Turbulent flows can be found in many industrial applications, and have a profound impact on the drag resistance and heat transfer characteristics of engineering equipment. While the behavior of turbulence is well-documented for canonical flow cases, its behavior under complex conditions is largely unknown. This poses a significant challenge when designing engineering equipment, since it is essential to accurately predict the impact of variable property flows and surface roughness on pressure losses and heat transfer rates. Yet such practical cases are beyond the scope of canonical flow data. During recent years, the combination of advances in machine learning and GPU-accelerated flow solvers has yielded promising results in the fields of turbulence and scalar transport. Using a few modern GPUs and state-of-the-art algorithms, it is now possible to simulate highly complex turbulent flows in short periods of time. This has enabled further progress in the field of machine learning for fluid mechanics, since high-fidelity turbulent flow databases can be easily generated, and new models can be trained to predict increasingly complex flow effects accurately. In this project, multiple challenges are addressed, such as optimizing GPU-accelerated DNS solvers for extreme-scale simulations, creating data-augmented RANS turbulence models for flows with strong variations in their thermophysical properties, and developing machine learning models for turbulent flows past rough surfaces.

Prior to working in GPU-accelerated simulations, the sub-project about machine learning for variable property flows focused on building data-augmented RANS turbulence models using a technique known as FIML (Field Inversion Machine Learning). In this framework, non-linear optimization is first performed to obtain an ideal set of corrections, after which a neural network (or another model) is trained to predict the observed corrections. When working with variable-property flows, several challenges arise, such as not knowing beforehand the local fluid properties due to thermal changes. This makes it difficult, for instance, to calculate the input features for the neural network. The solution created is a feedback loop, where CFD predictions are used to re-calculate the local fluid properties and to update the neural network input features, etc. The results show that the data-augmented RANS model can accurately improve the predictions for unique flow cases, which need unusually high corrections not observed in the training set. Additionally, a weighted relaxation factor methodology is proposed, to ensure convergence of the RANS models after inserting neural network corrections. The final results show that, for the most challenging CFD case identified, our machine learning system is able to reduce the L-infinity error on the velocity profile from 23.4% to 4.0%.

To generate the required amounts of data for machine learning studies regarding complex geometries like roughness, it was necessary to develop a GPU-accelerated DNS solver. This work focuses on the implementation of a parallel tridiagonal solver for extreme-scale simulations, and the creation of a new cross-platform communication library for supercomputers with either AMD or NVIDIA GPUs. In general terms, turbulent flow simulations in GPUs can be highly efficient, since all operations can be mapped to different GPU threads. However, large-scale data transfers are the main performance bottleneck of GPU-based simulations. While halo exchanges between GPU sub-domains have a minimal impact, the large-scale transpose operations needed for 3D arrays inside Poisson/Helmhotz solvers occupy most of the total running time. Therefore, any transpose operation avoided will drastically improve the running times for the entire DNS solver. By using a parallel tridiagonal solver, it is possible to reduce the number of transposes (for full 3D arrays) by 50% for 2D pencil decompositions inside the Poisson/Helmhotz solvers, and to replace these avoided transposes by simplified operations with a computational cost resembling halo exchanges. Additionally, in this work, a new opportunity to speed up simulations was found, by re-deriving the coefficients of parallel tridiagonal solvers to substantially improve the efficiency and the GPU parallelization of the DNS solver for implicit 1-D diffusion equations. Based on these improvements, it is observed in the results that the efficiency of the DNS solver is substantially improved for extreme-scale simulations in the LUMI and Leonardo supercomputers. Moreover, we show that the entire DNS solver can operate using 2D pencil decompositions without performance degradation compared to most optimal 1D decompositions available for smaller systems. Therefore, the parallel tridiagonal solver enables high-performance in extreme-scale simulations, where 1D decompositions are not feasible.

To enable extreme-scale simulations in AMD GPUs, a new cross-platform communication library was created, named diezDecomp. This library is able to achieve high performance working with both CPUs and GPUs in NVIDIA or AMD-based supercomputer. The underlying algorithm corresponds to an advanced implementation that works by directly intersecting the x/y/z bounds of all MPI tasks and scheduling data transfer operations. This allows the implementation of any-to-any transpose operations between mismatched 2D pencil decompositions, with complex communication patterns beyond the scope of traditional all-to-all operations. In extreme-scale simulations, direct x-to-z transposes can improve efficiency while solving for implicit 1D diffusion, but they are not available in existing libraries. Thanks to the flexibility of the diezDecomp library, x-to-z transposes can be easily implemented, and the running times of implicit 1D diffusion solvers were improved up to 55% for extreme-scale simulations in the LUMI supercomputer with 1024 GCDs.

