Robert A. Jones
AEM 620
Final Project
12/15/09
Computational Fluid Dynamics Modeling of Butterfly Scales as
Asymmetrically Shaped Cavities with Lid-Driven Flow
Introduction
There is currently a large amount of interest in reducing energy costs of large
aircraft and ships and a clear avenue for this is in minimizing drag and separation
over the surface of these vessels. Several geometries have been investigated, such
as the dimples used on golf balls, stream- or span-wise channels embedded in the
surface, and even biomimetic surfaces. This study falls in the last category and
specifically seeks to analyze the effect of a range of Reynolds numbers on cavities
shaped like those formed by butterfly scales. A microscopic view of a butterfly wing
is shown in Fig. 1, and by taking one of these cavities and modeling it in Gambit we
arrive at the geometry seen in Fig. 2.
Figure 1.
D. plexipus (Monarch)
butterfly wing cross section at 200x
magnification. Flow goes from left
to right.
Figure 2.
Simplified cavity modeled
after Monarch scales.
Using Fig. 2 as a model of a single cavity, the flow over the butterflies’ wing
can be roughly approximated by moving the upper surface, or lid, of the cavity at a
constant velocity to the right. In this study, the viscous and pressure forces on the
cavity walls are measured for several values of Re<100. The upper limit of Re=100
is chosen both to maintain laminar flow within the cavity and because higher Re
which correspond with higher free stream velocities are not physically
representative of conditions the butterfly is exposed to. The reason for studying a
range of Re as opposed to a specific value is that though we know butterflies fly at
speeds on the order of 1 m/s, and the depth of the cavities on their wing is roughly
100 µm, resulting in a Re = 8 at standard atmospheric conditions, where Re is given
by
Re 
VL

(1)
the actual speed of the flow as it passes over any given cavity varies wildly. This is
due to flow separation and reattachment, resulting in a stagnation point on the

upper surface of the wing along with reverse flow associated with a leading edge
vortex. Therefore a specific Re appropriate for all of the cavities over the wing is not
achievable, and range from 0.1<Re<100 will allow for the conditions mentioned
previously.
Procedure
The software used to create and mesh the geometry was Gambit, while its
counterpart, Fluent, was used to solve for the velocity and pressure using the semiimplicit method for pressure linked equations (SIMPLE) algorithm. First order
upwind differencing was used for the momentum and energy equations, and an
under-relaxation factor of 0.3 was used for both pressure and momentum
equations. This low value aids in reducing overshoot and oscillation between
iterations, which can inhibit convergence, at the cost of increasing the number of
iterations for cases that would converge for higher values of the under-relaxation
factor. Because of the range of Re chosen for this project, a single, low value proved
to work best and resulted in convergence for all cases.
In calculating Re for each case, density and viscosity were chosen to be unity,
and since the cavity has a height of 1 meter, the Reynolds number became a 1:1
function of lid velocity. The aspect ratio of the cavity is 2:1 length to height and the
angle of the walls is 45° with the horizontal.
To mesh this geometry, care had to be taken with the upper right corner of
the cavity, where the right wall forms an acute angle with the approaching lid.
Because of the no-slip condition, flow near the surface of the lid will be moving at
the same velocity as the lid, then suddenly encounter the right wall where there is
no velocity. Here the flow will turn downwards and form a vortex as with a
traditional cavity flow, but due to the acute angle involved, higher velocity gradients
are expected here. The other corners are likely areas of high velocity and pressure
gradients as well, so a grading scheme was applied to the mesh such that there were
more grid points near the corners than towards the middle of the cavity. The initial
mesh is shown in Fig. 2, and has 45 grid points along each edge, with a double-sided
grading of 1.1 which results in a large number of cells in each corner.
A grid refinement study was done to ensure the mesh would converge upon
accurate results. To do this, the shear force along the bottom wall was calculated for
the initial mesh, and then the number of grid points was increased by about 20%
and the shear force was recalculated. Because there was significant variation
between these two sets of data, the grid was refined again, increasing the number of
nodes by 20%. The last two data sets are in close agreement, which means the
lower of the two, 60 intervals, is a valid mesh size. The results of the grid
refinement study are shown in Fig. 3 below.
Figure 3.
Grid refinement study showing variation in shear force along the
bottom wall for 3 different grid sizes with Re=100.
The Reynolds number used for the grid refinement study was Re=100, the
highest value to be considered and therefore the one which should highlight the
differences most clearly. While the actual increase between refinements was more
than 20%, this gives confidence that the chosen mesh size is sufficient.
