Phase synchronisation between tests

For a host of diagnostics and in order to produce the results later in this chapter, zeroing and synchronisation of repeated tests was required. Repeated tests need to be synchronised in order that they can be compared, and in some cases, averaged to account for some of the uncertainty between tests. Other areas which require synchronisation are where data are captured at the same time, but from a separate acquisition system.

It can be quickly noticed by the reader when comparing two repeated time-series from experimental data, that variations due to several sources of uncertainty mean that they are not identical. This causes problems when attempting to synchronise two signals based on say, the time of peak. Unless there is a very sharp step-like signal with an instantaneous peak, when noise is introduced to a fairly flat underlying peak, the time of local maximum will vary between signals. Similarly, when using a maxima (or minima) in the time-derivative, similar problems are encountered.

Using combinations of these values and searching for several constraining points in the time signal can improve the performance, but results were unsatisfactory.

An alternative to looking at only single points in the time-domain is to utilise the entire signal. A technique for doing this is known as Cross Correlation. Cross correlation essentially compares one function with another by shifting one along by a lag, and testing the correlation between the two functions. A cross correlation is very similar to a convolution (see Kreyszig (1999) for example), a technique widely used in probability theory and signal processing. The difference being in a convolution, one signal is reversed during the process, which does not occur during a cross-correlation.

For two functions, f (t) and g(t), the cross correlation (denoted by the ? symbol) is

defined in (3.6).

(f ? g)(t) = Z ∞

−∞

f^∗(τ )g(t + τ )dτ (3.6)

where f^∗ indicates complex conjugate of f , where τ is the phase shift. For discrete data, as required for our purpose, a different form is used (3.7);

(y₁? y₂)[n] = X∞ n=−∞

y₁^∗[δ]y₂[n + δ] (3.7)

where y₁ and y₂ are time-series (sampled at the same rate), δ is the lag and n is the data point. When the cross-correlation is plotted against the lags, a peak in the correlation function corresponds to a maximum likelihood estimate of the phase shift between the two signals. The time lag can be identified in this way and the performance is found to be very good, even for signals with moderate levels of noise.

In order to compare signals from different tests and synchronise them in this manner, at least one reference signal is required that is present in all tests to be synchronised to. Within the data collected for the unsteady tests, three channels are common to all data throughout the test duration without any variation (movement of probes etc). Two wave probes, (“Offshore 1”, located just away from the mouth of the generator and “Toe”, which is located at the toe of the bathymetry where the flat propagation region finished) were common to all tests. In addition to this, a time-series is captured which logs the wave generator’s valve position, and this was also present across all tests.

An example set of profiles that has been synchronised to the time of the first test is shown in Figure 3.14. The profiles are for the wave probe at the toe of the bathymetry, but by synchronising this profile, the entire set of data for those waves are also synchronised that were captured using the same data acquisition board.

This enables velocity, pressures, load signals and other wave probes to be viewed for a given wave type; particularly useful as the velocity probes are moved around be-tween different tests to cover other regions. One remaining data set that cannot be

WaveID 133 Wave Probe at Toe of Bathymetry

Time(s)

SurfaceAmplitude(m)

0 20 40 60 80 100

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02 0.03

Figure 3.14: Free-surface elevation time series measurements of the wave probe located at the toe of the bathymetry. Plot shows four repeated tests, and the average of all four (black dashed line).

captured on the same acquisition board is the water elevation data around the struc-tures. This is instead captured using image processing of high speed photography and so is not synchronised with the rest of the data in its raw form. However, the pressure transducer tests with the pressures left in units of mm of water (p/ρg) are compared to the surface elevation at the front of the structure in mm. The height of the lowest transducer (36mm) is added to the pressure, and water depths lower than this were set to 36mm in the surface profile. In this way, the same cross-correlation technique could be used to calculate the necessary phase shift. For tests without pressure measurements, the water profile could be shifted by comparing to other tests with pressure measurements. In this way, all data regardless of the logging system are synchronised for each test along with any repeated tests. The data for subsequent repeated tests are synchronised to the timing of the first test.

By performing these analyses, it is possible to combine quantities like velocity and water height, an essential step in the following sections which rely heavily on the ability to do this with confidence.

3.4 Experimental results

3.4.1 Observations

Initially water levels in the flume are still and at a level close to the top of the sloping bathymetry at an offshore still water depth (SWD) of between 640 and 670mm, depending on the wave and valve position and wave being generated. SWD is achieved with the pump activated and control valve closed for our elevated wave tests, and with the valve open for the N-wave tests. This is because the elevated waves are created only by positive discharge from the wave generator, whereas for the N-waves the generator removes water from the flume firstly in the negative portion of the wave. To therefore maintain the required constant SWD, a different volume of water is required in the flume system for the different types of waves.

Once the generator control valve is actuated, water is either released or taken in to the tank through the gate, and a positive / negative fronted wave propagates along the flume towards the bathymetry. The waves in this portion of the flume

propagate with a constant profile shape, at a constant velocity which approximately corresponds to the shallow water wave celerity (gh)^1/2. When this wave interacts with the bathymetry, the free-surface profile changes. The amplitude increases, and the velocity reduces in line with the effects of shoaling. As the wave approaches the shoreline, the water depth initially increases for elevated waves causing the shoreline to advance, and reduces for the N-waves causing the shore to initially retreat. Once the positive portion of the N-waves arrives, the water depths increase and the shore advances back up the bathymetry. At this point for both elevated and N-waves, the shore advances up to the flat portion, over-topping at the point where the bathymetry slope becomes zero, and velocities increase. Longer period waves tend to advance more slowly and act like a slowly rising flood. Water inundates the shore, propagating across the flat region of the flume and onto the instrumented buildings.

When the water reaches the instrumented test structures, the depth increases slightly upstream of the block, and water flows around the structure. Short-period waves (here T . 10s) tend to make an impulsive impact on the front of the structure and water splashes upwards to heights occasionally in excess of the building height (300mm). For low flow rates, the upstream and downstream depths (h₁, h_d) are similar, but as the wave approaches its peak and flows increase, the free-surface profile around the structure changes. Levels drop away towards the rear of the structure, before the inundating water spills into the sump at the rear of the flume.

This is the same choked flow behaviour as observed for the incremental steady-flow tests of Chapter 2. For long-period waves, as would be expected for a tsunami, the transition between these states occurs within one time-series of the same wave.