Drift and generation of interfacial progressive waves in immiscible fluids

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Drift and generation of interfacial progressive waves in immiscible

fluids

Computational Science and Engineering Jon Alexander Pirolt

Master’s Thesis, Spring 2021

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The study programme is unspecified. Please consult the documentation for the packagemasterfrontpage in order to correctly print the colophon.

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Abstract

When periodic internal waves propagate in immiscible fluid, the fiction in the fluid spatially dampens the waves. The horizontal wave momentum decreases with the amplitude but cannot just disappear. Weber and Christensen suggest that the momentum gets redistributed to Eulerian mean currents that will increase with time.

We built and programmed a wavemaker to generate progressive internal waves between one layer of Isopar Vl (H1) and one layer of water (H2). We managed to create repeatable waves in the range with a frequency of 0.25hz to 0.75hz.

We used image processing to measure surface elevation, phase speed, and wavelength. We found the most efficient wavelength for the wavemaker to generate waves and made a table with a wide range of inputs and the waves they create.

To investigate Weber and Christensen’s theory, we used PTV to track the particle paths and find the drift. We focused on the more viscous top layer and saw a drift around the interface in all the cases. The theoretical drift profiles increase with time, and this appears in the measured results as well. When we compared the magnitude of the theory and the measurements, the longest and most linear wave had a close agreement. AskH1 and wave steepness, ak increased, we saw that the measurements increasingly surpassed the theoretical drift

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Acknowledgements

Many people deserve to be mentioned here.

First my loving husband for all the nights you were alone when I studied.

Fanny for being a champion before my deadline.

Atle Jensen for guiding me through the process.

Jean for showing patients when I did not understand his code.

Olav for all the nice conversations in the lab. Alt all the help!

Laila for stating as my partner in crime.

Yi Yi for all the good advise.

Thea for being a supporter when it got hard.

and finally to my sibling for all in their way supporting me!

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List of Figures

2.1 Reproduction of fig. 5 in [17]. H1 = H2 = 0.1m, ⁄ = 1m,

‹1 = 9.22◊10^≠5m²s^≠1 (olive oil), and ‹2 = 1.12◊10^≠6m²s^≠1 (water), assuminga= 2cm . . . 7 3.1 The wave tank, Field Of View (FoV) one, and FoV two are

respectively 1.1 meters and 2.9 meters away from the wavemaker.

The styrofoam plates extend to the glass on both sides and are held in place by a rack. . . 12 3.2 Using Progressive Automations model PA-15-6-11. Three in-

put signals (cyan line) with frequency 0.35hz and different amplitudes (from table A.9: f0_35a1200_r1, f0_35a900_r2, f0_35a600_r2). Plotted against measured horizontal position of the wave maker (magenta line). . . 14 3.3 1000 as input. First picture shows stating position. Second and

third shows the maximum displacement in both directions . . . 14 3.4 Timeseries, before and after tracking. Run with 0.35hz and

amplitude . . . 16 3.5 Run f0_35a400_r4, FoV loc_two, layers 10x10cm, 11. period . . . 18 4.1 Response plot. starts (*) is 10x10cm layers. Circles (¶) are

10x20cm layers. S = Stroke length, a = amplitude at approximately 1.1meter ,–=

2/ﬁ³ . . . 19 4.2 Three different waves. Frequency 0.25hz, 0.35hz and 0.45hz. 50

fps. The grey and pink line represent one separate run each. We fixed the axis for comparison. Only part of the total time-series is represented. 0.35hz is also zoomed in to one crest, to show the deviation better. . . 20 4.3 1.2 meters from the paddle. Two wavepackets plotted (soild dark

cyan, dashed light cyan). The magenta line shows the time-interval integrated to find the energy. . . 23 4.4 LS: Polynomial fit to back-flow. RS: Shifted to theory. Plotted

period 1, 5 and 9 from top to bottom. Run f0_35a400_r2, FoV loc_two, layers 10x10cm . . . 25 4.5 Showing full column, f = 0.35hz, a = 1.02cm , FoV location two,

layers 10x10cm (f0_35a400_r2) . . . 27 4.6 f = 0.35, a = 1.02 , FoV location two, layers 10x10cm

(f0_35a600_r10) . . . 29

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List of Figures

4.7 f = 0.35, a = 1.56cm, FoV location two, layers 10x10cm

(f0_35a600_r10) . . . 31

4.8 Timeseries showing chosen periods. f = 0.35, a = 1.56cm, FoV location two, layers 10x10cm (Run f0_35a600_r10) . . . 32

4.9 f = 0.35hz, a = 1.81, FoV location two, layers 10x10cm (f0_35a850_r4) . . . 34

4.10 Timeseries with plotted periods. f = 0.35hz, a = 1.81, FoV location two, layers 10x10cm (f0_35a850_r4) . . . 35

4.11 f = 0.35hz, a = 1.95cm. FoV location one, layers 10x10cm (f0_35a600_r9) . . . 35

4.12 Timeseries and period plotted. f = 0.45hz a = 0.92cm , FoV location two, layers 10x10cm (f0_45a300_r3) . . . 36

4.13 f = 0.45hz a = 0.92cm , FoV location two, layers 10x10cm (f0_45a300_r3) . . . 38

4.14 f = 0.45hz, a = 1.35cm, FoV location two, layers 10x10cm, (f0_45a500_r3) . . . 40

4.15 Timeseries with plotted periods. f = 0.45hz, a = 1.35cm, FoV location two, layers 10x10cm . . . 41

4.16 f = 0.45hz, a = 1.32, FoV location one, layers 10x10cm, (f0_45a300_r3) . . . 43

4.17 Timeseries for plotted periods,f = 0.45hz, a = 1.32, FoV location one, layers 10x10cm, (f0_45a300_r3) . . . 44

4.18 (f = 0.45hz, a = 2.04cm, FoV location one, layers 10x10cm, f0_45a500_r2) . . . 45

4.19 f = 0.35, a = 1.98cm. FoV loc location two layers 20x10cm: Periodes chosen by color plotted, (f0_35a750_r5) . . . 47

4.20 f = 0.35hz, a = 1.98cm. FoV loc location two layers 20x10cm . . . 49

4.21 f = 0.35hz, a = 2.38cm, FoV location one, layers 20x10cm, (f0_35a750_r4) . . . 50

4.22 f = 0.35hz, a = 2.38cm, FoV location one, layers 20x10cm, (f0_35a750_r4): Periodes chosen by color plotted . . . 51

4.23 f0.45hz, a = 1.81cm , FoV location two, layers 20x10cm (f0_45a600_r5): Periodes chosen by color plotted . . . 52

4.24 f0.45hz, a = 1.81cm , FoV location two, layers 20x10cm (f0_45a600_r5) . . . 54

