Calculation of the pressure on the side surface of bodies consisting of a spherical segment joined to an inverted cone in a supersonic stream

The results of investigations of inviscid flow over inverted cones with nose consisting of a spherical segment were published for the first time in Soviet literature in [1–4]. In the present paper, a numerical solution to this problem is obtained using the improved algorithms of [5, 6], which have proved themselves well in problems of exterior flow over surfaces with positive angles of inclination to the oncoming flow. It is shown that the Mach number 2 M , equilibrium and nonequilibrium physicochemical transformations in air (H = 60 km, V = 7.4 km/sec, R₀ = 1 m), and the angle of attack 0 40° influence the investigated pressure distributions. A comparison of the results of the calculations with drainage experiments for M = 6, = 0-25° confirms the extended region of applicability of the developed numerical methods. Also proposed is a simple correlation of the dependence on the Mach number in the range 1.5 M of the shape of the shock wave near a sphere in a stream of ideal gas with adiabatic exponent = 1.4.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 178–183, January–February, 1981.     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

Nonuniqueness of the Transonic Flow past an Airfoil

The inviscid transonic flow past a symmetric airfoil having a curvature minimum in the middle is numerically investigated. It is shown that at zero angle of attack both symmetric and asymmetric steady-state flow patterns can exist on a certain freestream Mach number range M_min < M_max. On this range, the asymmetric flows are stable against small perturbations, whereas the symmetric flows are stable only if M does not coincide with a singular Mach number at which small variations in M or can result in flow restructuring.     

Flow visualization and analysis of aerodynamically detuned supersonic rotors

The effect of aerodynamic detuning on the supersonic steady and unsteady blade passage flow field is experimentally investigated on a free surface water table by means of color Schlieren and shadowgraph flow visualization techniques. Two aerodynamic detuning mechanisms are considered: (1) alternate circumferential spacing of adjacent airfoils; (2) the replacement of alternate airfoils with splitters. The steady flow visualization demonstrates the significant effect of aerodynamic detuning on the passage flow field and, in particular, the shock wave-airfoil surface intersection locations. The unsteady flow visualization studies show the importance of the interblade phase angle. A mathematical model is also described and utilized to demonstrate the enhanced aeroelastic stability associated with the altered cascade passage shock wave structure due to these aerodynamic detuning mechanisms.List of symbols a dimensionless perturbation sonic velocity - C airfoil chord - I _x mass moment of inertia - k reduced frequency, k = C/2 - K spring constant - M _R dimensionless unsteady aerodynamic moment - M Mach number - P _split split splitter circumferential spacing - P _start splitter leading edge location - R reference full chord airfoil - R _s reference splitter - Sp _chord splitter chord length - u dimensionless perturbation chordwise velocity - v dimensionless perturbation normal velocity - amplitude of oscillation - interblade phase angle - level of aerodynamic detuning - undamped natural torsional frequency - ₀ reference frequency - flutter frequency     

Electrical charging of bodies as a consequence of charged-particle extraction by a hydrodynamic flow

We investigate the problem of electrical charging of bodies as a result of charged-particle extraction by a hydrodynamic flow. The analysis is performed in view of the application to the problem of motion electrification of aircraft caused by a stream of charged particles into the surrounding space. We formulate the appropriate system of nonstationary electrohydrodynamic equations. It is shown that in many applications the charging of electrically insulated bodies consists of two successive intermediate processes. The first process is the formation of charge Q on the body in time T₁ The second process consists of a change of the body potential (with a constant charge Q) as a consequence of the stream of charged particles into the outside space noted above. At the end of the second process (with duration T₂) the body potential is at . We also investigate the problem of charging a spherical source of neutral and charged particles. Using the analytical solution we find the quantities Q and and the characteristic times T₁ and T₂. It is shown that the time T₂ can exceed T₁ by several orders of magnitude. We formulate the problems of nonstationary electric fields during the extraction of several types of particles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 94–103, September–October, 1977.     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

Discrete vortices in the wake of various profiles in subsonic and supersonic two-dimensional gas flow

The Mach number dependence of the Strouhal number, the frequency of discrete vortices, the vortex velocity, and other parameters are determined in the wake of wedges and flat plates for low angles of attack. The studies were made using high-speed motion-picture photography through a Schlieren system and with photomultipliers. The results are presented in tabular and graphical form.Notation h transverse distance - l longitudinal distance between vortices - V freestream velocity in m/sec - n_v vortex frequency for one row of vortex street in sec - M freestream Mach number - S₁ Strouhal number based on projection of the model onto the plane perpendicular to the freestream direction - S₂ Strouhal number calculated from the wake neck width d₂ for M>1 - R Reynolds number calculated from d - R_* critical Reynolds number - model apex angle - angle of attack - L length in flow direction in mm The author wishes to thank G. I. Petrov for his interest in the study and his advice.     

