Influence of the viscous boundary layer on the deposition of particles from a gas suspension flowing past a sphere

The flow of incompressible gas containing particles past bodies of simple shapes at moderate and high velocities is investigated in [1–5], in which the flow of the carrier medium is assumed to be irrotational. The estimates made in [3] for the neighborhood of the stagnation point show that it is necessary to take into account the viscous boundary layer in the case of fine particles. In the present paper, the viscous flow of a gas suspension over the front surface of a sphere at Reynolds numbers R = 10³–10⁷ is considered. It is assumed that the carrier gas is incompressible and the particle concentra ion negligibly small. The influence of the boundary layer on the particle trajectories and the deposition of the disperse phase on the surface of the sphere is investigated. It is shown that there is a wide range of flow parameters for the gas suspension in which the influence of the boundary layer is important. The limits of this range are established.Translated from Izvestiya Akademli Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–66, January–February, 1982.I thank Yu. P. Savel'ev for a helpful discussion of the work.     

Detached flow over a step with the formation of a turbulent wake

A method of calculating the plane turbulent layer behind a step interacting with a free potential flow of incompressible fluid is developed. The method includes consideration of the initial boundary layer and injection (or suction) in the isobaric bottom region. Friction on the wall behind the step is neglected, which corresponds to symmetric quasisteady flow behind the straight edge of a plate. The inviscid flow is represented by the Keldysh-Sedov integral equations; the flow in the wake with a one-parameter velocity profile is represented by three first-order differential equations—the equations of momentum for the wake and motion along its axis and the equation of interaction (through the displacement thickness) of the viscous flow with the external potential flow. The turbulent friction in the wake is given, accurate to the single empirical constant, by the Prandtl equation. The different flow regions — on the plate behind the step, the isobaric bottom region, and the wake region — are joined with the aid of the quasi-one-dimensional momentum equation for viscous flow. The momentum equation for the flow as a whole serves as the closure condition. The obtained integrodifferential system of equations is approximated by a system of nonlinear finite-difference equations, whose solution is obtained on a computer by minimization of the sum of the squares of the discrepancies. The results of the calculations agree satisfactorily with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 17–25, May–June, 1977.We are grateful to V. I. Kuptsov for consultation and help in programming and to Z. A. Donskova who assisted in the calculations and preparation of the paper.     

Development of laminar jet flows with zero excess impulse

The problem of the development of a laminar jet of viscous incompressible fluid with zero excess impulse (the wake of a hydrodynamic motor) was investigated for the first time by Birkhoff and Zarantonello [1], who found a self-similar solution to the dynamical problem for the case of a two-dimensional laminar wake. The problem of the development of turbulent wakes of hydrodynamic motors in the near and far flow regions was solved by Ginevskii [2] on the basis of an integral method. In the present paper, the method of asymptotic expansions is used on the basis of the boundary layer equations to solve nonself-similar problems of the development of laminar jet flows of a viscous incompressible fluid with zero excess impulse. The obtained solution takes into account the influence of the details of the source (finite size of the body and its geometry) and the value of the Prandtl number on the velocity and temperature distribution. In the case of a laminar axi-symmetric wake, a self-similar solution is obtained to the thermal problem, the solution being valid in a wide range of Prandtl numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–33, May–June, 1984.     

Asymptotic theory of the flow near the rear end of a slender axisymmetric body

Steady high-Reynolds-number flow of a viscous incompressible fluid past a slender axisymmetric body is considered. The structure of the near wake and the boundary layer in the vicinity of the rear end of the body is studied. The relationship between the maximum relative body thickness and the rearend shape giving a local boundary layer — potential flow interaction zone in a small neighborhood of the rear end is found. The boundary value problem for this region is solved numerically.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 68–77, September–October, 1993.     

