Alain Lerat

Residual-based compact schemes for multidimensional hyperbolic systems of conservation laws

Abstract 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 robustness. High accuracy and efficiency are checked on 2-D and 3-D model problems. Various applications to the compressible Euler equations without and with shock waves are presented.

Spectral properties of high-order residual-based compact schemes for unsteady compressible flows

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 number associated to their spatial discretization operator, and their dispersive and dissipative behaviors are investigated as functions of a multidimensional wave number. For RBC schemes of higher orders (5 and 7), both dissipation and dispersion errors take very low values up to reduced wave numbers close to the grid resolvability limit, while higher frequencies are efficiently damped out. Thanks to their genuinely multidimensional formulation, RBC schemes conserve good dissipation and dispersion properties even for flow modes that are not aligned with the computational grid. Numerical tests support the theoretical results. Specifically, the study of a complex nonlinear problem dominated by energy transfer from large to small flow scales, the inviscid Taylor-Green vortex flow, confirms numerically the interest of a well-designed RBC dissipation to resolve accurately fine scale flow structures.

Vorticity-preserving schemes for the compressible Euler equations

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 preserving for the Euler equations, which is confirmed by vortex flow calculations.

On the design of high order residual-based dissipation for unsteady compressible flows

The numerical dissipation operator of residual-based compact (RBC) schemes of high accuracy is identified and analysed for hyperbolic systems of conservation laws. A necessary and sufficient condition (@g-criterion) is found that ensures dissipation in 2-D and 3-D for any order of the RBC scheme. Numerical applications of RBC schemes of order 3, 5 and 7 to a diagonal wave advection and to a converging cylindrical shock problem confirm the theoretical results.

Computation of Airfoil-Vortex Interaction Using a Vorticity-Preserving Scheme

We recently proved that a dissipative residual-based scheme of second-order accuracy is vorticity-preserving for the compressible Euler 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 convergence study and a comparison with a conventional scheme and with experimental measurements are presented. The new scheme shows a faster grid convergence, especially for the vortex trajectories and deformations during the interaction.

A third-order finite-volume residual-based scheme for the 2D Euler equations on unstructured grids

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 shock-capturing properties which motivated their extension to unstructured grids using a finite volume (FV) method. A second-order formulation of the FV–RB scheme for compressible flows on general unstructured grids was presented in a previous paper. The present paper describes the derivation of a third-order FV–RB scheme and its application to hyperbolic model problems as well as subsonic, transonic and supersonic internal and external inviscid flows.

Steady discrete shocks of high-order RBC schemes

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 Burgers equation and the oscillatory nature of the shock profiles is determined according to the scheme order and to the shock location with respect to the mesh.

Numerical Simulation of Shock/Boundary-Layer Self-Sustained Oscillations for External and Internal Flows

The simulation of external or internal transonic flows using a standard k − 2 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 production of turbulent kinetic energy. A weakly non-linear correction that makes the eddy viscosity coefficient dynamical and a so-called PANS approach that modifies the dissipation rate equation allow to improve the standard model so as to predict the appearance of selfsustained shock oscillations over an airfoil and in a diffuser.

Steady discrete shocks of 5th and 7th-order RBC schemes and shock profiles of their equivalent differential equations

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 Burgers 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 dissipative high-order schemes considered, discrete shocks in infinite domain or with boundary conditions at short distance (Rankine–Hugoniot relations) are found to be very close. Besides, the present analytical description of shock capturing in infinite domain is explicit and so simple that it could lead to a new approach for correcting parasitic oscillations of high order RBC schemes. In a second part of the paper, exact solutions are also derived for equivalent differential equations (EDE) approximating RBC2p−1 schemes (subscript denotes the accuracy order) at orders 2p and 2p+1. Although EDE involves Taylor expansions around steep structures, agreement between the exact EDE shock-profiles and the discrete shocks is remarkably good for RBC5 and RBC7 schemes. In addition, a strong similarity is demonstrated between the analytical expressions of discrete shocks and EDE shock profiles.

Transonic Flow Control Using a Navier-Stokes Solver and a Multi-Objective Genetic Algorithm

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 shock wave and the trailing edge on the upper airfoil surface.