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Abstract and Figures

The surface topology of the scale pattern from the European Sea Bass (Dicentrarchus labrax ) was measured using a digital microscope and geometrically reconstructed using Computer Assisted Design modelling. Numerical flow simulations and experiments with a physical model of the surface pattern in a flow channel mimic the flow over the fish surface with a laminar boundary layer. The scale array produces regular rows of alternating, streamwise low-speed and high-speed streaks inside the boundary layer close to the surface, with maximum velocity difference of about 9%. Low-velocity streaks are formed in the central region of the scales whereas the high-velocity streaks originated in the overlapping region between the scales. Thus, those flow patterns are linked to the arrangement and the size of the overlapping scales within the array. Because of the velocity streaks, total drag reduction is found when the scale height is small relative to the boundary layer thickness, i.e. less than 10%. Flow simulations results were compared with surface oil-flow visualisations on the physical model of the surface placed in a flow channel. The results show an excellent agreement in the size and arrangement of the streaky structures. From comparison to recent literature about micro-roughness effects on laminar boundary layer flows it is hypothesized that the fish scales could delay transition which would further reduce the drag.
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Streak formation in flow over Biomimetic Fish Scale Arrays
Muthukumar Muthuramalingam1, Leo S. Villemin1,2, Christoph Bruecker1
1School of Mathematics, Computer Science and Engineering
City, University of London
London, United Kingdom EC1V 0HB
2Former Student, School of Life Sciences
University of Keele
Staffordshire, United Kingdom ST5 5BJ
The surface topology of the scale pattern from the European Sea Bass (Dicentrarchus labrax) was measured using a digital
microscope and geometrically reconstructed using Computer Assisted Design modelling. Numerical flow simulations and
experiments with a physical model of the surface pattern in a flow channel mimic the flow over the fish surface with
a laminar boundary layer. The scale array produces regular rows of alternating, streamwise low-speed and high-speed
streaks inside the boundary layer close to the surface, with maximum velocity difference of about 9%. Low-velocity
streaks are formed in the central region of the scales whereas the high-velocity streaks originated in the overlapping
region between the scales. Thus, those flow patterns are linked to the arrangement and the size of the overlapping scales
within the array. Because of the velocity streaks, total drag reduction is found when the scale height is small relative to
the boundary layer thickness, i.e. less than 10%. Flow simulations results were compared with surface oil-flow visualisa-
tions on the physical model of the surface placed in a flow channel. The results show an excellent agreement in the size
and arrangement of the streaky structures. From comparison to recent literature about micro-roughness effects on lami-
nar boundary layer flows it is hypothesized that the fish scales could delay transition which would further reduce the drag.
KEY WORDS: Fish scale, Streaks, Hydrodynamics
All bodies, which move through a surrounding fluid, will generate a boundary layer over its surface because of the no-slip
condition at the wall (Schlichting and Gersten, 2017). This boundary layer is a region of concentrated vorticity, which
shears the fluid near the body surface and the work done to shear the fluid is the measure of the energy which is spent
in locomotion (Anderson et al., 2001). The shear stress near the surface depends on the velocity gradient at the wall
and the type of boundary layer, which exist near the surface (Schlichting and Gersten, 2017). If the boundary layer is
laminar, the drag will be lesser, but it is more prone to separation at adverse pressure gradients, which increases the
pressure drag. A turbulent boundary layer produces more skin friction because of the additional turbulent stress near the
surface, however, it can sustain much stronger adverse pressure gradient which allows operating on off-design conditions
(Schlichting and Gersten, 2017). There is always a trade-off in design to maintain the initial boundary layer laminar
for the maximum extent so that the skin friction drag is lesser (Selig et al., 1995) and changing quickly to turbulent
boundary layers in areas which are prone to separation. For marine vehicles, one may overcome larger friction by modi-
fying the surface with a hydrophobic coating so that the fluid slips along the surface in contrast to the no-slip condition
of an uncoated one. As a consequence, the skin friction reduces which in turn reduces the net drag of the body (Ou
et al., 2004; Daniello et al., 2009). This technology was motivated by the lotus-effect, reviewed recently in (Bhushan and