The benefits of machine learning to predict the thermal and hydrodynamic behavior of turbulent flows past rough surfaces are explored in Chapter 4. Due to the complexity of this task, a convolutional neural network was used to (independently) scan the input height maps of rough surfaces, and to generate detailed 2-D maps with the local skin friction factors and Nusselt numbers. The proposed neural network is optimized to have linear time complexity while creating 2D maps, instead of quadratic complexity as in naive approaches. The validation study using randomized surfaces with the Fourier spectrum of grit-blasted surfaces shows that machine learning can make accurate 2D predictions for both the local skin friction factors and Nusselt numbers of rough surfaces, with median deviations of 28.43% for the skin friction factors and 6.37% for the Nusselt numbers respectively. The averaged errors in the predictions for skin friction factors and Nusselt numbers were reduced from 24.9% and 13.5% using traditional correlations to only 8.1% and 2.9% thanks to machine learning.

Since the neural network predictions for the thermal behavior of rough surfaces were particularly promising, a further optimization study is presented in Chapter 5. Here, the objective is to combine the benefits of machine learning and GPU-accelerated simulations to improve convective heat transfer in dimpled-surfaces. Remarkably, it was found that machine learning can find a highly optimized dimpled-surface with a 53% higher Nusselt number using cross-aligned dimples after being trained with only random surfaces (displaying lower thermal performance). After the second iteration of the reinforcement learning loop, it was confirmed that the surface found by machine learning created a flow pattern with helical structures inside the dimples. All other tested parameters had lower thermal performance.

In summary, it is concluded that GPU-based DNS solvers can be optimized to enable extreme-scale simulations with minimal performance degradation. Creating highly flexible communication frameworks, such as the diezDecomp library, is also possible while keeping identical running times as traditional libraries. This research project also showcases how machine learning can be an effective tool in fluid mechanics, to predict the behavior of turbulent flows past complex geometries or to account for changes in flows with strong variations in their thermophysical properties.

We present a computational method for extreme-scale simulations of incompressible turbulent wall flows at high Reynolds numbers. The numerical algorithm extends a popular method for solving second-order finite differences Poisson/Helmholtz equations using a pencil-distributed parallel tridiagonal solver to improve computational performance at scale. The benefits of this approach were investigated for high-Reynolds-number turbulent channel flow simulations, with up to about 80 billion grid points and 1024 GPUs on the European flagship supercomputers Leonardo and LUMI. An additional GPU porting effort of the entire solver had to be undertaken for the latter. Our results confirm that, while 1D domain decompositions are favorable for smaller systems, they become inefficient or even impossible at large scales. This restriction is relaxed by adopting a pencil-distributed approach. The results show that, at scale, the revised Poisson solver is about twice as fast as the baseline approach with the full-transpose algorithm for 2D domain decompositions. Strong and weak scalability tests show that the performance gains are due to the lower communication footprint. Additionally, to secure high performance when solving for wall-normal implicit diffusion, we propose a reworked flavor of parallel cyclic reduction (PCR) that is split into pre-processing and runtime steps. During pre-processing, small sub-arrays with independent 1D coefficients are computed by parallel GPU threads, without any global GPU communication. Then, at runtime, the reworked PCR enables a fast solution of implicit 1D diffusion without computational overhead. Our results show that the entire numerical solver, coupled with the PCR algorithm, enables extreme-scale simulations with 2D pencil decompositions, which do not suffer performance losses even when compared to the best 1D slab configurations available for smaller systems.

Dimpled surface designs are known to be effective at enhancing convective heat transfer. However, optimizing these surfaces can be challenging due to the large parameter space created by the different combinations between geometrical features. In this paper, we combine a machine learning framework with a GPU-accelerated DNS solver to quickly assess the performance of a very large number of surface configurations, and to identify optimal designs. Our neural network can be trained to predict 2-D images with the local Nusselt numbers of rough surfaces within a few hours (in a single GPU), based on their original height maps. During evaluation, our neural network coupled with our parameterized geometrical formulation can evaluate one million dimpled surface designs in less than 45 min using a 64-core CPU architecture; with a low RAM memory footprint per core. Moreover, the GPU-accelerated DNS solver can calculate the Nusselt number of a rough surface within a few hours as well. The study considers a diverse parameter space including dimples with multiple depth profiles, major radiuses, corner effects, and inclination angles. To predict optimal designs, a basic reinforcement loop is created. In the first stage, only randomly chosen dimpled surface designs are selected as training data. The Nusselt numbers for each design are extracted from Direct Numerical Simulations (DNS), performed by the GPU-accelerated turbulent flow solver. Then, a convolutional neural network is trained, and different surface designs in our parameter space are evaluated. In order to advance the reinforcement learning loop, additional DNS cases are run for the optimal predicted surface, and other closely related geometrical variations. After adding these new DNS cases to the training set, the neural network is re-trained, and the process is repeated. Starting from the first iteration of the reinforcement learning loop, our results shows that machine learning can predict remarkably optimized dimpled surface designs, with high Nusselt numbers verified through DNS. Moreover, we find that machine learning chooses dimple configurations that enhance the interaction between roughness elements, even if other dimples with shorter radius (and equal depth) have more heat transfer area. The optimal surface has elongated dimples with opposite inclination angles, which create a zig-zag pattern for the flow near the walls. Additionally, we have shown that at different Reynolds numbers, the optimal geometry is different as well. We analyze other plausible optimal dimpled surface designs within our parameter space, and we find that machine learning correctly identified the adequate parameters to maximize heat transfer. Therefore, we conclude that machine learning is a highly effective tool to identify optimized designs for convective heat transfer enhancement.

