Development of immersed boundary methods for isothermal and thermal flows 9

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Chapter 9

Conclusions and Recommendations

9.1 Conclusions
In this thesis, several novel IBM solvers have been developed for fluid and
thermal flows involving complex geometries or moving objects/boundaries:
one for two dimensional incompressible viscous flows and two for thermal
flows respectively with Dirichlet and Neumann temperature conditions.
Additionally, a new version of boundary condition-enforced IBM in the
framework of NS solver is presented. Their abilities for dealing with fluid and
thermal problems have been well demonstrated by applications to various fluid
and thermal flows with geometric and dynamic complexities. Note that the
four methods proposed employ Dirac delta function in modeling the immersed
boundaries. In other words, the immersed boundaries are assumed to be have
finite thickness. Therefore they are truly immersed boundary methods.

The critical issue of the IBM lies in the accurate evaluation of body forces.
The conventional IBM generally calculates body forces explicitly, so the flow
penetration to the surface of the immersed objects happens. On the other hand,
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the velocity correction-based IBM, which was originally developed within the

framework of LBM solver, has proven to be effective in exactly enforcing the
no-slip condition on the immersed boundary. In this regard, the present thesis
firstly extends the method to a Navier-Stokes (NS) solver-based version where
the popular primitive variable formulation is solved. Combined with the
fractional step technique, the body force in the modified momentum equation
is implicitly determined so that the no-slip condition on the immersed
boundary is accurately satisfied. The good performance of the new IBM is
extensively verified, first through the classical problems of flow over a single
stationary circular cylinder and flow interference between two side-by-side
circular cylinders, and then by the moving boundary cases such as
vortex-structure interaction around a transversely oscillating cylinder and
vortex-induced-vibration of an elastically mounted circular cylinder.

Most IBM solvers are established under the framework of pressure-velocity
formulation. For incompressible flows, the pressure-velocity formulation
suffers from some difficulties and is usually rather expensive to solve. In
contrast, the stream function-vorticity formulation has been recognized to be
more efficient for two dimensional incompressible viscous flows. Inspired by
the merits of the stream function-vorticity formulation and IBM techniques, a
stream function-vorticity formulation-based IBM solver is proposed, in which
the immersed object is modeled as localized vorticity sources and plays the
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Chapter 9 Conclusions and Recommendations
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role of vorticity corrections in the stream function equation. The main idea of
the method is to accurately satisfy the governing equation and boundary
condition, which is realized through velocity correction and vorticity
correction procedures. The velocity corrections, in the present model, are
evaluated in the principle of guaranteeing the no-slip condition on the

immersed boundary, i.e., the velocity at the boundary (Lagrangian) point
interpolated from the physical velocity at the Eulerian points equals to the
given boundary velocity. The vorticity corrections, on the other hand, are
directly evaluated from the first-order derivatives of velocity corrections by
two proposed approximation methods. The whole fluid solver and the two
vorticity correction methods were first validated by their applications to a
stationary boundary problem of flow over a circular cylinder. The obtained
flow characteristics agree very well with the published data in the literature
both qualitatively and quantitatively. Additionally, no streamline penetration
appears on the cylinder surface, implying that the boundary condition is
accurately satisfied. Then the proposed solver was applied to examples of flow
over a left moving circular cylinder and flow over an inline oscillating circular
cylinder, to further examine its capability in handling moving boundary
problems. Furthermore, a fluid-structure interaction problem, sedimentation of
a circular particle between two parallel walls, was examined. All the obtained
numerical results for the considered moving boundary cases were in very good
agreement with the previous reported ones in the literature, indicating the
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promising potential of the stream function-vorticity formulation-based IBM
for solving two dimensional incompressible viscous flows involving complex
geometries and moving boundaries.

To extend the IBM for heat transfer problems which are suffering from
complex or moving/deforming boundaries, two thermal IBM solvers were
developed. Firstly, a boundary condition-enforced IBM solver was suggested
for heat transfer problems with Dirichlet-type boundary condition, where the
heated immersed boundary is modeled as a set of heat sources added to the

