Development of a Data-Driven Wall Model for Separated flows

Large Eddy Simulations (LES) are of increasing interest for turbomachinery design since they provide a more reliable prediction of flow physics and component behavior. However, they remain prohibitively expensive at high Reynolds numbers or actual complex geometries. Most of the cost is associated with the resolution of the boundary layer, and therefore, to save computational resources, wall-modeled LES (wmLES) has become a valued tool. However, wall models are not yet reliable in predicting the complex flow configurations occurring in turbomachinery passages. Most existing analytical wall models assume the flow to be fully turbulent, attached, flow aligned, and near-equilibrium. These assumptions no longer hold when different flow regimes and complex flow features coexist. Although significant progress has been made in recent years (e.g., non-equilibrium models using pressure gradients), they have not always brought a clear benefit for such realistic flows. This paper proposes an innovative data-driven wall model to treat separated flows. Among the many possibilities to solve this complex regression problem, deep neural networks have been selected for their universal approximation capabilities~\cite{hornik_approximation_1991}. In the present framework, the two-dimensional periodic hill problem is selected as a reference test case featuring the separation of a fully turbulent boundary layer. Gaussian Mixture Neural networks (GMN) and Convolutional Neural Networks (CNN) combined with a self-attention layer~\cite{Vaswani_sel_attention_2017} are trained to predict the wall-parallel components of the wall shear stress using instantaneous flow quantities and geometric parameters. The \textit{a priori} and \textit{a posteriori} validation of such data-driven wall models on the periodic hill problem will be presented.9. Industry, innovation and infrastructur

Machine Learning for wall modeling in LES of separating flows

Large Eddy Simulations (LES) are of increasing interest for turbomachinery design since they provide a more reliable prediction of flow physics and component behavior than standard RANS simulations. However, they remain prohibitively expensive at high Reynolds numbers or realistic geometries. The cost of resolving the near-wall region has justified the development of wall-modeled LES (wmLES), which uses a wall model to account for the effect of the energetic near-wall eddies. The classical assumptions of algebraic wall models do not hold for more complex flow patterns that frequently occur in turbomachinery passages (i.e., misalignment, separation). This work focuses on the extension of wall models to the separation phenomenon. Among possibilities to solve the complex regression problem (i.e., predicting the wall-parallel components of the shear stress from instantaneous flow data and geometrical parameters), neural networks have been selected for their universal approximation capabilities. Since DNS and LES perform well on academic and several industrial configurations, they are used to produce databases to train various neural networks. In the present work, we investigate the possibility of using neural networks to improve wall-shear stress models for flows featuring severe pressure gradients and separation. The database is composed of three building-blocks flows: (1) a flow aligned turbulent boundary layer at equilibrium; (2) a turbulent boundary layer subjected to a moderate pressure gradient; and (3) a turbulent boundary layer that separates and reattaches from a curved wall. These building blocks are referred to as a channel flow at a friction Reynolds number of 950 and the two walls (i.e., the flat upper surface and the curved lower one) of the two-dimensional periodic hill at a bulk Reynolds number of $10{,}595$, respectively. This work is constructed around three main questions: which input points should be considered for the data-driven wall model, how should one normalize the in- and output data to obtain a unified and consistent database, and which neural networks are considered

Development of Machine Learning based wall shear stress models for LES in the presence of adverse pressure gradients and separation

The Mixture Density Network (MDN), initially developed to predict uncertainty, is used as a wall shear stress model in wall-modeled Large Eddy Simulations (wmLES) of turbulent separated flows. Separation is a common phenomenon in turbomachinery (e.g., compressor and turbine blades), due to strong adverse pressure gradients and curvature effects. However, most standard wall shear stress (WSS) models are no longer applicable in non-equilibrium conditions because of their inherent modeling assumptions about the boundary layer (i.e., fully turbulent, at equilibrium, and attached). In this study, the MDN is trained on turbulent channel flows at various friction Reynolds numbers and on the two-dimensional periodic hill at the bulk Reynolds number of 10,595. The latter test case is designed to allow separation from the hill crest, followed by a massive recirculation bubble and reattachment of the free shear layer on the flat bottom surface. The model takes the velocity field, instantaneous and mean pressure gradients, and wall curvature as inputs. The model outputs the probability distribution of the two wall-parallel components of the wall shear stress. The databases are carefully non-dimensionalized using the kinematic viscosity and wall-model height for better generalizability. The model was successfully evaluated a priori on synthetic data generated from the law-of-the-wall. The relevance of the MDN-model was evaluated a posteriori by performing wmLES using the in-house flow solver Argo-DG on two channel flows and a separated flow

Outils numériques pour la simulation des grandes échelles d'écoulements incompressibles turbulents et application aux écoulements autour de capsules de rentrée