Jung, 2006). This phenomenon helps in even self-cleaning of the dirt on the surface which could reduce fouling in the
marine environment (Bhushan et al., 2009). For large fast aquatic swimmer such as sharks, there has been numerous
experimental and computational studies on the skin denticles (Wen et al., 2014; Oeffner and Lauder, 2012; Domel et al.,
2018). Those were found to manipulate the near skin flow to reduce turbulent drag. However, there is very little work
on smaller and slower fish with laminar or transitional boundary layer and the role of different arrangements and pat-
terns of fish scales on their swimming behaviour and hydrodynamics. Up to now, there are only hypotheses about the
role of fish scale in hydrodynamics, reported in a recent article by (Lauder et al., 2016) who claimed also that there is
still no detailed proof of their hydrodynamic function. The scale morphology of bluegill sunfish was measured success-
fully with GelSight technology and hinted about the possible hydrodynamic uses of the scales (Wainwright and Lauder,
2016). Later, using the same technology the surface topography of various fish species was measured with and without
the mucus layer (Wainwright et al., 2017). Some physical characteristics of scales from grass carp (Ctenopharyngodon
idellus ) were measured and manufactured as a bionic surface and the first indication of drag reduction of about 3% was
reported (Wu et al., 2018). They claimed a water-trapping mechanism to be responsible for this reduction, mainly due
to flow separation behind the scales. No further details were given on the flow structure. In addition, the scales were
not overlapping but treated as individual elements. The present paper aims to reproduce the fish surface in a more real-
istic way based on statistics of scale measurements and reproduction of the overlapping scale array along the body. We
focus our studies on the European Bass, which is a fish commonly found in Mediterranean, North African and North At-
lantic coastal water regions. The fish scale pattern and array overlap are almost homogeneous over the length of the body.
Anteroposterior axis
Figure 1: Microscope images and CAD replication of Sea bass fish scales (a) Picture of full Sea bass fish. Point
S and E represents the region from anterior to posterior where we took measurements of the scales. (b) Top view of
the scales (c) Topographical view from scanning with the Digital Microscope (d) Top view of replicated CAD model (e)
Isometric view of CAD model (f) Photograph of 3-D printed model from top.
Fish Samples
European bass was collected from a local fishmonger (Moxon’s Fishmonger – Islington, London). In total, five individuals
were analysed from both sexes and had a total length of 33 cm and above. Sampling occurred from the pectoral region
to the caudal region at ten equally spaced intervals between point S and E as shown in Fig.1a. The skin of the fish was
cleaned repeatedly with a 70% ethanol solution to remove the mucus layer in order to analyse the scale surface under
the microscope. Immediately after cleaning, the scale samples were removed from the skin and placed on object slides.
The samples were then analysed with a digital microscope (VHX-700FE series, Keyence) using the 3D mapping feature
of the built-in software. This allowed to scan the 3D contour and to store the coordinates for later replication of the scale
surface in Computer Aided Design (CAD) software. The 2D images and the 3D topographical scan from the microscope,
the replicated CAD design and the 3D printed surface of fish scale array are shown in Fig.1 b-f. The physical model was
scaled 10 times larger than the actual size, a practical scale for experimental studies in the flow channel. Experiments
with up or down-scaled models are a common strategy in hydrodynamic and aerodynamic research based on the boundary
layer scaling laws explained in Appendix 2.
Computational Methodology
The computational domain and the boundary conditions are shown in Fig.2a. For comparison with the experiments, the
length-scale of reference herein is the same as for the 10-times up-scaled physical model. The dimension in ’x’ (antero-
posterior axis), ’y’ (dorsoventral axis) and ’z’ (lateral axis) directions are 250mm, 200mm and 80mm. The array of scales
is designed with 10 rows along ’x’ direction and 5 rows in the ’y’ direction. The scale height from the base varies both in
’x’ and ’y’ direction. Hence, the height of the scale at a given position P(x,y) is defined as h(P), whereas the maximum
height of the scale in the centreline (hs) is about 1mm, which corresponds to a 10-times enlarged value compared to the
measured value of 100 microns. At the inlet to the domain, a laminar Blasius-type boundary layer velocity profile with a
boundary layer thickness (δ) of 10mm was imposed. This profile can be approximated according to Pohlhausen (Panton,
2013) as a second order polynomial profile given by the Eqn.1.