4.25 (f = 0.45, a = 2.34, FoV location one, layers 20x10cm, f0_45a600_r5) 56 4.26 f = 0.45, a = 2.34, FoV location one, layers 20x10cm, f0_45a600_r5) Periods chosen by color plotted . . . 57

B.1 Run f0_35a400_r2, FoV loc_one, layers 10x10cm . . . 76

B.3 Run f0_35a600_r9, FoV loc_two, layers 10x10cm . . . 78

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List of Tables

2.1 List of variables with description . . . 5 4.1 ¯‘ is the mean deviation comparing two runs. ‡‘ is standard

deviation of the deviation. . . 21 4.2 . . . 22 4.3 Overview of runs we selected for the analysis. h=H1/H2 . . . . 22 A.1 Overview of runs, Location one, 10cm oil, h = H1/H2, a =

amplitude, c = phase speed,⁄= wavelength . . . 64 A.2 Overview of runs, Location one, 20cm oil, h = H1/H2, a =

amplitude, c = phase speed,⁄= wavelength . . . 65 A.3 Overview of runs, Location two, 10cm oil, h = H1/H2, a =

amplitude, c = phase speed,⁄= wavelength . . . 66 A.4 Overview of runs, Location two, 20cm oil, h = H1/H2, a =

amplitude, c = phase speed,⁄= wavelength . . . 67 A.5 Overview of Amplitudes. s = stroke length for wave-maker away

from interface,a1, a2= amplitude at location one and two,‡1,‡2

= standard deviation when measuring amplitude, h = H1/H2 , 10cm oil . . . 68 A.6 Overview of Amplitudes. s = stroke length for wave-maker away

from interface,a₁, a₂= amplitude at location one and two,‡₁,‡₂

= standard deviation when measuring amplitude, h = H₁/H₂ , 20cm oil . . . 69 A.7 Overview of Phase speeds 1/2 . . . 71 A.8 Overview of Phase speeds 2/2 . . . 72 A.9 An overview of wave generated approximately 2.5 meters down-

stream from the paddle. Both oil and water level was set at 10cm.

Paddle stroke length (S) amplitude (a), phase speed (c), standard deviation (‡), wavelength (⁄). . . 73

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CHAPTER 1 Introduction

As surface waves propagate at the surface of a fluid, the internal waves oscillate within. In order for them to occur, there has to be a density difference, for instance, as a result of a sudden change in temperature and/or salinity. This stratification of density is called a pycnocline, and can be either narrow and sharp, or broader.

In nature, we find internal waves at continental plates close to large rivers, where the rivers create a less dense freshwater layer above the saltwater. We can also find internal waves elsewhere in the ocean and lakes. They can even be found in the atmosphere, then in the form of lee waves when cold streams of air rush over mountains.

During the last decade, internal waves and their properties have been studied extensively. A majority of the research examines cases with one layer of saltwater, and one layer of freshwater, with a stratification between them. The studies look at both solitary and periodic waves, with a particular focus on breaking.

In 1968 [12] Thorpe studied the shape of periodic internal waves. He performed two experiments; one two-layer experiment with a narrow pycnocline, as well as one with a continuous stratification through the entire water column. In the former, he used a plunger-type wavemaker with a triangle-shaped block, oscillating up and down at the interface. In the latter, he used a "flap-type"

wavemaker, rotating back and forth around an axis fixed at the interface. To investigate the shape, he added dye and photographed the waves.

Later, Thorpe published [11], where he continued to investigate the shape of, as well as the conditions for, breaking of non-linear internal waves with and without a slope. The method he used for the wave generation was different:

an airfoil was fixed at the interface, connected to a plate that got pushed up and down. The foil and plate separated the fluids with different densities. The movement expelled one fluid from one chamber, while it drew the other fluid into the other. The same method for generating progressive waves was later used in [4]. Long non-linear internal waves were examined, as they were sent over a slope going to a shelf. The data was then compared to theory. Many different stratifications gave a good match.

A variation of the "plunger-type", described above, was the method of choice to generate periodic waves in [2]. The triangle block attached to the plunger got traded in for a half-cylinder. The focus was on the conditions where wave instabilities occurred. Tetrachloride mixed with red oil dye was added to the

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1. Introduction

top layer, and would settle on the interface, visualizing the flow. To narrow the pycnocline, the water was slowly removed from the interface. The waves were killed offon the other side of the tank with a narrowing channel. At the lower amplitudes, there was little mixing around the wavemaker, but as the oscillations increased, so did the mixing.

Troy and Koseffalso use a linear actuator with a half-cylinder oscillating ver- tically at the interface (the "plunger-type") [5], [15] [13]. In these examples, we find a synthetic horsehair beach reducing reflections. The density profiled was measured using a microscale conductivity-temperature probe. For mono- chromatic waves, they investigated the shape with ultrasonic wave gauges at the interface. In the salty layer at the bottom, laser-florescent dye was added, and images were taken under the illumination of a laser sheet. The dye was assumed evenly distributed and diffused at the same rate as the rest of the salty water, giving a visualization of the stratification.

[15] gave a detailed overview of the generation of waves. They investigated the response of the wavemaker, comparing stroke length and measured amplitude.

The results show that the wavemaker is at peak performance when generating waves with⁄= 2D, where D is the diameter of the wavemaker.

Continuing, Troy and Kosefflooked at two different ways of provoking a wave to break. One method was focused waves, and the other was using a contract- ing channel. The latter method, also used in [13].

Not all internal waves are periodic. Solitary internal waves can be generated by trapping a volume behind a gate and releasing it [3] [7] [1].

The velocity fields induced by the internal waves are a natural point of interest.

In the previous decade, the measurements were mainly limited to pointwise observations. The recent development in image processing has given us particle image velocimetry (PIV), as well as particle tracking velocimetry (PTV). Both are tools allowing high-resolution and non-intrusive methods, allowing invest- igation of the velocity fields and particle paths by photographing a flow seeded with zero-bouncy particles. The former gives an eulerian representation of the flow [1] [3] [16], while the latter shows the path of individual particles giving a Lagrangian view [3].

Umeyama and Matsuki used PIV to visualize the velocity field induced by periodic internal waves [16]. A d-shaped "plunger-type" wavemaker generated the waves. Comparing 3. order stokes internal-wave theory, they found good agreement with the measurements. As well as the velocity field, they also traced the water particle trajectory (not by PTV but by converting the PIV).

The articles mentioned to this point have examined two-layer models with salt- and freshwater. However, internal waves can just as easily propagate in-between immiscible fluids. These waves cannot be found easily in nature, and this may be the reason it is difficult to find research on them.

