A technique for upwind differencing of the three-dimensional species continuity equations is presented which permits computation of steady flows in chemical equilibrium and nonequilibrium. The capabilities and shortcomings of the present approach for equilibrium and nonequilibrium flows is discussed. Modifications now being investigated to improve computational time are outlined.

Three-Dimensional AOTV Flowfields in Chemical Nonequilibrium

An approximate method for the efficient calculation of stagnation-streamline quantities in hypersonic flows about spheres or cylinders is suggested. Based on the local similarity of the flow field the two-dimensional Navier-Stokes equations are simplified to a one-dimensional approximation for the stagnation streamline. These equations are solved with an implicit finite-volume scheme. Comparisons with fully two–dimensional Euler and Navier–Stokes calculations for flows about spheres are presented, that include perfect gas flows and flows in chemical non-equilibrium. Comparisons with a number of experiments conclude this report.

Calculation of stagnation streamline quantities in hypersonic blunt body flows

Response of the barotropic western boundary current to typhoon passage is investigated by the use of the numerical models described in Part I. Steady states obtained in Part I are chosen as the initial conditions for undisturbed currents. In these models it is assumed that an axially symmetric typhoon (radius = 100 km, maximum wind speed = 27.4 m/sec) moves parallel to the western boundary.

A numerical experiment on the variations of western boundary currents: Part II. Response to a moving typhoon

An implicit, time-dependent approach is taken to solve the inviscid conservative equations in a vertex-based finite-volume formulation. The convective fluxes are estimated by a modified Riemann solver, and higher-order accuracy is obtained by means of the MUSCL. Implicit techniques are adopted to deal with the equation stiffness as a result of the clustered grid and chemical source term. The real-gas Roe matrix and the ADI and LU schemes have been tested on the sample problems: hypersonic flow over blunt configurations, such as the forebody of a double ellipse, and the near wake of a conic aerobrake. Temperature distribution and standoff distance are validated against the shock-fitting, central-differencing results. Equilibrium temperature may be recovered by artificially increasing the chemical rates at the wall. The present method satisfactorily resolves the flow structure, including the strong expansion behind the flat base. The relative merits and performance of the real-gas TVD and implicit schemes are discussed briefly.

Computational aspects of chemically reacting flows

A numerical method for the solution of nonequilibrium flows about blunt bodies is presented. The method is based on the splitting in two parts of the reactive Euler equations: the gasdynamic operator (mass and momentum equations) and the chemical operator (energy and species conservation equations). The gasdynamic operator is discretized on a body- and shock-litted grid, and integrated in diagonalized form by means of a semi-implicit technique. The chemical operator is integrated along the streamlines by means of an implicit technique with variable step size

Nonequilibrium hypersonic inviscid steady flows

: In nonequilibrium inviscid blunt-body flows, the state of the gas at the stagnation point is known to be in thermodynamic equilibrium for all finite relaxation times. The dependence of this state on the nonequilibrium processes and body geometry is investigated for the most general conditions. The stagnation-point state is always found to be in a narrow range bounded on one side by the state obtained in an equilibrium flow. The other bound, called the frozen limit, is far removed from the state obtained in an identically frozen flow (infinite relaxation times). For certain state variables, the frozen-limit value lies outside the range determined by frozen and equilibrium flow. Significant errors are found in several published predictions of the stagnation-point state, resulting from the nonanalytic approach to equilibrium in nearly frozen flow. The two bounds on the pressure are expressible in terms of the normal shock density ratios for equilibrium and frozen flow. The actual pressure for an arbitrary flow situation is found to depend only on the shock nose radius and the relation between density and time in the relaxation zone behind a normal shock wave. If the density law is given by a single relaxation model, a closed form expression for the pressure is obtained. The analysis is carried out for both plane and axisymmetric flows, and is also valid for nonequilibrium free-stream conditions. (Author)

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112070002239

Marcel Vinokur

Papers

On stagnation-point conditions in non-equilibrium inviscid blunt-body flows