ISSN 0567-7718, Volume 26, Number 3

This article was published in the above mentioned Springer issue.

The material, including all portions thereof, is protected by copyright;
all rights are held exclusively by Springer Science + Business Media.
The material is for personal use only;
commercial use is not permitted.
Unauthorized reproduction, transfer and/or use
may be a violation of criminal as well as civil law.
Acta Mech Sin (2010) 26:383–389
DOI 10.1007/s10409-009-0329-4
Author's personal copy
RESEARCH PAPER

Attitude dynamics of a cylinder floating in immiscible fluids

Jiann Lin Chen

Received: 3 September 2009 / Accepted: 10 September 2009 / Published online: 25 December 2009
© The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2009

Abstract This study elaborates the attitude dynamics of a Aerostatic aircraft, such as airships and hot air balloons,
cylinder floating in two immiscible fluids. A cubic polyno- remain aloft using buoyancy. Airship’s ability to hover in one
mial was derived based on the attitude angle, weight, center of place for an extended period outweighs the need for speed and
gravity, and the density ratio of fluids. The numerical solution maneuverability [4]. Some of these concepts can be explored
was validated by experimental data. Under prescribed con- for micro air vehicle applications, with the scaling laws and
straints for the physical model, we have found that multiple aerodynamics documented in Shyy et al. [5–7].
solutions exist for cases with no radially biased center of The magnitudes and locations of weight and buoyancy
gravity. When the center of gravity is biased, the attitude in bodies are important factors in their motion or floating
angles change abruptly around some critical values, which is characteristics in fluids [8,9]. In fact, some vehicles have
related to the density ratio. Moreover, the attitude angles are been designed to be operational for certain particular require-
less sensitive to the varying density ratios when the cylinder ments by virtue of the movement of their center of grav-
is heavier. The results also reveal that the cylinder tends to ity (CG) [10,11]. One application that takes advantage of
be vertical for nearly the whole range of density ratios when the attitudes of a body is the Floating Instrument Platform
the center of gravity is slightly biased radially. (FLIP), as described by Fisher and Spiess [12]. This was the
American research ship designed to float not only like a con-
Keywords Floating cylinder · Attitude angle · ventional surface vessel with its keel horizontal but also with
Density ratio its keel vertical after suitable flooding. Dugdale [13] showed
that a cylinder with uniform material having a length less
than 0.707 of its diameter will float with its axis vertical, and
1 Introduction when the length exceeds its diameter it will float with its axis
horizontal. Obviously, the relations among weight, buoyancy
It is easily found that bodies float in fluids in our environ- and their locations are strongly related to bodies’ states of
ment, whether they are organisms or man-made machines. equilibrium statically and dynamically [14,15]. Moreover,
For instance, waterfowls float on water. Marine organisms some authors have proposed techniques to measure phys-
float underwater. Plankton float or drift in great numbers in ical properties of a general body, e.g. the floating attitude
fresh or salt water, especially at or near the surface. Arthro- and CG [16,17]. Our previous work [18] modeled an ana-
pods are carried upwards into the atmosphere by air currents lytical expression for the inclinations of a cylinder floating
and may be found floating several thousand feet up [1,2]. on a water plane. The relations of the cylinder’s inclination
Francis [3] found that velella could float and sail as much versus its longitudinal location of CG were elaborated and
as 63◦ off the direction of the wind. Surface ships and sub- have been used as the benchmark solution for further study
mersibles are man-made vehicles and Archimedean floaters. in numerical simulation for attitudes of floating bodies with
arbitrary configurations [19].
J. L. Chen (B)
Bodies can float in fluid layers with different densities in
Department of Mechanical and Automation Engineering,
I-Shou University, Kaohsiung, Taiwan, China a real environment such as a pycnocline, a rapid change in
e-mail: james88@isu.edu.tw water density with depth. The following study extends an