δ) + B(y
δ(x) = 5·x
where A and B are the coefficients based on the free stream velocity (U= 0.1ms1) and the boundary layer thickness
(δ= 10mm) at the inlet (A = 2, B = 1). The boundary layer thickness given by Eqn.2 corresponds to a flat plate
Reynolds number of about Rexo = 33000 with an imaginary inlet length of xo= 333mm from the leading edge of a flat
plate until it reaches the inlet of the domain, where the Reynolds number (Rex) is defined by Eqn.3. Except for the floor
and the fish scale array, all the other side walls were specified with free slip conditions, i.e. zero wall-shear. The domain
was meshed with 18 million tetrahedral elements with 10 prism layers near the wall with a first cell value of 0.06mm.
To study the effect of the scale height relative to the boundary layer thickness on total drag, different boundary layer
thickness at the entrance were simulated. Therefore, the inlet domain was extended for 200mm upstream as shown in
Fig.2b and the boundary layer thickness at the new inlet was specified as 5,10 and 15mm. The problem was solved using
the steady state pressure based laminar solver in ANSYS Fluent 19.0 with a second-order upwind method for momentum
equation. Water was used as the continuum fluid in this CFD study with a density (ρ) of 1000 kgm3and a dynamic
viscosity (µ) of 0.001 kgm1s1.
Figure 2: Computational domain with boundary conditions. (a) Configuration of the fish-scale array in CFD
similar to the condition of the physical model of the scale array at the bottom wall of the wind tunnel. Note that
the velocity vector represents the inlet profile with free-stream velocity parallel to the ’x’ axis in positive direction
(Anteroposterior direction). ’y’ axis represents the spanwise (Dorsoventral direction) and ’z’ axis represents wall normal
direction. (b) CFD domain with symmetry conditions to simulate the drag variation with no end effects. In both the
figures ’xo’ is the imaginary length from the leading edge of the plate to the inlet of the domain.
Surface Flow Visualisation
The fish scale array with dimensions explained in the previous section was 3D printed with ABS plastic using Fused Depo-
sition Modeling (FDM)(Printing machine - Raise 3D). For manufacturing, the base layer thickness needed to be 4mm to
ensure stable handling. The model was placed on the floor of a wind tunnel (PARK Research Centre, Coimbatore, India)
in the test section (cross-section of 450mm and 600mm width). To reduce the disturbance of the step at the leading edge,
a chamfered flat plate (size 250mm x 200mm x 4mm) was placed upstream and downstream such that the region with
the scale array is flush with the wall. Surface oil-flow visualisation was performed with a mixture of Titanium-di-oxide,
kerosene and a drop of soap oil added to it to avoid the clustering of particles. More details of this visualization can
be found in (Merzkirch, 2012). Before starting the wind tunnel, the model was painted with the mixture in the region
downstream to the scale array. Thereafter, the tunnel flow was started to a free-stream velocity of 12ms1which gave
a boundary layer thickness of about 10mm at the entrance to the scales. Wind is transporting the dye according to the
local wall shear and a camera mounted on the top of the tunnel is capturing this process.
