HAL Id: hal-00922770
Submitted on 30 Dec 2013
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
An all-regime Lagrange-Projection like scheme for the gas dynamics equations on unstructured meshes
We propose an all-regime Lagrange-Projection like numerical scheme for the gas dynamics equations. By all-regime, we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization, i.e. a mesh size and time step much bigger than the Machnumber M. The key idea is to decouple acoustic and transport phenomenon and then alter the numerical ux in the acoustic approximation to obtain a uniform truncation error in term of M. This modied scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy. A discrete entropy inequality under a condition on the modication is obtained thanks to a reinterpretation of the modied scheme in the Harten Lax and van Leer formalism. A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed. Then a simple and ecient semi implicit scheme is also proposed. The resulting scheme is stable under a CFL condition driven by the (slow) material waves and not by the (fast) acoustic waves and so veries the all-regime property. Numerical evidences are proposed and show the ability of the scheme to deal with tests where the ow regime may vary from low to high Mach values.
3.2.5 Partial Conclusion on the Explicit Runge-Kutta Methods
Explicit Runge-Kutta methods are particularly appreciated for their stability domains which are larger than the majority of explicit methods. Nevertheless, high-order Runge-Kutta scheme may be very expensive in term of computational cost. Indeed the number of stages increases non-linearly with the order of accuracy. In this context, Williamson [ 42 ] designed low-storage Runge-Kutta schemes up to the fourth order with only two stages. High-order Runge-Kutta scheme with TVD properties was investigated by Jameson et al. [ 39 ] and Shou and Osher [ 43 ]. The explicit methods presented previously are conditionally stable, and of course they may have a very restrictive constraint. This is also a drawback for unsteady simulations for which the transient regime can be long and time-consuming and unconditionally stable methods should be preferred. Regardless issues of stability and computational cost, it is relevant that explicit schemes may be restricted to resolution of non-stiff problem since any stiff problem induces a significant decrease of the time step (as an example chemical effects have to be resolved with a smaller time step than aerodynamic ones). The following methods tends to extend explicit method to resolve these kinds of problem.
One possible answer to this problem consists in adopting a fully implicit algorithm for the original compressible system. Such approach has been developed in  within the framework of multigrid methods. It allows large CFL numbers but it needs a preconditioning procedure to avoid the large number iterations needed for the resolution of the non linear system which originates from the implicit time discretization of the compressible equations [41, 61, 64] Recently, it has been shown that a fully implicit discretization is not the only strategy which permits to get all Machnumberschemes. One alternative is represented by asymptotic preserving (AP) schemes [19, 17, 18, 16, 33, 71, 10, 54, 11, 22]. They deal with different models which share the common characteristic of describing a multi-scale dynamic: i.e. a dynamic in which fast and slow scales coexist. These techniques allow computing the solution of such stiff problems while avoiding time step limitations directly related to the fast scale dynamic. This fast scale, in the context of this work, appears in the lowMachnumber regime when the pressure waves become fast compared to the rest of the dynamic. In addition, these AP methods lead to consistent approximations of the limit model (here the incompressible model) when the parameter which describes the fast scale dynamics goes to zero (here the Machnumber). We stress that, even if the proposed methods in this work are specifically designed to avoid the fast scale resolution (remaining uniformly stable), if they are used with small time steps like those necessary for an explicit method, they are able to describe this fast dynamic with high accuracy. Thus, this approach also competitive compared to other methods designed to describe the fast pressure waves.
during the non linear stage of the instability. The number of square cells is 72 ˆ280 to mitigate the low accuracy of the first-order
accurate Lagrangian phase.
