NCMEH Research Project 2016

1

An experimental analysis of the effect of Capillary number on leading edge cavitation cell size

Dylan J Henness1

Abstract

An experimental hydrofoil cavitation set-up and procedure is utilised to analyse and explore the effects

of Capillary Number on the variation in the size of the cavity leading edge cell. This is undertaken

through the experimental modelling of viscid flow over a blunt-based section geometry centred on the

NACA 5-digit standard thickness distribution undertaken at the Australian Maritime College

Cavitation Research Laboratory, Launceston. The experimental study varies both cavitation number,

ranging from 0.25 to 0.52 and Reynolds number, ranging from 3.5∙106 to 7.0∙106 over a NACA 16-029

hydrofoil profile. This research is undertaken for the purpose of the greater understanding of global

cavitation physics including the transition to turbulence and the growth and collapse of cavity vapour

structures.

Keywords

Cavitation, Capillary Number, Saffman-Taylor, NACA, Cell Size, Hydrofoil, AMC

1 Introduction

Cavitation is the formation of vapour voids within a fluid body caused by the local static pressure falling below the

fluids vapour pressure. According to Bernoulli’s equation, this will often occur due to fluid acceleration around a

flow obstruction such as a hydrofoil or pump impeller. This cavity formation on the obstruction surface is also known

as ‘sheet’ or ‘attached’ cavitation and is further classified by its dynamic stability. If the formed cavity closes

downstream of the trailing edge, the cavity is deemed as ‘stable’ or ‘super’, featuring a relatively stable cavity length.

Instances where the formed cavity closes upstream of the trailing edge, the cavity is termed as ‘partial’, and often

exhibits a periodically varying cavity length as a result of the shedding of vapour voids within the body of the cavity.

The classification of a cavity is often determined by its Cavitation Number, a dimensionless value representing the

relationship between the difference in local absolute pressure, the vapour pressure and kinetic energy per unit volume.

Small divots are often present in the leading edge of attached cavitation causing the formation of cells perpendicular

to the direction of flow. Although very little known research has been conducted into cause and quantification of

these observed ‘cells’, their presence and structure suggests a possible relationship to the Saffman-Taylor instability.

The Saffman-Taylor instability represents the division of a lower viscosity, or ‘driving’ fluid body injected into a

1 Corresponding Author information:

Bachelor of Engineering (Ocean Engineering) (Honours)

email: dhenness@utas.edu.au

phone: (03) 6324 3999

NCMEH, University of Tasmania, Australian Maritime College

D. HENNESS - NCMEH Dissertation Proceedings 2015

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‘driven fluid’ of higher viscosity, caused by the morphologically unstable interface between the two fluids. The most

common example of this instability is air injected into corn syrup within a Hele-Shaw cell, shown below in Figure 1

is an illustration of this phenomenon. As shown, the driving fluid experiences ‘tip splitting’ as it is injected into the

driven fluid causing the formation of ‘fingers’ within the Hele-Shaw cell, this occurrence is more commonly known

as ‘viscous fingering’. It is this phenomenon that presents the possible link to the injection of vapour (driving fluid)

confined underneath the boundary layer, into the fluid body (driven fluid).

Figure 1: Viscous Fingering in a Hele-Shaw Cell, NetworkingCreatively (2014)

As the Saffman-Taylor instability is commonly studied with consideration to flow injection through a Hele-Shaw

cell, all current knowledge of the instability is founded based on the Hele-Shaw condition, thus for the comparative

analysis, the leading edge cavity condition will be explored and analysed against the current Hele-Shaw model.

The classic Saffman-Taylor model is based around the varying Hele-Shaw cell width, 𝑊ℎ, taken as the lateral cell

diameter of the predominantly circular or square test section, finger width, 𝑤𝑓, and the cell height, 𝑏ℎ, taken as the

spacing between the cell boundaries. For the analysis of the cavity leading edge cell with respect to the Hele-Shaw

model, characteristic finger width, 𝑤𝑓 cell width, 𝑊ℎ and cell height, 𝑏ℎ will be substituted with the leading edge cell

with, 𝑤𝑐, total cavity length, 𝑊𝑐, and the boundary layer displacement thickness at the leading edge cell, 𝛿∗,

respectively. The considered total cavity length and characteristic cell width are depicted below in Figure 1.

