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.     

Energy-balance equation for turbulent pulsations in the theory of the free turbulent boundary layer

Semiempirical expressions are proposed for the coefficient of turbulent viscosity and for the scale of turbulence in the equations for the free turbulent boundary layer in an incompressible fluid, these equations consisting of the equation of continuity, the equations of motion, and the equation for the average energy balance in the turbulent pulsations. The advantage of the expressions over the existing ones is that the two empirical constants in the equations have nearly the same values for circular and plane turbulent streams and also for a turbulent boundary layer at the edge of a semiinfinite homogeneous flow with a stationary fluid. The mean-energy distribution and the mean energy of the turbulent pulsations computed in this paper agree well with the experimental values.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 75–79, November–December, 1970.     

Mass transfer between bubbles and the continuous phase in a pseudofluidized layer

The problem of mass transfer between an isolated bubble and the continuous phase in a pseudofluidized layer is considered, when the rising velocity of the bubble exceeds the pseudofluidization rate. In this case the bubble with the surrounding region, a so-called two-phase system, is surrounded by a surface current impermeable to the liquid [1–3], and the problem reduces to determining the concentration field and the total flow on the material surface. The problem is solved for large and small Peclet numbers by a boundary layer diffusion method and by asymptotic expansion matching.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–49, July–August, 1973.     

The stability of the motion of a liquid,due to thermocapillary forces

A study is made of the stability of the plane-parallel flow of a viscous liquid in a layer with a free boundary, under weightless conditions. The motion of the liquid is due to the dependence of the surface tension on the temperature. An exact solution for an unperturbed boundary is obtained by the same method used in [1], but with a more general boundary condition for the temperature. A study of the stability was carried out by the method of small vibrations, taking account of the perturbation of the free boundary. The article discusses the asymptotic behavior of long waves at small Reynolds numbers, and the conditions for instability are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 94–98, November–December, 1971.     

Influence of polymer additives in a flow on the hydrodynamic characteristics of two-dimensional profiles

The problem of separationless flow of homogeneous dilute polymer solutions over two-dimensional profiles is considered. The complete flow is divided by the outer edge of the boundary layer and the wake into two regions: a region of irrotational flow and a region of viscous flow — the boundary layer and wake. The characteristics of the two regions are matched at their boundary. The problem is solved by successive approximation with allowance for the mutual influence of the two regions on each other. The influence of the irrotational region on the viscous region is taken into account through the distribution of the pressure on the boundary of the wake and the boundary layer. The influence of the viscous part of the flow is taken into account by the introduction of an associated vortex whose intensity is equal to the integral of the vorticity in the complete viscous region, and also by the introduction of additional velocities on the boundary of the wake and the boundary layer. These deform the streamlines in the irrotational part of the flow and ensure that they match the flow pattern in the real fluid. The results of the calculations of the hydrodynamic characteristics of a Zhukovskii profile are compared with experimental data. The influence of the introduction into the flow of polymer additives on the distributed and total characteristics of the flow at a number of Reynolds numbers is analyzed for the example of the modified profile NACA66.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 35–41, October–December, 1981.     

Determination of mechanical properties of fiber–matrix interface from pushout test

For ceramic matrix composites, the pushout test is the most widely used test for finding the two mechanical properties of the fiber–matrix interface – (1) the coefficient of friction and (2) the residual radial stress. Experimental measurements from the pushout test do not directly give the values of these two mechanical properties of the fiber–matrix interface, but need to be regressed to theoretical models. Currently, approximate theoretical models based on shear–lag analysis are used for regression. In this paper, the adequacy of the shear–lag analysis model in accurately finding the mechanical properties of the fiber–matrix interface is discussed. An elasticity solution of the pushout test based on boundary element method is developed. Regressing one set of available experimental data from a pushout test to both shear–lag analysis and boundary element method models gives values differing by 15% for the coefficient of friction but similar values for the residual radial stress. Parametric studies were also conducted to show the difference between the shear–lag analysis and boundary element method results for factors such as fiber to matrix elastic moduli ratios, coefficient of friction and fiber volume fractions.     

