Effect of pressure gradient of friction and heat transfer in a dusty boundary layer

Approximate analytic expressions for the local friction and heat transfer coefficients in a dusty laminar boundary layer are obtained and tested in the case of an incompressible carrier phase, power-law variation of the external gas flow velocity and small velocity and temperature phase disequilibrium. These expressions supplement the numerical analysis of the dusty boundary layer on a blunt body [1, 2] and the asymptotic calculation of the friction and heat transfer in a quasiequilibrium dusty gas boundary layer on a plate [3]. The combined effect of dustiness and pressure gradient on the friction and heat transfer coefficients is discussed. The results obtained can be used for the practical calculation of the friction and heat transfer in a quasiequilibrium dusty laminar boundary layer and for interpreting the corresponding experimental data. Tomsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 105–108, September–October, 1988.     

Quasiequilibrium Knudsen boundary layer on a nonisothermal porous body

The state of a gas near a permeable nonisothermal body with ultrathin pores, that is, pores in which the motion of molecules is not accompanied by intermolecular collisions, is studied. A boundary layer of a new type, namely, the quasiequilibrium Knudsen boundary layer on the porous body surface, is investigated. It is formed on condition that within the body there is a heat flux, even when the body is in an equilibrium gas. The statistical methods for solving the Boltzmann equation are used to determine the temperature and pressure jumps across the Knudsen layer near thin perforated and porous membranes.     

Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface

The mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent fluid is analyzed using similarity solution technique. Wall temperature and stretching velocity are assumed to have specific exponential function forms. The influence of buoyancy along with viscous dissipation on the convective transport in the boundary layer region is analyzed in both aiding and opposing flow situations. The flow is governed by the mixed convection parameter Gr/Re². The velocity and temperature inside the boundary layer are observed to be influenced by the parameters like Prandtl number Pr, Gebhart number Gb. Significant changes are observed in non-dimensional skin friction and heat transfer coefficients due to viscous dissipation in the medium. The flow and temperature distributions inside the boundary layer are analyzed and the results for non-dimensional skin friction and heat transfer coefficients are discussed through computer generated plots.     

Analytical study of heat transfer from circular cylinder in liquid metals

In this study the influence of a thin hydrodynamic boundary layer on the heat transfer from a single circular cylinder in liquid metals having low Prandtl number (0.004–0.03) is investigated under isothermal and isoflux boundary conditions. Two separate analytical heat transfer models, viscous and inviscid, are developed to clarify the discrepancy between previous results. For both models, integral approach of the boundary layer analysis is employed to derive closed form expressions for the calculation of the average heat transfer coefficients. For an inviscid model, the energy equation is solved using potential flow velocity only whereas for a viscous model, a fourth-order velocity profile is used in the hydrodynamic boundary layer and potential flow velocity is used outside the boundary layer. The third-order temperature profile is used inside the thermal boundary layer for both models. It is shown that the inviscid model gives higher heat transfer coefficients whereas viscous flow model gives heat transfer results in a fairly good agreement with the previous experimental/numerical results.     

Boundary layer on a blunt body in a flow of dusty gas

The problem of interaction of gas-dust flows with solid surfaces arose in connection with the study of the motion of aircraft in a dusty atmosphere [1–2], the motion of a gas suspension in power generators, and in a number of other applications [3]. The presence of a disperse admixture may lead to a significant increase in the heat fluxes [4] and to erosion of the surface [5]. These phenomena are due to the joint influence of several factors — the change in the structure of the carrier-phase boundary layer due to the presence of the particles, collisions of the particles with the surface, roughness of the ablating surface, and so forth. This paper continues an investigation begun earlier [6–7] into the influence of particles on the structure of the dynamical and thermal two-phase boundary layer formed around a blunt body in a flow. The model of the dusty gas [8] has an incompressible carrier phase. The method of matched asymptotic expansions [9] is used to obtain the equations of the two-phase boundary layer. In the frame-work of the refined classification made by Stulov [6], it is shown that the form of the boundary layer equations is different in the presence and absence of inertial precipitation of the particles. The equations are solved numerically in the neighborhood of the stagnation point of the blunt body. The temperature and phase velocity distributions in the boundary layer, and also the friction coefficients and the heat transfer of the carrier phase are found for a wide range of the determining parameters. In the case of an admixture of low-inertia particles that are not precipitated on the body, it is shown that even when the mass concentration of the particles in the undisturbed flow is small their accumulation in the boundary layer can lead to a sharp increase in the thermal fluxes at the stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–107, September–October, 1985.I thank V. P. Strulov for a discussion.     

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.     

Study on the equivalent convection coefficient of the hot surface of blast furnace stave

The calculation models of the equivalent convection coefficient between blast furnace gas flow and the hot surface of stave body, gas flow and in-laid brick were established by the combination of experiments and numerical calculation when the gas temperature is in the range of 505–1,248°C. The reason why the heat transfer coefficient between gas flow and in-laid brick is much more than that between gas and stave body was analyzed when the gas temperature is high. The opinion just to considering a kind of integration heat transfer coefficient while the numerical calculation of heat transfer model of stave will be changed. The division of above two heat transfer coefficients will increase the degree of heat transfer numerical calculation.     