energy equation as the source term. All the previous IBM solvers treated the
heat source term explicitly and no mechanism was implemented to enforce the
boundary condition for temperature. Thus the prescribed temperature on the
immersed boundary was not accurately satisfied and the accuracy of numerical
results was negatively affected. In the present boundary condition-enforced
IBM, the heat source/sink term is considered as unknown and determined
implicitly such that the energy equation and the corresponding thermal
boundary condition can be accurately satisfied. Furthermore, the critical issue
of how to effectively calculate the average Nusselt number in IBM was
properly addressed, as an additional contribution. Numerical experiment on
accuracy analysis showed a second-order accuracy of the proposed IBM solver.
The present method and proposed techniques to compute the Nusselt number
were then validated by both forced and natural convections where the obtained
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Chapter 9 Conclusions and Recommendations
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numerical results compared considerably well with available data in the
literature, implying that the suggested thermal solver and Nusselt number
evaluation techniques provide a wonderful tool for thermal problems with
Dirichlet condition. Secondly, a heat flux correction-based IBM was proposed
for thermal flows with prescribed heat flux condition. Note that this is the first
time IBM is applied to solve thermal flow problems with Neumann-type
boundary conditions. In the proposed solver, the heated immersed boundary
was regarded as localized heat sources whose evaluations and contributions
were considered through a heat flux correction procedure. To compensate the
differences between the calculated heat fluxes and the prescribed ones,
boundary heat sources arising from the heat flux differences were introduced
which were then distributed to the surrounding fluid as volumetric heat
sources to correct the temperature field. The whole process was intelligible

and easy to implement. The present solver, through a numerical analysis on a
model problem of heat conduction, was recognized to be of second order in
accuracy. Then its capability and efficiency were validated by well-established
examples like forced convection over a stationary heated circular cylinder and
natural convection in a horizontal concentric and eccentric annulus between
two circular cylinders. The numerical results obtained matched very well with
the reported ones, showing the high potential of the proposed IBM solver for
problems with Neumann conditions.

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Chapter 9 Conclusions and Recommendations
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Subsequently, the developed methods were applied to various two- and
three-dimensional fluid and thermal flows to further check their capabilities
for handling geometric complexity and moving boundaries. In two dimensions,
insect hovering flight where the insect wing undergoes a prescribed harmonic
translational and rotational motion near or away from the ground was firstly
simulated, which was followed by particulate flows where the motion of the
involved particle is not given prior but coupled and mutually determined with
the surrounding fluid. Obtained results showed that both types of moving
boundary problems are correctly predicted and good agreements with
available experimental and numerical results were observed. With the
confidence achieved from the above validations, the methods were utilized to
study the forced convective heat transfer from a transversely oscillating
circular cylinder in the wake of a stationary one, where it was found that the
flow behaviors and heat transfer characteristics were greatly affected by the
mutual influences of spacing and downstream cylinder excitation.

After successfully verifying their exciting performance in two-dimensional

applications, the proposed temperature correction-based and heat flux
correction-based IBM solvers are examined for their capabilities in dealing
with three-dimensional thermal flows involving complex or moving
boundaries. Examples related to forced convective heat and mass transfer from
stationary or streamwise rotating spheres in uniform cross flow were simulated.
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Chapter 9 Conclusions and Recommendations
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The good agreement between the obtained results and the published ones in
the literature clearly shows the easy implementation and accurate natures of
the proposed IBM solvers for three-dimensional heat transfer problems.
Finally, the simulation of complex moving boundary flows was conducted,
where they were either with complex geometry or in complex movement, or
both. One is the solid finite-span foil heaving and pitching in the air, and the
other is the flexible-body fish swimming in the water. Both problems were
accurately predicted as compared to the available numerical results, showing
the great potential of the immersed boundary method for three-dimensional
practical flow problems with complex solid or flexible moving boundaries.

9.2 Recommendations for the Future Work
The temperature correction-based IBM is proposed to improve the accuracy of
the existing IBM solvers for heat transfer problems subject to Dirichlet-type
boundary conditions and the developed heat flux correction-based IBM is
considered as an initial attempt to extend the IBM for solving problems with
Neumann-type boundary conditions. While validation tests indicate their
precious features such as accurate, efficient, simple and easy implementation,
they can be extensively adopted to solve a large number of thermal flows with
complex or moving geometries. However, both thermal solvers are limited to
the cases where the temperature condition on the immersed boundary, either in

Dirichlet or Neumann type, is prescribed in advance. From the viewpoint of
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Chapter 9 Conclusions and Recommendations
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
methodology development, it may be desirable to generalize the solvers to
more general cases where the temperature conditions change with time and are
to be determined during the solution process. From the viewpoint of practical
and engineering applications, the present study conducted simulations for
three-dimensional forced and natural convective flows with complex
geometries (Chapter 7) and three-dimensional moving boundary isothermal
problems with prescribed motion (Chapter 8). Fluid and thermal problems
with high dynamic complexity, such as mix convections, three-dimensional
moving boundary thermal flows or fully fluid-structure interactions, are not
included in the present thesis. All these would motivate the future
investigations.