The classic Saffman-Taylor problem describes that as the driving fluid body is injected into the driven fluid by the

imposed pressure gradient, ∇𝑃 the initially uniform interface between the two fluids begins to destabilize, thus

warping the inner face of the cavity, it is this warping that causes the cavity body to split and divide, leading to the

𝑊ℎ

Figure 2: A hydrofoil cavity describing the characteristic cavity length, 𝑾𝒄 and cell width, 𝒘𝒄. Australian

Maritime College (2014)

𝑊𝑐

𝑤𝑐

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formation of the commonly known, ‘viscous finger’. In a steady state condition, a finger of width, 𝑤 is found to

occupy a fraction of the total cavity width as represented by the relative finger width expression, 𝜆 = 𝑤 𝑊⁄ .

For Newtonian fluids, the two dimensional velocity field, u averaged through the height of the sample fluid body is

given by Darcy’s Law with consideration to the Hele-Shaw condition, expressing the relationship between pressure

gradient and velocity within the fluid as shown in Equation (1), (Chevalier, Amar, Bonn, & Lindner, 2005).

2

12

hbu p

(1)

Where µ denotes the ‘dynamic’ or ‘absolute’ viscosity of the injected fluid.

Following Equation (1) assuming an incompressible flow, the pressure field satisfies Laplace’s Equation,

0p (2)

The pressure field is calculated within the injected fluid including a pressure rise over the interface due to the

interfacial tension between the two fluids expressed as,

p R (3)

Where 𝜎 is the interfacial tension between the two fluids, with 𝑅 denoting the radius of curvature of the interface,

again assuming two-dimensional flow as justified in the limits of small Capillary Numbers Leger and Ceccio (1998).

The final boundary condition states the assumption of equal normal velocity at either side of the fluid interface with

𝑛 denoting the normal vector and 𝑈 the normal velocity at the outer boundary adjacent to the interface.

U n u n (4)

Solving the boundary conditions (1), (2) & (4) the finger shape and width can be solved for a given pressure gradient.

Many quantitative studies into the instability focus on the width of the finger, 𝑤 relative to the Hele-Shaw cell width,

𝑊ℎ as a function of the fluid injection velocity. It follows from Equations (1), (2) & (3) that the finger width is

bounded by the relative effect of viscous forces to surface tension across the liquid/gas interface, also known as the

Capillary Number, 𝐶𝑎. Equation (5) describes the general form of Capillary number with Equation (6) denoting the

Hele-Shaw modification for Capillary number from Kondic (2014).

Re

WeCa

v (5)

2

2

12Hele Shaw

h

URCa

b

(6)

Where 𝐯 denotes the characteristic velocity. Weber number, We and Reynolds number, Re are defined as follows.

Rel

v (7)

2lWe

v (8)

Where 𝜌 denotes the fluid density and l the characteristic length.

From Equation (6) above it is clear that the radius of curvature of the interface, and thus relative finger width is

governed by Capillary number, and is therefore expected that finger width and thus leading edge cell size will

decrease as velocity is increased. This prediction is validated from various experimental observations in Hele-Shaw

cell injections undertaken by Tabeling, Zocchi, and Libchaber (1988) and Chevalier et al. (2005).

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When inertial forces within the fluid body are considered in Darcy’s Law, both Reynolds number and Weber number

take effect, therefore an inertial correction must be made for the applicability of this problem.

In succession from Equations (1) & (3) Chevalier et al. (2005) predicts the modification of the Capillary number to

form the universal control parameter for the fingering problem, 1/B allowing the aspect ratio of various systems to

fall on a universal curve. This classical Saffman-Taylor control parameter is expressed as,

21

12 ar CaB (9)

Where, 𝑟𝑎 denotes the aspect ratio of the cell, defined as 𝑟𝑎 = 𝑊ℎ 𝑏ℎ⁄ .

