# A. Lerat's research while affiliated with Ecole Nationale Supérieure d'Arts et Métiers and other places

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## Publications (83)

Residual-Based Compact (RBC) schemes approximate the 3-D compressible Euler equations with a 5th- or 7th-order accuracy on a 5×5×5-point stencil and capture shocks pretty well without correction. For unsteady flows however, they require a costly algebra to extract the time-derivative occurring at several places in the scheme. A new high-order time...

Residual-Based Compact (RBC) schemes can approximate the compressible Euler equations with a high space-accuracy on a very compact stencil. For instance on a 2-D Cartesian mesh, the 5th- and 7th-order accuracy can be reached on a 5. ×. 5-point stencil. The time integration of the RBC schemes uses a fully implicit method of 2nd-order accuracy (Gear...

Recent developments about the extension of high-order Residual-Based Compact schemes to unsteady flows and complex configurations are discussed, with application to scale-resolving simulations and complex turbomachinery flows.

An exact expression of steady discrete shocks was recently obtained by the author in [9] for a class of residual-based compact schemes (RBC ) applied to the inviscid Bürgers equation in a finite domain. Following the same lines, the analysis is extended to an infinite domain for a scalar conservation law with a general convex flux. For the dissipat...

The paper discusses the design principles of a Finite-Volume Residual-Based Compact (RBC) scheme for the spatial discretization of the unsteady compressible governing equations of gas dynamics on general structured meshes. The scheme makes use of weighted approximations that allow to ensure high accuracy while taking benefit from the structured nat...

The validity of the equivalent differential equation (also called modified equation or differential approximation) for representing shock solutions of high order schemes is investigated through a comparison of exact analytical solutions of the discrete scheme and its equivalent equation, for steady shocks of the inviscid BRurgers equation. For a th...

The wave propagation (spectral) properties of high-order Residual-Based compact
(RBC) discretizations are analyzed to obtain information on the evolution
of the Fourier modes supported on a grid of finite size. For these genuinely
multidimensional and intrinsically dissipative schemes, a suitable procedure is
used to identify the modified wave numb...

Exact expressions of steady discrete shocks are found for a class of
dissipative compact schemes approximating a one-dimensional nonlinear
hyperbolic problem with a 3rd, 5th and 7th order of accuracy. A discrete
solution is given explicitly for the inviscid Bürgers equation and
the oscillatory nature of the shock profiles is determined according to...

The paper discusses the design principles of a Finite-Volume Residual-Based
Compact (RBC) scheme for the spatial discretization of the unsteady compressible
governing equations of gas dynamics on general structured meshes.
Our goal is to develop an accurate and robust approximation methodology,
well suited for complex problems of industrial interes...

The present study consists in an analysis of the DNS database of a flow overcoming a transitional separation induced by an adverse pressure gradient on a flat plate under a curved upper wall. This study takes place in the context of improving RANS models for the simulation of the stall phenomenon for rotor blades applications. To mimic the real flo...

In this paper, we investigate the accuracy and efficiency of a Residual-Based Compact (RBC) numerical scheme of fifth order accuracy for the computation of compressible flows on structured grids. After providing the design principles and main numerical properties of the proposed scheme, applications to steady and unsteady, inviscid and viscous test...

This paper describes how transonic buffet over a supercritical airfoil can be alleviated using a contour bump passive control
technique. Buffet control is formulated as a 4-parameter and 2-objective optimization problem which is solved using a Pareto-based
genetic algorithm. Buffet suppression is achieved thanks to optimal bumps located between the...

A residual-based (RB) scheme relies on the vanishing of residual at the steady-state to design a transient first-order dissipation, which becomes high-order at steady-state. Initially designed within a finite-difference framework for computations of compressible flows on structured grids, the RB schemes displayed good convergence, accuracy and shoc...

A flexible multiblock implementation of second and third-order accurate
residual-based compact (RBC) schemes is proposed and validated for turbomachinery flows of
increasing complexity. Applications to complex transonic unsteady flows in turbomachinery
show the advantages of using such schemes in terms of increased accuracy and better insight into...

A new fourth-order dissipative scheme on a compact 3 × 3 stencil is presented for solving 2D hyperbolic problems. It belongs to the family of previously developed residual-based compact schemes and can be considered as optimal since it offers the maximum achievable order of accuracy on the 3 × 3-point stencil. The computation of 2D scalar problems...