123
384 Author's personal copy J. L. Chen

analytical expression for the inclination of a floating slender below. Inside the upper fluid, the upper body can be expressed
body on a water plane by Chen and Chen [18]. Two fluids as two portions. The first is a semi-cylinder body with the
sharing an interface are considered to involve a cylinder of buoyancy force B1 acting upwards. The second is a cylin-
interest. The mathematical model was verified and validated. drical body above the interface with the buoyancy force B2
Results account for attitudes of the cylinder floating at the acting upwards. The portion of the body under the interface
interface between two fluid layers with different densities. thus is fully submerged inside the lower fluid. Similarly, one
The characteristics of the governing equation are first inves- portion of the lower body is a semi-cylinderical body with the
tigated for various parameters and the influences of density buoyancy force B3 acting upwards, and the other is a cylin-
ratio are elaborated. Finally, some conclusions are presented drical body with the buoyancy force B4 acting upwards. This
according to our findings. implies the identity, B = B1 + B2 + B3 + B4 . In order to
analyze this problem properly, the assumption has been made
that the top end of the cylinder is situated above the inter-
face and the other end is below the interface with L/D > 1.
2 Analysis The interface is considered to be stationary in this work. In
reality, the interfacial dynamics needs to be addressed as a
2.1 Physical model moving boundary problem [20], which outside the scope of
this investigation.
In some environments, bodies float in fluids in immiscible
layers having different densities. Figure 1 shows a schematic
representation of a cylinder floating between two fluids with 2.2 Mathematical model
an interface. The negative ratio in fluids, (dρ/dy), with y
pointing upward, is the vertical density gradient, which keeps In Fig. 1, the Cartesian coordinate, x yz, is arbitrarily
the layering stable. The lift for the cylinder during floatation employed at the interface and fixed with respect to the Earth.
is static lift; that is, only buoyancy force is considered herein. Also, a body-fixed coordinate system is defined at one end of
The cylinder settles until the buoyancy force is equal to the the cylinder with the axes ξ in the longitudinal direction and
weight and rotates until the centers of buoyancy (CB) and η in the radial direction. Thus, the condition for a fixed atti-
gravity lie at the same point, or in the same vertical line for tude can be easily stated from fundamental identities of static
certain special cases. The cylinder with height over diame- mechanics in lump system. The net force of the cylinder in
ter ratio L/D floats in the fluids at an attitude angle θ , with Fig. 1 is
weight W equal to the total buoyancy B. Moreover, if the fluid 
F = Bj − W j = B1 j + B2 j + B3 j + B4 j − W j = 0,
above the interface is air, it is reasonable to assume that the
buoyancy exerted by air can be neglected as in our previous (1)
work [18]. Before proceeding, the cylinder is divided into
two portions separated by an interface. The portion above where j is one of those orthogonal unit vectors i, j, k for the
the interface contains fluid that is less dense than the fluid x yz coordinate system. The net moment on the cylinder is

M = r B1 × (B1 j) + r B2 × (B2 j) + r B3 × (B3 j)
+r B4 × (B4 j) + r W × (−W j) = 0, (2)

where r B1 , r B2 , r B3 , r B4 , r W are position vectors of different

forces acting on the cylinder; for instance, r B2 = ξ B2 I+η B2 J
is the position vector of buoyancy force B2 , where I, J are
orthogonal unit vectors for the ξ, η coordinate system.
Figure 2 shows the sketch of the semi-cylinder in Fig. 1.
Physically, the buoyancy force B1 or B3 acts through the
centroid of the respective semi-cylinder.
From the fundamental principle of calculus, it follows that
 2R cot θ  R 
ξ tan θ−R 2 R − η dηξ dξ
2 2
0 5 
ξ̄ B1 =  2R cot θ  R  = L , (3)
ξ tan θ−R 2 R − η dηdξ
2 2 16
0
R 
2 R 2 − η2 (R + η)ηcotθ dη 1
Fig. 1 General sketch for a cylinder floating at the interface between η̄ B1 = −R
R
 = D, (4)
two fluids −R 2 R − η (R + η) cot θ dη
2 2 8

123
Attitude dynamics of a cylinder floating Author's personal copy 385

Constraint (1): r L/D < ω < L/D

The cylinder floats at the interface between two fluids. Thus
the total buoyancy force exerted by both fluids will follow
the relation

π D 2 Lγa π D 2 Lγb
<B< . (6)
4 4

Rewrite Eq. (6) to get constraint (1).

