Supersonic two-phase flow around a wedge

Supersonic two-phase flow around bodies is encountered in calculating the flow around the last stages of blades of condensing turbines, in studying the motion of airplanes under cloudy conditions, etc. In the latter case, there is, along with erosion of the forward edges of the wing profiles, a change in the wave structure and interference situation in the flow about the airplane, leading to off-design regimes of motion. Supersonic flow of a two-phase mixture around a wedge, without taking account of the influence of the particles on the flow, was investigated in [1–3]. In [4], also in this kind of simplified setting, a study was made of the interaction of particles with the surface of a wedge in which reflection of the particles from the wall was taken into account. Morganthaler [5] made an experimental study of the flow of a mixture of air and aluminum oxide particles around a wedge. In [6] a theoretical study was made of a supersonic two-phase flow around thin flat axially-symmetric bodies. In particular, for the flow around a wedge, closed form solutions were obtained for the form of the shock wave, the gas streamlines and particle paths, and the distribution of all the parameters along the surface of the wedge. On the basis of the equations given in [7] and the method of characteristics, which were developed for flows consisting of a mixture of a gas and heterogeneous particles in nozzles [8,9], we present below a study of a supersonic two-phase flow around a wedge.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 83–88, March–April, 1972.     

Asymptotic of the flow upon shock incidence on a wedge cavity

The asymptotic of the motion originating because of shock incidence on a wedge cavity in a metal is investigated as the wave amplitude tends to zero. It has been shown in [1] that the flow is hence divided into two domains. The principal term governing the flow in the first domain agrees with the acoustic approximation. The flow in the second domain is described by incompressible fluid equations in the principal term. Determination of the flow in the second domain is reduced herein to the solution of a singular nonlinear integral equation. A numerical solution is found for a series of values of the cavity aperture.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 129–138, May–June, 1972.     

A scheme of developed cavitation flow past a wedge

A scheme is proposed for studying the symmetric steady flow with developed cavitation of an ideal incompressible and weightless fluid past a wedge in a channel. This scheme generalizes the existing schemes of cavitation flow past bodies and describes flow that is intermediate between the flow past obstacles in the Riabouchinsky and Efros schemes. Detailed calculations have been made for the drag coefficient and the cavity size in the case of a plate in an unbounded flow. The results of the calculations are tabulated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 165–170, January–February, 1983.I thank P. M. Ivanov, Kh. Kh. Kalazhokov, and T. Sokhov for assistance in the computer calculations.     

Boundary-layer flow of Reiner–Philippoff fluids past a stretching wedge

This paper presents a numerical analysis of the steady boundary-layer flow of a Reiner–Philippoff fluid induced by a 90° stretching wedge in a variable free stream. The governing partial differential equations are converted into a set of two ordinary differential equations by the use of a similarity transformation. The flow is therefore governed by a stretching velocity parameter λ and two non-Newtonian fluid parameters γ and μ₀. The variation of the skin friction, as well as other flow characteristics, as a function of the governing parameters is presented graphically and tabulated. A stability analysis has also been performed for this self-similar flow based on linear disturbances to the steady similarity solutions. The results presented in this paper reveal that there are no multiple (dual) solutions for the present problem and the unique solution is stable.     

Drag of a flat wedge in a two-phase flow

The drag of a flat wedge in a subsonic two-phase flow is investigated. In contrast to earlier work of Balanin and Zlobin [1] particular attention is devoted to the influence of the particle size. Detailed investigations are made of the dependences of the forces and aerodynamic coefficients on the concentration of the solid phase, the opening angle of the wedge, and the particle size. It is established that the drag coefficients depend on the particle size only for particles with diameters less than 30 um.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 177–180, March–April, 1982.     

Supersonic flow past a blunt wedge

A study is made of the asymptotic solution of the problem of flow past a blunt wedge by a uniform supersonic stream of perfect gas. By separation of variables it is shown that at large distances the disturbance of the flow is damped exponentially. In the case of subsonic flow behind the shock wave the exponent of the leading correction term in the expansion of the shock front is calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1984.     

Dispersed phase motion in laminar boundary layer flow over a wedge

The motion of a dispersed phase in the laminar boundary layer on a wedge is considered with allowance for the effect of not only the Stokes force, which coincides in direction with the flow velocity, but also the transverse force (Saffman force) resulting from the transverse nonuniforrnity of the flow over the individual particle [1–3].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 34–42, November–December, 1993.In conclusion, the authors wishes to thank S. V. Manuilovich for assisting with the numerical calculations.     