Ruyer-Quil (2001) proposed an inertial correction to Darcy’s law originating from the three-dimensional Navier-

Stokes equation including a polynomial approximation of the velocity field. By averaging inertia in the third

dimension, Ruyer-Quil (2001) expressed the two-dimensional modification of Darcy’s Law as,

2

12uu u p

t b u

(10)

Where, 𝛼 = 1.2 𝑎𝑛𝑑 𝛽 = 1.5429.

*It should be noted that values for α and β may vary depending on how inertia is averaged in the third dimension.

Following Equation (10), by scaling length on characteristic cell width 𝑊ℎ, time on 𝑊ℎ 𝑈⁄ and pressure on

12𝑈𝜇𝑊ℎ/𝑏ℎ2, Plouraboué and Hinch (2002) derived the following dimensionless relationship,

*Re

uu u p u

t

(11)

Where,

* 1Re Re

12 12

b Ub b

W W

(12)

𝑅𝑒∗ denotes a Reynolds number modified in the same manner as the modified Capillary number previously described

in Equation (9).

Plouraboué and Hinch (2002) also proposed a similar modified Weber number, 𝑊𝑒∗ expressing the relative effect of

inertia and interfacial tension by combining the modified Reynolds number and Saffman-Taylor control parameter

expressed as,

2* * 1

ReU W W

We WeB b

(13)

2 Methods

The Australian Maritime College, Launceston Cavitation Research Laboratory (CRL) was used as the test facility for

the conducted research. The CRL tunnel is a 365m3 variable pressure water tunnel for the study of cavitation and

viscous flows, (Australian Maritime College, 2015). In order to control the dissolved gas content within the tunnel,

a rapid degasser is equipped with microbubble injection capable of reducing dissolved gas content within the tunnel,

OSI % to 20% of saturation at atmospheric pressure within 2 hours. The tunnel is capable at running at absolute

pressures ranging from 4 to 400 kPa and velocities up to 12 m/s through a 0.6 x 0.6 x 2.6 meter test section.

The transparent test section allows for the visual analysis of such testing including the quantitative analysis of leading

edge cavity cell size and the physics of cavity growth and collapse. The tunnel is also fitted with an array of injectors

that pierce the honeycomb from which generated nuclei are produced for the purpose of controlling the nuclei spectra.

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The produced nuclei sizes typically range from 10 to 100 µm with a concentration varying from 0.1 to 10/cm3. The

produced nuclei are later removed in a downstream tank through the process of coalescence and gravity separation,

this separation tank also has the capability to remove up to 200 litres of non-condensable gasses per second during

operation.

2.1 NACA0 16-029 Foil

For the purpose of theory validation, a model hydrofoil was tested varying both Reynolds and Cavitation number at

the Australian Maritime College, Launceston’s Cavitation Research Laboratory. Due to both its availability within

the facility and reputation through past studies, the NACA 5-digit foil profile was selected. The NACA 5-digit series

was generated using analytical equations that describe both the curvature of the mean-line and the thickness

distribution along the length of the air foil. The hydrofoil section geometry used is based on the NACA 5-digit

standard thickness distribution. As for the purpose of cavitation induction, no lift generation will be required thus a

foil shape with a symmetrical profile and no chamber at a 16% thickness to chord ratio was selected.

Figure 3: NACA 16-029 generated foil points, AirfoilTools (2015)

2.2 Experimental Procedure

2.2.1 Set-up and Calibration

The NACA 16-029 test foil was installed in the upper window of the cavitation tunnel test section at the longitudinal

and transverse centre under a semi-filled water level condition. Following the initial installation of the foil, under a

fully-filled condition, all air is bled from the foil connection cavity within the test section wall under a low velocity,

atmospheric pressure control state to ensure that no air is present within the cavity as it will likely be drawn into the

freestream under low pressure operating conditions.

