Session Q22: Turbulence: Machine Learning Methods for Turbulence Modeling II

A deep learning based closure model for the multiscale evolution of Burgers turbulence

We employed a sequence-to-sequence encoding-decoding mechanism with a long short-term memory (LSTM) integration to evolve the Burgers equation using a coarse projective integration multiscale scheme in which the energy spectrum was the coarse variable and the velocity field was the fine variable. This mechanism, commonly referred to as autoencoding, is mostly used in the domain of natural language processing (NLP). Using this autoencoder scheme, we trained our neural network model with a many-to-many type of mapping between the fine and coarse scale of the multiscale problem. This mapping provided a closure model for the lifting operator in the CPI scheme, allowing us to translate the coarse-scale information back to the fine scale accurately and enabled the evolution of the Burgers equation according to the multiscale scheme.   

Data-enabled, progressive recalibration of the Spalart-Allmaras model for general purposes

We develop data-enabled augmentation for the Spalart-Allmaras (SA) model. Our approach is progressive. We successively consider free shear flow, log layer, and wall layer. The training data is limited to channel flow and Couette-Poiseuille flows. The process makes use of Bayesian optimization and feed-forward neural networks. We test the resulting model in 10 flows, all outside the training dataset. We show that the data-enabled augmentation does not "break" the baseline model: Galilean invariance and the law of the wall are preserved. Furthermore, the resulting augmentation is "universal": it does not lead to degradation in free shear flows and gives more accurate results in wall-bounded flows. Specifically, the present augmentation gives more accurate predictions of the mean flow in the buffer layer and the skin friction coefficients in the wall-mounted hump and the back-facing step cases. Last, an attempt is made to explain the observed improvements in separated flows physically.

    

Reconstruction of Missing Flow Vectors Using Deep Image Inpainting Based on Partial Convolution Layer

Experimentally obtained velocity data from techniques such as Particle Image Velocimetry (PIV) suffer from various imperfections like reflections, background noise, and seed accumulation. Consequently, PIV data have significant numbers of randomly placed “holes” of various sizes. The current work utilizes deep image inpainting based on a partial convolution (PConv) layer (Liu et al. (2018)) allowing for mask convolution over irregularly shaped and randomly placed missing regions, closely resembling realistic PIV imperfections. To this end, PIV measurements for Mach 1.4, L/D = 6 open cavity flow are analyzed for reconstructing missing vectors. Preliminary testing with a model trained on natural images has shown promising results with overall reconstruction errors within 10% of U_∞, also showing robustness towards missing data percentage. Next, pre-training of the model is performed with existing open-source flow data to fine-tune model weights after which the final training on experimental cavity flow data is performed. The overall goal of this work is to develop an effective and robust data reconstruction tool that can be extended to other experimental and numerical flows.   

A data-driven approach using CNN for wall modeling in Large Eddy Simulation

Wall modeling in large eddy simulation (LES) is of great importance as it can make the computational cost of the LES of wall-bounded flows remarkably lower as there will be no need to resolve the near-wall region. One of the methods with computationally low cost for wall modeling in LES is the approximate boundary condition which applies the wall shear stress as the boundary condition at the wall. However, in this approach it is crucial to find an accurate model for the wall shear stress. Due to the ability of the data-driven approaches in extracting features, they can be considered as a proper candidate for deriving a model for the wall shear stress. Data-driven approaches have already been widely used for subgrid-scale (SGS) modeling in LES; however, there are not many attempts at wall-modeling in LES using a data-driven approach. In this work, a study has been performed on wall-modeling in LES using convolutional neural network (CNN). Initially, a wall model is developed using the data of channel flow at Re_τ= 400. Then, the model is tested for the channel flow at higher Reynolds number. The results show that the model has a reasonable accuracy in predicting the wall shear stress and establishing a wall model.

    

Data-driven closure modeling for scale resolving PANS simulations in flows with coherent structures

For complex turbulent flows with largescale instabilities and coherent structures, both traditional and data-driven Reynolds-averaged Navier-Stokes (RANS) methods are inherently unsuitable. Scale resolving simulations (SRS) such as the partially averaged Navier-Stokes (PANS) method are more appropriate as they resolve the unsteady and coherent scales of motion. The objective of this work is to develop a sub-filter stress neural network for SRS methods using high-fidelity data. The three main features of the new model development are: (i)  improved incorporation of the unsteady flow features of the high-fidelity data into the closure model; (ii) the features and input tensors used in the training are taken from model computations resulting in greater overall consistency; and (iii) explicit dependence of the closure on the filter size or degree of resolution. The closure development is performed in the context of PANS approach, but the technique can be extended to other SRS methods. The potential improvement of the proposed method over over existing approaches is clearly established.