Reference
316



Reference

Achenbach E., Vortex shedding from spheres, J. Fluid Mech., 62 (1974)
209-221.
Ahmad R.A., Qureshi Z.H., Laminar mixed convection from a uniform heat
flux horizontal cylinder in a crossflow, J. Thermophys Heat Transf., 6 (1992)
277-287.
Ahmed G.R., Yovanovich M.M., Approximate analytical solution of forced

convection heat transfer from isothermal spheres for all Prandtl numbers,
ASME, J. Heat Transf., 116 (1994) 838–843.
Anagnostopoulos P., Numerical study of the flow past a cylinder excited
transversely to the incident stream. Part 1: lock-in zone, hydrodynamic forces
and wake geometry, J. Fluid Struct., 14 (2000) 819-851.
Anderson J.M., Streitlien K., Barrett D.S., Truantafyllou M.S., Oscillating
foils of high propulsive efficiency, J. Fluid Mech., 360 (1998) 41–72.
Arthurs K.M., Moore L.C., Peskin C.S., Pitman E.B., Layton H.E., Modeling
Arteriolar Flow and Mass Transport Using the Immersed Boundary Method, J.
Comp. Phys., 147 (1998) 402-440.
Barrett D.S., Triantafyllon M.S., Yue D.K.P., Grosenbaugh M.A., Wolfgang
M., Drag Reduction in fish-like locomotion. J. Fluid Mech., 392 (1999)

Reference
317

183-212.
Beyer Jr. R.P., A computational model of the cochlea using the immersed
boundary method, J. Comp. Phys., 98 (1992) 145-162.
Bharti R.P., Chahabra R.P., Eswaran V., A numerical study of the steady forced
convection heat transfer from an unconfined circular cylinder, Heat Mass
Transf., 43 (2007) 639–648.
Birch J.M., Dickinson M.H., Spanwise flow and the attachment of the
leading-edge vortex on insect wings, Nature, 412 (2001) 729-733.
Bishop R.E.D., Hassan A.Y., The lift and drag forces on a circular cylinder
oscillating in a flowing fluid, Proceedings of the Royal Society of London,
Series A. 277 (1964) 51-75.
Blackburn H.M., Karniadakis G.E., Two- and three-dimensional simulations of
vortex-induced vibration of a circular cylinder, In 3
rd

Intl. Offshore & Polar
Engng Conf., Singapore, 7150-720.
Blondeaux P., Fornarelli F., Guglielmini L., Triantafyllou M.S., Verzicco R.,
Numerical experiments on flapping foils mimicking fish-like locomotion,
Phys. Fluids, 17 (2005) 113601.
Borazjani I., Sotiropoulos F., Numerical investigation of the hydrodynamics of
carangiform swimming in the transitional and inertial flow regimes. J. Exp.
Biol., 211 (2008) 1541-1558.
Borazjani I., Sotiropoulos F., Numerical investigation of the hydrodynamics of
anguiliform swimming in the transitional and inertial flow regimes. J. Exp.

Reference
318

Biol., 212 (2009) 576-592.
Braza M., Chassaing P., Minh H.H., Numerical study and physical analysis of
the pressure and velocity fields in the near wake of a circular cylinder, J. Fluid
Mech., 165 (1986) 79-130.
Breugem W.P., A second-order accurate immersed boundary method for fully
resolved simulations of particle-laden flows, J. Comput. Phys., 231 (2012)
4469-4498.
Buchholz J.H.J., Smits A.J., On the evolution of the wake structure produced
by a low-aspect-ratio pitching panel, J. Fluid Mech., 546 (2006) 433-443.
Chang K.S., Song C.J., Interactive vortex shedding from a pair of circular
cylinders in a transverse arrangement, Int. J. Numer. Meth. Fluids, 11 (1990)
317-329.
Chaviaropoulos P., Ciannakoglou K., A vorticity-streamfunction formulation
for steady incompressible two-dimensional flows, Int. J. Numer. Meth. Fluids,
23 (1996) 431-444.
Chen L., Yu Y., Hou G.X., Sharp-interface immersed boundary method lattice

Boltzmann method with reduced spurious-pressure oscillations for moving
boundaries, Phys. Rev. E, 87 (2013) 053306.
Cheng P.C., Wen R.C., Transient natural convection heat transfer between
concentric and vertically eccentric spheres, Int. J. Heat Mass Transfer, 39
(1996) 1439-1452.
Choi J.I., Oberoi R.C., Edwards J.R., Rosati J.A., An immersed boundary

