Pierre Sagaut

Comparison between lattice Boltzmann method and Navier-Stokes high order schemes for computational aeroacoustics

Computational aeroacoustic (CAA) simulation requires accurate schemes to capture the dynamics of acoustic fluctuations, which are weak compared with aerodynamic ones. In this paper, two kinds of schemes are studied and compared: the classical approach based on high order schemes for Navier-Stokes-like equations and the lattice Boltzmann method. The reference macroscopic equations are the 3D isothermal and compressible Navier-Stokes equations. A Von Neumann analysis of these linearized equations is carried out to obtain exact plane wave solutions. Three physical modes are recovered and the corresponding theoretical dispersion relations are obtained. Then the same analysis is made on the space and time discretization of the Navier-Stokes equations with the classical high order schemes to quantify the influence of both space and time discretization on the exact solutions. Different orders of discretization are considered, with and without a uniform mean flow. Three different lattice Boltzmann models are then presented and studied with the Von Neumann analysis. The theoretical dispersion relations of these models are obtained and the error terms of the model are identified and studied. It is shown that the dispersion error in the lattice Boltzmann models is only due to the space and time discretization and that the continuous discrete velocity Boltzmann equation yield the same exact dispersion as the Navier-Stokes equations. Finally, dispersion and dissipation errors of the different kind of schemes are quantitatively compared. It is found that the lattice Boltzmann method is less dissipative than high order schemes and less dispersive than a second order scheme in space with a 3-step Runge-Kutta scheme in time. The number of floating point operations at a given error level associated with these two kinds of schemes are then compared.

Error dynamics

The propagation of a signal in a continuous medium and the associated evolution of error is of prime importance in many applications of applied physics. There have been many efforts in analyzing error dynamics, using method attributed to von Neumann [1,2], that is readily applied for linear equations and in quasilinearized form for non-linear equations. The main assumption for linear problems is that the error and the signal follow the same dynamics. While this appears intuitively correct, the main aim behind this work is to show that this is not correct for discrete computing due to dispersion or phase error or when the numerical method is not strictly neutrally stable. We demonstrate the above with the help of the linear advection equation. For the analysis of space-time discretization schemes, the linear advection equation as a model that represents many flows and wave phenomena is used,

Large eddy simulations of aero-optical effects in a turbulent boundary layer

Large eddy simulations (LESs) of aero-optical effects in a turbulent boundary layer have been carried out at two different Mach numbers (0.9 and 2.3) for two different wall boundary conditions (adiabatic and isothermal). Moreover, in the adiabatic case, LESs were performed on two different meshes. First, aerodynamic fields are proved to compare favourably with theoretical and experimental results. Once validated, the characteristics of the boundary layer allow us to obtain information concerning optical beam degradation. The density field is then used to compute the phase distortion induced by turbulent fluctuations on a coherent optical beam. The Mach number effect on the phase distortion is evaluated by means of these computations and the link between index-of-refraction fluctuations and phase distortion is discussed. Moreover, LES allows us to study optical models and the validity of their assumptions. Finally, LES is proved to be considered as a reference tool to evaluate phase distortion.

An arbitrary Lagrangian-Eulerian approach for the simulation of immersed moving solids with Lattice Boltzmann Method

The flow-structure interactions streaming from the motion of an immersed solid body are investigated through an Arbitrary Lagrangian-Eulerian (ALE) approach applied to the Lattice Boltzmann method (LBM). The method is based on the use of a moving grid to describe the flow around the solid body, while the physical domain is resolved by the use of an Eulerian frame fixed grid. The moving grid displacements follows the same moving law of the body, and its shape does not change during the simulation. The communication between the moving grid and the fixed grid is performed at the beginning of each time step through interpolation.The ALE-LBM approach has been derived from the discretized Boltzmann equation by a Chapman-Enskog expansion procedure, the equivalence of the proposed method with the Navier-Stokes equations for a weakly compressible athermal flow being recovered.Numerical simulations of academical test cases have been performed in order to assess the method and to investigate the sensitivity of the error to the simulation parameters. Three different test cases have been considered, in order to perform a robust assessment of the ALE-LBM approach. More specifically, the Uniform Flow, the Poiseuille Flow and the Plane Wave test cases have been studied and the limits of application of the approach have been defined and discussed.Finally, the case of a rotating two dimensional square cylinder immersed in a Poiseuille Flow, Re = 80 , has been numerically investigated. The results confirm that the ALE-LBM approach is able to correctly represent the physical flow features and, in particular, the transition zone between the two grids used is smooth and continuous, a sign that the error due to the interpolation process is bounded and does not diverge in time.

A lattice Boltzmann method for nonlinear disturbances around an arbitrary base flow

In this paper we address the problem of the time evolution of a perturbation around a steady base flow with the use of the lattice Boltzmann method (LBM). This approach, named base flow lattice Boltzmann method, is of great interest in particular for aeroacoustic fields where the acoustic perturbation, on the one hand, is almost exclusively influenced by the large scale average structures of the underlying flow, and on the other hand, has a low effect on the large structures. The method is implemented for weakly compressible flows and the results of the base flow lattice Boltzmann are compared with the standard single relaxation time LBM. The boundary conditions for the base flow lattice Boltzmann method are discussed, as well as the implementation of outflow conditions for acoustic waves.