Constraint (2): 0 < r < 1
r indicates ra /rb and ra > rb indicates negative vertical
density gradient.
Constraint (3): 0.5L/D < ξW  < L/D

This means the CG is located at the lower part of the cyl-

inder in the longitudinal direction.
Constraint (4): −0.5 < ηW  <0

The CG is located between the centerline plane and the

Fig. 2 Sketch for the semi-cylinder with length L  surface of the lower part of the cylinder. This constraint
assures that the neutral stable situation is avoided.
where R is the radius of the cylinder and ξ̄ B1 , η̄ B1 are coordi- Constraint (5): θ = 90◦ as ηW  =0
nates of the centroid of the semi-cylinder above the interface. Two possible solutions can be found for the case of ηW = 0.
It is noteworthy that η̄ B1 is independent of the attitude angle However, only the solution of a 90 degree inclination is
of the cylindrical body. Similarly, we can obtain the centroid, considered meaningful. Hydrostatically, this means that a
ξ̄ B 3 , η̄ B 3 , for the semi-cylinder below the interface. Thus, cylinder with the CG lying along its centerline and apart
after using some geometric relations and using the identity from its middle floats upright.
L 2 = Dcotθ , we have the governing equation simulating the Constraint (6): 0 <  < L/D
floating cylinder in nondimensionalized form as The most extreme inclination of the floating cylinder is
   2 with its upper end of the cylinder just above the interface
1 L L
 + 16
3
ω +r
2
− 2ω and its lower end just below the interface. Combining the
(r − 1)2 D D facts of cotθ = L/D and θ ≤ 90◦ yields this constraint.
ω   ω
− 32 ξW + 2  = 32ηW , (5)
r −1 r −1
3 Numerical techniques
where  = cotθ, ξW  = ξ /D , η = η /D and ω =
W W W
W/τb and τb indicates the displacement of unit length below In this section, numerical techniques for finding the roots
the interface and is equal to π D 3 γb /4, with γb representing of the polynomial, Eq. (5), are employed. Those prescribed
weight density of the fluid below the interface. For clarity, we constraints are used to specify the meaningful boundary con-
have defined the ratio of displacement across the interface, ditions for implementing the root finding scheme.
r = τa /τb , where τa indicates the displacement of unit length
above the interface similar to τb . In other words, r indicates 3.1 Root-seeking techniques
the ratio of fluid densities across the interface. Equation (5)
shows that the attitude angle is a function of dimensionless Since some polynomials are ill-conditioned except in linear
terms ξW , η , ω and r , which represent the longitudinal loca-
W problems, root finding invariably proceeds by iteration. Start-
tion of CG from the top end, the biased location of CG in ing from some approximate trial solution, a useful algorithm
radial direction, the weight magnitude of the cylinder and will improve the solution until a predetermined convergence
the ratio of fluid densities across the interface, respectively. criterion is satisfied. Sometimes one’s knowledge of the phys-
Generally speaking, it is not easy to obtain the analytical ical problem will suggest a starting value [21]. In the present
solutions, so the following numerical method is employed. study, two schemes were used to approximately find the roots
of Eq. (5). The first scheme is to substitute a guessed value
2.3 Constraints for the physical model into the polynomial to see if the residue approaches zero,
starting incrementally from the lowest limit to the utmost
Before going further, several constraints are given based on limit in the interval of interest. If so, the approximate root
physical sense for solving Eq. (5). has been found and guessing continues to find another root.

