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.     

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 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.     

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.     

On the numerical modelling of the transitional flow in rarefied gases

The aim of this work is to simulate rarefied gas flow in complex geometries, under flow conditions that range from the hydrodynamic, through the transitional, to the molecular regimes. Existing computational models apply to molecular or viscous flow, but the treatment of the transitional flow is still underdeveloped.To deal with the difficult transitional flow, two models with overlapping ranges of applicability are introduced. A direct simulation Monte Carlo (DSMC) type model, which can be used in the molecular and up to the lower transitional flow, has been designed. For the viscous to the upper transitional flow, a numerical model using a particle method is proposed. The objective is to obtain a smooth transition between the probabilistic simulation of particle histories and the deterministic approach of the solution of partial differential equations.The DSMC model has been successfully applied to molecular and lower transitional flow in a complex geometry with stationary and moving boundaries. The test results agree well with published data. The particle method was tested using simplified Navier-Stokes equations in a channel. Preliminary results in the low viscous range seem to indicate that the approach is viable.     

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.     

On The Atmospheric Boundary Layer Flow Modelling

The paper presents a mathematical and numerical investigation of the flow in the atmospheric boundary layer (ABL) over complex topography. The flow is supposed to be viscous, incompressible, turbulent and stationary. Two different mathematical–numerical approaches are briefly mentioned. Both models have been used to simulate a flow and pollution dispersion over a complex surface coal field in the North Bohemia which is supposed to be partially covered by a high forest stand. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical and asymptotic study of non‐axisymmetric magnetohydrodynamic boundary layer stagnation‐point flows

Both numerical and asymptotic analyses are performed to study the similarity solutions of three‐dimensional boundary‐layer viscous stagnation point flow in the presence of a uniform magnetic field. The three‐dimensional boundary‐layer is analyzed in a non‐axisymmetric stagnation point flow, in which the flow is developed because of influence of both applied magnetic field and external mainstream flow. Two approaches for the governing equations are employed: the Keller‐box numerical simulations solving full nonlinear coupled system and a corresponding linearized system that is obtained under a far‐field behavior and in the limit of large shear‐to‐strain‐rate parameter (λ). From these two approaches, the flow phenomena reveals a rich structure of new family of solutions for various values of the magnetic number and λ. The various results for the wall stresses and the displacement thicknesses are presented along with some velocity profiles in both directions. The analysis discovered that the flow separation occurs in the secondary flow direction in the absence of magnetic field, and the flow separation disappears when the applied magnetic field is increased. The flow field is divided into a near‐field (due to viscous forces) and far‐field (due to mainstream flows), and the velocity profiles form because of an interaction between two regions. The magnetic field plays an important role in reducing the thickness of the boundary‐layer. A physical explanation for all observed phenomena is discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

On iterative and Monte Carlo solutions of the Boltzmann equation

The two most commonly used techniques for solving the Boltzmann equation, with given boundary conditions, are first iterative equations (typically the BGK equation) and Monte Carlo methods. The present work examines the accuracy of two different iterative solutions compared with that of an advanced Monte Carlo solution for a one-dimensional shock wave in a hard sphere gas. It is found that by comparison with the Monte Carlo solution the BGK model is not as satisfactory as the other first iterative solution (Holway's) and that the BGK solution may be improved by using directional temperatures rather than a mean temperature.     

On exponential ill-conditioning and internal layer behavior

In the limit of small difTusivity the internal layer behavior associated with the initial-boundary value problems for a viscous shock equation and a reaction diffusion equation is analyzed.As a result of the occurrence of exponentially small eigenvalues for the linearized problems the steady state internal layer solutions are shown to very sensitive to small perturbations.For the time dependent problems the small eigenvalues give rise to exponentially slow internal layer motion.Accurate numerical methods are used to compute the steady state internal layer solutions and the slow internal layer motion.The relationship between the viscous shock problem and some exponentially ill-conditioned linear singular perturbation problems is discussed.     

Flow and heat transfer of double fractional Maxwell fluids over a stretching sheet with variable thickness

This paper presents research on the fractional boundary layer flow and heat transfer over a stretching sheet with variable thickness. Based on the Caputo operators, the double fractional Maxwell model and generalized Fourier's law are introduced to the constitutive relationships. The governing equations are solved numerically by utilizing the finite difference method. The effects of fractional parameters on the velocity and temperature field are analyzed. The results indicate that the larger is the fractional stress parameter, the stronger is the elastic characteristic. However, fluids show viscous fluid-like behavior for a larger value of fractional strain parameter. Moreover, the numerical solutions are in good agreement with the exact solution and the convergence order can achieve the expected first order. The numerical method in this study is reliable and can be extended to other fractional boundary layer problems over a variable thickness sheet.     

Heat transfer and turbulent diffusion in one-dimensional hydrodynamics without pressure

A one-dimensional transient non-linear problem of continuum mechanics is considered, the possibility of an accurate analytic solution of which is later based on a general local analysis of singular solutions known as the Painlevé test. For one-dimensional non-linear hydrodynamic models without pressure, with the transfer of a passive impunity, which generalizes the well-known Burgers' model, it is shown that it is possible to reduce the problem to linear problems when the kinetic coefficients (viscosity and thermal conductivity) are equal. Using examples of their accurate solutions, the high sensitivity of the structure of shock waves with impurity fronts to the satisfaction of the law of conservation of impurity in the models is demonstrated. When it is satisfied, each steady propagating shock wave with a viscous structure of the velocity field is accompanied by an impurity soliton. When several such shock waves merge (the accurately solved problem), concentration of the impurity in one overall soliton occurs. It is shown that, when the action of time-dependent Gaussian random forces is taken into account, an additional diffusive spreading of the perturbations, with a time-dependent diffusion coefficient, is superimposed on the linearized viscous behaviour of the main models.     

Case of a Boltzmann gas leading to the Smoluchowski coagulation equation

A special model of a rarefied hard-sphere gas is considered. The hard-sphere particles undergo absolutely elastic collisions. It is assumed that particles can collide only if their nonzero velocities are orthogonal to each other. The model makes it possible to proceed from the Boltzmann equation to the Smoluchowski coagulation equation, where coagulation means that the kinetic energies of the colliding particles are added. A Monte Carlo scheme for simulation of the phenomenon is described, and the convergence of the simulation algorithm is proved. The convergence of numerical results to exact solutions of the Smoluchowski equation and to finite-difference solutions is tested.     

Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

In this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.     

Cross validation of discontinuous Galerkin method and Monte Carlo simulations of charge transport in graphene on substrate

The aim of this work is to simulate the charge transport in a monolayer graphene on a substrate. This requires the inclusion of the scatterings of the charge carriers with the impurities and the phonons of the substrate, besides the interaction mechanisms already present in the graphene layer. As physical model, the semiclassical Boltzmann equation will be assumed. Two approaches will be used for the simulations: a numerical scheme based on the Discontinuous Galerkin method for finding deterministic (non stochastic) solutions and a new Direct Monte Carlo Simulation formulated in Romano et al. (J Comput Phys 302:267–284, 2015) in order to deal in the appropriate way with the Pauli exclusion principle for degenerate Fermi gases. A cross validation of the deterministic and stochastic solutions shows the robustness and accuracy of both the approaches.     

Stagnation point flow and heat transfer over a non-linearly moving flat plate in a parallel free stream with slip

An analysis is presented for the steady boundary layer flow and heat transfer of a viscous and incompressible fluid in the stagnation point towards a non-linearly moving flat plate in a parallel free stream with a partial slip velocity. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation, which are then solved numerically using the function bvp4c from Matlab for different values of the governing parameters. Dual (upper and lower branch) solutions are found to exist for certain parameters. Particular attention is given to deriving numerical results for the critical/turning points which determine the range of existence of the dual solutions. A stability analysis has been also performed to show that the upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and, therefore, not physically possible.     

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.     

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.