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.

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.