Study of interpolation methods for high-accuracy computations on overlapping grids

Overset strategy can be an efficient way to keep high-accuracy discretization by decomposing a complex geometry in topologically simple subdomains. Apart from the grid assembly algorithm, the key point of overset technique lies in the interpolation processes which ensure the communications between the overlapping grids. The family of explicit Lagrange and optimized interpolation schemes is studied. The a priori interpolation error is analyzed in the Fourier space, and combined with the error of the chosen discretization to highlight the modification of the numerical error. When high-accuracy algorithms are used an optimization of the interpolation coefficients can enhance the resolvality, which can be useful when high-frequency waves or small turbulent scales need to be supported by a grid. For general curvilinear grids in more than one space dimension, a mapping in a computational space followed by a tensorization of 1-D interpolations is preferred to a direct evaluation of the coefficient in the physical domain. A high-order extension of the isoparametric mapping is accurate and robust since it avoids the inversion of a matrix which may be ill-conditioned. A posteriori error analyses indicate that the interpolation stencil size must be tailored to the accuracy of the discretization scheme. For well discretized wavelengthes, the results show that the choice of a stencil smaller than the stencil of the corresponding finite-difference scheme can be acceptable. Besides the gain of optimization to capture high-frequency phenomena is also underlined. Adding order constraints to the optimization allows an interesting trade-off when a large range of scales is considered. Finally, the ability of the present overset strategy to preserve accuracy is illustrated by the diffraction of an acoustic source by two cylinders, and the generation of acoustic tones in a rotor–stator interaction. Some recommandations are formulated in the closing section

On compressibility assumptions in aeroacoustic integrals: a numerical study with subsonic mixing layers

Two assumptions commonly made in predictions based on Lighthill’s formalism are investigated: a constant density in the quadrupole expression, and the evaluation of the source quantity from incompressible simulations. Numerical predictions of the acoustic field are conducted in the case of a subsonic spatially evolving two-dimensional mixing layer at Re = 400. Published results of the direct noise computation (DNC) of the flow are use as reference and input for hybrid approaches before the assumptions on density are progressively introduced. Divergence free velocity fields are obtained from an incompressible simulation of the same flow case, exhibiting the same hydrodynamic field as the DNC. Fair comparisons of the hybrid predictions with the reference acoustic field valid both assumptions in the source region for the tested values of the Mach number. However, in the observer region, the inclusion of flow effects in the Lighthill source term is not preserved, which is illustrated through a comparison with the Kirchhoff wave-extrapolation formalism, and with the use of a convected Green function in the integration process

Turbulent boundary layer noise : direct radiation at Mach number 0.5

Boundary layers constitute a fundamental source of aerodynamic noise. A turbulent boundary layer over a plane wall can provide an indirect contribution to the noise by exciting the structure, and a direct noise contribution. The latter part can play a significant role even if its intensity is very low, explaining why it is hardly measured unambiguously. In the present study, the aerodynamic noise generated by a spatially developing turbulent boundary layer is computed directly by solving the compressible Navier-Stokes equations. This numerical experiment aims at giving some insight into the noise radiation characteristics. The acoustic wavefronts have a large wavelength and are oriented in the direction opposite to the flow. Their amplitude is only 0.7 % of the aerodynamic pressure for a flat-plate flow at Mach 0.5. The particular directivity is mainly explained by convection effects by the mean flow, giving an indication about the compactness of the sources. These vortical events correspond to low-frequencies, and have thus a large life time. They cannot be directly associated with the main structures populating the boundary layer such as hairpin or horseshoe vortices. The analysis of the wall pressure can provide a picture of the flow in the frequency-wavenumber space. The main features of wall pressure beneath a turbulent boundary layer as described in the literature are well reproduced. The acoustic domain, corresponding to supersonic wavenumbers, is detectable but can hardly be separated from the convective ridge at this relatively high speed. This is also due to the low frequencies of sound emission as noted previously

A domain decomposition matrix-free method for global linear stability

This work is dedicated to the presentation of a matrix-free method for global linear stability analysis in geometries composed of multi-connected rectangular subdomains. An Arnoldi technique using snapshots in subdomains of the entire geometry combined with a multidomain linearized Direct Numerical Finite difference simulations based on an influence matrix for partitioning are adopted. The method is illustrated by three benchmark problems: the lid-driven cavity, the square cylinder and the open cavity flow. The efficiency of the method to extract large-scale structures in a multidomain framework is emphasized. The possibility to use subset of the full domain to recover the perturbation associated with the entire flow field is also highlighted. Such a method appears thus a promising tool to deal with large computational domains and three-dimensionality within a parallel architecture

Investigation of flow structures involved in sound generation by two- and three-dimensional cavity flows

Proper Orthogonal Decomposition and Stochastic Estimation are combined to shed some light on the link between organized flow structures and noise generation by turbulent flows. Proper Orthogonal Decomposition (POD) is firstly used to extract selected flow events. Based on the knowledge of these structures, the Quadratic Stochastic Estimation of the acoustic pressure field is secondly performed. Both procedures are successively applied to two- and three-dimensional numerical databases of a flow over a cavity. It is demonstrated that POD can extract selected aerodynamic events which can be associated with selected frequencies in the acoustic spectra. Reconstructed acoustic fields also indicate the aerodynamic events which are responsible of the main energy of the noise emission. Such mathematical tools offer new perspectives in analysing flow structures involved in sound generation by turbulent flows and in the experimental design of a flow control strategy

Study of interpolation methods for high-accuracy computations on overlapping grids

