fram nonlinear damping algorithms for the continuity equation abstract a nonlinear filtering methodology has been developed for the numerical solution of the convective terms in the eulerian form of the hydrodynamic equations fram filtering remedy and methodology introduces a strong local nonlinear dissipation into each of the governing equations to dampen non physical noise due to the properties of the solution algorithm selected for the convective terms in the equations the technique is developed for the multidimensional continuity equation and tested on several differencing schemes in both one and two spatial dimensions the technique is general in that there are no undetermined coefficients

a coupled finite volume solver for the solution of incompressible flows on unstructured grids this paper reports on a newly developed fully coupled pressure based algorithm for the solution of laminar incompressible flow problems on collocated unstructured grids the implicit pressure velocity coupling is accomplished by deriving a pressure equation in a procedure similar to a segregated simple algorithm using the rhie chow interpolation technique and assembling the coefficients of the momentum and continuity equations into one diagonally dominant matrix the extended systems of continuity and momentum equations are solved simultaneously and their convergence is accelerated by using an algebraic multigrid solver the performance of the coupled approach as compared to the segregated approach exemplified by simple is tested by solving five laminar flow problems using both methodologies and comparing their computational costs results indicate that the number of iterations needed by the coupled solver for the solution to converge to a desired level on both structured and unstructured meshes is grid independent for relatively coarse meshes the cpu time required by the coupled solver on structured grid is lower than the cpu time required on unstructured grid on dense meshes however this is no longer true for low and moderate values of the grid aspect ratio the number of iterations required by the coupled solver remains unchanged while the computational cost slightly increases for structured and unstructured grid systems the required number of iterations is almost independent of the grid size at any value of the grid expansion ratio recorded cpu time values show that the coupled approach substantially reduces the computational cost as compared to the segregated approach with the reduction rate increasing as the grid size increases

solution of the implicitly discretised reacting flow equations by operator splitting abstract a non iterative method for handling the coupling of the implicitly discretised time dependent fluid flow equations is described the method is based on the use of pressure and velocity as dependent variables and is hence applicable to both the compressible and incompressible versions of the transport equations the main feature of the technique is the splitting of the solution process into a series of steps whereby operations on pressure are decoupled from those on velocity at each step with the split sets of equations being amenable to solution by standard techniques at each time step the procedure yields solutions which approximate the exact solution of the difference equations the accuracy of this splitting procedure is assessed for a linearised form of the discretised equations and the analysis indicates that the solution yielded by it differs from the exact solution of the difference equations by terms proportional to the powers of the time step size by virtue of this it is possible to dispense with iteration thus resulting in an efficient implicit scheme while retaining simplicity of implementation relative to contemporary block simultaneous methods this is verified in a companion paper which presents results of computations carried out using the method

an asymptotic preserving all speed scheme for the euler and navier stokes equations we present an asymptotic preserving all speed scheme for the simulation of compressible flows valid at all mach numbers ranging from very small to order unity the scheme is based on a semi implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly this discretization which is the key to the asymptotic preserving property provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime the divergence free condition on the velocity in the incompressible regime is respected and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations the implicit treatment of the acoustic part allows the time step to be independent of the mach number the scheme is conservative and applies to steady or unsteady flows and to general equations of state one and two dimensional numerical results provide a validation of the asymptotic preserving all speed properties

application of time iterative schemes to incompressible flow the computation of steady incompressible flows by an euler implicit algorithm is studied using both the incompressible equations and the low mach number compressible equations the incompressible equations are handled by adding an artificial time derivative to the continuity equation this allows both the pressure and velocity to be obtained implicitly in one dimensional problems both systems converge rapidly even at low mach numbers where the eigenvalues are very stiff in two dimensions where approximate factorization is required the presence of stiff eigenvalues is highly detrimental stiffness can be avoided in the incompressible equations by selecting an appropriate pseudo mach number this insures reliable convergence and results in an efficient incompressible flow algorithm in the case of compressible equations the mach number cannot be chosen arbitrarily and the contamination introduced by approximate factorization must be removed a matrix preconditioning factor that accomplishes this is developed and demonstrated with this modification the convergence rate is the same as in the incompressible case and is independent of mach number rapid convergence is observed at mach numbers as low as 6 05