Still, some articles discuss these waves. [6] performed experiments with silicon oil and water (ﬂ1/ﬂ2= 0.85and‹1/‹2= 1.5). Small periodic waves were generated by oscillating a point-source-type cylindrical wavemaker. The cylinder was positioned at the interface as well as the surface, and inside the bottom layer of water. Using laser images, the elevations were looked at, and the results were confirmed with probes.

Another example of using silicon oil and water is [7]. The oil had similar

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1.1. Outline properties as [6], but this time solitary interval waves were created using the trapped volume method. Utilizing conductivity probes the wave profile was obtained, and compared with Korteweg-de Vires (KDV) and Miyata-Choi- Camassa (MCC) models.

1.1 Outline

In this master, we are going to generate period internal waves with two immiscible fluids and investigate the drift using PTV. The motivation is to compare to the theory derived by Weber and Christensen in [17]. In Chapter two we will describe this theory and look closer to the tracking method PTV in relation to the software digiflow as described in [9].

The choice of wavemaker is the plunger-type wave Troy used in [13], and we will also use the same synthetic horsehair beach.

Chapter 3 discusses how we build the wavemaker and the changes we made to the design. We also describe how we used image processing to find interfacial elevation and other wave properties. In the end, we explain the prosses of processing the PTV files from digiflow.

In Chapter 4, for we will look at the results. First, we will compare the efficiency with our wavemaker to Troy by making a response plot similar to his in [13], and [14]. Then we will continue to look into repeatability, reflections, and the return flow before taking a thorough discussion of 12 PTV results.

Chapter five gives a quick summary discussion and focus on a conclusion.

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CHAPTER 2 Theory

2.1 List of Variable

V ariable description

H₁, H₂ Upper and lower layer thickness, respectively

‹1,‹2 Upper and lower layer kinematic viscosity, respectively ﬂ1,ﬂ2 Upper and lower layer density, respectively

a Interfacial amplitude

÷ Interfacial elevation

› Surface elevation

x , y Horisontal and vertical position, respectively Ê Wave radian frequency

f Wave frequency

¯

u Mean drift for one wave period c Phase speed/wave celerity

⁄ Wave length

k Wave number

Table 2.1: List of variables with description

2.2 Lagrangian Drift Equations

When we generate periodic internal waves, we can see that the amplitude gets damped the further the wave propagates. This happens because of viscosity or, in other words, friction in the fluid. When the fluid loses its amplitude, there is a loss of horizontal momentum that does not simply disappear. Weber and Christiensen propose in [17] that the momentum gets redistributed to Eulerian mean currents at both sides of the interface. The Eulerian mean currents (¯uL) will develop in time and quickly grow beyond the Stokes drift(¯uS) and viscus boundary-layers terms (¯uB), and diffuse out in the into the fluid.

After they derive the Lagrangian mean drift (¯uL = ¯uS + ¯uB + ¯uE), they plot different cases and theoretically show how the Eulerian mean drift almost immediately dominates the Lagrangian mean drift in every case.

All equations are gotten from the article [17] if not specified otherwise.

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2. Theory

Lagrangian Drift (u¯L)

¯

u_L1,2= ¯u_S1,2+ ¯u_B1,2+ ¯u_E1,2 (2.1)

Stokes Drift (¯uS)

¯

u_S1=c₀”²/2, u¯_S2 =c₀h²”²/2 (2.2)

interfacial phase speed:

c0=Ê/k=

Û g(ﬂ2≠ﬂ1)

(ﬂ1coth(kH1) +ﬂ₂coth(kH2))k (2.3) We got this from the dispersion relation given by Troy in [14]. We chose this since this is for finite depth and not only long waves.

slowly-varying amplitude parameter:

”= (A/H1) exp(≠–˜x) (2.4)

˜

x is the distance from the where the wave is generated, and A is the initial amplitude at this position. We measure the amplitude and drift at the same location, so we setx˜= 0and A becomes a.

Spatial attenuation rate:

–=kG/[4H1“₁(1 +h)] (2.5)

Inverse boundary-layer thickness:

“1= [Ê/(2‹1)]^1/2, “2= [Ê/(2‹2)]^1/2 (2.6) the depth ratio:

h=H₁/H₂ (2.7)

Viscus Boundary-layers Terms (¯uB)

¯

u_B1=c₀”²Q₁ 53

4Q₁exp(≠2“1y)≠2 exp(≠“₁y) cos(“1y)6

(2.8)

¯

uB2=c0”²Q2h² 53

4Q2exp(2“2y)≠2 exp(“2y) cos(“2y)6

(2.9)

Dimensionless parameters:

Q₁= (H1+H₂)(1≠R)/H2, Q₂= (H1+H₂)R/H1 (2.10)

R=‹₁^1/2/(‹₁^1/2+‹₂^1/2)<1 (2.11)

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2.2. Lagrangian Drift Equations

Figure 2.1: Reproduction of fig. 5 in [17]. H1 = H2 = 0.1m, ⁄ = 1m,

‹1 = 9.22◊10^≠5m²s^≠1 (olive oil), and ‹2 = 1.12◊10^≠6m²s^≠1 (water), assuminga= 2cm

Eulerian Mean Drift (¯uE)

¯

u_E1=≠u¯_S1R 5

rFerfc3 y 2(‹1t)^1/2

4

≠G(2Êt)^1/2ierfc3 y 2(‹1t)^1/2

46 (2.12)

¯

uE2= ¯uS1R 5

F erfc3

≠y 2(‹1t)^1/2

4+G(2Êt)^1/2ierfc3

≠y 2(‹1t)^1/2

46 (2.13)

Dimensionless parameters:

r=‹₂^1/2/‹₁^1/2, G=r(1 +h)²/(1 +r) (2.14)

F = (1 +h)[(3 +r)h≠1≠3r]/(2 + 2r) (2.15)

To investigate if we coded the theory correctly, we reproduced fig. 5 in [17].

When we compare figure 2.1 and the original plot in the article, the two figures seem identical. It’s important to note that the theory is the time-averaged lagrangian drift over the inputted amount of time. For reasons explained in greater detail in section 3.8, we could not track one particle for several periods.

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2. Theory

Therefore we had to adjust the theory to show the average drift of only one period instead of several. We did this by finding the drift in length, from the beginning a period (t =t₁) to the end (t = t₂), then divide by the time difference,

¯

ut1≠t2= u¯₂t₂≠u¯₁t₁

t₂≠t₁ (2.16)

Assumptions

The assumptions made by Weber and Christiansen when they derived the lagrangian mean drift [17], were :

1. Linear longwave theory: kH₁π1, kH2π1 -will be discussed in the results

2. The interfacial elevation is much larger than the surface elevation: |›|π

|÷|

-Was verified by looking at the waves we generated.

