Self-sustained oscillations and bifurcations of transonic flow past simple airfoils

A turbulent flow past two symmetric airfoils, whose bow and aft portions are circular arcs, whereas midparts are flat, is studies numerically. The amplitude of lift coefficient oscillations versus the free-stream Mach number M _∞ is analyzed at zero angle of attack. Ranges of M _∞ in which there exist flow bifurcations are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 37–44, November–December, 2008     

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. Received 1 May 1998 and accepted 14 October 1998     

Reconsideration of inviscid shock interactions and transition phenomena on double-wedge geometries in a M ∞ = 9 hypersonic flow

Shock polar analysis as well as 2-D numerical computation technique are used to illustrate a diverse flow topology induced by shock/shock interaction in a M _∞ = 9 hypersonic flow. New flow features associated with inviscid shock wave interaction on double-wedge-like geometries are reported in this study. Transition of shock interaction, unsteady oscillation, and hysteresis phenomena in the RR ↔ MR transition, and the physical mechanisms behind these phenomena are numerically studied and analyzed.     

Effect of streamwise structures on heat transfer in a hypersonic flow in a compression corner

Coefficients of heat transfer to the surface in a laminar hypersonic flow (M _∞ = 21) over plane and axisymmetric models with a compression corner are presented. These coefficients are measured by an infrared camera. The parameters varied in the experiments are the angle of the compression corner and the distance to the corner point. Characteristics of the flow with and without separation in the corner configuration are obtained. The measured results are compared with direct numerical simulations performed by solving the full unsteady Navier-Stokes equations. Experiments with controlled streamwise structures inserted into the flow are described. A substantial increase in the maximum values of the heat-transfer coefficient in the region of flow reattachment after developed laminar separation is demonstrated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 112–120, July–August, 2009.     

On a Free Convection Problem Over a Vertical Flat Surface in a Porous Medium

The problem of the free convection boundary-layer flow over a semi-infinite vertical flat surface in a porous medium is considered, in which the surface temperature has a constant value T₁ at the leading edge, where T₁ is above the ambient temperature, and takes a value T₂ at a given distance L along the surface, varying linearly between these two values and remaining constant afterwards. Numerical solutions of the boundary-layer equations are obtained as well as solutions valid for both small and large distance along the surface. Results are presented for the three cases, when the temperature T₂ is greater, equal or less than the ambient temperature T_∞. In the first case, T₂ > T_∞, a boundary-layer flow develops along the surface starting with a flow associated with the temperature difference T₁ − T_∞ at the leading edge and approaching a flow associated with the temperature difference T₂ − T_∞ at large distances. In the second case, T₂ = T_∞, the convective flow set up on the initial part of the surface drives a wall jet in the region where the surface temperature is the same as ambient. In the final case, T₂ < T_∞, a singularity develops in the numerical solution at the point where the surface temperature becomes T_∞. The nature of this singularity is discussed.     

Experimental study of the reverse flow in the forward separation region in a pulsating flow around a spiked body

The flow with a free-stream Mach number M _∞ = 6 around a cylindrical body with a sharp spike is studied. The existence of a supersonic reverse flow for one of the phases of the pulsating flow regime is experimentally validated. A range of spike lengths is determined, which ensures a region of a supersonic reverse flow near the side surface of the spike. The time of existence of the supersonic reverse flow region is shown to be 0.15 of the period of pulsations if the spike length equals the model diameter. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 30–39, July–August, 2007.     

Investigation of Dynamic Stability Regimes for Axisymmetric Bodies in Supersonic Flow

The problem of supersonic flow past a slender blunt cone with allowance for the reverse boundary-layer effect on the outer flow is solved with the aim of studying the influence of the boundary layer on the damping coefficient of axisymmetric body oscillations. It is assumed that the body executes plane angular, both low-amplitude and low-velocity, oscillations about a center of rotation. A modified version of the method [1] is applied for calculating the time-dependent flow past a body with the viscosity effect taken into account. The high accuracy of the flow parameter determination provided by this technique is confirmed by wind- tunnel experiments on a large-scale cone model (L1 m) at Mach numbers M=4 and 6. The agreement between the calculated and measured data forms the basis for the numerical investigation of the blunt-cone damping coefficient over a wide range of freestream Mach (M=4–20) and Reynolds (Re _L =10⁶–10⁸) numbers. At moderate freestream Mach numbers (M=4 and 6) an appreciable Re _L effect on the damping coefficient was not detected. However, on the hypersonic range this effect manifests itself more strongly, especially when there is gas injection into the boundary layer from the vehicle surface.     

