PHYSICS OF FLUIDS 20, 093606 共2008兲
Symmetry unbreaking in the shapes of perfect projectiles
Marcus Roper,1,a兲 Todd M. Squires,2 and Michael P. Brenner1
1
School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
Department of Chemical Engineering, University of California Santa Barbara, Santa Barbara,
California 93106-5080, USA
2
共Received 4 September 2007; accepted 12 May 2008; published online 24 September 2008兲
We study the shapes of perfect projectiles: Bodies of prescribed volume that are designed to suffer
minimum fluid drag in steady flight. Perfect projectiles have a surprising property: Although the
flow of fluid around the body of the projectile is fore-aft asymmetric at moderate flow speeds, the
shape of the body that minimizes drag is nonetheless highly symmetrical. We show that perfect
projectiles are weakly asymmetric and that their asymmetry grows with the cube of the projectile
size for sufficiently small projectiles. The persistence of apparent fore-aft symmetry is speculated to
be a signature of the linearity and reciprocity of the drag-determining features of the flow around the
projectile. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2982500兴
I. INTRODUCTION
Drag minimization at high speeds favors slender shapes
with blunt noses and sharp trailing edges, as can be seen in
the body forms of fish evolved for high speed cruising such
as tuna or mackerel, or in the design of modern aerofoils.1
Drag minimization is not simply a selection pressure upon
large, fast moving predators with conspicuously streamlined
shapes but is also significant at the microscale. Fluid drag
encountered in flight severely limits the range of explosively
launched propagules, such as seeds or fungal spores; Vogel
estimated2 that the ascospores of the “fungal gun” Giberella
zeae lose 99.997% of their potential range to drag.
It is natural to ask whether, and in which respects, organism shapes have been influenced by selection pressure to
minimize drag while conserving propulsive capacity.1 Identification of drag-determining features of large rapidly swimming animals has been aided by comparison of animal body
shapes and measured drags with the shapes and reported
drags of high speed aerofoils or flat plates of matching volume or area. Such comparisons originate from von Kármán,3
who first noted the remarkably precise agreement between
the shapes of trout in dorsal-ventral section and the profile
of a modern aerofoil. We attempt to extend such comparisons
into the realm of moderate Reynolds numbers 共i.e.,
Reⱗ 100兲 inhabited by the smallest 共and most
drag-constrained2兲 bioprojectiles by computing the shapes of
perfect projectiles: Bodies with prescribed volume and speed
that are shaped so as to suffer minimum fluid drag in steady
flight.
Few of the properties of perfect projectiles are presently
known: It is not known how large an increase in range or
flight speed an organism can attain by redistributing mass to
attain a drag-minimizing shape, or whether such shapes are
undesirable according to some other criteria, such as poor
stability in flight or low stall angle. Here we make a first step
toward a full characterization of the properties of minimal
drag shapes by investigating the variation in fore-aft asyma兲
Electronic mail: [email protected].
1070-6631/2008/20共9兲/093606/13/$23.00
metry with Reynolds number. We find that although the flow
around the body and the pattern of surface stresses are both
highly asymmetric, perfect projectiles remain relatively foreaft symmetric up to moderate Reynolds numbers. We call
this surprising effect a “symmetry unbreaking:” An instance
where an asymmetric action 共the fluid drag functional兲 has a
highly symmetric minimum. Asymptotic studies illuminate
the slow increase in asymmetry of the minimum from vanishingly small Reynolds numbers. The persistence of symmetry at moderate Reynolds numbers is argued to be a signature of the underlying reciprocity and linearity of the dragdetermining features of the flow.
II. COMPUTATION OF MINIMUM DRAG SHAPES
We determine numerically the shapes of perfect projectiles for prescribed volumes and flight speed. The joint
effect of varying flight speed and body size is represented
by a single dimensionless group: The Reynolds number
Re= Ua / ␯, in which U is the speed of flight and a is a characteristic body dimension 共here we identify a with the radius
of a sphere of equivalent volume to the projectile兲. The minimal drag shape has already been determined for slowly
creeping flows 共Re= 0兲,4,5 and an algorithm for determining
the minimal drag shape has been proposed for flows at arbitrary Re 共Ref. 6兲 but previously implemented only for twodimensional bodies traveling at a small set of speeds.7 We
determine axisymmetric minimal drag body shapes using an
algorithm similar to that suggested by Pironneau.6
The core of our method for finding the shapes of perfect
projectiles is calculation of how an arbitrary infinitesimal
deformation to the boundary of a given projectile affects the
fluid drag upon the projectile. Once known, we may start
with an arbitrary initial projectile shape, and then use steepest descent to make successive optimal refinements to the
boundary until a drag minimum is achieved.
In order to compute the change in drag upon a projectile,
denoted by ⍀, immersed in an infinite body of fluid, due to a
small perturbation to the boundary of the shape ⳵⍀, we represent the drag upon the projectile by a functional
20, 093606-1
© 2008 American Institute of Physics
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093606-2
Phys. Fluids 20, 093606 共2008兲
Roper, Squires, and Brenner
4
5
x 10
4
3
M3
2
1
0
1
2
0
20
40
Re
60
80
100
FIG. 1. 共a兲 Minimum drag shapes obtained under a constant volume constraint. Key to shading: shapes computed for Re= 0.1 共darkest兲, Re= 5.0, Re= 25.0, and
Re= 100 共lightest兲. Flow approaches each body from the negative z-direction. 共b兲 Growth of the asymmetry parameter M 3 with Reynolds number. For
comparison, the M 3 values for a reference body of controllable asymmetry 关shown in figure 共c兲兴 are marked off. The number on each horizontal dotted line
reports the asymmetry ratio ratio l2 / l1 for the reference body. 共c兲 Axisymmetric reference body used to generate the fiducial lines in panel 共b兲. The reference
body is formed by union of two spheroids of major axes l1, l2, and common diameter 2d.
L关⍀兴 =
冕
⳵⍀
冕
冉冕
n · ␴ · UdS +
+ q ⵜ · u其dV + ␭
R3−⍀
⍀
兵w · 关ⵜ · ␴ − ␳共u · ⵜ兲u兴
冊
dV − 兩⍀兩 ,
共1兲
␣共x兲 ⬀ ␩
where the first term denotes the rate of working of the drag
force, and the adjoint velocity 共w兲 and pressure 共q兲 fields that
appear in the second term are Lagrange multipliers that we
introduce in order to enforce conservation of momentum and
incompressibility for the flow, respectively. A third Lagrange
multiplier ␭ constrains the volume of the body.8
Consider a small deformation to the bounding surface of
the projectile, in which each boundary point x is displaced to
a point x + ␣共x兲n for some function ␣共x兲 that remains small
over the entire of the boundary. Provided that the adjoint
fields satisfy an equation
− ⵜq + ␩ⵜ2w = − ␳共u · ⵜ兲w + ␳共U j + w j兲 ⵜ u j ,
共2兲
and ⵜ · w = 0, with w = 0 on the boundary of the shape and
w → −U in the far field, the change in the 共rate of working of
the兲 drag ␦L can be cast in the form of a single integral over
the unperturbed surface of the body,
␦L = −
冕
⳵⍀
冉
␣共x兲 ␩
冊
⳵u ⳵w
·
+ ␭ dS.
⳵n ⳵n
We now have means to calculate an optimal deformation
to the boundary of the shape,4 i.e., the shape perturbation that
achieves maximum drag benefit while fixing the volume of
the shape,
共3兲
We call Eq. 共2兲 and the solenoidal condition that follows
it the adjoint equations. A minimal drag shape must have
␦L vanishing for all volume-preserving 共i.e., having the
property that 兰⳵⍀␣dS = 0兲 boundary perturbations and must
therefore have J ⬅ ⳵u / ⳵n · ⳵w / ⳵n uniform over the entire of
the boundary.