A robust implementation of residual-based compact schemes of second- and third-order accuracy on structured multiblock grids of industrial use is proposed. Applications to complex transonic unsteady flows in turbomachinery show the advantages of using such schemes in terms of increased accuracy and better insight into the flow physics.

A residual-based compact (RBC) scheme originally designed on structured grids has been extended to unstructured grids. A second
and third-order finite-volume (FV) formulations of the residual-based scheme have been proposed, which rely on a linear or
quadratic least-square based solution reconstruction and an original dissipation flux. A simple mat...

The present study takes place in the context of improving RANS models for the simulation of the dynamic stall phenomenon for rotor blades applications. To reach this goal, a high quality reference data base is created through DNS simulations. Comparisons are made with two RANS k - ω Wilcox simulations using two different intermittency functions. Th...

Taking advantage of the notion of vorticity preserving schemes introduced by Morton and Roe for acoustics, and on the residual-based
schemes family proposed by Lerat and Corre, an implicit second order accurate residual-based vorticity preserving scheme is
presented and applied to blade vortex interaction.

A third-order mesh generation and adaptation method is presented for solving the steady compressible Euler equations. For interior points, a third-order scheme is used on Cartesian and curvilinear meshes. Concerning the mesh adaptation, the method of Meakin is also extended to third order. The accuracy of the new overset mesh adaptation method is d...

We recently proved that a dissipative residual-based scheme of second-order accuracy is vorticity-preserving for the compressible Enter equations. In the present paper, this scheme is extended to curvilinear grids and applied to the computation of the interaction between a Scully vortex and a NACA0012 airfoil at a Mach number of 0.5. A grid converg...

A residual-based compact scheme, previously developed to compute viscous compressible flows with 2nd or 3rd-order accuracy [Lerat A, Corre C. A residual-based compact scheme for the compressible Navier–Stokes equations. J Comput Phys 2001; 170(2): 642–75], is generalized to very high-orders of accuracy. Compactness is retained since for instance a...

I. Abstract I n the present article some high order finite difference schemes (seven and eleven points DRP schemes or high order compact schemes) are introduced for global stability problems. These schemes are compared to classical spectral collocation scheme usually employed in such stability problem. A complete comparative study of those schemes...

An efficient multidimensional scheme is constructed for solving the compressible Euler equations. It is deduced from a centered scheme of the Lax–Wendroff type by using a special time-step in the numerical flux, namely a matricial characteristic time-step. This produces a compact second-order upwinding in a very simple way and leads to accurate non...

The discretization of the viscous terms in a space discontinuous Galerkin method is investigated through a theoretical analysis
and numerical calculations. Two formulations are considered : the first one uses a shifted cell and the other one auxiliary
variables. Both are second order accurate on Cartesian meshes. They are applied to the interaction...

A second-order scheme designed to compute 3-D viscous flows is presented : it is based on a Lax-Wendroff solver modified by
a characteristic time-step technique. The scheme is driven to steady-state by various implicit treatments, the pros and cons
of which are discussed in terms of cost and memory requirement. The efficiency and accuracy of the pr...

A residual-based compact scheme, previously developed to compute d-dimensional inviscid compressible flows with third-order accu-racy on a 3 d -point stencil [Lerat A, Corre C. Residual-based compact schemes for multidimensional hyperbolic system of conservation laws. Comput Fluids 2002;31:639–61], is generalized to larger stencils allowing to reac...

The concept of vorticity-preserving scheme introduced by Morton and Roe is considered for the system wave equation and extended to the linearised and full compressible Euler equations. Useful criteria are found for a general dissipative conservative scheme to be vorticity preserving. Using them, a residual-based scheme is shown to be vorticity pres...

In the present article, a coupling between the mesh adaptation technique of Meakin and high-order numerical schemes is proposed. Several schemes for introducing an artificial dissipation into a high-order central finite volume approximation to the Euler equations are considered. Transfers in overlap regions are completed by high-order interpolation...

We consider the calculation of steady weak solutions of an hyperbolic system in one-space dimension. By using an implicit
centered scheme of second-order accuracy with an appropriate treatment of the boundary conditions, we obtain a quick convergence
to a non-oscillatory steady solution.