Reference
319

method for complex incompressible flows, J. Comput. Phys., 224 (2007)
757-784.
Chorin A.J., Numerical solution of the Navier-Stokes equations, Math. Comp.,
22 (1968) 745-762.
Chung M.H., Cartesian cut cell approach for simulating incompressible flows
with rigid bodies of arbitrary shape, Computers & Fluids, 35 (2006) 607-623.
Clift R., Grace J.R., Weber M.E., Bubbles, Drops and Particles, Academic
Press, New York, (1978).
Collett T.S., Land M.F., Visual control of flight behavior in the hoverfly syritta
pipiens, J. Comp. Physiol., 99 (1975a) 1-66.
Collett T.S., Land M.F., Visual spatial memory in a hoverfly, J. Comp. Physiol.,
99 (1975b) 59-84.
Dalton S., Kings J., Borne on the Wind: Extraordinary world of insects in
flight, Reader’s Digest Press, New York, 1975.
Deng J., Shao X.M., Ren A.L., A new modification of the immersed boundary
method for simulating flows with complex moving boundaries, Int. J. Numer.
Meth. Fluids, 52 (2006) 1195-1213.
Dennis S.C.R., Chang G.Z., Numerical solution for steady flow past a circular
cylinder at Reynolds number up to 100, J. Fluid Mech., 42 (1970) 471-489.
Dennis S.C.R., Hudson J.D., Smith N., Steady laminar forced convection from

a circular cylinder at low Reynolds numbers, Phys. Fluids, 11 (1968) 933–940.
Dennis S.C.R., Walker J.D.A., Hudson J.D., Heat transfer from a sphere at

Reference
320

low Reynolds numbers, J. Fluid Mech., 60 (1973) 273–283.
Dhiman A.K., Chhabra R.P., Sharma A., Eswaran V., Effects of Reynolds and
Prandtl numbers on the heat transfer across a square cylinder in the steady
flow regime, Numer. Heat Transfer, 49 (2006) 717–731.
Dhole S.D., Chhabra R.P., Eswaran V., A numerical study on forced
convection heat transfer from an isothermal and isoflux sphere in the steady
symmetric flow regime, Int. J. Heat Mass Transf., 49 (2006) 984-994.
Dickinson M.H., The effects of wing rotation on unsteady aerodynamic
performance at low Reynolds number, J. Exp. Biol., 192 (1994) 179-206.
Dickinson M.H., Lehmann F.O., Sane S.P., Wing rotation and the aerodynamic
basis of insect flight, Science, 284 (1999) 1954-1960.
Dickinson M.H., Gätz K.G., Unsteady aerodynamic performance of model
wings at low Reynolds numbers, J. Exp. Biol., 174 (1993) 45-64.
Ding H., Shu C., Yeo K.S., Xu D., Simulation of incompressible viscous flows
past a circular cylinder by hybrid FD scheme and meshless least square-based
finite difference method, Comput. Methods Appl. Mech. Engrg., 193 (2004)
727-744.
Ding H., Shu C., Yeo K.S., Xu D., Numerical simulation of flows around two
circular cylinders by mesh-free least square-based finite difference methods,
Int. J. Numer. Meth. Fluids, 53 (2007) 305-332.
Dong H., Mittal R., Najjar F.M., Wake topology and hydrodynamic
performance of low-aspect-ratio flapping foils, J. Fluid Mech., 566 (2006)

Reference

321

309-343.
Drucker E.G., Lauder G.V., Experimental hydrodynamics of fish locomotion:
Functional insights from wake visualization, Integr. Comp. Biol., 42 (2002)
243-257.
Du J., Guy R.D., Fogelson A.L., An immersed boundary method for two-fluid
mixtures, J. Comput. Phys., 262 (2014) 231-243.
Dütsch H., Durst F., Becker S., Lienhart H., Low-Reynolds-number flow
around an oscillating circular cylinder at low Keulegan-Carpenter numbers, J.
Fluid Mech., 360 (1998) 249–271.
Eggleton C.D., Popel A.S., Large deformation of red blood cell ghosts in a
simple shear flow, Phys. Fluids, 10 (1998) 1834
Eldredge J.D., Numerical simulation of the fluid dynamics of 2D rigid body
motion with the vortex particle method, J. Comp. Phys., 221 (2007) 626-648
Ellington C.P., The aerodynamics of hovering insect flight, I. the quasi-steady
analysis, Phil. Trans. R. Soc. Lond. B, 305 (1984) 1–15.
Ellington C.P., Berg C.V.D., Willmott A.P., Thomos A.L.R., Leading-edge
vortices in insect flight, Nature, 384 (1996) 626-629.
Fadlun E., Verzicco R., Orlandi P., Mohd-Yusof J., Combined
immersed-boundary finite-difference methods for three dimensional complex
flow simulations, J. Comp. Phys., 161 (2000) 35–60.
Feng J., Hu H.H., Joseph D.D., Direct simulation of initial value problems for
the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation, J. Fluid