123
386 Author's personal copy J. L. Chen

By doing so, a scheme called FORCE has been developed

to calculate brackets which contain one or more roots of the
present polynomial. There are some reasons that FORCE
successfully seeks out roots in the present study. First, the
meaningful range of the inclination angle is known to be
0◦ < θ < 90◦ . Therefore, the maximum interval and input
data can be clearly assigned as FORCE is employed. Sec-
ond, making a rough graph of the polynomial indicates that
no even roots existed in our present governing equation. This
means FORCE can always find roots if there are any. After the
approximate roots found by FORCE, the alternative scheme,
the secant method then was used to get more accurate esti-
mates of roots from the selected bracketed regions around
those estimates.
The rate of convergence for the secant method has been
proven to be much faster than the linear method. The secant
method, however, has the disadvantage that the root does not
necessarily remain bracketed. It may also shoot off to find a
root different from the expected bracket. Around the roots, Fig. 3 Comparisons of experimental data and theoretical prediction
in some cases where the secant method failed, as occurred
in the present study, FORCE was executed instead but in
were evaluated to be 4.756 by the classical root mean square
smaller increments. In our experience, running FORCE uses
error. Overall, the comparison is satisfactory and indicates
one tenth of the CPU time than executing the secant method,
a successful validation. Figure 3 also reveals that it is dif-
with convergence criteria set at the same order of magnitude.
ficult to accurately measure the location of a body’s CG,
especially in a radial direction. Obviously, minor errors in
3.2 Verification and validation measurement can cause significant deviations between ana-
lytical and experimental data in regions with steeper slope.
Lacasse et al. [22] stressed that any validation exercise looses
its significance and credibility if prior verifications of the
code and the calculations are not performed. Therefore, our
4 Results and discussions
model was verified numerically via the FORCE scheme by
two cases and validated experimentally by one case.  = 6
Cases with typical parameters, such as L/D = 8, ξW
and ω = 4, are used as a model, and one of them is chosen
Case 1: L/D = 8, ξW  = 7, η = −0.25, ω = 1 and to be a variable while the others are fixed in case studies.
W
r =0
Substituting the above parameters into Eq. (5), the exact 4.1 Cylinder without radially biased center of gravity
solution, θ , can be found as 14.301, while the correspond-
ing numerical solution by FORCE is 14.032. It is noteworthy that multiple solutions exist as ηW  equals
 = 6, η = 0, ω = 2 and r = 0 
zero. Substituting ηW = 0 to Eq. (5), one solution is θ = π/2,
Case 2: L/D = 8, ξW W
Similarly, substituting the above parameters into Eq. (5) which means the cylinder floats in the fluids vertically. But
gives the exact solution, 7.231 or 90, while the correspond- the other solutions are as follows:
ing numerical solutions are 7.238 and 89.99.   2
ω  1 L L
Case 3: Experiment in a water tank. = 32 ξ −2−16 ω2 +r −2ω .
r −1 W (r − 1)2 D D
(7)
The corresponding experiment was carried out in a water
tank and compared with the numerical results as shown in When the  approaches zero, the above identity can be
Fig. 3. The attitudes θ were obtained as CG moved respect rewritten and the critical value, ξC , with ω = 4 will be
to various ξW with given parameters ω = 3.515, L/D =
 1−r 2r
7.923, ηW = 0.116 and r = 0. Ninety measured data are ξC = 6 − − . (8)
plotted in Fig. 3, which reflects discrepancy between theory 64 1−r
and experiment to some extent, especially in the region with Figure 4 shows the relations of cases with the above condi-
steeper slope. The errors estimated between these data sets tions. Taking the case, r = 0, for example, the critical value,