3. Infinitely long tank, no back flow -Will be discussed in the results

4. Neglect the effect of surface, bottom and wall friction -Will be mentioned in the results

5. Only valid for relatively small times.

-We assume that less than 30s is small enough

6. The Stokes boundary layer are this: “₁H₁∫1,“₂H₂∫1 -Calculated to be within limit

2.3 PTV

We will look at PTV through the lenses of Digiflow, since this is the software we utilized to analyze the data in this experiment. The theory will be tied directly to the variable parameters in the program, as described in [9]

The basic concept is very straight forward. Imagine a series of pictures of only one particle. The position and time in each frame is noted, this is the way we track its movement both in space and time. When more particles are added, the tracking becomes more difficult, but as long as we move less than the particle’s diameter, the task of tracking is still reasonably straightforward.

The problem begins when the velocities become more significant, and it is no longer obvious which particle is the same in the different pictures. To overcome this challenge, we have to consider other characteristics, like shape, size and intensity to differentiate the various particles.

The location will be the most significant parameter, as well as how we balance the properties, depending on the images being tracked. One example is when the lighting is uneven, or the picture is overexposed; using the intensity as a matching parameter is not ideal in this situation. Another example is if

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2.3. PTV the particles are very small, the shape parameter is not going to work. In general, the tracking will work best if the seeding is not too high relative to the movement from one frame to the other. It will also help if the particles are larger, making it easier to differenciate one from the other.

When we look at the fluid velocity field, it is clearly advantageous to use particles with the same density as the fluid itself. There is room for some differences, but the tracking will not correctly represent the velocity field if there is too much buoyancy.

As PIV gives a Eulerian description of the flow, PTV follows the particle from a Lagrangian perspective. PIV is the more commonly used method. It gives high resolutions, and can, to some degree, function despite of noise in the picture.

It is possible to convert both methods to the two coordinate systems, but there are certain benefits to each method, due to how the tracking is done.

To mention some, PTV is considerably faster than PIV. Also, the individual tracking of particles allows PTV to separate velocity gradients, e.g. a moving reflection on the glass, from the particles’ movement. This a problem PIV cannot solve, as is finds the average displacement of the entire subwindow. On the other hand, the same averaging used in PIV smoothens the velocity field and makes it more even.

Other qualities we need to consider is the seeding of the fluid. PIVs correlation function works better where there are many particles, while the PTVs particle matching algorithm prefers a sparser amount. In the end, the more ideal choice will often be to consider what we are looking for. In our case, PTV was clearly the advantageous choice, seeing as we were examining the drift.

The first step of PTV is to locate all the particles. We scan through the picture and identify any areas within a given threshold of intensity. Then we compare eachblob against a set of requirements, like a specific size or shape.

We register the position of any area that qualifies as a particle, along with any other characteristics obtained.

After mapping all the particles P and Q in two sequential frames at t = tn

andt=tn+1, each particle in each set is assigned a labelpi for i = 1, 2, ..., M andqj for j = 1, 2, ..., N, where M and N is the amount of particles found at t=tn andt=tn+1. We create a matrixcij where each element is assigned a cost for two particles to be matched. The lower the cost, the more probable a match. Digiflow sets the basic cost by

cij = (pi) +ÿ

f

max(0,Êf(pi)’f(pi, qj)≠·f) (2.17)

, where f is a list of the different properties like location, size and intensity.

P hiis a joining fee for new particles. A positive value will make a preference towards matching particles with a velocity history. The unit costÊf also takes into account whether there is a matching history or not. For the particle’s location, the unit cost is

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2. Theory

Êx(pi) = Y_ _] __ [

1

L²₁ if no previous matches

1

L²₂ if one previous match

1

L²₃ if two or more previous matches

(2.18)

, where L1, L2 and L3 are a set maximum matching distance away from the predicted position. We recommended having more lenient matching requirements to find new particles, then place more stringent conditions when we have a velocity history, in order to weed out any mismatches. Later we can disregard any particles with less than three sequential matches, to remove potentially bad data.

While the unit cost is a set value based on matching history, the cost function

’f is calculated using the particle’s instantaneous properties. Again, we look at the location property.

’(pi, qj)x=|xi+ui”t≠xj|² (2.19) , wherexis the position,uis the velocity, and”tis the timestep between the two frames. Summed up, the function is the squared error from the predicted new position, if we disregard acceleration.

Lastly, we have·f, that is the threshold for a property. As we can see from the basic cost function, it is being subtracted. Therefore, a high threshold value tf will decrease the chance that any cost will be added at all.

Because the location is the most critical property used to determine the matching cost, we have focused on this above, but every attribute has its ownÊ, ’ andtau.

Continuing, we assign a new matrix–ij, where we define 1 as a match and 0 as a mismatch between two particles pi and qj. In the cases where N is not equal to M, we have to introduce dummy particles as –i0 and –0j. If we set the value to 1, it will describe a particle disappearing or appearing in the next frame. This happens if the flow carries the particle in or out of the field of view, or the illuminated area of the light sheet. Unlike actual particles,dummy particlescan be matched more than once.

Finally, we are ready to examine the actual matching. We set a function Z, that is the sum overi andj of aijcij. We chose the non-zero values of aij by minimising Z, using the graph theory approach. Without going into further detail, we have now matched the particles.

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CHAPTER 3 Experimental Setup and Method

3.1 General Setup

We performed the experiments in the hydro lab at the mechanic department at the University of Oslo. The tank itself is 7meter long, 25cm wide and 60cm deep (figure 3.1) We filled the tank with one layer of Isopar V and one layer of water. The water had a density of998kg/m³ and kinematic viscosity 1◊10^≠6m²/s (at 20 C^¶), while the density of the Isopar V was 815kg/m³ and kinematic viscosity1.44◊10^≠5. The latter was measure by a laboratory rheometer ( around 20C^¶)

Isopar V is a solvent used in industry and was chosen because it does not give off health hazard fumes. Later we experienced is that it also dissolves the silicone between the glass in the wave tank. We had the tank filled for several weeks, so it does not happen immediately, but we advise to find another oil or perform the experiments in a short timeframe.

We performed the experiment whereH₂ was set at 10 cm, while we used both 10 and 20cm for the oil level (H1) .

At one side of the tank, we generated periodic waves using a wavemaker. The controller was an Arduino Due. Jean Rabault created the code and is available open-sourced at https://github.com/jerabaul29/ArduinoPaddleControl. The basic principle: the Arduino gets parts of the wave signal from a computer with Python. The signal gets stored in a buffer, and the Arduino request the next part when needed the buffer has sufficient space. The benefit of having the Arduino store the signal is that it increases the signal’s accuracy as is

"travels" shorter.

The controls get the vertical position of the paddle from an LVDT. Then it calculates the output to give to the actuator, using a PID function. The PID function decides the output based on the set signal’s difference and the actual position.