Reference
322

Mech., 261 (1994) 95–134.
Feng Z G., Michaelides E.E., A numerical study on the transient heat transfer

from a sphere at high Reynolds and Peclet numbers, Int. J. Heat Mass Transfer,
42 (2000) 219-229.
Feng Z G., Michaelides E.E., The immersed boundary-lattice Boltzmann
method for solving fluid-particles interaction problem, J. Comput. Phys., 195
(2004) 602-628.
Feng Z G., Michaelides E.E., Proteus: A direct forcing method in the
simulations of particulate flows, J. Comput. Phys., 202 (2005) 20-51.
Feng Z G., Michaelides E.E., Inclusion of heat transfer computations for
particle laden flows, Phys. Fluids, 20 (2008) 040604.
Feng Z., Michaelides E.E., Heat transfer in particulate flows with Direct
Numerical Simulation (DNS), Int. J. Heat Mass Transf., 52 (2009) 777-786.
Fornberg B., A numerical study of steady viscous flow past a circular cylinder,
J. Fluid Mech., 98 (1980) 819-855.
Francois M., Shyy W., Computations of drop dynamics with the immersed
boundary method, Part 1: numerical algorithm and buoyance-induced effect,
Numer. Heat Transfer Part B, 44 (2003) 101-118.
Froszling N., Gerl. Beitr. Zur Geophysik, 170 (1938).
Gan H., Chang J.Z., Feng J.J., Hu H.H., Direct numerical simulation of the
sedimentation of solid particles with thermal convection, J. Fluid Mech., 481
(2003) 385-411.

Reference
323

Gao T., Lu X.Y., Insect normal hovering flight in ground effect, Phys. Fluids,
20 (2008) 087101.
Gau C., Wu J.M., Liang C.Y., Heat transfer enhancement and vortex flow
structure over a heated cylinder oscillating in the crossflow direction, J. Heat
Transfer, 121 (1999) 789-195.
Ghias R., Mittal R., Dong H., A sharp interface immersed boundary method

for compressible viscous flows, J. Comput. Phys., 225 (2007) 528-553.
Giacobello M., Ooi A., Balachandar S., Wake structure of a transversely
rotating sphere at moderate Reynolds numbers, J. Fluid Mech., 621 (2009)
103-130.
Gilmanov A, Sotiropoulos F, and Balarasf E, A general reconstruction
algorithm for simulating flows with complex 3D immersed boundaries on
Cartesian grids, J. Comp. Phys., 191 (2003) 660-669.
Glowinski R., Pan T., Hesla T., Joseph D., Periaux J., A fictitious domain
approach to the direct numerical simulation of incompressible viscous flow
past moving rigid bodies: application to particulate flow, J. Comp. Phys., 169
(2001) 363–426.
Glowinski R., Pan T W., Periaux J., A fictitious domain method for Dirichlet
problems and applications, Comp. Meth. Appl. Mech. Eng., 111 (1994)
283-303.
Glowinski R., Pan T W., Periaux J., A fictitious domain method for external
incompressible viscous flow modeled by Navier–Stokes equations, Comp.

Reference
324

Meth. Appl. Mech. Eng., 112 (1994) 133–148.
Glowinski R., Pan T W., Periaux J., A Lagrange multiplier/fictitious domain
method for the Dirichlet problem: generalization to some flow problems, Jpn.
J. Ind. Appl. Math., 12 (1995) 87–108.
Glowinski R., Pan T W., Periaux J., A Lagrange multiplier/fictitious domain
method for the numerical simulation of incompressible viscous flow around
moving rigid bodies (I): the case where the rigid body motions are known a
priori, CR Acad. Sci. Paris, 324 (1997) 361–369.
Glowinski R., Pan T W., Hesla T.I., Joseph D.D., A distributed Lagrange
multiplier/fictitious domain method for particulate flows, Int. J. Multiphase