Asymmetric flow of subsonic and sonic jets over an infinite wedge

A study is made of the flow of subsonic or sonic jets over an infinite wedge when the stagnation streamline bifurcates at the tip of the wedge. This regime can be realized only for a definite (previously unknown) relationship between the geometrical parameters. The problem is solved in the hodograph plane by the numerical method of [1] developed for the problem of a profiled Laval nozzle. A solution to the asymmetric problem obtained in the hodograph plane can be realized physically only for a definite relationship between the boundary values for the flow function. This relationship (which generalizes Prandtl's well-known formula [2] derived for asymmetric flow of incompressible jets over a plate on the basis of the momentum theorem) is obtained by analyzing the asymptotic behavior of the solution near the stagnation point. Examples of calculations are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 137–141, March–April, 1980.     

Arbitrary body,close to a wedge,in a supersonic flow

The problem of supersonic flow around bodies close to a wedge was first discussed in the two-dimensional case in [1]. The shock wave was assumed to be attached, and the flow behind it to be supersonic; taking this into account, the angle of the wedge was assumed to be arbitrary. The surface of the body was also arbitrary, provided that it was close to the surface of the wedge. In solution of the three-dimensional problem, there was first considered flow around two supporting surfaces with only slightly different angles of attack [2], and then around a delta wing [3, 4]. In all these articles, the Lighthill method was used to solve the Hilbert boundary-value problem [5, 6]. A whole class of surfaces of bodies with arbitrary edges, under the assumption that the surface of the body was cylindrical, with generatrices directed along the flow lines of the unperturbed flow behind an oblique shock wave, was discussed in [7]. In the present work, the problem is regarded for a broad class of surfaces of bodies, using a new method which generalizes the results of [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 109–117, July–August, 1974.The author thanks G. G. Chernyi for his direction of the work.     

On the continuous squeezing flow in a wedge

The paper is concerned with the continuous squeezing flow of Oldroyd-type fluids in a two-dimensional wedge. The flow mimics the lubrication action in a squeezing flow and is important in that there exists a similarity solution for any simple fluid. We are only concerned with Oldroyd-type fluids, however. It is shown by using a parameter continuation method that the Oldroyd-B model has a limiting Weissenberg number. The Phan Thien/Tanner model does not have this limiting Weissenberg number.     

Instability of a hypersonic flow over a wedge in the Newtonian approximation

At the present time, there are a number of works in the literature that treat unsteady hypersonic flows in the Newtonian approximation [1–4]. Since the angle of incidence of the shock wave _s coincides in the zero-order approximation with the angle of inclination of the bodys [1], the latter is usually used in the boundary conditions on the shock. However, in the zero-order approximation _b can be used with the same justification. Both approaches are equally justified and give similar results for a steady flow. For unsteady flows the results can differ radically. It will be shown below that for an investigation of a flow over a fixed wedge with constant conditions in the free stream a steady-state pattern is obtained in the first case and a solution growing in time, in the second case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 158–160, July–August, 1976.     

Free molecule flow over a nonconvex wedge at a finite mach number

We consider the problem of the effect of multiple molecular reflection on the aerodynamic characteristics of a nonconvex wedge with transverse free molecule flow. The molecular reflection from the body surface is assumed diffuse with partial thermal accommodation [1–7]. For these reflection conditions the aerodynamic characteristics are calculated and are then compared with the characteristics obtained using simpler assumptions.     

Ideal incompressible flow around a wedge tip

A planar analog of conical flows is considered: an inviscid incompressible fluid flow around a wedge tip. A class of conical flows is found where vorticity is transported along streamlines by the potential component of velocity. Problems of a wave “locked” in the corner region and of a flow accelerating along the rib of a dihedral angle are considered. By analogy with an axisymmetric quasi-conical flow, a planar quasi-conical flow of the fluid is determined, namely, the flow inside and outside the region bounded by tangent curves described by a power law. Conditions are found where vorticity and swirl produce a significant effect. An approximate solution of the problem of the fluid flow inside a “zero” angle is obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 57–65, March–April, 2007.     

Viscoelastic sink flow in a wedge for the UCM and Oldroyd-B models

The steady planar sink flow through wedges of angle π/α with α≥1/2 of the upper convected Maxwell (UCM) and Oldroyd-B fluids is considered. The local asymptotic structure near the wedge apex is shown to comprise an outer core flow region together with thin elastic boundary layers at the wedge walls. A class of similarity solutions is described for the outer core flow in which the streamlines are straight lines giving stress and velocity singularities of O(r⁻²) and O(r⁻¹), respectively, where r1 is the distance from the wedge apex. These solutions are matched to wall boundary layer equations which recover viscometric behaviour and are subsequently also solved using a similarity solution. The boundary layers are shown to be of thickness O(r²), their size being independent of the wedge angle. The parametric solution of this structure is determined numerically in terms of the volume flux Q and the pressure coefficient p₀, both of which are assumed furnished by the flow away from the wedge apex in the r=O(1) region. The solutions as described are sufficiently general to accommodate a wide variety of external flows from the far-field r=O(1) region. Recirculating regions are implicitly assumed to be absent.     