Calculation of the base flow behind the rounded trailing edge of a two-dimensional body separating two supersonic streams

A method of calculating the base pressure and the coordinates of the separation points on the curved surface of the trailing edge of a two-dimensional body separating two supersonic streams, taking into account the local overexpansion effect, has been developed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 170–173, November–December, 1991.     

Two-dimensional absolute instability of a supersonic boundary layer

The results of the author's earlier investigation of the stability of a partially viscous shock layer indicate that any plane-parallel flow may be absolutely unstable if for that flow there exists more than one normal instability mode. This assumption has been confirmed for a supersonic boundary layer at infinitely large Reynolds numbers. Two types of absolute instability, corresponding to two known types of branching of the dispersion relation, have been detected.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 176–179, January–February, 1988.     

Interaction between the boundary layer and a supersonic flow when part of the wall is rapidly heated

There have been many theoretical studies of aspects of the unsteady interaction of an exterior inviscid flow with a boundary layer [1–9]. The mathematical flow models obtained in these studies by the method of matched asymptotic expansions describe a wide range of phenomena observed experimentally. These include boundary layer separation near the hinge of a flap, the flow in the neighborhood of the trailing edge of an oscillating airfoil [1–2], and the development and propagation of perturbations in a boundary layer excited by an oscillating wall or some other way [3–5]. The present paper studies the interaction of an unsteady boundary layer with a supersonic flow when a small part of the surface of a body in the flow is rapidly heated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–70, January–February, 1984.     

DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow

The Lie group, or symmetry approach, developed by Oberlack (see e.g. Oberlack [26] and references therein) is used to derive new scaling laws for various quantities of a zero pressure gradient turbulent boundary layer flow. The approach unifies and extends the work done by Oberlack for the mean velocity of stationary parallel turbulent shear flows. From the two-point correlation (TPC) equations the knowledge of the symmetries allows us to derive a variety of invariant solutions (scaling laws) for turbulent flows, one of which is the new exponential mean velocity profile that is found in the mid-wake region of flat-plate boundary layers. Further, a third scaling group was found in the TPC equations for the one-dimensional turbulent boundary layer. This is in contrast to the Navier–Stokes and Euler equations, which have one and two scaling groups, respectively. The present focus is on the exponential law in the outer region of turbulent boundary layer corresponding new scaling laws for one- and two-point correlation functions. A direct numerical simulation (DNS) of a flat plate turbulent boundary layer with zero pressure gradient was performed at two different Reynolds numbers Re=750,2240. The Navier–Stokes equations were numerically solved using a spectral method with up to 140 million grid points. The results of the numerical simulations are compared with the new scaling laws. TPC functions are presented. The numerical simulation shows good agreement with the theoretical results, however only for a limited range of applicability. PACS 02.20.-a, 47.11.+j, 47.27.Nz, 47.27.Eq     

Model of a viscous nonequilibrium shock layer with a thin shock wave

It is shown that the concept of a viscous shock layer with boundary conditions specified in a thin shock wave is unsuitable for analyzing the flow of a chemically reacting gas, even in the case of high Reynolds numbers; it may produce a finite error when determining the parameters of the shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–178, September–October, 1973.     

Stability of a plane-parallel convective flow of a liquid in a horizontal layer

We consider the stationary plane-parallel convective flow, studied in [1], which appears in a two-dimensional horizontal layer of a liquid in the presence of a longitudinal temperature gradient. In the present paper we examine the stability of this flow relative to small perturbations. To solve the spectral amplitude problem and to determine the stability boundaries we apply a version of the Galerkin method, which was used earlier for studying the stability of convective flows in vertical and inclined layers in the presence of a transverse temperature difference or of internal heat sources (see [2]). A horizontal plane-parallel flow is found to be unstable relative to two critical modes of perturbations. For small Prandtl numbers the instability has a hydrodynamic character and is associated with the development of vortices on the boundary of counterflows. For moderate and for large Prandtl numbers the instability has a Rayleigh character and is due to a thermal stratification arising in the stationary flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–100, January–February, 1974.     

