Supersonic flow past blunt bodies screened by a permeable surface

A study is made of a steady flow of supersonic gas past the front part of blunt bodies with allowance for the screening effect of a thin perforated surface, whose presence is simulated by the introduction of a supplementary hydrodynamic discontinuity (an active surface [1, 2]).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 139–144, March–April, 1986.     

Viscous boundary layers in flows through a domain with permeable boundary

The effect of small viscosity on nearly inviscid flows of an incompressible fluid through a given domain with permeable boundary is studied. The Vishik–Lyusternik method is applied to construct a boundary layer asymptotic at the outlet in the limit of vanishing viscosity. Mathematical problems with both consistent and inconsistent initial and boundary conditions at the outlet are considered. It is shown that in the former case, the viscosity leads to a boundary layer only at the outlet. In the latter case, in the leading term of the expansion there is a boundary layer at the outlet and there is no boundary layer at the inlet, but in higher order terms another boundary layer appears at the inlet. To verify the validity of the expansion, a number of simple examples are presented. The examples demonstrate that asymptotic solutions are in quite good agreement with exact or numerical solutions.     

Turbulent boundary layer on a permeable surface

Several theoretical [1–4] and experimental [5–7] studies have been devoted to the study of the effect of distributed injection of a gaseous substance on the characteristics of the turbulent boundary layer. The primary study has been made of flow past a flat plate with gas injection. The theoretical methods are based primarily on the semiempirical theories of Prandtl [1] and Karman [2].In contrast with the previous studies, the present paper proposes a power law for the mixing length; this makes it possible to obtain velocity profiles which degenerate to the known power profiles [8] in the case of flow without blowing and heat transfer. This approach yields analytic results for flows with moderate pressure gradient.Notation x, y coordinates - U, V velocity components - density - T temperature - h enthalpy - H total enthalpy - c mass concentration - , , D coefficients of molecular viscosity, thermal conductivity, diffusion - c_p specific heat - adiabatic exponent - r distance from axis of symmetry to surface - boundary layer thickness - U velocity in stream core - friction - c_f friction coefficient - P Prandtl number - S Schmidt number - S_t Stanton number - M Mach number - j=0 plane case - j=1 axisymmetric case The indices 1 injected gas - 2 mainstream gas - w quantities at the wall - core of boundary layer - 0 flow of incompressible gas without injection - v=0 flow of compressible gas without injection - ^* quantities at the edge of the laminar sublayer - quantities at the initial section - turbulent transport coefficients     

Multicomponent three-dimensional viscous shock layer on blunt bodies with a catalytic surface at angles of attack and yaw

The three-dimensional supersonic flow of nonequilibrium dissociating air past smooth blunt bodies on whose surface heterogeneous chemical reactions are taking place is investigated within the framework of thin viscous shock layer theory. An economical numerical method of solving the equations with an improved order of approximation with respect to the normal coordinate is employed. This method does not require the preliminary solution of the Stefan-Maxwell relations for the diffusion fluxes and makes it possible to calculate flows that do not possess a plane of symmetry. The effect of the angles of attack and yaw, the catalytic reaction model and a number of other parameters of the problem on the pressure, heat flux and equilibrium surface temperature distributions is analyzed with reference to the example of flow past a triaxial ellipsoid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–150, January–February, 1990.Tha authors are grateful to G. A. Tirskii for useful discussions of their results.     

Linear stability of three-dimensional boundary layers

Stability of compressible three-dimensional boundary layers on a swept wing model is studied within the framework of the linear theory. The analysis based on the approximation of local self-similarity of the mean flow was performed within the Falkner-Skan-Cooke solution extended to compressible flows. The calculated characteristics of stability for a subsonic boundary layer are found to agree well with the measured results. In the case of a supersonic boundary layer, the results calculated for a Mach number M = 2 are also in good agreement with the measured spanwise scales of nonstationary vortices of the secondary flow. The calculated growth rates of disturbances, however, are substantially different from the measured values. This difference can be attributed to a high initial amplitude of disturbances generated in the experiment, which does not allow the linear stability theory to be applied. The evolution of natural disturbances with moderate amplitudes is fairly well predicted by the theory. The effect of compressibility on crossflow instability modes is demonstrated to be insignificant. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 3–14, March–April, 2008.     