We added a string of code for the Arduino to trigger the four cameras when the wavemaker starts.

At the other end, we had a "horsehair" beach, covered with synthetic grass, with a ration 1:7, to dampen the waves, without creating mixing to run more extended time series. On the top, we covered the fluids with fixed styrofoam plates to dampen the surface waves.

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3. Experimental Setup and Method

Figure 3.1: The wave tank, Field Of View (FoV) one, and FoV two are respectively 1.1 meters and 2.9 meters away from the wavemaker. The styrofoam plates extend to the glass on both sides and are held in place by a rack.

We investigated the velocity field at two separate locations, approximately at 1.1 and 2.9 meters from the paddle. To get a clear view of the interface, we tilted the cameras at an angle between 6 and 15 degrees. This would broaden the illuminated interface and make it visible in the pictures. Because of the different refraction in oil and water, we had to put one camera for each layer.

In the beginning, we only had about 10cm wide FoV but extended it because it limited how large drift we could measure.

As the wave propagated, the oil level would rise, and when it went down again, the oil would cling to the glass, creating distortions in the images. The angle on the camera helped to some degree to avoid this issue. The problem was that the camera tilt required to avoid the distortions altogether would be so steep that the illuminated interface’s broadness would hide particles closest to the interface. We decided to keep the angle somewhere in between because we wanted to see as much as we could of the velocities near the boundary layer.

Another side effect of a large tilt is that when particles are moving horizontally away from and towards the camera, it will be processed as vertical velocities by PIV and PTV.

3.2 Wave-maker design

In [15] they generated internal waves with a half-cylinder attached to a linear actuator creating an oscillating vertical motion at the internal interface. The diameter of the wave-maker was approximately half the total depth of the fluids. The range of the movement in both directions was therefor limited to half of the width of the paddle. In our case, the smallest layer would be

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3.3. Actuator 10cm, and the diameter of the half-cylinder was 11cm. The desired wave had a maximum amplitude of 2cm, and a wavelength of 1 meter, to ensure that we were in the shallow-water regime. Looking at figure 5 in [15], we can see that the described setup would limit us to 1cm amplitudes. Testing confirmed this.

To increase the energy put into the waves, we added a box (10cm x 11cm) before the half-cylinder to displace a more significant volume. We managed to make amplitudes up to 2.3cm with a wavelength of 1.2meter. We tried to generate the waves with and without a wall behind the wave-maker. Without we lost much energy, especially at specific frequencies.

3.3 Actuator

The first actuator we tested was a Progressive Automations (model PA04-8- 100). It had a stroke length of 8", Voltage 12VDC, 100lbs force and a max speed of 2.16"/sec when moving with a full load. When we started generating the waves, we quickly found out that speed was significantly lower than in the datasheet suggested. A slow actuator was a problem since it did not manage to follow the input signal when the amplitudes became too big. Therefore we changed to a different model (PA-15-6-11). This one had 6" stroke length, the same voltage, 11lbs force, and a max speed of 4.70"/sec. The speed was still lower than in datasheet, but now it was able to follow the input signal better.

We can see in table A.9, that this actuator was able to get close to create the waves we wished. In 3.2, we can see that the magenta lines sometimes are linear. This is when the actuator works at maximum capacity. When the amplitudes become lower, it manages to follow the lines better. Later we discovered that the wires were far to thin to deliver the current needed for the actuators operate at peak performance. We change back to PA04-8-100 to gain the added benefit of more force, allowing the paddle to work more stable.

3.4 Linear Position Sensor

For the Arduino to keep track of the actual displacement, we use a linear variable differential transformer (LVDT). At a given position, the sensor provides a voltage output that the Arduino translates to a value between 0 and 4096 (12bits). Where 0 is 0V and 4096 is 3.3V. We used the LVDT Omega LD621-150. The range is from 0 to 150mm, with an electrical output from 0 to 10V, and the linearity is ±0.2%. To be able to use the full range of the sensor, we converted the output voltage to meet the limitation of the Arduino.

Between the LVDT, the conversion of voltage and the Arduino, there is room for error. Therefore we calibrated the sensor by giving the Arduino the inputs 250, 500, 750 and 1000, and measured the displacement (figure 3.3). We found that an input of 1 provides 3.67e-08 meters oscillation around the starting position. Multiplying this with 4096 gives that 15.02cm, which is within the expected error.

3.5 Coordinate Transformation

When we run the particle tracking in Digiflow, it is possible to add the coordinate picture. However, this is a tedious prosses, where the world

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Figure 3.2: Using Progressive Automations model PA-15-6-11. Three input signals (cyan line) with frequency 0.35hz and different amplitudes (from table A.9: f0_35a1200_r1,f0_35a900_r2,f0_35a600_r2). Plotted against measured horizontal position of the wave maker (magenta line).

Figure 3.3: 1000 as input. First picture shows stating position. Second and third shows the maximum displacement in both directions

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3.6. Image processing coordinates of each point have to be manually added, without any possibility to automate the process. Therefore, we used the coordinate transformation function (createcoordsystem) provided in the HydrolabPIV package [8]. The input needed is the coordinate picture’s pixel positions, the corresponding world coordinates, and which transformation type we wish to use. The pixel position was obtained by the function pixel_p C.3, a slightly enhanced version of the script suggested in the HydrolabPIV tutorial [8].

Because the camera was positioned at an angle, we could not use a linear coordinate transformation [8], and therefore used a cubic.

Note that Matlab defines the horizontal position 1 at the top, while digiflow starts from the bottom. We needed to take this into account when creating the pixel input matrix.

3.6 Image processing

We used the time series to obtain wave parameters (amplitude, frequency, phase speed, and wavelength). To create this, we extract one column from each picture and stack them next to each other. The result is an overview of how the surface elevation develops at a specific position, as a function of time.

It is important to note that we cannot use this to look at the shape of the wave itself, as it is not a function of space in the horizontal direction.

When comparing the time series with a picture from the experiment, we can see in figure 3.4 that is it far easier to identify the crest’s position.

Digiflow has a function to create the picture, but we chose to write new code using Matlab. The benefits being that it went faster, we could extract as many columns as we wished, and we could loop through several runs in one click.

We made five time-series from each camera each run.

With the time-series found, we continue using Matlab to prosses the image.

Going through each column, we identified the brightest spot as the interface.

The function created takes the three inputs other than the picture:

-An interval to look for the line. We do this to avoid to follow other bright lines in the picture.

-A search range to limit the program’s possibility to deviate too far from the previously found brightest spot. A smaller value will smoothen the line, but it will not follow a steep curve. A higher value will also increase the program’s chance to identify spots outside the line as the brightest.