Prior to each testing session, the CRL tunnel was degassed ensuring dissolved gas saturation within the test flow is

reduced below a limiting value of 3.5%. It should also be noted that all testing conducted with respect to this document

was undertaken with a maximum allowable dissolved gas saturation within the test flow of 3.5%. Following the

degassing of the tunnel prior to testing, all free stream monitor lines for velocity, pressure, density etc. were purged

to ensure accurate working measurements of the test condition.

Following the execution of each testing procedure, the foil was uninstalled and removed from the tunnel test section

to allow a calibration ruler to be fitted. The calibration ruler was inserted into the test position of the foil and

photographed for the post process scaling of image data. The scaled calibration ruler and operating test foil are shown

below in Figure 4.

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2.2.2 Procedure

Prior to any testing procedure conducted within this document, it should be noted that initial tunnel conditions are as

per described in section 2.2.1 Set-up and Calibration.

The experimental test procedure is summarised in Table 1. When transitioning between test states, flow stabilisation

prior to data acquisition is imperative, thus free stream velocity and pressure were closely monitored and amended

during these periods to avoid target overshooting. Upon the stabilisation of flow states, the leading edge cavity cell

size was initially observed to randomly distribute along the cavity leading edge, soon converging towards a common

value via the process of cell division. Prior to data acquisition, once the free stream test flow is deemed to have

stabilised at the desired state, the visual convergence of the leading edge cell size distribution is essential for the

consistent recording of results. Figure 5 shows the cell division during the initial convergence stages following flow

state stabilisation.

Due to both time constraints and limited image storage space due to the requirement of high image resolution for the

purpose of batch processing and cell analysis, a limiting value of 200 still images and five intervals of two second

high-speed imaging per test state were obtained.

The test states selected for analysis were chosen based on their cavity classification as described in Section 1, with

an equal number of partially cavitating and super cavitating states. The selected test conditions for analysis are shown

below in Table 1. The test states selected for analysis were chosen based on their cavity classification as described in

Figure 4: Shows the calibration ruler (Left) and the operating test foil (Right).

Figure 5: Shows a partially developed leading edge cavity highlighting the dividing cells in the initial

convergence stages

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1, with an equal number of partially cavitating and super cavitating states. The selected test conditions for analysis

are shown below in Table 1.

Table 1: Testing schedule for ‘still’ and ‘high-speed’ data acquisition

V, [m/s] P, [kPa] σ Ret ∙ 106 OSI % T, °C ρ, [kg/m3] µ, [N. s/m2]∙ 10−3 k, [m2/s]∙ 10−6

6.16 11.2 0.52 3.5 3.355 17.82 998.658 1.06849 1.070054

6.2 6.1 0.25 3.5 3.208 17.67 998.685 1.073065 1.074599

9.27 23.5 0.52 5.25 3.352 17.76 998.669 1.07032 1.071872

9.29 12.3 0.25 5.25 3.2 17.67 998.685 1.073065 1.074599

12.36 41 0.52 7.0 3.248 17.69 998.682 1.072455 1.073993

12.39 20.7 0.25 7.0 3.144 17.62 998.694 1.07459 1.076114

*With regards to Table 1 only it should be noted that, 𝑉 and 𝑃 denote velocity and pressure in the free stream

respectively, and the corresponding values are the average over the specific test condition period. During testing,

Cavitation and Reynolds numbers were observed to fluctuate a maximum of ±5.6% of the mean value shown in

Table 1 due to variations in velocity and pressure in the free stream.

Data Acquisition System

For the analysis of cavity growth and shedding, a NACA 16-029 foil was utilised with a chord length, C of 0.14 m

corresponding to a 0.029 m total thickness, 𝑡 employing the testing regime shown above in Table 1. The still images

of the visual cell size data were documented using a Nikon D800E digital camera with a 105 mm lens and 36.3

effective megapixels mounted externally at foil mid span, perpendicular to the direction of free-stream flow. The

high-speed imagining utilised to analyse the cavity growth and shedding was undertaken using a LA Vision High-

speed Star 5 1024 x 1024 pixel resolution with a Nikon AF Nikkor 50mm 1:1.8D lens.

