Boundary Layer for the Navier-Stokes-alpha Model of Fluid Turbulence

We study boundary-layer turbulence using the Navier-Stokes-alpha model obtaining an extension of the Prandtl equations for the averaged flow in a turbulent boundary layer. In the case of a zero pressure gradient flow along a flat plate, we derive a nonlinear fifth-order ordinary differential equation, which is an extension of the Blasius equation. We study it analytically and prove the existence of a two-parameter family of solutions satisfying physical boundary conditions. Matching these parameters with the skin-friction coefficient and the Reynolds number based on momentum thickness, we get an agreement of the solutions with experimental data in the laminar and transitional boundary layers, as well as in the turbulent boundary layer for moderately large Reynolds numbers.     

Irreversible deformation and subsequent unloading of a spherical elastoviscoplastic layer

Analytical solutions of a number of one-dimensional quasi-static problems that describe the processes of elastic deformation of the material of a hollow sphere and of generation and development of the plastic flow in this material with increasing pressure on the external boundary are presented. The process of unloading during slow removal of the loading pressure is studied. Stress fields, fields of elastic and plastic strains in the material of the spherical layer, the law of motion of the elastoplastic boundary, and residual stresses are determined. It is demonstrated that (in contrast to the ideal plasticity case) the allowance for the viscous properties of the material during its plastic flow eliminates the possibility of plastic flow emergence during unloading.     

Prediction of periodic boundary layers

A relatively simple, yet efficient and accurate finite difference method is developed for the solution of the unsteady boundary layer equations for both laminar and turbulent flows. The numerical procedure is subjected to rigorous validation tests in the laminar case, comparing its predictions with exact analytical solutions, asymptotic solutions, and/or experimental results. Calculations of periodic laminar boundary layers are performed from low to very high oscillation frequencies, for small and large amplitudes, for zero as well as adverse time-mean pressure gradients, and even in the presence of significant flow reversal. The numerical method is then applied to predict a relatively simple experimental periodic turbulent boundary layer, using two well-known quasi-steady closure models. The predictions are shown to be in good agreement with the measurements, thereby demonstrating the suitability of the present numerical scheme for handling periodic turbulent boundary layers. The method is thus a useful tool for the further development of turbulence models for more complex unsteady flows.     

Termination of the starting problem of dynamic expansion of a spherical cavity in an infinite elastic-perfectly-plastic medium

Perhaps the simplest non-trivial problem in small deformation dynamic plasticity is expansion of a spherical cavity in an infinite elastic-perfectly-plastic medium. Here, example problems are considered with two boundary conditions at the cavity's surface: constant velocity and constant pressure. Attempts to obtain analytical solutions are complicated by the fact that, in general, the elastic-plastic boundary propagates with variable speed. However, it is known that the elastic-plastic boundary propagates at constant speed for the starting problem when the shocks due to the applied loads are large enough to cause inelastic response at the instant they are applied. When the value of the applied pressure equals the shock pressure due to the applied velocity the solutions of the two boundary value problems are initially identical and can be compared. The objective of this paper is to review the literature and to examine the termination conditions for the starting problem. Specifically, the starting problem terminates when either the jump in radial stress at the elastic-plastic boundary or the loading condition for plasticity vanishes there. These termination conditions depend on the applied load and on Poisson's ratio.     