123
Attitude dynamics of a cylinder floating Author's personal copy 387

ξC , is 5.984. Since the longitudinal location of the CG larger  for various η
4.2.1 Comparisons of θ with ξW W
than the critical value, the solution shows that the cylinder
floats close to vertically, that is θ = 90◦ . Although there is a The comparisons of θ with ξW  for different η with ω = 4
W
solution curve for a specific r , only the solution of θ = 90◦ and density ratio, r = 0 or r = 0.2, are shown in Fig. 5. This
is physically meaningful, which is indicated by a solid sym- shows the attitude approaches vertical as the CG approaches
bol for each r in Fig. 4. Also note that with larger density the end of the cylinder; that is, ξW approaches 8. It is note-

ratios, the critical ξC in Fig. 4 is smaller. Physically, a larger worthy that there is a steep rise in each curve for different
density ratio results in larger buoyancy forces by B1 and B2 , ηW , especially the case of η = −0.001 with the drastic
W
which will enhance the cylinder floating upright even if the rise. Obviously, the large variations are related to the corre-
longitudinal CG is close to the middle of the cylinder, which sponding cases of ηW  = 0, as revealed in Fig. 4. In other

is ξW  = 4 for the present case. Moreover, if the CG is very words, a greater change in attitude is expected as the CG is
close to the critical position, ξC , a slight longitudinal bias shifted closer to the critical location of ξC = 5.98 for the
to the CG can cause the attitude to vary abruptly. Although case of r = 0, or ξC = 5.49 for r = 0.2. To the right hand
the solutions indicated by hollow symbols in Fig. 4 have no side of the critical locations in Fig. 5, the smaller the ηW is,

physical meaning, the abrupt rise in attitude around the crit- the more vertical the cylinder appears. Moreover, when the
ical value helps to explain the following phenomenon. CG is located close to the bottom surface of the cylinder for
each case, the cylinder appears less vertical, especially for the
case of ηW = −0.49. In other words, a radial bias to the CG
4.2 Influences due to various density ratios
causes the attitude to depart from the vertical. Besides, the
attitude angles for r = 0.2 are larger than for cases of r = 0
Bodies can float in a pycnocline or any situation with a strat-  and η . Obviously,
with respect to the same positions of ξW W
ification of different densities. In this section, the influences
the fluid above the interface adds to the overall buoyant force
due to the different densities across the interface is elaborated
for the cylinder to float more vertically.
with respect to the specific conditions, L/D = 8, ω = 4 or
ξW = 6. When the density ratio, r , becomes zero, it indicates

that the density of the fluid above the interface is negligible 

4.2.2 Comparisons of θ with ω for various ηW
compared to the one below the interface. In fact, the cases
of r = 0 have already been thoroughly investigated [18] Figure 6 shows comparisons of θ with ω for different ηW  with
with bodies floating on the sea, where the density ratio of 
ξW = 6 and density ratio, r = 0 or r = 0.2. The cylinder’s
air to water is about 1/800. Moreover, the three conditions attitude appears more vertical as its magnitude of dimension-
of ηW = −0.01 , −0.1 or −0.49, representing CG slightly,
less weight approaches to the maximum, which is 8. Note that
moderately or sharply, respectively, biased radially from cen- there exists a minimal value for each curve either for cases of
tral line of the cylinder, have been discussed herein. r = 0 or r = 0.2. This means that there will be two solutions

 for various r with η = 0 and ω = 4

Fig. 4 Comparisons of θ with ξW  for various η and r with ω = 4
Fig. 5 Comparisons of θ with ξW
W W

123
388 Author's personal copy J. L. Chen

 with ξ  = 6 and ω = 4
Fig. 7 Comparisons of θ with r for various ηW
 and r with ξ  = 6
Fig. 6 Comparisons of θ with ω for various ηW W
W


