The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

A comparative analysis of approaches for investigating hypersonic flow over blunt bodies in a transitional regime

Hypersonic flows of a viscous perfect rarefied gas over blunt bodies in a transitional flow regime from continuum to free molecular, characteristic when spacecraft re-enter Earth's atmosphere at altitudes above 90-100 km, are considered. The two-dimensional problem of hypersonic flow is investigated over a wide range of free stream Knudsen numbers using both continuum and kinetic approaches: by numerical and analytical solutions of the continuum equations, by numerical solution of the Boltzmann kinetic equation with a model collision integral in the form of the S-model, and also by the direct simulation Monte Carlo method. The continuum approach is based on the use of asymptotically correct models of a thin viscous shock layer and a viscous shock layer. A refinement of the condition for a temperature jump on the body surface is proposed for the viscous shock layer model. The continuum and kinetic solutions, and also the solutions obtained by the Monte Carlo method are compared. The effectiveness, range of application, advantages and disadvantages of the different approaches are estimated.     

Continuum models in problems of the hypersonic flow of a rarefied gas around blunt bodiest

All possible continuum (hydrodynamic) models in the case of two-dimensional problems of supersonic and hypersonic flows around blunt bodies in the two-layer model (a viscous shock layer and shock-wave structure) over the whole range of Reynolds numbers, Re, from low values (free molecular and transitional flow conditions) up to high values (flow conditions with a thin leading shock wave, a boundary layer and an external inviscid flow in the shock layer) are obtained from the Navier-Stokes equations using an asymptotic analysis. In the case of low Reynolds numbers, the shock layer is considered but the structure of the shock wave is ignored. Together with the well-known models (a boundary layer, a viscous shock layer, a thin viscous shock layer, parabolized Navier-Stokes equations (the single-layer model) for high, moderate and low Re numbers, respectively), a new hydrodynamic model, which follows from the Navier-Stokes equations and reduces to the solution of the simplified (“local”) Stokes equations in a shock layer with vanishing inertial and pressure forces and boundary conditions on the unspecified free boundary (the shock wave) is found at Reynolds numbers, and a density ratio, k, up to and immediately after the leading shock wave, which tend to zero subject to the condition that (k/Re)^1/2 → 0. Unlike in all the models which have been mentioned above, the solution of the problem of the flow around a body in this model gives the free molecular limit for the coefficients of friction, heat transfer and pressure. In particular, the Newtonian limit for the drag is thereby rigorously obtained from the Navier-Stokes equations. At the same time, the Knudsen number, which is governed by the thickness of the shock layer, which vanishes in this model, tends to zero, that is, the conditions for a continuum treatment are satisfied. The structure of the shock wave can be determined both using continuum as well as kinetic models after obtaining the solution in the viscous shock layer for the weak physicochemical processes in the shock wave structure itself. Otherwise, the problem of the shock wave structure and the equations of the viscous shock layer must be jointly solved. The equations for all the continuum models are written in Dorodnitsyn--Lees boundary layer variables, which enables one, prior to solving the problem, to obtain an approximate estimate of second-order effects in boundary-layer theory as a function of Re and the parameter k and to represent all the aerodynamic and thermal characteristic; in the form of a single dependence on Re over the whole range of its variation from zero to infinity.

An efficient numerical method of global iterations, previously developed for solving viscous shock-layer equations, can be used to solve problems of supersonic and hypersonic flows around the windward side of blunt bodies using a single hydrodynamic model of a viscous shock layer for all Re numbers, subject to the condition that the limit (k/Re)^1/2 → 0 is satisfied in the case of small Re numbers. An aerodynamic and thermal calculation using different hydrodynamic models, corresponding to different ranges of variation Re (different types of flow) can thereby, in fact, be replaced by a single calculation using one model for the whole of the trajectory for the descent (entry) of space vehicles and natural cosmic bodies (meteoroids) into the atmosphere.     

两个非同心旋转圆柱间粘性流动的广义雷诺方程及其基本流

运用张量分析方法及修正双极坐标系,建立了轴承润滑流动所应满足的广义Reynolds方程.应用薄流层中的Navier-Stokes方程的渐近分析方法和张量分析工具,得到了两个非同心旋转圆柱之间粘性流动的基本流所应满足的方程.这个基本流可以表示为两个同心旋转圆柱之间的Taylor流加上一个扰动项,并且给出了数值计算例子.     

