Flow of a viscous fluid in an annular gap with intensive blowing from the walls

Self-similar solutions are obtained in [1, 2] to the Navier-Stokes equations in gaps with completely porous boundaries and with Reynolds number tending to infinity. Approximate asymptotic solutions are also known for the Navier-Stokes equations for plane and annular gaps in the neighborhood of the line of spreading of the flow [3, 4]. A number of authors [5–8] have discovered and studied the effect of increase in the stability of a laminar flow regime in channels of the type considered and a significant increase in the Reynolds number of the transition from the laminar regime to the turbulent in comparison with the flow in a pipe with impermeable walls. In the present study a numerical solution is given to the system of Navier-Stokes equations for plane and annular gaps with a single porous boundary in the neighborhood of the line of spreading of the flow on a section in which the values of the local Reynolds number definitely do not exceed the critical values [5–8]. Generalized dependences are obtained for the coefficients of friction and heat transfer on the impermeable boundary. A comparison is made between the solutions so obtained and the exact solutions to the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–24, January–February, 1987.     

Formula for the Flow Resistance Factor in a Pipe with a Sudden Expansion at Small Reynolds Numbers

A formula for the flow resistance factors in a pipe with a sudden expansion of the cross section at Reynolds numbers of 0.2 to 10 is obtained by numerical solution of the complete Navier–Stokes equations for incompressible fluids. The flow resistance factors obtained using the derived formula are compared to those found by numerical solution of the Navier–Stokes equations.     

Interference between a blunt edge shock layer and an impinging inclined shock at low Reynolds numbers

The interference between the shock layer on a cylinder modeling the leading edge of an air intake and an impinging plane inclined shock is investigated experimentally and numerically for a Reynolds number Re₀=32. The low-pressure wind tunnel experiments made it possible to visualize the flow and determine the local heat transfer in the presence of interference. The corresponding flow regimes were calculated numerically within the framework of the system of Navier-Stokes equations by the through-calculation method. The principal properties of the distribution of the flow characteristics for a low value of the Reynolds number were obtained for various types of interference and the differences with respect to the previously investigated interference regimes for high Reynolds numbers were examined.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–171, January–February, 1993.     

Turbulent air flow over a moving curved surface

A numerical model of turbulent air flow over a curved surface is described. The model is based on two-dimensional nonlinear Reynolds equations and continuity equations written in a coordinate system moving with the profile of the curved surface. The Reynolds stresses are represented in the form of the product of the isotropic turbulent viscosity coefficient, which increases linearly with height, and the deformation tensor of the mean velocity field. Flow over a stationary sinusoidal surface and a sinusoidal gravity wave on water is simulated. The structure of the velocity and pressure wave fields is obtained. The differences in flow over stationary and moving surfaces are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 20–24, September–October, 1986.     

Reynolds number dependence of two-dimensional laminar co-rotating vortex merging

The paper reports unsteady Navier–Stokes calculations of laminar two-dimensional co-rotating vortex merging for various Reynolds numbers. The unsteady, incompressible two-dimensional Navier–Stokes equations were solved with fourth-order Runge–Kutta temporal discretization and fourth-order symmetric compact schemes for spatial discretization. Calculations of the unsteady Taylor vortex benchmark showed that fourth-order accurate solutions for all primitive variables were indeed achieved. Calculations for a pair of equal-strength co-rotating vortices show good agreement with reported direct numerical simulation and experiments for the evolution of the separation distance and core radius. It is found that the time required for merging is inversely proportional to the square root of the Reynolds number. According to previous experimental research, it was also found that complete merging in laminar regime undergoes four stages with physical meaning. The physical mechanism responsible for the merging process is investigated and it is found that the antisymmetric vorticity dynamics plays an important role until full merging.     

Interaction of two spherical particles in a steady flow of viscous fluid when the Reynolds number is not small

The problem of the interaction of two or more particles moving in a viscous incompressible fluid at small Reynolds numbers (Re 1) has been well studied. The linearity of the Stokes equations makes it possible to develop effective methods of solution of the problem for two and many particles [1]. If the Reynolds number is not small, the inertia forces in the Navier-Stokes equations cannot be ignored, and the problem becomes nonlinear, i.e., much more complicated. The present note is devoted to the problem of the interaction of two spherical particles in a steady uniform flow of a viscous incompressible fluid when the Reynolds number is not small. Asymptotic expressions are obtained for the interaction forces between the particles when the distances between them are large compared with their radius.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 142–144, May–June, 1983.     