Influence of second-order boundary layer effects in hypersonic flow over slender blunt cones

The aim of this study is to determine the influence of second-order effects in the aggregate on supersonic axisymmetric flow over slender blunt cones and also to determine the region of applicability of approximate methods of taking into account the strongest of these second-order effects — entropy layer absorption. A system of complete viscous shock layer equations containing all the terms of the gas dynamic Euler equations and all the second-order terms of asymptotic boundary layer theory is chosen as the gas-dynamic model. Within the calculation domain the problem is solved in a unified manner.Translated from Izvestiya Rossiiskoi Akademii Nauk, Meknanika Zhidkosti i Gaza, No.4, pp. 129–134, July–August, 1992.     

Allowance for longitudinal diffusion in the neighborhood of a discontinuity of catalytic surface properties

In the investigation of flow near surfaces with discontinuous changes in the catalytic properties the question arises of the applicability of parabolic boundary and viscous shock layer equations in the neighborhood of the discontinuity. In the present paper, three types of problem are solved in which longitudinal diffusion is taken into account. In the first an insertion with different catalytic properties is placed in the neighborhood of the stagnation point, in the second the discontinuity lines of the catalytic properties are perpendicular to the oncoming flow, while in the third they are parallel. On the main surface and on the insertion surface the heterogeneous catalytic reactions are assumed firstorder reactions with various rate constants whose values vary in a wide range. The data of the solution are compared with the solution obtained using the boundary layer approximation and the regions of influence of the longitudinal diffusion are estimated. In [1–4] a problem similar to the second one was solved by the numerical method of [1] and the Wiener-Hopf method for the case of transition from a noncatalytic to a perfectly catalytic surface and the region of applicability of the boundary layer was estimated [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 99–105, July–August, 1986.     

Effect of Boundary-Layer Thickness on the Structure of a Near-Wall Flow with a Two-Dimensional Obstacle

Results of physical and numerical experiments on investigating the effect of the depth of immersion of a two-dimensional obstacle with a square cross section into a developed turbulent boundary layer on the length of the separated flow region are presented. The numerical simulation is based on solving averaged Navier–Stokes equations with the use of the k– model of turbulence. The near-wall flow is visualized in the experiments, and the fields of mean and fluctuating velocities are measured. Flow regions where the results of numerical simulation agree with experimental data are determined. It is shown that the length of the recirculation flow region in the near wake increases with decreasing depth of immersion of the two-dimensional obstacle into the turbulent boundary layer.     

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.     

Computation of base flow characteristics with uniform blowing

Blowing at bluff body base was considered under different conditions and for small amount of blowing this problem was solved using dividing streamline model [1]. The effect of supersonic blowing on the flow characteristics of the external supersonic stream was studied in [2–4]. The procedure and results of the solution to the problem of subsonic blowing of a homogeneous fluid at the base of a body in supersonic flow are discussed in this paper. Analysis of experimental results (see, e.g., [5]) shows that within a certain range of blowing rate the pressure distribution along the viscous region differs very little from the pressure in the free stream ahead of the base section. In this range the flow in the blown subsonic jet and in the mixing zones can be described approximately by slender channel flow. This approximation is used in the computation of nozzle flows with smooth wall inclination [6, 7]. On the other hand, boundary layer equations are used to compute separated stationary flows with developed recirculation regions [8] in order to describe the flow at the throat of the wake. The presence of blowing has significant effect on the flow structure in the base region. An increasing blowing rate reduces the size of the recirculation region [9] and increases base pressure. This leads to a widening of the flow region at the throat, usually described by boundary-layer approximations. At a certain blowing rate the recirculation region completely disappears which makes it possible to use boundary-layer equations to describe the flow in the entire viscous region in the immediate neighborhood of the base section.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 76–81, January–February, 1984.     

Flow of viscous gas out of a cylindrical channel into vacuum

A study is made of the exhausting of a jet of viscous gas from a cylindrical channel into vacuum in the presence of a flat bounding surface outside the channel in the plane of its exit section. The problem is solved numerically using the complete system of Navier—Stokes equations. The developed flow model makes it possible to take into account the influence of an external medium into which the jet exhausts on the structure of the flow in the exit section of the channel, and also the influence of the subsonic part of the boundary layer in the channel on the flow field of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–128, January–February, 1981.     

