Numerical methods for computations of nonequilibrium hypersonic flow around bodies

In this paper we present numerical methods for computations of nonequilibrium hypersonic flow of air around bodies including chemical reaction effects and present numerical result of the flow over concave corners. We developed implicit finite difference method to overcome numerical difficulties associated with the lack of resolution behind the shock and near the body. Using our method we were able to find details of the flow properties near the shock and body and were able to continue the computation of the flow for a long distance from the corner of the body.     

Axisymmetric bodies of minimum drag in hypersonic flow

The problem of determining the longitudinal contour of a slender, axisymmetric body in hypersonic flow which has minimum drag is considered. The pressure distribution is assumed to be Newtonian, while the skin-friction distribution is for laminar flow and depends on body geometry. This investigation is conducted with the method of steepest descent, whose feasiblity is demonstrated by solving minimum drag problems having known analytical solutions.     

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

Three-dimensional flow of supersonic jets past bodies

Translated from Aktual'nye Voprosy Prikladnoi Matematiki, pp. 228–235, 1989.     

Dissociation effect on the hypersonic flow past a circular cylinder

The effects of dissociation of air on hypersonic flow past a circular cylinder at zero angle of incidence are considered under the assumptions that the shock wave is in the shape of a circular cylinder, the density ratio across the shock is constant, the flow behind the shock is at constant density and dissociation occurs only behind the shock wave. In the present paper, the velocity, pressure and drag coefficients, vorticity, shock detachment distance, stagnation point velocity gradient and sonic points on the shock and the surface have been obtained in the presence of dissociation. The results have been compared with the corresponding results obtained in the case when dissociation dose not occur and the corresponding results in the case of the sphere in the presence of dissociation.     

Supercritical flow past blunt bodies in shallow water

The flow about blunt objects in shallow water which support a detached hydraulic jump is investigated. This work is an extension of the blunt body problem of gas dynamics to an analogous situation existing in free-surface flows. The problem is solved inversely, by expressing the flow variables as series expansions about the shock apex, and then determining the position of the body downstream. The presence of various singular lines in the solution is discussed.

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Zusammenfassung Es wird die Strömung in seichtem Wasser untersucht, welche bei der Umströmung von stumpfen Körpern einen abgelösten hydraulischen Sprung erzeugt. Dies ist eine Verallgemeinerung des gasdynamischen Problems der stumpfen Körper zur analogen Situation bei der Strömung mit freier Oberfläche. Es wird das inverse Problem gelöst, durch Reihenentwicklung der Strömungs-Variablen von der Stoß-Spitze ausgehend; damit wird der stromabwärts liegende Körper bestimmt. Das Auftreten von verschiedenen singulären Linien in der Lösung wird diskutiert.

     

Simplified Navier-Stokes equations for computing viscous gas flow past long bodies

Simplified Navier-Stokes equations are applied to analyze the flow of supersonic viscous gas at moderately large Reynolds numbers near the lateral surface of long bodies. Numerical integration of the equations is performed by the marching method, stabilizing in each step the solution of the nonstationary system of equations in the longitudinal coordinates. We consider the flow past cylinders and cones with a spherical blunt nose. We investigate the effect of the Reynolds number and the body shape on the flow field, the drag coefficient, and the heat flux. The numerical solutions of simplified and complete Navier-Stokes equations are compared.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 231–239, 1985.     

Comparison of the simplified and full MHD models for laminar incompressible flow past a circular cylinder

In the majority of research on incompressible magnetohydrodynamic (MHD) flows, the simplified model with the low magnetic Reynolds number assumption has been adopted because it reduces the number of equations to be solved. However, because the effect of flow on magnetic field is also neglected, the solutions of the simplified model may be different from those of the full model. As an example, the flow of an electrically conducting fluid past a circular cylinder under a magnetic field is investigated numerically using the simplified and full models in this paper. To solve the problems, two second-order compact finite difference algorithms based on the streamfunction-velocity formulation of the simplified model and the quasi-streamfunction-velocity formulation of the full model are developed respectively.Numerical simulations are carried out over a wide range of Hartmann number for steady-state laminar problems with both models. For the full model, magnetic Reynolds number (Re_m) is chosen from 0.01 to 10. The computed results show that solutions of the simplified MHD model are not exactly the same as those of the full MHD model for this flow problem in most cases even if Re_m in the full model is very low. Only in the special case that a strong external magnetic field is exerted perpendicular to the dominant flow direction, can the simplified MHD model be regarded as an approximation of the full MHD model with low Re_m.     