Surface shape of a magnetic liquid drop near the edge of a magnetic wedge

The two-dimensional problem of the shape of the free surface of a magnetic fluid in a gravity field, a uniform external magnetic field and the nonuniform field of a magnetized metal wedge is considered. The results of numerically calculating the shape of the free surface of a magnetic liquid drop retained on an inclined plane by the field of a magnetizing wedge are presented. The changes in the shape of the free surface of an infinite volume of magnetic liquid near the edge of a wedge with increase in the external field are investigated. It is shown that for a certain critical field some of the magnetic liquid separates and adheres to the edge of the wedge. Experimental data on the determination of the maximum cross-sectional area of a drop retained by the magnetic field of a wedge and the critical rise of the magnetic liquid relative to the level outside the field are presented. The experimental and theoretical results are in agreement.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 113–119, July–August, 1992.The authors wish to thank V. V. Gogosov for useful discussions and his interest in the work.     

非稳态分数阶Oldroyd-B流体绕楔形体流动研究

研究了非稳态分数阶Oldroyd-B流体在多孔介质中通过楔形拉伸板的驻点流动问题。基于分数阶Oldroyd-B流体的本构模型建立了动量方程,并在其中引入了浮升力和驻点流动特征。此外,考虑了具有热松弛延迟时间的修正的分数阶Fourier定律,并将其应用于能量方程和对流换热边界条件。接着,采用与L1算法相结合的有限差分法求解控制偏微分方程。最后,分析了相关物理参数对流动的影响。结果表明,随着楔角参数的增加,流体受到的浮升力增大,导致速度加快;达西数越大,介质的孔隙度变大,流体的流动越快;此外,温度分布先略有上升后明显下降,这表明Oldroyd-B流体具有热延迟特性。     

Non-Newtonian flow over a wedge with suction

A pseudo-similarity solution is obtained for the flow of an incompressible fluid of second grade past a wedge with suction at the surface. The non-linear differential equation is solved using quasi-linearization and orthonormalization. The numerical method developed for this purpose enables computation of the flow characteristics for any values of the parameters K, a and b, where K is the dimensionless normal stress modulus of the fluid, a is related to the wedge angle and b is the suction parameter. A significant effect of suction on the wall shear stress is observed. The present results match exactly those from an earlier perturbation analysis for Kx^2a ? 0·01 but differ significantly as Kx^2a increases.     

Nonlinear analysis of the flow initiated by the sudden motion of a wedge

Certain self-similar problems involving the sudden motion of a wedge which were treated in the linear approximation in [1–3] are studied by the method of matched asymptotic expansions. The nature of the wave boundary of the perturbed region is determined. Second-approximation solutions are constructed which describe flows behind weak shock fronts propagating in a stationary gas and behind fronts of weak discontinuity lines propagating by known uniform flows. A boundary-value problem is formulated whose solution describes, in first approximation, flows in the neighborhoods of points of interaction of the fronts. The existence of similarity rules of flows in these nieghborhoods is estimated. An approximate solution of the problems is given.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 37–47, May–June, 1976.     

Supersonic two-dimensional flow in a forward separation zone

Supersonic separated flow past a spiked body with a plane cap on the end of the spike (a flat plate or a wedge) is studied. The cases of both symmetric and nonsymmetric flow, with a rotating or nonrotating spike, are considered. The flow pattern, visualized by means of a Toepler instrument and a laser knife, and the limiting streamline patterns are analyzed. The reasons for the initiation of self-oscillations in the flow between the cap and the body are determined. The flow patterns for a rotating and nonrotating spike are compared.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 152–162, January–February, 1995.     

Laminar boundary layer on an oscillating wedge

A study is made of the nonstationary laminar boundary layer on a sharp wedge over which a compressible perfect gas flows; the wedge executes slow harmonic oscillations about its front point. It is assumed that the perturbations due to the oscillations are small, and the problem is solved in the linear approximation. It is also assumed that the thickness of the boundary layer is small compared with the thickness of the complete perturbed region. Then in a first approximation the influence of the boundary layer on the exterior inviscid flow can be ignored, and the parameters on the outer boundary of the boundary layer can be taken equal to their values on the body for the case of inviscid flow over the wedge. They are determined from the solution to the inviscid problem that is exact in the framework of the linear formulation. The wall is assumed to be isothermal. The dependence of the viscosity on the temperature is linear. Under these assumptions, the problem of calculating the nonstationary perturbations in the boundary layer on the wedge is a self-similar problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 146–151, July–August, 1980.