Velocity u/U
-15 -10 -5 0 5 10 15
Streak variation in % of U
y (mm)
z = 1.5mm
z = 2.5mm
z = 3.5mm
z = 4.5mm
z = 5.5mm
z = 6.5mm
z = 7.5mm
z = 8.5mm
z = 9.5mm
z = 10.5mm
0.0 510
(AST)Velocity variation in % of U
y (mm)
Figure 3: Velocity contour and velocity profiles (a) Normalised Velocity Contour at a wall-parallel plane at a distance
of z = 0.25δfrom the surface. The arrows indicate the flow direction. Note that the black arrows at the inlet are uniform
in length, while red and yellow arrows at the outlet differ in length. (b) Velocity variation in spanwise direction at various
wall-normal distances in the boundary layer. Scale array is shown in red color for better illustration. Blue line (Line-1)
represents a centreline of a row of scales. Green line (Line-2) represents the overlap region between the scales. Black
line represents a location in the ’x’ direction at 190mm from inlet. Location1 (P1) and Location2 (P2) are probe points
at 190mm from inlet on centreline region and overlap region. Black arrow indicates mean flow direction
Flow data obtained from the CFD results are first presented as velocity fields and profiles. Fig.3a shows colour-coded
contours of constant streamwise velocity (normalised with the free-stream velocity) in a wall-parallel plane at a distance
of 0.25δ. At the inlet, the velocity is uniform along the spanwise direction (’y’ direction), whereas, along the flow direction
over the scales, there is a periodic velocity variation in spanwise direction. Low-velocity regions have emerged in direction
of the centrelines of the scales, which is indicated with yellow arrows. In comparison, high-velocity region (Red Arrow
regions) are seen along the regions where the scales overlap each other. These high velocity and low-velocity regions are
referred in the following as streaks. These structures are linked in number, location and size with the overlap regions
along the dorsoventral axis over the surface.
Further information of the variation of the velocity in the streaks is demonstrated Fig.3b. It shows spanwise profiles of
the streamwise velocity at the location x=xo+ 190mm (8th scale in the row along streamwise direction from the inlet)
for different wall normal locations. At a wall normal location of 0.15δthe velocity variation is around 10% of Ubetween
the peak (local max) and valley (local min) in the profile. Given this difference, the streak amplitude is calculated using
the Eqn.4 from (Siconolfi et al., 2015).
AST =max
y{U(X, y, z )} − min
y{U(X, y, z )}(2U) (4)
As seen from the different profiles, the location of peaks and valleys do not change with wall normal position, therefore
the streaks extend over most of the boundary layer thickness in a coherent way. The streak amplitude AST is plotted
along the wall normal location in Fig.4 and it can be seen that the streak amplitude is maximum within the first 20% of
the boundary layer thickness with a value of 4.5% of U. As the distance from the wall increases, the streak amplitude
decreases monotonically until the displacement effect of the scales has died out at the outer edge of the boundary layer
(1.05δfrom the wall).
(AST ) Streak amplitude in % of U
0 1 2 3 4 5
Figure 4: Variation of streak amplitude along wall-normal direction at 190mm from inlet (refer black line
in Fig.3b
Figure 5: Surface flow visualisation picture from top view onto the scales. (a) Black arrow represents mean flow
direction. Note that the oil-mixture was painted in the region downstream of the scales to highlight the generation of
the streaks. 2-D top view of the CAD model is merged to get the impression of the arrangement of the scales. The red
arrows were added to illustrate the trace of the streaks relative to the arrangement of the scales. (b)Red arrow represents
the mean flow direction. Herein, the oil-mixture was painted directly onto the scales. Surface streamlines from CFD
simulation are overlaid to compare the results. Note that the regions of accumulated oil-patches match with the regions
of flow reversals from CFD simulation.
Experimental flow visualisation pictures of the streaks behind the fish scale array are shown in Fig.5a. As the particle
mixture coated on the surface moves according to the direction and the magnitude of wall-shear, the mixture moves
farther in the regions of high shear, than in regions of low shear. Therefore, the flow produces streaky patterns on the
surface with different length downstream of the scale array. This can be observed in Fig.5a from finger-like pattern of
which each finger represents a high-speed streak. The red lines depict the orientation of the streaks relative to the pattern
of the scale array. It is clearly seen that the high-speed streaks are formed in the overlap regions as previously claimed
from the CFD results. For better comparison with the CFD results, the surface flow visualisation over the scale array is
overlaid with surface streamlines from CFD (see Fig.5b), which is discussed later.
Figure.6 shows the variation of the normalised velocity profile at two locations along the span at the 8th scale row (probe
point location P1 and location P2 , compare Fig.3b). In the absolute coordinate system (Fig.6a) there is a shift in ’z’
direction because of the variation in the scale height h(P) along the span of the surface. When the profiles are plotted in
the body relative system (z=zh(P)) the difference along the wall normal direction (Fig.6b) becomes more obvious.