The context of this thesis is the numerical simulation of turbulent flows at moderate Reynolds numbers and the improvement of the capabilities of an in-house 3D unsteady and incompressible flow solver called SFELES to simulate such flows.In addition to this abstract, this thesis includes five other chapters.The second chapter of this thesis presents the numerical methods implemented in the two CFD solvers used as part of this work, namely SFELES and PHASTA.The third chapter concentrates on the implementation of a new library called FlexMG. This library allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods, which require much less memory to solve linear systems than a direct sparse LU solver available in SFELES. Multigrid is an iterative procedure that relies on a series of increasingly coarser approximations of the original 'fine' problem. The underlying concept is the following: low wavenumber errors on fine grids become high wavenumber errors on coarser levels, which can be effectively removed by applying fixed-point methods on coarser levels.Two families of algebraic multigrid preconditioners have been implemented in FlexMG, namely smooth aggregation-type and non-nested finite element-type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non-nested finite element-type multigrid so that our approaches can be considered as hybrid. Our aggregation-type multigrid is smoothed with either a constant or a linear least square fitting function, whereas the non-nested finite element-type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand-alone solvers or coupled with a GMRES (Generalized Minimal RESidual) method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics (unsteady flow past a circular cylinder at low Reynolds number), their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in this work.The fourth chapter is devoted to the study of subgrid scale models for the large eddy simulation (LES) of turbulent flows.It is well known that turbulence features a cascade process by which kinetic energy is transferred from the large turbulent scales to the smaller ones. Below a certain size, the smallest structures are dissipated into heat because of the effect of the viscous term in the Navier-Stokes equations.In the classical formulation of LES models, all the resolved scales are used to model the contribution of the unresolved scales. However, most of the energy exchanges between scales are local, which means that the energy of the unresolved scales derives mainly from the energy of the small resolved scales.In this fourth chapter, constant-coefficient-based Smagorinsky and WALE models are considered under different formulations. This includes a classical version of both the Smagorinsky and WALE models and several scale-separation formulations, where the resolved velocity field is filtered in order to separate the small turbulent scales from the large ones. From this separation of turbulent scales, the strain rate tensor and/or the eddy viscosity of the subgrid scale model is computed from the small resolved scales only. One important advantage of these scale-separation models is that the dissipation they introduce through their subgrid scale stress tensor is better controlled compared to their classical version, where all the scales are taken into account without any filtering. More precisely, the filtering operator (based on a top hat filter in this work) allows the decomposition u' = u - ubar, where u is the resolved velocity field (large and small resolved scales), ubar is the filtered velocity field (large resolved scales) and u' is the small resolved scales field. At last, two variational multiscale (VMS) methods are also considered.The philosophy of the variational multiscale methods differs significantly from the philosophy of the scale-separation models. Concretely, the discrete Navier-Stokes equations have to be projected into two disjoint spaces so that a set of equations characterizes the evolution of the large resolved scales of the flow, whereas another set governs the small resolved scales. Once the Navier-Stokes equations have been projected into these two spaces associated with the large and small scales respectively, the variational multiscale method consists in adding an eddy viscosity model to the small scales equations only, leaving the large scales equations unchanged. This projection is obvious in the case of a full spectral discretization of the Navier-Stokes equations, where the evolution of the large and small scales is governed by the equations associated with the low and high wavenumber modes respectively. This projection is more complex to achieve in the context of a finite element discretization. For that purpose, two variational multiscale concepts are examined in this work.The first projector is based on the construction of aggregates, whereas the second projector relies on the implementation of hierarchical linear basis functions.In order to gain some experience in the field of LES modeling, some of the above-mentioned models were implemented first in another code called PHASTA and presented along with SFELES in the second chapter.Finally, the relevance of our models is assessed with the large eddy simulation of a fully developed turbulent channel flow at a low Reynolds number under statistical equilibrium. In addition to the analysis of the mean eddy viscosity computed for all our LES models, comparisons in terms of shear stress, root mean square velocity fluctuation and mean velocity are performed with a fully resolved direct numerical simulation as a reference.The fifth chapter of the thesis focuses on the numerical simulation of the 3D turbulent flow over a re-entry Apollo-type capsule at low speed with SFELES. The Reynolds number based on the heat shield is set to Re=10^4 and the angle of attack is set to 180º, that is the heat shield facing the free stream. Only the final stage of the flight is considered in this work, before the splashdown or the landing, so that the incompressibility hypothesis in SFELES is still valid.Two LES models are considered in this chapter, namely a classical and a scale-separation version of the WALE model. Although the capsule geometry is axisymmetric, the flow field in its wake is not and induces unsteady forces and moments acting on the capsule. The characterization of the phenomena occurring in the wake of the capsule and the determination of their main frequencies are essential to ensure the static and dynamic stability during the final stage of the flight. Visualizations by means of 3D isosurfaces and 2D slices of the Q-criterion and the vorticity field confirm the presence of a large meandering recirculation zone characterized by a low Strouhal number, that is St≈0.15.Due to the detachment of the flow at the shoulder of the capsule, a resulting annular shear layer appears. This shear layer is then affected by some Kelvin-Helmholtz instabilities and ends up rolling up, leading to the formation of vortex rings characterized by a high frequency. This vortex shedding depends on the Reynolds number so that a Strouhal number St≈3 is detected at Re=10^4.Finally, the analysis of the force and moment coefficients reveals the existence of a lateral force perpendicular to the streamwise direction in the case of the scale-separation WALE model, which suggests that the wake of the capsule may have some preferential orientations during the vortex shedding. In the case of the classical version of the WALE model, no lateral force has been observed so far so that the mean flow is thought to be still axisymmetric after 100 units of non-dimensional physical time.Finally, the last chapter of this work recalls the main conclusions drawn from the previous chapters.Doctorat en Sciences de l'ingénieurinfo:eu-repo/semantics/nonPublishe