Weak Discontinuities in Solutions of Long‐Wave Equations for Viscous Flow

Nonlinear integrodifferential equations describing the propagation of disturbances in a thin layer of viscous liquid with free surface are studied. These equations admit solutions with weak discontinuities, which are located on the characteristics. The possibility of an unbounded increase in the amplitude of the weak discontinuity and the formation of the shock in the process of flow evolution is established. Differential balance laws approximating the integrodifferential model are proposed. These laws are used to perform numerical simulation of wave propagation in a fluid.     

The fluid-dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks

This paper studies the asymptotic equivalence of the Broadwell model of the nonlinear Boltzmann equation to its corresponding Euler equation of compressible gas dynamics in the limit of small mean free path ε. It is shown that the fluid dynamical approximation is valid even if there are shocks in the fluid flow, although there are thin shock layers in which the convergence does not hold. More precisely, by assuming that the fluid solution is piecewise smooth with a finite number of noninteracting shocks and suitably small oscillations, we can show that there exist solutions to the Broadwell equations such that the Broadwell solutions converge to the fluid dynamical solutions away from the shocks at a rate of order (ε) as the mean free path ε goes to zero. For the proof, we first construct a formal solution for the Broadwell equation by matching the truncated Hilbert expansion and shock layer expansion. Then the existence of Broadwell solutions and its convergence to the fluid dynamic solution is reduced to the stability analysis for the approximate solution. We use an energy method which makes full use of the inner structure of time dependent shock profiles for the Broadwell equations.     

Viscous Shock Motion for Advection-Diffusion Equations

This paper studies the asymptotic solution of the initial-boundary value problem for scalar convection-dominated evolution equations on a bounded spatial domain when initial and boundary conditions are such that the solution develops a single thin shock layer of steep change. The exponentially slow motion of the shock is determined for exponentially long times using an ansatz based on the solution for the special case of Burgers' equation, obtained through the Cole-Hopf transformation. Results obtained analytically are confirmed by numerical experiments.     

Generalized viscous shock layer equations with slip conditions on a surface in a flow field and on a head shock

The plane and axisymmetric problems of super- and hypersonic flow of a homogeneous viscous heat-conducting perfect gas over a blunt body are considered. Generalized viscous shock layer equations that take into account all the second-order effects of boundary-layer theory, i.e., the terms O(Re^?1/2), are derived from the Navier–Stokes equations by the asymptotic method, and all the out-of-order third-order terms O(Re^?1) and higher-order terms are also retained, except terms with second derivations in the marching coordinate (Re is Reynolds number, determined from the free-stream density and velocity the linear dimension, which is equal to the nose radius of the blunt Body, and the free-stream shear viscosity at the stagnation temperature). Thus, only the presence of terms with second derivatives in the marching coordinate, which specify the elliptical properties of the complete system of Navier–Stokes equations, distinguish it from the generalized viscous shock layer equations, which do not contain these terms. Slip and a temperature jump conditions on a body surface are presented with the same degree of accuracy, and generalized Rankine–Hugoniot conditions on a head shock, which take into account the effects of the viscosity and heat conduction, including their influence on the determination of the pressure, are derived. The incorrect and unfounded approximations used in preceding studies and the efficiency of iterative marching techniques for solving the generalized viscous shock layer equations, as well as the ability of the latter to provide a correct solution for the drag and heat-transfer coefficients in the transitional flow regime if the solution is constructed taking the slip and temperature jump on a surface and on a head shock into account, are noted.     

Nonexistence of Riemann solutions for a quadratic model deriving from petroleum engineering

The study presented in this work shows that the viscous profile entropy criterion is too selective in reducing the number of solutions to guarantee existence of stable weak self-similar Riemann solutions to conservation laws. This result is shown on a particular quadratic model derived from the three-phase flow equations used in petroleum engineering. The viscosity matrix considered in this work derives from capillary pressures. The Riemann initial data is hyperbolic and corresponds to a Lax 1-shock that does not admit a viscous profile. The nonexistence of a profile in this example is due to the presence of a limit cycle in the vector field associated with the viscous profile entropy criterion.To establish the main result of this work, a complete list of possibilities that could lead to a solution, is examined. This list includes solutions that consist of only classical waves and the solutions that contain at least one nonclassical (shock) wave. The construction of solutions breaks down because either the shock waves do not satisfy the viscous entropy criterion, or the speeds of the waves that comprise a solution are decreasing. To the author's knowledge, this is the first result on nonexistence of stable solutions for models that allow nonclassical (transitional) shock waves.The results presented in this paper are a combination of analytical and numerical work. The theoretical ideas and techniques derive from the bifurcation theory of vector fields and the theory of weak solutions of conservation laws. These are combined with numerical results when no theory is available.     

