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4.1.1.Introduction

Combinations of oscillatory period and stroke length were selected (Table 4.1 and Appendix B), to provide simulations of wind-induced waves (6s period) and larger storm (swell) waves (9s and 12s period) using the oscillating trolley equipment in Southampton (Section 3.2). Differences in stroke length were used to simulate variations in the orbital amplitude, representative of a range of theoretical water depth and wave height combinations.

Theoretical combinations of wave height and water depth for selected experiments, derived on the basis of small amplitude wave theory, are shown in Figure 4.1. Similar combinations of period and stroke length were used for the non-linear experiments, together with a superimposed asymmetry of R=0.55, 0.6 or 0.7. The range of R selected for these

experiments was not designed to represent specific theoretical conditions; rather, values were selected to test the robustness of the laminar flow field model. Flows were simulated over the smooth bed and two fixed-roughness beds, as described in Section 3.2.7.

0 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.25 0.5 0.5 0.75 0.5 0.9 Water Depth (m) Wave Height (m) T=6 T=9 T=12

Figure 4.1. Theoretical combinations of wave height and water depth relating to the simulated conditions listed in Table 4.1 (R=0.5). The values were derived on the basis of linear wave theory. Note: the text labelling indicates the stroke length (m).

Bed type R=0.5 R=0.55 R=0.6 R=0.7 Smooth 6s (25, 50cm) 1 6s (50cm) 1 9s (50cm) 1 9s (50, 75cm) 1 9s (50cm) 1 12s (50, 90cm) 1 12s (50cm) 1 275μm 6s (25, 50cm) 2 9s (50cm) 2 6s (50cm) 2 9s (50, 75cm) 2 12s (50cm) 2 9s (50cm) 2 12s (50, 90cm) 2 12s (50cm) 2 500μm 6s (252, 50cm2,3) 9s (50cm) 2 6s (50cm) 2 9s (50, 75cm) 2,3 12s (50cm) 2 9s (50cm) 2 12s (50, 90cm) 2,3 12s (50cm)2

Table 4.1. Summary of the velocity profiles measured (wave period and stroke length). Key: (1), high-resolution data set; (2), low-resolution data set; (3), medium-resolution data set. (see also, Section 4.1.2 and Appendix B).

4.1.2.General methodology

Oscillatory flows were simulated using the oscillating trolley equipment described in Section 3.2. Coincident time-series of the horizontal (u) and vertical (w) components of velocity were measured in a vertical profile, using the LDV system described in Section 3.3. Measurements were obtained at 200Hz for 2 min duration at each of the heights. The phase referencing of individual velocity time-series, relative to the plate motion, was undertaken manually at the time of data collection. This operation was achieved by the operator initiating data collection at the point of plate reversal. Automation of this operation was not possible, due to limitations of the LDV equipment, i.e. the system did not support an external trigger to initiate data collection.

In total, 42 velocity profiles were obtained: 10 over a smooth bed; 11 over the 275μm fixed roughness bed; and 21 over the 550μm fixed roughness bed. All of the smooth bed profiles were collected at ‘high-resolution’, where measurements were obtained at 50 positions above the bed. Depending upon the (wave) period, 32-38 of these measurements were obtained within the boundary layer. In addition, 5 of the 550μm profiles were collected at ‘medium- resolution’ where measurements were obtained at 32 positions above the bed. Depending upon the period, 23-26 of these measurements were made within the boundary layer. Spatial

resolution for all types of measurement was greatest close to the bed; this was reduced progressively with height above the bed. However, the ‘high-‘ and ‘medium-resolution’ data sets required some considerable time for their collection and exhibited important phase lag errors (described in more detail in Section 5.5) due to the methodology; as such, they had no significant advantage over data collected at a lower resolution. Hence, subsequent

investigations utilised a ‘low-resolution’ profile, measuring at only 14 heights above the bed; 9-10 of these were within the boundary layer. In order to further improve experimental efficiency, phase referencing of the time-series for measurement of phase lag was discontinued in the ‘low-resolution’ experiments. The precise locations of measurements made using high-, medium- or low-resolution profiles are listed in Table B.3.

4.1.3.Data analysis

The velocity time-series in each of the experiments were analysed; this was to derive a single representative (mean) velocity cycle, at each of the heights above the bed. Using the

oscillatory period for the observed flow conditions, together with the sampling frequency (200Hz), the full length time-series were bin-averaged; the bin width was defined as T/100. Corresponding bins (phases), within each wave period, were then averaged (mean of bins [1, 101, 201, 301… etc.]), to generate a single representative wave cycle, 100 values in length. Within the ‘high-‘ and ‘medium-resolution’ data sets, all of the resulting velocity cycles were phase-referenced, relative to the plate motion (and to each other, see Section 4.1.2). On the basis of improving experimental efficiency, data collection was undertaken continuously, without pause between the time-series. Hence, the ‘low-resolution’ observations were not phase-referenced.

In addition to the horizontal time series, the phase-referenced data could be analysed further in order to yield: a time-series of vertical velocity profiles; vertical profiles of phase lag; and vertical profiles of velocity amplitude. The non-phase referenced data could be analysed to yield only vertical profiles of velocity amplitude. These data were compared subsequently with the theoretical distribution, as predicted by

Eq. 2.14.

4.2.Transition to turbulence

4.2.1.Introduction

A broad range of oscillatory parameters was simulated over the three prepared beds, as described in Section 3.2.7 (one hydraulically smooth bed surface; together with two of fixed

granular roughness, D=275μm and 550μm); a full list of the conditions simulated can be

found in Appendix B. The flow variable combinations of T, R and S simulatedwere in the same range as, and were representative of, those used later in the observation of the threshold of sediment motion (Chapter 7). In addition, the combinations were comparable to those used by other major contributors to the field of transition to turbulence included herein (Li, 1954; Manohar, 1955; Vincent 1957; Lhermitte, 1958; and Tanaka et al., 2000) (Section 7.1).