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.     

Multiparameter boundary layer method for slender bodies of revolution

The problem of the laminar and steady incompressible boundary layer on elongated slender bodies of revolution in an axisymmetric flow is examined in those cases in which the ratio of the boundary layer thickness to the radius of the body's cross section should not be neglected. For this purpose the boundary layer equation for slender bodies of revolution is universalized on the basis of Loitsyanskii's method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 22–27, November–December, 1991.     

Separated inviscid gas flow past a disk and a body with maximum critical Mach numbers

The problem of the separated axisymmetric subsonic flow of an inviscid perfect gas with the specific heat ratio 1.4 past a disk in accordance with the Riabouchinsky scheme is solved using the method developed in [1]. Formulas relating the main parameters with the base pressure coefficient and the Mach number at the free boundary are presented. Formulas which make it possible to determine the shape of the body of revolution giving the maximum critical Mach numbers are also derived.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 166–172, May–June, 1996.     

Theory of the energy separation of a compressible gas flow

Optimum gas parameters which ensure the maximum heat transfer across a flat plate separating two streams with different Mach numbers are found on the basis of an exact self-similar solution for a laminar boundary layer.     

Hypersonic viscous flow around extended bodies

The characteristic feature of flow around an extended body is the interaction of the thickened boundary layer with the external nonviscous flow. This phenomenon becomes more significant at low Reynolds numbers and high Mach numbers. Theoretical investigation of this interaction is difficult because of the presence of shock waves, which are characteristic of hypersonic velocities; the position and curvature of these shock waves depend on the state of the boundary layer developing in conditions of pronounced vorticity of the external flow. With increasing rarefaction of the flow, the problem begins to take on an elliptic character, and this necessitates the use of methods of investigation of more general form than the classical boundary-layer theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 164–166, March–April, 1976.The authors thank V. G. Farafonov and V. N. Arkhipov for guidance and assistance in the work.     

Ablation of Teflon in a stream of dissociated air

A numerical solution to the coupled problem of heat and mass transfer on an ablating Teflon surface is used to analyze the influence of nonequilibrium physicochemical processes in the boundary layer on ablation. The results of the numerical calculations are compared with the experimental data in the literature on the ablation of Teflon subject to high heat fluxes and with data on measurements of the concentrations of the components in a boundary layer containing Teflon ablation products. It is shown that an important factor that must be taken into account in interpreting experimental data, particularly at low pressures and high stagnation enthalpies, is the influence of the catalytic properties of the surface on the heat transfer. An approximate expression is derived for calculating the ablation rate; it is valid in the range of free-stream velocities 3 km/sec < V < 8 km/sec.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 103–109, May–June, 1984.     

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.     

Non-Darcian Effects on the Onset of Convection in a Porous Layer with Throughflow

The Brinkman extended Darcy model including Lapwood and Forchheimer inertia terms with fluid viscosity being different from effective viscosity is employed to investigate the effect of vertical throughflow on thermal convective instabilities in a porous layer. Three different types of boundary conditions (free–free, rigid–rigid and rigid–free) are considered which are either conducting or insulating to temperature perturbations. The Galerkin method is used to calculate the critical Rayleigh numbers for conducting boundaries, while closed form solutions are achieved for insulating boundaries. The relative importance of inertial resistance on convective instabilities is investigated in detail. In the case of rigid–free boundaries, it is found that throughflow is destabilizing depending on the choice of physical parameters and the model used. Further, it is noted that an increase in viscosity ratio delays the onset of convection. Standard results are also obtained as particular cases from the general model presented here.