With the scales on the surface, the gradient of the velocity near the wall gets steeper in the location discussed here (at the
probe points P1 and P2). This is concluded from the comparison to the theoretical velocity profile for a smooth flat plate
(dashed black line, from Eqn.1). However, the boundary layer thickness is approximately the same. Hence, it could be
concluded that the scales change the profile shape inside the boundary layer region but would not change the boundary
layer thickness (nevertheless affecting the displacement and momentum thickness).
The surface streamline picture generated from the CFD results is shown in Fig.7a. In the centreline of the scales, the
flow mostly follows the direction of the main flow. Section X-X is enlarged and the cross-sectional flow in the centre of
0 0.2 0.4 0.6 0.8 1 1.2
0 0.2 0.4 0.6 0.8 1 1.2
z / δ
in mm
(z - h(p)) / δ(x)
Figure 6: Boundary layer profiles at Location 1 and Location 2 (probe points P1 and P2 in Fig.3b).
(a)Normalised velocity profiles in the absolute coordinate system. Note that the shift in velocity profiles along ’z’
direction is because of the change in scale height h(P) for the different probe points P (b)Normalised velocity in the body
coordinate system with theoretical boundary layer profile along a smooth flat plate.
the scales is shown in Fig.7b. It is seen that the flow follows the small slope caused by the tilt angle of the scale until it
separates from the sharp edge on the scales and reattaches further downstream at approximately 2.5 times the scale height
(hs) on the surface as a laminar boundary layer. This non-dimensional reattachment length is very similar to the value
reported in horizontal backward facing step flows if the Reynolds number defined with the step height and free stream
velocity is around 100 for the given flow situation(Goldstein et al., 1970). This separated flow region behind the step is
visible from the dividing streamline (shown as thick dotted line in Fig.7b). Also, from the surface flow visualisation, the
separated flow region behind the edge of the scales can be observed by the white patches due to the accumulation of the
particles (see Fig.5b). These white-patched regions match in size and locations with the flow reversal zones in the CFD.
When the fluid moves along the scales, the streamwise component of velocity is reduced in the central region of the scale
by the large separated zone as explained above. This causes a spanwise pressure gradient and forces the fluid to move
from the central region of the scales to the overlapping region. This movement is seen in the zig-zag pattern (shown in
blue arrows in Fig.7a) with larger spanwise components of fluid motion. The spanwise flow towards the overlapping region
produces a higher streamwise velocity because of mass conservation. This causes high-speed streaks in these regions. In
addition, it is evident that the flow reversal is reduced compared to the cross-section at the central region of the scales.
This is the root cause of producing low speed and high-speed streaks.
Figure.8 shows the surface streamlines on the scale array along with cores of intense vortices visualised by isosurfaces of
the ’Q’ value (Jeong and Hussain, 1995). The colours of the isosurfaces indicate the streamwise helicity which is defined as
(Uxx), where, ωxis the vorticity component along ’x’ direction. The yellow colour region defines the region in which the
vortex direction is Counter Clockwise (CCW) with respect to the ’x’ axis direction (i.e. mean flow direction represented
by a white straight arrow in Fig.8.), similarly, the blue colour region defines the vortex direction in Clockwise (CW).
It also displays the cross-flow velocity fields on planes parallel to the Y-Z plane near the scale overlap region for two
consecutive scales. The vortex in the central region of the scales (i.e. white colour vortex core) reflects the reversed flow
region behind the step. There the flow direction remains nearly aligned with the mean flow. In comparison, when the
flow moves downstream in the overlap region it is affected by successive vortices with alternating direction switching from
CCW to CW and vice versa. This causes the streamlines in the overlap region to generate a zig-zag pattern as already
illustrated in Fig.7a.
