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.     

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 simulation of the performance of a snow fence with airfoil snow plates by FVM

The aim of this work is to analyze the efficiency of a snow fence with airfoil snow plates to avoid the snowdrift formation, to improve visibility and to prevent blowing snow disasters on highways and railways. In order to attain this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (snow particles) and fluid phase (air). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, we have used a particle transport model termed as Lagrangian particle tracking model, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 15,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. Finally, the conclusions of this work are exposed.     

Steady state numerical simulation of the particle collection efficiency of a new urban sustainable gravity settler using design of experiments by FVM

The aim of this work is to analyze the efficiency of a new sustainable urban gravity settler to avoid the solid particle transport, to improve the water waste quality and to prevent pollution problems due to rain water harvesting in areas with no drainage pavement. In order to get this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (sand particles) and fluid phase (water). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, a particle transport model termed as Lagrangian particle tracking model is used, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 2,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. The entire FVM model is built and the design of experiments (DOE) method was used to limit the number of simulations required, saving on the computational time significantly needed to arrive at the optimum configuration of the settler. Finally, conclusions of this work are exposed.     

Numerical modeling of the movement of a rigid particle in viscous fluid

Modeling the movement of a rigid particle in viscous fluid is a problem physicists and smathematicians have tried to solve since the beginning of this century. A general model for an ellipsoidal particle was first published by Jeffery in the twenties. We exploit the fact that Jeffery was concerned with formulae which can be used to compute numerically the velocity field in the neighborhood of the particle during his derivation of equations of motion of the particle. This is our principal contribution to the subject. After a thorough check of Jeffery's formulae, we coded software for modeling the flow around a rigid particle based on these equations. Examples of its applications are given in conclusion. A practical example is concerned with the simulation of sigmoidal inclusion trails in porphyroblast.     

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.     

Mathematical and Numerical Study of the 3D Atmospheric Boundary Layer Flows Including Pollution Dispersion

The paper presents a mathematical and numerical investigation of the atmospheric boundary layer (ABL) flow over 3D complex terrain part of which is represented by the real topography of the Krkonose mountains located in the Czech Republic. The flow is supposed to be turbulent, non‐stratified, viscous, incompressible and stationary. Two mathematical models have been formulated. The first model is based upon the RANS equations in the conservative form and the second one uses the Boussinesq approximation of RANS equations and takes the non‐conservative form. Also pollution dispersion over the complex 3D terrain has been considered in both models. The problem closure is achieved by an algebraic turbulence model and given boundary conditions.     

Computational fluid dynamics applied to the aerodynamic design of a land‐based supersonic vehicle

The BLOODHOUND SSC project was publicly announced in October 2008, with a primary engineering objective of designing, constructing and running a vehicle capable of achieving a speed of 1000 mph on land. The aerodynamic design of this vehicle is to be accomplished using computational simulation only and this paper describes the development and application of the approach adopted. The computational model employs a cell vertex finite volume algorithm for the solution of compressible viscous flow problems on unstructured hybrid meshes. A one equation turbulence model is adopted and the solution of the steady flow equations is obtained by explicit relaxation. For the combination of high Mach number, complex geometry and complex boundary conditions, involving rotating surfaces and a rolling ground, a consistent HLLC numerical flux function is adopted to ensure a stable procedure. To illustrate the impact of the approach upon the final configuration, a number of simulations undertaken to aid the aerodynamic design are described. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Limits for Detailed Description of Problems in Continuous Media Mechanics and Numerical Algorithm for Viscous Gas Flow Simulation

Doklady Mathematics - A mathematical model for viscous gas flow simulation with time scales bounded by the characteristic time between molecular collisions is considered. This approach is a variant...     

A particle method for 1-D compressible fluid flow

This paper proposes a novel particle scheme that provides convergent approximations of a weak solution of the Navier-Stokes equations for the 1-D flow of a viscous compressible fluid. Moreover, it is shown that all differential inequalities that hold for the fluid model are preserved by the particle method: mass is conserved, mechanical energy is decaying, and a modified mechanical energy functional is also decaying. The proposed particle method can be used both as a numerical method and as a method of proving existence of solutions for compressible fluid models.     

On the direct connection of rheological elements in nonlinear continuum mechanics

The structure of complicated phenomenological material models at finite strains is often exemplified with the help of rheological elements. Thereby, simple material behaviour, i.e. elasticity or viscous and plastic flow, are composes by components. In our approach, we directly apply this concept to obtain material models at finite strains. Towards this end, the thermodynamically consistent material behaviour of single elements is defined first. Subsequently, the elements are connected by evaluation of stress equilibria equations formulated on interconnecting configurations. The basic equations of this concept are presented using the example of nonlinear viscoelasticity of Maxwell type. The model results from a series connection of an elastic and a viscous element, whereas both are formulated in a thermodynamically consistent way within the framework on nonlinear continuum mechanics. Furthermore, an approach of numerical implementation using the stress equilibria is suggested. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

弹性动脉血管中血液流动特性的模拟和分析

研究了两个不同的非牛顿血液流动模型:低粘性剪切简单幂律模型和低粘性剪切及粘弹性振荡流的广义Maxwell模型．同时利用这两个非牛顿模型和牛顿模型,研究了磁场中刚性和弹性直血管中血液的正弦型脉动．在生理学条件下,大动脉中血液的弹性对其流动性态似乎并不产生影响,单纯低粘性剪切模型可以逼真地模拟这种血液流动．利用高剪切幂律模型模拟弹性血管中的正弦型脉动流,发现在同一压力梯度下,与牛顿流体相比较,幂律流体的平均流率和流率变化幅度都更小．控制方程用Crank-Niclson方法求解．弹性动脉中血液受磁场作用是产生此结果的直观原因．在主动脉生物流的模拟中,与牛顿流体模型比较,发现在匹配流率曲线上,幂律模型的平均壁面剪切应力增大,峰值壁面剪切应力减小．讨论了弹性血管横切磁场时的血液流动,评估了血管形状和表面不规则等因素的影响．     