The simulation of external or internal transonic flows using a standard k − turbulence model, relying on the Boussinesq assumption which states a linear dependence of the turbulent stresses on the mean shear stress, does not allow the successful prediction of unsteady flow phenomena such as self-sustained shock oscillations, because of an excessive...

One of the purposes of the present study was to show how the analysis of the equivalent system corresponding to the general class of schemes,
( µ = 1 + \tfracÖ5 2,b = \tfrac12)( \propto = 1 + \tfrac{{\sqrt 5 }}{2},\beta = \tfrac{1}{2})
minimizes the
Maxh Î [ - 1,1]\mathop {Max}\limits_{\eta \in [ - 1,1]}
E2. with the constraint E2O , the scheme...

In this paper the scalar equation ut + f(u) = 0 is approximated by three-point difference schemes in conservation form with an order of accuracy p equal to one or two. The partial differential equations approximated by the schemes with (p + 1)-th order accuracy are derived and their schock structures are analytically obtained and found to be close...

A residual-based compact scheme, previously developed to compute d-dimensional inviscid compressible flows with 3(rd)-order accuracy on a 3(d)-point stencil, is generalized to larger stencils allowing to reach a very high order of accuracy. Compactness is retained since for instance a 7(th)-order accurate dissipative approximation can be achieved o...

A dissipative compact scheme is developed within a dual time stepping framework for the computation of unsteady compressible flows. The design of the scheme relies on the vanishing, at steady state with respect to the dual time, of a residual that includes the physical time-derivative. High-order accuracy and numerical dissipation are obtained in a...

A residual-based compact scheme initially developed for steady compressible flows computation is extended to the unsteady case. Within a dual time step framework, the full residual, including the physical time derivative, is used to obtain high-accuracy and build the numerical dissipation in a compact way. The resulting scheme is compared with a co...

Dissipative compact schemes are constructed for multidimensional hyperbolic problems. High-order accuracy is not obtained for each space derivative, but for the whole residual, which avoids any linear algebra. Numerical dissipation is also residual based, i.e. constructed from derivatives of the residual only, which provides simplicity and robustne...

We present a scheme located between Lax-Wendroff and Roe schemes for the multidimensional compressible Euler and Navier-Stokes equations. The scheme is second-order accurate at steady state with a low internal dissipation. It is easy to code and involves no tuning parameter. Accuracy and efficiency are demonstrated for several flow problems, includ...

After looking for a convenient definition of accuracy for finite-volume schemes on structured meshes, a high-order accurate scheme is constructed for the Euler equations. Thanks to suitably weighted discretization operators, the proposed scheme is third-order on mildly deformed grids and second-order on highly deformed grids. The influence of mesh...

A simple and efficient time-dependent method is presented for solving the steady compressible Euler and Navier–Stokes equations with third-order accuracy. Owing to its residual-based structure, the numerical scheme is compact without requiring any linear algebra, and it uses a simple numerical dissipation built on the residual. The method contains...

In the present study a high-order accurate numerical method is used to investigate problems related
to helicopter rotors in high-speed forward flight. Numerical results are presented for compressible
turbulent flows past oscillating airfoils both in transonic regime (advancing blade tip), including
shock-stall conditions, and for a difficult dynami...

A dissipative compact scheme is presented for solving the steady compressible Euler and Navier-Stokes equations with a third-order accuracy. High-order accuracy is not obtained for each space derivative but for the whole residual, which avoids any linear algebra and provides compactness. Numerical dissipation is also constructed from derivatives of...

A comparison is made of two approaches for constructing high-order accurate schemes: one is Directional Non Compact (DNC) and the other is Residual-Based Compact (RBC). Both methods are simple and tuning-parameter free. Real accuracy and computational efficiency are discussed for various 2-D hyperbolic problems including compressible flows with sho...

In the present work, a centered third-order accurate scheme is developed for the simulation of unsteady compressible flows. In order to ensure a truly high-order accuracy of the scheme on general curvilinear grids, the spatial discretization is constructed by using suitably weighted formulae, which take into account mesh deformations. Time integrat...

Introduction In most domain decomposition methods for Computational Fluid Dynamics, the interface conditions are treated explicitly. When using implicit schemes, this causes stability or efficiency limitations (see [Rai86, Jen94]). For implicit schemes leading to the solution of block-tridiagonal linear systems, one can use the parallel factorisati...

