Jack Weatheritt

A novel evolutionary algorithm applied to algebraic modifications of the RANS stress-strain relationship

This paper presents a novel and promising approach to turbulence model formulation, rather than putting forward a particular new model. Evolutionary computation has brought symbolic regression of scalar fields into the domain of algorithms and this paper describes a novel expansion of Gene Expression Programming for the purpose of tensor modeling. By utilizing high-fidelity data and uncertainty measures, mathematical models for tensors are created. The philosophy behind the framework is to give freedom to the algorithm to produce a constraint-free model; its own functional form that was not previously imposed. Turbulence modeling is the target application, specifically the improvement of separated flow prediction. Models are created by considering the anisotropy of the turbulent stress tensor and formulating non-linear constitutive stress-strain relationships. A previously unseen flow field is computed and compared to the baseline linear model and an established non-linear model of comparable complexity. The results are highly encouraging.

Use of Symbolic Regression for construction of Reynolds-stress damping functions for Hybrid RANS/LES

A novel approach to turbulence model development is applied to formulate a Hybrid RANS/LES methodology suitable on coarse meshes. The Reynolds-stress damping function in the Flow Simulation Methodology (FSM) framework is di cult to formulate rigorously from first principles, however it is critical to the success of the model. The damping function dictates locally and instantaneously the contribution level of a RANS model required to supplement structures resolved by the grid. The current formulation is a damping of the turbulent length scale in a nonlinear explicit algebraic stress RANS closure. This turns the eddy viscosity into a sub-grid scale model. In this proposal Symbolic Regression, an evolutionary process according to survival of the fittest, is used to build a new damping function. A population of randomly generated damping functions is evolved by measuring how closely they represent an idealized form. This ideal dataset is generated by filtering DNS to represent a pseudo FSM flow field. In the present study DNS of a turbulent pipe flow is used. Hybrid RANS/LES using the new damping function is performed on two di↵erent test cases. Results are presented on coarse meshes for two dimensional periodic hills and inline tandem cylinders, for which the modified FSM performs very well.

Hybrid Reynolds-Averaged/Large-Eddy Simulation Methodology from Symbolic Regression: Formulation and Application

A unified hybrid Reynolds-averaged Navier–Stokes/large-eddy simulation closure is presented that is built from data-driven methods. This is a novel way to construct such models that does not impose constraints. Direct numerical simulation data is filtered, and the ratio of resolved to unresolved energy is used to fit an ideal length scale damping function for the unified framework. This study shows the viability of using high-fidelity data, not just for a priori testing but for the complete creation of lower-fidelity methods. Alongside the physical model, a convection scheme is proposed that marries well to it. This numerical scheme ensures that the damped turbulence model is provided the appropriate ratio of stability to accuracy. Furthermore, an additional function guarantees that the large-eddy simulation mode is only active in vortical flow. This hybrid closure is then applied to two industrially relevant yet very different geometries for which reliable reference data exist. The periodic hills test ca...

Machine Learning for Turbulence Model Development Using a High-Fidelity HPT Cascade Simulation

The validity of the Boussinesq approximation in the wake behind a high-pressure turbine blade is explored. We probe the mathematical assumptions of such a relationship by employing a least-squares technique. Next, we use an evolutionary algorithm to modify the anisotropy tensor a priori using highly resolved LES data. In the latter case we build a non-linear stress-strain relationship. Results show that the standard eddy-viscosity assumption underpredicts turbulent diffusion and is theoretically invalid . By increasing the coefficient of the linear term, the farwake prediction shows minor improvement. By using additional non-linear terms in the stress-strain coupling relationship, created by the evolutionary algorithm, the near-wake can also be improved upon. Terms created by the algorithm are scrutinized and the discussion is closed by suggesting a tentative non-linear expression for the Reynolds stress, suitable for the wake behind a high-pressure turbine blade. NOMENCLATURE ai j Anisotropy tensor. k Turbulent kinetic energy. k′ Normalized turbulent kinetic energy: k/kmax ∗Address all correspondence to this author. N Population size; number of training data points (context specific) M Number of GEP solutions Pk Turbulent kinetic energy production: τi j∂x jUi Si j Strain rate: 1 2 (∂x jUi +∂xiU j) S′ i j Deviatoric component of strain rate: Si j − 3 δi jSkk. tI Turbulent time scale: 1/ω . Ui Time-averaged velocity vector s Intrinsic coordinate along the wake center line, normalized by axial chord. x, y Cartesian coordinates, normalized by axial chord. β Optimization parameter in least-squares regression. ∂φ Shorthand for ∂ ∂φ . As a differential operator, it acts on everything to the right within a term. ε Least squares model error. μt Eddy-viscosity. ρ Time-averaged density. τi j Reynolds stress: ρuiuj. ω Specific dissipation rate. Ωi j Rotation rate tensor: 2 (∂x jUi −∂xiU j)