4.2.3 Comparisons of θ with r for various ηW
of weight with respect to a specific attitude angle for the  with
Figure 7 shows comparisons of θ with r for different ηW
given conditions in Fig. 6. It is significant that critical value 
ξW = 6 and ω = 4. As the density ratio approaches zero, the
for dimensionless weight exists only for the case of r = 0,  that can
attitude angle becomes a specific value for each ηW
where the critical ω can be obtained from Eq. (5) by setting
 = 6, L/D = 8 and r = 0. Two critical values of be derived from Eq. (5). It can be reasonably assumed that in
 = 0, ξW
this situation the fluid above the interface is air and the other
ω can thus be found, one is 3.97 and the other one is 0.032 for
is water. On the other hand, the cylinder appears vertical in
r = 0. But for the case of r = 0.2, no solutions for critical
fluids no matter how radially biased the CG is as the density
ω can be found from Eq. (5). For cases of r = 0, substantial
 . That is, ratio approaches unity. This situation, where r is close to 1,
changes occur around two locations for various ηW
exists as the cylinder is fully submerged in a fluid. For the
a greater attitude change is expected when the weight mag-
case of a slight radial bias to CG, the attitude of the floating
nitude increases toward the critical region where ωC is 3.97.
cylinder appears upright for almost whole range of density
At around 3.97 for the case of r = 0, slightly larger mag-
ratios except when r is close to zero. We can conclude that
nitudes of weight cause substantial changes, especially for
 = −0.001. But when ω is between 1 and the radially biased weight plays a more straightforward role
less biased CG, ηW
than the density ratio in the body’s attitude. Other than the
3, attitude variations caused by radially biased CG are neg-  = −0.001, it is obvious that the fluid above the
case of ηW
ligible. Moreover, as weight magnitudes approach the other
interface help enhance the cylinder’s buoyant force, making
critical value, ω = 0.032, for cases of r = 0, each case of dif-
 gives almost the same variations. Abrupt changes its attitude more upright.
ferent ηW
in attitude occur as weight magnitudes approach the critical
ω for cases of r = 0. Generally speaking, the overall trends of
the curves in Fig. 6 are similar for cases of r = 0 and r = 0.2. 5 Concluding remarks
As the weight magnitude approaches to the maximal value,
ω = 8, the attitude deviates more from being upright if the A cubic polynomial with nondimensionalized parameters,
radial CG is more biased. Either r = 0 or r = 0.2 gives including the attitude angle, weight, center of gravity, and
the attitude angles indistinguishably for the same radically density ratio, has been developed to describe the attitudes of
biased CG as ω close to the maximum. But the deviations of a cylinder floating at the interface between two fluids. After
the cylinder’s attitudes between cases of r = 0 and r = 0.2 verification and validation, we obtain several conclusions.
become large when the dimensionless weights are less than
the critical value. The attitude variations for cases of r = 0.2
are moderate, unlike cases of r = 0. In fact this phenomenon (1) The governing polynomial of degree 3 is expressed
is revealed by Eq. (5) that no solutions for 90◦ attitude angle in terms of dimensionless attitude, weight magnitude
are found for cases of r > 0. and location as well as the density ratio of two fluids.