Outils numériques pour la simulation des grandes échelles d'écoulements incompressibles turbulents et application aux écoulements autour de capsules de rentrée

The context of this thesis is the numerical simulation of turbulent flows at moderate Reynolds numbers and the improvement of the capabilities of an in-house 3D unsteady and incompressible flow solver called SFELES to simulate such flows.In addition to this abstract, this thesis includes five other chapters.The second chapter of this thesis presents the numerical methods implemented in the two CFD solvers used as part of this work, namely SFELES and PHASTA.The third chapter concentrates on the implementation of a new library called FlexMG. This library allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods, which require much less memory to solve linear systems than a direct sparse LU solver available in SFELES. Multigrid is an iterative procedure that relies on a series of increasingly coarser approximations of the original 'fine' problem. The underlying concept is the following: low wavenumber errors on fine grids become high wavenumber errors on coarser levels, which can be effectively removed by applying fixed-point methods on coarser levels.Two families of algebraic multigrid preconditioners have been implemented in FlexMG, namely smooth aggregation-type and non-nested finite element-type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non-nested finite element-type multigrid so that our approaches can be considered as hybrid. Our aggregation-type multigrid is smoothed with either a constant or a linear least square fitting function, whereas the non-nested finite element-type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand-alone solvers or coupled with a GMRES (Generalized Minimal RESidual) method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics (unsteady flow past a circular cylinder at low Reynolds number), their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in this work.The fourth chapter is devoted to the study of subgrid scale models for the large eddy simulation (LES) of turbulent flows.It is well known that turbulence features a cascade process by which kinetic energy is transferred from the large turbulent scales to the smaller ones. Below a certain size, the smallest structures are dissipated into heat because of the effect of the viscous term in the Navier-Stokes equations.In the classical formulation of LES models, all the resolved scales are used to model the contribution of the unresolved scales. However, most of the energy exchanges between scales are local, which means that the energy of the unresolved scales derives mainly from the energy of the small resolved scales.In this fourth chapter, constant-coefficient-based Smagorinsky and WALE models are considered under different formulations. This includes a classical version of both the Smagorinsky and WALE models and several scale-separation formulations, where the resolved velocity field is filtered in order to separate the small turbulent scales from the large ones. From this separation of turbulent scales, the strain rate tensor and/or the eddy viscosity of the subgrid scale model is computed from the small resolved scales only. One important advantage of these scale-separation models is that the dissipation they introduce through their subgrid scale stress tensor is better controlled compared to their classical version, where all the scales are taken into account without any filtering. More precisely, the filtering operator (based on a top hat filter in this work) allows the decomposition u' = u - ubar, where u is the resolved velocity field (large and small resolved scales), ubar is the filtered velocity field (large resolved scales) and u' is the small resolved scales field. At last, two variational multiscale (VMS) methods are also considered.The philosophy of the variational multiscale methods differs significantly from the philosophy of the scale-separation models. Concretely, the discrete Navier-Stokes equations have to be projected into two disjoint spaces so that a set of equations characterizes the evolution of the large resolved scales of the flow, whereas another set governs the small resolved scales. Once the Navier-Stokes equations have been projected into these two spaces associated with the large and small scales respectively, the variational multiscale method consists in adding an eddy viscosity model to the small scales equations only, leaving the large scales equations unchanged. This projection is obvious in the case of a full spectral discretization of the Navier-Stokes equations, where the evolution of the large and small scales is governed by the equations associated with the low and high wavenumber modes respectively. This projection is more complex to achieve in the context of a finite element discretization. For that purpose, two variational multiscale concepts are examined in this work.The first projector is based on the construction of aggregates, whereas the second projector relies on the implementation of hierarchical linear basis functions.In order to gain some experience in the field of LES modeling, some of the above-mentioned models were implemented first in another code called PHASTA and presented along with SFELES in the second chapter.Finally, the relevance of our models is assessed with the large eddy simulation of a fully developed turbulent channel flow at a low Reynolds number under statistical equilibrium. In addition to the analysis of the mean eddy viscosity computed for all our LES models, comparisons in terms of shear stress, root mean square velocity fluctuation and mean velocity are performed with a fully resolved direct numerical simulation as a reference.The fifth chapter of the thesis focuses on the numerical simulation of the 3D turbulent flow over a re-entry Apollo-type capsule at low speed with SFELES. The Reynolds number based on the heat shield is set to Re=10^4 and the angle of attack is set to 180º, that is the heat shield facing the free stream. Only the final stage of the flight is considered in this work, before the splashdown or the landing, so that the incompressibility hypothesis in SFELES is still valid.Two LES models are considered in this chapter, namely a classical and a scale-separation version of the WALE model. Although the capsule geometry is axisymmetric, the flow field in its wake is not and induces unsteady forces and moments acting on the capsule. The characterization of the phenomena occurring in the wake of the capsule and the determination of their main frequencies are essential to ensure the static and dynamic stability during the final stage of the flight. Visualizations by means of 3D isosurfaces and 2D slices of the Q-criterion and the vorticity field confirm the presence of a large meandering recirculation zone characterized by a low Strouhal number, that is St≈0.15.Due to the detachment of the flow at the shoulder of the capsule, a resulting annular shear layer appears. This shear layer is then affected by some Kelvin-Helmholtz instabilities and ends up rolling up, leading to the formation of vortex rings characterized by a high frequency. This vortex shedding depends on the Reynolds number so that a Strouhal number St≈3 is detected at Re=10^4.Finally, the analysis of the force and moment coefficients reveals the existence of a lateral force perpendicular to the streamwise direction in the case of the scale-separation WALE model, which suggests that the wake of the capsule may have some preferential orientations during the vortex shedding. In the case of the classical version of the WALE model, no lateral force has been observed so far so that the mean flow is thought to be still axisymmetric after 100 units of non-dimensional physical time.Finally, the last chapter of this work recalls the main conclusions drawn from the previous chapters.Doctorat en Sciences de l'ingénieurinfo:eu-repo/semantics/nonPublishe

Performance of a hybrid spectral/finite element solver for incompressible flows with parallel direct solver

The development of parallel direct solvers enabled many CFD codes to apply on finer and finer meshes. Since a decade, our department has expanded the capacities of an in-house hybrid spectral/finite element solver for 3D unsteady incompressible flow problems with a direction of periodicity. The use of a spectral development along the direction of periodicity leads to a set of 2D decoupled systems of equations within each time step to determine the modal components of the nodal unknowns. The parallelization within the code is made both in physical and spectral domains, trying to keep as few communications between the processes as possible. We chose to couple the parallel direct solver MUMPS to our CFD code in order to solve each mode by several cores. Doing so, we can now perform computations on meshes containing tens of millions of nodes, which is now a standard in the CFD community. In the present work, we first present the nature of the parallelization made in our code. Then, on the basis of a standard validation case for magnetohydrodynamic flows, we compare the time and memory requirements between our former direct solver and the new one.info:eu-repo/semantics/publishe

On the performance of a parallel direct solver used within a hybrid spectral/finite element solver for incompressible flows

The development of parallel direct solvers enabled many CFD codes to apply on finer and finer meshes. Since a decade, our department has expanded the capacities of an in-house hybrid spectral / finite element solver for 3D unsteady incompressible flow problems with a direction of periodicity. The use of a spectral development along the direction of periodicity leads to a set of 2D decoupled systems of equations within each time step to determine the modal components of the nodal unknowns. The parallelization within the code is made both in physical and spectral domains, trying to keep as few communications between the processors as possible. We chose to couple the parallel direct solver MUMPS to our CFD code in order to solve each mode by several processors. Doing so, we can now perform computations on meshes containing tens of millions of nodes, which is now a standard in the CFD community. We first present the nature of the parallelization made in our code. Then, on the basis of a magnetohydrodynamic flow encountered in axisymmetric geometries, we compare the time and memory requirements between our former direct solver and the new one.info:eu-repo/semantics/publishe

Challenges for the (scale-resolved) simulation of transonic turbines

The talk resumes three current challenges for the use of DNS and LES on transonic turbomachinery passages for the training of turbulence models : non-dissipative shock capturing, generating model-unaware reference data sets and quantifying statistical averaging error