Oscillatory flows were simulated using the oscillating trolley equipment described in Section 3.2. The hydrodynamic conditions were observed and interpreted subsequently using three methods, namely: ‘visual observation’; ‘velocity time-series’; and ‘turbulent intensity’ (Sections 4.2.3, 4.2.4 and 4.2.5, respectively).

4.2.2.General methodology

The oscillatory period and degree of asymmetry were set (over one of the artificially prepared beds); these then remained constant throughout the experimental run. Initial stroke lengths were determined on the basis of relationships proposed previously (Sleath, 1984); these were such that initial conditions would be within the laminar flow regime, but close to the point of transition. The initial flow conditions were confirmed as being laminar, either by visual observation (for the visual observation method), or by inspection of the initial velocity time-series (for the velocity time-series and turbulent intensity methods, described below).

During each experimental run, the stroke length was increased incrementally, by between 1- 2cm (corresponding to an increase of ∼1cms-1

in velocity amplitude), until the ‘transition to turbulence’ occurred. At least three further increments and observations were undertaken, beyond the first signs of transition occurring. At each of the stroke lengths, the flow was allowed to adjust over three full oscillatory cycles, prior to observations being made or to the initiation of the data collection. For the ‘velocity time-series’ and ‘turbulent intensity’ methods, flow velocity measurements were undertaken at a height of 3mm above the bed, over 1 min, following each stroke length increment and allowing time for flow adjustment. A minimum of six time-series were collected, during each of the experiments.

4.2.3.Visual observation method

Visual observations of the transition to turbulence were performed by initially placing a small crystal of potassium permanganate on the central part of the oscillating plate. Under laminar flow conditions, a ‘stable’, longitudinal streak of dye was formed close to the bed,

i.e. initial dye dispersion from the streak was by molecular diffusion processes only. As the flow (stroke and/or velocity) amplitude was increased, instability waves were observed to develop at the top of the dye streak. Latterly, tongues of dye were ejected from within the boundary layer at the point of flow deceleration and reversal; these became more regular and widespread with further incremental increases in the flow amplitude. Immediately following this condition, eddy diffusion processes dispersed the dye streak rapidly. The observation of this dispersion was designated as the transition to turbulence. The progressions of such instability events under oscillatory flows have been described previously elsewhere (e.g. Bagnold, 1949, Li, 1954; Vincent, 1957; George and Sleath, 1978).

4.2.4.Velocity time-series method

The measured time-series of the horizontal (u) and vertical (w) components of velocity, corresponding to laminar flow, appeared as a smooth harmonic signal (Figure 4.2a) and as a small-amplitude (uniform variability) signal within a narrow envelope (±0.005ms-1

) (Figure 4.2b), respectively. These observations were in accordance with laminar flow theory (Appendix A). As the flow was increased, the signal began to deviate from the laminar solution. Both components of velocity exhibited evidence of high-frequency fluctuations (±0.05ms-1

) (Figure 4.2c&d); these were attributed to the periodic ejection of eddies, from within the boundary layer. Such a condition was designated as the ‘point of transition’; as such, it is consistent with the ‘visual observation’ method (Section 4.2.3). The transition occurred, typically, over a narrow range of velocity amplitudes (1-2cms-1).

0 10 20 30 −0.3 0 0.3 Time (s) u (ms −1 ) (a) 0 10 20 30 −0.3 0 0.3 Time (s) u (ms −1 ) (c) 0 10 20 30 −0.05 0 0.05 Time (s) w (ms −1 ) (b) 0 10 20 30 −0.05 0 0.05 Time (s) w (ms −1 ) (d)

Figure 4.2. Recorded velocity time-series, corresponding to: (a) & (b) laminar (stroke length = 66cm, and RE = 573); and, (c) & (d) transitional/turbulent, (stroke length = 82cm, and RE = 712) flows, over a smooth bed (T = 6s, R=0.6, z=3mm).

4.2.5.Turbulent intensity method.

Using the measured velocity data (Section 4.2.2), the vertical component of the turbulent intensity, E, was calculated for each time-series, using,

2

w

E

=

ρ

′

where, ρ is the density of the fluid and w′ is the fluctuating component of w. The horizontal component of velocity, u, was not utilised as the mean of the fluctuating component varies with asymmetry, even in the case of pure laminar flow, and would therefore present additional unnecessary complication. E was plotted then against stroke length, for each combination of bed type, oscillating period and asymmetry. A particular case [D=275μm, T=6s and R=0.55] is presented as an example of this approach, in Figure 4.3.

0.5 0.55 0.6 0.65 0.7 0.75 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Stroke Length, S (m) Turbulent Intensity, E Baseline value of E Fitted line Designated point of transition

Figure 4.3. Turbulent intensity, E, at z=3mm, with increasing stroke length, demonstrating the transition to turbulence. (D=275μm, T=6s and R=0.55).

The computed value of E, derived from the experimental runs, was observed to remain constant (a ‘baseline value’, with only slight variability) until increasing suddenly. A straight line was fitted to those data exhibiting elevated E (taken as where, E>baseline value+0.005). The ‘critical’ stroke length, corresponding to the transition to turbulence, was defined as the point where the fitted curve increased by more than 0.005 units, above the mean baseline value of E, observed during each of the experiments. The critical stroke length was expressed to the nearest centimetre. This method was consistent with the observation of ‘significant eddy formation’, as used in the visual observation and velocity time-series methods (Sections 4.2.3 and 4.2.4, respectively).