Unsteady turbulent transonic flows over oscil-lating airfoils are investigated by using a novel third-order accurate scheme on block-structured meshes. Two turbu-lence models (algebraic and two-transport equation mod-els) are studied in order to generate flow fields in agree-ment with experimental data. Numerical results are pre-sented for 2-D prob...

An efficient multidimensional scheme is constructed for solving the compressible Euler equations. It is deduced from a centered scheme of the Lax–Wendroff type by using a special time-step in the numerical flux, namely a matricial characteristic time-step. This produces a compact second-order upwinding in a very simple way and leads to accurate non...

A third-order scheme for compressible
ows is constructed in by correcting the
dispersive second-order error of a purely centered scheme. The scheme is extended to curvilinear 2-D and
3-D meshes using a �nite-volume formulation. Mesh cell deformations are taken into account
through suitably weighted numerical flux formulae�.

Based on artificial compressibility and dual time-stepping, an implicit scheme is developed for solving the steady and unsteady incompressible Navier-Stokes equations for one and two-phase flows. The scheme is centered but, due to its internal dissipation, it needs no staggered grid or upwinding to be stable. Its stability with respect to pseudo an...

Multidomain treatments are studied in order to solve the steady compressible Euler equations using implicit time-dependent finite volume methods on block-structured grids. Unconditionally GKS-stable and conservative treatments are proposed for continuous and discontinuous 1D matchings and extended to 2D patched grids. Efficiency of the present inte...

For the numerical solution of the Euler equations in aerodynamics dissipative third-order accurate schemes are constructed using a dispersive error correction technique. Depending on the means of correcting, compact or non-compact schemes are obtained that are easy to implement and give very accurate solutions.

This paper is concerned with the solution of the steady Euler equations by an implicit unsteady method of Lax-Wendroff type without artificial viscosity or upwinding. Stability and efficiency are discussed for various treatments of the implicit operator: direct solution, approximate factorization or a few iterations of a line-relaxation method. Com...

When solving the compressible Navier-Stokes equations, any method requires some numerical dissipation. This dissipation, internal in upwind and Lax-Wendroff-type centered schemes, and artificial in Runge-Kutta centered scheme, is necessary to avoid instability and spurious oscillations. However, this numerical dissipation should be significantly sm...

In this contribution to the Workshop on Hypersonic Flows for Reentry Problems, we are interested in showing that an implicit method without upwinding or flux limiter can calculate external flows at large Mach number, large angle of attack and large Reynolds number.

An implicit Eulersolver is presented for the calculation of steady flows. This Euler solver is centered in space but does not require an artifical viscosity. It converges fastly to an accurate steady-state while capturing shock waves over one or two mesh cells. Applications are described for several test cases of subsonic and transonic flows over a...

For a hyperbolic system of conservation laws, the general form of conservative difference schemes involving two time-levels
in an explicit or implicit way is obtained under natural assumptions. General results are shown on the schemes and this framework
is used to study implicit schemes of second-order accuracy.

A centered Euler solver based on an implicit method of second-order
accuracy is described. Consideration is given to various applications to
transonic aerodynamics, including the internal flow in a channel with a
bump and several external flows over an airfoil at low and high angles
of attack. The efficiency and shock-capturing capabilities of the...

The present investigation is concerned with noniterative implicit
finite-difference methods for hyperbolic systems, taking into account
new space-centered methods involving two-time levels. The conducted
analysis leads to the selection of a space-centered method which is
expressed as an implicit correction of an explicit scheme of
second-order accu...

The present chapter is devoted to a detailed description of the implicit Euler solver that we have used and to a discussion of its accuracy and efficiency. First, this discussion is carried out for quasi-one-dimensional flows in a nozzle. Several forms of the algorithm with or without block inversion are compared for moderate and large CFL numbers....

Discusses finite volume methods to solve a pseudo-unsteady system deduced from the unsteady Euler equations by using the condition of constant total enthalpy. This simplification is consistent with the steady-state solution in the present case of iso-energetic flows. The reason for the choice of finite volume methods is their property of being exac...

