This thesis explores fast simulation of time-periodic flows, characterized by behavior that repeats at regular intervals. These flows are present in both natural and engineered systems. Examples include flows past wind turbines, in the heart and arteries, and around propellers. Due to their long time domains, time-periodic flows pose challenges in both experimental and numerical studies.
Many engineering tasks, like optimization, require numerous model evaluations across a wide range of inputs. We need a fast and accurate model to directly interact with the model in the design process. To contribute to this goal, we first develop a high-fidelity method specifically designed for time-periodic flows. Using this model, we then create a time-periodic reduced-order model to enhance simulation efficiency.

In the high-fidelity model, we employ the isogeometric analysis framework to achieve higher-order smoothness in both space and time. The discretization is performed using residual-based variational multiscale modelling and weak boundary conditions are adopted to enhance the accuracy near the moving boundaries of the computational domain. We enforce the time-periodic boundary condition within the isogeometric discretization spaces, which converts the two-dimensional time-dependent problem into a three-dimensional boundary value problem. The motion is known a priori and we restrict ourselves to two spatial dimensions. Application of the computational setup to heaving and pitching hydrofoils displays very accurate and exactly periodic results for lift and drag.

We use the high-fidelity model to develop a POD-Galerkin reduced-order model, which retains inherits the features of the high-fidelity model while reducing the number of variables in both space and time. We evaluate the reduced-order model with numerical experiments on moving hydrofoils. Reduced-order model solutions agree well with those of the high-fidelity model. The errors over the entire time period of the computed forces are less than 0.2%. Our time-periodic reduced-order model offers speed-ups ranging from O(10^2) - O(10^3) compared to the full-order model.

The non-linear nature of the Navier-Stokes equations creates a computational bottleneck in the reduced-order model. We explore hyper-reduction techniques to mitigate these challenges. We focus on empirical interpolation methods, which have shown promise in reducing the complexity of non-linear operators. The model performed well for the experiment with steady flow, with force and solution errors ranging from O(0.01% − 10%), depending on the sampling method. The hyper-reduced model achieves a speed-up of O(10^5) compared to the full-order model, and O(10^3) compared to the reduced-order model. This enables real-time computations with direct interation of the user. However, the hyper-reduced model could not provide a solution for the time-periodic flow experiment.

To show the applicability of the reduced-order model to an industrial problem, we apply the the model to a vertical axis wind turbine. Vertical-axis wind turbines offer significant advantages for urban applications over conventional wind turbines due to their lower noise levels. However, their performance is highly sensitive to various factors requiring high-fidelity simulations to optimize its performance. The model was used to determine the optimal operating point of the turbine, maximizing energy production per cycle under the given conditions. It was observed that the turbine's energy output was negative, likely due to the low Reynolds number (Re = 1000) used in this study.

Future research can expand on this thesis in several ways. Extending the model to three spatial dimensions would require solving four-dimensional boundary value problems, potentially benefiting from mesh adaptivity techniques. Further exploration of hyper-reduction methods, such as empirical quadrature procedures or neural network-based models, could enhance the model's efficiency. Applying hyper-reduction directly within isogeometric discretizations may also offer significant advantages. Additionally, further verification for higher Reynolds numbers and adaptation to other industrial applications, like wind farm and ship propeller optimization, could bridge the current theoretical advances with practical use. Finally, using time-periodic data from standard models to develop time-periodic reduced-order models could expand their applicability.@en

The computation of periodic flows is typically conducted over multiple periods. First, a number of periods is used to develop periodic characteristics, and afterwards statistics are collected from averages over multiple periods. As a consequence, it is uncertain whether the numerical results are exactly time-periodic, and additionally, the time domain might be needlessly long. In this article, we circumvent these concerns by using a time-periodic function space. Consequently, the boundary conditions and solutions are exactly periodic. We employ the isogeometric analysis framework to achieve higher-order smoothness in both space and time. The discretization is performed using residual-based variational multiscale modeling and weak boundary conditions are adopted to enhance the accuracy near the moving boundaries of the computational domain. We enforce the time-periodic boundary condition within the isogeometric discretization spaces, which converts the two-dimensional time-dependent problem into a three-dimensional boundary value problem. Furthermore, we determine the boundary velocities of moving hydrofoils directly from the computational mesh and use a conservation methodology for force extraction. Application of the computational setup to heaving and pitching hydrofoils displays very accurate and exactly periodic results for lift and drag.

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Simulating forced time-periodic flows in industrial applications presents significant computational challenges, partly due to the need to overcome costly transients before achieving time-periodicity. Reduced-order modelling emerges as a promising method to speed-up computations. We extend upon the work of Lotz et al. (2024) where a time-periodic space–time model is introduced. We present a time-periodic reduced-order model that directly finds the time-periodic solution without requiring extensive time integration. The reduced-order model gives a reduction in variables in both space and time. Our approach involves a POD-Galerkin reduced-order model based on a time-periodic full-order model that employs isogeometric analysis, residual-based variational multiscale turbulence modelling and weak boundary conditions. The projection-based reduced-order model inherits these features. We evaluate the reduced-order model with numerical experiments on moving hydrofoils. The motion is known a priori and we restrict ourselves to two spatial dimensions. In these experiments we vary the Strouhal and Reynolds numbers, and the motion profile respectively. Reduced-order model solutions agree well with those of the full-order model. The errors over the entire time period of thrust and lift forces are less than 0.2%. This includes complex scenarios such as the transition from drag to thrust production with increasing Strouhal number. Our time-periodic reduced-order model offers speed-ups ranging from O(10²) to O(10³) compared to the full-order model, depending upon the basis size. This makes it an appealing solution for prescribed time-periodic problems, with potential for additional speedup through nonlinear reduction techniques such as hyper-reduction.

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