time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

spectral lagrangian methods for collisional models of non equilibrium statistical states we propose a new spectral lagrangian based deterministic solver for the non linear boltzmann transport equation bte in d dimensions for variable hard sphere vhs collision kernels with conservative or non conservative binary interactions the method is based on symmetries of the fourier transform of the collision integral where the complexity in its computation is reduced to a separate integral over the unit sphere s d 1 the conservation of moments is enforced by lagrangian constraints the resulting scheme implemented in free space is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro reversibility inelastic interactions or elastic models of slowing down process our simulations are benchmarked with available exact self similar solutions exact moment equations and analytical estimates for the homogeneous boltzmann equation both for elastic and inelastic vhs interactions benchmarking of the simulations involves the selection of a time self similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for maxwell molecules as studied first in bobylev et al a v bobylev c cercignani g toscani proof of an asymptotic property of self similar solutions of the boltzmann equation for granular materials journal of statistical physics 111 2003 403 417 and generalized to a wide range of related models in bobylev et al a v bobylev c cercignani i m gamba on the self similar asymptotics for generalized non linear kinetic maxwell models communication in mathematical physics in press url the method also produces accurate results in the case of inelastic diffusive boltzmann equations for hard spheres inelastic collisions under thermal bath where overpopulated non gaussian exponential tails have been conjectured in computations by stochastic methods t v noije m ernst velocity distributions in homogeneously cooling and heated granular fluids granular matter 1 57 1998 m h ernst r brito scaling solutions of inelastic boltzmann equations with over populated high energy tails journal of statistical physics 109 2002 407 432 s j moon m d shattuck j swift velocity distributions and correlations in homogeneously heated granular media physical review e 64 2001 031303 i m gamba s rjasanow w wagner direct simulation of the uniformly heated granular boltzmann equation mathematical and computer modelling 42 2005 683 700 and rigorously proven in gamba et al i m gamba v panferov c villani on the boltzmann equation for diffusively excited granular media communications in mathematical physics 246 2004 503 541 39 and a v bobylev i m gamba v panferov moment inequalities and high energy tails for boltzmann equations with inelastic interactions journal of statistical physics 116 2004 1651 1682

a fourth order finite difference solver for nerve conduction equation in rectangular domains a fourth order finite difference solver for the nerve conduction equation is presented that has mesh independent convergence rate and unconditional convergence for any time step size numerics are performed for a number of test problems which shows that the features of the nerve conduction equation are preserved the computational loads are roughly same as ones of normal second order scheme such as the crank nicolson scheme

symmetry breaking in vortical flows over cones theory and numerical experiments a stability analysis suggests that inviscid incompressible flow independent from angle of attack and regardless whether attached or separated over slender cones is only marginally stable in regions of decelerating circumferential flow reducing slenderness or surface curvature lowers the frequency spectrum of the harmonic perturbations and thus reduces their impact on the overall stability of flows over slender cones associating the notion of instabilities in such flows with the onset of vortex asymmetries provides a model for explaining a variety of flow phenomena in navier stokes simulations of laminar ret 0 6 1 2 and 2 4 x 106 is body length incompressible flows over three right circular cones cone half angle 8 2 5 5 and 10 deg at moderate to high angles of attack a 5 1 2 and 3

navier stokes computation of flow around a round edged double delta wing computations of three dimensional vortical flows over a thin round edged double delta wing with an aspect ratio of 2 05 are performed using an implicit upwind relaxation finite difference scheme the effects of grid and angle of attack on the navier stokes computations are studied coarse grid calculations can not predict the detailed structures of the vortical flowfields for lack of grid resolution on the contrary fine grid computations show that key features of vortex formation interaction and breakdown are simulated furthermore computed lift coefficients and spanswise surface static pressure distributions are in good agreement with the experimental data up to alpha 25 deg