An adjoint-based lattice Boltzmann method for noise control problems

In this paper optimal control of acoustic problems is addressed within the lattice Boltzmann method framework. To this end, an adjoint-based lattice Boltzmann method is proposed to solve the adjoint problem. The adjoint state provides an easy access to the optimization gradients. The line search step in Newtons descent method is performed through a combination of complex differentiation and adjoint problem in the lattice Boltzmann method. The implementation of an active noise reduction method for two-dimensional weakly compressible (low Mach number) flows is discussed, and the applicability of the method is assessed.

Large Eddy Simulation Study of Synthetic Jet Frequency and Amplitude Effects on a Rounded Step Separated Flow

Numerical simulations of active separation control by means of synthetic jet are carried out to study the forcing frequency and amplitude effects on the flow. The chosen test case is a rounded ramp at a Reynolds number based on the step height of Re h = 30192. The incoming flow is fully turbulent with Re uf071 uf020 = 1350 at the separation point. The whole flow in the synthetic jet cavity is computed to ensure an accurate description of the actuator effect on the flow-field. Through 21 LES simulations, it is shown that the optimal frequency is relative to the considered objective. In the present case, no frequency simultaneously improves every evaluated criterion. The effect of amplitude seems to be independent of the frequency one.

NARX Modeling and Adaptive Closed-Loop Control of a Separation by Synthetic Jet in Unsteady RANS computations

A numerical study concerning a SISO active closed-loop separation control on a rounded step is presented. A first study of the synthetic jet frequency effect on the separation shows that the mean separation bubble surface is minimized if the mean pressure of a unique wall pressure sensor is maximized. With the aim of designing a closed-loop strategy for the control of the recirculation bubble, a NARX black-box model of the pressure signal is identified thanks to a unique unsteady RANS simulation. The basic extremum-seeking algorithm is improved with an adaptive gain to guarantee algorithm performance and validated against the nonlinear black-box model of the forced flow. Then, the robust adaptive closed-loop is applied in real-time with an unsteady RANS simulation. Closed-loop results demonstrate the ability of the extremum-seeking control with adaptive gain to automatically control the separation by minimizing the recirculation bubble surface thanks to an unsteady RANS simulation.

An adaptive numerical method for solving EDQNM equations for the analysis of long-time decay of isotropic turbulence

A new numerical formulation for the Eddy-Damped Quasi-Normal Markovian (EDQNM) model is proposed in the present paper. This formulation is based on an adaptive procedure that progressively modifies the spectral mesh at the large scales, forcing a resolution requirement set by the user.The resulting adaptive numerical method for the EDQNM model has been systematically tested by comparison with the classical model, covering a wide range of initial conditions. In particular, the sensitivity of the results to the initial Reynolds number Re λ and to the slope of the energy spectrum at the large scales ? ( E ( k ? 0 ) ? k ? ) has been investigated. For all the initial conditions prescribed, the adaptive numerical method recovers exactly the same solution of the classical version. This result has been observed in the analysis of the main statistical quantities of interest, such as the turbulent kinetic energy K and the energy dissipation rate e. The same conclusions have been drawn by the direct comparison of the energy spectra E.An extension of the adaptive numerical method for the analysis of sheared/rotating turbulence is as well proposed and successfully assessed.The numerical algorithm proves to be progressively more efficient when long-time simulations are performed. In particular, a reduction up to one order of magnitude in the computational resources required is observed. Due to its efficiency and precision, the adaptive formulation of the EDQNM model promises to be an optimal tool to be blended with the methods of the Uncertainty Quantification (UQ) theory, in order to provide new insights about isotropic turbulence decay.

Thermal lattice Boltzmann simulation with immersed solid boundary model

f Isothermal continuous distribution function g Isothermal discrete distribution function q Thermal continuous distribution function h Thermal discrete distribution function f Isothermal equilibrium distribution function g Isothermal equilibrium distribution function q Thermal equilibrium distribution function h Thermal equilibrium distribution function τf Relaxation time for distribution function f τg Relaxation time for distribution function g τq Relaxation time for distribution function q τh Relaxation time for distribution function h g Non-equilibrium part of g h Non-equilibrium part of h xi Spatial vector, m u Velocity, m.s ci Discrete velocity m Mass, kg ρ Density, kg.m T Temperature, K R Universal constant of perfect gaz, J.K.mol r Constant of gaz, J.kg.K D Dimension kB Boltzmann constant, J.K −1 λ Caracteristic time between two collision ǫ Knudsen number μ Dynamic viscosity, kg.m.s β Thermal dilatation coefficient, K wα Weight of discrete velocity E Total energy, J Pij Total stress tensor