Results
The same 60-interval mesh was used for each case, and the Reynolds
numbers chosen were 0.1, 1, 5, 10, 25, 50,75, and 100. For each case the total force
was calculated by Fluent, separated into x- and y-components, and characterized as
either pressure or viscous in nature. The variation of shear stress along each of the
three walls- left, bottom, and right- have also been plotted so a trend can be seen.
Additionally, for the Re=10 case, the energy equation was used to show the steady
state temperature distribution when the walls were set to 273 K and the lid was set
to 298 K. For this case, the distribution of the Nusselt number along the right wall
was calculated and an average value was found.
For each case, the contours of the stream function were plotted along with
velocity vectors, but there is very little variation among these so only Re=0.1, 1, 10,
and 100 are shown.
Figure 4.
Contours of the stream function, Re=0.1
Figure 5.
Contours of the stream function, Re=1
Figure 6.
Contours of the stream function, Re=10
Figure 7.
Contours of the stream function, Re=100
The effect of Reynolds number on the stream function is clear- the center of
the embedded vortex has shifted downstream, and the magnitude of the stream
function values has increased. Next, the velocity vectors are considered, shown
here in Fig. 8
Figure 8.
Velocity vectors for Re=10, in (m/s)
Because the image is so small, it is helpful to understand that the general
rotation is clockwise, and contours of velocity magnitude are shown, seen in Figs. 912 below.
Figure 9.
Contours of velocity magnitude (m/s), Re=0.1
Figure 10.
Contours of velocity magnitude (m/s), Re=1
Figure 11.
Contours of velocity magnitude (m/s), Re=10
Figure 12.
Contours of velocity magnitude (m/s), Re=100
The following Figs. 13-15 show the distribution of shear stress along each of
the walls. It should be noted that the shear stress for the left and right walls are
plotted on a semi-log scale to show the variations, otherwise no difference would be
seen.
Figure 13.
Shear stress along the left wall for the whole range of Re
Figure 14.
Shear stress along the right wall for the whole range of Re
Figure 15.
Shear stress along the bottom wall for the whole range of Re
From these graphs there is an obvious increase in shear stress with the
increase in Re, and this is reflected in the total force on these three walls as shown
in Table 1 and 2.
Table 1.
X-Forces
Re 0.1
Re 1
Re 5
Re 10
Re 25
Re 50
Re 75
Re 100
Table 2.
Y-Forces
Re 0.1
Re 1
Re 5
Re 10
Re 25
Re 50
Re 75
Re 100
X-components of force for each Re
Pressure Force
3.92948
39.305739
196.90038
396.02252
1007.8207
2072.8371
3198.2613
4384.5448
Viscous Force
-0.33197396
-3.3138546
-16.598611
-34.89111
-101.48255
-236.78092
-395.06956
-573.40506
Y-components of force for each Re
Pressure Force
-4.1171056
-41.150496
-205.49047
-411.91351
-1037.2116
-2102.5791
-3212.6714
-4374.8306
Viscous Force
-0.5717342
-5.71314908
-29.001988
-59.495617
-161.2116
-360.6753
-591.98351
-850.96231
The x- and y-components were combined into a total force and plotted in Fig.
16 below, which shows nearly linear trend between Re and forces.
Figure 16.
pressure forces.
Total force vs. Reynolds number, separated into viscous and
This total force was then taken and normalized by the Reynolds number, and
from this the linearity is quite
clear.
Figure 17.
Normalized force vs. Reynolds Number
Figure 18, below, shows the steady state temperature distribution for the
Re=10 case. The velocity is high enough that the fluid near the lid is warmed up but
not all the way to the wall temperature.
Figure 18.
Steady state temperature distribution, Re=10
The Nusselt number distribution along the right wall is shown below, where
the average Nusselt number for the surface is 475.61. This shows there is a much
higher temperature gradient near the lid, and lower levels toward the bottom of the
cavity.
Figure 19.
Nusselt number distribution along the right wall
Discussion
From the plots of the stream function and velocity distribution, we see that
the flow in cavity is fairly constant in orientation, where the Reynolds number
determines the magnitude. From Fig. 17 it is obvious the forces experienced by the
cavity are linear functions of Reynolds number for this low range of values. The
static pressure variation, not shown, was minimal as expected for this low Re flow,
and that brings up the conclusion. For the range of Re tested in the given geometry,
the velocities and shear stresses vary linearly with Reynolds number. Further
knowledge about exactly what flow velocities are occurring along the butterfly wing
coupled with these findings will allow better calculations of the lift and drag
experienced by a butterfly.
References
No works are cited, all data comes from current research conducted by the
author.