Unified unsteady supersonic/hypersonic theory of flow past double wedge airfoils

A unified supersonic/hypersonic theory is given of flow past a pitching oscillating double wedge airfoil at arbitrary mean angle of attack. The amplitude and the (reduced) frequency parameter of the oscillation are assumed small and a perturbation method is employed. Closed form formulae are obtained for the stiffness and damping-in-pitch derivatives. They are exact with respect to the free stream Mach number, angle of attack and body thickness etc., provided only that the bow shock is attached to the leading edge. The theory predicts negative damping (instability) if the angle of attack, or the airfoil thickness is sufficiently large, or if the free stream Mach number is sufficiently low. It is shown to be in good agreement with experiments of Scruton et. al. Comparisons with Van Dyke second order potential theory and with Lighthill piston theory are also given. Finally the theory may easily be extended to arbitrary smooth airfoils.     

A new approximate analytical solution for arbitrary Stokes flow past rigid bodies

A method of computing general Stokes flows in the presence of rigid boundaries of arbitrary shape is proposed. The solution satisfies the governing field equations exactly and the boundary conditions approximately. The method has been illustrated with three examples. The advantage of the method lies in the ease of implementation for rigid bodies of arbitrary shape, providing an approximate but analytical solution throughout the domain.     

Flow simulation past axisymmetric bodies using four different turbulence models

A comparative assessment is presented of three different versions of high-Re k- models of turbulence for simulation of the flow past the entire length of axisymmetric bodies: standard, Chen and Kim-modified, and RNG-derived versions. Comparisons show that the standard model with wall functions can predict the flow characteristics at least as accurately as the other two. Application of the low-Re Lam and Bremhorst model further improves the predictive capability of wall-related quantities by removing the need for wall functions.     

A comparative analysis of approaches for investigating hypersonic flow over blunt bodies in a transitional regime

Hypersonic flows of a viscous perfect rarefied gas over blunt bodies in a transitional flow regime from continuum to free molecular, characteristic when spacecraft re-enter Earth's atmosphere at altitudes above 90-100 km, are considered. The two-dimensional problem of hypersonic flow is investigated over a wide range of free stream Knudsen numbers using both continuum and kinetic approaches: by numerical and analytical solutions of the continuum equations, by numerical solution of the Boltzmann kinetic equation with a model collision integral in the form of the S-model, and also by the direct simulation Monte Carlo method. The continuum approach is based on the use of asymptotically correct models of a thin viscous shock layer and a viscous shock layer. A refinement of the condition for a temperature jump on the body surface is proposed for the viscous shock layer model. The continuum and kinetic solutions, and also the solutions obtained by the Monte Carlo method are compared. The effectiveness, range of application, advantages and disadvantages of the different approaches are estimated.     

Supercomputer simulation of viscous noncompressible flow past bodies of complex shape

The article reports the results of numerical analysis of viscous thermally conducting noncompressible flow past bodies of complex shape. The effect of physico-chemical processes is ignored. Flow past bodies with various nose cone designs is examined. The process is described using a system of Navier–Stokes equations augmented with the energy equation. The continuity equation is used to control the numerical accuracy. The simulation results are reported in the form of vector fields and surface components of the velocity vector in various channel sections. The results are analyzed for various body configurations.     

Dissociation effects in hypersonic flow past a circular cone at an angle of attack

The analytical solution to the steady, compressible, non-viscous, inviscid hypersonic flow past a circular cone at an angle of incidence, with an attached Shockwave, in the presence of dissociation of air in the shock layer, has been obtained here under the assumption of thermal equilibrium. Expression for the velocity, pressure, temperature, density, velocity of air, Mach number, pressure, drag and lift coefficients have been obtained both in the shocklayer outside the vortical layer and on the surface of the cone inside the vortical layer.     

Supersonic/hypersonic flow past an oscillating flat plate at high angles of attack

Summary The problem of inviscid supersonic/hypersonic flow over the expansion surface of an oscillating flat plate at an angle of attack is studied. The amplitude and reduced frequency of the oscillation are assumed small and the unsteady flow regarded as a small perturbation to the steady Prandtl-Meyer flow. By imposing appropriate upstream boundary conditions the perturbation problem is properly formulateds and solved analytically. The solution, together with the known oscillating wedge solution of Hui, is used to study the aerodynamic stability of a flat plate. In particular, it is found that for large enough angles of attack the pitching motion of a flat plate is aerodynamically unstable, and that at high angles of attack the specific heat ratio of the gas has very significant effects and can cause aerodynamic instability.