Flow, 25 (1999) 755-794.
Glowinski R., Pan T.W., Hesla T.I., Joseph D.D., Priaux J., A distributed
Lagrange multiplier/fictitious domain method for flows around moving rigid
bodies: application to particulate flow, Int. J. Numer. Meth. Fluids, 30 (1999)
1043-1066.
Griffin O.M., Ramberg S.E., The vortex-street wakes of vibrating cylinder, J.
Fluid Mech., 66 (1974) 553-576
Griffin O. M., Votaw C.W., The vortex street in the wake of a vibrating
cylinder, J. Fluid Mech., 55 (1972) 31-48.
Griffith B.E., Peskin C.S., On the order of accuracy of the immersed boundary
method: higher order convergence rates for sufficiently smooth problems, J.
Comp. Phys., 208 (2005) 75-105.

Reference
325

Grove A.S., Shair F.H., An experimental investigation of the steady separated
flow past a circular cylinder, J. Fluid Mech., 19 (1964) 60-80.
Goldstein D., Hadler R., Sirovich L., Modeling a no-slip flow boundary with
an external force field, J. Comp. Phys., 105 (1993) 354–366.
Goldstein D., Hadler R., Sirovich L., Direct numerical simulation of turbulent
flow over a modelled riblet covered surface, J. Comp. Phys., 302 (1995) 333.
Gu W., Chyu C., Rockwell D., Timing of vortex formation from an oscillating
cylinder, Phys. Fluids, 6 (1994) 3677-3682.
Guilmineay E., Queutey P., A numerical simulation of vortex shedding from
an oscillating circular cylinder, J. Fluid Struct., 16 (2002) 773-794.
Gustavson K., Leben R., Computation of dragonfly aerodynamics, Comput.
Phys. Commun., 65 (1991) 121-132.
Han T., Yang K.S., Lee K., Heat transfer characterization of two isothermal
circular cylinders in proximity, J. Heat Transf., 132 (2010) 034504-1.

He X., Doolen G.D., Lattice Boltzmann method on curvilinear coordinates
system: flow around a circular cylinder, J. Comput. Phys., 134 (1997)
306-315.
Hess F., Videler J.J., Fast continuous swimming of saithe (Pollachius virens): a
dynamic analysis of bending moments and muscle power, J. Exp. Biol., 109
(1984) 229-251.
Ho C.J., Lin Y.H., Chen T.C., A numerical study of natural convection in
concentric and eccentric horizontal cylindrical annuli with mixed boundary

Reference
326

conditions, Int. J. Heat Fluid Flow., 10 (1989) 40-47.
Hoskin N.E., The laminar boundary layer on a rotating sphere, Fifty years of
Boundary Layer Research, (1955) 127-131.
Hu H.H., Direct simulation of flows of solid-liquid mixtures, Int. J.
Multiphase Flow, 22 (1996) 335-352.
Hu H.H., Joseph D.D., Crochet M.J., Direct simulation of fluid particle
motions, Theoret. Comput. Fluid Dyn., 3 (1992) 285–306
Hu H.H., Patankar N.A., Zhu M.Y., Direct numerical simulations of fluid-solid
systems using the arbitrary Lagrangian-Eulerian technique, J. Comput. Phys.,
169 (2001) 427-462.
Igarashi T., Characteristics of the flow around two circular cylinders arranged
in tandem (1st report), Bulletin of the JSME, 24 (1981) 323–331.
Jeong J., Hussain F., On the identification of a vortex, J. Fluid Mech., 285
(1995) 69-94.
Je H., Lien F.S., Yee E., A robust and efficient hybrid cut-cell/ghost-cell
method with adaptive mesh refinement for moving boundaries on irregular
domains, Comput. Methods Appl. Mech. Engrg., 198 (2008) 432-448.
Ji C., Munjiza A., Williams J.J.R., A novel iterative direct-forcing immersed

boundary method and its finite volume applications, J. Comput. Phys., 231
(2012) 1797-1821.
Johnson T.A., Patel V.C., Flow past a sphere up to a Reynolds number of 300,
J. Fluid Mech., 378 (1999) 19-70.