The foil itself was artificially illuminated during both high-speed and still imaging conditions. During high-speed

imaging two upstream spotlights perpendicular to the direction of flow, above and below the camera were employed

with an upward facing constellation LED downstream of the foil as shown in Figure 6. During still imaging two

strobing lights were utilised beneath the foil, upstream and downstream to ensure total cell wall illumination also

shown in Figure 6. It should be noted that ‘strobing’ lights were used for all still imaging as to eliminate the blurring

of images taken at insufficient shutter speeds. As the tunnel test section is elevated, each camera and light was

individually mounted to a fixed tripod to ensure a constant image framing through all testing regimes. The high-speed

and still imaging set-ups are shown below in Figure 6.

Figure 6: Shows the high-speed imagining setup, (Right) and the still imagining setup,

(Left).

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3 Results and Discussion

3.1 Boundary Layer Analysis

In order to produce, prepare and analyse the proposed testing regime for the foil, a range of boundary layer conditions

required establishment. A 2D Panel Code developed by Brandner (2016) modified from Katz and Plotkin (2001) was

utilised to predict the Pressure Coefficient, 𝐶𝑝 distribution along the foil with a user input foil shape specified as

NACA generated points, AirfoilTools (2015). The resultant pressure coefficient plot, as shown below in Figure 7 is

then transformed into the pressure distribution along the foil with the implication of Equation (14).

21

2

p pCp

v

(14)

Where subscript ∞ indicates the freestream component.

The adverse pressure gradient featured in Figure 7 suggests a point of laminar boundary layer separation at the

inflection point of Cp at 𝑥 = 0.0615𝑚. It is predicted that this point of boundary layer separation will induce a cavity

confined within the boundary layer displacement thickness. This is in turn what is observed experimentally from the

observations of Sampson (2008) and Franc and Michel (1988).

Utilizing the pressure coefficient distribution, 𝐶𝑝 shown above in Figure 7, the boundary layer displacement thickness

is estimated by employing Thwaite’s method, Gerhart and Gross (1985). Thwaite’s method for laminar boundary

layers with pressure gradient is an empirical method based on the assumption that all boundary layer flows are bound

by the following relationship,

2

2e eu dudA B

v dx v dx

(15)

Where 𝑢𝑒 denotes edge velocity at the stagnation point and x the horizontal position along the foil downstream of the

leading edge. Thwaite’s recommends 𝐴 = 0.45 and 𝐵 = 6.0 in Equation (15) from quantitative analytical methods.

By method of integration, Equation (15) becomes,

6

2 5 2 5

6 6 6

0 0

00.45 0.450

x x

e

e e

e e ex x

u xv vu dx x u dx

u u x u

(16)

Figure 7: Pressure Coefficient, Cp distribution over the NACA 16-029 foil, Brandner (2016) indicating an adverse

pressure gradient

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*It should be noted that Thwaite’s method assumes edge velocity, 𝑢𝑒 at the stagnation point is equal to zero for blunt

nosed foil shapes and momentum thickness at the stagnation point, 𝛷0 is also equal to zero for all foil shapes.

Once the momentum thickness at the area of interest, 𝑥0 is known, the dimensionless pressure gradient parameter,

𝜆1 can be obtained,

2

1

dU

v dx

(17)

This dimensionless pressure gradient parameter can then be used to calculate the shape factor, 𝐻 from the following

relationships,

2

1 1 1

1

1

1

2.61 3.75 5.24

0.07312.088 0;

0.15

H

(18)

The boundary layer displacement thickness is then calculated as follows,

*

1H (19)

Although Equation (19) represents boundary layer displacement thickness for the laminar boundary layer separation

over the foil, the proportionality of the numerical model can be partially verified by plotting the boundary layer

thickness vs. Reynolds number for laminar flow over a flat plate as predicted by Cimbala (2015) where 𝑥 denotes the

location of interest along the plate, taken as the expected point of cavity formation, 𝑥 = 0.0615𝑚 as previously

described. The boundary layer displacement thickness estimation derived by Cimbala (2015) is expressed below in

Equation (20).