⳵u ⳵w
␩
−
⳵ n ⳵ n 兩⳵ ⍀兩
冕
⳵u ⳵w
·
dS,
⳵⍀ ⳵ n ⳵ n
共4兲
where we arrive at the second term by choosing ␭ to ensure
that 兰⳵⍀␣dS = 0. Now we imagine parametrizing the sequence of shapes that the body must pass through from the
initial shape to the minimal drag shape by some parameter ␮
and identify dx / d␮ as ␣n. Previous implementations of the
gradient-descent method for drag minimization have integrated the equation for the sequence of shapes passed
through by the body using a fixed step-size Euler scheme and
encountered instabilities in the boundary shape, which must
then be removed by filtering.7 We avoid such instabilities by
integrating the equation using a low-order variable step-size
routine 共the MATLAB function “ode23”兲 and by permitting
very small step sizes, at the cost of more, costly, evaluations
of the function ␣. Numerical solution of the coupled Navier–
Stokes and adjoint equations using finite element software is
described in Appendix A.
III. PRESERVATION OF SYMMETRY
UP TO MODERATE REYNOLDS NUMBERS
In Fig. 1共a兲 shapes of perfect projectiles are plotted for
Reynolds numbers ranging from 0.1 up to 100. This interval
in Reynolds number corresponds to flight speeds of 0.1 up to
100 ms−1 for a 20 ␮m long body traveling through air
共␯ = 1.8⫻ 10−5 m2 s−1兲, which covers the full range of estimated launch conditions of explosively launched fungal
propagules.2
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093606-3
Phys. Fluids 20, 093606 共2008兲
Symmetry unbreaking
FIG. 2. Asymmetric flow field around a perfect projectile at Re= 50: 共a兲 Left pane: vorticity contours. Right
pane: streamlines 共curves兲 and flow speed 共shading兲. 共b兲
Latitudinally integrated stress field plotted around the
boundary of the perfect projectile. We separate the drag
into 共a兲 viscous shear stress and 共b兲 pressure
contributions.
We observe that each of the minimal drag shapes is very
fore-aft symmetric: Resourceful readers are challenged to
guess, without consulting the figure caption, whether projectiles are designed for flight in the positive or in the negative
z-direction! A more careful quantification of the slow emergence of asymmetry is given in panel 共b兲 of the figure, in
which we chart the growth of projectile asymmetry with Re
via the third moment
M3 =
3兰⍀共z − z̄兲3dV
,
4␲L3
共5兲
where L is the total length of the body and z̄ = 3兰⍀zdV / 4␲ is
the first moment of the projectile. We make this measure of
asymmetry more intelligible by plotting, on the same figure,
values of M 3 for a reference shape of volume equal to that of
our perfect projectile and with controllable asymmetry. We
form this shape by compounding two hemispheroids of different major axes l1, l2 共and with the length of minor axis, d,
fixed by the constraint upon the volume of the shape兲. For
such a body, a little calculation gives
共q − 1兲共7 + 18q + 7q 兲
,
640共q + 1兲3
2
M3 =
共6兲
in which q ⬅ l2 / l1, independently of the total length of the
body.
The asymmetry must be regarded as imperfectly resolved in the limit of vanishingly small Re but remains small
until Reⲏ 25, whereupon it grows linearly but slowly with
Re.
Fore-aft symmetry of the perfect projectile is expected in
the case Re= 0. For such flows, the pattern of streamlines
around any body remains unchanged whether it travels forward or backward through the fluid: Flow everywhere is perfectly reversed by reversal in the direction of body motion.
The fluid drag is therefore the same whether the body travels
forward or backward. Hence if we assume that the perfect
projectile is unique, then it must have a fore-aft symmetric
shape at Re= 0. Our numerical experiments seem to exclude
the possibility of multiple perfect projectiles since searches
starting from different initial shapes all ultimately converged
upon the same drag minimum.
We visualize the flow field and boundary stress upon a
perfect projectile for Re= 50 in Fig. 2. Inertial effects give
the flow two well-characterized asymmetric features:9 Vorticity generated at the boundary of the projectile remains confined within a thin boundary layer that grows with distance
from the front of the body and is shed into an approximately
paraboloidal wake behind the body. The stress field around
the body is similarly asymmetric, with both viscous and
pressure drag contributions largest in the extensional flow at
the front of the body, and declining with distance from the
front stagnation point. That this high degree of flow asymmetry is not impressed upon the drag-minimizing body produces a peculiar example of symmetry unbreaking: Although
the action 共1兲 is asymmetric, the action-minimizing shape is
apparently symmetric.
IV. SYMMETRY BREAKING IN WEAKLY
INERTIAL FLOWS
Asymptotic study of the limit Re→ 0 allows us to fortify
our numerical results, which cannot well-resolve the growth
of asymmetry with Reynolds number below Re⬇ 5. The authors may as well admit that they believed that the symmetry
of the perfect projectile would be broken at a finite value of
Re, mirroring the hierarchy of finite Re bifurcations that are
seen in both steady and unsteady flows around bluff bodies,
including the onset of boundary layer separation, breaking of
lateral symmetry in the vortex wake, and the transition from
steady to unsteady vortex shedding. Our asymptotic study
rules this out, showing that perfect projectiles have weak
asymmetry all the way down to Re= 0. We start by showing
that leading-order weak inertial effects are not sufficient for
symmetry breaking.
A. Leading-order inertia effects
do not break symmetry
We take a frame of reference comoving with the body
and identify the direction of the stream approaching the body
with the z-direction. Upon nondimensionalizing velocities u
by the far field velocity U, lengths by the volumetric radius
a, and stresses by ␩U / a, the flow field around the body
may be seen to be governed by the steady Navier–Stokes
equations,
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093606-4
Phys. Fluids 20, 093606 共2008兲
Roper, Squires, and Brenner
Re u · ⵜu = − ⵜp + ⵜ2u
共7兲
in addition to incompressibility ⵜ · u = 0 and a no-slip boundary condition u = 0 imposed upon the surface of the body, and
uniform stream u → ẑ, p → 0 at infinity.
If inertial effects are weak 共i.e., the Reynolds number is
sufficiently small兲 then the leading-order effect of inertia on
the fluid drag encountered by the projectile can be effectively
modeled by linearizing the left-hand side of Eq. 共7兲 about the
uniform stream to give Oseen’s approximation,
Re
⳵u
= − ⵜp + ⵜ2u
⳵z
and
ⵜ · u = 0,
共8兲
and this pair of equations can be solved by straightforward
application of Green’s function techniques.9
We show that the drag predicted by the Oseen equations
is unaffected by the direction of travel of the body. First we
denote by u and û the two flow setups when the incident
flow approaches from the negative and positive z-directions,
respectively. Now Oseen’s equations may be written in a
conservative form for the two flows, i.e., as ⵜ · T = ⵜ · T̂ = 0 if
we define an effective stress tensor 共which includes both viscous stresses and advection of momentum by the uniform
stream兲 for the forward flow, T ⬅ −p1 + ⵜu + 共ⵜu兲T − Re共uẑ
+ ẑu兲, and similarly for the reversed flow. Then observe that
ˆ · u兲 = ⵱ · 关Re共u · û兲ẑ兴,
ⵜ · 共␴ · û − ␴
B. Fore-aft symmetry is broken at O„Re3…
Inertial contributions up to O共Re3兲 must be included
before the asymmetry of the perfect projectile becomes
detectable. Our proof of this statement has two-halves: First
we show that the drag upon an arbitrary body is unaffected
when its direction of travel is reversed, up to terms of
O共Re3 log Re兲. Then we show that contributions at O共Re3兲
are sensitive to the direction of travel. We take an indirect
route to the second result by explicitly computing the first
drag-minimizing step for a sphere, and showing that this step
includes a nonsymmetric component at O共Re3兲. Both calculations are somewhat lengthy, and we will consign the details
to Appendix B. A physically motivated examination of the
structure of the perturbation expansion explains the weak
contribution of body asymmetry to drag.