We represent the volume fraction a 1 at time t “ 0.115 (CFL=0.5) in Figure 15: on the left part it is with the VOF-ML flux,
on the right part it is with the SLIC/Downwind flux. The dynamics is the same even if they are some differences. The structure is narrower with VOF-ML. Actually the quality of results is also very sensitive to the CFL number: if it is too small, extra-diffusion occurs; if it is too large, some oscillations appear. Nevertheless these results shows that the VOF-ML flux has the ability to capture the dynamics of the interface even at the transition between the linear regime and the non linear regime. The symmetry of VOF-ML result is perfectly preserved by our implementation. The number of flagged cells displayed on the bottom-right part of the Figure is clearly related to the length of the interface. Indeed one recognizes the classical strong compression at iteration « 150 followed by a growth of the size of the interface in the linear range. Until the strong compression, the number of flagged cells is approximatively constant as in the right part of Figure 6. Then the number of flagged cells increases linearly, following
Table 1 provides the ﬁnal computational times for diﬀerent schemes and CF L numbers. The values are normalized with respect to the time required by the RK4 scheme with CF L = 1. For CF L = 1 the RK4 is the cheapest scheme, whereas BW2 is by far the more expensive one, due to the inner subiteration of the quasi-exact Newton procedure. The RK6 scheme is about 40% more expensive than RK4 but it is also more accurate. Adding the IRS treatment increases the cost of the RK4 scheme by a bit more than 50%, both with IRS2 and IRS4. The similar ovecost obtained for IRS2 and IRS4 proves the eﬃciency of the pentadiagonal solver. For the RK6 scheme, the cost of adding IRS is lower, namely, a bit more than 30%, since the computational cost of the baseline solver is higher. These comparisons are made using the same time step for all of the schemes. For higher values of the CF L number, the overall cost of the implicitschemes becomes lower than that of the explicit ones, due to the reduced number of time steps required to complete the simulation. The BW2 scheme is more than twice more expensive than the IRS schemes, and is less accurate. The RK6IRS4 scheme, which provides the more accurate results for the hump advection problem at CF L = 5, leads to an overall cost that is 70% lower than the RK4 at CF L = 1, while keeping a similar accuracy. Even if the RK4 can be run at CF L greater than 2 (albeit with lower accuracy), the overall cost of RK6IRS4 remains lower.
9 School of Mathematics & Statistics, University of Sydney, NSW 2006, Australia Received 2014 August 18; accepted 2014 October 14; published 2014 December 3
Thermal conduction is an important energy transfer and damping mechanism in astrophysical flows. Fourier’s law, in which the heat flux is proportional to the negative temperature gradient, leading to temperature diffusion, is a well-known empirical model of thermal conduction. However, entropy diffusion has emerged as an alternative thermal conduction model, despite not ensuring the monotonicity of entropy. This paper investigates the differences between temperature and entropy diffusion for both linear internal gravity waves and weakly nonlinear convection. In addition to simulating the two thermal conduction models with the fully compressible Navier–Stokes equations, we also study their effects in the reduced “soundproof” anelastic and pseudoincompressible (PI) equations. We find that in the linear and weakly nonlinear regime, temperature and entropy diffusion give quantitatively similar results, although there are some larger errors in the PI equations with temperature diffusion due to inaccuracies in the equation of state. Extrapolating our weakly nonlinear results, we speculate that differences between temperature and entropy diffusion might become more important for strongly turbulent convection.
A low-di ffusion self-adaptive flux-vector splitting method is presented for the Euler equations. The flux-vector is here split into convective and acoustic parts following the formulation recently proposed by the authors. This procedure is based on the Zha-Bilgen (or previously Baraille et al. for the Euler barotropic system) approach enriched by a dynamic flow-dependent splitting parameter based on the local Machnumber. As a consequence, in the present self-adaptive splitting, the convective and acoustic parts decouple in the low-Machnumber regime whereas the complete Euler equations are considered for the sonic and highly subsonic regimes. The low di ffusive property of the present scheme is obtained by adding anti-di ffusion terms to the momentum and the energy components of the pressure flux in the acoustic part of the present splitting. This treatment results from a formal invariance principle preserving the discrete incompressible phase space through the pressure operator. Numerical results for several carefully chosen one- and two-dimensional test problems are finally investigated to demonstrate the accuracy and robustness of the proposed scheme for a wide variety of configurations from subsonic to highly subsonic flows.
Keywords: LowMachnumber limit, Asymptotic preserving schemes, Euler sys- tem, stability analysis.