Reattachment line
Figure 7: Surface streamline and vector plots from CFD simulation over the scales. (a)Top view of surface
streamline over the scales. Note the zig-zag motion along the overlap region compared to the parallel flow at the central
regions of the scales. (b)Vector field in the x-z cross sectional plane along the central region of the scale at line X-X
(vectors indicate only direction and not magnitude). hs= 1mm and α= 3 degrees. S and R represents the separation
and reattachment of the flow streamlines. Thick dashed line indicates the region of recirculating flow with an arrow
indicating the direction of rotation. In both drawings the black arrow indicates the mean flow direction.
Figure 8: Surface streamline plot with direction of vortices. Vector plots at the overlap region for two consecutive
scale rows. Helicity coloured in yellow for positive (Vortex direction CCW) and blue for negative (Vortex direction CW).
Other rotational vectors are based on the colouring of vortices. Vortices are identified with ’Q’ criterion. White straight
arrow represents the mean flow direction.
0 50 100 150 200 250
x-xo (mm)
Line -1
Flat plate theory
Mid profile
0 50 100 150 200 250
x-xo (mm)
Line -2
Flat plate theory
Overlap profile
x-xo (mm)
0 50 100 150 250 250 0 50 100 150 250 250
x-xo (mm)
Flat plate theory
Line - 1
Mid profile Flat plate theory
Line - 2
Overlap profile
Figure 9: Variation of skin friction coefficient (Cfx )along ’x’ direction at two locations. (a)Blue line represents
the Cfx variation along (Line-1) (refer Blue line in Fig.3b) (b)Green line represents the Cfx variation along (Line-2)
(refer Green line in Fig.3b). Red lines represents variaion of ’z’ coordinate in ’x’ direction along corresponding locations
(not to scale). Black line represents the variation of Cf x theory for flat plate boundary layer by Eqn.5. xo(333mm) is the
imaginary length before the inlet of the domain.
Skin Friction and Total Drag
As previously mentioned, the scales modulate the near wall flow with streaks which will change the wall shear stress (τw)
distribution on the surface when compared with flow over smooth flat plate. To analyse this effect, skin friction coefficient
Cfx defined by Eqn.5 is plotted along the centreline (see Blue line (Line-1) in Fig.3b) together with the surface profile
variation in Fig.9a. In addition, the figure shows the profile of the theoretical skin friction coefficient (Cfx theor y) for a
smooth flat plate case, given in Eqn.6. Along the initial smooth part of the surface until 25mm the skin friction coefficient
follows the theoretical skin friction coefficient Cfx theory . As it enters the scale region, initially the skin friction drops
because of the adverse pressure gradient caused by the first wedge. Over the scale, it increases again because of the
local acceleration until it reaches the maximum at the edge of the scale. Then, Cf x drops to a negative value because of
the recirculation region explained in Fig.7b. Once the flow reattaches, the skin friction gets positive again and increases
until it reaches the peak as it approaches the edge of the next scale. This process repeats itself in flow direction with
the succession of scales. The same process happens in the overlap region, but here, for a single scale length, the process
happens twice because of two small steps formed by the adjacent scales in the lateral overlap region (note the difference
in the scale profile in the central region in Fig.9a and the scale profile in the overlap region in Fig.9b). Additionally, the
streamwise wall shear does not reach negative values in the valleys as there is no flow reversal in these zones. The shear
drag along the central region (determined by the integration of wall shear in the streamwise direction along Blue line
(Line-1) in Fig.3b) gives a 12% reduced value compared to the theoretical drag for a smooth flat plate. In contrast, the
overlap region (determined by the integration of wall shear in the streamwise direction along Green line (Line-2) in
Fig.3b) gives a 5% increase in shear drag. This tendency along the span correlates with the low and high-velocity regions
as the wall shear stress is directly proportional to the velocity gradient. The integral over the total surface leads to the
total friction drag, which is a net effect of the streaks. As we introduce a surface which is not smooth, the total drag is
the sum of the friction and the pressure drag. The latter depends on the wake deficit behind the step of the scale because
of the separated flow regions. Both need to be taken into account from the CFD results to investigate the net effect on
possible total drag reduction.