Introduce a system of spherical polar coordinates
共r , ␪ , ␾兲 with ␪ = 0 coinciding with the z-axis. We define a
stream function ␺ for the velocity field so that
ur =
共9兲
or with a little manipulation
ⵜ · 关T · û − T̂ · u + Re共ẑu · û + uûz + ûuz兲兴 = 0.
Oseen equations correctly predict the O共Re兲 correction to the
drag upon a projectile even though they do not correctly
capture the pattern of streamlines13,14 but that they do not
capture higher order inertial corrections.
共10兲
Integrating the left-hand side over the entire of the fluid domain, applying the divergence theorem, and carefully accounting for the rate of decay of the velocity field at infinity
to eliminate the contributions from the far field, we obtain
ˆ 兲 · ẑdS = 0, where the integration is
the result, 兰⳵⍀n · 共␴ + ␴
taken over the boundary of the body ⳵⍀. It follows that,
according to Oseen’s approximation, the total drag force on
the body is identical in magnitude but opposite in sign when
the direction of the far field flow is reversed.
It follows that leading-order inertial effects, which can
be modeled using Oseen’s equations, are not sufficient to
break the fore-aft symmetry of the projectile in the limit as
Re→ 0. Moreover, we note here that our result shows that
the Oseen equations produce completely fore-aft symmetric
perfect projectiles even when contributions to all orders of
Re are included. We thank S. Childress for pointing out to us
that the symmetry of the drag upon a body in Oseen flow
was previously proven by Olmstead and Gautesan.10,11
We have no reason, however, to expect the Oseen equation to accurately reproduce the drag upon the projectile beyond terms of O共Re兲. Oseen’s equation only properly approximates the Navier–Stokes equations at distances greater
than O共a / Re兲 from the projectile. Closer to the body the
perturbation to the uniform stream cannot be said to be
small. A more careful treatment of the low Reynolds number
limit requires that Oseen’s approximation in the far field be
matched to a regular perturbation expansion in the near field
of the body.12 From these analyses, it can be shown that the
1 ⳵␺
,
r2 sin ␪ ⳵ ␪
u␪ = −
1 ⳵␺
,
r sin ␪ ⳵ r
共11兲
and follow Ref. 12 by defining a transformed angle ordinate:
␤ ⬅ cos ␪. The vorticity equation 关formed by taking the curl
of the Navier–Stokes equation 共7兲兴 may then be written in a
form12
Er4␺ = Re Hr关␺, ␺兴,
共12兲
in which we have defined operators
Er2 ⬅
and
⳵2 1 − ␤2 ⳵2
+
,
⳵ r2
r2 ⳵ ␤2
冉
Lr ⬅
␤ ⳵ 1 ⳵
+
1 − ␤2 ⳵ r r ⳵ ␤
冊
2
1 ⳵ 共␺,Er ␾兲 2 2
+ 2 E r ␾ L r␺ .
H r关 ␺ , ␾ 兴 ⬅ 2
r ⳵ 共r, ␤兲
r
共13兲
共14兲
Equation 共12兲 is supplemented by boundary conditions associated with regularity on the symmetry axis 共␺ = 0 on
␤ = ⫾ 1兲, no slip on the surface of the body 关␺ = 共n · ⵜ兲␺ = 0兴
and matching to the uniform stream at infinity
关␺ → 共1 / 2兲r2 sin2 ␪ as r → ⬁兴.
In the limit of Re→ 0, the flow field can be divided into
two regions. There is a Stokes layer close to the body, whose
thickness scales with the size of the body, and an Oseen layer
for the flow at O共1 / Re兲 distances from the body. In the
Stokes layer, the leading-order part of the vorticity equation
represents a diffusive steady state attained as vorticity is introduced from some parts of the boundary of the body and
eliminated at others. In the Oseen layer, we must introduce a
rescaled length variable: r = ␭ / Re and stream function
␺ = ⌿ / Re2, Er2 = Re2 E␭2 , and Lr = Re L␭. The vorticity equation takes the form
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093606-5
Phys. Fluids 20, 093606 共2008兲
Symmetry unbreaking
E␭4 ⌿ = H␭关⌿,⌿兴,
共15兲
and the dominant balance in this layer features both diffusion
of vorticity and advection by the uniform stream. Matching
of the two solutions, Oseen and Stokes, imposes the condition that
lim ⌿ = Re2 lim ␺ .
␭→0
共16兲
r→⬁
We seek an asymptotic expansion for the stream function
within the Stokes region,
␺ = ␺0 + Re ␺1 + Re2 log Re ␺2L + Re2 ␺2
+ Re3 log Re ␺3L + Re3 ␺3 + O共Re4 log2 Re兲,
共17兲
and similarly within the Oseen region. At each step of the
calculation, we find one new term in each of the Oseen and
Stokes regions. Solutions in both solutions contain arbitrary
constants, corresponding to undetermined multiples of solutions to the homogeneous Stokes’ and Oseen’s equations.
Matching the solutions in the two regions using Eq. 共16兲
fixes the homogeneous components of the stream functions.
We pay particular attention to the form of the leadingorder stream function within the Stokes region ␺0 which satisfies Er4␺0 = 0. At sufficiently large distances from the body
共1 Ⰶ r Ⰶ Re−1兲 the leading-order stream function may be expanded as a sum of force and source-multipole contributions,
which we terminate after the force-octupole and sourcequadrupole contributions,
冉
冊
冉
冊
1
D1
C1
␺ 0 = − r 2 − D 0r +
Q 1共 ␤ 兲 + C 0 + 2 Q 2共 ␤ 兲
2
r
r
+
冉冊
1
C2
D2
Q 3共 ␤ 兲 + 2 Q 4共 ␤ 兲 + O 3 .
r
r
r
共18兲
Dependence of the stream function on meridional position is
compactly expressed in terms of modified Gegenbauer poly␤
nomials: Qn共␤兲 = 兰−1
Pn共␤⬘兲d␤⬘.12 Here we have distinguished between symmetric and antisymmetric contributions, Q2n−1共␤兲 = Q2n−1共−␤兲 and Q2n共␤兲 = −Q2n共−␤兲, so that
all of the asymmetry of the flow is contained, to this order, in
the O共1兲 共stresslet兲 and O共1 / r2兲 共source-quadrupole and
force-octupole兲 contributions. More specifically, if the direction of flight of the projectile were to be reversed and new
coordinates 共r⬘ , ␤⬘兲 defined in the same way 共equivalent to
reflecting the body in plane z = 0 so that the incident stream
continues to approach from the negative z-direction兲, then
the new flow would be given by a stream function ␺−, where
at O共1兲: ␺−共r⬘ , ␤⬘兲 = ␺共r⬘ , −␤⬘兲, so that the multipole expan−
−
= −Cm and Dm
= D m.
sion for ␺− would have coefficients Cm
The Stokes drag force upon the particle is fixed by the
strength of the O共r兲 共Stokeslet兲 term: F = 2␲D0.15 Moreover,
by choice of our coordinate origin, it is possible to eliminate
the stresslet 关O共1兲兴 component of the stream function, permitting us to set C0 = 0 without any loss of generality.