Turbulent flows past rough surfaces can create substantial energy losses in engineering equipment. During the last decades, developing accurate correlations to predict the thermal and hydrodynamic behavior of rough surfaces has proven to be a difficult challenge. In this work, we investigate the applicability of convolutional neural networks to perform a direct image-to-image translation between the height map of a rough surface and its detailed local skin friction factors and Nusselt numbers. Additionally, we propose the usage of separable convolutional modules to reduce the total number of trainable parameters, and PReLU activation functions to increase the expressivity of the neural networks created. Our final predictions are improved by a new filtering methodology, which is able to combine the results of multiple neural networks while discarding non-physical oscillations likely caused by over-fitting. The main study is based on a new DNS database formed by 80 flow cases at a friction Reynolds number of Re_τ=180 obtained by applying random shifts to the Fourier spectrum of the grit-blasted surface originally scanned by Busse et al. (2015). The results show that machine learning can accurately predict the skin friction values and Nusselt numbers for a rough surface. A detailed comparison with existing correlations in the literature revealed that the maximum errors generated by deep learning were only 8.1% for the global skin friction factors C_f¯ and 2.9% for the Nusselt numbers Nu¯, whereas the best classical correlations identified reached errors of 24.9% and 13.5% for C_f¯ and Nu¯ respectively. The deep learning results also proved stable with respect to rough surfaces with abrupt changes in their roughness elements, and only presented a minor sensitivity with respect to variations in the dataset size.

This paper presents a machine learning methodology to improve the predictions of traditional RANS turbulence models in channel flows subject to strong variations in their thermophysical properties. The developed formulation contains several improvements over the existing Field Inversion Machine Learning (FIML) frameworks described in the literature. We first showcase the use of efficient optimization routines to automatize the process of field inversion in the context of CFD, combined with the use of symbolic algebra solvers to generate sparse-efficient algebraic formulas to comply with the discrete adjoint method. The proposed neural network architecture is characterized by the use of an initial layer of logarithmic neurons followed by hyperbolic tangent neurons, which proves numerically stable. The machine learning predictions are then corrected using a novel weighted relaxation factor methodology, that recovers valuable information from otherwise spurious predictions. Additionally, we introduce L2 regularization to mitigate over-fitting and to reduce the importance of non-essential features. In order to analyze the results of our deep learning system, we utilize the K-fold cross-validation technique, which is beneficial for small datasets. The results show that the machine learning model acts as an excellent non-linear interpolator for DNS cases well-represented in the training set. In the most successful case, the L-infinity modeling error on the velocity profile was reduced from 23.4% to 4.0%. It is concluded that the developed machine learning methodology corresponds to a valid alternative to improve RANS turbulence models in flows with strong variations in their thermophysical properties without introducing prior modeling assumptions into the system.

Turbulent flows past rough surfaces can create substantial energy losses in engineering equipment. During the last decades, developing accurate correlations to predict the thermal and hydrodynamic behavior of rough surfaces has proven to be a difficult challenge. In this work, we develop a convolutional neural network architecture to perform a direct image-to-image translation between the height map of a rough surface and its detailed local drag resistance and heat transfer rates. Various techniques are discussed to improve the computational efficiency of the machine learning architecture proposed, and even to reduce its time and space complexity. The main study is based on a new DNS database formed by 24 flow cases at a friction Reynolds number of Re_τ = 180 obtained by applying a random shift to the Fourier spectrum of the grit-blasted surface scanned by Busse et al. (2015,). The results show that machine learning can accurately predict the global values of the drag resistance and heat fluxes across a rough surface. The local predictions for both momentum and heat transfer also show a considerable improvement upon increasing the dataset size. A detailed analysis of the global skin friction values and Stanton numbers predicted by deep learning further reveals that the results surpass the accuracy of traditional correlations by a substantial margin in the dataset analyzed.