-The size of the brightest spot. If there are not enough particles at the interface, the line might become non-continuous. Especially at high frequency and amplitudes, the particles get pushed forward, so there is nothing to light up at the interface. To help with these gaps, we can increase how many pixels we test. As an unintended side effect, the line also became smoother. Note when the seeding density is high in the fluid relative to a narrowly lit interface, a larger test spot will make the program lose the line.

In the end, we further polish the line with the smooth function in Matlab. We are using default settings of a 5 point moving average.

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Figure 3.4: Timeseries, before and after tracking. Run with 0.35hz and amplitude

3.7 Sources of error

At location one, we are very close to the wavemaker, and the waver is not fully developed.

At location two, we are very close to the beach, so there will be more reflections.

The reflected energy magnitude cannot be measured because we could not send a wavepacket and measure it one way before it returned.

The PTV can track wrong and match the wrong particles.

Other movements in the fluids from the water being heated by the light sheat.

We can get errors from the image processing method when we measure wave properties.

3.8 Plotting PTV Results

After we did the tracking in Digiflow, we get a dft files that contain all the tracking data on the particles. We read the .dft files using the function dfttracker, made by Jostein Kolaas, to extract the data to a .mat file we could use in Matlab (found in C.5). It is important to add that dfttracker only works properly on a Linux machine. The function also processed the coordinate transformation file that we made with HydrolabPIV and gives the result in world coordinates. The files contain data on every particle tracked, regardless if it’s two frames or five hundred. Therefore we wrote a function C.1 that sifts through the particles and picks out any that has a certain minimum length in a given interval.

The function C.2 takes the chosen particles , and returns the horizontal displacement. It can focus on a specific area on the x-axis, and figure 3.5 a) we chose particles from an interval ±3cm around 2.91 meters downstream from the wavemaker. When we picked particles in a narrow band, it gave us less spread in how far the particles drifted, so we chose this for all the plots where we had a sufficient amount of particles tracked for the full period. In figure 3.5 b) we plotted the path of the chosen particles. To give a clearer view of the paths, we only chose particles with a minimum distance between them in the y-direction, while in section all the particles in the interval were chosen.

The theory presented in [17] gives a time-averaged drift from the beginning to a specific time. This becomes problematic to measure for too many periods

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3.8. Plotting PTV Results at the time. As the drift becomes more substantial, and the particle goes out of our field of view, we lose the tracking without any possibility to pick it up again as the particle re-enters the frame. To deal with this, we tracked each period individually and adjusted the theory.

It is necessary to mention that we have two choices when we find the drift in a period. One is the known wave period which is the duration for two crests to pass by the same point. The other option is the "particle period", which is when a particle oscillates full circle back to the vertical position it started. Note that these are not the same when we have a drift. To use our case an example: when a wave period has passed at one point, the particle that used to be on the crest has now drifted forward and will not reach a full

"particle period" before later, when the particle again reaches a crest further downstream. This thesis consistently used the wave period and measured it from where the elevation wasy= 0.

After we had found the particles that was tracked for the full period, we calculated the Lagrangeian average drift velocity by finding the horizontal displacement and divide it by the period.

¯

uL= xend≠xstart tend≠tstart

(3.1) To scale the vertical position y, we used the oil layer’s depth (H2). When it came to the drift velocities, we scaled the theoretical drift with the theoretical phase speed, and the measured drift with the measured phase speed.

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(a) Cyan dots are the starting positions for the particles tracked. Magenta dots shows the chosen particles plotted in b). Magenta lines mark the interval we found particles to plot

(b) Cyan line is the path of a particle. The light cyan line marks the starting point, and the magenta stars the endpoint

Figure 3.5: Run f0_35a400_r4, FoV loc_two, layers 10x10cm, 11. period

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CHAPTER 4 Experimental Results

4.1 Response

Troy and Koseffinvestigate the efficiency of their wavemaker by plotting the relation between stroke length of the paddle and the resulting wave amplitude.

They name this relation response and plot it against wavenumber scaled with the diameter of the wavemaker (kD/2ﬁ) (fig 5 in [15]). The peak performance was found to be whenkD/2ﬁ= 0.5. We cannot non-dimensionalize k with D because our paddle has a different shape. Instead, we propose to scale with the square root of the area of the cross-section of the paddle. To make it comparable to Troy and Koseff, we introduce a constant–we determined by solving the equation–Ô

A=D/2ﬁ. The resulting plot 4.1 show peak efficiency of the wavemaker around the non-dimensionalize wavenumber 0.5.

Figure 4.1: Response plot. starts (*) is 10x10cm layers. Circles (¶) are 10x20cm layers. S = Stroke length, a = amplitude at approximately 1.1meter ,–=

2/ﬁ³

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4. Experimental Results

(a) Frequency 0.25hz

(b) Frequency 0.35hz

(c) Frequency 0.45hz

(d) Frequency 0.35hz. Zoomed in to 4. crest

Figure 4.2: Three different waves. Frequency 0.25hz, 0.35hz and 0.45hz. 50 fps. The grey and pink line represent one separate run each. We fixed the axis for comparison. Only part of the total time-series is represented. 0.35hz is also zoomed in to one crest, to show the deviation better.

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4.2. Repeatability

4.2 Repeatability

To ensure that the experiment is reproducible, we checked the repeatability.

We ran every wave at least two times. In the same way, as we estimated at the possible error in the measurement (see Appendix A), we looked at the deviation between crests and crests and troughs and troughs. We were now comparing two different runs, but with the same wave parameters.

Looking at Table 4.1, we can see that there is approximately a 0.5mm±0.25mm mean deviation with a standard deviation around the same value. There is a vague indication that lower frequencies have a smaller variation. We can explain this by that the Arduino gets less accurate with faster movements. More likely its an error in measurements since steeper waves "washes" the particles from the interface, making it more difficult to trace.

When we compare the result with the possible error in measurements, we see that it is around the same magnitude, and we can conclude that the waves are repeatable. When we look at figure 4.2, it becomes clear that this is reasonable.

The pink line and grey line are so close that it almost looks like the one. When we zoom in, we more clearly see the deviation.

W ave ¯‘[mm] ‡‘[mm] H₂/H₁

f0_25a600 0.28 0.23 1

f0_25a900 0.26 0.19 1

f0_35a400 0.60 0.40 1

f0_35a600 0.64 0.37 1

f0_35a850 0.63 0.47 1

f0_45a300 0.27 0.19 1

f0_45a500 0.27 0.25 1

f0_55a300 0.58 0.49 1

f0_75a200 0.74 0.54 1

Table 4.1: ¯‘is the mean deviation comparing two runs. ‡‘is standard deviation of the deviation.