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

upwind relaxation algorithms for the navier stokes equations analyse des erreurs de troncature de la differenciation spatiale et analyse de la stabilite de l algorithme de relaxation calcul pour une serie d ecoulements a l aide de la forme couche mince des equations calcul du decollement de couche limite induit par chocs

on fully discrete galerkin approximations for partial integro differential equations of parabolic type the subject of this work is the application of fully discrete galerkin finite element methods to initial boundary value problems for linear partial integro differential equations of parabolic type we investigate numerical schemes based on the pade discretization with respect to time and associated with certain quadrature formulas to approximate the integral term a preliminary error estimate is established which contains a term related to the quadrature rule to be specified in particular we consider quadrature rules with sparse quadrature points so as to limit the storage requirements without sacrificing the order of overall convergence for the backward euler scheme the crank nicolson scheme and a third order pade type scheme the specific quadrature rules analyzed are based on the rectangular the trapezoidal and simpson s rule for all the schemes studied optimal order error estimates are obtained in the case that the solution of the problem is smooth enough since this is important for our error analysis we also discuss the regularity of the exact solutions of our equations high order regularity results with respect to both space and time are given for the solution of problems with smooth enough data 17 refs

new implicit boundary procedures theory and applications analytical techniques for the application of implicit boundary conditions for inviscid flows to shock and body boundary layer conditions involving the euler equations are presented the theory of characteristics is used to update boundary points with spatial second order accuracy the method is useful for implicit schemes which feature approximate factorization and as such is incorporated into an existing pns code examples are provided in terms of flows over a cone over a maneuverable reentry vehicle and over a finned vehicle improvements in the convergence rate are demonstrated for the conical flow solutions

high order finite difference methods for a second order dual phase lagging models of microscale heat transfer two adi solvers are given for 2d and 3d heat conduct equations at microscale they are uniquely solvable it is shown that they are of order 2 in time and order 4 in space numerical results exhibit the efficiency of the solvers in this paper a compact alternating direction implicit adi method which combines the fourth order compact difference for the approximations of the second spatial derivatives and the approximation factorizations of difference operators is firstly presented for solving two dimensional 2d second order dual phase lagging models of microscale heat transfer by the discrete energy method it is shown that it is second order accurate in time and fourth order accurate in space with respect to l2 norms additionally the compact adi method is successfully generalized to solve corresponding three dimensional 3d problem also the convergence result of the solver for 3d case is given rigorously finally numerical examples are carried out to testify the computational efficiency of the algorithms and exhibit the correctness of theoretical results

high resolution upwind schemes for the three dimensional incompressible navier stokes equations based on flux difference splitting implicit high resolution schemes are constructed for efficient computations of steady state solution to the three dimensional incompressibl e navier stokes equations in curvilinear coordinates these schemes use first order accurate euler backward time differencing and second order central differencing for the viscous shear fluxes up to third order accurate upwind differencing is achieved through a reconstruction of the solution from its cell averages the reconstruction is accomplished by linear interpolation where the node stencils are selected such that in regions of smooth solution the flow is highly resolved while spurious oscillations in regions of rapid changes in gradient are still suppressed fairly rapid convergence to steady state solutions is attained with a vectorizable hybrid time marching method flows around a sharp edged delta wing are computed with the maximum accuracy of the upwind differencing restricted to first second and third order to illustrate the effect of accuracy oh the global and on the local vortical flowfields the results are validated with experimental data i introduction t here are basically two options to compute threedimensional low speed flows as steady state solutions to the navier stokes equations numerical approximations 1 to the compressible navier stokes equations for low mach numbers m or 2 to the incompressible navier stokes equations in view of the numerous time dependent methods both explicit1 and implicit 2 4 for the compressible navier stokes equations the first option appears to be a viable choice however the efficiency of these schemes deteriorates considerably with decreasing subsonic mach number due to the stiffness of the physical problems associated with lowspeed viscous flow the stiffness arises from both disparate characteristic speeds and length scales furthermore the accuracy of low speed flowfield computations with implicit approximate factorization af schemes e g those in refs 2 4 becomes questionable for low mach numbers since the factorization errors increase in an unbounded fashion for a decreasing mach number 5 that suggests to remove the sound speed time scale and to assume the flow to be truly incompressible aziz and heliums6 proposed to solve the navier stokes equations for incompressible fully three dimensional flow in a vector potential vorticity formulation because of its storage requirements and the necessity to solve three poisson equations at each time level that method failed to become popular that leaves solving the navier stokes equations for threedimensional incompressible flow in their primitive variable formulation common methods for this formulation advance the solution for the velocity field by explicitly or implicitly solving the equations of motion the updated velocity field is used to compute the corresponding pressure field by solving a poisson equation7 or by an iterative procedure with the equation of continuity as a compatibility condition 8 such schemes

