Hypersonic Small-disturbance Theory Research Articles

Physical and computational aspects of shock waves over power-law leading edges

Computations using the direct simulation Monte Carlo (DSMC) method are presented for hypersonic flow on power-law shaped leading edges. The primary aim of this paper is to examine the geometry effect of such leading edges on the shock-wave structure. The sensitivity of the shock-wave shape, shock-wave thickness, and shock-wave standoff distance to shape variations of such leading edges is investigated by using a model that classifies the molecules in three distinct classes: (1) undisturbed freestream, (2) reflected from the boundary, and (3) scattered, i.e., molecules that had been indirectly affected by the presence of the leading edge. The analysis showed that, for power-law shaped leading edge with exponent between 2∕3 and 1, the shock wave follows the body shape. It was found that, at the vicinity of the nose, the shock-wave power-law exponent is 1∕2. Far from the nose, calculations showed that the shock-wave shape is in surprising qualitative agreement with that predicted by the hypersonic small disturbance theory for the flow conditions considered.

Comparison between methods of generation of waveriders derived from conical flows

Waveriders are supersonic or hypersonic lifting configurations which are generated from a known flowfield, preventing leakage from the high-pressure undersurface to the uppersurface, and allowing high lift-to-drag ratios. A method based on an Euler code is used to study waveriders generated from the flow field around axisymmetric bodies. In the case of a cone, the results are compared with those given by the Taylor–Maccoll system and inviscid hypersonic small-disturbance theory. In the present study conditions at Mach 5, the last method paradoxically works at high cone angles (greater than 10°) while the best lift-to-drag ratio are obtained with small angles. To cite this article: B. Mangin et al., C. R. Mecanique 334 (2006).

Calculation of Shock Wave Structure over Power Law Bodies in Hypersonic Flow

A numerical study is conducted of power law shaped leading edges situated in a rarefied hypersonic flow. The motivation is interest in investigating the flowfield structure of power law shaped leading edges as possible candidates for blunting geometries of hypersonic leading edges. The primary aim is to examine in detail the effect of geometry of such leading edges on the shock wave structure. The sensitivity of the shock wave shape, standoff distance, and shock wave thickness to shape variations of the power law class of leading edges is calculated by using the direct simulation Monte Carlo method. Calculations show that the shape of the shock wave is in agreement with that predicted by the analytic hypersonic small disturbance theory for the flow conditions considered.

Optimization of Conical Wings in Hypersonic Flow

A method of calculation is presented to determine conical wing shapes that minimize the coefficient of (wave) drag, C D, for a fixed coefficient of lift, C L, in steady, hypersonic flow. An optimization problem is considered for the compressive flow underneath wings at a small angle of attack δ and at a high free-stream Mach number M ∞ so that hypersonic small-disturbance (HSD) theory applies. A figure of merit, F=C D/C L 3/2, is computed for each wing using a finite volume discretization of the HSD equations. A set of design variables that determine the shape of the wing is defined and adjusted iteratively to find a shape that minimizes F for a given value of the hypersonic similarity parameter, H= (M ∞δ)−2, and planform area. Wings with both attached and detached bow shocks are considered. Optimal wings are found for flat delta wings and for a family of caret wings. In the flat-wing case, the optima have detached bow shocks while in the caret-wing case, the optimum has an attached bow shock. An improved drag-to-lift performance is found using the optimization procedure for curved wing shapes. Several optimal designs are found, all with attached bow shocks. Numerical experiments are performed and suggest that these optima are unique.

Nonlinear asymptotic theory of hypersonic flow past a circular cone

The hypersonic small-disturbance theory is reexamined in this study. A systematic and rigorous approach is proposed to obtain the nonlinear asymptotic equation from the Taylor-Maccoll equation for hypersonic flow past a circular cone. Using this approach, consideration is made of a general asymptotic expansion of the unified supersonic-hypersonic similarity parameter together with the stretched coordinate. Moreover, the successive approximate solutions of the nonlinear hypersonic smalldisturbance equation are solved by iteration. Both of these approximations provide a closed-form solution, which is suitable for the analysis of various related flow problems. Besides the velocity components, the shock location and other thermodynamic properties are presented. Comparisons are also made of the zeroth-order with first-order approximations for shock location and pressure coefficient on the cone surface, respectively. The latter (including the nonlinear effects) demonstrates better correlation with exact solution than the zeroth-order approximation. This approach offers further insight into the fundamental features of hypersonic small-disturbance theory.