1. Introduction. Almost all fluids can be said to be compressible. However, there are many situations in which the changes in density are so small to be considered negligible. We refer to these situations saying that the fluid is in an incompressible regime. From the mathematical point of view, the difference between compressible and incompressible situations is that, in the second case, the equation for the conservation of mass is replaced by the constraint that the divergence of the velocity should be zero. This is due to the fact that when the Machnumber tends to zero, the pressure waves can be considered to travel at infinite speed. From the theoretical point of view, researchers try to fill the gap between those two different descriptions by determining in which sense compressible equations tend to incompressible ones [2, 20, 21, 22, 33]. In this article we are interested in the numerical solution of the Euler system when used to describe fluid flows where the Machnumber strongly varies. This causes the gas to pass from compressible to almost incompressible situations and consequently it causes most of the numerical methods build for solving compressible Euler equations to fail. In fact, when the Machnumber tends to zero, it is well known that classical Godunov type schemes do not work anymore. Indeed, they lose consistency in the incompressible limit. This means that when close to the limit, the accuracy of theses schemes is not sufficient to describe the flow. Many efforts have been done in the recent past in order to correct this main drawback of Godunov schemes, for instance by using preconditioning methods  or by splitting and correcting the pressure on the collocated meshes , [9, 10], , [13, 14, 30], , [23, 24], [26, 27, 28], or instead by using staggered grids like in the famous MAC scheme, see for instance , , , , , . Unfortunately, even if these approaches permit to bypass the consistency problem of Godunov methods, they all need to resolve the scale of
The fractional step strategy involving an elliptic pressure correction step has been recognized to yield algo- rithms which are not limited by stringent stability conditions (such as CFL conditions based on the celerity of the fastest waves) since the first attempts to build ”all flow velocity” schemes in the late sixties  or in the early seventies ; these algorithms may be seen as an extension to the compressible case of the celebrated MAC scheme, introduced some years before . These seminal papers have been the starting point for the develop- ment of numerous schemes, using staggered finite volume space discretizations [4, 6, 34, 35, 38, 41, 47, 64–69, 71], colocated finite volumes [2, 10, 32, 33, 36, 37, 39, 43, 48–51, 54, 57, 59, 61, 70] or finite elements [3, 46, 52, 72]. Al- gorithms proposed in these works may be essentially implicit-in-time, and the pressure correction step is then an ingredient of a SIMPLE-like iterative procedure, or only semi-implicit, with a single (or a limited number of) prediction and correction step(s), as in projection methods for incompressible flows (see [7, 60] for seminal works and  for a review of most of the variants). The schemes which we propose in the present paper fall in this latter class.
Finite volume schemesfor the solution of hyperbolic problems such as the system (1) generally use a collocated arrangement of the unknowns, which are associated to the cell centers, and apply a Godunov-like technique for the com- putation of the fluxes at the cells faces: the face is seen as a discontinuity line for the beginning-of-time-step numerical solution, supposed to be constant in the two adjacent cells; the value of the solution of the so-posed Riemann problem on the discontinuity line is computed, either exactly or approximately; the nu- merical solution at the end-of-time-step is computed with these values, and is a piecewise constant function (see e.g. [39, 3] for the development of such solvers). In one space dimension, this method consists, at least for exact Riemann solvers, in a projection of the exact solution. Then, thanks to the properties of the pro- jection, this process applied to the Euler equations yields consistent schemes which preserve the non-negativity of the density and the internal energy and, for first-order variants, satisfy an entropy inequality. The price to pay is the computational cost of the evaluation of the fluxes, and the fact that this issue is intricate enough to put almost out of reach implicit-in-time formulations, which would allow to relax CFL time step constraints. In addition, preserving the ac- curacy forlowMachnumberflows is a difficult task (see e.g.  and references herein).
A Weighted Splitting Approach ForLow-MachNumberFlows
David Iampietro 1,3 , Fr´ed´eric Daude 1 , Pascal Galon 4 , and Jean-Marc H´erard 2,3
Abstract In steady-state regimes, water circulating in the nuclear power plants pipes behaves as a lowMachnumber flow. However, when steep phenomena occur, strong shock waves are produced. Herein, a fractional step approach allowing to decouple the convective from the acoustic effects is proposed. The originality is that the split- ting between these two parts of the physics evolves dynamically in time according to the Machnumber. The first one-dimensional explicit and implicit numerical re- sults on a wide panel of Mach numbers show that this approach is as accurate and CPU-consuming as a state of the art Lagrange-Projection-type method.
Abstract Large-Eddy Simulation (LES) becomes a more and more demanded tool to improve the design of aero-engines. The main reason for this request stems from the constraints imposed on the next generation low-emission engines at the industrial development level and the ability for LES to provide information on the instantaneous turbulent flow field which greatly contributes to improving the prediction of mixing and combustion thereby offering an improved prediction of the exhaust emission. The work presented in this thesis discusses two recurring issues of LES. For one, numerical schemesfor LES require certain properties, i.e. low-diffusion schemes of high order of accuracy so as not to interfere with the turbulence models. To meet this purpose in the context of fully unstructured solvers, a new family of high-order time-integration schemes is proposed. With this class of schemes, the diffusion implied by the numerical scheme become adjustable and built-in. Second, since fully unsteady by nature, LES is very consuming in terms of CPU time. Even with today's supercomputers complex problems require long simulation times. Due to the low flow velocities often occurring in industrial applications, the use of a low-Machnumber solver seems suitable and can lead to large reductions in CPU time if comparable to fully compressible solvers. The impact of the incompressibility assumption and the different nature of the numerical algorithms are rarely discussed. To partly answer the question, detailed comparisons are proposed for an experimental swirled configuration representative of a real burner that is simulated by LES using a fully explicit compressible solver and an incompressible solution developed at CORIA.