Numerical investigation of laminar separation in the case of supersonic flow of viscous gas past spiked bodies

The complete Navier-Stokes equations for a compressible viscous perfect heat conducting gas have been used in a numerical investigation of laminar separation in the case of supersymmetric axisymmetric flow past cylinders with a conical nose and a spike at the front of finite thickness. The flow structure has been studied in its dependence on the length of the spike and the half-angle of the conical tip. For the considered free-stream parameters (2 M 6, 100 Re 500) and spike lengths, which do not exceed the diameter of the cylinder, the existence of steady flow regimes has been established and it has been shown that the spike in front of the body reduces its total drag and the heat flux to its surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–131, March–April, 1984.     

A turbulence model for the heat transfer near stagnation point of a circular cylinder

A one-equation low-Reynolds number turbulence model has been applied successfully to the flow and heat transfer over a circular cylinder in turbulent cross flow. The turbulence length-scale was found to be equal 3.7y up to a distance 0.05 and then constant equal to 0.185 up to the edge of the boundary layer (wherey is the distance from the surface and is the boundary layer thickness).The model predictions for heat transfer coefficient, skin friction factor, velocity and kinetic energy profiles were in good agreement with the data. The model was applied for Re 250,000 and Tu0.07.Nomenclature µ,C _D Constants in the turbulence kinetic energy equation - C ₁,C ₂ Constants in the turbulence length-scale equation - Skin friction coefficient atx - D Cylinder diameter - F Dimensionless flow streamwise velocityu/u _e - k Turbulence kinetic energy =1/2 the sum of the squared three fluctuating velocities - K Dimensionless turbulence kinetic energyk/u _e ^/2 - I Dimensionless temperature (T–T _w )/(T –T _w ) - l Turbulence length-scale - l _e Turbulence length-scale at outer region - Nu _D Nusselt number - p Pressure - Pr Prandtl number - Pr _t Turbulent Prandtl number - Pr _k Constant in the turbulence kinetic energy equation - R Cylinder radius - Re _D Reynolds number u D/µ - Re _x Reynolds number u x/µ - R _K Reynolds number of turbulence - T Mean temperature - T Mean temperature at ambient - T _s Mean temperature at surface - Tu Cross flow turbulence intensity, - u Mean flow streamwise velocity - u Fluctuating streamwise velocity - u _e Mean flow velocity at far field distance - u Mean flow velocity at ambient - u* Friction velocity - v Mean velocity normal to surface - V Dimensionless mean velocity normal to surface - x,x ₁ Distance along the surface - y Distance normal to surface - Dimensionless pressure gradient parameter - Boundary layer thickness atu=0.9995u _e - Transformed coordinate iny direction - Fluid molecular viscosity - _t Turbulent viscosity - _eff + _t - µ Fluid molecular viscosity at ambient - Kinematic viscosity/ - Density - Density at ambient - _w Wall shear stress - _w,0 Wall shear stress at zero free stream turbulence     

Flow of a viscous gas in a shock layer with equilibrium chemical reactions

Steady flow of supersonic air over a sphere is examined, allowing for viscosity, heat conduction, and actual physical and chemical processes. Flow in the shock layer at flight speeds in the range 3 km/sec V10 km/sec (10⁴R10⁶) is investigated, under the assumption of local thermodynamic equilibrium. The flow is described by simplified Navier-Stokes equations, which are solved by a finite difference method. The case of a cooled surface is examined. The distribution of gasdynamic parameters is obtained in different flow regimes. The distribution of heat flux and friction coefficient is investigated as a function of the oncoming-stream parameters and the sphere radius. The shape and position of the shock wave are determined, and the stream lines and sonic lines are constructed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 150–153, July–August, 1970.The authors thank Yu. P. Lun'kin and F. D. Popov for their help in formulating the problem and their constant interest.     

On the Asymptotic Properties of Solutions of Linear Functional Differential Equations with Constant Coefficients and Linearly Transformed Argument

We establish new properties of C ¹(0, +)-solutions of the linear functional differential equation in the neighborhood of the singular point t = +.     