a multiscale kinetic fluid solver with dynamic localization of kinetic effects this paper collects the efforts done in our previous works 7 9 10 to build a robust multiscale kinetic fluid solver our scope is to efficiently solve fluid dynamic problems which present non equilibrium localized regions that can move merge appear or disappear in time the main ingredients of the present work are the followings ones a fluid model is solved in the whole domain together with a localized kinetic upscaling term that corrects the fluid model wherever it is necessary this multiscale description of the flow is obtained by using a micro macro decomposition of the distribution function 9 the dynamic transition between fluid and kinetic descriptions is obtained by using a time and space dependent transition function to efficiently define the breakdown conditions of fluid models we propose a new criterion based on the distribution function itself several numerical examples are presented to validate the method and measure its computational efficiency

a direct method for the boltzmann equation based on a pseudo spectral velocity space discretization a deterministic method is proposed for solving the boltzmann equation the method employs a galerkin discretization of the velocity space and adopts as trial and test functions the collocation basis functions based on weights and roots of a gauss hermite quadrature this is defined by means of half and or full range hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space the resulting semi discrete boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches the spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of maxwell molecules as an application the two dimensional cavity flow of a gas composed of hard sphere molecules is studied for different knudsen and mach numbers although computationally demanding the proposed method turns out to be an effective tool for studying subsonic slightly rarefied gas flows

lattice boltzmann models based on half range gauss hermite quadratures we discuss general features of thermal lattice boltzmann models based on half range gauss quadratures specialising to the half range gauss hermite and gauss laguerre cases the main focus of the paper is on the construction of high order half range hermite lattice boltzmann hhlb models the performance of the hhlb models is compared with that of laguerre lattice boltzmann llb and full range hermite lattice boltzmann hlb models by conducting convergence tests with respect to the quadrature order on stationary profiles of the particle number density macroscopic velocity temperature and heat fluxes in the two dimensional couette flow the bhatnagar gross krook bgk collision term is used throughout the paper to reduce the computational costs of the numerical simulations we use mixed lattice boltzmann models constructed using different quadrature methods on each cartesian axis for kn 0 01 the hlb models require the least number of velocities to satisfy our convergence test when kn 0 05 the hlb models are outperformed in terms of number of velocities employed by both the llb and the hhlb models moreover we find that the hhlb models require less quadrature points than the llb models at all tested values of kn which we attribute to the maxwellian form of the weight function for the half range hermite polynomials

a gaussian quadrature procedure for use in the solution of the boltzmann equation and related problems abstract a new gaussian quadrature procedure is developed for integrals of the form 0 e y 2 y p y dy for p 0 1 and 2 recursion relations are derived for the coefficients in the general three term recurrence relation for the polynomials whose roots are the quadrature abscissae a comparison with the gauss laguerre quadrature procedure is presented solutions of the chemical kinetic boltzmann equation are obtained with a discrete ordinate method based on this gaussian quadrature procedure the results are compared with previous solutions obtained with a polynomial expansion method

solving the boltzmann equation in n log 2 n in c mouhot and l pareschi fast algorithms for computing the boltzmann collision operator math comp to appear c mouhot and l pareschi c r math acad sci paris 339 2004 pp 71 76 fast deterministic algorithms based on spectral methods were derived for the boltzmann collision operator for a class of interactions including the hard spheres model in three dimensions these algorithms are implemented for the solution of the boltzmann equation in two and three dimensions first for homogeneous solutions then for general nonhomogeneous solutions the results are compared to explicit solutions when available and to monte carlo methods in particular the computational cost and accuracy are compared to those of monte carlo methods as well as to those of previous spectral methods finally for inhomogeneous solutions we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium which was suggested in the work l desvillettes and c villani invent math 159 2005 pp 245 316

