Boundary layer development in an MHD duct

This study presents a method of calculation for two-dimensional, steady-state, laminar flow in the entrance region of an MHD duct. The electrically conducting fluid in the free stream is compressible whereas the medium in the boundary layer itself is taken to be incompressible. Thus, the density is variable in the axial direction of the duct only, and the momentum and energy equations for the boundary layer are uncoupled. These equations are solved using an extended Von Kármán-Pohlhausen method as described by U. P. Hwang for a compressible MHD flow with zero electric field. In this study, however, the electric field is essentially not zero and the MHD duct can work as a generator. The equations of the insulator boundary layer are solved in the assumption that the displacement thickness of the electrode boundary layer equals that of the insulator boundary layer, so the total influence of the varying effective crossection on the free stream is taken into account. In this way a quick method of calculating the MHD flow in the entrance region of a duct is obtained.     

Boundary layer transition on the surface of a delta wing in supersonic flow

Laminar-turbulent transition on the surface of a delta wing has been experimentally investigated in a supersonic wind tunnel at Mach numbers M_t8=3–5. It is shown that when M,=3, Re_L=6.5·10⁶, and =–5.5° much of the upper surface of the wing in the neighborhood of the line of symmetry is occupied by a wedge-shaped region of turbulent flow. In this region the heat fluxes reach the same values as at the heat transfer maxima induced here by separated flows and may exceed the turbulent heat flux level on the windward surface of the wing. Changing the shape of the under surface of the wing from plane to pyramidal leads to acceleration of the boundary layer transition on the under surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 87–92, May–June, 1989.     

Optimal supersonic nozzles with power-law generators

The problem of designing supersonic nozzles that provide maximum exhaust thrust is considered. An analytical solution is derived on the basis of a local analysis of the aerodynamic force distribution. It is shown that near-optimal thrust characteristics are realized by nozzles with a power-law dependence of the radius on the longitudinal coordinate with an exponent 2/3. The study is performed within the framework of the Euler model of an inviscid, non-heat-conducting gas.     

Boundary layer development from transition provoking devices

The results of an experimental investigation of four different, incompressible, transitional boundary layer situations are presented. The experiments were carried out in zero pressure gradient conditions and transition was initiated from two- and from three-dimensional provoking agents.

The measurements of transitional intermittency from two-dimensional tripping agents showed a trend consistent with that reported elsewhere in the literature, with the development of mean and fluctuating component velocity profiles and local skin friction coefficient exhibiting approximate similarity through the transition region.

Disturbance frequency and spread angles for turbulent wedge growth behind isolated roughness elements were similar to those reported by others.

Computer predictions using a transition model based on the present correlations show reasonable agreement with the data.     

Excitation and development of unstable disturbances in a supersonic boundary layer

The initial-boundary value problem of the development of two-dimensional inviscid disturbances excited by an external unsteady local action, turned on at time t=0, is examined. The spectrum of the problem is investigated by means of the WKB method and numerical calculations, and the asymptotic expansions of the wave packets as t are found. It is shown that, contrary to the conclusions of [4], the inviscid instability of the supersonic boundary layer is convective. The reasons for this discrepancy are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 21–29, May–June, 1990.     

Boundary layer development of pulsatile blood flow in a tapered vessel

Assuming that the tapered angle is small,the problems of developing flow under unsteady oscillatory condition are studied in this paper.The formula of velocity distribution is obtained.The analyses for the results show that the blood flow in a converging tapered vessel remains a developing flow throughout the length,and the effects of tapered angle on the developing flow are increased with the increment of the tapered angle.     

Mode switching in a supersonic boundary layer

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 6, pp. 67–72, November–December, 1992.     

Influence of nonparallel flow on the development of disturbances in a supersonic boundary layer

The linear theory is used to solve the problem of the development of two-dimensional disturbances in the boundary layer of compressible fluid. In contrast to the stability theory of plane-parallel flows, the present paper takes into account the presence in the boundary layer of transverse (at right angles to the flow direction) motions, the dependence of the averaged flow parameters on the longitudinal coordinate, and also the deformation of the amplitude distribution profile of the disturbances as a function of the longitudinal coordinate. The calculations are made for Mach number M = 4.5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 26–31, March–April, 1980.     

Instability of a three-dimensional supersonic boundary layer

This study was conducted with financial support from the Russian Fund for Basic Research (Grant No. 94-01-000497).     

Numerical simulation of supersonic reacting mixing layer

In this paper, the supersonic chemically reacting mixing layer is simulated with the third order ENN scheme, based on the Navier-Stokes equations, containing transport equations of all species. The numerical results show that the thickness of mixing layer increases gradually along the flow direction, and that the Kelvin-Helmholtz instabilities may not exist in mixing layer flows. The project supported by the National Natural Science Foundation of China     

Boundary conditions on a shock wave in a supersonic flow

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 49–56, January–February, 1989.     

The development of acoustic generators and their application as a boundary layer transition control device

An experimental investigation of a controllable artificial boundary layer transition by means of electro-acoustic generators was carried out in flat-plate boundary layers. The acoustic generators were flush mounted with the model surface in order to minimize local surface roughnesses which may cause flow instabilities in the laminar boundary layer. The dependence of the input power, pulse ratio, and input frequency of the acoustic generators on the transition threshold values of the input power were determined with surface hot films. In addition, the functional application of the acoustic generators for a controllable artificial boundary layer transition was examined qualitatively by flow visualizations applying the liquid crystal technique.List of symbols A fluctuating component of the hot film anemometer output voltage - time mean hot film anemometer output voltage - ₀ time mean hot film anemometer output voltage at zero velocity - B spectral component of any measured quantity (the used dB-scale is referred to 1 Volt) - time mean hot wire anemometer output voltage - ₀ time mean hot wire anemometer output voltage at zero velocity - f frequency - I _av average input current of the acoustic generator - P mean input power of the acoustic generator - p pressure - Re Reynolds number, Re=U x t/ - t time - t _i period of pulse signal - t _p pulse width of input power - t _s time after switching off heating lamp - U freestream velocity - blowing or sucking velocity produced by the acoustic generator - x longitudinal coordinate from the leading edge (Fig. 2) - x _t distance from the flat plate to the transition location - y coordinate normal to the wall (Fig. 2) - z spanwise coordinate (Fig. 2) - angle of incidence - pulse ratio t _p /t _I - kinematic viscosity - density - ₀ wall shear stress     

Self-similar solutions of non-Newtonian fluid boundary layer in MHD

The self-similar solutions of the boundary layer for a non-Newtonian fluid in MHD were considered in [1, 2] for a power-law velocity distribution along the outer edge of the layer and constant electrical conductivity through the entire flow. However, the MHD flows of many conducting media, which are solutions or molten metals, cannot be described by the MHD equations for non-Newtonian fluids.The self-similar solutions of the boundary layer for a non-Newtonian fluid without account for interaction with the electromagnetic field were studied in [3].In the following we present the self-similar solutions for the boundary layer of pseudoplastic and dilatant fluids with account for the interaction with an electromagnetic field for the case of a power-law velocity distribution along the outer edge of the layer, when the conductivity of the fluid is constant throughout the flow and the magnetic Reynolds number is small.Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 2, No. 6, pp. 77–82, 1967The author wishes to thank S. V. Fal'kovich for his interest in this study.     

Boundary layer interaction in three-dimensional flows

An approach known from the theory of matched asymptotic expansions involving the isolation of subregions with different scales is used to study flows which are assumed to be described by the boundary layer equations almost everywhere near the surface except for a fairly narrow zone in which the inflowing boundary layers interact. Two characteristic types of interaction are identified. An approximate theory describing the flow in the interaction zone, which makes it possible to locate the position of the interaction zone on the surface, is proposed. The interaction flow on the end wall of a vane channel is calculated subject to certain simplifications. The results of an experimental investigation of this flow are presented and it is shown that the theoretical model proposed describes the three-dimensional corner separation which occurs in the neighborhood of the line of intersection of the end wall and the convex edge of the vane.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 116–123, May–June, 1988.     

Theory of laminar boundary layer separation in supersonic flow

A method of calculation is constructed for the Navier-Stokes equations using asymptotic expansions for high Reynolds numbers and matching techniques to join solutions (for example, [1]). The equations and boundary conditions for the first approximation, the method, and the results of numerical integration for the region lying above the separation point, including the separation point as well, are presented. A comparison is made with experimental data, and corrections corresponding to the second approximation are estimated. On the basis of these results, the limits of applicability of the approximate theories that utilize the boundary layer equations are discussed.     