In order to investigate the relative contributions of friction and pressure drag over the skin, we varied the boundary layer
thickness (δ) relative to fish scale height (hs) as reported in Table 1. The inlet boundary layer thickness in the CFD
domain was increased in steps from δ= 5mm, 10mm and 15mm respectively with a free stream velocity (U) value of
0.1ms1. Drag coefficients were calculated using the drag force values obtained from CFD. The change in friction drag
and total drag coefficients is given in Eqn.7. The theoretical drag coefficient (Cd theory ) is calculated by integrating the
skin friction coefficient (Cfx theory ) along the ’x’ direction.
Cfx =τw
Cfx theory =0.73
Cdf (%) = (Cdf Cd theory )
Cd theory ×100 ∆Cd tot(%) = (Cd tot Cd theory )
Cd theory ×100 (7)
Table 1: Dependence of drag force with boundary layer thickness to fish scale height ratio
δ/hsCdp Cdf Cd tot Cd theory Cdf (%) ∆Cd tot (%)
5 0.000277 0.00448 0.00476 0.00453 -1.03 5.08
10 0.000193 0.00301 0.00320 0.00316 -4.68 1.43
15 0.000129 0.00214 0.00226 0.00236 -9.31 -3.84
For all the three cases the change in friction drag (∆Cdf (%)) relative to the smooth flat plate is negative indicating that
the scales are efficient in reducing skin friction. This effect increases with increasing boundary layer thickness to scale
height ratio. However, the total drag is only reduced for the third case (∆Cd tot =3.84%) when δ/hsratio is 15. This
is the typical ratio between the boundary layer thickness and the scale height in cruising conditions of the flow around
the fish and will be explained in the discussion.
In this paper, 3D microscopic measurements of the scales on the European bass fish are presented. Based on the data
statistics, a biomimetic scale array was replicated with the use of Computer Aided Design and 3D printing. The study
differs from previous ones on biomimetic scales (Dou et al., 2012; Wainwright et al., 2017; Wainwright and Lauder, 2017)
that it is the first for European bass and the first using a typical 3D curvature of the scales with an additional overlap
pattern. Flow over the scale array was analysed using Computational Fluid Dynamics and experimental results were
obtained from the surface flow visualisation. Excellent qualitative agreement was found, showing the formation of alter-
nating high-speed and low-speed streaks along the span, which concludes that the location, size and arrangement of the
streaks are linked with the overlap pattern of the scales. The experimentally validated CFD data further allows drawing
conclusions about the total drag of the surface, which is relatively difficult to obtain. The derived drag values show that
the overlapping scale arrays are able to reduce the body drag if their characteristic step height is sufficiently small (at
least one order of magnitude) compared to the local boundary layer thickness. If this conclusion holds for typical flow
conditions and size of the scale for European bass, the consequence would be a reduction of total drag, hence costing less
energy to the fish in cruising. In the following, we discuss the possible relevance of this finding to the situation of sea
bass in steady swimming conditions, including a critical review of the limitations of the study.
Mucus layer and transport:
Any mucus on the scales needed to be washed away for optical reasons before the scales could be measured in the
microscopy. It is known for similar fish species that the mucus only covers the microstructures of the scales such
as circulae and the ridges which connect the ctenii, therefore the overall shape of the scales is not affected by the
wash-out procedure (see also the conclusion by (Wainwright et al., 2017)). Thus the flow dynamics is representative
for the natural situation of the scales in the flow.
The observed recirculating flow near the central region of the scales might be helpful in retaining the mucus and
reducing the mucus secretion rate. This inference is supported from the fact that in the surface flow visualisation
experiments the mixture was largely trapped in these regions. This is comparable with the results on flow over
grass carp fish scales(Wu et al., 2018).