*

0.5

1.72

Rex

(20)

Figure 8: Reynolds number vs. Boundary layer displacement thickness as per Gerhart and Gross (1985) for the

selected test conditions shown in Table 1

Where the Reynolds number of the cavitation research tunnel, 𝑅𝑒𝑡 is expressed as,

Re tt

vD

(21)

𝐷𝑡 denotes the characteristic test section diameter. The Cavitation number of the flow regime is calculated as,

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21

2

r vp p

v

(22)

From the observations conducted by Bakker (2006) the increased steepness of the Thwaite’s predicted curve in

comparison to the flat plate estimation shown in Figure 8 is expected due to the increase in flow turbidity associated

with a foil based profile, therefor suggesting the produced Thwaite’s boundary layer displacement thickness

estimation will act as a satisfactory monitor point for this analysis.

3.2 Image Processing

For the efficient examination of the recorded images, a MATLAB code was produced employing the following cell

size analysis procedure:

- The initial image is cropped to a portion exhibiting minimal tip effects throughout the considered image set.

This section was taken at the centre foil span.

- The image contrast is then increased, weighting to darker intensities as to segregate cell edges.

- An image ‘erosion’ is then performed, returning the value of every pixel to the minimum value of its

neighbouring pixels within the image array, creating a consistent cell edge definition.

- The image is now dilated, returning the value of every pixel to the maximum value of its neighbouring pixels,

thus removing imperfections in the cell edge.

- Following the reconstruction of the dilated image, a complementation is performed, producing the negative

of the original image.

- By applying a ‘sobel’ filter to the dilated image, the image returned features detected edges based upon a

user specified threshold value. As the utilised NACA 16-029 foil featured minor imperfections, the threshold

value specified for sobel edge detection was minimised as to avoid the detection of surface scratched within

the foil material.

- Due to the limited sobel threshold, a follow up image dilation was performed for the further definition of the

detected cell edges.

- A binary gradient mask is then applied to the image removing linear gaps between the dilated image pixels

at 0 and 90 degrees emphasizing vertical and horizontal lines respectively within the detected edges.

- By morphologically closing the image, the partial horizontal and vertically produced lines are joined to form

complete cell edges.

- An ‘imfill’ function is then utilised to flood any contained pixel regions, thus filling any holes within the

cell wall.

The complete filtering process described above is shown below in Figure 9.

*It should be noted that the position along the foil span at which cell size is measured is manually fluctuated

depending on the test condition due to the shift of the dynamic cavity structure between test states.

Figure 9: The complete image filtering process undertaken by the MATLAB processing

code.

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Once the cell boundaries have been clearly defined, the individual cell sizes are measured through the implication of

a MATLAB ‘peaks’ analysis of the image intensity plot. Figure 10 shows the image intensity plot of the final filtered

image.

Once the intensity array of the image has been established, a ‘maxima’ or ‘peaks’ MATLAB function was utilised to

determine the lateral spacing between intensity peaks. As highlighted in Figure 10, due to the general cell structure

of the leading edge cavity, the intensity plot was often comprised of multiple peaks, in order to prevent the ‘peaks’

analysis detecting multiple peaks within a single cell wall, the minimum allowable peak prominence was defined as

approximately at 50% of the minimum observed cell size from each data set. Due to the detection of the upstream

face of the leading edge cells, minor peaks are present adjacent to each cell wall, to ensure these are not detected as

cell boundaries, the minimum allowable peak distance is defined at approximately 50% of the minimum peak value

for each set.

Following the establishment of the cell size distribution, the attained data is then filtered to remove any values outside

of two standard deviations either side of the local mean. This eliminates the incuded analysis of any multi-peak cell

walls outside of the defined minimum allowable peak prominence and the merging of cells beneath the minimum

allowable peak distance threshold.

As highlighted in Figure 11, the leading edge cavity has shifted downstream from the upstream foil edge with the

decrease in free stream velocity, this is expected and occurs due to the increased resistance to flow separation as the

boundary layer becomes sequentially more turbulent at the cavity leading edge with the increase in momentum

transport due to turbulence from the free stream flow, (Bakker, 2006). Due to this variation in leading edge cell

position, for consistent data acquisition, it was imperitive that the image cropping process described in Section 3.2

be constantly monitored and anemded during its implication.