Our freedom to omit the stresslet term from the expansion of the stream function provides a route to establishing
the absence of symmetry breaking below O共Re3兲 without any
further detailed calculation. In the absence of the stresslet
term, any fore-aft asymmetry in the shape of the projectile is
communicated to an observer in the far field only through
force-octupole and source-quadrupole or weaker contributions to the flow field, giving a velocity field that decays like
O共1 / r4兲 or a contribution to the stress tensor that decays like
O共1 / r5兲. By contrast, the strongest flow disturbance associated with the body is a Stokeslet velocity field that decays
like O共1 / r兲, and leads to an O共Re兲 departure from the uniform stream in the Oseen layer. The O共1 / r3兲 weaker asymmetric contribution to the multipole expansion leads to an
O共Re3兲 weaker correction to the flow in the Oseen layer
关since the two flow fields must be matched on a sphere of
O共1 / Re兲 radius兴. The drag on the body can be extracted
merely from determining the flow in the Oseen region by
enlarging the sphere on which the integral 共33兲 is evaluated
to some ⬃1 / Re radius.16 Since the contribution of projectile
asymmetry to the flow field is only O共Re4兲 within the Oseen
region, the effect of asymmetry on drag will only be felt
when O共Re3兲 inertial effects are accounted for.
This argument glosses both the appearance of novel multipole terms among the higher order 共in Re兲 components of
the stream function and the possibility of additional cancellations or symmetries that could allow the postponement of
symmetry breaking effects to yet higher order in Re. Excluding such effects requires the continuation of the asymptotic
expansion up to terms of O共Re3兲. We proceed with this calculation, although the uninterested reader will lose nothing
by continuing directly to Sec. V.
In order to determine higher order 共in Re兲 contributions
to the stream function, it will turn out to be necessary to
know the stream function for the flow induced by placing the
body in pure straining flow 关␺S ⬃ r3Q2共␤兲 + O共1兲 as r → ⬁兴,
in addition to the stream function ␺0 representing the slowly
creeping flow around the body in a uniform stream. ␺S includes no uniform flow terms proportional to r2Q1共␤兲 共since
these are forbidden by the far field conditions兲 or Stokeslet
contribution 关proportional to rQ1共␤兲兴 since our choice of origin ensures that an axial straining flow exerts no force on the
body. For, in general, the force, torque, and stresslet upon the
particle are related to the flow fields imposed at infinity via a
grand-resistance matrix 共see Ref. 17, pp. 108–112兲, in particular for a uniform stream at infinity the stresslet exerted by
the fluid upon the particle is given by
S = G̃ · U⬁ ,
共19兲
and for uniform straining flow at infinity, the force upon the
particle is given by
F = G:E⬁ .
共20兲
Here G and G̃ are both rank 3 tensors, and it can be shown
via the reciprocal theorem that G̃ijk = Gkij. Thus by selecting
our coordinates to eliminate the stresslet under uniform
streaming conditions, we have ensured that G̃ij3 = 0, and thus
guaranteed that under arbitrary symmetric straining flows
there is no drag force upon the particle.
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093606-6
Phys. Fluids 20, 093606 共2008兲
Roper, Squires, and Brenner
This set of observations gives us enough information to
compute the drag force exerted by the fluid on the particle up
to O共Re2 log Re兲 corrections simply by following the matching procedure procedure developed by Proudman and
Pearson.12
The first two terms for the stream function within the
Oseen layer are shown as
⌿0 = 21 ␭2共1 − ␤2兲,
⌿1 = −
共21兲
D0
共1 + ␤兲共1 − e−␭共1−␤兲/2兲.
2
共22兲
The O共Re兲 contribution to the stream function within the
Stokes layer can be separated into a particular integral ␺1P
and a complementary function, which must be some multiple
of ␺0, with the prefactor chosen to ensure that the matching
condition 共16兲 is satisfied up to terms of O共r2兲. We may
additionally require of the particular integral that
␺−1P共r⬘ , ␤⬘兲 = −␺1P共r⬘ , −␤⬘兲. We may then make a multipole
expansion of the solution
␺1 ⬃ −
冉
冊
C3
D0 2 D0
D0
r −
␺0
rQ1 +
r + D3 Q2 +
2
8
2
8
冉冊
1
+O
,
r
共23兲
D2
␣1
␣1
␭共1 − ␤2兲 − ␭2共1 − ␤2兲共1 − ␤兲 + 0 ␭␤共1
2
8
32
− ␤ 2兲 +
−
冉
冊
ER4 ␺2P2 = Hr关␺1P, ␺0兴,
共25兲
Clearly, we may take ␺2P1 = D0 / 4␺1P, and demand that the
other components have the reversal property ␺−2P2共r⬘ , ␤⬘兲
= ␺2P2共r⬘ , −␤⬘兲 and ␺−2P3共r⬘ , ␤⬘兲 = ␺2P3共r⬘ , −␤⬘兲. We must add
to this multiples of our two homogeneous solutions ␺0 and
␺S. The solution within the Stokes layer can be shown to
have the form
␺2 =
冉
+
冊
D 2r 2
D0r3 D20r2 log r
+ D 4R Q 1 + 0 Q 2
−
40
32
40
冉
冊
D0r3 43D20r2
Q3 + A1␺0 + B1␺S + O共1兲
−
60
2400
共24兲
up to O共␭3兲 corrections. Here ␥ is the Euler–Mascheroni
constant and the coefficient ␣1 共which corresponds to addition of a multiple of the homogenous solution ⌿1兲 need to be
determined by application of the matching condition.18
Proceeding to terms of O共Re2兲 in the Stokes region, we
can construct a particular integral having three components,
共26兲
with unknown constants A1 and B1 together with ␣1 from the
Oseen layer solution all chosen to satisfy the matching condition 共16兲at O共r3兲,
␣1 = −
D20
,
16
B1 = −
A1 = − D20
冉
冊
1
29
␥
−
− log 2 ,
800 40 24
D0
.
24
共27兲
Notice that on matching to the Oseen layer solution on a
sphere of radius O共1 / Re兲, we appear to pick up a uniform
stream contribution at O共Re2 log Re兲. Demonstrably, no such
terms appear in the Oseen layer expansion, so it is necessary
to introduce a canceling term at intermediate order in the
perturbation expansion,
␺2L =
D20
␺0 .
40
共28兲
The appearance of a term of this order forces the introduction
of terms at O共Re3 log Re兲 in both Stokes and Oseen layers,
having respective forms
␺3L ⬃ A2␺0 +
冉
冊
冉冊
D30 2 D0
1
r −
,
r Q2 + O
160
2
r
⌿3L = ␣2共1 + ␤兲共1 − e−␭共1−␤兲/2兲,
D20 2
83
1
␥ 1
␭ 共1 − ␤2兲 log ␭ + + log 2 −
16
5
200
5 3
43
␤
共5␤2 − 1兲
+
8 1200
D0
Hr关␺0, ␺0兴,
4
ER4 ␺2P3 = Hr关␺0, ␺1P兴.
and
and
where the coefficients 共which transform into D−3 = D3 and
C−3 = −C3 when the direction of flight is reversed兲 must be
determined by applying the no-slip condition upon the surface of body. Now D3 is independent of whether the body
moves forward or backward and it can be shown that
C3 = 0: There is no Stokeslet contribution because the drag
suffered by the projectile is reversible up to O共Re兲
corrections.14
The correction at O共Re2兲 to the flow within the Oseen
region is most easily obtained by Green’s function techniques and has no simple closed form. However, by careful
asymptotic study of the integral expression, it can be determined that as ␭ → 0,
⌿2 ⬃
ER4 ␺2P1 =
共29兲
共30兲
and in order to match with to the Oseen solution, it is also
necessary to isolate the far field contribution of size r2 log r
arising from ␺3
␺3 = ¯ −
3D30 2
r 共log r兲Q1 + ¯ ,
320
共31兲
matching gives ␣2 = −D30 / 80 and A2 = D30 / 80.