the application of preconditioning in viscous flows a time derivative preconditioning algorithm that is effective over a wide range of flow conditions from inviscid to very diffusive flows and from low speed to supersonic flows has been developed the algorithm uses a preconditioning matrix that introduces well conditioned eigenvalues while simultaneously avoiding nonphysical time reversals for viscous flows the resulting algorithm also provides a mechanism for controlling the inviscid and viscous time step parameters at very diffusive flows thereby ensuring rapid convergence for very viscous flows as well as for inviscid flows computational capabilities are demonstrated through computation of a wide variety of problems convergence rates are shown to be accelerated by as much as two orders of magnitudes while providing solutions that are indentical to those obtained without preconditioning method

the convergence of shooting methods for singular boundary value problems we investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems particular attention is paid to the well posedness of the process it is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available moreover convergence results for a perturbed newton iteration are discussed

a note on the sturmian theorem for singular boundary value problems in this paper we study properties of the eigenfunctions of singular boundary value problems the main results show that for a class of singular boundary value problems the sturmian oscillation theorem holds it is a generalization of the standard case to the singular case

a conservative discrete compatibility constraint low mach pressure correction algorithm for time accurate simulations of variable density flows in this paper we develop the discrete compatibility constraint pressure correction algorithm for transient simulations of variable density flows at low mach numbers the constraint for the velocity field is constructed from a combination of the discrete equations of continuity and scalar e g energy transport imposing that the newly predicted state must be compatible in agreement with the equation of state this way mass and scalar conservation are guaranteed and the equation of state is exactly fulfilled at every time step for comparison reasons two other types of well known pressure correction algorithms are also used the first class denoted as continuity constraint pressure correction is based on a constraint for the velocity field that is derived solely from the continuity equation the second class denoted as analytical compatibility constraint pressure correction constructs the constraint from an analytical combination of the material derivative of the equation of state and the continuity and scalar equations the algorithms are tested for three example fluid configurations a single fluid ideal gas a two fluid inert mixture and a two fluid reacting mixture the latter is special in the sense that the equation of state is non linear and not everywhere differentiable the continuity constraint pressure correction algorithm yields unstable solutions if density ratios are high the analytical compatibility constraint pressure correction algorithm yields stable results but the predicted states do not correspond to the equation of state the discrete compatibility constraint pressure correction algorithm performs well on all test cases the simulation results are stable and exactly match the equation of state

exponential runge kutta for the inhomogeneous boltzmann equations with high order of accuracy we consider the development of exponential methods for the robust time discretization of space inhomogeneous boltzmann equations in stiff regimes compared to the space homogeneous case or more in general to the case of splitting based methods studied in dimarco pareschi 7 a major difficulty is that the local maxwellian equilibrium state change with respect to time and thus needs a proper numerical treatment we show how to derive asymptotic preserving ap schemes of arbitrary order and in particular by using the shu osher representation of runge kutta methods we explore the monotonicity properties of such schemes like strong stability preserving ssp and positivity preserving several numerical results confirm our analysis

application of a new control volume based finite element formulation to the shock tube problem introduction i this article the performance of a newly developed pressurebased method for incompressible compressible flow calculation is investigated by solving the shock tube problem this problem has served as a benchmark case in the literature since it has many transient flow features pressure based methods have been increasingly used for the solution of compressible flow the present method is a generalized version of the method of ref 5 which was developed for incompressible flow computation using primitive variables primitive variable methods are widely used in the incompressible range here it is demonstrated that the present method is also very capable of solving transient and compressible flows incorporating strong discontinuities the governing equations are discretized using a control volumebased finite element cvbfe method the cvbfe method was initially developed for the solution of heat transfer and fluid flow problems later its application to compressible flow computation was presented in the literature with the use of a new formulation for the mass conserving velocity or convecting velocity proposed by karimian and schneider a strong coupling between pressure and velocity is created which does not allow any kind of checkerboard oscillations the effect of three different methods for energy flux evaluation on the results are also examined in this article the present results agree very well with the exact solution no explicit artificial viscosity is added to the solution