Construction of a method for calculating a plane boundary layer in weak solutions of polymers with laminar,transitional, and turbulent flow zones

The investigation of the effect of small polymer additives on the characteristics of the flow of a viscous liquid is, at the present time, one of the most promising approaches to the lowering of the friction resistance. One interesting question in this connection is the study of the effect of small polymer additives on the characteristics of the transitional region of flow in a boundary layer, as well as on the value of the friction resistance with the presence of laminar, transitional, and turbulent sections in the boundary layer. The article sets forth a possible method for calculation of a plane boundary layer and the friction resistance for the case of the motion of a body in weak polymer solutions with a constant concentration, taking account of the change in the flow conditions in the layer and based on the use of integral relationships. Questions connected with the development of a boundary at a body, with the feeding of a polymer in it, as well as with the effect of degradation or destruction of the polymer in the solution, are not discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–48, May–June, 1977.     

Calculation of three-dimensional boundary layer on a triangular plate of finite length in a hypersonic stream

The results are given of the calculation of a three-dimensional boundary layer on a triangular plate of finite length in a regime of strong viscous interaction with an external hypersonic stream for both symmetric flow as well as in the presence of an angle of slip. The influence of the change in the pressure on the trailing edge of the plate on the boundary layer characteristics is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 46–52, September–October, 1982.     

Investigation of hypersonic flow of rarefied gas past wings of infinite span

In the framework of the locally self-similar approximation of the Navier-Stokes equations an investigation is made of the flow of homogeneous gas in a hypersonic viscous shock layer, including the transition region through the shock wave, on wings of infinite span with rounded leading edge. The neighborhood of the stagnation line is considered. The boundary conditions, which take into account blowing or suction of gas, are specified on the surface of the body and in the undisturbed flow. A method of numerical solution of the problem proposed by Gershbein and Kolesnikov [1] and generalized to the case of flow past wings at different angles of slip is used. A solution to the problem is found in a wide range of variation of the Reynolds numbers, the blowing (suction) parameter, and the angle of slip. Flow past wings with rounded leading edge at different angles of slip has been investigated earlier only in the framework of the boundary layer equations (see, for example, [2], which gives a brief review of early studies) or a hypersonic viscous shock layer [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 150–154, May–June, 1984.     

The Development of the Asymptotic Viscous Dissipation Profile in a Vertical Free Convective Boundary Layer Flow in a Porous Medium

The effect of viscous dissipation on the development of the boundary layer flow from a cold vertical surface embedded in a Darcian porous medium is investigated. It is found that the flow evolves gradually from the classical Cheng–Minkowycz form to the recently discovered asymptotic dissipation profile which is a parallel flow.     

Boundary layer of a body of revolution in a drag-reducing polymer solution

A possible method is presented for calculating an axisymmetric boundary layer when a body moves in weak polymer solutions with constant concentration. The method is based on the use of a velocity profile and system of integral equations which take into account most fully the effects of the transverse curvature of the body's surface. The computational scheme makes it possible to take into account the change in the flow regimes in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–48, May–June, 1981.     

Multicomponent chemically-reacting turbulent viscous shock layer on a catalytic surface

Turbulent flows past blunt bodies at high supersonic speeds are mainly investigated within the framework of the boundary layer model. However, even at large Reynolds numbers owing to the strong entropy gradient on the lateral surface it becomes necessary to take boundary layer corrections into account in the higher approximations [1]. The use of viscous shock layer theory makes it possible to obtain fairly accurate results over a broad interval of variation of the Reynolds numbers without organizing iterations with respect to vorticity and displacement thickness. The nonequilibrium nature of both homogeneous and heterogeneous catalytic reactions is taken into account. The results obtained are compared with the experimental data [2, 3]. Previously, in [4, 5] turbulent flow was investigated within the framework of viscous shock layer theory in the case of equilibrium homogeneous reactions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–149, March–April, 1989.     

