extension of finite volume compressible flow solvers to multi dimensional variable density zero mach number flows when attempting to compute unsteady variable density flows at very small or zero mach number using a standard finite volume compressible flow solver one faces at least the following difficulties i spatial pressure variations vanish as the mach numberm

mixed stabilized finite element methods based on backward difference adams bashforth scheme for the time dependent variable density incompressible flows in this paper we consider a second order mixed stabilized finite element method based on pressure projection method for variable density incompressible flows the originality of the proposed approach is to use a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for variable density incompressible flows approximated by the lowest equal order finite element pairs a second order backward difference bdf for the temporal term and a second order adams bashforth ab for the explicit treatment of the nonlinear term lead to the presented second order bdf ab scheme the stability of the method was proved and the performance of the method is numerically illustrated finally comparison with some established methods a series of numerical experiments are given to show that this method has better stability and accuracy

low mach number flows in time dependent domains we perform a multiple time scale single space scale analysis of a compressible fluid in a time dependent domain when the time variations of the boundary are small with respect to the acoustic velocity we introduce an average operator with respect to the fast time the averaged leading order variables satisfy modified incompressible equations which are coupled to linear acoustic equations with respect to the fast time we discuss possible initial boundary data for the asymptotic equations inherited from the initial boundary data for the compressible equations

an efficient scheme for solving steady incompressible navier stokes equations abstract the steady incompressible navier stokes equations in a 2d driven cavity are solved in primitive variables by means of the multigrid method the pressure and the components of the velocity are discretized on staggered grids a block implicit relaxation technique is used to achieve a good convergence and a simplified fmg fas algorithm is proposed special focus on the finite differences scheme used to approach the convection terms is made and a large discussion with other schemes is given results in a square driven cavity are obtained for reynolds numbers as high as 15 000 on fine uniform meshes and the solution is in good agreement with other studies for re 5000 the secondary vortices are very well represented showing the robustness of the method for reynolds numbers higher than 5000 the loss of stability for the steady solution is discussed moreover some computations on a rectangular cavity of aspect ratio equal to two are presented in addition the method is very efficient as far as cpu time is concerned for instance the solution for re 1000 on a 128 128 grid is obtained within 24 s on a siemens vp 200

pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations a calculation procedure for viscous flows at all mach numbers has been described the scheme has been developed for a nonorthogonal coordinate system in order to handle both incompressible and compressible flows pressure has been selected as a primary dependent variable in preference to density the pressure field is evaluated using the compressible form of the simpler algorithm the accuracy of the proposed scheme has been assessed by comparing the results with experimental data or other numerical results available in the literature

semi implicit extension of a godunov type scheme based on low mach number asymptotics i one dimensional flow a single scale multiple space scale asymptotic analysis provides detailed insight into the low mach number limit behavior of solutions of the compressible euler equations we use the asymptotics as a guideline for developing a low mach number extension of an explicit higher order shock capturing scheme this semi implicit scheme involves multiple pressure variables large scale differencing and averaging procedures that are discretized versions of standard operations in multiple scales asymptotic analysis advection and acoustic wave propagation are discretized explicitly and upwind and only one scalar elliptic equation is to be solved implicitly at each time step this equation is a pressure correction equation for incompressible flows when the mach number is zero in the low mach number limit the time step is restricted by a courant number based essentially on the maximum flow velocity for moderate and large mach numbers the scheme reduces to the underlying explicit higher order shock capturing algorithm

radiation models for thermal flows at low mach number simplified approximate models for radiation are proposed to study thermal effects in low mach flow in open tunnels the governing equations for fluid dynamics are derived by applying a low mach asymptotic in the compressible navier stokes problem based on an asymptotic analysis we show that the integro differential equation for radiative transfer can be replaced by a set of differential equations which are independent of angle variable and easy to solve using standard numerical discretizations as an application we consider a simplified fire model in vehicular tunnels the results presented in this paper show that the proposed models are able to predict temperature in the tunnels accurately with low computational cost

higher order temporal integration for the incompressible navier stokes equations in bounded domains author s minion ml saye ri abstract 2018 elsevier inc this paper compares and contrasts higher order semi implicit temporal integration strategies for the incompressible navier stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations particular focus is placed on the imposition of boundary conditions in the semi implicit formulation the accurate treatment of the pressure term and the smoothness of the numerical solution for different formulations the main result presented here is the formulation and numerical validation of a single step semi implicit method called spectral deferred pressure corrections which can in theory obtain arbitrary formal order of accuracy in time numerical results demonstrate up to eighth order accuracy scenarios where optimal temporal accuracy is attained and scenarios where order reduction is observed due to time dependent boundary conditions