Figure 11: Highlights the downstream shift of the leading edge cavity at 𝑹𝒆 = 𝟑. 𝟓 ∙ 𝟏𝟎𝟔, 𝝈 =𝟎. 𝟓𝟐 (Bottom), and 𝑹𝒆 = 𝟓. 𝟐𝟓 ∙ 𝟏𝟎𝟔, 𝝈 = 𝟎. 𝟓𝟐 (Top)

Figure 10: The filtered cell size image (Left) and the image intensity plot (Right) highlighting

multiple peaks within the defined cell wall

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3.3 Results Analysis

Figure 12 shows the filtered average leading edge cell size for the test conditions displayed in Table 1. From the

preliminary analysis, it is evident that the average leading edge cell size shares an essentially linear relationship with

the variation in Thwaite’s calculated boundary layer displacement thickness.

Figure 13 shows the relative cell width against the Saffman-Taylor control parameter 1/B as described in Equation

(9) varying both the Reynolds number and Cavitation number of the free-stream flow. Figure 13 clearly shows that

for low values of 1/B, the decrease in relative finger width as parameter 1/B is increased. From the experimental

observation of the Saffman-Taylor finger width within a Hele-Shaw cell conducted by Chevalier et al. (2005), the

decrease in relative finger width with increasing 1/B was clearly observed for low values of control parameter 1/B.

Although Chevalier et al. (2005) also consistently observed the gradual increase in the relative finger width for large

values of 1/B past a point of inflection often characterised by the specific Hele-Shaw cell height where all condition

curves tend towards a common ‘master curve’, this phenomena is evidently absent in the collated data presented in

Figure 13 due to the limited data sample provided, however a similar steadily increasing variance between varying

test condition curves is clearly evident in Figure 13 as also observed by Chevalier et al. (2005).

Figure 14 displays the relative cell width, λ against the Saffman-Taylor modified Reynolds number, 𝑅𝑒∗ as described

in Equation (12). It is visually evident that similarly to Figure 13, the relative leading edge cell width shown in Figure

14 is not characterised by the modified Reynolds number for low values of 𝑅𝑒∗, however begins to tend towards a

common trend approaching values for 𝑅𝑒∗ in excess of the data range considered. This tendency to progress towards

a single curve for high values of 𝑅𝑒∗ is also apparent in the investigation of inertial effects on the Saffman-Taylor

instability carried out by Chevalier et al. (2005). This consistent correlation suggests the possible quantification of

the cavity leading edge cell size through implication of the Saffman-Taylor instability.

Figure 12:Shows the filtered average cell size against the Thwaite’s boundary layer displacement thickness

Figure 13: Represents the cell finger width, λ vs. the Saffman-Taylor control parameter, 1/B

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Figure 14: Displays the relative cell width, λ vs. the Saffman-Taylor modified Reynolds number, 𝑹𝒆∗

Figure 15 represents the relationship between relative cell width, λ and the Saffman-Taylor modified Weber number,

𝑊𝑒∗ described in Equation (13). From Figure 13 and Figure 14 it is observed that the leading edge cell width is

governed by two limiting factors, for high velocity conditions modified Reynolds number, 𝑅𝑒∗ tends towards a

common curve, and for low velocity conditions, control parameter 1/B exhibits a similar trend. The crossover between

these two regimes can be characterised by the combination of the modified Reynolds number, 𝑅𝑒∗ and the Saffman-

Taylor control parameter, 1/B as described in Equation (13).

From the visual analysis of Figure 15, it is evident that a consistent variance between the two test condition curves

is present, thus although only limited data is presented, the initial trends suggests a similar inclination to the

observations of Chevalier et al. (2005).

Figure 15: Presents the relative cell width, λ vs. the Saffman-Taylor control parameter, 𝑾𝒆∗

Although modified Weber number, 𝑊𝑒∗ governs the crossover between the two regimes, due to the limited data

sample provided, the point of crossover between the two regimes cannot clearly be identified. However, from the

initial trend shown in Figure 15, the relative cell width appears to rescale with cavitation number proposing that

independent of flow regime, modified Weber number will not characterise relative cell width.