Stability of a disperse-mixture flow in a boundary layer

The hydrodynamic stability of a dilute disperse mixture flow in a quasi-equilibrium region of a boundary layer with a significantly nonuniform particle concentration profile is investigated. The mixture is described by a two-fluid model with an incompressible viscous carrier phase. In addition to the Stokes drag, the Saffman lifting force is taken into account in the interphase momentum exchange. On the basis of a numerical solution of the boundary-value problem for a modified Orr-Sommerfeld equation, neutral stability curves are analyzed and the dependence of the critical Reynolds number on the governing parameters is studied. It is shown that taking into account the particle concentration nonuniformity in the main flow and the Saffman lifting force significantly changes the stability limits of the two-phase laminar boundary layer flow. The effect of these factors on the boundary layer stability is considered for the first time.     

A bypass wake induced laminar/turbulent transition

The process of laminar to turbulent transition induced by a von Karman vortex street wake, was studied for the case of a flat plate boundary layer. The boundary layer developed under zero pressure gradient conditions. The vortex street was generated by a cylinder positioned in the free stream. An X-type hot-wire probe located in the boundary layer, measured the streamwise and normal to the wall velocity components. The measurements covered two areas; the region of transition onset and development and the region where the wake and the boundary layer merged producing a turbulent flow. The evolution of Reynolds stresses and rms-values of velocity fluctuations along the transition region are presented and discussed. From the profiles of the Reynolds stress and the mean velocity profile, a ‘negative' energy production region along the transition region, was identified. A quadrant splitting analysis was applied to the instantaneous Reynolds stress signals. The contributions of the elementary coherent structures to the total Reynolds stress were evaluated, for several x-positions of the near wall region. Distinct regions in the streamwise and normal to the wall directions were identified during the transition.     

Simulation of ordered structures in open turbulent flows

Extensive experimental material [1–4] indicates that ordered (coherent) structures play an important part in determining the nature of the flow, the generation of Reynolds stresses and turbulence energy, and the transport of heat, momentum, and passive admixtures in a turbulent flow. In the present paper, a model is constructed for describing coherent structures in which, given the profile of the mean velocity, one can determine the characteristic sizes, the propagation velocities, and also the frequency and amplitude characteristics of these ordered motions. The model is based on the analogy between the ordered formations and secondary flows in a subsidiary laminar flow whose velocity profile is the same as the turbulent profile of the mean velocity. The influence of small-scale pulsations is described by the introduction of the coefficient of turbulent viscosity. In the framework of the model, numerical calculations are made for two-dimensional turbulent flows in a mixing layer, a jet, and a wake behind a cylinder. The results of the calculations are compared with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 45–52, July–August, 1981.     

Calculation of supersonic separation flow in circular nozzles with abrupt expansion

A study is made of the interaction of a circular supersonic jet with a turbulent layer of the near-wake kind formed behind a circular step on the end part of a nozzle with abrupt expansion. The flow in the viscous layer is calculated by the integral method, and in the inviscid flow by a through-computation method using a monotonic implicit difference scheme of first order of accuracy. The interaction between the inviscid and turbulent flows is determined by the displacement thickness of the viscous layer. The initial conditions for the flow in the layer are determined from the integral conditions of its matching to the isobaric mixing flow in the base region behind the step. The computed interaction flows are determined as a function of the length of the end part and the counter pressure simultaneously or separately by the boundary condition that the pressure at the end of the end part be equal to the pressure of the external medium and by a singular solution of the equations passing through a saddle singular point — the throat of the wake. In a conical nozzle with profiled attachments of different lengths, calculations were made of separation flows with open and closed base regions and with allowance for secondary separation. The obtained solutions are multivalued in a certain range of lengths of the end part and values of the counter pressure. The solutions realized in reality are selected by analyzing the regimes of operation of the nozzles, and the calculated regions of hysteresis and associated low-frequency nonstationary separation flows are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 120–128, July–August, 1979.We are grateful to M. Ya. Ivanov for consultation on the use of the program of [6] and to Z. A. Donskovaya for assistance in preparing the paper.