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

low mach number limit of the full navier stokes equations the low mach number limit for classical solutions of the full navier stokes equations is here studied the combined effects of large temperature variations and thermal conduction are taken into account in particular we consider general initial data the equations lead to a singular problem depending on a small scaling parameter whose linearized system is not uniformly well posed yet it is proved that solutions exist and they are uniformly bounded for a time interval which is independent of the mach number ma 0 1 the reynolds number re 1 and the peclet number pe 1 based on uniform estimates in sobolev spaces and using a theorem of g metivier s schochet 30 we next prove that the penalized terms converge strongly to zero this allows us to rigorously justify at least in the whole space case the well known computations given in the introduction of p l lions book 26

the l2 l2 bumpless transfer problem for linear plants its definition and solution a novel characterization of bumpless transfer among alternative controllers both in continuous and discrete time is introduced the bumpless transfer problem is to design a compensation scheme guaranteeing an l 2 respectively l 2 bound on the mismatch after the switching time between the actual plant output and a particular target ideal response minimization of the gain from initial plant state mismatch to l 2 respectively l 2 plant output mismatch provides improved transients a solution to the bumpless transfer problem is given both for plants without input saturation and for plants with input saturation the solution guarantees anti windup features in the latter case

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

unsteady solution of incompressible navier stokes equations abstract a new time accurate finite difference numerical method for solving incompressible navier stokes equations is presented with primitive variables as the unknowns the numerical scheme is a crank nicolson implicit treatment of all terms of the equations with central differencing for space derivatives the pressure and the nonlinearities in convection terms are not lagged to obtain the incompressible solution at the advanced time level a continuous auxiliary system is introduced in pseudo time with artificial compressibility time accurate solutions are presented for two dimensional fluid flows in a square cavity with an impulsively starting lid and with an oscillating lid

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

asymptotics for symmetric hyperbolic systems with a large parameter the asymptotic behavior as is analysed for solutions u of quasilinear symmetric hyperbolic systems of the form a0ut ajuxi cjuxj fu 0 x u0 x where the ak and f depend on t x u and u but the cj are constant the results are applied to the equations of slightly compressible fluid dynamics to obtain a generalized form of the acoustics equations that describe the first order correction to incompressible flow the special case of a barotropic fluid was analyzed previously by klainerman and majda

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

a new fractional time stepping method for variable density incompressible flows this paper describes a new method for solving variable density incompressible viscous flows we have dealt with the momentum equation and the divergence free constraint in a new manner by rewriting the original equations the originality of the proposed approach is that we have used different numerical methods to evaluate the evolution of the velocity and pressure compared with some established methods the proposed approach is parameter free more flexible and can avoid the difficulties caused by the original equations the stability analysis of the method is performed to show that our method is stable finally numerical experiments are given to show the accuracy efficiency and validness of this method for variable density incompressible flows

a unified method for computing incompressible and compressible flows in boundary fitted coordinates a unified method for computing incompressible and compressible flows with mach uniform accuracy and efficiency is described the method is equally applicable to stationary and nonstationary flows a pressure based discretisation on a staggered grid in general boundary fitted coordinates is used for the euler equations extension to navier stokes is straightforward dimensionless variables that remain finite for all mach numbers are used mach number independent accuracy and efficiency is shown by numerical experiments

a sharp interface finite volume method for elliptic equations on cartesian grids we present a second order sharp interface finite volume method for the solution of the three dimensional elliptic equation x u x f x with variable coefficients on cartesian grids in particular we focus on interface problems with discontinuities in the coefficient the source term the solution and the fluxes across the interface the method uses standard piecewise trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27 point stencil singularities associated with vanishing partial volumes of intersected grid cells are removed by a two term asymptotic approach in contrast to the 2d method presented by two of the authors in m oevermann r klein a cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces journal of computational physics 219 2006 749 769 we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz this simplifies the treatment of the different cut cell types and avoids additional special operations for degenerated interface topologies the resulting set of linear equations has been solved with a bicgstab solver preconditioned with an algebraic multigrid in various testcases including large ratios and non smooth interfaces the method achieves second order of accuracy in the l and l 2 norm