123
Attitude dynamics of a cylinder floating Author's personal copy 389

Practical solutions have been found according to the 7. Shyy, W., Lian, Y., Tang, J., Liu, H., Trizila, P., Stanford, B., Bernal,
prescribed constrains. L.P., Cesnik, C.E.S., Friedmann, P., Ifju, P.: Computational aero-
dynamics of low Reynolds number plunging, pitching and flexible
(2) The attitude angles of the floating cylinder in fluids wings for MAV applications. Acta Mech. Sin. 24, 351–373 (2008)
change sharply with its parameters, such as weight 8. Clayton, B.R., Bishop, R.E.D.: Mechanics of Marine Vehicles.
magnitude and location of CG, around certain ranges E. & F. N. SPON, London (1982)
near their critical values. These critical values vary with 9. Burcher, R., Rydill, L.: Concepts in Submarine Design. Cambridge
University Press, Cambridge (1999)
respect to the corresponding density ratios. Besides, the 10. Light, R.D., Morison, J.: The autonomous conductivity–tem-
fluid above the interface helps to enhance the total buoy- perature vehicle: First in the seashuttle family of autonomous
ancy, causing the cylinder to be more upright. underwater vehicle’s for scientific payloads. Proc. Oceans ’89 3,
(3) In the condition of a vertically floating cylinder, two 793–798 (1989)
11. Woolsey, C.A., Leonard, N.E.: Moving mass control for underwa-
critical weights can be found for cases of assuming the
ter vehicles. Proc. 2002 Am. Control Conf. 4, 2824–2829 (2002)
fluid’s density above the interface is ignorant. There- 12. Fisher, F.H., Spiess, F.N.: FLIP—floating instrument platform.
fore, two solutions of weight with respect to one atti- J. Acoust. Soc. Am. 35, 1633–1644 (1963)
tude angle for all cases can be obtained. Furthermore, 13. Dugdale, D.S.: Stability of a floating cylinder. Int. J. Eng. Sci. 42(7),
691–698 (2004)
the attitude angles are less sensitive to varying density 14. Barlow, J.J., Nicholson, K.: An introduction to submarine longitu-
ratios when the cylinder is heavier or its center of gravity dinal static and dynamic stability concepts. International Sympo-
is biased slightly. The attitude angle is clearly affected sium on Naval Submarines, London, UK (1983)
by the density ratio when the cylinder is lighter or its 15. Zubaly, B.R.: Applied Naval Architecture. Cornell Maritime
Press, Centreville (1996)
center of gravity is more radially biased. 16. Buyanov, E.V.: A method and device for determining a center of
gravity. Meas. Tech. 35(8), 919–922 (1992)
17. Schechter, S.E., Leyenaar, A.R.: Center of gravity locating method.
References United States Patent 5081865, USA (1992)
18. Chen, J.L., Chen, C.I.: The modeling and numerical solutions for
the inclinations of a floating cylinder on a liquid surface. J. Chin.
1. Thurman, H.V.: Introductory Oceanography. Prentice-Hall, New
Soc. Mech. Eng. 27(2), 241–248 (2006)
Jersey (1997)
19. Chen, J.L., Wu, C.H., Hsu, C.C.: The modeling and simulation for
2. Vogel, S.: Life in Moving Fluids. Princeton University Press, New
attitudes of floating bodies with arbitrary configurations. In: Pro-
Jersey (1994)
ceedings of 2005 Spring Simulation Multiconference, SCS, San
3. Francis, L.: Sailing downwind: aerodynamic performance of the
Diego, CA., USA, 179–185 (2005)
Velella sail. J. Exp. Biol. 158, 117–132 (1991)
20. Shyy, W., Udaykumar, H.S., Rao, M.M., Smith, R.W.: Com-
4. Khoury, G.A., Gillett, J.D. (eds.): Airship Technology (Cambridge
putational Fluid Dynamics with Moving Boundaries. Taylor &
Aerospace Series). Cambridge University Press, UK (2004)
Francis, Washington, DC (1996)
5. Shyy, W., Lian, Y., Tang, J., Viieru, D., Liu, H.: Aerodynamics of
21. Gerald, C.F., Wheatley, P.O.: Applied Numerical Analysis.
Low Reynolds Number Flyers. Cambridge University Press, New
Addison–Wesley, Harlow (1984)
York (2008)
22. Lacasse, D., Turgeon, E., Pelletier, D.: On the judicious use of the
6. Shyy, W., Berg, M., Ljungqvist, D.: Flapping and flexible wings
k-e model, wall functions and adaptivity. Int. J. Therm.
for biological and micro air vehicles. Progr. Aero. Sci. 35, 155–
Sci. 43, 925–938 (2004)
205 (1999)

123