High-resolution velocimetry in energetic tidal currents using a convergent-beam acoustic Doppler profiler

The Edinburgh Subsea Instrument Platform (ESIP-1) was designed by the University of Edinburgh in 2012 as a rugged, modular and removable instrumentation platform for integration with the Alstom DeepGen IV 1MW tidal turbine. Containing multiple oceanographic instruments including up to 12 single-beam acoustic Doppler profilers (s-ADP), ESIP-1 was mounted atop the nacelle of the turbine installed in the Fall of Warness, Orkney, UK at the European Marine Energy Center (EMEC) tidal test site. Throughout 2013 multiple flow measurement campaigns were conducted using non-convergent acoustic Doppler profilers in coordination with tidal turbine commissioning and operation. In June 2013 modifications were carried out to include C-ADP functionality and preliminary tests were conducted. Following the recent completion of primary flow characterisation activities in October 2014, further C-ADP specific tests were conducted. ESIP-1 installation position on the turbine nacelle is shown in figure 2. Instruments were connected to the platform's central computing hub which in turn communicated in real-time with the Internet via the turbine's optical fibre connection to shore. The turbine provided a high power, 24 V dc, uninteruptible power supply (UPS). Hard-wired communications and UPS enabled long data measurement campaigns, on-the-fly configuration of instrumentation and real-time data capture and analysis.

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The deployment site features 45 m water depth, flows regularly exceeding 3 m s⁻¹ and an energetic wave field, particularly in winter months. The commercial-scale turbine (1 MW rated power, 20 m diameter rotor plane) was developed, installed and is operated as part of the ongoing ReDAPT Project (Reliable Data Acquisition Platform for Tidal); a project commissioned and funded by the Energy Technologies Institute (ETI), UK. In addition to environmental flow field measurement, the focus of this paper, comprehensive turbine performance data sets were produced by the turbine developer and represent an opportunity for future investigation.

Being installed on the turbine the coordinate system of the each instrument was defined in terms of the turbine coordinate system, with the x-direction along the turbine axis in the principal flow direction, the y-direction in the cross-flow direction, and the z-direction as upwards to the water surface, as shown in figure 3.

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The instantaneous Cartesian velocity components aligned with the turbine coordinate system are denoted (u,v,w). The mean velocity components, calculated over a time averaging window of t_a, are denoted (U,V,W).

Flow metrics calculated by the C-ADP are compared with the reference instruments introduced in sections 3.1 and 3.2. The error between a flow metric calculated by the C-ADP and a reference instrument is defined by equation (1), where subscript R denotes the reference measurement.

$$\begin{eqnarray}&&{{e}_{U}}={{U}_{\text{C}-\text{ADP}}}-{{U}_{\text{R}}}\end{eqnarray} \tag{ 1 }$$

Equivalent expressions for measurement error in other flow metrics are achieved by substition of U in equation (1).

The focal point of the C-ADP is located at z = 4 m for the set of experiments presented. In order to make meaningful comparisons at this location, all the acoustic Doppler profiling instruments must have a measurement location available at this point. This is achieved by adjusting the blanking distance and profiling bin size of each instrument [22].

The convergence of the acoustic beams at the focal point of the C-ADP obscures the reflected signal to each of the respective instruments if the acoustic signals are fired concurrently, referred to herein as synchronous operation. The C-ADP was developed with controllable time offsets for each single-beam instrument, with micro-second accuracy through the use of a timing signal provided by a GPS Grandmaster clock. Instrument offset times were varied and the resulting influence on the measurements (signal return amplitude and velocity) were observed in mono-static mode. In this mode, each instrument operates independently to receive the reflected acoustic signal that was transmitted from itself.

Bi-static sampling is available by using the vertically orientated s-ADP as the transmitter and the C-ADP (comprising four convergent s-ADP instruments) as receivers. Bi-static modes of operation involving a large test-matrix of configuration settings were also tested. Analysis is ongoing and results are not presented in this paper.

The C-ADP comprises four single-beam Doppler instruments developed around an early variant of the Nortek AS AD2CP platform (recently released commercially) which allows a high level of online user configuration over TCP/IP. These early-release models were progressively upgraded with firmware, software and hardware updates as they became available by the manufacturer.

