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.     

Continuum models of rarefied gas flows in problems of hypersonic aerothermodynamics

The limits of applicability of continuum flow models in the problem of the hypersonic rarefied gas flow over blunt bodies are determined by an asymptotic analysis of the Navier–Stokes equations, the numerical solution of the viscous shock layer equations and the numerical and asymptotic solution of the thin viscous shock layer equations for low Reynolds numbers. It is shown that the thin viscous shock layer model gives correct values of the skin friction coefficient and the heat transfer coefficient in the transitional to free-molecule flow regime. The asymptotic solutions, the numerical solutions obtained within the framework of different continuum models, and the results of a calculation by Direct Simulation Monte Carlo method are compared.     

Numerical study of a transverse streamlined flow of an infinite series of cylinders

The problem of transverse streamlined flow of an infinite series of cylinders by a flow of viscous incompressible liquid is considered. The problem is solved numerically by using the method of difference approximation of the Navier-Stokes equations with use of Arakawa scheme of the second degree of accuracy. Computations were given for a series of cylinders with the step L=2.2, 2.8, 3.2, 3.6 and with Reynolds numbers Re equal to 40, 100, 200, 400. Relations between the hydrodynamic characteristics of the series and the distance between the axes of the cylinders are obtained for different Reynolds numbers.Translated from Dinamicheskie Sistemy, No. 7, pp. 53–57, 1988.     

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.     

Wall effects on the slow steady motion of a particle in a viscous incompressible fluid

In this paper, asymptotic expansions with respect to small Reynolds numbers are proved for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid past a single plane wall. The flow problem is modelled by a certain boundary value problem for the stationary, nonlinear Navier-Stokes equations. The coefficients of these expansions are the solutions of various, linear Stokes problems which can be constructed by single layer potentials and corresponding boundary integral equations on the boundary surface of the particle. Furthermore, some asymptotic estimates at small Reynolds numbers are presented for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid between two parallel, plane walls and in an infinitely long, rectilinear cylinder of arbitrary cross section. In the case of the flow problem with a single plane wall, the paper is based on a thesis, which the author has written under the guidance of Professor Dr. Wolfgang L. Wendland.     

Improved stability techniques for the solution of Navier-Stokes equations

When solving the Navier-Stokes equations for transient incompressible viscous flow problems, one normally encounters a decrease in numerical stability of the time integration algorithm with an increase in Reynolds number. This instability cannot be easily overcome due to the non-linearities present. The present paper, using the finite element method to integrate the equations in the spacial dimension, incorporates a time-staggered semi-implicit fractional step technique to improve stability at the higher Reynolds numbers. Unlike the upwind or directional differencing schemes normally employed to increase numerical stability, the present scheme does not introduce numerical damping or artificial viscocity, and becomes more stable as the Reynolds number increases. Results for this scheme are compared with various explicit integration schemes for the case of flow around a circular cylinder at Reynolds numbers of 100 to 400. For comparable accuracy the time-staggered semi-implicit fractional step technique was found to be up to 25 times more efficient than the other explicit integration schemes.     

Comparison of gas-dynamic models for hypersonic flow past bodies

Various gas-dynamic models for describing chemically non-equilibrium flows are compared using the example of the steady flow past the blunt nose of the “Buran” [1] and “Space Shuttle” vehicles during their descent from orbit. Models of locally self-similar approximations of the Navier-Stokes equations [2], of a chemically equilibrium and non-equilibrium complete viscous shock layer (CVSL) [3] and a model of a thin viscous shock layer (TVSL) [4] are considered. In all the models the occurrence of physicochemical processes was taken into account in the same way using fixed values of the constants for the gas-phase chemical reactions (their effect has been considered in [5]). Good agreement between the results of calculations of the heat flux at the critical point is found.

Chemically non-equilibrium flows have been considered earlier using the approximate Navier-Stokes equations [6], within the framework of a TVSL [7] and a CVSL [8, 9](for more detail, see the review [10]). The TVSL and CVSL models were compared in [11] in the case of flows of a uniform gas.     

BOUNDARY LAYER ASYMPTOTIC BEHAVIOR OF INCOMPRESSIBLE NAVIER-STOKES EQUATION IN A CYLINDER WITH SMALL VISCOSITY

The object of this article is to study the boundary layer appearing at large Reynolds number （small viscosity ε） incompressible Navier Stokes Equation in a cylinder in space dimension three. These are Navier-Stokes equations linearized around a fixed velocity flow： the authors study the convergence as ε →0 to the inviscid type equations, the authors define the correctors needed to resolve the boundary layer and obtain convergence results valid up to the boundary and the authors also study the behavior of the boundary layer when, simultaneously, time and the Reynolds number tend to infinity, in which case the boundary layer tends to pervade the whole domain.     

Nonlinear stability of a parabolic velocity profile in a plane periodic channel

An inviscid or viscous incompressible flow with a general parabolic velocity profile in an infinite plane periodic channel with parallel walls that can move is considered with the impermeability conditions (for the Euler equations) or the no-slip conditions (for the Navier-Stokes equations). The nonlinear (for the original equations) and nonlocal (for all Reynolds numbers) stability of the unperturbed flow with respect to arbitrary two-dimensional smooth perturbations of the initial velocity field is established.     

Boundary layer growth on two spheres

An asymptotic solution of the unsteady Navier-Stokes' equations is derived for the problem of the mutual hydrodynamic interaction between two solid spheres immersed in a viscous fluid moving at infinity in a direction parallel to their line of centers. It is assumed that the Reynolds and Strouhal numbers are much larger than one but small enough for the flow to remain stable. The influence of (i) the separation distance between the two bodies, (ii) the ratio of their radii, (iii) the Reynolds number and (iv) the acceleration parameter of the flow on the formation and the initial stage of development of the two boundary layers around the two spheres are investigated. The movement of the detachment points and hydrodynamic forces experienced by the two bodies are calculated. Some typical relations and findings are shown graphically.     