Nonexistence of Riemann solutions for a quadratic model deriving from petroleum engineering

The study presented in this work shows that the viscous profile entropy criterion is too selective in reducing the number of solutions to guarantee existence of stable weak self-similar Riemann solutions to conservation laws. This result is shown on a particular quadratic model derived from the three-phase flow equations used in petroleum engineering. The viscosity matrix considered in this work derives from capillary pressures. The Riemann initial data are hyperbolic and correspond to a Lax 1-shock that does not admit a viscous profile. The nonexistence of a profile in this example is due to the presence of a limit cycle in the vector field associated with the viscous profile entropy criterion.To establish the main result of this work, a complete list of possibilities that could lead to a solution, is examined. This list includes solutions that consist of only classical waves and the solutions that contain at least one nonclassical (shock) wave. The construction of solutions breaks down because either the shock waves do not satisfy the viscous entropy criterion or the speeds of the waves that comprise a solution are decreasing. To the author's knowledge, this is the first result on nonexistence of stable solutions for models that allow nonclassical (transitional) shock waves.The results presented in this paper are a combination of analytical and numerical work. The theoretical ideas and techniques derive from the bifurcation theory of vector fields and the theory of weak solutions of conservation laws. These are combined with numerical results when no theory is available.     

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.     

The motions of liquid films in a gas

The motions in a gas of thin films of a viscous incompressible liquid acted upon by capillary forces are considered. The surface tension depends on the impurity concentration of a surface-active material, soluble or insoluble in the liquid, and the liquid is non-volatile. The inertia of the liquid, viscous stresses, the Laplace pressure and the surface-tension gradients, impurity transfer and also the particular properties of super-thin films are taken into account. The motions of the films are described using the model of quasi-steady viscous flow. Systems of equations are obtained in the approximation of an ideal compressible medium and for small Mach numbers. The conditions for the incompressible film surface approximation to hold are obtained. The severe limitations of the gas-dynamic approximation in the case of a soluble impurity due to attenuation of the waves related to diffusion are investigated. A continuum model of the film as a compressible medium with a non-equilibrium pressure is constructed. The asymptotic form of the solutions of unsteady problems of impurity transfer in the limit of weak non-equilibrium is obtained. Integrals of the equations of motion of the films in steady one-dimensional problems are derived. Integral forms of the equations of momentum and its moment for an arbitrary contour of the film are presented, which hold for steady flows in a film and in quasi-statics. The boundary conditions for the solutions of the system of equations of motion of films are given.     

Theory of the hypersonic viscous shock layer at high Reynolds numbers and intensive injection of foreign gases : PMM vol. 38, n 6, 1974, pp. 1015–1024

