On the outer boundary conditions for the fluid dynamics simulation of vertical-axis turbines

Many computational fluid dynamics simulations of isolated vertical-axis turbines use a 2D, rectangular computational domain and slip or symmetry boundary conditions (BCs) along the domain’s lateral outer boundaries or side walls. These BCs prevent any flux of mass and momentum across the side walls and so can cause the velocity at the domain inlet to be less than the freestream velocity at infinity. With further simplification that the flow is steady, an equation for the difference between these velocities is derived from the impulse form of the axial momentum equation for a control volume that coincides with the outer boundaries. The difference depends on the turbine thrust and the distance to the side walls. Corrections are derived for the power and thrust coefficients for isolated turbines and estimates provided for the domain size needed to reduce the correction to a specified level. When multiple turbines are arranged normally to the flow in close proximity, symmetry or periodic BCs are appropriate, but the difference between the inlet and freestream velocity can be large enough to invalidate recent claims that proximity increases the power output. We argue that both isolated and multiple turbine simulations should use BCs that include a point vortex for consistency with the turbine side force and a point source for consistency with the thrust. Nevertheless, it is not possible to ensure consistency with the moment equation for the control volume, and this may affect the accuracy of the calculated power output.