A Comparative Study of Contrasting Machine Learning Frameworks Applied to RANS Modeling of Jets in Crossflow

Classical RANS turbulence models have known deficiencies when applied to jets in crossflow. Identifying the linear Boussinesq stress-strain hypothesis as a major contribution to erroneous prediction, we consider and contrast two machine learning frameworks for turbulence model development. Gene Expression Programming, an evolutionary algorithm that employs a survival of the fittest analogy, and a Deep Neural Network, based on neurological processing, add non-linear terms to the stress-strain relationship. The results are Explicit Algebraic Stress Model-like closures. High fidelity data from an inline jet in crossflow study is used to regress new closures. These models are then tested on a skewed jet to ascertain their predictive efficacy. For both methodologies, a vast improvement over the linear relationship is observed.

Reynolds Stress Structures in the Hybrid RANS/LES of a Planar Channel

The near-wall cycle of hybrid RANS/LES is studied by calculating the flow through a planar channel. Statistical results are commented on and related to instantaneous structures which are extracted from the flow field. The problematic structures in the artificial near-wall cycle, well known to be super-streaks, are identified and quantified. The calibration of such closures provides a correct mixing length argument in the logarithmic layer. However because of these overly intense streamwise streaks, it is impossible to simultaneously predict the Reynolds streamwise normal and shear stress components correctly. Further, because the location of the RANS-LES interface changes spatially and temporally, we see these structures are more free to move vertically and this further worsens statistical results.

Applying Machine Learnt Explicit Algebraic Stress and Scalar Flux Models to a Fundamental Trailing Edge Slot

Machine learning was applied to large-eddy simulation (LES) data to develop nonlinear turbulence stress and heat flux closures with increased prediction accuracy for trailing-edge cooling slot cases. The LES data were generated for a thick and a thin trailing-edge slot and shown to agree well with experimental data, thus providing suitable training data for model development. A gene expression programming (GEP) based algorithm was used to symbolically regress novel nonlinear explicit algebraic stress models and heat-flux closures based on either the gradient diffusion or the generalized gradient diffusion approaches. Steady Reynolds-averaged Navier–Stokes (RANS) calculations were then conducted with the new explicit algebraic stress models. The best overall agreement with LES data was found when selecting the near wall region, where high levels of anisotropy exist, as training region, and using the mean squared error of the anisotropy tensor as cost function. For the thin lip geometry, the adiabatic wall effectiveness was predicted in good agreement with the LES and experimental data when combining the GEP-trained model with the standard eddy-diffusivity model. Crucially, the same model combination also produced significant improvement in the predictive accuracy of adiabatic wall effectiveness for different blowing ratios (BRs), despite not having seen those in the training process. For the thick lip case, the match with reference values deteriorated due to the presence of large-scale, relative to slot height, vortex shedding. A GEP-trained scalar flux model, in conjunction with a trained RANS model, was found to significantly improve the prediction of the adiabatic wall effectiveness.

Application of an evolutionary algorithm to LES modelling of turbulent transport in premixed flames

Abstract Recently the concept of Gene Expression Programming (GEP) has been introduced with very encouraging results for the purpose of modelling the unclosed tensors in the context of RANS (Reynolds Averaged Navier–Stokes) turbulence modelling. This paper extends the previous framework to modelling subgrid stresses (SGS) in the context of Large Eddy Simulation (LES). In order to achieve this goal the GEP algorithm was coupled with an external multiprocessor postprocessing tool that evaluates a cost function based on a-priori analysis of explicitly filtered DNS data of turbulent premixed planar flames. Although LES of combustion systems is becoming increasingly popular, the closures for sub-grid scale (SGS) stresses have mostly been derived assuming constant density flows. However, it has been shown recently that depending on the relative strength of heat release and turbulence, counter-gradient transport can occur for the stress tensor if the isotropic part is not properly accounted for. The focus of this work is not to put forward a particular new model but to demonstrate that evolutionary algorithms can successfully be used in the framework of LES modelling. To achieve this purpose the GEP software is used for modelling the deviatoric stress, the trace of the SGS tensor and the stress tensor itself. Although the functional form of the model was not imposed, the evolutionary algorithm did find a well known model from the literature with even the model constants comparable to values reported in the literature.