a numerical method for two phase flow consisting of separate compressible and incompressible regions abstract we propose a numerical method for modeling two phase flow consisting of separate compressible and incompressible regions this is of interest for example when the combustion of fuel droplets or the shock induced mixing of liquids is numerically modeled we use the level set method to track the interface between the compressible and incompressible regions as well as the ghost fluid method gfm to create accurate discretizations across the interface the gfm is particularly effective here since the equations differ in both number and type across the interface the numerical method is presented in two spatial dimensions with numerical examples in both one and two spatial dimensions while three dimensional extensions are straightforward

a multiscale pressure splitting of the shallow water equations abstract direct representation of the free surface in ocean circulation models leads to a number of computational difficulties that are due to the fast time scales associated with free surface waves these fast time scales generally result in severe time step restrictions when the free surface is advanced using an explicit scheme and may result in large phase errors when the free surface is treated implicitly with a large time step a multiple scale analysis of the shallow water equations is used to analyze this stiffness and to guide the construction of a computational methodology that overcomes the associated difficulties specifically we explore a class of fractional step methods that utilize coarsened grids in the propagation of long wave data the behavior of the corresponding schemes is examined in detail in light of one dimensional model problems based on finite difference or spectral element discretizations

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

stability of a cartesian grid projection method for zero froude number shallow water flows in this paper a godunov type projection method for computing approximate solutions of the zero froude number incompressible shallow water equations is presented it is second order accurate and locally conserves height mass and momentum to enforce the underlying divergence constraint on the velocity field the predicted numerical fluxes computed with a standard second order method for hyperbolic conservation laws and applied to an auxiliary system are corrected in two steps first a mac type projection adjusts the advective velocity divergence in a second projection step additional momentum flux corrections are computed to obtain new time level cell centered velocities which satisfy another discrete version of the divergence constraint the scheme features an exact and stable second projection it is obtained by a petrov galerkin finite element ansatz with piecewise bilinear trial functions for the unknown height and piecewise constant test functions the key innovation compared to existing finite volume projection methods is a correction of the in cell slopes of the momentum by the second projection the stability of the projection is proved using a generalized theory for mixed finite elements in order to do so the validity of three different inf sup conditions has to be shown the results of preliminary numerical test cases demonstrate the method s applicability on fixed grids the accuracy is improved by a factor four compared to a previous version of the scheme

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 high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

a numerical algorithm for the stokes problem based on an integral equation for the pressure via conformal mappings in this paper we present an algorithm for solving numerically the stokes problem in the plane the known algorithms are all based on certain discretization schemes for the analytic equations in contrast to this recent work our algorithm uses an explicit analytic solution of a certain approximating problem which can easily be solved numerically up to machine accuracy on the one hand this analytic formula is based on a complex representation of all solutions of the stokes differential equations and on the other hand it is based on the conformal mapping of the given domain on the unit disc therefore a central prerequisite of our corresponding program is a program for computing this conformal mapping

incompressible moving boundary flows with the finite volume particle method abstract mesh free methods offer the potential for greatly simplified modeling of flow with moving walls and phase interfaces the finite volume particle method fvpm is a mesh free technique based on interparticle fluxes which are exactly analogous to intercell fluxes in the mesh based finite volume method consequently the method inherits many of the desirable properties of the classical finite volume method including implicit conservation and a natural introduction of boundary conditions via appropriate flux terms in this paper we describe the extension of fvpm to incompressible viscous flow with moving boundaries an arbitrary lagrangian eulerian approach is used in conjunction with the mesh free discretisation to facilitate a straightforward treatment of moving bodies non uniform particle distribution is used to concentrate computational effort in regions of high gradients the underlying method for viscous incompressible flow is validated for a lid driven cavity problem at reynolds numbers of 100 and 1000 to validate the simulation of moving boundaries flow around a translating cylinder at reynolds numbers of 20 40 and 100 is modeled results for pressure distribution surface forces and vortex shedding frequency are in good agreement with reference data from the literature and with fvpm results for an equivalent flow around a stationary cylinder these results establish the capability of fvpm to simulate large wall motions accurately in an entirely mesh free framework

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer