Separated flow prediction and assessment using LES and machine learning

Large Eddy Simulation is a predictive technology that has the potential to revolutionise CFD. Significant effort is now being put into improving lower order models based on high fidelity data. The current work contrasts LES and RANS for a low Reynolds number ribbed channel flow relevant to turbine and electronics cooling. The anisotropy of turbulence is chosen as a starting point to compare RANS modelling deficiencies, and it is found that there are significant differences between the anisotropy predicted by RANS and LES. In the LES, a spreading shear layer introduces anisotropic content into the passage. Downstream of the rib, scouring eddies shed from the rib destroy the classical boundary layer flow. A machine learning classifier trained on a database of similar flows is used to predict the anisotropy in the ribbed passage. The classifier is shown to be capable of predicting many of the flow features identified in the LES, demonstrating the potential of such approaches for application to this category of flows

Effect of Mesh Quality on Flux Reconstruction in Multi-dimensions

Theoretical methods are developed to understand the effect of non-uniform grids on Flux Reconstruction (FR) in multi-dimensions. The analysis reveals that the same effect of expanding and contracting grids is seen in two dimensions as in one dimension. Namely, that expansions cause instability and contractions cause excess dissipation. Subsequent numerical experiments on the Taylor-Green Vortex with jittered elements show the effect of localised regions of expansion and contraction, with an initial increase in the kinetic energy observed on non-uniform meshes. Some comparison is made between second-order FR and second-order finite volume (FV). FR is found to be more resilient to mesh deformation, however, FV is found to be more resolved when operated at second order on the same mesh. In both cases, it is recommended that a kinetic energy preserving/conservation formulation should be used as this can greatly increase resilience to mesh deformation

Numerical investigation of three-dimensional separation in an axial flow compressor

Regions of three-dimensional separations are an inherent flow feature of the suction surface-endwall corner in axial compressors. These corner separations can cause a significant total pressure loss and reduce the compressor's efficiency. This paper uses wall-resolved LES to investigate the loss sources in a corner separation, and examines the influence of the inflow turbulence on these sources. Different subgrid scale (SGS) models are tested and the choice of model is found to be important. The σ SGS model, which performed well, is then used to perform LES of a compressor endwall flow. The time-averaged data are in good agreement with measurements. The viscous and turbulent dissipation are used to highlight the sources of loss, with the latter being dominant. The key loss sources are seen to be the 2D laminar separation bubble and trailing edge wake, and the 3D flow region near the endwall. Increasing the freestream turbulence (FST) intensity changes the suction surface boundary layer transition mode from separation induced to bypass. However, it does not significantly alter the transition location and therefore the corner separation size. Additionally, the FST does not noticeably interact with the corner separation itself, meaning that in this case the corner separation is relatively insensitive to the FST. The endwall boundary layer state is found to be significant. A laminar endwall boundary layer separates much earlier leading to a larger passage vortex. This significantly alters the endwall flow and loss. Hence, the need for accurate boundary measurements is clear

Uncertainty quantification for data-driven turbulence modelling with Mondrian forests

Data-driven turbulence modelling approaches are gaining increasing interest from the CFD community. Such approaches generally aim to improve the modelled Reynolds stresses by leveraging data from high fidelity turbulence resolving simulations. However, the introduction of a machine learning (ML) model introduces a new source of uncertainty, the ML model itself. Quantification of this uncertainty is essential since the predictive capability of a data-driven model diminishes when predicting physics not seen during training. In this work, we explore the suitability of Mondrian forests (MF's) for data-driven turbulence modelling. MF's are claimed to possess many of the advantages of the commonly used random forest (RF) machine learning algorithm, whilst offering principled uncertainty estimates. An example test case is constructed, with a turbulence anisotropy constant derived from high fidelity turbulence resolving simulations. A number of flows at several Reynolds numbers are used for training and testing. MF predictions are found to be superior to those obtained from a linear and non-linear eddy viscosity model. Shapley values, borrowed from game theory, are used to interpret the MF predictions. Predictive uncertainty is found to be large in regions where the training data is not representative. Additionally, the MF predictive uncertainty is found to exhibit stronger correlation with predictive errors compared to an a priori statistical distance measure, which indicates it is a better measure of prediction confidence. The MF predictive uncertainty is also found to be better calibrated and less computationally costly than the uncertainty estimated from applying jackknifing to random forest predictions. Finally, Mondrian forests are used to predict the Reynolds discrepancies in a convergent-divergent channel, which are subsequently propagated through a modified CFD solver. The resulting flowfield predictions are in close agreement with the high fidelity data. A procedure for sampling the Mondrian forests' uncertainties is introduced. Propagating these samples enables quantification of the uncertainty in quantities of interest such as velocity or a drag coefficient, due to the uncertainty in the Mondrian forests' predictions. This work suggests that uncertainty quantification can be incorporated into existing data-driven turbulence modelling frameworks by replacing random forests with Mondrian forests. This would also open up the possibility of online learning, whereby new training data could be added without having to retrain the Mondrian forests

Uncertainty quantification for data-driven turbulence modelling with Mondrian forests

Data-driven turbulence modelling approaches are gaining increasing interest from the CFD community. However, the introduction of a machine learning (ML) model introduces a new source of uncertainty, the ML model itself. Quantification of this uncertainty is essential since the predictive capability of a data-driven model diminishes when predicting physics not seen during training. In this work, we explore the suitability of Mondrian forests (MF's) for data-driven turbulence modelling. MF's are claimed to possess many of the advantages of the commonly used random forest (RF) machine learning algorithm, whilst offering principled uncertainty estimates. An example test case is constructed, with a turbulence anisotropy constant derived from high fidelity turbulence resolving simulations. Shapley values, borrowed from game theory, are used to interpret the MF predictions. Predictive uncertainty is found to be large in regions where the training data is not representative. Additionally, the MF predictive uncertainty is found to exhibit stronger correlation with predictive errors compared to an a priori statistical distance measure, which indicates it is a better measure of prediction confidence. The MF predictive uncertainty is also found to be better calibrated and less computationally costly than the uncertainty estimated from applying jackknifing to random forest predictions. Finally, Mondrian forests are used to predict the Reynolds discrepancies in a convergent-divergent channel, which are subsequently propagated through a modified CFD solver. The resulting flowfield predictions are in close agreement with the high fidelity data. A procedure for sampling the Mondrian forests' uncertainties is introduced. Propagating these samples enables quantification of the uncertainty in output quantities of interest

All Equalities Are Equal, but Some Are More Equal Than Others: The Effect of Implementation Aliasing on the Numerical Solution to Conservation Equations

We investigate the effect of aliasing when applied to the storage of variables, and their reconstruction for the solution of conservation equations. In particular, we investigate the effect on the error of storing primitives versus conserved variables for the Navier-Stokes equations. It was found that storing the conserved variables introduces less dissipation and that the dissipation caused by constructing the conversed variable from the primitives grows factorially with the order. Hence, this problem becomes increasingly important with the continuing move towards higher orders. Furthermore, the method of gradient calculation is investigated, as applied to the viscous fluxes in the Navier-Stokes equations. It was found that in most cases the difference was small, and that the product rule applied to the gradients of the conserved variables should be used due to a lower operation count. Finally, working precision is investigated and found to have a minimal impact on free-stream-turbulence-like flows when the compressible equations are solved, except at low Mach numbers

Effect of Mesh Quality on Flux Reconstruction in Multi-dimensions

Theoretical methods are developed to understand the effect of non-uniform grids on Flux Reconstruction (FR) in multi-dimensions. A better theoretical understanding of the effect of wave angle and grid deformation is established. FR is shown to have a smaller variation in properties than some finite difference counterparts. Subsequent numerical experiments on the Taylor–Green Vortex with jittered elements show the effect of localised regions of expansion and contraction. The effect this had on Nodal DG-like schemes was to increase the dissipation, whereas for more typical FR schemes the effect was to increase the dispersion. Some comparison is made between second-order FR and a second-order finite volume (FV) scheme. FR is found to be more resilient to mesh deformation, however, FV is found to be more resolved when operated at second order on the same mesh