Boundary layer development in a gas flow with stagnation enthalpy varying across the streamlines

We consider a laminar boundary layer for which the stagnation enthalpy specified in the initial section is variable with height. Such problems arise, for example, for bodies located in the wake behind another body, for hypersonic flow past slender blunted bodies (as a result of the large transverse entropy gradients in the highentropy layer), for stepwise variation of the temperature of a surface on which there is an already developed boundary layer, for sudden expansion of the boundary layer as a result of its flow past a corner of the surface, etc.Strictly, we should in such cases solve the boundary layer equations (if the longitudinal gradients are much smaller than the transverse) with the specified initial distribution of the quantities. However, from the physical point of view, the distributed region may be broken down into two regions, the near-wall boundary layer and an outer region which is a gas flow with constant velocity and the specified initial temperature profile, whose calculation yields the edge conditions for the boundary layer. The boundary between the regions is determined from the condition of adequately smooth matching of the solutions. This approach is much preferable to the first, since it permits avoiding (within the framework of boundary layer theory) the difficulties associated with the presence of a possible singularity at the initial point of the surface due to the discontinuity of the boundary conditions at this point, and also permits using conventional boundary layer theory if the effect of the viscosity in the outer region is not significant. However, this partition requires additional justifications of the possibility of independent determination of the solution in the outer region and the determination of the edge of the boundary layer, considered as the region of influence of the wetted surface. The boundary layer in a nonuniform flow has been considered in several works for a linear initial velocity or temperature profile [1–3].It should be noted that the linear initial enthalpy or velocity profiles for constant gas properties do not undergo changes under the influence of viscosity or thermal conductivity. Thus the fundamental characteristic features noted above which are associated with the presence of the two regions and their interaction in essence cannot be investigated using these examples.In this study we obtain and analyze the exact solutions of the equations of the compressible boundary layer for a power-law variation of the initial stagnation enthalpy profile as a function of the stream function for a constant initial velocity. Here it is shown that the influence of the boundary conditions at the wall are actually localized in the near-wall boundary layer, which is similar in dimensions to the conventional velocity or thermal boundary layers. In the region which is external with relation to this layer, in accordance with the physical picture described above, the solution coincides with the solution of the Cauchy problem for the heat conduction equation, which describes the development of the initial temperature profile in an infinite steady-state flow with constant velocity.It is shown that for the sufficiently smooth initial profiles which are of interest in practice the outer flow undergoes practically no changes until we reach the inner boundary layer, and it may be calculated using the perfect gas laws.     

Multiple distortion of a supersonic turbulent boundary layer

Two experiments were performed to study the response of a supersonic turbulent boundary layer to successive distortions. In the first experiment (Case 1), the flow passed over a forward-facing ramp formed by 20° compression corner followed by a 20° expansion corner located about 4_o downstream, where _o is the incoming boundary layer thickness. In the second experiment (Case 2), the forward-facing ramp was constructed of curved compression and expansion surfaces with the same turning angles and total step height as in Case 1. The radii of curvature for the compression and expansion surfaces were equal to 12_o. In both experiments, the flow relaxation was observed over a distance equal to 12_o. In this relaxation region, the mean and turbulent flow behavior of the boundary layer was measured. The mean velocity profile was found to be altered by the distortion. Recovery of the profile began near the wall and occurred rapidly, but in the outer part of the boundary layer, recovery proceeded slowly. Turbulence measurements revealed a dramatic reduction in the turbulence shear stress and a progressively decaying streamwise Reynolds stress profile.     

Boundary element method solution of MHD flow in a rectangular duct

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with an external magnetic field applied transverse to the flow has been investigated. The walls parallel to the applied magnetic field are conducting while the other two walls which are perpendicular to the field are insulators. The boundary element method (BEM) with constant elements has been used to cast the problem into the form of an integral equation over the boundary and to obtain a system of algebraic equations for the boundary unknown values only. The solution of this integral equation presents no problem as encountered in the solution of the singular integral equations for interior methods. Computations have been carried out for several values of the Hartmann number (1 ? M ? 10). It is found that as M increases, boundary layers are formed close to the insulated boundaries for both the velocity and the induced magnetic field and in the central part their behaviours are uniform. Selected graphs are given showing the behaviours of the velocity and the induced magnetic field.