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Sommaire On étudie le problème de l'écoulement supersonique/hypersonique non visqueux au-dessus de la surface d'expansion d'une plaque plane oscillante, avec un angle d'attaque donné. On suppose que l'amplitude et la fréquence réduite de l'oscillation sont petites, et que l'écoulement non stationnaire peut être considéré comme une petite perturbation de l'écoulement stationnaire de Prandtl et Meyer. En imposant les conditions aux limites applicables en amont, on formule convenablemen le problème perturbatif et on le résout analytiquement. La solution de ce problème aussi bien que la solution du problème du dièdre oscillant trouvée par Hui, est utilisée pour l'étude de la stabilité aérodynamique d'unè plaque plane. On montre en particulier que lorsque l'angle d'attaque est assez grand, le mouvement depitching d'une plaque plane (c'est à dire le mouvement de rotation autour d'un axe passant par son bord d'attaque) devient aérodynamiquement instable. De plus, pour de grands angles d'attaque, la valeur du rapport des chaleurs spécifiques du gas est très significative et peut causer une instabilité aérodynamique.

     

Modelling two-dimensional flow past arbitrary cylindrical bodies using boundary element formulations

This paper presents an efficient method of solving Queen's linearized equations for steady plane flow of an incompressible, viscous Newtonian fluid past a cylindrical body of arbitrary cross-section. The numerical solution technique is the well known direct boundary element method. Use of a fundamental solution of Oseen's equations, the ‘Oseenlet’, allows the problem to be reduced to boundary integrals and numerical solution then only requires boundary discretization. The formulation and solution method are validated by computing the net forces acting on a single circular cylinder, two equal but separated circular cylinders and a single elliptic cylinder, and comparing these with other published results. A boundary element representation of the full Navier-Stokes equations is also used to evaluate the drag acting on a single circular cylinder by matching with the numerical Oseen solution in the far field.     

Analyzing the longitudinal effect of hypersonic flow past a conical cone via the perturbation method

To analyze the hypersonic flow past a conical cone, the variations of gasdynamic properties subjected to the longitudinal curvature effect by using the perturbation method. An outer perturbation expansion has been carried out by recent researchers, but a problem occurred, the outer expansion solutions are not uniformly valid in the shock layer, however, the outcome near the conical body surface called vortical layer remains deflective. This study intends to discover uniformly valid analytical solutions in the shock layer by applying the inner perturbation expansions matching with the out expansions to analyze the characteristics in the whole region including shock layer and vortical layer. Starting from the zero-order approximate solutions for hypersonic conical flow is then applied as the basic solutions for the outer perturbation expansions of a flow field. The governing equations and boundary conditions are also expanded via outer perturbations. Using an approximate analytical scheme in the derivation process, first-order perturbation equations can be simplified and the approximate closed-form solutions are obtained; furthermore, the various flow field quantities, including the normal force coefficient on the cone surface, have been calculated. According to the variations of gasdynamic properties, the longitudinal curvature effect for the hypersonic flow past a conical cone can be determined. Thicknesses of shock layer and vortical layer can be predicted as well. The physical phenomena inside both layers can be investigated carefully, the conditions for an elliptic cone with longitudinal curvature, m = 1 and n = 2 and other conditions of parameters; the perturbation parameter, ε_m2 = 0.1, semi-vertex angle of the unperturbed cone, δ = 10°, and hypersonic similarity parameter, K_δ = M_∞δ = 1.0, the thickness of vortical layer, η_VL, can be calculated at the position angle of conical cone body, ? = 30° was demonstrated here. Results show how very thin the vortical layer is approximately only 10% of the shock layer close to the body, the pressure in the whole shock layer is verified to be uniformly valid which agrees with previous studies. Large gradient changes in entropy and density are found when the flow approaches the cone surface, the most important is, this method provides a benchmark solution to the hypersonic flow past a conical cone and to assist the grids and numerics for numerical computation should be fashioned to accommodate the whole flow field region including the vortical layer of rapid adjustment, and let the analysis become more effective and low cost.