Unstructured Cartesian refinement with sharp interface immersed boundary method for 3D unsteady incompressible flows

Abstract

A novel numerical method is developed for solving the 3D, unsteady, incompressible Navier-Stokes equations on locally refined fully unstructured Cartesian grids in domains with arbitrarily complex immersed boundaries. Owing to the utilization of the fractional step method on an unstructured Cartesian hybrid staggered/non-staggered grid layout, flux mismatch and pressure discontinuity issues are avoided and the divergence free constraint is inherently satisfied to machine zero. Auxiliary/hanging nodes are used to facilitate the discretization of the governing equations. The second-order accuracy of the solver is ensured by using multi-dimension Lagrange interpolation operators and appropriate differencing schemes at the interface of regions with different levels of refinement. The sharp interface immersed boundary method is augmented with local near-boundary refinement to handle arbitrarily complex boundaries. The discrete momentum equation is solved with the matrix free Newton-Krylov method and the Krylov-subspace method is employed to solve the Poisson equation. The second-order accuracy of the proposed method on unstructured Cartesian grids is demonstrated by solving the Poisson equation with a known analytical solution. A number of three-dimensional laminar flow simulations of increasing complexity illustrate the ability of the method to handle flows across a range of Reynolds numbers and flow regimes. Laminar steady and unsteady flows past a sphere and the oblique vortex shedding from a circular cylinder mounted between two end walls demonstrate the accuracy, the efficiency and the smooth transition of scales and coherent structures across refinement levels. Large-eddy simulation (LES) past a miniature wind turbine rotor, parameterized using the actuator line approach, indicates the ability of the fully unstructured solver to simulate complex turbulent flows. Finally, a geometry resolving LES of turbulent flow past a complete hydrokinetic turbine illustrates the potential of the method to simulate turbulent flows past geometrically complex bodies on locally refined meshes. In all the cases, the results are found to be in very good agreement with published data and savings in computational resources are achieved.

An approach of dynamic mesh adaptation for simulating 3‐dimensional unsteady moving‐immersed‐boundary flows

“…According to the topological connection of the subdomains, Angelidis et al classified the numerical methods using mesh adaptation techniques into 3 categories: overlapping decomposition, nonoverlapping decomposition, and nondecomposed methods. The present method is nondecomposed because the adapted mesh is monolithic and fully unstructured.…”

Multiresolution Large‐Eddy Simulation of an Array of Hydrokinetic Turbines in a Field‐Scale River: The Roosevelt Island Tidal Energy Project in New York City

“…The freesurface boundary of the flow channel is treated as a rigid lid, an assumption that has been successfully applied in the previous works of Kang et al, (, , ) and Yang, Kang, et al () among others, for low Froude number flows ( F r ≈0.2 for the flow in the East River). More details about numerical method can be found in Angelidis et al ().…”

“…The actuator line model can also predict the significant levels of time‐averaged spanwise velocity induced by the rotating turbines, which is more intense in the wake of the first and second turbines (Figure b). The distribution of the time‐averaged velocity profiles provides evidence of the smooth transition of the flow properties across cells with different levels of refinement (Angelidis et al, ). Figure c depicts contours of the streamwise turbulence intensity computed on a two‐level refined grid.…”

“…Such a grid resolution is known to give acceptable results for the turbine modeled with an actuator line model (Yang, Kang, et al, ). Actuator line models have been previously used successfully to predict the far wake dynamics of axial hydrokinetic turbines (Churchfield et al, ; Kang et al, ) and wind turbines (Angelidis et al, ; Porté‐Agel et al, ; Yang, Sotiropoulos, et al, ). Also, it is evident from prior numerical studies (Angelidis et al, ), and the results we present herein that the proposed numerical framework allows turbine simulations on locally refined grids without introducing any artificial reflection of the vortical structures at the coarse/fine interface of the computational mesh.…”

Large-Eddy Simulations of Two In-Line Turbines in a Wind Tunnel with Different Inflow Conditions

“…The computational cost of these simulations is still too high for real-time applications for owners and operators. Research efforts are being directed to make LES computationally more affordable by using advanced numerical techniques, as in the recent example of [12]. LES provide rich detailed data, which can be of great value, for example, for tuning the parameters of engineering models, or can provide physical insights to guide wind turbine or wind farm design.…”