### Abstract

THE high-energy flow at potential tidal and wind energy sites generates turbulent boundary layers that extend across the height of turbines. Understanding and characterizing the vertical profiles of the flow in the boundary layer is important for optimizing turbine performance and improving flow modelling. In this work we examine two models of the vertical profiles: the standard power law given by equation (1) and the law of the wake, formed by adding a wake function to the standard law of the wall, given by equation (2). Most previous tidal studies, for example [1] and [2], have focused on the applying the power law for this purpose. More recently, Milne et al. [3] used the “law of the wake” first introduced by Coles [4]. In [5], we compare the two approaches by examined the vertical profiles of ADCP data from the Minas Passage, Bay of Fundy. Flow in Minas Passage routinely exceeds 5 m/s and is highly turbulent (Re~108 ) [6], an ideal location to examine the theories of turbulent boundary layers. Here we recount some of those results and extend the comparison to a wind data set to determine. Our goal is to better understand the two models, and how they can be applied to ensemble averages.

*ADCP and Wind Data*

The tidal current velocity data considered here are derived from three Acoustic Doppler current profiler (ADCP) deployments at the FORCE CLA in the northern portion of the Minas Passage. A single five-beam Nortek Signature 500 kHz ADCP was deployed three times on a stationary platform. The ADCPs were deployed within 25 m of each other, on a volcanic platform that is relatively flat with a mean water depth between 35 to 40m. A full description of the deployments and analysis of the data can be found in [6]. The data analysed here are 5-minute ensemble averages, taken every 15 minutes. Table 1 lists the details of the three deployments, including the number of ensemble-mean vertical profiles used in the analysis.

For the wind data, we use three years of data (2013, 2019, 2020) from the Cabauw Experimental Site for Atmospheric Research [7]. We choose this dataset since it is well studied and easy to access. The data set records the average wind speed of 10-minute ensembles, at 6 heights up to a maximum of 200 m. One year of data roughly matches the time span of the ADCP data set and gives over 17,000 high-speed profiles, see Table 1. We choose to examine 3 years of data to see if the analysis would vary over different years.

*Previous Studies*

Determining the form of the mean vertical profile of turbulent flow has a long history, going back to the work of von Kármán and Prandtl. But there is still considerable discussion about whether the laws are “universal” or dependent on Reynolds number, and whether the laws can be rigorously derived or established empirically (for example, see the discussion in [8]).

The “law of the wall” is often used to describe the velocity near the bottom, in what is called the logarithmic boundary layer. The law of the wall has been shown to accurately model the bottom boundary layer in tidal flows [9], [10], but it generally does not extend through the entire water column.

On the other hand, the “power law,” given in (1), is used to model the entire water column. Even though it satisfies the no-slip condition at the bottom, it is often thought that the “validity of this power formula ceases in the immediate neighborhood of the wall.”[11] Recent application of the power law to tidal flows include Lewis et al. [1], who examined the vertical profiles of two ADCPs in the Irish Sea, and Sentchev et al. [2], who examined the vertical profiles from a single ADCP in Alderney Race.

There is also a less-known model introduced by Coles [4]: the “law of the wake,” given in (2). Coles found that the velocity deficit outside the boundary layer had a consistent form, with dynamics similar to that of a wake. For simplicity we will refer to this as the “wake law.” The wake law can be seen as an extension of the law of the wall beyond the logarithmic boundary layer, to the entire water column. More recently, the wake law has been discussed by Schultz and Flack [12] and a number of works by Guo [13]. In terms of tidal flows, Milne et al. [3] used the wake law to examine the vertical profile of ADCP data deployed in Pentland Firth.

In terms of the atmospheric boundary layer, both the law of wall (or log law) and the power law are standard tools for analyzing vertical profiles [14]. Often the log law is adapted to add a stability correction function related to the thermal stability of the boundary layer [15]. In practice, this function plays a similar role to the wake function extending of the law of the wall beyond the logarithmic boundary layer. However, the wake law has not been used extensively in atmospheric studies.

*Paper Summary*

In this work, we examine the vertical profiles of the flow from tidal and wind data sets discussed above. We examine both short and long time means of the data. We compare the power law to wake law in terms of the quality of fit and the variation in the model parameters.

*Canada*.