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.Comment: Accepted for publication as a Regular Article in Physical Review

A linear stability analysis of compressible hybrid lattice Boltzmann methods

An original spectral study of the compressible hybrid lattice Boltzmann method (HLBM) on standard lattice is proposed. In this framework, the mass and momentum equations are addressed using the lattice Boltzmann method (LBM), while finite difference (FD) schemes solve an energy equation. Both systems are coupled with each other thanks to an ideal gas equation of state. This work aims at answering some questions regarding the numerical stability of such models, which strongly depends on the choice of numerical parameters. To this extent, several one- and two-dimensional HLBM classes based on different energy variables, formulation (primitive or conservative), collision terms and numerical schemes are scrutinized. Once appropriate corrective terms introduced, it is shown that all continuous HLBM classes recover the Navier-Stokes Fourier behavior in the linear approximation. However, striking differences arise between HLBM classes when their discrete counterparts are analysed. Multiple instability mechanisms arising at relatively high Mach number are pointed out and two exhaustive stabilization strategies are introduced: (1) decreasing the time step by changing the reference temperature $T_{ref}$ and (2) introducing a controllable numerical dissipation $\sigma$ via the collision operator. A complete parametric study reveals that only HLBM classes based on the primitive and conservative entropy equations are found usable for compressible applications. Finally, an innovative study of the macroscopic modal composition of the entropy classes is conducted. Through this study, two original phenomena, referred to as shear-to-entropy and entropy-to-shear transfers, are highlighted and confirmed on standard two-dimensional test cases.Comment: 49 pages, 23 figure

Delineating loss sources within a linear cascade with upstream cavity and purge flow

Purge air is injected in cavities at hub of axial turbines to prevent hot mainstream gas ingestion into inter-stage gaps. This process induces additional losses for the turbine due to an interaction between purge and mainstream flow. This paper investigates the flow in a low-speed linear cascade rig with upstream hub cavity at a Reynolds number commonly observed in modern low pressure turbine stages by the use of numerical simulation. Numerical predictions are validated by comparing against experimental data available. Three different purge mass flow rates are tested using three different rim seal geometries. Numerical simulations are performed using a Large Eddy Simulation (LES) solver on structured grids. An investigation of the different mechanisms associated to turbine flow including cavity and purge air is intended through this simplified configuration. The underlying mechanisms of loss are tracked using an entropy formulation. Once described for a baseline case, the influence of purge flow and rim seal geometry on flow mechanisms and loss generation are described with the emphasis to obtain design parameters for losses reduction. The study quantifies loss generation due to boundary layer on wetted surfaces and secondary vortices developing in the passage. The analysis shows different paths by which purge flow and rim seal geometry can change loss generation including a modification of the shear layer between purge and mainstream, interaction with secondary vortices and a modification of the flow behavior close to hub compared to a smooth configuration. The study shows the influence of purge flow rate and swirl on the strengthening of secondary vortices in the passage and the ability of axial overlapping rim seal to delay the development of secondary vortices compared to simple axial gaps

Revisiting the spectral analysis for high-order spectral discontinuous methods

The spectral analysis is a basic tool to characterise the behaviour of any convection scheme. By nature, the solution projected onto the Fourier basis enables to estimate the dissipation and the dispersion associated with the spatial discretisation of the hyperbolic linear problem. In this paper, we wish to revisit such analysis, focusing attention on two key points. The first point concerns the effects of time integration on the spectral analysis. It is shown with standard high-order Finite Difference schemes dedicated to aeroacoustics that the time integration has an effect on the required number of points per wavelength. The situation depends on the choice of the coupled schemes (one for time integration, one for space derivative and one for the filter) and here, the compact scheme with its eighth-order filter seems to have a better spectral accuracy than the considered dispersion-relation preserving scheme with its associated filter, especially in terms of dissipation. Secondly, such a coupled space–time approach is applied to the new class of high-order spectral discontinuous approaches, focusing especially on the Spectral Difference method. A new way to address the specific spectral behaviour of the scheme is introduced first for wavenumbers in [0,π][0,π], following the Matrix Power method. For wavenumbers above π, an aliasing phenomenon always occurs but it is possible to understand and to control the aliasing of the signal. It is shown that aliasing depends on the polynomial degree and on the number of time steps. A new way to define dissipation and dispersion is introduced and applied to wavenumbers larger than π. Since the new criteria recover the previous results for wavenumbers below π, the new proposed approach is an extension of all the previous ones dealing with dissipation and dispersion errors. At last, since the standard Finite Difference schemes can serve as reference solution for their capability in aeroacoustics, it is shown that the Spectral Difference method is as accurate as (or even more accurate) than the considered Finite Difference schemes

