Starting of an axisymmetric convergent-divergent nozzle in a hypersonic flow

The starting of an axisymmetric convergent-divergent nozzle, with the result that supersonic flow is formed within almost the entire channel, is modeled, as applied to the hypersonic aerodynamic setup of the Institute of Mechanics of Moscow State University. A successful starting is realized when the nozzle is thrown in a uniform supersonic air flow at a fairly high Mach number. The steady flow structure is studied. It is numerically shown that in the convergent section of the channel there arises an oblique shock wave whose interaction with the nozzle axis leads to the formation of a reflected shock and a curvilinear Mach disk with a region of unsteady subsonic flow in the vicinity of the throat. The mathematical model is based on the two-dimensional Euler equations for axisymmetric gas flows.     

Axisymmetric-conical and locally conical flows without swirling

Axisymmetric steady conical and locally conical non-swirled flows of an ideal (inviscid and non-heat-conducting) gas are considered. Like two-dimensional conical flows, the examined onedimensional (axisymmetric) flows can be conically subsonic and supersonic. If the uniform flow is not considered as a conical flow, then the type of one-dimensional conical flows can change only on the shock wave, except for the junction of two one-dimensional conical flows of different types on the C ⁺ characteristic. C ^± characteristics and streamlines are constructed for a number of locally conical flows and some already known and new conical flows.     

Numerical solution of viscous flows using integral equation methods

A formulation of the boundary element method for the solution of non-zero Reynolds number incompressible flows in which the non-linear terms are lumped together to form a forcing function is presented. Solutions can be obtained at low to moderate Reynolds numbers. The method was tested using the flow of a fluid in a two-dimensional converging channel (Hamel flow) for which an exact solution is available. An axisymmetric formulation is demonstrated by examining the drag experienced by a sphere held stationary in uniform flow. Performance of the method was satisfactory. New results for an axisymmetric free jet at zero Reynolds number obtained using the boundary element method are also included. The method is ideal for this type of free-surface problem.     

Influence of a Longitudinal Magnetic Field on the Hall Effect in the Plasma Accelerator Channel

An analytic model of steady-state two-dimensional flows in coaxial plasma-accelerator channels in the presence of a longitudinal magnetic field is proposed. The solution of the problem is found in the smooth channel approximation for the MHD equations of an ideal two-component plasma. An example of the developing axisymmetric flows is given and the features of the plasma-dynamic processes are investigated. It is found that the Hall effect and the anode flow zone can be reduced using a longitudinal field and plasma rotation.     

Method of approximate account for the wall effect in cavitation flow around bodies in water tunnels

One of the basic questions in the study of advanced cavitation in water tunnels of the closed-circuit type is the establishment of the correspondence between the flow patterns observed in the channel and in an unbounded stream. The objective of the study of the wall effect must be the determination of a connection between the basic characteristics of the phenomenon, i. e., the cavitation numbers, the cavity dimensions, the drag coefficients, etc., for the unbounded flow and the channel flow. A large number of works devoted to this question are known [1–7], but in the majority of them only two-dimensional flows are considered. These studies contain either exact solutions obtained with the aid of the apparatus of functions of a complex variable or solutions in the linearized formulation.At the present time there is urgent need to obtain at least approximate solutions for axisymmetric cavitation flows in a tunnel.In several studies [1, 2, 4] it has been shown that in the case of two-dimensional flows the presence of solid boundaries influences the drag coefficient only through the mechanism of a change of the magnitude of the cavitation number, while the variation of the drag coefficient itself with the cavitation number is not changed in comparison with the unbounded flow. It may be assumed that an analogous situation obtains for the axisymmetric case as well. Then the question of the wall effect may be reduced to establishing the connection between the corresponding cavitation numbers.The present paper makes an attempt to establish the correspondence between the cavitation numbers in the unbounded flow and in the tunnel for which the cavities behind the same body have the same areas of the maximal cross section.     

NUMERICAL SIMULATION OF STEADY FLOW IN A COMPLIANT TUBE OR CHANNEL WITH TAPERED WALL THICKNESS

Flow through compliant tubes with linear taper in wall thickness is numerically simulated by finite element analysis. Two models are examined: a compliant channel and an axisymmetric tube. For verification of the numerical method, flow through a compliant stenotic vessel is simulated and compared to existing experimental data. Steady two-dimensional flow in a collapsible channel with initial tension is also simulated and the results are compared with numerical solutions from the literature. Computational results for an axisymmetric tube show that as cross-sectional area falls with a reduction in downstream pressure, flow rate increases and reaches a maximum when the speed index (mean velocity divided by wave speed) is near unity at the point of minimum cross-sectional area, indicative of wave-speed flow limitation or “choking” (flow speed equals wave speed) in previous one-dimensional studies. For further reductions in downstream pressure, the flow rate decreases. Cross-sectional narrowing is significant but localized. For the particular wall and fluid properties used in these simulations, the area throat is located near the downstream end when the ratio of downstream-to-upstream wall thickness is 2; as wall taper is increased to 3, the constriction moves to the upstream end of the tube. In the planar two-dimensional channel, area reduction and flow limitation are also observed when outlet pressure is decreased. In contrast to the axisymmetric case, however, the elastic wall in the two-dimensional channel forms a smooth concave surface with the area throat located near the mid-point of the elastic wall. Though flow rate reaches a maximum and then falls, the flow does not appear to be choked.     