The laminar boundary layer near a body corner point

A method is developed for calculating the characteristics of a laminar boundary layer near a body contour corner point, in the vicinity of which the outer supersonic stream passes through a rarefaction flow. In the study we use the asymptotic solution of the Navier-Stokes equations in the region with large longitudinal gradients of the flow functions for large values of the Reynolds number, the general form of which was used in [1].The pressure, heat flux, and friction distributions along the body surface are obtained. For small pressure differentials near the corner the solution of the corresponding equations for small disturbances is obtained in analytic form.The conventional method for studying viscous gas flow near body surfaces for large values of the Reynolds number is the use of the Prandtl boundary layer theory. Far from the body the asymptotic solution of the Navier-Stokes equations in the first approximation reduces to the solution of the Euler equations, while near the body it reduces to the solution of the Prandtl boundary layer equations. The characteristic feature of the boundary layer region is the small variation of the flow functions in the longitudinal direction in comparison with their variation in the transverse direction. However, in many cases this condition is violated.The necessity arises for constructing additional asymptotic expansions for the region in which the longitudinal and transverse variations of the flow functions are quantities of the same order. The general method for constructing asymptotic solutions for such flows with the use of the known method of outer and inner expansions is presented in [1].In the following we consider the flow in a laminar boundary layer for the case of a viscous supersonic gas stream in the vicinity of a body corner point. Behind the corner the flow separates from the body surface and flows around a stagnant zone, in which the pressure differs by a specified amount from the pressure in the undisturbed flow ahead of the point of separation. A pressure (rarefaction) disturbance propagates in the subsonic portion of the boundary layer upstream for a distance which in order of magnitude is equal to several boundary layer thicknesses. In the disturbed region of the boundary layer the longitudinal and transverse pressure and velocity disturbances are quantities of the same order. In this study we construct additional asymptotic expansions in the first approximation and calculate the distributions of the pressure, friction stress, and thermal flux along the body surface.     

Irreversible deformation with subsequent unloading of a spherical viscoelastoplastic layer

Analytic solutions are obtained for a sequence of one-dimensional quasistatic problems describing viscoelastic deformation processes in the material of a hollow ball and the plastic flow nucleation and evolution processes occurring in the ball as the pressure on the outer boundary increases. The unloading process under slow removal of the loading pressure is considered as well. The stress fields and the elastic and plastic strain fields in the spherical layer material, the law of motion of the elastoplastic boundary, and the residual stress level and distribution are computed. It is assumed that at the stage preceding the plastic flow the material obeys the viscoelastic Voigt model and the loading surface is determined by the von Mises plastic flow condition.     

Wake layer in a thermal turbulent boundary layer with pressure gradient

The open equations of thermal turbulent boundary layer subjected to pressure gradient have been analysed by method of matched asymptotic expansions at large Reynolds number. The flow is divided into outer wake layer and inner wall layer. The asymptotic expansions are matched by Millikan-Kolmogorov hypothesis. The temperature profile in overlap region yields composite law which reduce to log. law for moderate pressure gradient and inverse half power law for strong adverse pressure gradient. In case of a shallow thermal wake, the matching result of outer wake layer reduces to composite temperature defect law, which is more general than the classical log. law. The comparison of data for thermal boundary layer with strong adverse pressure gradient is also considered. Received on 26 May 1998     

On Disturbance Propagation in Internal-Flow Boundary Layers

The unsteady interaction of plane-channel wall boundary layers with a supersonic inviscid flow is investigated. The flow regimes in which disturbances introduced by the boundary layer developing on one wall influence the boundary layer on the other wall are considered. The regime of relatively large pressure disturbance amplitudes generated near the nozzle outlet or by deforming the channel walls is studied. In these conditions, the interaction process is described by a system of Burgers equations with retarded arguments. Numerical solutions of this system are obtained for symmetric and antisymmetric perturbations of the channel walls.     

Global pressure relaxation for laminar two-dimensional internal flow

This study extends the reduced Navier–Stokes (RNS) global pressure relaxation procedure developed by Rubin and co-workers for external flow to internal flow applications. The streamwise pressure gradient is split into a backward-differenced or initial value component, as in boundary layer marching, and a forward-differenced or boundary value component that represents the elliptic downstream effects. The streamwise convection terms are upwind-differenced and all other streamwise derivatives are backward-differenced. We thus obtain a standard boundary layer marching technique imbedded in a conventional line relaxation technique. For compressible flow the pressure iteration determines the interior flow interation as well as the inlet mass flux that is consistent with the outflow pressure boundary condition. Results have been computed for incompressible flow in both rectangular and curved channels, and for subsonic compressible flow in the simulation of an aerofoil in a wind tunnel. Converged solutions were obtained over a range of Reynolds numbers generating small to moderately large separation bubbles.     