Flow visualization of compressible vortex structures using density gradient techniques

Mathematical results are derived for the schlieren and shadowgraph contrast variation due to the refraction of light rays passing through two-dimensional compressible vortices with viscous cores. Both standard and small-disturbance solutions are obtained. It is shown that schlieren and shadowgraph produce substantially different contrast profiles. Further, the shadowgraph contrast variation is shown to be very sensitive to the vortex velocity profile and is also dependent on the location of the peak peripheral velocity (viscous core radius). The computed results are compared to actual contrast measurements made for rotor tip vortices using the shadowgraph flow visualization technique. The work helps to clarify the relationships between the observed contrast and the structure of vortical structures in density gradient based flow visualization experiments.Nomenclature a Unobstructed height of schlieren light source in cutoff plane, m - c Blade chord, m - f Focal length of schlieren focusing mirror, m - C _T Rotor thrust coefficient, T/( ² R ⁴) - I Image screen illumination, Lm/m ² - l Distance from vortex to shadowgraph screen, m - n _b Number of blades - p Pressure,N/m ² - p Ambient pressure, N/m ² - r, , z Cylindrical coordinate system - r _c Vortex core radius, m - Non-dimensional radial coordinate, (r/r _c ) - R Rotor radius, m - Tangential velocity, m/s - Specific heat ratio of air - Circulation (strength of vortex), m ²/s - Non-dimensional quantity, ² 8²p r _c ² - Refractive index of fluid medium - ₀ Refractive index of fluid medium at reference conditions - Gladstone-Dale constant, m ³/kg - Density, kg/m ³ - Density at ambient conditions, kg/m ³ - Non-dimensional density, (/ ) - Rotor solidity, (n _b c/ R) - Rotor rotational frequency, rad/s     

Jet methods of gas injection into fluid boundary layer for drag reduction

Two jet methods for saturating the fluid boundary layer with microbubbles for drag reduction in contrust with gas injection through porous materials are considered. The first method is the gas injection through the slot under a special fluid wall jet. The second method is the saturation of boundary layer by microbubbles via the gas-water mixture injection through the slot. Experimental data, reflecting the skin friction drag reduction on the flat plate and total drag reduction of axisymmetric bodies, are presented. The comparison between a jet methods of gas injection and gas injection through porous materials is made.Nomenclature v free-stream velocity - v _j mean velocity of a water through slot - v _g mean velocity of a gas through slot - h width of slot for realizing water jet - h ₁ width of slot for gas injection - incidence angle - Q volume airflow rate - C _Q airflow rate coefficient (v _g/v ) - C _f skin friction coefficient - v _j/v - C _f0 C _f ifQ=0 andv _j=0 - _f C _f/C _{f 0} - d diameter of an axisymmetric body - L length of body - C _Q 4 · ·Q/d ² v - C _D 4 ·D/1/2v ² ·d ² - C _Q 4 ·Q/d ² v - Q _j volume flow rate of water jet - C 8 ·Q _jv_j/d ² v ² - 1 fluid density of main flow - 2 fluid density of wall jet - B ₁ main stream total pressure - B ₂ wall jet total pressure - v ₁ main stream velocity - Be (B ₂ –B ₁)/1/2₁ v ₁ ² = Bernoulli number - ₂ v ₂/₁ v ₁ - p _st static pressure - p _at atmospheric pressure - p _st/p _at - D hydrodynamic drag of body     

Hyperbolic phenomena in a strongly degenerate parabolic equation

We consider the equation u _t =((u) (u _x )) _x , where >0 and where is a strictly increasing function with lim _s = <. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u _t = ((u)) _x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.     

Asymptotic behavior of solutions of nonlinear elliptic equations

We study and obtain formulas for the asymptotic behavior as ¦x¦ of C ² solutions of the semilinear equation u=f(x, u), x (*) where is the complement of some ball in ⁿ and f is continuous and nonlinear in u. If, for large x, f is nearly radially symmetric in x, we give conditions under which each positive solution of (*) is asymptotic, as ¦x¦, to some radially symmetric function. Our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign. Examples when f has power-like growth or exponential growth in the variables x and u usefully illustrate our results.     

Free convection heat transfer from an isothermal plate with arbitrary inclination

This paper presents a new formulation for the laminar free convection from an arbitrarily inclined isothermal plate to fluids of any Prandtl number between 0.001 and infinity. A novel inclination parameter is proposed such that all cases of the horizontal, inclined and vertical plates can be described by a single set of transformed equations. Moreover, the self-similar equations for the limiting cases of the horizontal and vertical plates are recovered from the transformed equations by setting=0 and=1, respectively. Heated upward-facing plates with positive and negative inclination angles are investigated. A very accurate correlation equation of the local Nusselt number is developed for arbitrary inclination angle and for 0.001 Pr .