4.3 Reflections

To investigate the efficiency of the beach, we sent one wavepacket and measured the wave energy going out and then again when it returned. The experiment was performed for the frequency 0.25hz, 0.35hz, 0.45 and 0.55 at which point we stopped since higher frequencies gave no visible reflections. We generated large amplitudes kept it similar for all the runs. The wave energy E can be estimated by

E =gc ﬂ

⁄ t2

t1

÷²(t)dt (4.1)

where c is the phase speed, ﬂis the density difference, and÷is the interfacial displacement at a given position over time [5]. For the estimate to be valid, the energy must propagate in one direction. Therefore the water had to become still before the reflections returned. Because of trailing waves, it was not possible

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to measure the wave energy at the FOV closes to the beach. As we can see in figure 4.3 is was possible with a small margin 1.2 meters downstream from the wavemaker. The result shows that 1.5% of the energy travelled back for the longest waves. Even less was reflected for the higher frequencies. It is important to note that not only the beach dampened the waves. From the measured position, the waves had to travel 3.5meters each way back and forth to the beach, so viscous dampening in the tank has contributed (table A.5).

Troy reports in [14] that he had less then 1% reflections (table 4.2). His tank was shorter, and the beach had a tilt of 1:6, but he did not specify which frequencies he tested or at what position. He had one layer of salt water and one layer of freshwater. The increased viscosity should have raised the dampening for us. Our beach had a 1:7 slope, and the tank was longer so it would be natural to think we would have less energy reflected. Possible reasons might be he tested shorter waves, or that the surface on the beach was more efficient to dampen the wave.

Frequency 0.25hz 0.35hz 0.45hz 0.55hz Reflected energy 1.5% 1% 0.5% 0.2%

Table 4.2

W ave a c ⁄ kH

₁

ak h

cm m/s m

f 0_35a400_r2_loc_two 1.02 0.29 0.83 0.76 0.08 1 f 0_35a600_r10_loc_two 1.56 0.29 0.82 0.77 0.12 1 f 0_35a850_r4_loc_two 1.81 0.29 0.82 0.76 0.14 1 f 0_35a600_r9_loc_one 1.95 0.27 0.78 0.81 0.16 1 f 0_45a300_r3_loc_two 0.92 0.28 0.63 1.00 0.09 1 f 0_45a500_r3_loc_two 1.35 0.27 0.60 1.04 0.14 1 f 0_45a300_r3_loc_one 1.32 0.27 0.60 1.05 0.14 1 f 0_45a500_r2_loc_one 2.04 0.27 0.59 1.06 0.22 1 f 0_35a750_r5_loc_two 1.98 0.33 0.95 1.33 0.13 2 f 0_35a750_r4_loc_one 2.38 0.31 0.89 1.41 0.17 2 f 0_45a600_r5_loc_two 1.81 0.32 0.70 1.78 0.16 2 f 0_45a600_r5_loc_one 2.34 0.30 0.66 1.89 0.22 2

Table 4.3: Overview of runs we selected for the analysis. h=H1/H2 Troy said that if the thinner layer has akh <0.88and kaπ(kh)², then the wave celerity for non-linear waves internal waves will decrease as the amplitude increases [14]. We can see from A.4 that these conditions apply for us, and the measurements confirm a decreasing phase speed for increased amplitude.

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4.3. Reflections

(a) Frequency 0.25hz

(b) Frequency 0.35hz

(c) Frequency 0.45hz

(d) Frequency 0.55hz

Figure 4.3: 1.2 meters from the paddle. Two wavepackets plotted (soild dark cyan, dashed light cyan). The magenta line shows the time-interval integrated to find the energy.

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4.4 Back-flow

In [17], Weber and Christiansen point out that the theory assumes an infinitely long channel; however, this was not the case when we performed the experiments. As the waves are being generated, the induced drift has a net flux in the same direction the waves propagate. Because the tank has a limited length, a pressure gradient occurs at the far side, creating a backflow. This effect is evident when we look at the experimental results plotted in the fol- lowing subchapter. Rabault suggests in [10] that the backflow velocity profile would have a symmetrical parabolic shape going to zero at the surface and bottom. The magnitude of the parabolic profile was adjusted, so that when it was subtracted from theoretical drift, the net flux would become zero.

When we tried follow Rabault suggests in this experiment’s post-processing, but it did not give a drift profile that seemed reasonable. The parabolic profile size seemed to be too small, suggesting the theory was undershooting. We could have adjusted this with the measured flux, but we did not manage to track the entire column’s drift. Therefore, we tried different magnitudes, but none seemed to give a reasonable shape. Just looking at figure 4.5, there was some hope that the parabolic shape would fit in the oil layer, but in the lower water layer, it was apparent that it would never fit.

Going forward, we tested a different approach to remove the backwards flow.

We subtracted the theory from the measured drift in the top half of the column, added zero at the interface and surface, and fitted both a second (P2) and fourth-degree polynomial (P4). In figure 4.4 LS. we have plotted the mean drift for four different periods in the same run. P4 fits better to the backflow in the first period, and hits zero at the surface and the interface (y = 0and y= 0.1). P2 does not appear to resemble the backflow profile and misses zero by a little, both at the top and bottom. When we look at the later periods, we can see how P2 becomes a better fit as time progresses and gradually gets closer to zero around the interface. In the later periods, P4 continues to follow the backflow and hit zero, but the overall shape resembles an "s", and does not seem like a realistic representation of the backflow.

We implemented similar investigations for all runs presented. The pattern described above is found several times, especially in the 10cm oil layer at location two, both with the frequency of 0.35Hz and 0.45Hz. When we go to the field of view closer to the paddle, iP2 hits zero at the interface more sporadic.

This pattern becomes even more inconsistent in the experiments with 20cm oil.

Overall, we are more content with P2, because it does not overcompensate with a strange shape to hit the marks necessary. The effect we are looking for is close to the interface, so there is no immediate risk of the results being manipulated when we subtract a profile that is close to zero around the boundary layer. Looking at the figure 4.4 RS. we can see how the adjustment is minimal near the interface, while only 2cm into the fluid the differences between the three sets of data points are a lot larger. Thus, we have to be very careful to draw any conclusions in this area, even though we can see how the P2 adjustment gives a close agreement with the theory.

We do not propose that this is the actual shape of the backward flow. However,

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4.4. Back-flow

Figure 4.4: LS: Polynomial fit to back-flow. RS: Shifted to theory. Plotted period 1, 5 and 9 from top to bottom. Run f0_35a400_r2, FoV loc_two, layers 10x10cm

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it still serves to filter away "noise" when comparing experimental data to the theory, mainly when focusing on the effects close to the boundary layer. We will consistently use P2 to shift the drift, and show both the original and adjusted version.

