Numerical method for steady ideal 2-D flows of finite vorticity with applications to vertical-axis wind turbine aerodynamics

Abstract

This paper presents a new method for approximately modeling 2-D ideal steady fluid flows with finite vorticity induced by actuator curves of arbitrary shape. An actuator curve is an infinitesimally thin region containing a body force density acting on the flow, while the rest of the flow is force-free. The approach can be used for any flow satisfying this description, but, like other actuator methods, it is naturally suited for modeling turbines or propellers. We derive a weak formulation of the governing equations in the Lagrange streamfunction (Formula presented.) and solve it using finite elements. Compared to related methods such as the actuator cylinder (AC) approach, our formulation is uniquely suited for computations involving wake–wake or wake–turbine interactions. We validate the method by computing the flow through a single Darrieus vertical-axis wind turbine (VAWT) and comparing with previous work. To demonstrate the ability to simulate interacting actuators of arbitrary configuration, we simulate a three-VAWT array. The turbines are modeled in a freestream, and the loading is chosen to represent ideal airfoils. The standard VAWT results are consistent with previous work, validating the method. The three-VAWT array demonstrates a higher efficiency than the single VAWT (0.56 vs. 0.52), with differing optmal tip speed ratios for the upwind and downwind turbines (upwind: 3.9, downwind: 3.1. The optimum for a single turbine is 3.6). The flow field of the three-VAWT array shows expected features such as an acceleration of flow between the two counter-rotating upwind turbines.