Master of Science Thesis

Residual based turbulence modeling of Variational Multiscale Method (VMM) formulations eliminates the closure problem plaguing traditional wall modeling techniques in Large Eddy Simulations (LES) which lack universality due to their inherently phenomenological nature. Moreover, coupling Artificial Neural Networks (ANNs) with current state-of-the-art VMM algebraic models has resulted in robust unresolved scale models, but gradually begins to unravel in cases of high non-linearity present in the unresolved scales [8, 18, 35]. As a result, these VMM-ANNs are increasingly being used as surrogate models in predicting the so called turbulence interaction terms required to stabilize the flow [31, 68]. However, standalone neural networks have been studied to succumb to short-term and long-term instabilities in an online simulation, primarily due to the unphysical nature of data encountered during the Newton iterations of a time-step. An effective solution reported popularly in literature is to re-train a neural network by adding noise to the training data, or its weights and biases [2, 10]. Noise-augmented neural networks have shown to possess greater stability and higher generalizability than neural networks trained only on information obtained from a true solution (Direct Numerical Simulation (DNS)) [51]. The scope of this thesis is to investigate the methodology of introducing White noise, a random signal of uniform strength, and Gaussian noise, a normal distribution scaled with error in interaction terms, as two noise sources in perturbing base training data and facilitating the noise-augmented training of a Base-ANN (B-ANN), thereby creating a White-ANN (W-ANN) and Gaussian-ANN (G-ANN) respectively. Three strategies of introducing noise were studied by sampling data (1) near the walls {{x ≤ 0.2} ∪ {x ≥ 0.8}}, (2) during the 1st phase of the simulation (t ≤ 0.1T ), and (3) intermittently in the complete space-time domain with varying proportions {20%,40%,70%}; to find out that the performance of G-ANNs was found to be generally more robust across the parameter space when compared with W-ANNs. The range of parameters for which noise-augmented networks trained on FS2 (feature set 2 of 3 total) consistently exhibited robust behaviour without using the 99−percentile limiting method [68] are (1) k ∈ (O (10−1) ,O (101)), and (2) α= 2.0 (40%). The strategy of noise-augmentation varied according to the forcing. In the case of Problem A (uniform forcing), noise was added to data points near the right-wall since the magnitude of u′ behaviour is stronger in that region on account of the rightward characteristics of the solution when subjected to a constant positive bias. In the case of Problem B (unsteady forcing), noise-augmentation focused on intermittently sampled data throughout the space-time domain was more beneficial than choosing to perturb data points only near the walls. The analysis was repeated after changing the time-march scheme to second-order implicit Euler on a subset of the first-order parameters focused at the stable portion of the 2RMS (RMS of solution error) curve. For the uniform forcing case, the comparison of the steady state solution of several Wand G-ANNs with the L2 projection of the DNS revealed an accurate representation of the behaviour in the region immediately to the left of x = 0.8, without requiring limiting in the online simulations. This observation acts as proof to the concept that noise-augmentation inherently improves ANN performance, since the solution of the corresponding B-ANN had a negative slope in that region and showed a kink in the solution. Moreover, stability was achieved for the B-ANN after the application of limiting, which betrays that the network had inherently inaccurate behaviour in that particular region. In the case of Problem B, the percentage of intermittent sampling of training data was varied between three levels, low(20%)-medium(40%)-high(70%), and it was found that the proportion of data sampling did not effect the solution. The comparison of statistically averaged solution profiles with the L2 projection of the averaged DNS solution shows a good resemblance in the bulk of the flow and the ability of G-ANNs to closely replicate near-wall behaviour slightly better than W-ANNs which required limiting to achieve greater accuracy.