general strong conservation formulation of navier stokes equations in nonorthogonal curvilinear coordinates the selection of primary dependent variables for the solution of navier stokes equations in the curvilinear body fitted coordinates is still an unsettled issue reported formulations with primitive variables involve contravariant velocity components cartesian components and velocity projections also known as resolutes most of the formulations result in a weak conservation form of the momentum equations which contain grid line curvature and divergence related coriolis and centrifugal terms this paper presents a general strong conservation formulation of the momentum equations allowing the flexibility in choosing the various forms of the velocity components as the dependent variables ambiguous issues relating geometrical topology and forms of governing equations are discussed and clarified computational results obtained with both strong and weak forms are presented and compared to known analytical experimental data the results confirm the soundness of the formulation 19 refs

a mach uniform algorithm coupled versus segregated approach a mach uniform algorithm is an algorithm with a good convergence rate for any level of the mach number in this paper the severe time step restriction for low speed flows is removed by treating the acoustic and diffusive terms implicitly after identification of these terms in the conservative set we end up with a semi implicit system the way to solve this system can be chosen three different solution techniques are presented a fully coupled algorithm the coupled pressure and temperature correction algorithm k nerinckx j vierendeels e dick mach uniformity through the coupled pressure and temperature correction algorithm journal of computational physics 206 2005 597 623 and a fully segregated pressure correction algorithm we analyse the convergence behavior of the considered algorithms for some typical flow problems moreover a fourier stability analysis is done for inviscid flow the fully segregated and the fully coupled algorithm need about as much time steps to reach steady state therefore the more segregation is introduced the faster the calculation can be done in case of heat transfer the fully segregated pressure correction algorithm suffers from a diffusive time step limit this is not the case for the semi segregated coupled pressure and temperature correction algorithm finally when the gravity terms play an important role only the fully coupled algorithm can avoid an additional time step restriction

a high resolution pressure based algorithm for fluid flow at all speeds abstract a new collocated finite volume based solution procedure for predicting viscous compressible and incompressible flows is presented the technique is equally applicable in the subsonic transonic and supersonic regimes pressure is selected as a dependent variable in preference to density because changes in pressure are significant at all speeds as opposed to variations in density which become very small at low mach numbers the newly developed algorithm has two new features i the use of the normalized variable and space formulation nvsf methodology to bound the convective fluxes and ii the use of a high resolution scheme in calculating interface density values to enhance the shock capturing property of the algorithm the virtues of the newly developed method are demonstrated by solving a wide range of flows spanning the subsonic transonic and supersonic spectrum results obtained indicate higher accuracy when calculating interface density values using a high resolution scheme

a novel approach to the numerical solution of boundary value problems on infinite intervals the classical numerical treatment of two point boundary value problems on infinite intervals is based on the introduction of a truncated boundary instead of infinity where appropriate boundary conditions are imposed then the truncated boundary allowing for a satisfactory accuracy is computed by trial motivated by several problems of interest in boundary layer theory here we consider boundary value problems on infinite intervals governed by a third order ordinary differential equation we highlight a novel approach to define the truncated boundary the main result is the convergence of the solution of our formulation to the solution of the original problem as a suitable parameter goes to zero in the proposed formulation the truncated boundary is an unknown free boundary and has to be determined as part of the solution for the numerical solution of the free boundary formulation a noniterative and an iterative transformation method are introduced furthermore we characterize the class of free boundary value problems that can be solved noniteratively a nonlinear flow problem involving two physical parameters and belonging to the characterized class of problems is then solved moreover the falkner skan equation with relevant boundary conditions is considered and representative results obtained by the iterative transformation method are listed for the homann flow all the obtained numerical results clearly indicate the effectiveness of our approach finally we discuss the possible extensions of the proposed approach and for the question of a priori error analysis