The hypersonic flow around smooth blunted bodies in the presence of intensive injection from the surface of these is considered. Using the method of external and internal expansions the asymptotics of the Navier-Stokes equations is constructed for high Reynolds numbers determined by parameters of the oncoming stream and of the injected gas. The flow in the shock layer falls into three characteristic regions. In regions adjacent to the body surface and the shock wave the effects associated with molecular transport are insignificant, while in the intermediate region they predominate. In the derivation of solution in the first two regions the surface of contact discontinuity is substituted for the region of molecular transport (external problem). An analytic solution of the external problem is obtained for small values of parameters ₁ = ρ_∞/ρs^* and δ = ρ_ω^*1/2ν_ω^*/ρ_∞1/2ν_∞, in the form of corresponding series expansions in these parameters. Asymptotic formulas are presented for velocity profiles, temperatures, and constituent concentration across the shock layer and, also, the shape of the contact discontinuity and of shock wave separation. The derived solution is compared with numerical solutions obtained by other authors. The flow in the region of molecular transport is defined by equations of the boundary layer with asymptotic conditions at plus and minus infinity, determined by the external solution (internal problem). A numerical solution of the internal problem is obtained taking into consideration multicomponent diffusion and heat exchange. The problem of multicomponent gas flow in the shock layer close to the stagnation line was previously considered in [1] with the use of simplified Navier-6tokes equations.The supersonic flow of a homogeneous inviscid and non-heat-conducting gas around blunted bodies in the presence of subsonic injection was considered in [2–7] using Euler's equations. An analytic solution, based on the classic solution obtained by Hill for a spherical vortex, was derived in [2] for a sphere on the assumption of constant but different densities in the layers between the shock wave and the contact discontinuity and between the latter and the body. Certain results of a numerical solution of the problem of intensive injection at the surface of axisymmetric bodies of various forms, obtained by Godunov's method [3], are presented. Telenin's method was used in [4] for numerical investigation of flow around a sphere; the problem was solved in two formulations: in the first, flow parameters were determined for the whole of the shock layer, while in the second this was done for the sutface of contact discontinuity, which was not known prior to the solution of the problem, with the pressure specified by Newton's formula and flow parameters determined only in the layer of injected gases. The flow with injection over blunted cones was numerically investigated in [5] by the approximate method proposed by Maslen. The flow in the shock layer in the neighborhood of the stagnation line was considered in [6, 8], and intensive injection was investigated by methods of the boundary layer theory in [8–12].     

Density-driven compactional flow in porous media

In the mathematical modelling of compactional flow in porous media, the constitutive relation is typically modelled in terms of a nonlinear relationship between effective pressure and porosity, and compaction is essentially poroelastic. However, at depths deeper than 1 km where the pressure is high, compaction becomes more akin to a viscous one. Two mathematical models of compaction in porous media are formulated and the nonlinear equations are then solved numerically. The essential features of numerical profiles of poroelastic and viscous compaction are thus compared with asymptotic solutions. Two distinguished styles of density-driven compaction in fast and slow compacting sediments are analysed and shown in this paper.     

Calculation of radial supersonic gas flow past sharp-nosed cones

The paper presents the results of numerical analysis of supersonic source flow of a gas past sharp-nosed cones. Axisymmetric and three-dimensional flows are considered. The flow geometry and the distributions of the gas-dynamic parameters in the shock layer are analyzed; their asymptotic properties are established. The numerical solutions are compared with solutions for uniform gas flow past a cone.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 32–47.     

Integral representations of solutions for two-dimensional viscous flow problems

This paper concerns the steady flow of a viscous, incompressible fluid past a cylinderical obstacle. Solutions are represented in terms of simple layer potentials. Existence and uniqueness of solutions to the corresponding integral equations of the first kind are established. Emphasis is given on the asymptotic behaviors of the solutions as well as their connections to the singular perturbation results.     

Dynamics and Coarsening of Interfaces for the Viscous Cahn—Hilliard Equation in One Spatial Dimension

In one spatial dimension, the metastable dynamics and coarsening process of an n -layer pattern of internal layers is studied for the Cahn–Hilliard equation, the viscous Cahn–Hilliard equation, and the constrained Allen–Cahn equation. These models from the continuum theory of phase transitions provide a caricature of the physical process of the phase separation of a binary alloy. A homotopy parameter is used to encapsulate these three phase separation models into one parameter-dependent model. By studying a differential-algebraic system of ordinary differential equations describing the locations of the internal layers for a metastable pattern for this parameter-dependent model, we are able to provide detailed comparisons between the internal layer dynamics for the three models. Layer collapse events are studied in detail, and the analytical theory is supplemented by numerical results showing the different behaviors for the different models. Finally, an asymptotic-numerical algorithm, based on our asymptotic information of layer collapse events and the conservation of mass condition, is devised to characterize the entire coarsening process for each of these models. Numerical realizations of this algorithm are shown.     

N-S方程的拟序扰动序列与改善的渐近展开匹配法

提出了以拟序扰动序列逼近N-S方程定解问题渐近解的一种方法。对N-S方程及其边界条件的渐近拟序扰动序列解进行了讨论,并应用此方法对球坐标系中的圆球绕流进行求解,改善了渐近展开匹配方法,使匹配函数更容易确定。改善后的阻力曲线与实测资料相比在雷诺数小于等于4×104以前完全吻合。