The vortex structures generated in the wake of swimming oblate jellyfish
with paddlingbased locomotion

The structure denoted by 1* is generated from the shear layer at the inner wall
(I.W); 2* is generated by the previous recovery stroke (Previous R.S) and shed
from the bell outer wall (O.W); 3* is generated by the present power stroke
(Present P.S) and shed from the bell inner wall (I.W); 4* is generated by the
present recovery stroke (Present R.S) and shed from the bell outer wall (O.W)
Variations in the propulsive efficiency as a function of the flapping frequency and the force duration.
 The
propulsive efficiency increased as the force duration increased for a given
flapping frequency, given the same impulse of the body force.
 The
propulsive efficiency also increased with the increasing of the flapping
frequency, due to the effect of the added mass. When a body is accelerating or
decelerating in fluid, the body must move some volume of surrounding fluid, in
which the inertia is added to a system. For swimming jellyfish, the
intermittent acceleration and deceleration of the fluid surrounding the
jellyfish lead to added cost of locomotion. In the case of the swimming fish,
the effect of added mass is less important since they are able to produce
thrust almost continuously, which implies that the added mass which the animal carries
in a forward motion become less significant for a uniform locomotion.
 For
the case of continuous swimming, the acceleration reaction problem is not as
significant as the intermittent swimming in which the coasting phase exists.
 Despite
an increase in the power input, the propulsive efficiency increased as the
flapping frequency increased.
An improved version of the immersed boundary method for simulation of the interaction between fluid and flexible structure
 In the present forinmulation, the fluid motion defined on an Eulerian grid and the filament motion defined on a Lagrangian grid are solved independently.
 Governing equations for the incompressible viscous fluid flow:
 The governing equations for an inextensible filament:
Simulation of flow over flexible filaments by the immersed boundary method
 Flow over a filament: The filament is flexible, inextensible and massive. As the Reynolds number increases, the vortex structure becomes smaller, and the number of small vortices of each sign in one shedding period is increased.
 Flow over two sidebyside filaments: The two filaments flap in phase for small inter distance and out of phase for large inter distance
Instantaneous vorticity contours of a uniform flow over a filament
Instantaneous vorticity contours of a uniform flow over two sidebyside filaments
3D simulation of a flapping flag in a uniform flow
 The instantaneous flag motion is analyzed under different conditions and surrounding vortical structures are visualized.
 The Strouhal number defined in terms of the flapping amplitude increases slightly with increasing Reynolds number and is between 0.15 and 0.25, consistent with the general value of a flying or swimming animal.
 A linear stability analysis for a flag of infinite spanwise width in a 3D flow is carried out.
Instantaneous flag positions without gravity force, time history of the transverse position of the trailing edge of the flag, and the streamwise and transverse drag forces of the flag
Vortical structures shedding from the flapping flag
Flag motion with gravity force
Vortical structures shedding from the flapping flag
3D simulation of a valveless pump
 A net flow is generated inside the valveless pump through the periodic pinching of the elastic tube at a position that is asymmetric with respect to its ends.
 Two valveless pumps are chosen, a single valveless pump and a double valveless pump. The effects on the average flow rate of varying the pinching frequency and the pinching position were investigated.
 The interaction between the wave dynamics and the inertia of the returning flow was examined for a closed loop system.
Schematic diagram of the valveless pump. (a) single pump. (b) double pump.
 Single valveless pump
¡¤ The positive average flow rate (f=10Hz) and negative average flow rate (f=30Hz) are obtained respectively, due to the complex interactions between the pincher and flow waves.
¡¤ When the pinching position is shifted, the phase delay between the reflected wave and the source wave is also affected. Therefore, high positive and zero average flows are generated by different pinching frequencies for different pinching positions.
(a) Evolutions of the flow rate for f=10Hz and 30Hz. (b) Effects on the average flow rate of varying the pinching frequency and position
 Double valveless pump
¡¤ The average flow rate has a nonlinear relation with the pinching frequency and the phase difference. When the pinching frequency is fixed, the average flow rate reaches a maximum in the range of .
¡¤ The maximum average flow for the double valveless pump is almost two times larger than that of the single valveless pump. The larger average flow for the double pump can be obtained by optimizing the pinching frequency and the phase difference.
(a) Effects on the average flow rate of varying the phase difference and the pinching frequency for x_{pc}=2d. (b) Comparison of the average flow rate.