We now have sufficiently many terms to be able to compute the drag 共D兲 upon the body up to O共Re3 log Re兲 contributions. Note that, similar to the Oseen equations, the
Navier–Stokes equations may be written in a conservative
form,
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093606-7
Phys. Fluids 20, 093606 共2008兲
Symmetry unbreaking
ⵜ · ␶ = 0,
where
␶ = − p1 + 关ⵜu + 共ⵜu兲T兴 − Re uu.
共32兲
By appealing to the divergence theorem, we may then
convert the integral of the surface stress around the body to
an integration around a sphere of very large 共but nonetheless
Ⰶ1 / Re兲 radius,
D=
冕
n · ␶ · ẑdS = −
S
冕
n · ␶ · ẑdS = F p + Fs + Fm , 共33兲
S⬁
in which n is the outward pointing normal on S and equal to
−er on S⬁. We have separated the integral into three distinct
contributions, a contribution from the pressure force,
Fp = −
冕
p cos ␪dS = ␲
S⬁
冕
⳵p
共1 − ␤2兲1/2r2d␤ ,
⳵
␪
−1
1
共34兲
where ⳵ p / ⳵␪ may be read directly off from the meridional
component of the Navier–Stokes equation, plus a contribution from viscous stresses,
冕冋
1
Fs = 2␲
2
−1
冉
⳵ ur
⳵ u␪ u␪
−
␤ − 共1 − ␤2兲1/2
⳵r
⳵r
r
共1 − ␤ 兲
−
r
2 1/2
⳵ ur
⳵␤
冊册
冕
r d␤ ,
1
ur关ur␤ − u␪共1 − ␤2兲1/2兴r2d␤ .
共36兲
Summing these three contributions for some O共1 / Re兲
value of r, we arrive at an expression relating the total drag
upon the body to the coefficients of the multipole expansion
for the Stokes layer stream function,
D2
D0
D
Re + 0 Re2 log Re
=1+
8
40
DS
冋冉
+
1 ⳵␾
,
r2 sin ␪ ⳵ ␪
w␪ = −
冊
冉
冊
⳵ 共␺, ␾兲
,
⳵ 共r, ␤兲
册
共38兲
共39兲
and this equation can be analyzed in the limit Re→ 0 in the
same manner as the vorticity equation 共32兲. Sufficient information about the velocity stream function ␺ is provided by
the earlier analysis of Chester and Breach.18 Just as for the
vorticity equation 共12兲, it is necessary to separate the flow
field into Stokes and Oseen layers, which entails rescaling
the flux function and radial coordinate for the Oseen layer.
Within the Oseen region, the leading-order flow is a uniform stream directed in the opposite sense to the physical
flow around the body,
⌽0共␭, ␤兲 = − 21 ␭2共1 − ␤2兲.
共40兲
Picking out terms of O共Re兲 from Eq. 共39兲 gives
13
log2 ␥
D3 2D4
+
+
+
−
Re2
24
20 D0
40 600
D30 3
Re log Re + O共Re3兲.
80
1 ⳵␾
.
r sin ␪ ⳵ r
By taking the curl of the adjoint Eq. 共2兲, we arrive at
共35兲
2
−1
+ D20
wr =
− Er4␾ = Re Hr关␾, ␺兴 + Re Er2
and a contribution from transport of momentum through the
surface S⬁,
Fm = − 2␲
the O共1 / r兲 components of the velocity field around an arbitrary axisymmetric projectile under reversal of its direction
of flight. The drag upon the projectile may be explicitly related to these velocity field components and shown to depend, in fact, only on the symmetric components. Antisymmetric flow field components start to contribute to the drag
upon the projectile when O共Re3兲 inertial corrections are
included.
We show that O共Re3兲 contributions to the drag are affected by reversal in the direction of travel of the projectile
by direct computation of the first step made if the dragminimizing algorithm is started with a spherical body. This
drag-minimizing step includes an antisymmetric component,
which we exhibit. In order to do this, we construct an analog
of the vorticity equation for the adjoint field w by introducing a flux function ␾ so that
共37兲
Here we have followed Chester and Breach18 by scaling the
total drag force upon the projectile by the Stokes drag
共DS = 2␲D0兲. Note that all of the coefficients that appear in
this expression remain invariant when the direction of flight
is reversed: the terms appearing in the stream function for
reversed flow ␺− are identical to their counterparts in ␺:
−
= Dm. It follows that up to O共Re3兲 contributions the drag
Dm
upon the projectile is independent of whether it travels backward or forward through the fluid.
It is therefore possible to determine, up to O共Re3 log Re兲
inertial corrections, the symmetry or antisymmetry of all of
E␭2 关E␭2 ⌽1 + 共1 − ␤2兲L␭⌽1兴 = 0,
共41兲
which is identical to Oseen’s equation with the direction of
far field advection reversed. The resemblance to the physical
flow field enables us to write down the second term in the
perturbation expansion of ⌽,
⌽1共␭, ␤兲 = − ⌿1共␭,− ␤兲 = − 23 共1 − ␤兲共1 − e−␭共1+␤兲/2兲,
共42兲
while similar treatment of Eq. 共39兲 within the Stokes region
yields
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093606-8
Phys. Fluids 20, 093606 共2008兲
Roper, Squires, and Brenner
冉
␾0共r, ␤兲 = − ␺0共r,− ␤兲 = r2 −
冊
3r 1
+
Q1共␤兲,
2 2r
冉
共43兲
tion version of the matching condition 共16兲 and asymptotic
study of our Green’s function solution produces
冊
⌽2共r, ␤兲 ⬃
3
3r 1
␾1共r, ␤兲 = − ␺1共r,− ␤兲 = r2 − +
Q 1共 ␤ 兲
8
2 2r
冉
冊
1 1
3
2r2 − 3r + 1 − + 2 Q2共␤兲.
+
16
r r
− E␭2
冋
册
1 ⳵ 共⌿1,⌽1兲
.
␭2 ⳵ 共␭, ␤兲
␾2 =
冋 冉
冊冉
− Er2
冉
冊册 冋 冉
冊 冉
+
9
16r3 22r2 14r 16 8 log r 573 8 2669
1
+
−
−
+
−
+ −
−
+
Q3 .
320
9
15
3
15
35r
70r r2 630r3 3r4
冉
Q1 +
共47兲
from which we deduce
3
3␥ 229
log 2 +
−
4
5 200
3r 1
+
2 r
冊
1 ⳵ 共 ␺ 0, ␾ 1兲 1 ⳵ 共 ␺ 1, ␾ 0兲
+ 2
,
r2 ⳵ 共r, ␤兲
r ⳵ 共r, ␤兲
+
r2 −
共46兲
Er4␾2 = − Hr关␾0, ␺1兴 − Hr关␾1, ␺0兴
8 log r 1111 1
16r3
287r
2
4
9
+1+
+
− −
−
+
+ 32r2 log r + 24r log r −
640
3
15
5r
45r r2 3r3 9r4
冉
冊
Matching to this solution then gives us enough boundary
conditions to completely solve for the adjoint field within the
Stokes region,
共45兲
We follow Chester and Breach18 by solving this equation
using Green’s function techniques, consigning the details of
the calculation to Appendix B. Application of the flux func-
冉
⫻ ␭2共1 − ␤2兲.