pressure based control volume finite element method for flow at all speeds we present a collocated pressure based method for the solution of viscouslinviscid fluid flow problems incorporating compressible and incompressible regimes the solution domain is discretized using a control volume based finite element method the fully implicit method does not experience any stability difficulties due to compressibility effects at low mach numbers the pressure velocity decoupling problem is fully resolved herein for flow regimes from incompressible through compressible flow transonic and supersonic flows are covered by including the pressure density coupling which arises in high speed flows in the continuity equation this is done in such a manner that the continuity equation remains a constraint equation for pressure in all flow regimes test problems with wide variations in geometry and fluid physics have been successfully solved demonstrating the generality and computational capability of the method the scheme is both efficient and robust the accuracy of the method has been checked by comparing the results with other numerical results available in the literature and with the exact solution

pressure based compressible calculation method utilizing total variation diminishing schemes a pressure based implicit procedure to solve the euler and navier stokes equations on a nonorthogonal mesh with collocated finite volume formulation is described the boundedness criteria for this procedure are determined from total variation diminishing tvd schemes which are based on characteristic variables and are applied to the fluxes of the convected quantities including mass flow rate the procedure incorporates the e eddy viscosity turbulence model the algorithm is first tested for inviscid flows at different mach numbers ranging from subsonic to supersonic on a bump in a channel geometry where the results are compared with other existing numerical solutions the method is then validated against experiment and another numerical solution for the case of turbulent transonic flow through a gas turbine rotor blade cascade for which wind tunnel experimental data exist

preconditioned methods for solving the incompressible low speed compressible equations abstract acceleration methods are presented for solving the steady state incompressible equations these systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state we also consider the compressible equations in conservation form with slow flow two arbitrary functions and are introduced in the general preconditioning an analysis of this system is presented and an optimal value for is determined given a constant it is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed several generalizations to the compressible equations are presented which extend previous results

low diffusion flux splitting methods for flows at all speeds methods for extending the advective upwind splitting method ausm family of low diffusion flux splitting schemes to operate effectively at all flow speeds are developed the extensions developed are designed for use with time derivative preconditioning and are based on the idea that the speed of sound should cease to be an important scaling parameter for the diffusive contributions to the interface flux as the mach number becomes small using this criterion alternative definitions for the interface mach numbers are developed and methods for ensuring pressure velocity coupling at low speeds are formulated results are presented for inviscid flows through a channel at various mach numbers developing viscous flow in a two dimensional duct driven cavity flows at various mach and reynolds numbers flow over a backward facing step and hydrogen nitrogen mixing layers

pressure based multigrid algorithm for flow at all speeds a full multigrid full approximation storage fmg fas method is developed along with a unified pressurebased algorithm to handle the complex fluid flow problems encompassing both compressible and incompressible regimes besides highlighting the key elements of the present algorithm on a curvilinear staggered grid system several flow problems ranging from incompressible to hypersonic speeds are computed in conjunction with an adaptive grid method to demonstrate its computational capability with wide variations in geometry fluid physics and grid distribution speedup between the single and multigrid procedures can be obtained because different pressure boundary conditions are needed for incompressible subsonic and supersonic inlet and outlet the number of grid levels that can be effectively used by the present multigrid method appears flow dependent the implications of this observation are discussed i introduction u se of pressure correction methods for solving incompressible viscous flow problems in cartesian or cylindrical coordinates is well established and a number of methods of this type such as simple 1 2 simpler 2 and piso3 have been developed the basic similarity of these methods is that the momentum equations are first solved using a guessed pressure field resulting in a tentative velocity field an equation for the pressure corrections is obtained via manipulation of the momentum and continuity equations this equation is solved to obtain the pressure corrections and the velocities are then corrected to satisfy the continuity equation the extension of the simple algorithm to a general nonorthogonal curvilinear coordinate system has been described in refs 4 6 and has been successfully applied to various problems as documented in ref 6 when solving a viscous flow problem using a pressure based method the computer time required to solve the pressure correction equation is often a sizable fraction of the total computational effort 7 8 van doormaal and raithby7 report that this fraction can be as high as 80 of the total cpu time this observation is also qualitatively confirmed in ref 8 which finds that for high reynolds number flows the pressure correction equation requires many more sweeps of a point symmetric successive over relaxation point ssor or line ssor method to converge than either the momentum equation or other scalar transport equations this is because the pressure correction equation which has the form of an anisotropic elliptic equation is diffusion dominated whereas the other equations are convection dominated it is well known that point and line iterative methods converge rather slowly on elliptic problems such as the poisson equation particularly as the number of mesh points becomes large to expedite the convergence rates the multigrid mg method9 10 has been found to be very useful for improving the performance of these single grid sg solvers for elliptic equations