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4.5. PTV results

4.5 PTV results

Figure 4.5: Showing full column, f = 0.35hz, a = 1.02cm , FoV location two, layers 10x10cm (f0_35a400_r2)

Frequency 0.35hz - 10cm oil, 10cm water

Weber and Christiansen suggested, in [17], to test the theory by generating a wave with a wavelength of 1 meter and amplitude 2cm. With a stabile

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shape at all layers, the longest waves we managed to create was the ones with a frequency of 0.35hz. With a 10cm oil layer, this created a wavelength of approximately 80cm. Looking at table A.4 we can see that kH₁ ¥ 0.75, which is not a lot smaller than 1, which means that even if we are close the recommended wavelength, we cannot say that we are well within the shallow water regime [17]. Although, Troy sets the limit for dispersive deep water waves atkH1, kH2'1.75[14] and we were not in this regime either. Looking at the 0.35hz waves we saw no visible signs of dispersion, and we are assuming we in an intermediate area leaning against long waves.

To enhance the non-linearities, we increased the amplitudes three times. We mainly looked at location two, where the waves were more developed, and we managed to fulfil a better PTV tracking. However, we did include one run from the field of view closed to the wavemaker, in order to get the largest amplitude.

The theory describes a wavetrain that suddenly "switches on". In reality, this does not happen, and there will be a transient wavefront before the waves are fully developed. In figure 4.5, we can see an example on the periods we chose to plot. The choice was based on two reasons: the wave had reached full amplitude, and the drift had started do develop close to the interface. For comparability, we start plotting the same wave period in all the different runs.

In general, it was problematic to prosses the data from the water layer. Usually, we did not get close enough to the interface to observe the effect we were looking for, and overall, the tracking did not give a consistent, repeatable result. Because these problems are not encountered in the oil layer, we are going to set our focus there and only show the full column once, in figure 4.5.

Amplitude 1.02 cm, FoV two (f0_35_a400_r2_loc_two)

We start by looking at the longest wave (⁄ = 83cm), with the smallest amplitude (1.02cm). With a steepness ak = 0.08, this is the run that most closely meets the assumptions of the theory. In figure 4.5 we see the both the oil and water layer plotted without any adjustments to the backwards flow.

The less viscous water layer is from the scaled depth of 0 to -1. From -0.15 to -0.9, the profiles are relatively straight vertical lines that start aroundu/c¯ = 0, and slightly decrease for each period. The exception being from the first and the second period, where the overall profile grew. At the very bottom, we see a positive drift that seems constant as time progresses. Looking at this area in raw data, we can see a jet occurring after the trough passed, going in the opposite direction of the rest of the flow. This jet explains the positive drift we see from -0.95 to -1. We found similar results in other runs we managed to track sufficiently close to the tank’s base. As this is not our focus, we will not look into the reason this jet appears.

At the top of the water column, we managed to track the flow to a little less than 0.05 away fromy/H₁= 0. Even this close to the interface, we did reach the drift’s theoretical maximum around -0.2. Still, we see a clear tendency that the drift is increasing as time passes. One exception is that the velocity seems to lessen in the 11. period, but this is only based on one data point, and can easily be explained as faulty tracking. Compared to the theory in this area, we see that the experimental results consistently undershoot the theoretical drift, but has the same development from one period to the next.

When we shift our focus to the oil layer, we see that the profile is very different.

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4.5. PTV results

(a) Profile adjusted for backflow

(b) Timeseries showing chosen periods

Figure 4.6: f = 0.35, a = 1.02 , FoV location two, layers 10x10cm (f0_35a600_r10)

The drift from 0.25 and above is almost a straight line around zero in the first period, but quickly the backflow increases and creates a parabolic shape. The steady rise in backflow stops at the 7th or 8th period and then alternates between turning weaker or stronger. We repeated this run with the same parameters, but discovered frame drops, so we cannot verify if this would happen again.

In the area below 0.25, the periods get an increasingly positive drift. The drift’s magnitude is much larger than what we observed in the water layer, even though we did not manage to track as close to the interface. This is consistent with what Weber and Christensen [17] said about how the drift at the interface would diffuse additionally into the more viscous layer.

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As we see in theory, there is a steady increase in mean velocity drift close to the interface, but it does not increase at the same rate. The shape seems to dip, suggesting a maximum value are being approached giving a good agreement with the theoretical profiles.

In figure??we have adjusted for the backward flow as described in section 4.4, and now the measurements follow the theory, except for the later periods where it slightly overshot closest to the interface. Because we do not know in what way the backflow affects the drift around the interface, we cannot say if this adjustment is for the better, regardless it highlights the observations already made about figure??.

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4.5. PTV results

(a) Unaltered drift

(b) Profile adjusted for backflow

Figure 4.7: f = 0.35, a = 1.56cm, FoV location two, layers 10x10cm (f0_35a600_r10)

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Amplitude 1.56 cm, FoV two (f0_35_a600_r10_loc_two)

We are now increasing the amplitude from 1.02cm to 1.56cm, and non- linearities gets stronger as the steepness ak increases from 0.08 to 0.12. The backflow develops in a similar way as in figure 4.5, where it gradually increases, before it starts shifting back and forth after the 7th to 8th period. When comparing to figure B.3, showing a repetition of the current run we are discussing, we see the same pattern. A possible explanation is that this paddle stops generating new waves around the 9. period.

We managed to track slightly closer to the interface compared to the previous run, but we do not see the same dip, indicating that we are approaching maximum drift. The 4th and 5th as well as the 9th and 10th period pair up and has the same drift, and this pattern is recognized in the repetition (figure B.3). Regardless, we can see a clear trend that the velocities are increasing as time progresses. We did expect this development, as shown in the theory plotted, but the measured results have a larger magnitude. In figure 4.7 we get a clear picture of how the measured results increasingly exceed the theoretical profiles, without the "noise" of the backflow.

Figure 4.8: Timeseries showing chosen periods. f = 0.35, a = 1.56cm, FoV location two, layers 10x10cm (Run f0_35a600_r10)

Amplitude 1.81cm, FoV two (f0_35a850_r2_loc_two)

Again, we increased the amplitude to 1.81cm, and the steepness ak becomes 0.14.

From the first glance at the figure 4.9, we quickly see that the drift is a lot greater than before. The backwards flow is being pushed upwards as the drift at the interface diffuses into the layer, but the magnitude’s development follows the same pattern as before.

The maximum drift we managed to track was a little less than 10cm/s. Two reasons for this is that it is more difficult for the PTV to match particles with large movements, relative to the seeding [9]. This is also why the amount of tracked particles decrease as the velocities grow. We could counter this by tweaking the input parameters given to digiflow. More important for the maximum trackable drift velocity is the size of the FoV. To track the drift we need to be able to fit the entire particle path in the picture. If the particle

Drift and generation of interfacial progressive waves in immiscible fluids