Asymptotic solution of the equations for a three-dimensional multicomponent laminar boundary layer with large injection

The asymptotic method of outer and inner expansions is used to analyze the flow of a multicomponent gas in a three-dimensional boundary layer on a smooth blunt body with large injection. Asymptotic expressions are derived for the friction coefficients, the heat and diffusion fluxes of the components on the surface of the body, and the velocity, temperature, and concentration profiles of the components across the layer of injected gases. It is shown that with large injection the limiting (bottom) streamlines on the surface of the body coincide in the first approximation with the vectorial lines of the pressure gradient.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 47–56, March–April, 1975.The author is indebted to G. A. Tirskii for a discussion of the work.     

Hypersonic viscous shock layer on sweptback wings of infinite span at different angles of attack

A study is made of the flow of a viscous compressible gas in a hypersonic shock layer on sweptback wings of infinite span with blunt leading edge at different angles of attack. The equations of the hypersonic viscous shock layer with modified Rankine-Hugoniot relations across the shock wave and boundary conditions on the surface of the body that take into account slip and discontinuity of the temperature are solved by a method of successive approximation which yields not only an analytic solution for the first approximations but also an exact numerical solution when the method is implemented on a computer. The analytic solution of the problem is found in the first approximation. Expressions are obtained for the coefficients of friction and heat transfer on the surface of the body, and also for the profiles of the velocities and the temperature across the shock layer. Comparison of the analytic solution with the numerical solution reveals a satisfactory accuracy of the analytic solution for not too large Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 91–102, March–April, 1979.We thank G. A. Tirskii for his interest in the work and valuable discussions.     

Unsteady Boundary Layer on a Blunt Body in a Hypersonic Flow of Non-Uniform Dusty Gas

The unsteady heat transfer at the stagnation point on a blunt body traveling at hypersonic velocity through a layer of nonuniform dusty gas with low-inertia particles (not deposited on the body surface) is investigated. Using the matched asymptotic expansion method, the equations of the two-phase unsteady boundary layer near the symmetry axis of the body are derived with account for the polydispersity of the particles. The structure of the unsteady boundary layer and the variation of the friction and heat transfer coefficients at the stagnation point are studied numerically. Layered nonuniformities of the particle concentration and size are considered, the limits of variation of the thermal and mechanicals loads are found, and the effect of the dust polydispersity on the heat transfer is investigated.     

The turbulent boundary layer on the moving surface of a cylindrical body

A study is made of the problem of a two-dimensional turbulent boundary layer on the moving surface of a cylindrical body (a Rankine oval with a relative elongation of four) moving at constant velocity in an incompressible fluid. For the numerical simulation of the turbulent flow of the fluid, the boundary layer is divided into exterior and interior regions in accordance with a two-layer model, using different expressions for the coefficients of turbulent transfer for each region. A study was nade of the development of the boundary layer on the body at different speeds of the body surface and different Reynolds numbers. The following integral characteristics were found by numerical calculation: the work of friction as the body is displaced; the work expended on the movement of its surface; and, for a flow regime with separation, the work of the pressure force. In this case the following model of separation flow is assumed: beyond the singular point in the solution of the boundary layer equations that indicates the appearance of a region of reverse flow, the pressure and friction stress on the wall are constant and are determined by their values at the singular point.Translated from Izvestiya Akademii Nauk SSSH, Mekhanika Zhidkosti i Gaza, No. 5, pp. 61–67, September–October, 1984.Finally, the author would like to thank G. G. Chernyi and Yu. D. Shevelev for useful discussions and for their interest in this work.     

Investigation of the heat transfer mechanism in supersonic turbulent boundary layers

A method is described for calculating turbulent Prandtl numbers from Mach number and total temperature profiles in supersonic boundary layers. The calculations are based on boundary layer measurements in the Mach number range from 3.5 to 5. The investigations clearly indicate that in addition to accurate profile measurements reliable values of shear stress and heat flux at the wall must exist, in order to be able to calculate the turbulent Prandtl number in the viscous regime of the boundary layer. For flow conditions with and without heat transfer, the derived turbulent Prandtl numbers indicate that the turbulent transport of heat decreases much faster towards the wall than the turbulent transport of momentum. The results of the analysis show that only the unequivocal qualitative result of increasing turbulent Prandtl numbers in the viscous region of the boundary layer, can be expected. The variation of the turbulent Prandtl number can be described successfully using a simple approximation, based on the mixing length concept, and is applied to the calculation of total temperature distribution using the law of the wall for compressible flow.     

The noneouilibrium dissociation of diatomic molecules in a boundary layer

A study is made of the influence of the process of diffusion of diatomic molecules on the rate of their thermal dissociation. On the basis of the ladder model of the dissociation of molecules — truncated harmonic oscillators — an expression is obtained for the macroscopic reaction rate which depends explicitly and exponentially on the square of the gradient of the gas temperature. The molecules constitute a small admixture in a monatomic inert gas. The effect of the diffusion on the rate of dissociation is illustrated by the results of numerical calculation of the flow in the boundary layer around the stagnation point of a blunt body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 1, pp. 161–170, January–February, 1985.     