Linear stability of two-dimensional steady flow in wavy-walled channels

Linear stability of fully developed two-dimensional periodic steady flows in sinusoidal wavy-walled channels is investigated numerically. Two types of channels are considered: the geometry of wavy walls is identical and the location of the crest of the lower and upper walls coincides (symmetric channel) or the crest of the lower wall corresponds to the furrow of the upper wall (sinuous channel). It is found that the critical Reynolds number is substantially lower than that for plane channel flow and that when the non-dimensionalized wall variation amplitude is smaller than a critical value (about 0.26 for symmetric channel, 0.28 for sinuous channel), critical modes are three-dimensional stationary and for larger , two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of sinuous channel flows are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point, while the critical two-dimensional mode resembles closely the Tollmien–Schlichting wave for plane Poiseuille flow.     

Nozzle startup in an oncoming flow

Three variants of the startup of an axisymmetric convergent-divergent nozzle are considered with the static pressures at the entry and exit of the nozzle being the same at the beginning of the process. The subsonic startup corresponds to open nozzle acceleration in air. The supersonic startup simulates the sudden opening of a cover at the nozzle inlet under supersonic flight conditions. A successful nozzle startup with the formation of steady supersonic flow along the whole channel is realized in the third variant of supersonic startup with gas injection through a small region of the wall of the divergent nozzle section. The investigation is performed numerically, on the basis of the Euler equations for axisymmetric gas flows.     

Investigation of the influence of the conditions of swirling on the structure of a two-phase flow in an expanding channel

Both experimental and theoretical investigations show that the main feature in the structure of sufficiently strongly swirling gas flows is the presence in then of reverse circulation regions whose configuration depends very strongly both on the law of swirling of the flow and the conditions at the entrance as well as on the channel geometry [1–6]. In expanding channels, the occurrence of such regions is most probable in the axial region [7, 8]. In short annular channels for which the characteristic transverse and longitudinal dimensions are of the same order, reverse flow arises in the exit part of the channel along its inner wall [6, 9]. Hitherto, the investigations have been made for single-phase gas flows. The present paper reports a numerical investigation of the influence of particles of a condensed phase on the intensity of the reverse flow and the structure of the gas flow in an annular expanding channel under conditions of thermal, mass, and mechanical interaction of the phases. The method of stabilization was used to solve the boundary-value problem. The system of equations describing the axisymmetric unsteady flow of the two-phase medium was integrated by means of Godunov's difference scheme [10, 11]. The calculations were made for different conditions of injection of the particles of the condensed phase into the channel.     

Nonlinear stability of two-dimensional steady flows of an ideal fluid with constant vorticity and their axisymmetric analogs

Flows with constant vorticity are widely used as local models of more complicated flows [1]. In many cases, such flows are stable against finite two-dimensional perturbations. In particular, the inviscid plane-parallel Couette flow has the property of nonlinear stability. Similar treatment of a class of axisymmetric flows yields nonlinear stability of a spherical Hill vortex and inviscid Poiseuille flow in a circular tube with respect to axisymmetric perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 16–21, October–December, 1981.     

Rarefied hypersonic flow in a plane channel with a surface glow discharge

The gasdynamic structure of a hypersonic molecular nitrogen flow in a plane channel whose opposite surfaces are segmented electrodes for generating a continuous surface glow discharge is investigated using a two-dimensional computational model. The electrodynamic structure of the surface glow discharge in the hypersonic rarefied gas flow (distributions of the charged particle concentrations, current density, and electric potential) is studied. A two-dimensional conjugate electrical-gasdynamic model consisting of the continuity, Navier-Stokes, and energy conservation equations and the chargedparticle continuity equations in the ambipolar approximation is proposed. The real thermophysical and transport properties of molecular nitrogen are taken into account. It is shown that using a surface glow discharge in a hypersonic rarefied gas flow makes it possible effectively to modify the shock-wave flow structure and hence to consider this type of discharge as additional tool for controlling rarefied gas flows.     

Calculations of two-dimensional nonstationary flows of a viscous gas blown into a short cylindrical channel

An investigation was made of the nonstationary problem of laminar axisymmetric flow of a viscous gas in a short cylindrical channel at one end of which the gas is blown in at an increasing rate with parabolic flow-rate profile; the other end is open. The problem was solved numerically on the basis of the complete two-dimensional system of Navier-Stokes equations for an ideal gas at the characteristic numbers M = 0.2, Pr = 1, Re = 20, 200, 2000. An investigation was made into the nonlinear process of pulsed excitation of longitudinal acoustic oscillations, which for Re = 2 000 give rise to separation flows. In this case, secondary oscillatory phenomena are observed in the separated viscous shear layers. The damping of the oscillations is treated on the basis of the proposed nonlinear model of the sound absorbing end wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. I, pp. 9–17, January–February, 1979.I thank G. I. Petrov for his interest in the work and valuable comments.     