Chevalier et al. (2005) suggested a control parameter considering the crossover point between the two regimes of

𝑊𝑒𝑐∗, tending towards 1/B for 𝑊𝑒∗ < 𝑊𝑒𝑐

∗ and (𝑊𝑒𝑐∗ 𝑅𝑒∗⁄ ) for 𝑊𝑒∗ > 𝑊𝑒𝑐

∗ expressed as,

* *

11 ' 1

1 c

B BWe We

(23)

D. HENNESS - NCMEH Dissertation Proceedings 2015

14

Figure 16: Expresses the Chevalier et al. (2005) control parameter 1/B’ fitting classical 1/B control parameter

curve utilizing a crossover point of, 𝑾𝒆𝒄∗ = 𝟕 ∙ 𝟏𝟎𝟕

Although Figure 16 presents a limited data sample, with comparison to Figure 13 variance between conditions

remains moderately consistent and minimal suggesting that the modified control parameter 1/B’ could present a

universal curve capable of bounding relative cell width for varying flow regimes, with an estimated crossover point

of 𝑊𝑒𝑐∗ = 7 ∙ 107 from Equation (23).

4 Conclusion

From the experimental analysis of the leading edge cavity cell size distribution varying both Reynolds and Cavitation

number, with comparison to the classical Saffman-Taylor fingering problem, average cell size evidently increases

with the increase in boundary layer thickness at a virtually linear rate. It is clearly observed that as velocity is

increased, and sequentially boundary layer thickness is decreased, the gradient of the relative finger width tends to

zero suggesting an inflection point to which a second regime is apparent where relative finger width will begin to

increase. The visual analysis of Figure 13 and Figure 14 suggests that the explicit behaviour of these two regimes is

characterised by control parameter 1/B for low velocities, and modified Reynolds number, 𝑅𝑒∗ for high velocities as

also observed by Chevalier et al. (2005). This is expected as considering the relative influence of internal forces on

velocity, capillary forces scale as 𝑽0, viscous forces as 𝑽1 and inertial forces as 𝑽2, therefore the predominant

influence at low velocities will be capillary forces (1/B), and the ratio of inertial to viscous forces (𝑅𝑒∗) at high

velocities.

By introducing the modified weber number, 𝑊𝑒∗ including both the classical Saffman-Taylor control parameter, 1/B

and the modified Reynolds number, 𝑅𝑒∗ the crossover between these two regimes was modelled displaying a

satisfactory correlation to the Hele-Shaw model observations of Chevalier et al. (2005). Although an insufficient data

range was supplied, by employing the modified control parameter 1/B’, the fitting of the initial curve predicted a

crossover point of 𝑊𝑒∗ = 7 ∙ 107.

The initial results presented clearly show that boundary layer thickness acts as the major influence governing the

relative cavity leading edge cell width, λ indicating cell division due to an unstable interface between the vapour

cavity and boundary layer interface as a result of Capillary effects, suggesting a relationship to and possible

quantification through the implication of the Saffman-Taylor instability.

It is recommended that a larger data sample be analysed to verify the presence of the inflection point defining the

regime crossover to confirm the applicability of the modified control parameter 1/B’ and in succession, the presence

of the Saffman-Taylor instability.

Acknowledgements

The author would like to extend his sincere appreciation to the following for their support and encouragement

throughout the duration of this research; Post-Doctoral Fellow Dean Giosio, for his crucial guidance, patience and

D. HENNESS - NCMEH Dissertation Proceedings 2015

15

countless hours of assistance throughout the entire duration of this project. Supervisor A/Prof. Paul Brandner, for his

profound knowledge and collaboration throughout the course of this work. Above all, to my family for their endless

support throughout my academic endeavours. Finally, to the AMC Engineering class of 2016, for their constant banter

and fuckwithery to which this paper would not have been made possible nor bearable without.

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