Investigation of the influence of blowing on the flow in a hypersonic viscous shock layer near the stagnation streamline on a blunt body

In the framework of the approximation of local similarity to the Navier-Stokes equations, an investigation is made of the axisymmetric flow of homogeneous gas in a hypersonic shock layer, this including the region of transition through the shock wave. Boundary conditions, which take into account blowing of gas, are specified on the surface of the body and in the undisturbed flow. A numerical solution to the problem is obtained in a wide range of variation of the Reynolds number and the blowing parameter. Expressions are found for the dependences on the blowing parameter usually employed in boundary layer theory of the coefficients of friction and heat transfer on the surface of the body, which are divided by their values obtained for blowing parameter equal to zero. It is shown that these dependences are universal and the same as the dependences obtained from the solution of the equations of a hypersonic viscous shock layer with modified Rankin-Hugoniot relations across the shock wave and from the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 199–202, January–February, 1980.     

Decomposition of an axisymmetric body of revolution made of a material of complex chemical composition in a partially ionized airstream

The problem of the decomposition of an axisymmetric body of revolution in an ionized airstteam is formulated and solved numerically. Equations are presented for numerical calculation of the diffusional fluxes for generalized Schmidt numbers for the case of ambipolar diffusion in a multicomponent partially ionized mixture with ions of various sorts which differ in mass and charge. For materials consisting of the elements H, C, N, and O we discuss the kinetic decomposition condition, which is used to close the system of boundary conditions, in the form of the dependence of the linear decomposition rate on the surface temperature, gas pressure, and body dimensions. As an example we calculate the decomposition of a sphere made from a material which is similar in properties to textolite in an airstream with stagnation temperature of 14 000° K. It is found that the distribution of the decomposition rate over the sphere surface is nearly independent of the type of material and freestream parameters and coincides to within 10 percent with the distribution of the thermal fluxes to the nondecomposed body. We study a generalized analogy for the mass transfer coefficients in the vicinity of the forward stagnation point.In conclusion the authors wish to thank O. N. Suslov for numerous valuable comments.     

Asymptotic solution of the three-dimensional multicomponent laminar boundary-layer equations with large suction

The flow of a multicomponent compressible gas in a three-dimensional laminar boundary layer is investigated for large values of the suction parameter. Asymptotic expressions are derived for the profiles of the velocities, temperatures, and concentration of the components across the boundary layer, as well as for the friction, heat-, and mass-transfer coefficients on the surface of the body.The authors wish to thank G. A. Tirskii for a discussion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 41–45, July–August, 1975.     

Investigation of supersonic three-dimensional flow past pointed axisymmetric bodies

The flow pattern near bodies of revolution with very long cylindrical and pointed nose sections is studied in the framework of an ideal gas model by means of a numerical method based on MacCormack's difference scheme. The existence of internal shock waves, oriented in both the longitudinal and the transverse directions, in the shock layer is established. The variation of the aerodynamic coefficients of the configuration with its length, angle of attack, and free stream Mach number is investigated. The calculated and experimental data are compared, and the connection between the flow parameters on the body surface and the position of the separation line of the boundary layer on its lateral face is established. A method of calculating the influence of the boundary layer on the values of the aerodynamic coefficients of bodies of revolution of large aspect ratio at small angles of attack is proposed. Axisymmetric flow near blunt bodies has been studied in detail in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–133, September–October, 1986.The author expresses his gratitude to A. N. Pokrovskii for his help in calculating the boundary layer parameters on the surfaces of the considered configurations.     

Three-dimensional turbulent boundary layers on bielliptic bodies in a stream of compressible gas at an angle of attack

The results are given of a numerical investigation of the three-dimensional turbulent boundary layer formed on bielliptic bodies in a stream of compressible gas at an angle of attack. The investigation was made on the basis of the finite-difference method of calculation. The influence of a number of determining parameters on the development of the three-dimensional flows is analyzed. The characteristic flow regions in the boundary layer are found: lines of flow divergence and convergence, the region of “separation,” and flow division surfaces. The positions of the maximal values of the heat flux and the friction on the surface are determined, and the behavior of the limiting streamlines on the body is described.     

Numerische Berechnung des Partikeltransportes zu einer laminar angeströmten Kreisscheibe

Increasingly process steps become important, in which particles as product particles or contaminants are deposited on substrates out of the gas phase. In this paper the particles transport processes are investigated close to the surface of a circular plate surrounded by a laminar flow. The analogy between the governing equations of momentum, energy and mass is applied to the extended diffusion equation. In the nondimensional form the results of the numerical calculations give informations about velocity, temperature and particle concentration boundary layer thickness as well as their distributions. Especially the impact of external forces on particle concentration boundary layer thickness and profile is discussed. The transport of submicron particles to the surface due to convection, diffusion, gravity and thermophoretic forces acting independently is investigated. In the used normalized form the different forces are acting as one resulting force independently of their origin. Their resulting effect in comparison to the effect due to convective diffusive transport is important for particle deposition.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.