Lattice Boltzmann method for computational aeroacoustics on non-uniform meshes: a direct grid coupling approach

The present study proposes a highly accurate lattice Boltzmann direct coupling cell-vertex algorithm, well suited for industrial purposes, making it highly valuable for aeroacoustic applications. It is indeed known that the convection of vortical structures across a grid refinement interface, where cell size is abruptly doubled, is likely to generate spurious noise that may corrupt the solution over the whole computational domain. This issue becomes critical in the case of aeroacoustic simulations, where accurate pressure estimations are of paramount importance. Consequently, any interfering noise that may pollute the acoustic predictions must be reduced. The proposed grid refinement algorithm differs from conventionally used ones, in which an overlapping mesh layer is considered. Instead, it provides a direct connection allowing a tighter link between fine and coarse grids, especially with the use of a coherent equilibrium function shared by both grids. Moreover, the direct coupling makes the algorithm more local and prevents the duplication of points, which might be detrimental for massive parallelization. This work follows our first study (Astoul~\textit{et al. 2020}) on the deleterious effect of non-hydrodynamic modes crossing mesh transitions, which can be addressed using an appropriate collision model. The Hybrid Recursive Regularized model is then used for this study. The grid coupling algorithm is assessed and compared to a widely-used cell-vertex algorithm on an acoustic pulse test case, a convected vortex and a turbulent circular cylinder wake flow at high Reynolds number.Comment: also submitted to Journal of Computational Physic

In-Plane Forces Prediction and Analysis in High-Speed Conditions on a Contra-Rotating Open Roto

Due to the growing interest from engine and aircraft manufacturers for contra-rotating open rotors (CROR), much effort is presently devoted to the development of reliable computational fluid dynamics (CFD) methodologies for the prediction of performance, aerodynamic loads, and acoustics. Forces transverse to the rotation axis of the propellers, commonly called in-plane forces (or sometimes 1P forces), are a major concern for the structural sizing of the aircraft and for vibrations. In-plane forces impact strongly the stability and the balancing of the aircraft and, consequently, the horizontal tail plane (HTP) and the vertical tail plane (VTP) sizing. Also, in-plane forces can initiate a flutter phe- nomenon on the blades or on the whole engine system. Finally, these forces are unsteady and may lead to vibrations on the whole aircraft, which may degrade the comfort of the passengers and lead to structural fatigue. These forces can be predicted by numerical methods and wind tunnel measurements. However, a reliable estimation of in-plane forces requires validated prediction approaches. To reach this objective, comparisons between several numerical methods and wind tunnel data campaigns are necessary. The primary objective of the paper is to provide a physical analysis of the aerodynamics of in-plane forces for a CROR in high speed at nonzero angle of attack using unsteady simulations. Confidence in the numerical results is built through a code-to-code comparison, which is a first step in the verification process of in-plane forces prediction. Thus, two computa- tional processes for unsteady Reynolds-averaged Navier–Stokes (URANS) simulations of an isolated open rotor at nonzero angle of attack are compared: computational strategy, open rotor meshing, aerodynamic results (rotor forces, blades thrust, and pressure distributions). In a second step, the paper focuses on the understanding of the key aerodynamic mechanisms behind the physics of in-plane forces. For the front rotor, two effects are predominant: the first is due to the orientation of the freestream velocity, and the second is due to the distribution of the induced velocity. For the rear rotor, the freestream velocity effect is reduced but is still dominant. The swirl generated by the front rotor also plays a major role in the modulus and the direction of the in-plane force. Finally, aerodynamic interactions are found to have a minor effect