Overset strategy can be an efficient way to keep high-accuracy discretization by decomposing a complex geometry in topologically simple subdomains. Apart from the grid assembly algorithm, the key point of overset technique lies in the interpolation processes which ensure the communications between the overlapping grids. The family of explicit Lagrange and optimized interpolation schemes is studied. The a priori interpolation error is analyzed in the Fourier space, and combined with the error of the chosen discretization to highlight the modification of the numerical error. When high-accuracy algorithms are used an optimization of the interpolation coefficients can enhance the resolvality, which can be useful when high-frequency waves or small turbulent scales need to be supported by a grid. For general curvilinear grids in more than one space dimension, a mapping in a computational space followed by a tensorization of 1-D interpolations is preferred to a direct evaluation of the coefficient in the physical domain. A high-order extension of the isoparametric mapping is accurate and robust since it avoids the inversion of a matrix which may be ill-conditioned. A posteriori error analyses indicate that the interpolation stencil size must be tailored to the accuracy of the discretization scheme. For well discretized wavelengthes, the results show that the choice of a stencil smaller than the stencil of the corresponding finite-difference scheme can be acceptable. Besides the gain of optimization to capture high-frequency phenomena is also underlined. Adding order constraints to the optimization allows an interesting trade-off when a large range of scales is considered. Finally, the ability of the present overset strategy to preserve accuracy is illustrated by the diffraction of an acoustic source by two cylinders, and the generation of acoustic tones in a rotor–stator interaction. Some recommandations are formulated in the closing section

Global and Koopman modes analysis of sound generation in mixing layers

It is now well established that linear and nonlinear instability waves play a significant role in the noise generation process for a wide variety of shear flows such as jets or mixing layers. In that context, the problem of acoustic radiation generated by spatially growing instability waves of two-dimensional subsonic and supersonic mixing layers are revisited in a global point of view, i.e., without any assumption about the base flow, in both a linear and a nonlinear framework by using global and Koopman mode decompositions. In that respect, a timestepping technique based on disturbance equations is employed to extract the most dynamically relevant coherent structures for both linear and nonlinear regimes. The present analysis proposes thus a general strategy for analysing the near-field coherent structures which are responsible for the acoustic noise in these configurations. In particular, we illustrate the failure of linear global modes to describe the noise generation mechanism associated with the vortex pairing for the subsonic regime whereas they appropriately explain the Mach wave radiation of instability waves in the supersonic regime. By contrast, the Dynamic Mode Decomposition (DMD) analysis captures both the near-field dynamics and the far-field acoustics with a few number of modes for both configurations. In addition, the combination of DMD and linear global modes analyses provides new insight about the influence on the radiated noise of nonlinear interactions and saturation of instability waves as well as their interaction with the mean flow

Comparison of Subgrid-scale Viscosity Models and Selective Filtering Strategy for Large-eddy Simulations

Explicitly filtered large-eddy simulations (LES), combining high-accuracy schemes with the use of a selective filtering without adding an explicit subgrid-scales (SGS) model, are carried out for the Taylor-Green-vortex and the supersonic-boundary-layer cases. First, the present approach is validated against direct numerical simulation (DNS) results. Subsequently, several SGS models are implemented in order to investigate if they can improve the initial filter-based methodology. It is shown that the most accurate results are obtained when the filtering is used alone as an implicit model, and for a minimal cost. Moreover, the tests for the Taylor-Green vortex indicate that the discretization error from the numerical methods, notably the dissipation error from the high-order filtering, can have a greater influence than the SGS models

Space-dependent aggregation of data-driven turbulence models

A machine-learning approach for data-driven Reynolds-Averaged Navier--Stokes (RANS) predictions of turbulent flows including estimates of turbulence modelling uncertainties is developed by combining a Bayesian symbolic identification methodology for learning customised RANS model corrections for selected classes of flows and a space-dependent model-aggregation algorithm that combines the predictions of a set of competing machine-learned RANS models by means of weighting functions depending on a vector of local flow features. The customised model corrections are learned by using the SBL-SpaRTA algorithm, recently proposed by \citet{cherroud2022sparse}, which delivers sparse model correction terms in analytical form and whose parameters are described by probability distribution functions. This makes the learned models naturally interpretable and endowed with a measure of uncertainty. The learned models are subsequently aggregated by training Random Forests Regressors (RFR), which associates a model performance score with a set of local flow features. The scores can be interpreted as the probability that a candidate model performs better than its competitors, given the flow behavior at a given location. Predictions of new flows are then formulated as a locally weighted average of the solutions of a set of machine-learned models. An uncertainty measure is obtained by propagating through the models their posterior parameter distributions and by combining competing model predictions to estimate the inter-model variance. The proposed space-dependent model aggregation procedure (X-MA) is applied to flows of varying complexity, showing significant improvement with respect to the baseline model and good generalizability to unseen flows. The results make X-MA an attractive candidate for the development of a generalizable data-driven turbulence modelling framework with quantified uncertainty

A domain decomposition matrix-free method for global linear stability

This work is dedicated to the presentation of a matrix-free method for global linear stability analysis in geometries composed of multi-connected rectangular subdomains. An Arnoldi technique using snapshots in subdomains of the entire geometry combined with a multidomain linearized Direct Numerical Finite difference simulations based on an influence matrix for partitioning are adopted. The method is illustrated by three benchmark problems: the lid-driven cavity, the square cylinder and the open cavity flow. The efficiency of the method to extract large-scale structures in a multidomain framework is emphasized. The possibility to use subset of the full domain to recover the perturbation associated with the entire flow field is also highlighted. Such a method appears thus a promising tool to deal with large computational domains and three-dimensionality within a parallel architecture