Unite´ de recherche INRIA Lorraine, Technopoˆle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LE`S NANCY Unite´ de recherche INRIA Rennes, Iri[r]

Unite´ de recherche INRIA Lorraine, Technopoˆle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LE`S NANCY Unite´ de recherche INRIA Rennes, Iri[r]

HAL Id: hal-00922770
https://hal.archives-ouvertes.fr/hal-00922770
Submitted on 30 Dec 2013
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An all-regime Lagrange-Projection like scheme **for** the gas dynamics equations on unstructured meshes
Abstract
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 **Mach** **number** 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.

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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.

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One possible answer to this problem consists in adopting a fully **implicit** algorithm **for** the original **compressible** system. Such approach has been developed in [39] 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 **Mach** **number** **schemes**. 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 **low** **Mach** **number** 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 **Mach** **number**). 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.

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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

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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 **implicit** **schemes** 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.

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9 School of Mathematics & Statistics, University of Sydney, NSW 2006, Australia Received 2014 August 18; accepted 2014 October 14; published 2014 December 3
ABSTRACT
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.

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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 **Mach** **number**. As a consequence, in the present self-adaptive splitting, the convective and acoustic parts decouple in the **low**-**Mach** **number** 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**.

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Keywords: **Low** **Mach** **number** 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 **Mach** **number** 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 **Mach** **number** 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 **Mach** **number** 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 [34] or by splitting and correcting the pressure on the collocated meshes [5], [9, 10], [12], [13, 14, 30], [15], [23, 24], [26, 27, 28], or instead by using staggered grids like in the famous MAC scheme, see **for** instance [3], [16], [17], [18], [19], [31]. Unfortunately, even if these approaches permit to bypass the consistency problem of Godunov methods, they all need to resolve the scale of

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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 [23] or in the early seventies [24]; these algorithms may be seen as an extension to the **compressible** case of the celebrated MAC scheme, introduced some years before [25]. 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 [19] **for** a review of most of the variants). The **schemes** which we propose in the present paper fall in this latter class.

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Finite volume **schemes** **for** 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 **for** **low** **Mach** **number** **flows** is a difficult task (see e.g. [18] and references herein).

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A Weighted Splitting Approach **For** **Low**-**Mach** **Number** **Flows**
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 **low** **Mach** **number** 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 **Mach** **number**. 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.

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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 **schemes** **for** 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**-**Mach** **number** 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.

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Abstract
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 **low** **Mach** **number** **flows**. The resulting **implicit**-explicit fractional step approach leans on a dynamic splitting designed to react to the time fluctuations of the maximal flow **Mach** **number**. 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 **Mach** **number** is **low**. One-dimensional **low** **Mach** **number** 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 **Mach** **number** if the Courant **number** related to the convective subsystem arising from the splitting is of order unity.

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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 **low** **Mach** **number** [ 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 **low** **Mach** **number**. 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 **low** **Mach** **number** problem this loss of accuracy in the spatial periodic case **for** ( 1 ) or ( 2 ). And we study this **low** **Mach** **number** 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.

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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 **Upwind** **Schemes** in the **Low** **Mach**

p′ (14)
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 [37] . 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

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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 [14] have demon- strated that receptivity coefficients and the approach based on adjoint equations [13] 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 [20] 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.

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