2
E␭2 关E␭2 ⌽2 + 共1 − ␤2兲L␭⌽2兴 = − H␭关⌽1,⌿1兴
冊
9 359 8
9␤ 9
+
␭共1 − ␤2兲 +
− log ␭
64 100 5
32 32
1
8
11
1
8
− ␥ − log 2 − ␤ − ␤2 − 共1 + ␤兲
5
3
2
60
2
共44兲
Evaluating O共Re 兲 contributions within the Oseen layer, we
arrive at an equation,
冉
冊
1 1
5 3 5
3
9
2r2 − 3r + 1 − + 2 +
r − + 2
64
40
2 2r
r r
冊
冊册
Q2
共48兲
Additionally, because there is no term at O共Re log Re兲 in the Oseen-region expansion, we must introduce an additional term
at O共Re2 log Re兲 for matching,
␾2L =
冉
冊
9 2 3r 1
r −
Q1 .
+
20
2 2r
共49兲
In order to determine the effect of O共Re3兲 inertial terms on the first drag-minimizing step, we must solve for both the
stream function and adjoint flux function since the analysis of Chester and Breach18 does not extend to this order. It suffices
for demonstration that symmetry is indeed broken to compute the Q3 and Q4 components of both of these functions. In order
to calculate the Q2-component of the stream function we would need to extend the expansion of ⌿2 up to O共␭3兲, and in order
to compute the Q1-component, we would need to make a lengthy asymptotic study of ⌿3. For the stream function ␺, we obtain
␺3 = ¯ +
冋 冉
冉
冊 冉
5
3
7
9 16r3 129r2 377r 76 32 621 144 log r 221 12 977 6 log r
−
+
−
+ 3
−
r3 −
−
+
+
2 +
3 +
3
640 3
10
3
160
18
2520r
35r
36r
2520r
7r
2r 2r
冊册
+
3 5r4 67r3 677r2 237r 24 log r 2627 72 log r 821 45 log r 640 019 211 39 log r
−
+
−
−
−
+
−
+
+
−
+
224 3
7
30
112
8
7
7r
336r
28r2
73 920r2 32r3
14r3
−
155 log r 306 839
1
Q4 .
4 −
4 +
132r
73 920r
6r5
冊
Q3
共50兲
In order to compute the contribution at O共Re3 log Re兲, we must also extract all of the terms proportional to r2 log r from ␺3
and this can be done without knowing the full functional form of the Q1 and Q2 components,
␺3 = ¯ + 809 r2 log r共− 49 Q1 + Q2兲 + ¯ .
共51兲
18
We can also solve the O共Re3 log Re兲 problem directly to arrive at
冉
␺3L = d r2 −
冊
冉
冊
3r 1
27 2 3r 1 1
1
Q1 +
r −
+
+ −
+ 2 Q2 ,
160
2 2r
2 2 2r 2r
共52兲
with d as an unknown constant that must be deduced from matching to the Oseen layer solution,
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093606-9
Phys. Fluids 20, 093606 共2008兲
Symmetry unbreaking
⌿3L = D共1 + ␤兲共1 − e−␭共1−␤兲/2兲.
共53兲
27
d = − 320
27 18
D = − 80
.
and
Application of the matching condition 共16兲 then gives
A similar sequence of steps allows us to pick out the O共Re3 log Re兲 contributions to the adjoint flux function,
␾3 = ¯ +
冋 冉
冊 冉
12 977
3
7
221 9 log r
5
3 3 387r2 377r 19 27 log r 32 621
r −
+
+
r4 −
+
+
−
+
−
+
40
160
4
160
96
35r
13 440r 192r2 56r3
13 440r3
2r 2r
冉
+
31 989 486 log r 3269 1 911 443 495 log r
1
511r2 1323r
+ 36 log r +
−
+
140r4 − 132r3 +
−
−
−
31 360
5
20
2
r
2r
880r2
r2
−
109 857 2835 log r 1 101 527 12 log r 503 105
+
+
+
− 5 +
Q4 .
40r3
11r4
880r4
r5
5r
11r6
冊
In order to determine ␾3L, we isolate the component of the
adjoint flux function that varies with r as r2 log r,
␾3 = ¯ +
81r2 log r
Q1 + ¯
160
matching then yields
␾3L =
冉
冊
共55兲
6561 2 3r
81
1
r −
+ 2 Q1 +
51 200
320
2 2r
冉
⫻ r2 −
冊
1
3r 1 1
+ −
+ 2 Q2 .
2 2 2r 2r
共56兲
In terms of the stream and flux functions, the 共dimensional兲 shape derivative 共4兲 takes a form
␣⬀
冉 冊冉
␩U2
a2
−
Er2␺Er2␾
r2 sin2 ␪
+
冓 冔冊
Er2␺Er2␾
r2 sin2 ␪
.
共57兲
952 431
Re3Q4 .
2 759 680
Q3
共54兲
that under the Oseen approximation, the drag upon a projectile is independent of its degree of asymmetry; perfect projectiles in Oseen flow remain exactly fore-aft symmetric up
to arbitrarily large Reynolds numbers. It has previously been
shown that weak 关O共Re兲兴 inertial corrections to the drag
upon a moving body are independent of whether it moves
forward or backward through the fluid.14,19 Because of this,
such weak corrections may be directly computed by solving
Oseen’s equations throughout the entire of the fluid-filled
domain, allowing the different physical balances that operate
within Stokes and Oseen layers to be overlooked.13 We postulate that the drag-determining features of the Navier–
Stokes equations up to moderate Reynolds numbers can
similarly be captured by an approximation that has the same
features of linearity and reciprocity under reversal in the direction of flight11 as Oseen’s equations.
One candidate set of equations with these properties has
Inputting our expansions for ␾ and ␺ we see that asymmetric
terms start to appear in this expression only at O共Re3兲, and
include 共in addition to a Q2 component that we have not
determined兲 a term that provides the requisite symmetry
breaking effect,
␣as =
冊册
共58兲
V. DISCUSSION
We have presented a surprising property of perfect
projectiles—bodies of prescribed volume that are shaped to
experience the minimum possible drag when traveling
steadily through a fluid—although the flow around such bodies becomes markedly asymmetric at moderate Reynolds
numbers, the projectiles themselves remain quite fore-aft
symmetric.
We have shown, by asymptotic study of the Navier–
Stokes drag upon an arbitrarily shaped projectile, that there
are small 关O共Re3兲兴 asymmetries in the perfect projectile
shape even in the limit of arbitrarily weak inertia. However,
the persistence of near-symmetry up to moderate Reynolds
numbers remains unexplained.
Our study of the Oseen equations offers one possible
explanation for the symmetry “unbreaking.” We have shown
FIG. 3. Sketch of the numerical domain showing boundary conditions applied on far field boundaries 共gray兲 and on the projectile boundary 共black兲.
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093606-10
Phys. Fluids 20, 093606 共2008兲
Roper, Squires, and Brenner
been proposed by Carrier 共cited in Refs. 20 and 21兲, who
sought an approximation that correctly captured the flow
field within the immediate neighborhood of the body and in
the far field without requiring fidelity at intermediate scales.
He fastened upon a variant of the Oseen approximation in
which the Reynolds number featuring in the Oseen equation
共8兲 is allowed to differ from the value that is put into the
Navier–Stokes equations. By fitting to experimental data,
Carrier determined that setting ReO = 0.43 ReNS gives fairly
good accord for the steady fluid drag upon flat plates,
spheres, and cylinders from ReNS = 5 up to around 20.