Inflow boundary condition for DNS of turbulent boundary layers on supersonic blunt cones

For direct numerical simulation (DNS) of turbulent boundary layers, gen- eration of an appropriate inflow condition needs to be considered. This paper proposes a method, with which the inflow condition for spatial-mode DNS of turbulent boundary layers on supersonic blunt cones with different Mach numbers, Reynolds numbers and wall temperature conditions can be generated. This is based only on a given instant flow field obtained by a temporal-mode DNS of a turbulent boundary layer on a flat plate. Effectiveness of the method is shown in three typical examples by comparing the results with those obtained by other methods.     

Investigation of surface instability of stratified bodies in a three-dimensional formulation

Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 2, pp. 3–24, February, 1990.     

Near-wall structure of three-dimensional turbulent boundary layers

 Modifications to near-wall turbulent boundary layer structure with increased three-dimensionality have been investigated through the use of hydrogen bubble wire flow visualization. Results indicate that three-dimensionality does not influence the strength or sign of near-wall streamwise vortices. Increased three-dimensionality does stabilize the near-wall structure resulting in less ejection type activity. The spanwise spacing between low-speed streaks also decreased slightly with increased cross-flow. Received: 15 October 1996/Accepted: 2 April 1997     

Self-similar thermal boundary layers on plane and axisymmetric bodies

This paper proposes an approximate solution procedure for the prediction of the forced convection heat transfer through self-similar laminar boundary layers. The differential equations governing the viscous and thermal boundary layers have been reduced to a pair of algebraic equations for the boundary layer shape factor and the boundary layer thickness ratio. The local Nusselt number predicted under various pressure gradients turns out to be in excellent agreement with that of the exact solution over a wide range of the Prandtl number.     

Receptivity of boundary layers to three-dimensional disturbances

The 3D receptivity of 2D laminar boundary layers to localized surface vibrations has been investigated both experimentally and theoretically for two types of basic flow: (i) the Blasius boundary layer and (ii) a boundary layer with a negative streamwise pressure gradient (Hartree parameter β_H=0.10). For the boundary-layer excitation, a specially designed surface vibrator was used. The development of the excited wave-trains was measured by means of hot-wire anemometry and decomposed into oblique normal Tollmien–Schlichting-modes. The initial spectra of the excited perturbations at the position of the vibrator was obtained by two different techniques. The first used an additional source which was mounted upstream and provided the amplification curves for the instability modes in the vicinity of the vibrator, the second was based on linear stability calculations. The receptivity coefficients were defined as the ratio of the initial wavenumber spectrum of the excited TS-waves and the corresponding resonant spectrum of the surface vibrations. They were determined for each fixed frequency as a function of the spanwise wavenumber.The boundary value problem for the disturbance produced by the vibrating membrane was solved theoretically for the same conditions as in the experiments in the framework of the classical hydrodynamic stability theory. The Navier–Stokes equations were linearized around a incompressible basic flow described by a solution of the Falkner–Skan equation. Comparisons of the theoretical and experimental results on the 3D receptivity show good quantitative agreement. It is concluded that the favorable pressure gradient increases the boundary-layer receptivity to surface vibrations.     