Flight simulation of a waverider-based hypersonic vehicle

This work presents three-dimensional, hypersonic flow computations for a waverider-based hypersonic vehicle with finlets. The waverider geometry, as composed of an airframe and a reflexed trailing-edge, is derived from a perturbed hypersonic flow past a cone with combined transverse and small longitudinal curvature. The flowfield simulations associated with the design and off-design flight conditions are performed by solving the three-dimensional compressible Navier-Stokes equations that are incorporated with an implicit upwind-biased finite volume algorithm. Also, the lift, drag, skin friction, lift-to-drag ratio, and static margin are calculated. In the on-design conditions ( M ∞ = 4.0), the inviscid and viscous numerical results correlate well with the estimates of the hypersonic small-disturbance theory. In addition, the influence on aerodynamic performance and stability of adding finlets is discussed here. Furthermore, the variations of aerodynamic performance caused by altering the angle-of-attack from −10 to +10 ° are investigated.

Application of Approximate Equations for Waveriders with Multi-Directional Curvature

This study thoroughly analyses the parameters affecting aerodynamic performance and the shapes of the multi-directional curvature waverider. This hypersonic vehicle is developed from a hypersonic flowfield of a perturbed cone with longitudinal and transverse curvatures, as derived by combining the hypersonic small disturbance theory (HSDT) with the perturbation method. A range both theoretically and practically sound is also determined to alter the parameters. Numerical methods are then used to find the vehicle's optimal shape having a maximum lift, lift–drag ratio, or volumetric ratio when the viscous effect is considered. Furthermore, the curve fitting method is utilized to establish a set of approximate equations capable of speedily estimating the aerodynamic performance and the volumetric ratio. In addition several samples are presented to illustrate how to use this set of equations to generate the optimal shape under any combination of constraints in the lift–drag ratio, lift, and volumetric ratio. Notably, the viscous effect is approximate and takes no account of displacement corrections. Moreover, the base drag is neglected here.

Aerodynamic forces of elliptic-cone derived waveriders at hypersonic velocities

The aerodynamic data for two kinds of elliptic-cone derived waveriders are presented and analyzed for a wide range of Mach numbers and angles of attack. The basic shapes of the waveriders are obtained by means of hypersonic small-disturbance theory. Various off-design performances of the waveriders as well as on-design performances are obtained and fully discussed. These aerodynamic data are informative for the design of hypersonic airplanes which are being investigated as future planes. Also from the entropy distributions across the shock layers for elliptic-cones and waveriders, the vortical layer is analyzed and discussed.

Hypersonic inviscid and viscous flow over a wedge and cone

An analysis of hypersonic flow over a wedge and cone is developed that is based on hypersonic smalldisturbance theory in combination with laminar boundary-layer theory. The parameters that determine the flow are the ratio of specific heats ( = 1.4), the freestream Mach number, which ranges from 4 to 10, the wall half-angle, equal to 5, 10, or 15 deg, a wall to freestream temperature ratio, equal to 1 or 3, a Reynolds number (= 106), and a geometric parameter that is zero for a wedge and unity for a cone. All results are nondimensional and in a convenient form for establishing trends or for comparisons with experimental or CFD data. Results are provided for the inviscid surface values of the pressure, temperature, and Mach number; wave and viscous drag coefficients and several heat transfer coefficients; the maximum temperature and its location inside the boundary layer; and a variety of viscous and thermal boundary-layer thicknesses. The independent parameter is the freestream Mach number; the dependence on the wall to freestream temperature ratio and a Reynolds number are explicitly given.

Waveriders with finlets

A generalized analytical method for designing waverider shapes is presented. Waverider configurations that incorporate finlets are of primary interest. These finlets resemble fins on conventional aircraft and might imaginably be used as control surfaces. The design methodology determines the waverider shape by alternatively specifying the planform shape, the freestream-surface trailing-edge shape, the compression-surface trailing-edge shape, or various combinations of the three. A wide variety of shapes is presented, and the aerodynamic properties are calculated by means of hypersonic small-disturbance theory. The effects of finlets are to shift the c.p. aft and to improve static stability. The results are novel, and the analysis is readily usable for a variety of design studies.