Aerodynamic experimentation with ducted models as applied to hypersonic air-breathing vehicles

 A methodology of experimentation in high supersonic wind tunnels for studying aerodynamic characteristics of hypersonic flying vehicles powered by air-breathing engines is discussed. Investigations of such total aerodynamic forces as drag, lift and pitching moment at testing the models are implicit when the air flow through the model ducts is accomplished so that to provide the simulation of the external flow around the airplane and flow over the inlets, but the operating engines and, hence, the exhaust jets are not modeled. The methods used for testing such models are based on the measurement of duct stream parameters alongside with the balance measurement of aerodynamic forces acting on the models. In the tests, aerometric tools are used such as narrow metering nozzles (plugs), pitot and static pressure probes, stagnation temperature probes and pressure orifices in walls of the model duct. The aerometric data serve to determine the flow rate and momentum of the stream at the duct exit. The internal non-simulated forces of the model ducts are also determined using the conservation equations for energy, mass flow and momentum, and these forces are eliminated from the aerodynamic test results. The techniques of the said model testing have been well developed as applied to supersonic aircraft, however their application for hypersonic vehicles whose models are tested at high supersonic speeds, Mach number M _∞>4, implies some specific features. In the present paper, the results of experimental and theoretical study of these features are discussed. Some experimental data on aerodynamics of hypersonic aircraft models received in methodological tests are also presented. The tunnel experiments have been carried out in the Mach number range M _∞=2–6. Received: 25 July 1996/ Accepted: 14 December 1998     

The Effect of Sweep-Angle Variation on the Turbulence Structure in a Separated,Three-Dimensional Flow

A three-dimensional separated flow behind a swept, backward-facing step is investigated by means of DNS for Re _H = C _∞ H/ν = 3000 with the purpose to identify changes in the statistical turbulence structure due to a variation of the sweep angle α from 0° up to 60°. With increasing sweep angle, the near-wall turbulence structure inside the separation bubble and downstream of reattachment changes due to the presence of an edge-parallel mean flow component W. Turbulence production due to the spanwise shear ∂W/∂y at the wall becomes significant and competes with the processes caused by impingement of the separated shear-layer. Changes due to a sweep angle variation can be interpreted in terms of two competing velocity scales which control the global budget of turbulent kinetic energy: the step-normal component U _∞ = C _∞cosα throughout the separated flow region and the velocity difference C _∞ across the entire shear-layer downstream of reattachment. As a consequence, the significance of history effects for the development into a two-dimensional boundary layer decreases with increasing sweep angle. For α ≥50°, near-wall streaks tend to form inside the separated flow region. Received 7 November 2000 and accepted 9 July 2002 Published online 3 December 2002 RID="*" ID="*" Part of this work was funded by the Deutsche Forschungsgemeinschaft within Sfb 557. Computer time was provided by the Konrad-Zuse Zentrum (ZIB), Berlin. Communicated by R.D. Moser     

Numerical Investigation of Off-Design Regimes of Flow Past Power-Law Waveriders Based on the Flows Behind Plane Shocks

The results of a numerical investigation of supersonic off-design flow past waveriders at the freestream Mach numbers M = 4 and 8 are presented. Flow regimes with M both greater and smaller than the design value M _d are analyzed. Configurations based on the flows behind plane shocks and described by power-law functions are considered. The results are obtained by the finite-volume solution of the Euler equations using higher-order TVD Runge-Kutta schemes.     

Bifurcations of Heteroclinic Orbits

The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ ₁ as t → −∞ and u(t) → γ ₂ as t → ∞ where γ ₁, γ ₂ are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.     

Effects of Mach number on turbulent separation behaviours induced by blunt fin

 An experimental study of the interaction between shock wave and turbulent boundary layer induced by blunt fin has been carried out at M _∞=7.8 using oil flow visualization and simultaneous measurements of fluctuating wall pressure and heat transfer. This paper presents the effects of Mach number on turbulent separation behaviours induced by blunt fin. Received: 21 July 1996/Accepted: 4 February 1998     

Numerical calculations of supersonic underexpanded jets in a cocurrent flow with the use of parabolized Navier-Stokes equations

Results of a numerical study of three-dimensional supersonic jets propagating in a cocurrent flow are described. Averaged parabolized Navier-Stokes equations are solved numerically on the basis of a developed scheme, which allows calculations in supersonic and subsonic flow regions to be performed in a single manner. A jet flow with a cocurrent flow Mach number 0.05 ⩽ M_∞ ⩽ 7.00 is studied, and its effect on the structure of the mixing layer is demonstrated. The calculated results are compared with available experimental and numerical data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 54–63, May–June, 2008.     

Lp-Lq Estimate of the Stokes Operator and Navier–Stokes Flows in the Exterior of a Rotating Obstacle

We consider the motion of a viscous fluid filling the whole three-dimensional space exterior to a rotating obstacle with constant angular velocity. We develop the L _p -L _q estimates and the similar estimates in the Lorentz spaces of the Stokes semigroup with rotation effect. We next apply them to the Navier–Stokes equation to prove the global existence of a unique solution which goes to a stationary flow as t → ∞ with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in L _3,∞ (weak-L ₃ space).     

Finite difference analysis of laminar free convection flow past a non isothermal vertical cone

A finite-difference analysis for the transient free convection flow of an incompressible viscous fluid past a vertical cone with variable wall surface temperature T _w^′ (x) = T _∞^′ + a x ⁿ varying as power function of distance from the apex (x = 0) is presented here. The dimensionless governing equations of the flow that are unsteady, coupled and non-linear partial differential equations are solved by an efficient, accurate and unconditionally stable finite difference scheme of Crank-Nicolson type. The velocity and temperature fields have been studied for various parameters such as Prandtl number and n (exponent in power law variation in surface temperature). The local as well as average skin-friction and Nusselt number are also presented and analyzed graphically. The present results are compared with available results in literature and are found to be in good agreement.     

