The effect of uncertain bottom friction on estimates of tidal current power

Abstract

Uncertainty affects estimates of the power potential of tidal currents, resulting in large ranges in values reported for a given site, such as the Pentland Firth, UK. We examine the role of bottom friction, one of the most important sources of uncertainty. We do so by using perturbation methods to find the leading-order effect of bottom friction uncertainty in theoretical models by Garrett & Cummins (2005 Proc. R. Soc. A 461, 2563–2572. (doi:10.1098/rspa.2005.149410.1098/rspa.2005.1494); 2013 J. Fluid Mech. 714, 634–643. (doi:10.1017/jfm.2012.51510.1017/jfm.2012.515)) and Vennell (2010 J. Fluid Mech. 671, 587–604. (doi:10.1017/S002211201000619110.1017/S0022112010006191)), which consider quasi-steady flow in a channel completely spanned by tidal turbines, a similar channel but retaining the inertial term, and a circular turbine farm in laterally unconfined flow. We find that bottom friction uncertainty acts to increase estimates of expected power in a fully spanned channel, but generally has the reverse effect in laterally unconfined farms. The optimal number of turbines, accounting for bottom friction uncertainty, is lower for a fully spanned channel and higher in laterally unconfined farms. We estimate the typical magnitude of bottom friction uncertainty, which suggests that the effect on estimates of expected power lies in the range −5 to +30%, but is probably small for deep channels such as the Pentland Firth (5–10%). In such a channel, the uncertainty in power estimates due to bottom friction uncertainty remains considerable, and we estimate a relative standard deviation of 30%, increasing to 50% for small channels.

A note on the tuning of tidal turbines in channels

“…Garrett and Cummins (2005) have shown that there is a maximum amount of power that can be extracted from a tidal channel, and that both this maximum extractable power and the optimal turbine drag, λ 1 , depend solely on the value of λ 0 . A recent study by Kreitmair et al (2019) has used this simple channel model to show how uncertainty in the value of specific inputs (the bed roughness coefficient, in this case) can propagate through the model to produce uncertain optimal tunings and maximum power estimates. Uncertainty will be an important consideration for studies of real tidal power sites, where bed roughness and other parameters must be carefully calibrated prior to undertaking resource assessments, but is not considered in this study, where the aim is simply to compare between different idealised models of turbine performance.…”

The effect of bed roughness uncertainty on tidal stream power estimates for the Pentland Firth

“…The effect of uncertainty in C d on these parameters is now explored. [9]. For all values of m C d , uncertainty in C d acts to increase the expected power slightly above the deterministic value.…”

“…The results shown on the right-hand axis of figure 5 are in good agreement with results shown in fig. 5 of Kreitmair et al [9], again dependent on choice of value for λ 0 .…”

“…It may be noted that the results shown in figure 4 are in good qualitative agreement with fig. 4a of [9], suggesting that the analytic models of Garrett & Cummins [10] and Vennell [11] used for analysis may give good approximations to a more in-depth numerical analysis of bed roughness uncertainty effects, provided an appropriate value for the non-dimensional channel drag parameter λ 0 is determined. Figure 5 shows the standard deviation in power σ P (left-hand axis) and the ratio of relative standard deviation in power to that in bed roughness, i.e.…”

“…Kreitmair et al [9] examined the effect of uncertainty in bottom friction on tidal power estimates by introducing uncertainty as a (narrow) symmetric distribution about the mean bed roughness coefficient in three analytical models, namely those of Garrett & Cummins [10], Vennell [11] and Garrett & Cummins [12]. Using perturbation methods, Kreitmair et al found that introduction of uncertainty in bed roughness coefficient produces changes to expected power in the range of −5 to 30% for a strait connecting two large oceans, depending on the relative strength of inertial to drag forces, and tidal farm geometry.…”