Universal Velocity Defect Law for the Turbulent Boundary Layer

Turbulent plane boundary layer flows of an incompressible fluid are considered. A refinement of the known Coles wake law is proposed. This refinement makes it possible to ensure the smooth matching of the turbulent boundary layer velocity profile with the outer flow and to extend the range of validity of the law to the case of large positive pressure gradients. The accuracy of the analytical approximation obtained is verified by comparison with the known experimental equilibrium velocity profiles. Using the approximation proposed, a relation for calculating the cross-sectional distribution of the Reynolds stress in the equilibrium boundary layer is derived. The pressure distributions for which the equilibrium turbulent boundary layer flows are single- and two-valued are distinguished.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 89–101.Original Russian Text Copyright © 2005 by Mikhailov.     

Viscid contributions to the hydrodynamic flows past a rising spherical gas bubble with a time-dependent radius

We study the effect of the viscosity on the hydrodynamic flow fields past the interface of a spherical deforming gas bubble impulsively started at a constant velocity in a viscous liquid of large extent at rest. Exact solutions for the unsteady inner and outer flow fields within the boundary layers are obtained making appropriate scalings on the position, velocity and time variables in the non-linear Navier-Stokes equations. These theoretical results apply to any slowly deforming fluid sphere, whatever the time-dependence of its radius, provided that the internal circulation is complete, the flow separation is negligible, the Reynolds numbers are large and the bubble retains its spherical shape. A numerical application to the case of deforming air bubble in water is discussed.     

Analytical Solutions for Solute Transport in a Spherically Symmetric Divergent Flow Field

Exact analytical solutions for an equation describing advection, dispersion, first-order decay, and rate-limited sorption of a solute in a steady, hemispherical or spherically symmetric, divergent flow field are presented for constant concentration and constant flux boundary conditions in a porous medium. The partial differential equation describing transport is a confluent hypergeometric equation that may be solved with variable substitution and Laplace transform, and the solutions are expressed by parabolic cylindrical functions. The novel solutions derived here may be applied to predict concentration distributions in space and time for porous media transport in a spherically symmetric flow field or for the special case where injection is just below a confining layer (hemispherical flow). The analytical solutions can be used to simulate wastewater injection from short-screened wells into thick formations or to analyze tracer tests that use short-screened wells to create approximately spherical flow fields in thick formations.     

Influence of viscosity on the flow over the sharp edge of bodies consisting of a spherical segment joined to an inverted cone

The results are given of an experimental investigation of the supersonic axisymmetric flow over a body consisting of a spherical segment joined to an inverted cone in the neighborhood of the point of inflection of the profile (Fig. 1a). For the limiting case of a cylinder with a flat end and M = 3, a study was made of the influence of the Reynolds number and the state of the boundary layer on the parameters of the local separation region formed near the inflection (Fig. 1b). It was found that there is an appreciable decrease in the length of the separation region and the pressure in it when the Reynolds number increases in the range Re = 10⁵– 10⁷ in the case of a laminar boundary layer on the flat end near the inflection point. A low level of the pressure on the surface of the body was achieved — of the order of thousandths of the pressure behind a normal shock. There was found to be a sharp increase in the pressure in the separation region when the boundary layer on the end becomes turbulent with transition to a flow regime that is self-similar with respect to the Reynolds number. Under conditions of a turbulent boundary layer, systematic experimental data on the pressure on the inverted cone near the point of inflection of such bodies were obtained and generalized.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 154–157, January–February, 1981.     

Viscous boundary layers in flows through a domain with permeable boundary

The effect of small viscosity on nearly inviscid flows of an incompressible fluid through a given domain with permeable boundary is studied. The Vishik–Lyusternik method is applied to construct a boundary layer asymptotic at the outlet in the limit of vanishing viscosity. Mathematical problems with both consistent and inconsistent initial and boundary conditions at the outlet are considered. It is shown that in the former case, the viscosity leads to a boundary layer only at the outlet. In the latter case, in the leading term of the expansion there is a boundary layer at the outlet and there is no boundary layer at the inlet, but in higher order terms another boundary layer appears at the inlet. To verify the validity of the expansion, a number of simple examples are presented. The examples demonstrate that asymptotic solutions are in quite good agreement with exact or numerical solutions.     