Reference
327

Jones K.D., Dohring C.M., Platzer M.F., Experimental and computational
investigation of the Knoller-Betz effect, AIAA J., 36 (1998) 1240-46.
Kalitzin G., Iaccarino F., Toward immersed boundary method simulation of
high Reynolds number flows, Center for Turbulence Research Annual
Research Briefs, 2003.
Kang S., Characteristics of flow over two circular cylinders in a side-by-side
arrangement at low Reynolds numbers, Phys. Fluids, 15 (2003).
Kempe T., Frohlich J., An improved immersed boundary method with direct
forcing for the simulation of particle laden flows, J. Comput. Phys., 231 (2012)
3663-3684.
Kern S., Koumoutsakos P., Simulations of optimized anguiliform swimming, J.
Exp. Biol., 209 (2006) 4841-4857.
Kim B.S., Lee D.S., Ha M.Y., Yoon H.S., A numerical study of natural
convection in a square enclosure with a circular cylinder at different vertical
locations, Int. J. Heat Mass Transfer, 51 (2008) 1888–1906.
Kim D., Choi H., Laminar flow past a sphere rotating in the streamwise
direction, J. Fluid Mech., 461 (2002) 365-386.
Kim H.J., Durbin P.A., Observations of the frequencies in a sphere wake and
of drag increase by acoustic excitation, Phys. Fluids, 31 (1988) 3260-3265.
Kim J., Kim D., Choi H., An immersed-boundary finite-volume method for
simulations of flow in complex geometries, J. Comp. Phys., 171 (2001)
132–150.


Reference
328

Kim Y.S., Peskin C.S., 2-D parachute simulation by the immersed boundary
method, SIAM J. SCI. Comput., 28 (2006) 2294-2312.
Kim Y.S., Peskin C.S., Penalty immersed boundary method for an elastic
boundary with Mass, Phys. Fluids., 19 (2007).
Kim Y.S., Peskin C.S., 3-D parachute simulation by the immersed boundary
method, Computers & Fluids, 38 (2009) 1080-1090.
Kim Y.S., Seol Y.C., Numerical simulations of two-dimensional wet foam by
the immersed boundary method, Computers & Structures, 122 (2013)
259-269.
Koopmann G.H., The vortex wakes of vibrating cylinders at low Reynolds
number, J. Fluid Mech., 27 (1967) 501-512.
Kostic Z.G., Oka S.N., Fluid flow and heat transfer with two cylinders in cross
flow, Int. J. Heat Mass Transfer, 15 (1972) 279-299.
Kuehn T.H., Goldstein R.J., An experimental and theoretical study of natural
convection in the annulus between horizontal concentric cylinders, J. Fluid
Mech., 74 (1976) 695–719.
Lai M.C., Peskin C.S., An immersed boundary method with formal
second-order accuracy and reduced numerical viscosity, J. Comp. Phys., 160
(2000) 705–719.
Lange C.F., Durst F., Breuer M., Momentum and heat transfer from cylinders
in laminar crossflow at
200Re10
4
≤≤
−
, Int. J. Heat Mass Transf., 41(1998)

3409–3430.

Reference
329

Le D.V., Khoo B.C., Lim K.M., An implicit-forcing immersed boundary
method for simulating viscous flows in irregular domains, Comput. Methods
Appl. Mech. Engrg., 197 (2008) 2119-2130.
Lee D.S., Ha M.Y., Yoon H.S., Balachandar S., A numerical study on the flow
patterns of two oscillating cylinders, J. Fluids Struct., 25 (2009) 263-283.
Lee J., Kim J., Choi H., Yang K., Sources of spurious force oscillations from
an immersed boundary method for moving-body problems, J. Comput. Phys.,
230 (2011) 2677-2695.
Lee K.C., Aerodynamic interaction between two spheres at Reynolds numbers
around
4
10
, Aerosp. Q., 30 (1979) 371-385.
Le D.V., Khoo B.C., Peraire J., An immersed interface method for viscous
incompressible flows involving rigid and flexible boundaries, J. Comput.
Phys., 220 (2006) 109-138.
LeVeque R.J., Li Z.L., The immersed interface method for elliptic equations
with discontinuous coefficients and singular sources, SIAM J. Numer. Anal,
31 (1994) 1019-1044.
LeVeque R.J., Lin Z.L., Immersed interface methods for Stokes flow with
elastic boundaries or surface tension, SIAM J. Sci. Comput., 18 (1997)
709-735.
Li J., Chambarel A., Donneaud M., Martin R., Numerical study of laminar
flow past one and two circular cylinders, Comput. Fluids, 19 (1991) 155-170.
Li Y., Jung E., Lee W., Lee H.G., Kim J., Volume preserving immersed


