  Encouraged by recent studies on the performance of tidal turbine arrays, we
extend the classical momentum actuator disc theory to include the free surface
effects and allow the vertical arrangement of turbines. Most existing
literatures concern one dimensional arrays with single turbine in the vertical
direction, while the arrays in this work are two dimensional (with turbines in
both the vertical and lateral directions) and also partially block the channel
which width is far larger than height. The vertical mixing of array scale flow
is assumed to take place much faster than lateral one. This assumption has been
verified by numerical simulations. Fixing the total turbine area and utilized
width, the comparison between two-dimensional and traditional one-dimensional
arrays is investigated. The results suggest that the two dimensional
arrangements of smaller turbines are preferred to one dimensional arrays from
both the power coefficient and efficiency perspectives. When channel dynamics
are considered, the power increase would be partly offset according to the
parameters of the channel but the optimal arrangement is unchangeable.
Furthermore, we consider how to arrange finite number of turbines in a channel.
It is shown that an optimal distribution of turbines in two directions is
found. Finally, the scenario of arranging turbines in infinite flow, which is
the limiting condition of small blockages, is analysed. A new maximum power
coefficient 0.869 occurs when $Fr=0.2$, greatly increasing the peak power
compared with existing results.