On hypersonic flow over two‐dimensional aerofoils

AbstractThis paper considers the problem of hypersonic flow past two‐dimensional thin aerofoils with sharp leading edges and attached shock waves. The analysis relies on another recent formulation of the hypersonic small disturbance theory, HSDT. The equations are reduced to a complicated functional‐differential equation for which an iterative numerical scheme is employed. In each iteration both the functional and differential parts of the functional‐differential equation are eliminated, thus reducing it to an algebraic equation. The iterative scheme is found to be stable and rapidly convergent, and the results obtained for the surface pressure and shock wave compare well with other exact and approximate methods.

Effects of Pressure Mismatch on Slot Injection in Supersonic Flow

In inany applications of slot injection in supersonic or hypersonic flow, the static pressures of the external and injectant streams are not matched. We have developed a simplified analysis of cases with initial pressure mismatch for tangential slat injection into a hypersonic ( o r supersonic) external flow having a thick, turbulent initial boundary layer including tho effects of turbulentmiuing, viscid/inviscid'interaction, skin friction and heat transfer and heat release for reactive injectants. The treatment uses hypersonic (or supersoni.c) small disturbance theory, R power law shape for the initial boundary layer velocity profile and a Crscco integral to get the temperature profile, a turbulent entrainment model including the effect oE a Convective Mach Number, a quasi-1D assumption in the miring and burning region, an instantaneous, one-step heat release modo1 and simple models of skin friction and heat transfer. 'The initial adjustments at the slot lip and further interactions with the viscous regions by waves reflected off the lower wall are treated with weak, locally inviscid waves. Laminar cases can also he handled. Good agreement of predictions with experiment for several experimental cases for gross flayfield variables such as p(x), A ( r ) , M(x) , T,(x), ete. was achieved. Both overand underexpanded cases were treated. The calculations need only about 1-2 seconds of CPU time on an Amdahl 5890 c lass csmputer. Lastly, calculations for combustor flowfields for a generic Scramjet vehicle at a flight Mach number of 10 with 0 . 8 5 pj/p, 2 1.25 are @en and discussed . I

Optimum hypersonic airfoil with power law shock waves

In the present paper the flow field over a class of two-dimensional lifting surfaces is examined from the viewpoint of inviscid, hypersonic small-disturbance theory (HSDT). It is well known that a flow field in which the shock shape S(x) is similar to the body shape F(x) is only possible for F(x) = x{sup k} and the freestream Mach number M{sub {infinity}} = {infinity}. This self-similar flow has been studied for several decades as it represents one of the few existing exact solutions of the equations of HSDT. Detailed discussions are found for example in papers by Cole, Mirels, Chernyi and Gersten and Nicolai but they are limited to convex body shapes, that is, k {le} 1. The only study of concave body shapes was attempted by Sullivan where only special cases were considered. The method used here shows that similarity also exists for concave shapes and a complete solution of the flow field for any k > 2/3 is given. The effect of varying k on C{sub L}{sup 3/2}/C{sub D} is then determined and an optimum shape is found. Furthermore, a wider class of lifting surfaces is constructed using the streamlines of the basic flow field and analysed with respectmore » to the effect on C{sub L}{sup 3/2}/C{sub D}. 9 refs., 3 figs.« less

A first order theory for Newtonian flow over two-dimensional airfoils

SummaryA simple closed-form formula for the coefficient of surface pressure, Cp is given in this paper. The formula is based on first and second approximations of the full problem and can be used for the Newtonian flow past two-dimensional thin airfoils at small angles of attack and with attached shock waves. It thus extends Cole's zero order theory to the effects of non-zero (γ – 1) and finite values of Mx where γ is the ratio of the specific heats of the gas and Mx is the free stream Mach number. The analysis presented relies on a previous recent formulation of the hypersonic small disturbance theory which has proved to be more advantageous than the hypersonic small disturbance theory. The results are compared with other approximate methods and the agreement is found to be generally good.

Cone-derived waveriders with longitudinal curvature

Lifting-body waverider configurations with longitudinal curvature are derived by means of known flowfields past ogival bodies. The flowfields are generated by means of perturbations of the supersonic flow past a circular cone. Approximate analytical solutions for the shock-layer flowfield are found within the framework of hypersonic small-disturbance theory. Perturbations due to longitudinal curvature produce reduced drag, increased lift-to-drag ratios, and provide design advantages for a more efficient volume distribution related to packaging of guidance, propulsion, and payload units. Nomenclature Ab =base area of waverider Cp = pressure coefficient D = drag / =body perturbation function Gm = shock perturbation function for power law g = shock perturbation function Kd =M00d, hypersonic similarity parameter; d in radians L =lift t = length of basic cone MO, =freestream Mach number m = perturbation index factor Q -Poo^oo/2