Estimating the statistical error in calculating velocity and temperature components by the direct simulation Monte Carlo method

The direct simulation Monte Carlo (DSMC) method is widely used to solve problems in rarefied gas dynamics. In solving steady state problems, a special feature of the method is in using dependent sample values of random variables to calculate the macroparameters of gas flow. In this paper, the possibilities of methods of statistical physics to estimate the statistical error of the DSMC method are theoretically analyzed. A simple approach to approximate estimation of the statistical error in calculating temperature and velocity components is proposed. The approach is tested on a number of problems.     

Simulation of Separating Flows over a Backward-Facing Step

Simulation results are reported for plane two-dimensional viscous incompressible flow in a channel with an abrupt expansion. The mathematical model is provided by the quasi-hydrodynamic equations in the incompressible fluid approximation. The computations are carried out in a range of Reynolds numbers including both laminar and turbulent flow. As the Reynolds number increases, the solution bifurcates and the steady laminar flow changes to time-dependent flow. The computation results are consistent with known experimental data. Turbulence models were not used for large Reynolds number computations.     

Application of finite element method in aeroelasticity

The subject of this paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil, which can rotate around the elastic axis and oscillate in the vertical direction. The numerical simulation consists of the finite element approximation of the Navier–Stokes equations coupled with the system of ordinary differential equations describing the airfoil motion. The arbitrary Lagrangian–Eulerian (ALE) formulation of the Navier–Stokes equations, stabilization the finite element discretization and coupling of both models is discussed. Moreover, the Reynolds averaged Navier–Stokes (RANS) system of equations together with the Spallart–Almaras turbulence model is also discussed. The computational results of aeroelastic calculations are presented and compared with the NASTRAN code solutions.     

Aerodynamic focusing of inertial particles in supersonic micronozzles

A particle-laden flow in a supersonic micronozzle is studied using a one-way coupled two-fluid approach. The carrier gas parameters are obtained from the numerical solution of the Navier-Stokes equations, rarefaction effects are taken into account by imposing velocity slip and temperature jump boundary conditions on the nozzle walls. Under conditions considered, the flow around particles is transitional and free-molecular. As a result of numerical solution of the dispersed-phase equations in Lagrangian variables, two types of particle motion in the expanding part of the nozzle are detected: particle spraying and particle accumulation. The particle focusing effect is most pronounced for particles of about 1–2 µm in size. The particle number density fields contain singularities appearing on the envelopes of particle trajectories. However, the model of non-colliding particles remains valid because the mean distance between the particles near the singularities remains much greater than the particle size. The aerodynamic scheme of aerosol particle focusing proposed may be used in various technologies (microthrusters, needle-free drug injection, microfabrication, etc.). (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling and numerical aspects in 2d river flow simulation

Flow simulation in rivers is used for the prediction of waterlevel and runoff quantities. The presently used simulation by the German Federal Institute of Hydrology is based on a network approach with coupled 1d models in which the river flow is described by the St. Venant‐equations. Work is underway to extend this by 2d submodel elements using the shallow water equations to allow simulations in greater detail. The wetting and drying of parts of the river bed at varying waterlevels force us to formulate the flow simulation as free boundary problem. The constraints of embedding the 2d model into the existing network approach suggest to use a fixed grid approach. This leads to the problem of solving the shallow water equations on the whole grid. However, the shallow water equations degenerate for dry beds and therefore do not allow a straight forward discretisation.     

Numerical investigation of blood flow. Part I: In microvessel bifurcations

In some diseases there is a focal pattern of velocity in regions of bifurcation, and thus the dynamics of bifurcation has been investigated in this work. A computational model of blood flow through branching geometries has been used to investigate the influence of bifurcation on blood flow distribution. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier–Stokes equations for Newtonian fluids. The governing equations of mass and momentum conservation were solved to calculate the pressure and velocity fields. Movement of blood flow from an arteriole to a venule via a capillary has been simulated using the volume of fluid (VOF) method. The proposed simulation method would be a useful tool in understanding the hydrodynamics of blood flow where the interaction between the RBC deformation and blood flow movement is important. Discrete particle simulation has been used to simulate the blood flow in a bifurcation with solid and fluid particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, and translational and angular momenta. These kinds of simulations open a new way for modeling the dynamics of complex, viscoelastic fluids at the micro-scale, where both liquid and solid phases are treated with discrete particles.     

Conservation laws of turbulence models

The conservation of mass, momentum, energy, helicity, and enstrophy in fluid flow are important because these quantities organize a flow, and characterize change in the flow's structure over time. In turbulent flow, conservation laws remain important in the inertial range of wave numbers, where viscous effects are negligible. It is in the inertial range where energy, helicity (3d), and enstrophy (2d) must be accurately cascaded for a turbulence model to be qualitatively correct. A first and necessary step for an accurate cascade is conservation; however, many turbulent flow simulations are based on turbulence models whose conservation properties are little explored and might be very different from those of the Navier-Stokes equations.We explore conservation laws and approximate conservation laws satisfied by LES turbulence models. For the Leray, Leray deconvolution, Bardina, and Nth order deconvolution models, we give exact or approximate laws for a model mass, momentum, energy, enstrophy and helicity. The possibility of cascades for model quantities is also discussed.