Description of the Flow in a Two-Stage Low-Pressure Turbine With Hub Cavities

In gas turbine, multi-stage row blading and technological effects can exhibit significant differences for the flow compared with isolated smooth blade rows. Upstream stages promote a non-uniform flow field at the inlet of the downstream rows that may have large effects on mixing or boundary layer transition processes. The rows of current turbines (and compressors) are already very closely spaced. Axial gaps between adjacent rows of approximately 1/4 to 1/2 of the axial blade chord are common practice. Future designs with higher loading and lower aspect ratios, i.e., fewer and bigger blades, and the ever present aim at minimizing engine length or compactness, will aggravate this condition even further. Interaction between cascade rows will therefore keep increasing and need to be taken into account in loss generation estimation. Also the cavities at hub platform induce purge flow blowing into main annulus and additional losses for the turbine. A robust method to account for the loss generated due to these different phenomena needs to be used. The notion of exergy (energy in the purpose to generate work) provides a general framework to deal with the different transfers of energy between the flow and the gas turbine. This study investigates the flow in a two-stage configuration representative of a low-pressure turbine including hub cavities based on large eddy simulation (LES). A description of the flow in the cavities, the main annulus, and at rim seal interface is proposed. The assessment of loss generated in the configuration is proposed based on an exergy analysis. The study of losses restricted to boundary layer contributions and secondary flows show the interaction processes of secondary vortices and wake generated in upstream rows on the flow in downstream row

Description of the flow in a linear cascade with an upstream cavity part 2: Assessing the loss generated using an exergy formulation (draft)

Purge air is injected in cavities at hub of axial turbines to prevent hot mainstream gas ingestion into in- terstage gaps. This process induces additional losses for the turbine due to an interaction between purge and mainstream flow. To deal with this issue, this paper is devoted to the study of a low speed linear cas- cade with an upstream cavity at a Reynolds number representative of a low-pressure turbine using RANS and LES with inlet turbulence injection. Different rim seal geometries and purge flow rates are studied. Details about numerical methods and comparison with experiments can be found in a companion paper. The analysis here focuses on the loss generation based on the description of the flow and influence of the turbulence introduced in the companion paper. The measure of loss is based on an exergy analy- sis (i.e. energy in the purpose to generate work) that extends a more common measure of loss in gas turbines, entropy. The loss analysis is led for a baseline case by splitting the simulation domain in the contributions related to the boundary layers over the wetted surfaces and the remaining domain (i.e. the complementary of boundary layers domains) where secondary flows and related loss are likely to occur. The analysis shows the strong contribution of the blade suction side boundary layer, secondary vortices in the passage and wake at the trailing edge on the loss generation. The study of different pur ge flow rates shows increased secondary vortices energy and subsequent loss for higher purge flow rates. The rim seal geometry with axial overlapping promotes a delayed development of secondary vortices in the passage compared to simple axial gap promoting lower levels of loss

Description of the flow in a linear cascade with an upstream cavity Part 1: Influence of turbulence (draft)

In gas turbines, transitional flows are likely to occur over many components depending on the geometri- cal arrangement, inlet turbulence and Reynolds number. In the case of a low-pressure turbine, the transi- tion from a laminar to a turbulent boundary layer is generally either a bypass process due to free stream turbulence or a separation-induced transition due to the adverse pressure gradient on the blade. The overall blade losses and the operating point are strongly dependent on the ability to predict this bound- ary layer state, the size and length of the separation bubble. Therefore, turbomachinery designers require tools which accurately predict the laminar-turbulent transition. The Reynolds Averaged Navier–Stokes (RANS) formalism is currently commonly used due a to relatively low computational cost. Except partic- ular developments, this approach is not suited to predict transition processes. The Large Eddy-Simulation (LES) approach is able to predict transition processes at a higher computational cost making it suitable for low-pressure turbine applications in conjunction with inlet turbulence injection since the free-stream turbulence is generally non-negligible and affect near-wall flow behavior. The present study introduces a description of the flow in a linear cascade with an upstream hub cavity at a Reynolds number represen- tative of low-pressure turbines by three different approaches (RANS, LES and LES with inlet turbulence injection). This study shows the influence of turbulence modelling and turbulence injection at the inlet of the domain on the boundary layer state at hub and shroud modifying the secondary vortices radial migration in the blade passage and the cancelling of suction side separation bubble at high free-stream turbulence. The Kevin–Helmholtz instability at the rim seal interface is also cancelled at high free-stream turbulence