Optimization of waverider configurations generated from axisymmetricconical flows

Analytic results from inviscid hypersonic small-disturbance theory for axisymmetric conical flow are used to generate waverider configurations valid for all values of Kb ^A/^5, where d is the semi vertex angle of the basic cone. By means of the calculus of variations, those configurations are sought that yield the maximum lift-todrag ratio subject to suitable constraints, such as fixed lift, fixed volume, fixed base area, fixed planform area, and so on. The inviscid analysis accounts for wave drag only, but the effects of viscous drag are discussed. For the fixed-lift constraint, two types of optimized configurations are found: types with pointed noses, and types with round noses with sharp leading edges. The results are analytic in nature and particularly suitable for studying the various tradeoffs that are involved in missile design. Comparison of the results with other types of lifting bodies suggests that properly selected cone-derived waveriders are among the best producers of large liftto-drag ratios.

Hypersonic Flow Past an Axisymmetric Body with Small Longitudinal Curvature

The problem of steady supersonic flow past an axisymmetric body with small longitudinal curvature is solved by means of a perturbation technique. Assuming a slender body within the framework of hypersonic smalldisturbance theory, the resulting differential equations are solved explicitly in closed form using the weak polar crossflow approximation. The perturbation quantities are expressed as series expansions involving powers of the radial distance from the nose of the body and functions of a single angular variable 0. The various flowfield quantities including the surface pressure coefficient have been calculated explicitly in terms of the series. The present analytical results compare favorably with available experimental data and Van Dyke's numerical calculations for hypersonic flow past an ogival-shaped slender body.

Newtonian flow theory for slender bodies in a dusty gas

Hypersonic small-disturbance theory is extended to consider the problem of dusty-gas flow past thin two-dimensional bodies. The mass fraction of suspended particles is assumed to be sufficiently large that the two-way interaction between particle phase and gas phase must be considered. The system of eight governing equations is further reduced by considering the Newtonian approximation γ → 1 andM∞→ ∞. The Newtonian theory up to second order is studied and the equations are solved for the case of a thin wedge at zero angle of attack. Expressions for the streamlines, dust-particle paths, shock-wave location and all flow variables are obtained. It is seen that the presence of the dust increases the pressure along the wedge surface and tends to bend the shock wave towards the body surface. Other effects of the interaction of the two phases are also discussed.

Approximation for hypersonic flow past an inclined cone.

An approximate analytical solution is obtained for hypersonic flow past an inclined circular cone. The governing equations are simplified by means of the constant-density approximation, and the ensuing quadrature solution is evaluated in closed form by approximations based on hypersonic small disturbance theory. This solution corresponds to the outer solution in a matched asymptotic expansion scheme and describes the region outside the thin vortical layer that lies adjacent to the surface of the cone. The results are in good agreement with tabulated numerical solutions and are also valid for values of the hypersonic similarity parameter that lie outside the range of the tabulated results. The present approximation is also applicable to much broader flow regimes than previous analytical descriptions, such as that of the Newtonian approximation. The results of this analysis are useful whenever analytical expressions are desired and can be coupled with one of the existing inner solutions for the vortical layer to form a composite uniformly-valid approximation to first order in angle of attack, valid for a wide range of hypersonic similarity parameters.

High Lift-Drag Ratio Double Wedges of Given Volume which Support Two-Dimensional Supersonic Flow Fields

Several authors have considered the lifting characteristics of various simple shapes at hypersonic speeds. Non-weiler has dealt with shapes derived from wedge flows, and Townend has considered a class of two-dimensional shapes with concave undersurfaces. Cole and Aroesty and Pike have both considered two-dimensional shapes which are perturbations of simple wedge flow. They show that slightly higher values of the lift-to-drag ratio than that of the wedge are obtainable. Cole and Aroesty apply hypersonic small disturbance theory, under the constraint of constant lift, and suggest that the optimum shape is a “multi-wedge”. They show further that the optimum of a more restricted class of shapes having undersurfaces made of two plane surfaces, has a slightly concave surface. Pike has applied a more accurate (second-order) perturbation theory to a class of shapes which consist of small deviations from a wedge. He finds that the optimum undersurface consists of two plane surfaces, and confirms Cole and Aroesty's result that it must be slightly concave, except perhaps at low Mach numbers below the hypersonic range. It must be remembered that these results depend upon the assumed constraint of constant lift.