In the construction of the C-ADP, four s-ADPs were installed at the corners of the ESIP-1, all angled towards a point directly above the center of the platform. This was achieved by yawing each instrument by a target angle of $\phi ={{26.5}^{{}^\circ}}$ from the x–z plane. The alignment of the diagonally paired instruments were verified visually by aligning consumer-grade laser pointers mounted on removable instrument alignment brackets. The authors have successfully operated s-ADPs onboard the ESIP-1 using remotely controlled actuators with pan and tilt motion capabilities. The results presented herein, however, are for beam angles pitched at constant angle, $\theta ={{20}^{{}^\circ}}$ from vertical. Angles reported represent design angles. Geometric errors are expected to arise from installation and operation in the harsh marine environment.

An s-ADP is installed on the frame with the instrument axis aligned in the positive z-direction (upwards). This instrument is used as a relatively high resolution reference instrument (when compared to traditional divergent-beam profilers) for comparing with the w-velocity component measured by the C-ADP. A lateral offset from the instrument platform center by $\Delta x=-150$ mm was necessary in order to avoid interference with a central structural component of the ESIP-1 (figure 3(a)).

A divergent-beam acoustic Doppler profiler (D-ADP) was also installed on the instrumentation platform. This device was a 1 MHz Nortek AWAC instrument which uses three acoustic beams to calculate the velocity profile from the platform to the water surface. The D-ADP was orientated to minimise acoustic interaction of the three primary beams—neglecting side-lobes—with those of the C-ADP or vertical s-ADP, as shown in figure 3. The properties of the acoustic instruments installed on the instrumentation platform are summarised in table 2.

A sample time series of stream-wise velocity measurements is shown in figure 4(a) for three tidal cycles. The mean velocity measured by the D-ADP is shown with a line, with the corresponding mean velocity measurement of the C-ADP shown with a circular marker. Sample rates for the C-ADP and D-ADP were 2 Hz and 1 Hz respectively at an ensemble averaging period of 5 min.

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The mean horizontal distance from the sample volume to the instrument axis, h_SV, is shown for the C-ADP and D-ADP in figure 4(b). In conjunction with figure 2, this demonstrates the significant improvement in spatial resolution which is achieved by the C-ADP, relative to existing divergent acoustic Doppler profilers. The single vertical s-ADP, whilst providing the highest spatial resolution velocity profile, is restricted to measurement of an individual vertical velocity component, w.

C-ADP testing was conducted on an opportunity basis as a subset of an extensive site characterisation test programme during 2013–2014. C-ADP triggering tests were routinely conducted during turbine operation where the assessment of relative changes in performance is sufficient. Ambient flow conditions were measured wherever possible by conducting tests during periods of turbine non-operation or through scheduled changes to the turbine orientation relative to the tidal flow, positioning the blades downstream of the ESIP-1.

The maximum profile range of the sensor array is determined by the individual s-ADP power settings which were optimised for analysing agreement at the focal point. Instrument configuration can be adapted through bin size, blanking distance, timing offset and instrument power in order to measure the specific flow property and location under investigation. For example, when sea surface elevation is the required measurement parameter, power can be increased to each of the s-ADP instruments ensuring an adequate profiling range. In the following experiments the bin size was set to 1.0 m for the D-ADP and 0.5 m for the C-ADP and s-ADP.

The ADP instruments return the component of velocity aligned with the direction of the beam axis. This beam-wise velocity for instrument i is denoted b_i and defined as positive for flow directions away from the transducer. The numbering convention used for the C-ADP instruments is shown by the circled labels in figure 3(b).

Expressing the velocity component in the direction from $\text{s}-\text{AD}{{\text{P}}_{1}}$ to $\text{s}-\text{AD}{{\text{P}}_{3}}$ as $u\text{cos}\left(\phi \right)+v\text{sin}\left(\phi \right)$, and the velocity component in the direction from $\text{s}-\text{AD}{{\text{P}}_{2}}$ to $\text{s}-\text{AD}{{\text{P}}_{4}}$ as $u\text{cos}\left(\phi \right)-v\text{sin}\left(\phi \right)$, the components of the Cartesian velocity captured by the Doppler shift in each beam is given by equation (2). This formulation is a modification of that presented for a D-ADP in [23].

