned Missile Configurations,” Journal of Spacecraft and Rockets, Vol. 30, Nov.-Dec. 1993, pp. 674-681; see also AIAA Paper 92-0753.

(6) Bhutta, B.A., and Lewis, C.H., “Three-Dimensional Hypersonic Nonequilibrium Flows at Large Angles of Attack,” Journal of Spacecraft and Rockets, Vol. 26, May-June 1989, pp. 158-166. See also AIAA Paper 88-2568, June 1988.

(7) Bhutta, B.A., and Lewis, C.H., “New Technique for Low-to-High Altitude Predictions of Ablative Hypersonic Flowfields,” Journal of Spacecraft and Rockets, Vol. 29, Jan.-Feb. 1992, pp. 35-50; see also AIAA Paper 91-1392, June 1991.

(8) Bhutta, B.A., and Lewis, C.H., “Improved Nonequilibrium Viscous Shock-Layer Scheme for Hypersonic Blunt-Body Flowfields,” Journal of Spacecraft and Rockets, Vol. 29, Jan.-Feb. 1992, pp. 24-34.

(9) Blottner, F.G., Johnson, M., and Ellis, M., “Chemically Reacting Viscous Flow Program for Multi-Component Gas Mixtures,” Report No. SC-RR-70-754, Sandia Laboratories, Albuquerque, NM, Dec. 1971.