    

Regression-based projection for learning Mori-Zwanzig operators for isotropic turbulence

The Mori-Zwanzig (MZ) formalism provides a mathematically rigorous procedure for constructing reduced-order representations of high-dimensional dynamical systems, where the effect due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. Our previous work on data-driven learning of MZ operators based on Mori's projection operator demonstrated successful extraction of these operators for homogeneous isotropic turbulence. However, the linearity of Mori's projection operator significantly limits the applicability of the algorithm to turbulence modeling. To bridge the gap between Mori's linear and Zwanzig's projection operators, our group developed a more general MZ learning algorithm that adopts statistical regression as a projection operator. We experiment with a range of regression-based models, including those connected with existing turbulence modeling frameworks, for learning non-Markovian models based on MZ formalism from DNS datasets. We show that the extracted operators exhibit improved performance compared to previous linear projection-based MZ operators, depending on the complexity of the regression models. The regression-based MZ learning algorithm provides a promising framework for developing turbulence models with memory effects.   

Frame invariance and scalability of vector cloud neural network for partial differential equations

Solving partial differential equations (PDEs) often requires prohibitively high computational costs, especially when multiple evaluations are to be made for different parameters or conditions. After training, neural operators can provide PDEs solutions significantly faster than traditional PDE solvers. In this work, a recently proposed vector cloud neural network (VCNN) has been assessed to emulate the invariance properties and non-local dependencies of transport PDEs. First, the invariance properties and computational complexity of VCNN have been examined for transport PDE of a scalar quantity. For comparison purposes, an alternate neural operator based on graph kernel network (GKN) is considered, for which a modified formulation of GKN has been presented to ensure frame invariance. GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN. However, GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN. Furthermore, VCNN is presented as a robust tool to emulate transport equations for tensorial quantities. We demonstrate its performance on Reynolds stress transport equations, showing that the VCNN can effectively emulate the Reynolds stress transport model for Reynolds-averaged Navier–Stokes (RANS) equations. The VCNN respects all the invariance properties desired by constitutive model and faithfully reflects the region of influence in physics.   

Self-Similar Stochastic Excitations For Linear Models In Turbulent Channel Flow

A physics aware data-driven method is formulated to determine an optimal forcing structure for an eddy viscosity enhanced, linearised Navier-Stokes model of turbulent channel flow. By restricting the forcing to be white-in-time and spatially decorrelated, an optimisation problem is solved to determine the forcing spectra which drives the linear model's velocity spectra to match that of a high Reynolds direct numerical simulation. Through exploiting the self-similarity within the attached eddy hypothesis, this optimisation problem is further reduced to a single spanwise length scale, allowing a rapid approximation of the entire forcing spectra. By exploiting the linear nature of the model, the isolated effects of each of the forcing components and how they can be extrapolated to a global forcing structure is analysed. Continuing with the utilisation of linearity, the role of the amplification mechanisms in the linear model (Orr mechanism and lift-up effect) in mimicking the DNS velocity spectra is evaluated. Interestingly, the linear model can reproduce all qualitative features of the DNS spectra, with the scale-dependent contribution of either mechanism resembling the underlying physics of the attached eddies, as well as phenomenologically mimicking energy cascade features.   

Model-free forecasting of large partially observable spatiotemporally chaotic systems

We implement a reservoir-computing-based recurrent neural network (RC-RNN) in which we first expand the system observables into a high dimensional space using radial basis functions (RBFs), then connect the expanded input to a reservoir from which the output (system forecast) is obtained via a linear readout. The RBFs enable the network to capture the system nonlinearities robustly without any knowledge of the governing equations. With this approach, we achieve the first successful application of an RC-RNN to forecast a large partially observable spatiotemporally chaotic system, namely the Gray-Scott reaction-diffusion system for which only one species concentration is observable. Our method does not require any model reduction and can deal with noisy and sparse measurements. The required reservoir dimension is only two orders of magnitude larger than the system dimension itself, indicating our method's ability to handle large turbulent systems, such as for weather forecasting. We also find that the use of RBFs as nonlinear projectors is interpretable in terms of their nonlinear approximation properties, thus suggesting their generalization to other RNNs.