$$\begin{eqnarray}&&{{b}_{1}}=\left({{u}_{1}}\ \text{cos}\ \left(\phi \right)+{{v}_{1}}\ \text{sin}\ \left(\phi \right)\right)\ \text{sin}\ \left(\theta \right)+{{w}_{1}}\ \text{cos}\ \left(\theta \right)\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}&&{{b}_{2}}=\left({{u}_{2}}\ \text{cos}\ \left(\phi \right)-{{v}_{2}}\ \text{sin}\ \left(\phi \right)\right)\ \text{sin}\ \left(\theta \right)+{{w}_{2}}\ \text{cos}\ \left(\theta \right)\end{eqnarray} \tag{ 2b }$$

$$\begin{eqnarray}&&{{b}_{3}}=\left(-{{u}_{3}}\ \text{cos}\left(\phi \right)-{{v}_{3}}\text{sin}\left(\phi \right)\right)\text{sin}\left(\theta \right)+{{w}_{3}}\ \text{cos}\left(\theta \right)\end{eqnarray} \tag{ 2c }$$

$$\begin{eqnarray}&&{{b}_{4}}=\left(-{{u}_{4}}\ \text{cos}\left(\phi \right)+{{v}_{4}}\text{sin}\left(\phi \right)\right)\text{sin}\left(\theta \right)+{{w}_{4}}\ \text{cos}\left(\theta \right)\end{eqnarray} \tag{ 2d }$$

As with the D-ADP, it is necessary to assume homogeneity of the flow between the measurement locations of the sample bins such that (u_i,v_i,w_i) = (u,v,w). This assumption is valid at the focal point of the C-ADP even for instantaneous velocity measurements, as the sample bins are co-located. The three unknown Cartesian velocity components can be solved from the four sub-equations of equation (2) as shown in equation (3).

$$\begin{eqnarray} &&b_{1}+b_{2}-b_{3}-b_{4} = 4u\sin(\theta)\cos(\phi) \end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray} &&b_{1}-b_{2}-b_{3}+b_{4} = 4v\sin(\theta)\sin(\phi) \end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray}&&{{b}_{1}}+{{b}_{2}}+{{b}_{3}}+{{b}_{4}}=4w\text{cos}\left(\theta \right)\end{eqnarray} \tag{ 3c }$$

The vector transformation matrix from the four acoustic beam directions to the C-ADP instrument coordinate system in matrix form is therefore given by equation (4).

$$\begin{eqnarray}&&\left[\begin{array}{c} u \\ v \\ w \\ \end{array}\right]=\left[\begin{array}{c} a & a & -a & -a \\ b & -b & -b & b \\ c & c & c & c \\ \end{array}\right]\times \left[\begin{array}{c} {{b}_{1}} \\ {{b}_{2}} \\ {{b}_{3}} \\ {{b}_{4}} \\ \end{array}\right]\end{eqnarray} \tag{ 4 }$$

The geometric scaling factors of equation (4) are $a=1/\left(4\text{sin}\left(\theta \right)\text{cos}\left(\phi \right)\right)$, $b=1/\left(4\text{sin}\left(\theta \right)\text{sin}\left(\phi \right)\right)$ and $c=1/\left(4\text{cos}\left(\theta \right)\right)$. For the experiments presented herein, the fixed angles of $\theta ={{20}^{{}^\circ}}$ and $\phi ={{26.5}^{{}^\circ}}$ are used as described in section 3.1.

Tests were conducted to allow an assessment of the C-ADPs performance relative to an industry-standard D-ADP and to the high resolution s-ADP.

4.4.1. C-ADP versus D-ADP.

4.4.2. C-ADP versus vertically orientated s-ADP.

The stream-wise velocity was divided into velocity bins of 1.0 m s⁻¹ over the mean velocity range of 0–3 m s⁻¹, and the error velocity is shown in figure 5(b). The C-ADP agrees well with the reference velocity in the region of the focal point, with error magnitudes increasing above and below this location. The maximum absolute error within the 10 m profile shown is 0.05 m s⁻¹ at flow speeds in the range of 2–3 m s⁻¹.

At distances of z > 4 m, the C-ADP acts in the same way as a divergent-beam ADP, as shown in figure 4(b), enabling the quantification of bulk flow velocities. The agreement between the mean flow velocities calculated using the C-ADP and D-ADP indicates that the mean velocity field is relatively homogeneous over the averaging period of t_a = 300 s at the turbine test site.