The persistence of near symmetry in perfect projectiles
up to much larger Reynolds numbers than this is indirect
evidence that if we consider a broader class of possible dependencies of ReO upon ReNS, then an approximation in the
same spirit may correctly render the drag-determining features of the flow. This regime of Reynolds numbers, which
spans a number of flow transitions including the onset of
vortex shedding from a bluff body and breaking of wake
symmetry, has been notoriously inaccessible to asymptotic
study 共Ref. 9, p. 327兲, and linearity and reciprocity under
flow reversal must hitherto have been considered severe
impediments to any theory that aspires to capture the
drag-determining characters of the flow. Modifications in
Oseen’s equation provide us with a new way of thinking
about flow in this most difficult of regimes and we discuss
their strengths and limitations more fully in a follow-up
paper.
Although the mechanisms underlying the persistence
of near symmetry in the shape of the perfect projectile
require additional elucidation, fore-aft symmetry provides us with one robust, though unexpected, signature
of drag minimization against which the shapes of small,
rapidly moving, bioprojectiles may be compared.
ACKNOWLEDGMENTS
This work was supported by the Kao and Kodak Fellowships 共to M.R.兲 and by the NSF Division of Mathematical
Sciences. Discussions with Howard Stone and Silas Alben
are gratefully acknowledged.
lengths of adjacent triangles not permitted to exceed 1.18
away from these boundaries. N ⬇ 60 equally spaced mesh
points are selected on the shape boundary and evolved according to the update rule dx / d␮ = ␣n, with alpha the
optimal-shape perturbation 共4兲. The system of equations was
integrated with a low-order variable step-size explicit
scheme 共the MATLAB function “ode23”兲. Convergence of the
projectile to its perfect shape was tested for by measuring the
convergence of the integrand J to a uniform value along the
shape boundary and from the computed total drag upon the
projectile. A typical sequence of intermediate shapes between initial guess and optimal shape is shown in Fig. 4共a兲.
In panel 共b兲 of the figure we show the approach to optimality
via the convergence in the drag and of a measure M ␣ of the
uniformity of the shape gradient 共4兲 over the boundary of the
shape defined by
M␣ ⬅
冏冕
冕
⳵⍀
兩␣兩dS
⳵u ⳵w
␩ ·
dS
⳵⍀ ⳵ n ⳵ n
冏
共A1兲
.
Finally, the convergence of ␣ to zero over the entire of the
body is shown explicitly in Fig. 4共c兲 at selected values of the
integration parameter ␮.
APPENDIX B: SENSITIVITY OF DRAG TO DIRECTION
OF FLIGHT AT O„Re3…
In order to fully compute the adjoint flux function ␾2
within the Stokes layer, we must solve, in the limit as
␭ → 0, for the corresponding Oseen flux function from
E␭2 关E␭2 ⌽2 + 共1 − ␤2兲L␭⌽2兴
= − H␭关⌽1,⌿1兴 − E␭2
=
冋
1 ⳵ 共⌿1,⌽1兲
␭2 ⳵ 共␭, ␤兲
册
9
共1 − ␤2兲兵− e−␭共1+␤兲/2关8 + 4␭共1 + 2␤兲
16␭4
+ ␭2共1 + ␤兲兴 + 2e−␭共1−␤兲/2关2 + ␭共1 + 3␤兲兴
APPENDIX A: NUMERICAL CALCULATION
OF PERFECT PROJECTILE SHAPES
Computation of the perfect projectile shape requires that
for each of the intermediate projectile shapes the coupled
Navier–Stokes and adjoint Eq. 共2兲 be solved numerically. To
do this, the physical and adjoint velocity fields, u and w, are
discretized by quadratic finite elements and the pressure and
adjoint pressure, p and q, by linear elements, and the complete finite element model is solved using COMSOL
Multiphysics. We impose far field boundary conditions upon
a cylindrical surface with radius R⬁ and length 2L⬁ 共see
Fig. 3兲. Over the analyzed range in Re it was found to be
sufficient to take R⬁ = L⬁ = 400. An irregular triangular mesh
is constructed using the COMSOL built-in function
“meshinit,” with approximately 300 triangle vertices distributed along the boundary of the projectile, the same number
along each of the symmetry axes, and the ratios of side
+ e−␭关4 + 4␭共1 + 2␤兲 + ␭2共3 + 5␤兲 + ␭3共1 + ␤兲兴其.
共B1兲
18
We follow Chester and Breach by solving this equation
using Green’s function techniques. Specifically, if we abbreviate the right hand side of 共B1兲 by F共␭ , ␪兲, then a particular
solution to the equation may be written as
⌽2共␭, ␪兲 = −
␭ sin ␪
4␲
冕冕冕
冕
⫻F共␭1, ␪1兲
␲
⬁
0
0
2␲
␭1 cos ␸1d␭1d␪1d␸1
0
1/2关共␭21 + ␭2 − 2t␭␭1兲1/2+␭ cos ␪−␭1 cos ␪1兴
0
⫻
−␣
1−e
d␣ ,
␣
共B2兲
where we have defined
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Phys. Fluids 20, 093606 共2008兲
Symmetry unbreaking
1
0.5
0.96
0.4
0.92
0.3
M
D/D0
093606-11
0.88
0.2
0.84
FIG. 4. Convergence from an initial
shape guess to the perfect projectile
shape at Re= 10. 共a兲 Sequence of
shapes at ␮ = 0 共lightest兲 0.1, 0.2, 0.4,
0.65 共darkest兲. The initial shape 共␮
= 0兲 is a spindle formed by revolving a
segment of a circle around its chord to
form an axisymmetric body with aspect ratio 1.5. 共b兲 Convergence in projectile drag 共solid curve, left axis兲 and
shape gradient uniformity measure,
M ␣ 共A1兲, 共dotted curve, right axis兲.
The drag on projectile D is nondimensionalized by D0, the drag upon a
sphere of equal radius a. 共c兲 Shape
gradient ␣ along a body meridian 共s
= 0 is the front of the nose s = 1 is the
rear of the tail兲 for ␮ = 0 共䊊兲, 0.1 共〫兲,
0.2 共䉱兲, 0.4 共䉲兲, 0.65 共䊏兲.
0.1
0.8
0
0.1
0.2
0.3
µ
0.4
0.5
0
0.6
t ⬅ cos ␪ cos ␪1 + sin ␪ sin ␪1 cos ␸1 .
共B3兲
Now in order to approximate this integral in the limit of
␭ → 0, we carve the integration domain into two regions: a
near field 0 ⬍ ␭1 ⬍ k and far field ␭1 ⬎ k, where we impose
a separation of scales ␭ Ⰶ k Ⰶ 1. In the near field we may
approximate
冕
␮
0
−␣
1−e
1
d␣ = ␮ − ␮2 + O共␮3兲,
4
␣
冕
=−
冕
2␲
共␭21 + ␭2 − 2t␭␭1兲1/2cos ␸1d␸1
0
=−
冕
2␲
0
␭␭1 sin ␪ sin ␪1
共␭21 + ␭2 − 2t␭␭1兲1/2
sin2 ␸1d␸1 ,
共B6兲
共B4兲
⌽共n兲
2 共␭, ␪兲 =
␭2 sin2 ␪
8␲
冕冕冕
k
0
␲
0
⫻␸1d␭1d␪1d␸1
冉
2␲
␭21 sin ␪1 sin2
0
F1共␭1, ␪1兲
共␭21
+ ␭2 − 2t␭␭1兲1/2
1
⫻ 1 − 关共␭21 + ␭2 − 2t␭␭1兲1/2 + ␭ cos ␪
4
共␭21 + ␭2 − 2t␭␭1兲cos ␸1d␸1
0
2␲
on integrating by parts. Thus the near field integral may be
approximated as
where we have defined a place-holder variable ␮ ⬅ 21
关共␭21 + ␭2 − 2t␭␭1兲1/2 + ␭ cos ␪ − ␭1 cos ␪1兴. We can then
greatly simplify the work required for the evaluation of the
integral by noting that
2␲
冕
冊
− ␭1 cos ␪1兴 .