Reference
330

boundary methods for two-phase fluid flows, Int. J. Numer. Meth. Fluids, 69
(2012) 842-858.
Liao C.C., Lin C.A., Simulations of natural and forced convection flows with
moving embedded object using immersed boundary method, Comput.
Methods Appl. Mech. Engrg., 213 (2012) 58-70.
Lighthill M.J., Hydrodynamics of aquatic animal propulsion, Annu. Rev. Fluid
Mech., 1 (1969) 413-445.
Lima E Silva L.F., Silveira-Neto A., Damasceno J.J.R., Numerical simulation
of two-dimensional flows over a circular cylinder using the immersed
boundary method, J. Comput. Phys., 189 (2003) 351-370.
Lin Z.L., Lai M.C., The immersed interface method for the Navier-Stokes
equations with singular forces, J. Comput. Phys., 171 (2001) 822-842.
Lindsey C.C., Form, function and locomotory habits in fish, Fish Physiology,
New York: Academic Press, (1978) 1-100.
Liu C., Zheng X., Sung C.H., Preconditioned multigrid methods for unsteady
incompressible flows, J. Comput. Phys., 139 (1998) 35-57.
Liu H., A numerical study on jet-stream propulsion of oscillating bodies, J.
Soc. Naval Architects Jpn., 178 (1995) 101.
Liu H., Ellington C.P., Kawachi K., Van Den Berg C., Willmott A.P., A
computational fluid dynamic study of hawkmoth hovering, J. Exp. Biol., 201
(1998) 461-477.
Liu H., Kawachi K., A numerical study of undulatory swimming, J. Comp.

Reference
331


Phys., 155 (1999) 223-247.
Liu H., Wassersug R., The three-dimensional hydrodynamics of tadpole
locomotion, J. Exp. Biol., 200 (1997) 2807-2819.
Liu H., Wassersug R.J., Kawachi K., A computational fluid dynamics study of
tadpole swimming, J. Exp. Biol., 199 (1996) 1245-1260.
Ljachowski D.N., J. techn. Phys., Leningrad, 999 (1940).
Luo K., Wang Z., Fan J., A modified immersed boundary method for
simulations of fluid-particle interactions, Comput. Meth. Appl. Mech. Eng.,
197 (2007) 36-46.
Luthander S., Rydberg A., Experimental investigation of the drag of a rotating
body with the wind direction parallel to the axis, Phys. Z., 36 (1935) 552-558.
Luo K., Wang Z.L., Fan J.R., A modified immersed boundary method for
simulations of fluid-particle interactions, Comput. Methods Appl. Mech.
Engrg., 197 (2007) 36-46.
Magarvey R.H., Bishop, R.L., Transition ranges for three-dimensional wakes,
Can. J. Phys., 39 (1961) 1418-1422.
Magnaudet J., Rivero M., Fabre J., Accelerated flows past a rigid sphere or a
spherical bubble. Part 1. Steady straining flow, J. Fluid Mech., 284 (1995)
97-135.
Mahir N., Altac Z., Numerical investigation of convective heat transfer in
unsteady flow past two cylinders in tandem arrangement, Int. J. Heat Fluid
Flows, 29 (2008) 1309-1318.

Reference
332

Majumdar S., Laccarino G., Durbin P., RANS solvers with adaptive structured
boundary non-conforming grids, Center for Turbulence Research Annual
Research Briefs, (2001).
Maury B., Direct simulations of 2D fluid-particle flows in biperiodic domains,

J. Comput. Phys., 156 (1999) 325-351.
McQueen D.M., Peskin C.S., Heart simulation by an immersed boundary
method with formal second-order accuracy and reduced numerical viscosity,
Proceedings of the International Conference on Theoretical and Applied
Mechanics, (2000).
Miller L.A., Peskin C.S., A computational fluid dynamics study of ‘clap and
fling’ in the smallest insects, J. Exp. Biol., 208 (2005) 195-212.
Mittal R., Dong H., Bozkurttas M., Najjar F.M., Vargas A., Von Loebbecke A.,
A versatile sharp interface immersed boundary method for incompressible
flows with complex boundaries, J. Comput. Phys., 227 (2008) 4825-4852.
Mittal R., Iaccarino G., Immersed Boundary Methods, Ann. Rev. Fluid Mech.,
37 (2005) 239-261.
Mizushima J., Ino Y., Stability of flows past a pair of circular cylinders in a
side-by-side arrangement, J. Fluid Mech., 595 (2008) 491-507.
Mohd-Yusof J., Combined immersed boundaries/B-Splines methods for
simulations of flows in complex geometries, CTR Annual Research Briefs,
NASA Ames/Stanford University, 1997.
Morse T.L., Williamson C.H.K., Fluid forcing, wake modes, and transitions