The method presented below focuses on the numerical approximation of the Euler compressible system. It pursues a two-fold objective: being able to accurately follow slow material waves as well as strong shock waves in the context of lowMachnumberflows. The resulting implicit-explicit fractional step approach leans on a dynamic splitting designed to react to the time fluctuations of the maximal flow Machnumber. When the latter rises suddenly, the IMEX scheme, so far driven by a material-wave Courant number, turn into a time-explicit approximate Riemann solver constrained by an acoustic-wave Courant number. It is also possible to enrich the dynamic splitting in order to capture high pressure jumps even when the flow Machnumber is low. One-dimensional lowMachnumber test cases involving single or multiple waves confirm that the present approach is as accurate and e fficient as an IMEX Lagrange-Projection method. Besides, numerical results suggest that the stability of the present method holds for any Machnumber if the Courant number related to the convective subsystem arising from the splitting is of order unity.
In order to capture rarefaction and/or shock waves, a classical numerical approach is to dis- cretize ( 1 ) or ( 2 ) by using a Godunov type scheme. In this paper, a Godunov type scheme is a finite volume type scheme whose numerical fluxes are constructed by using an exact or an approximate 1D Riemann solver in the normal direction of the edges of the mesh (e.g. the Roe scheme [ 36 ] and the VFRoe scheme [ 5 ]). Nevertheless, it is now well known that first order Godunov type schemes applied to ( 1 ) or ( 2 ) are most of the time not accurate at lowMachnumber [ 3 , 7 , 9 , 39 , 21 , 19 , 20 , 32 ]. It is also shown in [ 21 ] that the second order Roe scheme suffers from a similar inaccuracy at lowMachnumber. In the same way, it is shown in [ 1 ] that this is also the case for the second and third orders discontinuous Galerkin scheme using Roe-type fluxes although the results are improved by increasing the order. For the sake of simplicity, we name in the sequel lowMachnumber problem this loss of accuracy in the spatial periodic case for ( 1 ) or ( 2 ). And we study this lowMachnumber problem in the case of first order Godunov type schemes although the proposed theoretical tools could also be applied to higher orders. Nevertheless, due the non-linearities introduced by slope limiters and to larger stencils of high order schemes, the proposed analysis will be much more difficult at orders greater than one.
archive for the deposit and dissemination of sci- entific research documents, whether they are pub-
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, On the Behavior of UpwindSchemes in the LowMach
Eq. (10) suggests that the ‘natural’ variables for a Helmholtz solver are p′ and u′ while Eq. (12) indicates that J′ and m′ are more attune for LEE. The choice of this latter set of fluctuating state variables is commonly used in the aeroacoustic community [35 , 36] because they are themselves independent of the flow when placed in a wave equation, contrary to the pressure or the velocity which are locally dependent on the flow velocity  . Moreover, state variables J′ and m′ are more suitable because their product represents the acoustic power in the presence of a mean flow. However one can notice that these two sets of independent variables are strictly equivalent and one may write the LEE or Helmholtz equations with any set. A reduced impedance, Z, and reflection coefficient, R, may be defined for each set of variables. Using p′ and u′ one has
kind of ’energy’ definition. They are set respectively to identity I and to the radius r in the following. Sensitivity coefficients can be therefore explained as how the response of any variation in the output of a system expressed as a mathematical functional can be apportioned to different sources of variation in the input of the model. Such analysis is com- mon in different fields of engineering and in the field of fluid dynamics since it is closely related to opti- mization problems and optimal control [12, 13]. In the last 45 years the physical problem of receptivi ty and sensitivity of boundary layers flows were investi- gated in different theoretical, experimental and com- putational manners. Airiau et al  have demon- strated that receptivity coefficients and the approach based on adjoint equations  can be associated to an optimization problem and therefore they were strongly closed to sensitivity coefficients. Later it was used to perform optimal control in the lami- nar boundary layer flow [12, 15]. The use of adjoint equations in flow instability dates back to the early 1990s [16, 17], but did not become widespread until the late 2000s [18, 19] and finally the  where the concepts of senstivity analysis with adjoint are spell out in details. Sensitivity analysis based on the ad- joint of compressible Navier-Stokes equations have been recently derived [11,21] and applied to optimal control studies of the two dimensional shear layer in the aeroacoustic framework. Some other examples of sensitivity can be found in the mesh optimization and in the optimization of structures.