Similarity of three-dimensional and axisymmetric chemically-nonequilibrium flows in the neighborhood of a plane of symmetry

Three-dimensional chemically-nonequilibrium flow past blunt bodies in the neighborhood of the plane of symmetry is investigated within the framework of viscous shock layer theory. The similarity of three-dimensional and axisymmetric flows, previously established in [1] for a uniform gas, is extended to chemically-nonequilibrium gas flows. It is shown that the problem of determining the heat fluxes and friction stress in the neighborhood of the line of flow divergence can be reduced to the problem of determining these quantities for the axisymmetric body. The validity of the axisymmetric analogy is verified by carrying out numerical calculations for bodies of different shapes re-entering the earth's atmosphere along a gliding trajectory. Various models of surface catalytic activity are considered. The use of similarity relations makes it possible to apply existing programs for calculating axisymmetric flows to the solution of three-dimensional problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 115–120, March–April, 1990.     

Self-similar turbulent compressible gas flows in a channel

In this study we establish for turbulent compressible gas flow (to within a constant factor) the laws governing the variation of the height (radius) and the static pressure along the length of a planar or axisymmetric channel for which the longitudinal velocity component and gas temperature are functions only of the transverse dimensionless coordinate. In such channels the gas density decrease due to friction is compensated by the increase of the cross-sectional area so that the velocity and temperature profiles remain unchanged at all sections of the channel.The results obtained are a generalization to the gas case of the known laws governing the turbulent flow of an incompressible fluid in a cylindrical channel.The author wished to thank A. P. Byrkin for helpful discussions.     

Calculation of laminar flow of compressible gas in the presence of heat transfer in two-dimensional curvilinear channels

Numerical solution of the complete system of Navier-Stokes equations is used to investigate laminar (Re ? 1000) subsonic flows of a compressible gas in the presence of heat transfer (cooled walls) in two-dimensional channels containing a bend section (for different curvature parameters). The appearance of closed separation regions of the flow on the channel walls, their deformation as the parameters of the problem are changed, and the loss of pressure are studied. The sections of the channel walls with maximal and minimal heat fluxes are determined, and the connection between these sections and the separation regions is elucidated.     

Asymptotic analysis of transonic flows

In this paper we derive the equations of the second and third approximations for the stream function of two-dimensional and axisymmetric potential transonic flow of an inviscid gas and find their particular solutions corresponding to certain transonic flows.A similar study concerning the second approximation of subsonic and supersonic flow was made by Van Dyke [1] and Hayes [2]. The second approximation for the velocity potential of transonic flow has been examined in detail by Hayes [3]. Euvrard [4, 5] has investigated the asymptotic behavior of transonic flow far from a body, while Fal'kovich, Chernov, and Gorskii [6] have studied the flow in a nozzle throat.The transonic asymptotic analysis for the stream function is presented in this paper.     

Two-dimensional subcritical and supercritical open channel flow calculation using a time-marching method

A time-marching method is presented for the calculation of two-dimensional, high-speed channel flow, including the usually neglected terms of slope and bottom friction. Time-marching methods are potentially the most flexible means of calculating flow through geometrically complex channel passages, since they can readily deal with subcritical and supercritical flows. The adopted numerical scheme comes straight from gas flow computations in turbomachines. The flow is assumed to be fully mixed in the vertical direction, so that vertical variations may be neglected. Comparisons with other numerical solutions for various open channel configurations show that the proposed approach is a comparatively accurate, reliable and fast technique. It can be utilized for open channel designs.     

Experimental and Numerical Investigation of Flows in Complex Shaped Axisymmetric Channels with Mass Injection

The results of an experimental and numerical investigation of the flow in an axisymmetric channel with radial cavities are presented. The experiments were performed with air in a wind tunnel. The theoretical results obtained by numerically solving the problem of viscous axisymmetric gas flow agree with the experimental data on the longitudinal velocity profiles in a channel with a radial cavity.     

Algorithm for transient growth of perturbations in channel Poiseuille flow

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the OrrSommerfeld and Squire(OSS) equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a reorthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the Krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes(N-S) equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is proposed. The algorithm is then applied to a one-dimensional base flow(the plane Poiseuille flow) and a two-dimensional base flow(the plane Poiseuille flow with a low-speed streak)in a channel. For one-dimensional cases, the effects of the spanwise width of the channel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is characterized by high-and low-speed streaks and the corresponding streamwise vortical structures.The lift-up mechanism that induces the transient growth of perturbations is discussed.The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one-and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the crosscheck method.     

Stagnation flow towards a shrinking sheet

The stagnation flow towards a shrinking sheet is studied. A similarity transform reduces the Navier-Stokes equations to a set of non-linear ordinary differential equations which are then integrated numerically. Both two-dimensional and axisymmetric stagnation flows are considered. It is found that solutions do not exist for larger shrinking rates and may be non-unique in the two-dimensional case. The non-alignment of the stagnation flow and the shrinking sheet complicates the flow structure. Convective heat transfer decreases with the shrinking rate due to an increase in boundary layer thickness.