While close agreement is shown between the C-ADP and the D-ADP over the depth profile range of $2\text{m}\leqslant z\leqslant 10$ m, the velocity at the focal point of z = 4 m is of particular interest, as the location of maximum spatial resolution of the convergent-beam instrument. At this location the mean velocity components measured by the C-ADP exhibit close agreement with those of the divergent-beam reference instrument. This is shown by figure 6, where the mean velocity component measured by the C-ADP is plotted against that of the D-ADP at the focal point elevation. These results show 36 h of 5 min mean velocity with the turbine aligned with the flow direction.

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The vertical velocity component measured by the C-ADP was compared against the centrally located reference vertical single beam instrument. The instrument profiling configuration was designed to provide a reference velocity for every sample elevation of the C-ADP. While only the single velocity component of w was able to be compared, this presented a unique verification of the C-ADP measurement quality against an instrument at a relatively high spatial and temporal resolution. Such spatial resolution is not possible using a D-ADP due to the inherent divergent geometry of the acoustic beams. The results presented in this section were recorded at an increased sample frequency of 4 Hz enabled via upgrades to instrument firmware.

5.2.1. Instrument noise.

The coordinate transform of the C-ADP for w combines the four acoustic beam velocity in a way that averages the signals. The averaging process reduces the standard deviation of white Doppler noise in the beam-wise velocity by a factor $1/\sqrt{N}$, where N is the number of signals averaged at each time step. As such, the signal to noise ratio from the C-ADP instrument is greater than the single-beam reference instrument. This effect is relatively independent of measurement elevation within the operating range of the instrument.

Accounting for the attitude of the individual s-ADPs, the standard deviation of the C-ADP signal is reduced by a factor of $\text{cos}\left(\theta \right)/\sqrt{N}$ through the coordinate transform described in section 4.3. For the four-beam array (N = 4) with $\theta ={{20}^{{}^\circ}}$, this corresponds to a 47% reduction in Doppler noise relative to the single vertical s-ADP instrument.

A spectral analysis using a fast Fourier transform shows the power of the fluctuating vertical velocity component of the C-ADP and s-ADP at each frequency level, shown in figure 7. This data set was acquired during a period of wave activity and mean flow speeds in the range 0.75–1.25 m s⁻¹ at a sample rate of 4 Hz. The significant wave height at the site during this period was calculated using wave spectra from the deployment site as H_m0 = 2.2 m. The spectra presented in figure 7 represents the mean power spectral density from 93 Hanning-windowed, stationary time series of 1024 points with 50% overlap.

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The presence of large waves at the test site is indicated by the energy concentration in the frequency range of 0.07–0.17 Hz. The spectra of the vertical velocity measured using the CADP follows the theoretical cascade of energy predicted by classical theories at a rate of f ^−5/3 [24] in the frequency range of 0.2 < f < 2 Hz [25]. The Doppler noise floor of the s-ADP instrument is indicated by the region of zero gradient at frequencies greater than approximately 0.3 Hz.

5.2.2. Cross-correlation of vertical velocity.

Cross-correlation can be used as a measure of the similarity between two signals. In this case, the maximum cross-correlation coefficient of the vertical velocity measured by the s-ADP and C-ADP is calculated to indicate the comparability of two instrument measurements with varying durations of temporal averaging [26]. The results are presented in figure 8. An averaging period of t_a = 0.25 s corresponds to the raw 4 Hz data.

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Again, a peak in the maximum cross-correlation coefficient was observed near the focal point of the convergent beam system (z = 4 m). The beam separation of the C-ADP increases with distance from this focal point and the correlation is seen to decrease as a result. The width of the correlation peak increases with averaging period, as the magnitude of flow perturbations with length scales less than the beam separation are reduced. The peak cross-correlation of 0.8 is calculated at the focal point using the raw 4 Hz data, which increases to 0.96 when a moving average of t_a = 4 s is applied.

This result further demonstrates the ability of the C-ADP to resolve velocity perturbations with a relatively high spatial and temporal resolution. The results at the focal point are evidence of the importance of using an instrument with a reduced spatial resolution, compared to existing divergent beam configurations, when high-frequency velocity measurements are required.