Turbulence Energy Distribution in the Stokes Layer

Using the method of matched asymptotic expansions, an analytical solution of the balance equation for turbulence energy is constructed for a shallow basin (sea) in which the fluid depth does not exceed the Stokes layer thickness. In this case, a gradient-viscous balance is established with the turbulent viscosity being balanced mainly by the pressure gradient. It is shown that nonlinear boundary layers attributable to turbulence energy diffusion are formed near the bottom and the free surface (or ice). In the neighborhood of the point of maximum flow velocity (if this maximum is attained inside the flow), a nonlinear internal boundary layer also develops. Outside these layers, the turbulence energy generation is in the first approximation balanced by the energy dissipation. Asymptotic solutions for the boundary layers are constructed.     

Formation and evolution of vortex rings induced by interactions between shock waves and a low-density bubble

The deformation and instability of a low-density spherical bubble induced by an incident and its reflected shock waves are studied by using the large eddy simulation method. The computational model is firstly validated by experimental results from the literature and is further used to examine the effect of incident shock wave strength on the formations and three-dimensional evolutions of the vortex rings. For the weak shock wave case (Ma?=?1.24), the baroclinic effect induced by the reflected shock wave is the key mechanism for the formation of new vortex rings. The vortex rings not only move due to the self-induced effect and the flow field velocity, but also generate azimuthal instability due to the pressure disturbance. For the strong shock wave case (Ma?=?2.2), a boundary layer is formed adjacent to the end wall owing to the approach of vortex ring, and unsteady separation of the boundary layer near the wall results in the ejection and formation of new vortex rings. These vortex rings interact in the vicinity of the end wall and finally collapse to a complicated vortex structure via azimuthal instability. For both shock wave strength cases, the evolutions of vortex rings due to the instability lead to the formation of the complicated structure dominated by the small-scale streamwise vortices.     

Thin shear boundary layers in flow of hot aluminium

The flow of hot aluminium in channels is investigated. The constitutive relation considered for the flow stress is the inverse hyperbolic sine function of the Zener–Hollomon parameter. Analytical solutions for the flow are derived. At high strain rates, an exponential velocity profile close to the channel walls is predicted indicating the existence of a thin shear boundary layer characterized by strong shear. The characteristic length scale for the exponential velocity profile is a function of material parameters in the constitutive relation and the inverse of the local pressure gradient. The analytical prediction of a thin shear boundary layer close to the channel walls for large strain rates is consistent with the observed microstructure in an extruded section.     

奇异摄动在层流预混火焰理论研究中的应用

奇异摄动被广泛应用于求取力学问题的近似解.一个典型问题就是流体力学中的边界层.郭永怀先生曾发展了适用于平板黏性流动边界层问题的奇异摄动理论.类似于流体力学中的边界层,燃烧研究中的层流预混火焰也可以通过奇异摄动理论进行分析,在燃烧研究中通常称其为大活化能渐近分析.本文介绍了大活化能渐近分析在一维平面预混火焰和球形传播火焰中的应用及相关研究进展.首先介绍了预混火焰结构及其涉及的不同特征尺度,分析了大活化能条件下出现的特征尺度分离,并给出了关于平面预混火焰大活化能渐近分析的详细推导,讨论了热辐射对火焰传播的影响;然后介绍了大活化能渐近分析在点火与球形传播火焰方面的应用,指出了只有能够同时描述点火与球形火焰传播的理论才能准确地预测临界点火条件,并讨论了考虑链式反应的点火与火焰传播理论,分析了热辐射对球形火焰传播的影响,给了关于火焰稳定性理论研究的发展趋势.最后,基于当前研究进展对未来的研究方向进行了展望,其中涉及多步化学反应、低温冷火焰、复杂流动、辐射重吸收等.