computation of turbulent supersonic flows around pointed bodies having crossflow separation abstract a recently reported thin layer parabolized navier stokes method has been used to compute turbulent supersonic flows around pointed bodies at large incidence these flow fields are complex and contain extensive regions of crossflow separation extensive investigations were carried out to assess the effects of grid resolution in the viscous region and inviscid region of the leeward side vortices and the effects of the algebraic eddy viscosity turbulence model comparisons between computed and experimentally measured flow fields of several pointed bodies show significant improvement in the computed flow fields obtained using a properly modified turbulence model in the crossflow separation region and adequate spatial grid resolution the effect of adding the circumferential and cross viscous terms was found to be insignificant for the present cases

comparison of baldwin lomax turbulence models for two dimensional open cavity computations a study of several modified versions of the baldwin lomax turbulence model applied to two dimensional supersonic open cavity flowfields are performed the results are clearly affected by the levels of turbulent viscosity imposed in the cavity

a hybrid method for hydrodynamic kinetic flow part i a particle grid method for reducing stochastic noise in kinetic regimes in this work we present a hybrid particle grid monte carlo method for the boltzmann equation which is characterized by a significant reduction of the stochastic noise in the kinetic regime the hybrid method is based on a first order splitting in time to separate the transport from the relaxation step the transport step is solved by a deterministic scheme while a hybrid dsmc based method is used to solve the collision step such a hybrid scheme is based on splitting the solution in a collisional and a non collisional part at the beginning of the collision step and the dsmc method is used to solve the relaxation step for the collisional part of the solution only this is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution performing collisions as in a standard dsmc method and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution the latter is added to a piecewise constant reconstruction of the non collisional part of the solution which in fact remains unchanged during the relaxation step numerical results show that the stochastic noise is significantly reduced at large knudsen numbers with respect to the standard dsmc method indeed in this algorithm the particle scheme is applied only on the collisional part of the solution so only this fraction of the solution is affected by stochastic fluctuations but since the collisional part of the solution reduces as the knudsen number increases stochastic noise reduces as well at large knudsen numbers

a low variance deviational simulation monte carlo for the boltzmann equation we present an efficient particle method for solving the boltzmann equation the key ingredients of this work are the variance reduction ideas presented in baker and hadjiconstantinou l l baker n g hadjiconstantinou variance reduction for monte carlo solutions of the boltzmann equation physics of fluids 17 2005 art no 051703 and a new collision integral formulation which allows the method to retain the algorithmic structure of direct simulation monte carlo dsmc and thus enjoy the numerous advantages associated with particle methods such as a physically intuitive formulation computational efficiency due to importance sampling low memory usage no discretization in velocity space and the ability to naturally and accurately capture discontinuities in the distribution function the variance reduction achieved by simulating only the deviation from equilibrium results in a significant computational efficiency advantage for low signal flows e g low flow speed compared to traditional particle methods such as dsmc in particular the resulting method can capture arbitrarily small deviations from equilibrium at a computational cost that is independent of the magnitude of this deviation the method is validated by comparing its predictions with dsmc solutions for spatially homogeneous and inhomogeneous problems

fast algorithms for computing the boltzmann collision operator the development of accurate and fast numerical schemes for the five fold boltzmann collision integral represents a challenging problem in scientific computing for a particular class of interactions including the so called hard spheres model in dimension three we are able to derive spectral methods that can be evaluated through fast algorithms these algorithms are based on a suitable representation and approximation of the collision operator explicit expressions for the errors in the schemes are given and spectral accuracy is proved parallelization properties and adaptivity of the algorithms are also discussed

a spectral transform method for singular sturm liouville problems with applications to energy diffusion in plasma physics we develop a spectrally accurate numerical method to compute solutions of a model pde used in plasma physics to describe diffusion in velocity space due to fokker planck collisions the solution is represented as a discrete and continuous superposition of normalizable and nonnormalizable eigenfunctions via the spectral transform associated with a singular sturm liouville operator we present a new algorithm for computing the spectral density function of the operator that uses chebyshev polynomials to extrapolate the value of the titchmarsh weyl m function from the complex upper half plane to the real axis the eigenfunctions and density function are rescaled and a new formula for the limiting value of the m function is derived to avoid amplification of roundoff errors when the solution is reconstructed the complexity of the algorithm is also analyzed showing that the cost of computing the spectral density function at a point grows less rapidly than any fractional inverse power of the desired acc