Investigation of the transitional process of the startup of a gas bearing

In this article we derive equations for transitional conditions of a gas lubricant under nonisothermal conditions. A numerical solution of the Reynolds equation is described for the problem of the pulsed startup of a radial bearing with a fixed geometry, and the results of this solution are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 156–160, September–October, 1977.     

Energy analysis of the stability of MHD flows

The analog of Orr's problem is formulated for MHD flows. Arbitrary three-dimensional disturbances satisfying the continuity equations are considered. It is established that direct interaction of the disturbances of the magnetic field and the velocity field cannot increase the energy estimate of the critical Reynolds number. Numerical calculations for Hartmann flow and modified Couette flows are made for the particular case of small magnetic Reynolds numbers, The minimum value of R is attained for disturbances with a wave vector perpendicular to the velocity vector of the main flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–9, July–August, 1971.The authors thank M. A. Gol'dshtik for his interest in their work.     

Heat transfer in the neighborhood of the critical point or line in the presence of hydrogen blowing and combustion

The results of numerical calculations of the thin viscous shock layer equations are presented. The dependence of the heat exchange on the generalized blowing parameter, which depends on the blown gas flow rate and the Reynolds number, is obtained.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 144–149, May–June, 1995.     

Localized Coherent Structures in the Boundary Layer

A Blasius laminar boundary layer and a steady turbulent boundary layer on a flat plate in an incompressible fluid are considered. The spectral characteristics of the Tollmien—Schlichting (TS) and Squire waves are numerically determined in a wide range of Reynolds numbers. Based on the spectral characteristics, relations determining the three–wave resonance of TS waves are studied. It is shown that the three–wave resonance is responsible for the appearance of a continuous low–frequency spectrum in the laminar region of the boundary layer. The spectral characteristics allow one to obtain quantities that enter the equations of dynamics of localized perturbations. By analogy with the laminar boundary layer, the three–wave resonance of TS waves in a turbulent boundary layer is considered.     

Comparative analysis of viscous flows in cavities and channels containing axisymmetric obstacles

The flow past axisymmetric bodies of various shapes in expanding cavities and cylindrical channels is studied on the basis of a numerical solution of the Navier-Stokes equations. For each body shape velocity, pressure and shear stress distributions are obtained. These data are then used for the purposes of a comparative flow analysis in terms of body shape and Reynolds number. The properties characteristic of flows around bodies in channels and cavities with moving boundaries are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 60–66, May–June, 1988.     

Numerical investigation of supersonic viscous gas flow over long blunt cones with allowance for equilibrium physicochemical effects

Supersonic axisymmetric viscous heat-conducting gas flow over long spherically biunted cones is considered over a broad range of Reynolds numbers on the basis of the complete system of viscous shock layer equations. An economical numerical method based on global iterations is used to solve the viscous shock layer equations. The general influence of the second-approximation effects of boundary layer theory and the influence of equilibrium physicochemical processes on the heat loads are determined for bodies with a large aspect ratio.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 202–205, January–February, 1993.The author wishes to thank S. A. Vasil'evskii and I. A. Sokolova for providing the tables used to calculate the transport coefficients and G. A. Tirskii for his constant interest and useful discussions.     

Nonisothermal Couette flow of a non-Newtonian fluid under the action of a pressure gradient

Nonisothermal Couette flow has been studied in a number of papers [1–11] for various laws of the temperature dependence of viscosity. In [1] the viscosity of the medium was assumed constant; in [2–5] a hyperbolic law of variation of viscosity with temperature was used; in [6–8] the Reynolds relation was assumed; in [9] the investigation was performed for an arbitrary temperature dependence of viscosity. Flows of media with an exponential temperature dependence of viscosity are characterized by large temperature gradients in the flow. This permits the treatment of the temperature variation in the flow of the fluid as a hydrodynamic thermal explosion [8, 10, 11]. The conditions of the formulation of the problem of the articles mentioned were limited by the possibility of obtaining an analytic solution. In the present article we consider nonisothermal Couette flows of a non-Newtonian fluid under the action of a pressure gradient along the plates. The equations for this case do not have an analytic solution. Methods developed in [12–14] for the qualitative study of differential equations in three-dimensional phase spaces were used in the analysis. The calculations were performed by computer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 26–30, May–June, 1981.     