¡¤ To understand the physics of the double valveless pump, the progressions of the flow waves along the tube are illustrated for different phase differences ( and at f=15Hz and x_{pc}=2d)
Progressions of the flow waves for f=15Hz, x_{pc}=2d and .
Progressions of the flow waves for f=15Hz, x_{pc}=2d and .
3D simulation of an elastic capsule in shear flow
 Deformation of 3D elastic capsules in a linear shear flow is studied in detail.
 For an initially spherical capsule the tanktreading motion is formed under various dimensionless shear rates and reduced bending moduli.
 For a nonspherical capsule, with the initial shape of the oblate spheroid or the biconcave circular disk as a model of the red blood cell, the swinging motion is observed due to the shape memory effect.
Schematic of the capsule deformation in a simple shear flow.
 A spherical capsule in a linear shear flow
¡¤ As the dimensionless shear rate ¥ç increases the capsule is more deformed and is more aligned with the xaxis due to the enhanced viscous force over the elastic force.
(a) Instantaneous shapes of the capsule and streamlines of the surrounding fluid flow in the center plane (z=0) at the steady state: (a) ¥ç=0.6; (b) ¥ç=0.3; (c) ¥ç=0.15; (d) ¥ç=0.075.
 A nonspherical capsule in a linear shear flow
¡¤ In the moderate dimensionless shear rate, swinging motion occurs
¡¤ The dimples on the biconcave capsule are vanished as the capsule is stretched due to the high dimensionless shear rate.
Instantaneous shapes of the initially biconcave capsule with the SK membrane of C=50, ¥ç=1.2 and ¥æ=0: (a) t=0; (b) t=4; (c) t=8; (d) t=12; (e) t=16; (f) t=20. The inset denotes a side view into the cut in the center plane (z=0).
¡¤ At the reduced bending modulus ¥æ=0, buckling occurs obviously on the capsule surface. Wrinkles are suppressed and the tumbling motion is resulted by including the bending rigidity for ¥æ=0.09.
Instantaneous shapes of the initially biconcave capsule with the SK membrane of C=50, ¥ç=0.1 and ¥æ=0: (a) t=0; (b) t=4; (c) t=8; (d) t=12; (e) t=16; (f) t=20. The inset denotes a side view into the cut in the center plane (z=0).
Instantaneous shapes of the initially biconcave capsule with the SK membrane of C=50, ¥ç=0.1 and ¥æ=0.09: (a) t=0; (b) t=4; (c) t=8; (d) t=12; (e) t=16; (f) t=20. The inset denotes a side view into the cut in the center plane (z=0).
Flexible ring flapping in a uniform flow
 Deformation of 2D flexible ring in a uniform flow is studied in detail.
 The boundary of the ring consists of a flexible filament with tension and bending stiffness.
 A
penalty method derived from fluid compressibility was used to ensure
the conservation of the internal volume of the flexible ring.
Schematic of the ring deformation in a uniform flow
Flexible ring in a uniform flow
 Volume conservation
¡¤ Recently,
Peng, Asaro & Zhu (2010) proposed a penalty method derived from
fluid compressibility that aims to improve the conservation of cell
volume during deformation. Here we adopt a similar approach to that of
Peng et al. due to its simplicity in implementation.
Changes in the volume of the flexible ring with no control, P control, and PI control
 Flapping modes
¡¤ Most curves can be divided into two region.
¡¤ C_{D} decreasing mode, C_{D} increasing mode
¡¤ Ordinary mode, energetic mode
Variation
of the mean drag coefficient of a flexible ring as a function of the
tension coefficient for various bending coefficients.
 Bistable property
¡¤ We
observed the hysteresis property of the flexible ring, which exhibits
the bistable states over a range of flow velocities depending on the
initial inclination angle, i.e. one is a stationary stable state and the
other a selfsustained periodically flapping state.
¡¤ The
Reynolds number range of the bistability region and the flapping
amplitude were determined for various aspect ratios a/b. For a/b=0.5,
the hysteresis region arises at the highest Reynolds number and the
flapping amplitude in the selfsustained flapping state is minimized.
Variations
of the flapping amplitude of the midpoint of a flexible ring with the
Reynolds number for various aspect ratios. The shapes of the flexible
ring for a/b=0.5 with ¥è_{0}=0 and ¥è_{0}=¥ð/24 are depicted.
Phase maps of the midpoint of a flexible ring of a/b=0.67 at four different Reynolds numbers: (a) ¥è_{0}=0; (b) ¥è_{0}=¥ð/24.
Flapping dynamics of an inverted flag in a uniform flow