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

a new high order algorithm for a class of nonlinear evolution equation we derive a new finite difference scheme which is easily extended to fourth order accurate in both temporal and spatial dimensions it is shown through a discrete fourier analysis that the method is unconditionally stable for a 2d problem it requires only a regular seven point difference stencil similar to that used in the standard second order algorithms such as the crank nicolson algorithm numerical experiments are conducted to test its high accuracy and efficiency of the new algorithm

numerical solution of the boltzmann equation i spectrally accurate approximation of the collision operator in this paper we show that the use of spectral galerkin methods for the approximation of the boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost we prove that the spectral algorithm preserves the total mass and approximates with infinite order accuracy momentum and energy consistency of the method is also proved and a stability result for a smoothed positive scheme is given we demonstrate that the fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method

error estimates for semidiscrete finite element methods for parabolic integro differential equations application des methodes d elements finis aux equations integro differentielles de type parabolique avec un noyau integral forme par un operateur aux derivees partielles d ordre 2

the spatially non uniform convergence of the numerical solutions of flows abstract the spatial distribution of the numerical disturbances that are generated during a numerical solution of a flow is examined it is shown that the distribution of the disturbances is not uniform in regions where the structure of a flow is simple the magnitude of the generated disturbances is small and their decay is fast however in complex flow regions as in separation and vortical areas large magnitude disturbances appear and their decay may be very slow the observed nonuniformity of the numerical disturbances makes possible the reduction of the calculation time by application of what may be called the partial grid calculation technique in which a major part of the calculation procedure is applied in selective subregions where the velocity disturbances are large and not within the whole grid this technique is expected to prove beneficial in large scale calculations such as the flow about complete aircraft configurations at high angle of attack also it has been shown that if the navier stokes equations are written in a generalized coordinate system then in regions in which the grid is fine such as near solid boundaries the norms become infinitesimally small because in these regions the jacobian has very large values thus the norms unless they are unscaled by the jacobian reflect only the changes that happen at the outer boundaries of the computation domain where the value of the jacobian approaches unity and not in the whole flow field inc

high order numerical methods for the space non homogeneous boltzmann equation in this paper we present accurate methods for the numerical solution of the boltzmann equation of rarefied gas the methods are based on a time splitting technique the transport is solved by a third order accurate in space positive and flux conservative pfc method the collision step is treated by a fourier approximation of the collision integral which guarantees spectral accuracy in velocity coupled with several high order integrators in time strang splitting is used to achieve second order accuracy in space and time several numerical tests illustrate the properties of the methods

thin vs full navier stokes computation for high angle of attack aerodynamics numerical methods than can simulate the asymmetry phenomenon must account for the fluid viscosity objections have been raised regarding the justification for utilizing the thin layer navier stokes tlns model to study flow asymmetries because the viscosity effect of the circumferential and streamwise flow components might become important at the high attack angles of interest our objective is to test this question by comparing the three dimensional beam warming thin layer navier stokes bw tlns algorithm and beam warming full navier stokes bw fns algorithm where all viscous terms are kept the three dimensional flux splitting thin layer navier stokes fs tlns algorithm will be used as an additional reference

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

a modified backward time discretization for nonlinear parabolic equations using patch approximations a modified backward difference time discretization is presented for galerkin approximations for the nonlinear parabolic equation in mathbb r d q x u u t sum i j 1 d frac partial partial x i left a ij x u frac partial u partial x j right sum i 1 d b i x u frac partial u partial x i f x t u this procedure uses a local approximation of the coefficients based on patches of finite elements to update the coefficient matrix at each time step and allows for efficient use of such direct methods for solving linear systems of equations as nested dissection optimal order error estimates and almost optimal order work requirements are derived an alternating direction variant is also presented which has both optimal order error estimates and optimal order work requirements

the finite element numerical analysis for a class nonlinear evolution equations a complete stability and convergence analysis is given for two level scheme for the approximate solution of a class generalized nerve conduction equation defining a kind of truncation error and using a kind of technique optimal order estimates in l 2 is constructively obtained

