Flexible body 

 


Simulation of swimming oblate jellyfish with a paddling based locomotion

 

- The simulation of the oblate swimming jellyfish was performed by the penalty immersed boundary method.

- The fluid governing equation was solved by the fractional step method, while the solid motion equation was solved by  subdivision finite element method.


(a) Schematic diagram of the jellyfish bell in a quiescent fluid and the computational domain; (b) Schematic diagram showing the triangular mesh and the body force applied to the jellyfish bell. Points 1, 2, and 3 denote the top point, the representative center point (the height of which is the same as the heights of the force points), and the force point, respectively; (c) Time histories of the body force required to contract the bell. of the triangular mesh and the body force applied on the jellyfish bell.

 


Swimming oblate jellyfish in a quiescent flow

 



The vortex structures generated in the wake of swimming oblate jellyfish with paddling-based 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 side-by-side 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 side-by-side 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 xpc=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 xpc=2d)  

Progressions of the flow waves for f=15Hz, xpc=2d and .

 Progressions of the flow waves for f=15Hz, xpc=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 tank-treading motion is formed under various dimensionless shear rates and reduced bending moduli.

 

-  For a non-spherical 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 increases the capsule is more deformed and is more aligned with the x-axis 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) =0.6; (b) =0.3; (c) =0.15; (d) =0.075.

 

-  A non-spherical 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, =1.2 and =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 =0, buckling occurs obviously on the capsule surface. Wrinkles are suppressed and the tumbling motion is resulted by including the bending rigidity for =0.09.

 

Instantaneous shapes of the initially biconcave capsule with the SK membrane of C=50, =0.1 and =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, =0.1 and =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.

CD decreasing mode, CD 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 self-sustained 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 self-sustained flapping state is minimized.

 

Variations of the flapping amplitude of the mid-point of a flexible ring with the Reynolds number for various aspect ratios. The shapes of the flexible ring for a/b=0.5 with 0=0 and 0=/24 are depicted.

 

 

Phase maps of the mid-point of a flexible ring of a/b=0.67 at four different Reynolds numbers: (a) 0=0; (b) 0=/24.




Flapping dynamics of an inverted flag in a uniform flow