mach uniformity through the coupled pressure and temperature correction algorithm we present a new type of algorithm the coupled pressure and temperature correction algorithm it is situated in between the fully coupled and the fully segregated approach and is constructed such that mach uniform accuracy and efficiency are obtained the essential idea is the separation of the convective and the acoustic thermodynamic phenomena a convective predictor is followed by an acoustic thermodynamic corrector for a general case the corrector consists of a coupled solution of the energy and the continuity equations for both pressure and temperature corrections for the special case of an adiabatic perfect gas flow the algorithm reduces to a fully segregated method with a pressure correction equation based on the energy equation various test cases are considered which confirm that mach uniformity is obtained

a general non orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second moment turbulence transport closure part 1 computational implementation abstract a computational procedure has been developed for predicting separated turbulent flows in complex two dimensional and three dimensional geometries the procedure is based on the fully conservative structured finite volume framework within which the volumes are non orthogonal and collocated such that all flow variables are stored at one and the same set of nodes to ease the task of discretization and to enhance the conservative property of the scheme a cartesian or datum line adapted decomposition of the velocity field has been used the solution algorithm is iterative in nature approaching the steady solution with the aid of a pressure correction scheme convection is approximated with a range of schemes among them higher order upstream weighted approximations and a tvd type muscl form the last applied principally to the transport equations governing turbulence properties effects of turbulence are represented either by two equation eddy viscosity models or by a full reynolds stress transport closure the former category includes both high and low reynolds number variants in two as well as three dimensional conditions to achieve a stable implementation of the reynolds stress equations a special interpolation practice analogous to that of rhie and chow for momentum 1 has been introduced within the general framework the procedure has been formulated so as to apply to both incompressible and compressible flows the latter may contain shocks and highly supersonic regions to achieve this range of applicability the retarded density concept has been combined with the basic pressure based algorithm to capture shock waves a full approximation multigrid scheme for convergence acceleration has been incorporated and applied in conjunction with all turbulence models including second moment closure the present first part of a twin paper focuses on numerical and turbulence modelling issues in part 2 computational results are presented for six representative applications out of fifteen recently predicted with the algorithm within an extensive validation exercise

computation of flows with arbitrary equations of state the extension of time marching computations to fluids with arbitrary equations of state is demonstrated by means of stability analyses simplified problems and practical applications most of the examples use the properties of supercritical hydrogen for which the density varies by more than an order of magnitude for small changes in pressure and temperature but representative computations for incompressible fluids and perfect gases are also given to demonstrate the generality of the procedure because representative flow velocities in typical supercritical fluids applications are much lower than the speed of sound convergence enhancement through eigenvalue control is often necessary this is accomplished through a generalization of earlier preconditioning methods that enables efficient computation of arbitrary equation of state fluids perfect gases and incompressible fluids by a single procedure the present approach thus provides a single method that is uniformly applicable to all equations of state

pressure based navier stokes solver using the multigrid method apressure based implicit procedure to solve steadystate navier stokes equations is developed a multistep pressure correction procedure with an implicit density treatment is used to establish the pressure and velocity fields a multigrid relaxation scheme is used to solve the scalar matrices resulting from the finite volume formulation the algorithm is valid for all mach number flows ranging from incompressible to supersonic flow regimes

a general non orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second moment turbulence transport closure part 2 application abstract a general non orthogonal finite volume procedure documented in a companion paper presented earlier is applied to six flows one 2d laminar one 2d turbulent two 3d turbulent one inviscid transonic and one turbulent transonic the principal aim is to convey the capabilities of the method and this aim is pursued by choosing flows which feature a wide range of phenomena and processes in some cases a comparison is presented between two or more turbulence modelling practices and in these cases considerable emphasis is put on an analysis of model performance and associated physical implications