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Wärmeübertragung bei freier Konvektion an einer isothermen Platte mit beliebiger Neigung

Zusammenfasssung Diese Untersuchung stellt eine neue Formulierung der laminaren freien Konvektion von Flüssigkeiten mit einer Prandtl-Zahl zwischen 0,001 und unendlich an einer beliebig schräggestellten isothermen Platte dar. Ein neuer Neigungsparameter wird eingeführt, so daß alle Fälle der horizontalen, geneigten oder vertikalen Platte von einem einzigen Satz transformierter Gleichungen beschrieben werden können. Die unabhängigen Gleichungen für die beiden Fälle der horizontalen and vertikalen Platte wurden für=0 und=1 aus den transformierten Gleichungen wieder abgeleitet. Es wurden erwärmte aufwärtsgerichtete Platten mit positiven und negativen Neigungswinkeln untersucht. Eine sehr genaue Gleichung wurde für die lokale Nusselt-Zahl bei beliebigen Neigungswinkeln und für 0,001 Pr entwickelt.

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Nomenclature C _p specific heat - f reduced stream function - g gravitational acceleration - Gr local Grashof number,g(T _w –T _w ) x³/v² - h local heat transfer coefficient - k thermal conductivity - n constant exponent - Nu local Nusselt number,hx/k - p pressure - Pr Prandtl number, v/ - Ra local Rayleigh number,g(T _w –T )J x³/v - T fluid temperature - T _w wall temperature - T temperature of ambient fluid - u velocity component in x-direction - v velocity component in y-direction - x coordinate parallel to the plate - y coordinate normal to the plate Greek symbols thermal diffusivity - thermal expansion coefficient - (Ra¦sin¦)^1/4/( Ra cos()^1/5 - pseudo-similarity variable, (y/) - dimensionless temperature, (T–T )/(T _w–T ) - ( Ra cos)^1/5+(Rasin)^1/4 - v kinematic viscosity - 1/[1 +(Ra cos)^1/5/( Ra¦sin)^1/4] - density of fluid - Pr/(1+Pr) - _w wall shear stress - angle of plate inclination measured from the horizontal - stream function - dimensionless dynamic pressure     

Investigation of supersonic viscous flow past a sphere in the presence of subsonic and sonic injection

Supersonic flow past a sphere with a given rate of gas injection along the generator is investigated numerically on the range Re=10²–10⁴. Calculations have been made on the interval 0 90°, where is the angle between the axis of symmetry and the normal to the surface. It is shown that for high subsonic and sonic injection rates it is possible to observe qualitatively new features in the flow structure and in the distribution of the local supersonic flow characteristics around the perimeter of the sphere not previously noted in [9]. In the case of sonic injection the changes in flow structure occur only in the supersonic zone. In the neighborhood of the transition from a subsonic to sonic injection velocity the heat flux has a local maximum, which in absolute value does not exceed the heat flux in the absence of injection. It is shown that there may be qualitative differences in the pressure distribution over the surface of the body with increase in the injection parameter depending on the distribution and value of the injected gas flow rate and, moreover, the number Re.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 83–89, January–February, 1988.     

Existence and regularity for solutions of the Korteweg-de Vries equation

The construction suggested by an inverse-scattering analysis establishes the existence of solutions u(x, t) of the Korteweg-de Vries equation subject to an initial condition u(x, 0)=U(x), where U has certain regularity and decay properties. It is assumed that UC³(), that U is piecewise of class C ⁴, and that U ^(j) decays at an algebraic rate for j4. The faster the decay of U ^(j) the smoother the solution will be for t0. If U and its first four derivatives decay faster than ¦x¦^–n for all n, then the solution will be infinitely differentiable for t0. For t>0, the decay rate of u(x, t) as x + increases with the decay rate of U; but the decay rate as x - depends on the regularity of U. A solution u ₁ of the Korteweg-de Vries equation such that u ₁(·, 0)C() may fail to remain in class C for all time if u ₁(x, 0) does not decay fast enough as ¦x¦.This research was performed in part as a Visiting Member of the Courant Institute of Mathematical Science.     

Stress analysis of plates with a circular hole reinforced by flange reinforcing member

This paper deals with the problem of stress analysis of plates with a circular hole reinforced by flange reinforcing member. The so called flange reinforcing member here means that the reinforcing member is built up by setting shapes or bars with any section shape on both sides of the plates along the edge of the hole. Two cases of external loads are considered. In one case the external loads are stressesσ_X^(∞),σ_Y^(∞),and τ_XY^(∞) acting at infinite point of the plate, and in the other the external loads are linear distributed normal stresses. The procedure of solving the problems mentioned above consists of three steps. Firstly, the reinforcing member is taken out from the plates and considered to be a circular bar being solved to determine its deformation under the action of radial force q₀(θ) and tangential force t₀(θ) which are forces acting upon each other between reinforcing member and plate. Secondly, the displacements of plate with a circular hole under the action of q₀(θ) and t₀(θ) and external loads are determined. Finally, forces q₀(θ) and t₀(θ) are obtained by the compatibility of deformations between reinforcing member and plate. Then the internal forces and displacements of reinforcing member and plate are deduced from q₀(θ) and t₀(θ) obtained.