finite difference solutions of the euler equations in the vicinity of sharp edges attempts have been made to explain why finite difference solutions of the euler equations can describe flows with large vortical structures around sharp edged bodies the present paper is concerned with the influence of a singular sharp edge on the truncation error for a set of discretized euler equations an analysis is conducted of the distribution of the truncation error of one finite difference approximation of the euler equations near a sharp edge of a thin plate the analysis leads to a determination of the size of the region of the neighborhood of such a singularity attention is given to the consistency of a discretization of the euler equations and numerical experiments

navier stokes solutions for vortical flows over a tangent ogive cylinder reynolds number re effects on low speed vortical flows over 3 5 caliber tangent ogive cylinders at two angles of attack 20 and 30 deg are computationally assessed for re d 0 2 3 million d maximum diameter the flowfield results are steady state solution to the three dimensional incompressible navier stokes equations in their thin layer approximation for the first time a crossflow separation pattern which consists of a laminar separation bubble with a subsequent transition in the separating shear layer that forms a primary vortex has been modeled

accurate spectral numerical schemes for kinetic equations with energy diffusion we examine the merits of using a family of polynomials that are orthogonal with respect to a non classical weight function to discretize the speed variable in continuum kinetic calculations we consider a model one dimensional partial differential equation describing energy diffusion in velocity space due to fokker planck collisions this relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach it also allows us to integrate in time exactly and to focus entirely on the effectiveness of the discretization of the speed variable we show that for a fixed number of modes or grid points the non classical polynomials can be many orders of magnitude more accurate than classical hermite polynomials or finite difference solvers for kinetic equations in plasma physics we provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials for the non classical polynomials if the initial condition is not smooth at the origin when interpreted as a three dimensional radial function the exact solution leaves the polynomial subspace for a time but returns up to roundoff accuracy to the same point evolved to by the projected dynamics in that time by contrast using classical polynomials the exact solution differs significantly from the projected dynamics solution when it returns to the subspace we also explore the connection between eigenfunctions of the projected evolution operator and non normalizable eigenfunctions of the full evolution operator as well as the effect of truncating the computational domain

comparison between navier stokes and thin layer computations for separated supersonic flow in the numerical simulation of high reynolds number flow one can frequently supply only enough grid points to resolve the viscous terms in a thin layer as a consequence a body or stream aligned coordinate system is frequently used and viscous terms in this direction are discarded it is argued that these terms cannot be resolved and computational efficiency is gained by their neglect dropping the streamwise viscous terms in this manner has been termed the thin layer approximation the thin layer concept is an old one and similar viscous terms are dropped for example in parabolized navier stokes schemes however such schemes also make additional assumptions so that the equations can be marched in space and such a restriction is not usually imposed on a thin layer model the thin layer approximation can be justified in much the same way as the boundary layer approximation it requires therefore a body or stream aligned coordinate and a high reynolds number unlike the boundary layer approximation the same equations are used throughout so there is no matching problem furthermore the normal momentum equation is not simplified and the convection terms are not one sided differenced for marching consequently the thin layer equations are numerically well behaved at separation and require no special treatment there nevertheless the thin layer approximation receives criticism it has been suggested that the approximation is invalid at separation and more recently that it is inadequate for unsteady transonic flow although previous comparisons between the thin layer and navier stokes equations have been made these comparisons have not been adequately documented

evaluation of algebraic turbulence models for pns predictions of supersonic flow past a sphere cone les previsions de navier stokes parabolises sont comparees aux mesures pour des ecoulements adiabatiques a mach 3 et des ecoulements sur paroi froide a mach 5 et 8

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods

numerical investigation of crossflow separation on a three caliber tangent ogive cylinder a parabolized navier stokes code was used to calculate laminar crossflow separation on a simple body of revolution convergence lines were defined as lines on which skin friction lines converge and were determined visually from the computed skin friction patterns these convergence lines were shown to be limit lines and not envelopes examination of crossflow cuts of stream tubes that encircled the body indicated that crossflow separation defined by a fold in the stream tube began after the onset of negative crossflow velocities but before a convergence line could be determined from the skin friction patterns there were no rapid changes or extrema in the pressure skin friction vorticity or flow angle distributions that indicated where separation occurred instead crossflow separation developed gradually and involved an extended region of the surface