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Pierre Sagaut

Publications and source records attributed to Pierre Sagaut.

32 records · Page 2Linked to original sources
arXiv2017

Recursive regularization step for high-order lattice Boltzmann methods

A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of non-equilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second (and higher) order non-hydrodynamic contributions in under-resolved conditions. This is first shown in the isothermal case where the simulation of the doubly periodic shear layer is performed with a Reynolds number ranging from $10^4$ to $10^6$, and where a thorough analysis of the case at $Re=3\times 10^4$ is conducted. In the latter, results obtained using both regularization steps are compared against the BGK-LBM for standard (D2Q9) and high-order (D2V17 and D2V37) lattice structures, confirming the tremendous increase of stability range of the proposed approach. Further comparisons on thermal and fully compressible flows, using the general extension of this procedure, are then conducted through the numerical simulation of Sod shock tubes with the D2V37 lattice. They confirm the stability increase induced by the recursive approach as compared with the standard one.

physics.comp-ph↗
arXiv2013

Quantification of errors in large-eddy simulations of a spatially-evolving mixing layer

A stochastic approach based on generalized Polynomial Chaos (gPC) is used to quantify the error in Large-Eddy Simulation (LES) of a spatially-evolving mixing layer flow and its sensitivity to different simulation parameters, viz. the grid stretching in the streamwise and lateral directions and the subgrid scale model constant ($C_S$). The error is evaluated with respect to the results of a highly resolved LES (HRLES) and for different quantities of interest, namely the mean streamwise velocity, the momentum thickness and the shear stress. A typical feature of the considered spatially evolving flow is the progressive transition from a laminar regime, highly dependent on the inlet conditions, to a fully-developed turbulent one. Therefore the computational domain is divided in two different zones (\textit{inlet dependent} and \textit{fully turbulent}) and the gPC error analysis is carried out for these two zones separately. An optimization of the parameters is also carried out for both these zones. For all the considered quantities, the results point out that the error is mainly governed by the value of the $C_S$ constant. At the end of the inlet-dependent zone a strong coupling between the normal stretching ratio and the $C_S$ value is observed. The error sensitivity to the parameter values is significantly larger in the inlet-dependent upstream region; however, low error values can be obtained in this region for all the considered physical quantities by an ad-hoc tuning of the parameters. Conversely, in the turbulent regime the error is globally lower and less sensitive to the parameter variations, but it is more difficult to find a set of parameter values leading to optimal results for all the analyzed physical quantities.

physics.flu-dyn↗
arXiv2013

A stochastic view of isotropic turbulence decay

A stochastic EDQNM approach is used to investigate self-similar decaying isotropic turbulence at high Reynolds number ($400 \leq Re_λ\leq 10^4$). The realistic energy spectrum functional form recently proposed by Meyers & Meneveau is generalised by considering some of the model constants as random parameters, since they escape measure in most experimental set-ups. The induced uncertainty on the solution is investigated building response surfaces for decay power-law exponents of usual physical quantities. Large-scale uncertainties are considered, the emphasis being put on Saffman and Batchelor turbulence. The sensitivity of the solution to initial spectrum uncertainties is quantified through probability density functions of the decay exponents. It is observed that initial spectrum shape at very large scales governs the long-time evolution, even at high Reynolds number, a parameter which is not explicitly taken into account in many theoretical works. Therefore, a universal asymptotic behavior in which kinetic energy decays as $t^{-1}$ is not detected. But this decay law is observed at finite Reynolds number with low probability for some initial conditions.

physics.flu-dyn↗
arXiv2012

Analysis of the absorbing layers for the weakly-compressible lattice Boltzmann schemes

It has been demonstrated that Lattice Boltzmann schemes (LBSs) are very efficient for Computational AeroAcoustics (CAA). In order to handle the issue of absorbing acoustic boundary conditions for LBS, three kinds of damping terms are proposed and added into the right hand sides of the governing equations of LBS. From the classical theory, these terms play an important role to absorb and minimize the acoustic wave reflections from computational boundaries. Meanwhile, the corresponding macroscopic equations with the damping terms are recovered for analyzing the macroscopic behaviors of the these damping terms and determining the critical absorbing strength. Further, in order to detect the dissipation and dispersion behaviors, the linearized LBS with the damping terms is derived and analyzed. The dispersive and dissipative properties are explored in the wave-number spaces via the Von Neumann analysis. The related damping strength critical values and the optimal absorbing term are addressed. Finally, some benchmark problems are implemented to assess the theoretical results.

physics.comp-ph↗
arXiv2011

Sensitivity analysis and determination of free relaxation parameters for the weakly-compressible MRT-LBM schemes