a low dispersion and bounded convection scheme abstract the paper presents an effective procedure for approximating convective transport in steady state flow conditions the procedure combines three simple discretization schemes that is the second order upwind central and first order upwind differencing the switch between them being controlled by a convection boundedness criterion the resultant composite scheme is stable conservative and easy to implement for the simulation of multi dimensional flows three test cases two linear and one nonlinear are used to assess the performance of the method and comparisons are made with quick and the first order upwind scheme the test problem results demonstrate that the present scheme strictly preserves the boundedness of solutions while maintaining low dispersion of steep gradients and it is applicable to both linear and nonlinear problems

a new approach for building bounded skew upwind schemes abstract a new methodology for the construction of streamline bounded high resolution schemes is presented the route followed in the development is to combined existing streamline schemes with the normalized variable formulation nvf bounding approach the procedure is applied to the skew upwind differenting scheme suds to yield a new composite high resolution bounded convective scheme nvf suds the increase in computational cost with nvf suds is very small as compared to the unbounded suds the new scheme is tested and compared with the composite high resolution stoic scheme the upwind scheme and the unbounded suds by solving for three pure convection test problems results generated demonstrate the capability of the new upwind scheme in accurately resolving steep gradients as accurate as the ones obtained with the third order stoic scheme while retaining the solution bounded

a pressure based algorithm for multi phase flow at all speeds a new finite volume based numerical algorithm for predicting incompressible and compressible multi phase flow phenomena is presented the technique is equally applicable in the subsonic transonic and supersonic regimes the method is formulated on a non orthogonal coordinate system in collocated primitive variables pressure is selected as a dependent variable in preference to density because changes in pressure are significant at all speeds as opposed to variations in density which become very small at low mach numbers the pressure equation is derived from overall mass conservation the performance of the new method is assessed by solving the following two dimensional two phase flow problems i incompressible turbulent bubbly flow in a pipe ii incompressible turbulent air particle flow in a pipe iii compressible dilute gas solid flow over a flat plate and iv compressible dusty flow in a converging diverging nozzle predictions are shown to be in excellent agreement with published numerical and or experimental data

primitive variable strongly implicit calculation procedure for viscous flows at all speeds a coupled solution procedure is described for solving the compressible form of the time dependent twodimensional navier stokes equations in body fitted curvilinear coordinates this approach employs the strong conservation form of the governing equations but uses primitive variables m v p t rather than the more traditional conservative variables p pw pv et as unknowns a coupled modified strongly implicit procedure cmsip is used to efficiently solve the newton linearized algebraic equations it appears that this procedure is effective for mach numbers ranging from the incompressible limit mx 0 01 to supersonic generally smoothing was not needed to control spatial oscillations in pressure for subsonic flows despite the use of central differences dual time stepping was found to further accelerate convergence for steady flows sample calculations including steady and unsteady low mach number internal and external flows and a steady shock boundarylayer interaction flow illustrate the capability of the present solution algorithm

momentum variable procedure for solving compressible and incompressible flows navier stokes equations are solved for both compressible and incompressible flows using momentum component variables instead of the usual velocity variables as the dependent variables the numerical procedure is developed in a control volume based finite element context the procedure is determined in a pressure based algorithm rather than the density based algorithms which compressible methods normally use the proper selection of the connections between the variables on control volume surfaces and the main nodal values allow the use of a collocated grid arrangement the compressible and incompressible results of this algorithm are investigated by testing a number of test cases including the driven cavity an entrance region flow and a converging diverging nozzle flow the results indicate that the momentum component procedure is quite successful for solving compressible and incompressible flows within a single algorithm

preconditioning applied to variable and constant density flows a time derivative preconditioning of the navier stokes equations suitable for both variable and constant density fluids is developed the ideas of low mach number preconditioning and artificial compressibility are combined into a unified approach designed to enhance convergence rates of density based time marching schemes for solving flows of incompressible and variable density fluids at all speeds the preconditioning is coupled with a dual time stepping scheme implemented within an explicit multistage algorithm for solving time accurate flows the resultant time integration scheme is used in conjunction with a finite volume discretization designed for unstructured solution adaptive mesh topologies this method is shown to provide accurate steady state solutions for transonic and low speed flow of variable density fluids the time accurate solution of unsteady incompressible flow is also demonstrated