Swimming speed and Reynolds number:
The swimming speed of European bass is proportional to its body length (Carbonara et al., 2006). For the fishes
considered in this study, the swimming speed lies in the range from 1.2ms1and 1.4ms1corresponding to a
Reynolds number (calculated with the full body length L) in the range between 4 ×105and 6 ×105. This is in
classical fluid dynamics when transition from laminar to turbulent boundary layer flow sets in. As the reference
length is the tail end, we can conclude that the boundary layer over the sea bass for most of the body length remains
laminar. Direct measurements of the boundary layer on sea bass are not known so far, however, such data exist for
comparable fish such as a scup (Stenotomus chrysops), a carangiform swimmer, and rainbow trouts (Oncorhynchus
mykiss). Scup have mostly an attached laminar boundary layer over its body for most of the time and incipient
separation appears only for short time intervals in the swimming cycle (Anderson et al., 2001). PIV analysis on
swimming rainbow trout at a Reynolds number of 4 ×105revealed a laminar boundary layer with transition to
turbulence in the caudal region (Yanase and Saarenrinne, 2015). Hence, the laminar CFD analysis performed in
this study is representative for the effect of fish scales on typical European bass.
For the total drag of the biomimetic surface, a drag reduction was only observed when the scale step-height was
sufficiently small relative to the local boundary layer thickness (one order of magnitude). At a swimming speed of
1.2ms1and for a fish length of 300mm the boundary layer thickness is 1.5mm at the mid of the fish body (from
Eqn.2) measured from the snout of the fish to the begin of the caudal fin. In this region, the scale height measured
from the microscope was about 0.1mm which gives a boundary layer thickness to scale height ratio (δ/hs) of 15 and
has proven reduction in drag. Interestingly, the boundary layer thickness of scup is also in the same range discussed
here. Hence, the study shows, at least for steady swimming conditions, valid implications on total drag reduction
due to the presence of overlapping scale arrays.
Relevance of streaks in boundary layer transition:
In studies of a boundary layer flow over a flat plate it has been shown that placing arrays of micro-roughness
elements on the plate can delay transition (Fransson and Talamelli, 2012; Siconolfi et al., 2015). The effect of those
elements is that they produce low speed and high-speed streaks inside the laminar boundary layer, which delay the
non-linear growth of the Tollmien–Schlichting waves (Fransson et al., 2004). Although the mechanisms to generate
the streaky pattern might be different (lift-up mechanism of streamwise vortices versus alternating vortices in the
overlap regions), the fish-scale array producing streaks could also lead to the delay in transition.
To summarize, the biomimetic fish scale array produces steady low and high-speed streaks, which are arranged in spanwise
direction in the same pattern as the rows of the overlapping scales. Those regular arrangement of streaky structures are
known from flow studies on generic boundary layer flows to stabilize the laminar steady state and delay transition to
turbulence. As already mentioned, the swimming Reynolds number of the fish considered here lies in the transitional
range. Thus, we conclude that steady streaks similar as those observed for the biomimetic scale array are indeed produced
by the scales of fish and help to maintain laminar flow over the fish body. The presented biomimetic surfaces can be
engineered by purpose to reduce skin friction and delay transition in engineering application. However, this only refers
to steady swimming conditions. Undulatory motion of the body during active propulsion plays an additional role in the
boundary layer transition. Experiments with undulatory moving silicone wall in flow show an alternating cycle between
re-laminarization and transition in the trough and at the crest of the body wave (Kunze and Bruecker, 2011). As the fish
surface can also undergo bending motion, the overlapping scales can move relative to each other and deploy in regions of
strong curvature. From previous measurements of the boundary layer over swimming scup, it is known that the boundary
layer remains laminar for most of the body without flow separation even in the adverse pressure gradient region (i.e. aft
part of the fish) (Anderson et al., 2001). If the scales therefore also take part in any manipulation of flow separation is
still an open question (Duriez et al., 2006). From a technological perspective, artificial surfaces with scales can even be
built from flexible material, addressing also the issue of local flow separation.
The position of Professor Christoph Bruecker is co-funded by BAE SYSTEMS and the Royal Academy of Engineering
(Research Chair No. RCSRF1617\4\11, which is gratefully acknowledged. The position of MSc Muthukumar Muthura-
malingam was funded by the Deutsche Forschungsgemeinschaft in the DFG project BR 1494/32-1, which largely supported
the work described herein. We would like to thank Mr. Avin Alexander Jesudoss for providing help on performing surface
flow visualisation on the fish scale model.
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