It is well-known that there exist several free relaxation parameters in the MRT-LBM. Although these parameters have been tuned via linear analysis, the sensitivity analysis of these parameters and other related parameters are still not sufficient for detecting the behaviors of the dispersion and dissipation relations of the MRT-LBM. Previous researches have shown that the bulk dissipation in the MRT-LBM induces a significant over-damping of acoustic disturbances. This indicates that MRT-LBM cannot be used to obtain the correct behavior of pressure fluctuations because of the fixed bulk relaxation parameter. In order to cure this problem, an effective algorithm has been proposed for recovering the linearized Navier-Stokes equations from the linearized MRT-LBM. The recovered L-NSE appear as in matrix form with arbitrary order of the truncation errors with respect to $δt$. Then, in wave-number space, the first/second-order sensitivity analyses of matrix eigenvalues are used to address the sensitivity of the wavenumber magnitudes to the dispersion-dissipation relations. By the first-order sensitivity analysis, the numerical behaviors of the group velocity of the MRT-LBM are first obtained. Afterwards, the distribution sensitivities of the matrix eigenvalues corresponding to the linearized form of the MRT-LBM are investigated in the complex plane. Based on the sensitivity analysis and the recovered L-NSE, we propose some simplified optimization strategies to determine the free relaxation parameters in the MRT-LBM. Meanwhile, the dispersion and dissipation relations of the optimal MRT-LBM are quantitatively compared with the exact dispersion and dissipation relations. At last, some numerical validations on classical acoustic benchmark problems are shown to assess the new optimal MRT-LBM.

physics.flu-dyn↗
arXiv2011

Optimal low-dispersion low-dissipation LBM schemes for computational aeroacoustics

Lattice Boltmzmann Methods (LBM) have been proved to be very effective methods for computational aeroacoustics (CAA), which have been used to capture the dynamics of weak acoustic fluctuations. In this paper, we propose a strategy to reduce the dispersive and disspative errors of the two-dimensional (2D) multi-relaxation-time lattice Boltzmann method (MRT-LBM). By presenting an effective algorithm, we obtain a uniform form of the linearized Navier-Stokes equations corresponding to the MRT-LBM in wave-number space. Using the matrix perturbation theory and the equivalent modified equation approach for finite difference methods, we propose a class of minimization problems to optimize the free-parameters in the MRT-LBM. We obtain this way a dispersion-relation-preserving LBM (DRP-LBM) to circumvent the minimized dispersion error of the MRT-LBM. The dissipation relation precision is also improved.And the stability of the MRT-LBM with the small bulk viscosity is guaranteed. Von Neuman analysis of the linearized MRT-LBM is performed to validate the optimized dispersion/dissipation relations considering monochromatic wave solutions. Meanwhile, dispersion and dissipation errors of the optimized MRT-LBM are quantitatively compared with the original MRT-LBM . Finally, some numerical simulations are carried out to assess the new optimized MRT-LBM schemes.

physics.comp-ph↗
arXiv2008

Spurious caustics of Dispersion Relation Preserving schemes

A linear dispersive mechanism leading to a burst in the $L_\infty$ norm of the error in numerical simulation of polychromatic solutions is identified. This local error pile-up corresponds to the existence of spurious caustics, which are allowed by the dispersive nature of the numerical error. From the mathematical point of view, spurious caustics are related to extrema of the numerical group velocity and are physically associated to interactions between rays defined by the characteristic lines of the discrete system. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes.

math.AP↗
arXiv2008

Structural stability of finite dispersion-relation preserving schemes

The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurance of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to has a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solution of the original continuous equations. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes.

math.AP↗
arXiv2008

Towards optimal DRP scheme for linear advection

Finite difference schemes are here solved by means of a linear matrix equation. The theoretical study of the related algebraic system is exposed, and enables us to minimize the error due to a finite difference approximation, while building a new DRP scheme in the same time.

math.AP↗
arXiv2007

Lie group stability of finite difference schemes

Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference approximation, and leading to inaccurate numerical results. This paper proposes a method that enables us to build a scheme that preserves some of those symmetries.

math.NA↗
arXiv2006

DRP scheme optimization

A new DRP scheme is built, which enables us to minimize the error due to the finite difference approximation, by means of an equivalent matrix equation.

math.AP↗
arXiv2006

Theoretical Optimization of Finite Difference Schemes

The aim of this work is to develop general optimization methods for finite difference schemes used to approximate linear differential equations. The specific case of the transport equation is exposed. In particular, the minimization of the numerical error is taken into account. The theoretical study of a related linear algebraic problem gives general results which can lead to the determination of the optimal scheme.

math.AP↗
arXiv2006

A linear dispersive mechanism for numerical error growth: spurious caustics

A linear dispersive mechanism for error focusing in polychromatic solutions is identified. This local error pile-up corresponds to the existence of spurious caustics, which are allowed by the dispersive nature of the numerical error. From the mathematical point of view, spurious caustics are related to extrema of the numerical group velocity. Several popular schemes are analyzed and are shown to admit spurious caustics. It is also observed that caustic-free schemes can be defined, like the Crank-Nicolson scheme.

math.AP↗