On the minimum of extended dissipation in viscous nematodynamics

This paper develops the variational principle of minimum extended dissipation for slow (low Reynolds number) flows of nematic liquids as described by the five parametric Leslie–Ericksen–Parodi (LEP) constitutive equations. It is shown that the Eulers equations for minimizer of the extended dissipative functional are the Stokes equations for the LEP fluid. When the molecular (including magnetic) field is absent, the extended dissipative functional coincides with the true dissipative functional, whose Euler equations are the Stokes equations for the Ericksen fluid.     

Investigation of the hypersonic viscous shock layer around blunt bodies in a nonuniform flow

Unseparated viscous gas flow past a body is numerically investigated within the framework of the theory of a thin viscous shock layer [13–15]. The equations of the hypersonic viscous shock layer with generalized Rankine-Hugoniot conditions at the shock wave are solved by a finite-difference method [16] over a broad interval of Reynolds numbers and values of the temperature factor and nonuniformity parameters. Calculation results characterizing the effect of free-stream nonuniformity on the velocity and temperature profiles across the shock layer, the friction and heat transfer coefficients and the shock wave standoff distance are presented. The unseparated flow conditions are investigated and the critical values of the nonuniformity parameter a_k [10] at which reverse-circulatory zones develop on the front of the body are obtained as a function of the Reynolds number. The calculations are compared with the asymptotic solutions [10, 12].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–159, May–June, 1987.     

Optimization of body shape at small reynolds numbers

The problem of the optimization of the shape of a body in a stream of viscous liquid or gas was treated in [1–5]. The necessary conditions for a body to offer minimum resistance to the flow of a viscous gas past it were derived in [1], The necessary optimality conditions when the motion of the fluid is described by the approximate Stokes equations were derived in [2], The shape of a body of minimum resistance was found numerically in [3] in the Stokes approximation. The optimality conditions when the motion of the fluid is described by the Navier—Stokes equations were derived in [4, 5], and in [4] these conditions were extended to the case of a fluid whose motion is described in the boundary-layer approximation. The necessary optimality conditions when the motion of the fluid is described by the approximate Oseen equations were derived in [5] and an asymptotic analysis of the behavior of the optimum shape near the critical points was performed for arbitrary Reynolds numbers.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp, 87–93, January–February, 1978.     

Numerical investigation of axisymmetric flows in the wake of blunt bodies in a supersonic stream of viscous gas

The axisymmetric flow in the near wake of spherically blunted cones exposed to a supersonic stream of viscous perfect heat-conducting gas is numerically investigated on the basis of the complete Navier-Stokes equations. The free-stream Mach numbers considered M = 2.3 and 4 were such that the gas can be assumed to be perfect, and the Reynolds numbers such that for these Mach numbers the flow in the wake is laminar but close to laminar-turbulent transition [1–4]. The flow structure in the near wake is described in detail and the effect of the Mach and Reynolds numbers on the base pressure, the total drag and the wake geometry is investigated. The results of calculating the flow in the wake of spherically blunted cones are compared with the experimental data [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–47, July–August, 1988.     

Behavior of secondary flows in a rotating pipe

The later stages of laminar-turbulent transition are studied by direct numerical integration of the Navier-Stokes equations. A sequence of flow regimes which replace each other as the Reynolds number increases is obtained. In the coordinate system moving with the traveling wave this sequence includes steady, periodic, quasiperiodic and chaotic motions. The chaotic motion is preceded by synchronization of the frequencies of the quasiperiodic regime.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 29–35, November–December, 1992.The author is grateful to E. B. Rodichev for his assistance.     

The boundary layer on a rapidly rotating cylinder

The study considers plane steady flow of an incompressible fluid around a circular cylinder rotating in a homogeneous free stream. On the basis of an asymptotic analysis of the Navier-Stokes equations for high Reynolds numbers, it is shown that at a certain value of the angular velocity of the cylinder an interaction arises between the flow in the boundary layer and the external potential flow. A solution is obtained numerically which describes the flow in the region of interaction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 36–45, September–November, 1987.     

Influence of temperature factor on the aerodynamic characteristics of a circular cylinder in a supersonic perfect gas flow

The flowfield and aerodynamic characteristics of a circular cylinder at Mach number M=5 and zero yaw angle are calculated using Navier-Stokes equations. Calculations were carried out at several Reynolds numbers for both thermally insulated and isothermal surfaces; in the latter case the temperature factor was varied.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 156–162, May–June, 1994.