Laminar boundary layer on blunt bodies with account for vorticity of the external flow

This paper presents the technique for and results from numerical calculations of the hypersonic laminar boundary layer on blunted cones with account for the vorticity of the external flow caused by the curved bow shock wave. It is assumed that the air in the boundary layer is in the equilibrium dissociated state, but the Prandtl number is assumed constant, =0.72. The calculations were made in the range of velocities 3–8 km/sec, cone half-angles _k=0°–20°. With account for the vortical interaction of the boundary layer with the external flow, the distribution of the thermal flux and friction will depend on the freestream Reynolds number (other conditions being the same). In the calculations the Reynolds number R, calculated from the freestream parameters and the radius of the spherical blunting, varies from 2.5·10³ to 5.10⁴. For the smaller Reynolds numbers the boundary layer thickness on the blunting becomes comparable with the shock standoff, and for R<2.5·10³ it is apparent that we must reconsider the calculation scheme. With R>5·10⁴ for cones which are not very long the vortical interaction becomes relatively unimportant. The results of the calculations are processed in accordance with the similarity criteria for hypersonic viscous gas flow past slender blunted cones [1, 2].     

Investigation of hypersonic three-dimensional flow past blunt bodies within the framework of the parabolized Navier-Stokes equations

The flow of a homogeneous gas in a three-dimensional hypersonic viscous shock layer, which includes the shock wave structure, is examined within the framework of the parabolic approximation of the Navier-Stokes equations. The Navier-Stokes equations are simplified on the basis of the asymptotic analysis carried out in [1], are written in variables of the Dorodnitsyn type [2] and are solved by the method proposed in [3, 4] extended to the case of three-dimensional flows. The flow at zero angle of attack past elliptic paraboloids, two-sheeted hyperboloids and triaxial ellipsoids is calculated. Some results of investigating the flow past such bodies are presented. Flow past a sphere in the analogous approximation was considered in [5], where a comparison was also made with the solution of the complete Navier-Stokes equations [6, 7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 134–142, July–August, 1987.In conclusion, the authors wish to express their warm thanks to V. V. Lunev and G. A. Tirskii for useful discussion and valuable comments.     

Flow past blunt bodies with nose spikes and surface injection

Axisymmetric supersonic ideal-gas flow past a blunt body with a forward-projecting spike is numerically investigated with allowance for injection from the surface. The effect of the length and shape of the spike, the parameters of the injected gas and the position of the permeable zone on the flow pattern and drag is studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 128–133, July–August, 1987.     

Propagation of disturbances in three-dimensional supersonic boundary layers

The propagation of disturbances in three-dimensional boundary layers under the conditions of a global and a local strong inviscid-viscous interaction is analyzed. A system of subcharacteristics is found based on the condition for the pressure-related subcharacteristic, and an algebraic relation that gives the propagation velocity of disturbances is obtained. The velocity of propagation of disturbances is calculated for two- and three-dimensional flows. The studied problem is of great importance for accurately formulating problems for three-dimensional unsteady boundary-layer equations and for constructing adequate computational models. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 116–127, May–June, 1999.     

Stability of a supersonic boundary layer on a permeable surface with heat transfer

The effect of cooling of a permeable surface on the stability of a supersonic boundary layer on it is investigated. As distinct from the case of an impermeable surface, deep cooling can reduce the critical Reynolds number. Common points of the continuous and discrete spectra are found in the region of the disturbance parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 41–46, January–February, 1977.The work was carried out at the Department of Applied Mathematics of the University of Western Ontario in Canada. I am grateful to the Head of the Department Professor J. Blackwell for suggesting the investigation of this problem.     

Hydrodynamic stability of boundary layers with surface mass transfer

The linear stability of the flat plate boundary layer with surface blowing and suction is investigated by the application of numerical techniques. Complete neutral stability curves, critical Reynolds numbers and wave numbers, and other stability characteristics are determined for a wide range of surface mass transfer intensities. The critical Reynolds number, based on the displacement thickness, is found to vary from 59 to 32500 between the extreme limits of blowing and suction that are investigated. Comparisons are made between the present results and available linear stability information for boundary layers with surface mass transfer and with free-stream pressure gradients. The universal stability bound of Joseph is evaluated and compared with the corresponding numerically exact neutral stability curve.