  This paper solves an approximate form of conservation of mass and momentum
for a turbine in a wind farm array. The solution is a fairly simple explicit
relationship that predicts the streamwise velocity distribution within a wind
farm with an arbitrary layout. As this model is obtained by solving flow
governing equations directly for a turbine that is subject to upwind turbine
wakes, no ad hoc superposition technique is needed to predict wind farm flows.
A suite of large-eddy simulations (LES) of wind farm arrays is used to examine
self-similarity as well as validity of the so-called conservation of momentum
deficit for turbine wakes in wind farms. The simulations are performed with and
without the presence of some specific turbines in the wind farm. This allows us
to systematically study some of the assumptions made to develop the analytical
model. A modified version of the conservation of momentum deficit is also
proposed to provide slightly better results at short downwind distances, as
well as in the far wake of turbines deep inside a wind farm. Model predictions
are validated against the LES data for turbines in both full-wake and
partial-wake conditions. While our results highlight the limitation in
capturing the flow speed-up between adjacent turbine columns, the model is
overall able to acceptably predict flow distributions for a moderately sized
wind farm. Finally, the paper employs the new model to provide insights on the
accuracy of common wake superposition methods.