Inviscid transonic flows over an airfoil in motion are calculated by solving the Euler equations in integral conservation-law form with the finite-volume method on a moving mesh. The method is ″in conservation form″ in a sense specified by a definition generalizing the P. D. Lax and B. Wendroff's definition to the case of several space variables an...

Numerical techniques applicable to several types of unsteady inviscid flows are reported. The unsteady flow categories include three-dimensional incompressible flows with vortex sheets, and compressible two-dimensional flows which either remain close to steady flow or depart considerably from steady flow. The three-dimensional incompressible flows...

Résumé : Dans certaines conditions d'écoulement en régime transsonique, une interaction choc/couche limite sur un profil d'aile peut conduire au tremblement de la structure. Nous mettons en oeuvre un outil de simulation numérique efficace, qui permet de prédire rapidement l'apparition du tremblement, afin d'évaluer les effets de techniques de contr...

Le couplage aéroélastique entre la structure évidée d'une aile de missile de croisière et le fluide en écoulement à vitesse transsonique sur cette aile est exploité pour augmenter la portée du missile. Une boucle de calcul permet de calculer la finesse du missile sous charge à l'équilibre des forces et de vérifier sa tenue structurale. Une explorat...

## Citations

... the Lax-Wendroff centered scheme [9] and the Roe upwind scheme [20] can be expressed in form (2) with numerical fluxes respectively defined as ...

... There is also similarities with the methods developed by A. Lerat and his co-authors, see e.g. [43,36]. They are residual methods, formally the look like the SUPG scheme with the difference that they introduce a non linear stabilisation to deal with the flow discontinuities. ...

... Other examples of the use of higher-order schemes can be found in Ducros et al. [63], Smirnov et al. [229], Lerat et al. [151], Rezgui et al. [200] and Yee et al. [261]. ...

... Lerat [37,38] investigated, by means of both exact solutions of the difference equations and modified equation analysis for a steady 1D problem, the discrete shock profiles of RBC schemes. Results show that the RBC scheme of 3rd-order accuracy always produces monotone shock profiles; RBC schemes of higher-order may provide non-oscillatory shock profiles if the shock is aligned with a cell face, whereas oscillatory solutions are obtained for shocks aligned with a cell center. ...

... Ce solveur dispose en effet de schémas numériques d'ordre élevé en espace, nécessaire pour l'advection du tourbillon. On utilisera un schéma d'ordre 3 pour les maillages curvilignes et d'ordre 5 pour les maillages cartésiens (Saunier, Péron et al. 2007 ;Saunier, Benoit et al. 2008 ;Saunier 2008). Le schéma utilisé pour les grilles curvilignes est discrétisé sous une formulation de type volumes finis avec des termes de correction des flux numériques tenant compte de la géométrie locale du maillage. ...

... These can be discretized without altering the difference operators applied to convective terms, by means of a careful selection of Pade formulas with the same denominators as in the Euler terms (see [27]). Preliminary calculations of a viscous Taylor-Green vortex [28] seem to support the preceding conclusion. Further studies are currently in progress. ...

... Remark 4.2 The idea of defining the numerical diffusion operator for finite volume fluxes in terms of entropy variables is not new but was proposed in Tadmor (1986Tadmor ( ),(1987 and subsequently, was used in Hughes Franca and Mallet (1986), Khalfallah and Lerat (1989) and others. The specific structure of the diffusion operator in (4.39), however, is novel for the shallow water equations. ...

... There are many physical phenomena such as electromagnetics, structural vibration, acoustical propagation, ultrasonic cavitation, ocean wave propagation and fluid dynamics, which can be described via hyperbolic partial differential equations [1][2][3][4][5][6][7]. Formulating robust numerical schemes for solving hyperbolic equations are critical in simulating these complex phenomena, and it is still the interest of many researchers even today. ...

... The numerical flux depends only on and is supposed to be conservative. The scheme (7) is said to be consistent if : then then (10) Observe that, in the case of hyperbolic systems -for which conservativity is important-, we do not have to ensure any conservativity property of the operator in order to have the convergence of the numerical solution to a weak solution of the system of conservation law. In fact, we have the following result : ...

... Residual-Based Compact (RBC) schemes were developed by A. Lerat and C. Corre [124][125][126] for the calculation of compressible flows. Although they were originally developed for structured grids, they are related -in principle -to RD schemes. ...