不同马赫数Navier-Stokes方程计算方法的研究

为了在低马赫数到高马赫数范围内求解可压缩Navier-Stokes方程,给出了基于预处理算法的PLU-SGS方法.将高分辨率AUSMPW格式与三阶MUSCL格式融合,将其扩展到三阶精度,并采用特征边界条件.为了验证该方法的有效性,通过求解曲线坐标系可压缩Navier-Stokes方程,对几个典型流动问题进行了数值计算.计算结果与文献计算结果或实验数据比较表明,该方法对不同马赫数Navier-Stokes方程的计算,具有较高的计算精度和收敛速度以及良好的稳定性.     

Lattice Boltzmann simulation for high-speed compressible viscous flows with a boundary layer

In this study, the lattice Boltzmann method is employed for simulating high-speed compressible viscous flows with a boundary layer. The coupled double-distribution-function lattice Boltzmann method proposed by Li et al. (2007) is employed because of its good numerical stability and non-free-parameter feature. The non-uniform mesh construction near the wall boundary in fine grids is combined with an appropriate wall boundary treatment for the finite difference method in order to obtain accurate spatial resolution in the boundary layer problem. Three typical problems in high-speed viscous flows are solved in the lattice Boltzmann simulation, i.e., the compressible boundary layer problem, shock wave problem, and shock boundary layer interaction problem. In addition, in-depth comparisons are made with the non-oscillatory and non-free-parameter dissipation (NND) scheme and second order upwind scheme in the present lattice Boltzmann model. Our simulation results indicate the great potential of the lattice Boltzmann method for simulating high-speed compressible viscous flows with a boundary layer. Further research is needed (e.g., better numerical models and appropriate finite difference schemes) because the lattice Boltzmann method is still immature for high-speed compressible viscous flow applications.     

一维粘性守恒律初边值问题粘性激波解的渐近稳定性

本文考虑一维可压缩Navier-Stokes方程有关初边值问题粘性激波解的渐近稳定性,通过L~2-能量估计,证明了在小扰动情况下,粘性激波是稳定的。     

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.     

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.     

Fourth-order difference methods for the system of 2D nonlinear elliptic partial differential equations

We present the fourth-order finite difference methods for the system of 2D nonlinear elliptic equations using 9-grid points on a square region R subject to Dirichlet boundary conditions. The method has been tested on viscous, incompressible 2D Navier-Stokes equations. The numerical results show that the proposed methods produce accurate and oscillation-free solutions for large Reynolds numbers.     

用CT-VOF法在数值波试验槽中生成线性和非线性波

为渡水槽中波的模拟和传播提出了二维的数值模型．假设流动的流体为粘性、不可压缩的,并将Navier-Stokes方程和连续性方程作为控制方程．用标准的k-ε模型来模拟紊流流动;用交错网格的有限差分法,离散化Navier-Stokes方程;并用简化的标识和单元(SMAC)方法进行求解．使用活塞型波发生器生成并传播波;数值渡水槽的端部采用敞开式的边界条件．为了证明模型的有效性,进行了一些标准的试验,如顶盖驱动的方腔测试试验、单向的常速度场试验以及干燥河床上的溃坝试验．为了论证方法的性能及其精度,将所生成波的结果与已有波理论的结果进行比较．最后,采用群集技术(CT)生成网格,并提出最佳的网格生成条件．     

Numerical modelling of turbulent flow over rough walls

The numerical simulation of turbulent flow over rough walls was carried out for various types of roughness. The mathematical model was based on the Reynolds averaged Navier-Stokes equations for incompressible flow. The two-equation SST and oneequation Spalart-Allmaras turbulence models were used. Boundary conditions on rough walls were prescribed directly on the wall using the SST model modified to account for wall roughness by Hellsten and Laine (1997) and the SA model modified by Aupoix and Spalart (2003). Turbulence models were tested for the constant pressure turbulent boundary layer on the rough wall formed by commercial abrasive paper and by tightly packed spheres. The effect of wall roughness on the decelerated flow over a smoothly contoured ramp with flow separation was investigated. Obtained results were compared with experimental data. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simplified Navier-Stokes equations for computing viscous gas flow past long bodies

Simplified Navier-Stokes equations are applied to analyze the flow of supersonic viscous gas at moderately large Reynolds numbers near the lateral surface of long bodies. Numerical integration of the equations is performed by the marching method, stabilizing in each step the solution of the nonstationary system of equations in the longitudinal coordinates. We consider the flow past cylinders and cones with a spherical blunt nose. We investigate the effect of the Reynolds number and the body shape on the flow field, the drag coefficient, and the heat flux. The numerical solutions of simplified and complete Navier-Stokes equations are compared.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 231–239, 1985.     

An application of a boundary element method to natural convection

The direct boundary element method is applied to the numerical modelling of thermal fluid flow in a transient state. The Navier-Stokes equations are considered under the Boussinesq approximation and the viscous thermal flow equations are expressed in terms of stream function, vorticity, and temperature in two dimensions. Boundary integral equations are derived using logarithmic potential and time-dependent heat potential as fundamental solutions. Boundary unknowns are discretized by linear boundary elements and flow domains are divided into a series of triangular cells. Charged points are translated upstream in the numerical evaluation of convective terms. Unknown stream function, vorticity, and temperature are staggered in the computational scheme.

Simple iteration is found to converge to the quasi steady-state flow. Boundary solutions for two-dimensional examples at a Reynolds number 100 and Grashoff number 10⁷ are obtained.