Residual-Based Compact Schemes Research Articles

An efficient high-order compact scheme for the unsteady compressible Euler and Navier–Stokes equations

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 formulation has been recently proposed [13] for simplifying the RBC schemes and increasing their temporal accuracy. The present paper goes much further in this direction and deeply reconsiders the method. An avatar of the RBC schemes is presented that greatly reduces the computing time and the memory requirements while keeping the same type of successful numerical dissipation. Two and three-dimensional linear stability are analyzed and the method is extended to the 3-D compressible Navier–Stokes equations. The new compact scheme is validated for several unsteady problems in two and three dimension. In particular, an accurate DNS at moderate cost is presented for the evolution of the Taylor–Green Vortex at Reynolds 1600 and Prandtl 0.71. The effects of the mesh size and of the accuracy order in the approximation of Euler and viscous terms are discussed.

A high-order time formulation of the RBC schemes for unsteady compressible Euler equations

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 method) usually solved by a dual-time approach. This method is efficient for computing compressible flows in slow unsteady regimes, but for quick unsteady flows, it may be costly and not accurate enough. A new time-formulation is proposed in the present paper. Unusually, in a RBC scheme the time derivative occurs, through linear discrete operators due to compactness, not only in the main residual but also in the other two residuals (in 2-D) involved in the numerical dissipation. To extract the time derivative, a space-factorization method which preserves the high accuracy in space is developed for reducing the algebra to the direct solution of simple linear systems on the mesh lines. Then a time-integration of high accuracy is selected for the RBC schemes by comparing the efficiency of four classes of explicit methods. The new time-formulation is validated for the diagonal advection of a Gaussian shape, the rotation of a hump, the advection of a vortex for a long time and the interaction of a vortex with a shock.

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 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 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.

Toward improved simulation tools for compressible turbomachinery: assessment of residual-based compact schemes for the transonic compressor NASA Rotor 37

Residual-based compact schemes of second- and third-order accuracy are applied to a challenging three-dimensional flow through a transonic compressor. The proposed schemes provide almost mesh-converged solutions in good agreement with experimental data on relatively coarse grids, which allow achieving a given accuracy level with computational cost reductions by a factor between 2 and 4 with respect to standard solvers.

An accurate finite-volume formulation of a Residual-Based Compact scheme for unsteady compressible flows

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 interest. The scheme makes use of weighted approximations that allow to ensure high accuracy while taking benefit from the structured nature of the grid. The accuracy and Cauchy stability properties of the proposed spatial approximation are discussed in detail. Numerical applications to unsteady compressible flows of increasing complexity demonstrate the advantages of the proposed formulation with respect to straightforward extensions of RBC schemes to curvilinear meshes.

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.

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 condit...

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 (χ-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.

High-order residual-based compact schemes for aerodynamics and aeroacoustics

Residual-Based Compact (RBC) schemes are reviewed and applied to realistic compressible flow problems. The principle and advantages of high-order RBC schemes are first presented. Then, an implementation in the elsA code of the 3rd-order RBC scheme is used for RANS calculations of the transonic flow over the ONERA M-6 wing and of the flow in a CME2 compressor stage. Finally, the 7th-order RBC scheme is applied to the complete Euler equations for computing a planar aeroacoustic problem and the propagation of a spinning acoustic mode in an axisymetric aeroengine inlet. Numerical results demonstrate the accuracy and efficiency of the RBC approach.

Multiblock residual-based compact schemes for the computation of complex turbomachinery flows

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 the flow physics.

A residual-based compact scheme of optimal order for hyperbolic problems

A new fourth-order dissipative scheme on a compact 3 × 3 stencil is presented for solving 2 D 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 2 D scalar problems demonstrates the excellent accuracy and efficiency properties offered by this new RBC scheme with respect to existing second- and third-order versions.

High-order residual-based compact schemes for advection–diffusion problems

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 5th-order accurate dissipative approximation of a d-dimensional advection–diffusion problem can be achieved on a 5 d stencil, without requiring the linear system solutions associated with usual compact schemes. Applications to 1D and 2D model problems are presented and demonstrate that the theoretical orders of accuracy can be achieved in practice.

High-order residual-based compact schemes for compressible inviscid flows

A residual-based compact scheme, previously developed to compute d-dimensional inviscid compressible flows with third-order accuracy 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 reach a very high order of accuracy. Compactness is retained since for instance a seventh-order accurate dissipative approximation can be achieved on a 5 d-point stencil, without requiring the linear system solutions associated with usual compact schemes. The high-order generalization is also performed for unsteady flows. Applications to model problems and unsteady inviscid flows are presented.

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

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.