The hypersonic Mach number independence principle in the case of viscous flow

The hypersonic Mach number independence principle of Oswatitsch is important for hypersonic vehicle design. It explains why, above a certain flight Mach number (M _∞ ≈ 4−6, depending on the body shape), some aerodynamic properties become independent of the flight Mach number. For ground test facilities this means that it is sufficient for the Mach number in the test section to be high enough, that Mach number independence exists. However, the principle was derived for calorically perfect gas and inviscid flow only. In this paper a theoretical study for blunt bodies in the case of viscous flow is presented. We provide numerical results which give insight into how attached viscous flow behaves at high Mach numbers. The flow past an axisymmetric configuration is analysed by applying a coupled Euler/second-order boundary-layer method. Wall boundaries are treated by assuming an adiabatic or radiation-adiabatic wall for laminar flow. Calorically perfect or equilibrium air is accounted for. Lift, drag, and moment coefficients, and lift-to-drag ratios are given for several combinations of flight Mach number and altitude, i.e. Reynolds number. For blunt bodies considered here, which are pressure dominated, Mach number independence occurs for the adiabatic wall, but not for the radiation-adiabatic wall assumption.     

Outflow of an unexpanded jet into counter supersonic and subsonic flows

The article gives the results of an experimental investigation of the geometric structure of an opposing unexpanded jet. It discusses flow conditions with interaction between the jet and sub- and supersonic flows. It is shown that, with the outflow of an unexpanded jet counter to a supersonic flow, there are unstable flow conditions. For stable flow conditions with one roll, dependences are proposed determining the form of a jet in a supersonic opposing flow. A generalized dependence is obtained for the distribution of the pressure at the surface of a body with a jet, flowing out counter to a subsonic flow. The range of change in the determining parameters are the following: Mach numbers at outlet cross section of nozzle, M _a = 1 and 3; Mach numbers of opposing flow, M = 0.6–0.9 and 2.9; degree of effectiveness of jet, n = p _a /p = 0.5–800 (p _a and p are the static pressures at the outlet cross section of the nozzle and in the opposing flow); the ratios of the specific heat capacities, _a = = 1.4; the drag temperatures of the jet and the flow, T_o = T_oa = 290°K.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 89–96, January–February, 1977.     

Numerical simulation of stabilization of the boundary layer on a surface with a porous coating in a supersonic separated flow

Stability of a supersonic (M _∞ = 5.373) boundary layer with local separation in a compression corner with a passive porous coating partly absorbing flow perturbations is considered by solving two-dimensional Navier-Stokes equations numerically. The second mode of disturbances of a supersonic boundary layer is demonstrated to be the most important one behind the boundary-layer reattachment point. The possibility of effective stabilization of these disturbances behind the reattachment point with the use of porous coatings is confirmed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 39–47, March–April, 2007.     

Interference between a centered rarefaction wave and an inclined shock wave

The gas flow in the zone of interaction between an oblique shock and a centered isentropic rarefaction wave is studied using the direct statistical simulation method for solving the Boltzmann equation. The data of calculations of the shock and rarefaction wave structures, flow fields, and streamlines are given for the free-stream Mach number M_∞ = 6, 4 and 2. The formation of the interaction zone is simulated by a gas flow past a double-plane wedge in which the break of the generating line leads to formation of the centered isentropic rarefaction wave. The results of calculations of this flow in solving the Boltzmann equation are given in the Euler approximation.     

Correlation of the separation region length in shock wave/channel boundary layer interaction

 Results of an experimental investigation of the characteristics of a separation region induced by the interaction of an externally generated oblique shock with the turbulent boundary layer formed in a rectangular half channel are discussed. The experiments were carried out in the supersonic wind tunnel of the Institute of Theoretical and Applied Mechanics SB RAS at a free-stream Mach number M _∞=3.01 over a range of Reynolds numbers Re ₁=(9.7–47.5)×10⁶ m^-1 and at zero incidence and zero yaw of the model. Particular attention is paid to the size of the zone of the upstream propagation of disturbances (upstream influence region) under different experimental conditions: with varied values of the shock wave strength, half channel width, and Reynolds number. It is shown, in particular, that the normalized upstream influence region length as a function of inclination angle of the shock generator in a rectangular half channel is readily approximated by a simple exponential function. In support of the known reference data obtained for supersonic numbers M _∞ and moderate Re in other configurations, it is also shown that the upstream influence region length decreases with increasing Reynolds number. Generalization of experimental data on the length of the upstream influence region formed in similar geometric configurations is possible using an additional reference linear scale which is the distance from the leading edge of the shock generator to the exposed surface. A substantial dependence of the reference dimensions of separation region on the half channel width is also established. Received: 20 January 1997/Accepted: 30 June 1997