2␭␭1 sin ␪ sin ␪1 cos ␸1d␸1
2
共B5兲
共B7兲
0
while
We may, moreover, make a small argument approximation to
F共␭1 , ␪1兲,
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093606-12
Phys. Fluids 20, 093606 共2008兲
Roper, Squires, and Brenner
F共␭1, ␪1兲 = sin2 ␪1
冋
27
cos
4␭31
␪1 +
册
9
共1
32␭21
␸1 integration is performed we are left with a single nonvanishing contribution,
− 6 cos ␪1
+ 13 cos2 ␪1兲 ,
共B8兲
⌽共f兲
2 =
in which we have retained terms up to O共1 / ␭1兲. Picking a
typical term, in order to evaluate the leading-order contribution 关at O共␭兲兴 to ⌽共n兲
2 , we must compute
共␭, ␪兲 =
⌽共n,1兲
2
冉
冊 冕冕冕
27␭2 sin2 ␪ 1
8
4␲
⫻
␲
k
0
0
2␲
0
1
d␭1d␪1d␸1
␭1
sin3 ␪1 cos ␪1 sin2 ␸1
⫻
冕冕
⬁
⫻
1 2
sin ␪ cos ␪ sin2 ␸
␭3
共B10兲
evaluated at ␸ = 0. It is not difficult to solve this equation by
decomposing the forcing function into a sum of spherical
harmonics:
冉冑
1 1
␹=
␭ 5
1
−
12
␲ 0 1
Y −
3 1 12
冑 冊
冑
2␲ −2 1
Y −
105 3 30
冑
⌽共f兲
2 共␭, ␪兲 ⬃ −
⫻
9
32 ␭
and we may compute the other integrals that arise from expanding the sum 共B7兲 by the same method,
⌽共n兲
2 共␭, ␪兲 =
冋 冉冊
9
9 16
127
k
+
␭ sin2 ␪ cos ␪ +
log
32
128 5
50
␭
− cos ␪ −
册
11 2
cos ␪ ␭2 sin2 ␪ .
30
共B13兲
In the far field we instead approximate
冕
␮
0
1 − e −␣
d␣ =
␣
冕
␭1/2共1−cos ␪1兲
0
⫻
1 − e −␣
d␣ − ␭共t − cos ␪兲
␣
共1 − e−␭1/2共1−cos ␪1兲兲
,
␭1
共1 − cos ␪1兲
2
共B14兲
continuing our expansion as far as terms of O共␭21兲. When the
261
3
9
9
log k + log 2 + ␥ −
10
2
10
200
冊
共B16兲
共B17兲
and combining these three expressions then gives us
⌽2共r, ␤兲 ⬃
共B12兲
冉
−␭共1+␤兲/2
兲,
⌽共h兲
2 共␭, ␪兲 = C共1 − ␤兲共1 − e
共B11兲
sin2 ␪ cos ␪ ,
␭2 sin2 ␪
4
up to terms of O共␭3兲. Finally, we may also add an arbitrary
multiple of the solution to the homogeneous equation
for ␭ ⬍ k. Thus
共␭, ␪兲 =
⌽共n,1兲
2
共B15兲
This integral may be directly evaluated in the limit of
k → 0, yielding
␲ 0
Y
7 3
2␲ 2
Y + O共␭兲
105 3
d␭1d␪1共1 + cos ␪1兲共1 − e−␭1/2共1−cos ␪1兲兲
F共␭1, ␪1兲
.
sin ␪1
and 共omitting the prefactor in parenthesis兲 we recognize this
to be the Green’s function solution to Poisson’s equation
ⵜ 2␹ = −
␲
0
k
共B9兲
共␭21 + ␭2 − 2t␭␭1兲1/2
␭2 sin2 ␪
4
冉
9␤ C
+
␭共1 − ␤2兲
32 2
+
9 359 8
8
8
1
− log ␭ − ␥ − log 2 − ␤
64 100 5
5
3
2
−
11 2 8C
共1 + ␤兲 ␭2共1 − ␤2兲.
␤ −
60
9
冉
冊
冊
共B18兲
By matching with the Stokes layer expansion at O共r兲 we
9
, which completes our derivation of Eq. 共46兲.
infer that C = 16
1
S. Vogel, Life in Moving Fluids: The Physical Biology of Flow, 2nd ed.
共Princeton University Press, Princeton, 1994兲.
S. Vogel, “Living in a physical world II: The bio-ballistics of small projectiles,” J. Biosci. 30, 167 共2005兲.
3
T. von Kármán, Aerodynamics 共McGraw-Hill, Columbus, 1954兲.
4
O. Pironneau, “On optimum profiles in Stokes flow,” J. Fluid Mech. 59,
117 共1973兲.
5
B. Bourot, “Numerical computation of optimal profile in Stokes flow,” J.
Fluid Mech. 65, 513 共1974兲.
6
O. Pironneau, “On optimum design in fluid mechanics,” J. Fluid Mech.
64, 97 共1974兲.
7
D. W. Kim and M.-U. Kim, “Minimum drag shape in two-dimensional
viscous flow,” Int. J. Numer. Methods Fluids 21, 93 共1995兲.
8
B. Mohammedi and O. Pironneau, “Shape optimization in fluid mechanics,” Annu. Rev. Fluid Mech. 36, 11.1 共2004兲.
9
G. Batchelor, Inroduction to Fluid Dynamics 共Cambridge University
Press, Cambridge, 1967兲.
10
W. Olmstead and A. Gautesen, “A new paradox in viscous hydrodynamics,” Arch. Ration. Mech. Anal. 29, 58 共1968兲.
11
W. Olmstead, “Reciprocal relationships in viscous hydrodynamics,” Acta
Mech. 21, 289 共1975兲.
12
I. Proudman and J. R. A. Pearson, “Expansions at small Reynolds numbers
for the flow past a sphere and a circular cylinder,” J. Fluid Mech. 2, 237
共1957兲.
13
W. Chester, “On Oseen’s approximation,” J. Fluid Mech. 13, 557 共1962兲.
2
Downloaded 29 Dec 2008 to 136.152.210.10. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
093606-13
14
Phys. Fluids 20, 093606 共2008兲
Symmetry unbreaking
H. Brenner and R. G. Cox, “The resistance to a particle of arbitrary shape
in translational motion at small Reynolds numbers,” J. Fluid Mech. 17,
561 共1963兲.
15
J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, 2nd ed.
共Martinus Nijhoff, The Hague, 1973兲.
16
S. Goldstein, “The forces on a solid body moving through a viscous fluid,”
Proc. R. Soc. London, Ser. A 123, 216 共1929兲.
17
S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected
Applications 共Dover, Mineola, 2005兲.
18
W. Chester and D. R. Breach, “On the flow past a sphere at low Reynolds
number,” J. Fluid Mech. 37, 751 共1969兲.
19
R. Cox, “The steady motion of a particle of arbitrary shape at small Reynolds numbers,” J. Fluid Mech. 23, 625 共1965兲.
20
H. Brenner, “The Oseen resistance of a particle of arbitrary shape,” J.
Fluid Mech. 11, 604 共1961兲.
21
B. P. Le Clair, A. E. Hamielec, and H. R. Pruppacher, “A numerical study
of the drag on a sphere at low and intermediate Reynolds numbers,” J.
Atmos. Sci. 27, 308 共1970兲.
Downloaded 29 Dec 2008 to 136.152.210.10. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp