Fish swimming & maneuvering hydrodynamics

Brenden Epps

Thayer School of EngineeringMay 10, 2013

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questions?

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How does a fish swim?

It flaps its tail!

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What more is there?

performance

forceskinematics

robot fish: body and tail form, actuation, control, cost, manufacturability, service, battery life, etc.

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performance

forceskinematics

Why is this difficult?

robot fish: body and tail form, actuation, control, cost, manufacturability, service, battery life, etc.

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Outline

maneuvering

swimming

flapping foil propulsion

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maneuvering

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maneuvering (slow motion)

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vortex rings

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experimentsparticle image velocimetry (PIV)

• typical maneuver lasts 0.25 seconds

• acquire images at 500 fps

• need camera, laser, software, and patience

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• Vector field and vorticity contours shown

• Anticlockwise vorticity in red

• Clockwise vorticity in blue

• Static vorticity due to motion of ambient fluid

• Ignore data in the fish shadow

example maneuver

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kinematics

stage 1: preparatory stroke stage 2: propulsive stroke

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maneuvering dynamics

• By conservation of momentum:

� =Z

⇤dA

T ⇠r

L2

�=

s(10�2 m)2

10�6 m2/s= 10 s

Z⇥ufluiddV = ⇥mVfish

Ifluid = ⇥mV fish

I = ⇥��D2

4 , � = V D

1 Vortex Dynamics

Vortex linesLaw of Biot & SavartVortex impulse 3D vortex ring breakdown

1

�1 �2 V1 V2 D1 D2

⇤(V2D2)�D2

24 � ⇤(V1D1)

�D21

4 = m |⇥V fish|

⇤(V2D2)�D2

24 = m |⇥V fish|

Vfish

LD = 4

u(x) =�4⇥

�ds⇥ (x� x0)

|x� x0|3

� =⇤

⌅dA

T ⇤ L2

�=

(10�2 m)2

10�6 m2/s= 100 s

⇤⇤ufluiddV = ⇥mVfish

Ii = ⇤�i�D2

i4 , �i = ViDi

⇤(V D)�D2

4 = m |⇥V fish|

M =mfish |⇥V fish|

⇤(VwakeDwake)�D2

wake4

I =12⇤

⇤x⇥ ⌅ dx

time[s]

� =⇥

⌅dA[cm2/s]

� = V D[cm2/s]

% Di⇤erence[ ]

0.24 34 33 3 %0.25 35 34 3 %0.26 31 35 13 %0.27 34 32 6 %

1

� =Z

⇤dA

T ⇠r

L2

�=

s(10�2 m)2

10�6 m2/s= 10 s

Z⇥ufluiddV = ⇥mVfish

Ifluid = ⇥mV fish

I = ⇥��D2

4 , � = V D

1 Vortex Dynamics

Vortex linesLaw of Biot & SavartVortex impulse 3D vortex ring breakdown

1

V1

V2

D2

D1

Vfish

2⇤

i=1

⌅Ifluid = ��m ⌅Vfish

⇥

�1 �2 V1 V2 D1 D2

⇤(V2D2)�D2

24 � ⇤(V1D1)

�D21

4 = m |⇥V fish|

⇤(V2D2)�D2

24 = m |⇥V fish|

Vfish

LD = 4

u(x) =�4⇥

�ds⇥ (x� x0)

|x� x0|3

� =⇤

⌅dA

T ⇤ L2

�=

(10�2 m)2

10�6 m2/s= 100 s

⇤⇤ufluiddV = ⇥mVfish

Ii = ⇤�i�D2

i4 , �i = ViDi

⇤(V D)�D2

4 = m |⇥V fish|

M =mfish |⇥V fish|

⇤(VwakeDwake)�D2

wake4

I =12⇤

⇤x⇥ ⌅ dx

time[s]

� =⇥

⌅dA[cm2/s]

� = V D[cm2/s]

% Di⇤erence[ ]

0.24 34 33 3 %0.25 35 34 3 %0.26 31 35 13 %0.27 34 32 6 %

1

• For a vortex ring:

BP Epps, AH Techet (2007), Impulse generated during unsteady maneuvering of swimming fish. Experiments in Fluids 43:691–700

81 gcm/s

172°

78 gcm/s

133°

Monday, May 13, 13

related research

• Gharib: “formation number” of piston-cylinder vortex rings

A universal time scale for vortex ring formation 125

(a)

(b)

(c)

Figure 3. Visualization of vortex rings at X/D � 9 for (a) Lm/D = 2, Re � �/⇥ � 2800;(b) Lm/D = 3.8, Re � 6000; and (c) Lm/D = 14.5. Picture is taken at Upt/D = L/D = 8. All threecases were generated by an impulsive piston velocity depicted in figure 2.

4. Flow visualizationFigures 3(a), 3(b) and 3(c) show three vortex rings generated by three di�erent

maximum stroke ratios (Lm/D). The vortex rings shown in these pictures are at anapproximate axial position of X � 9D from the nozzle exit. In figure 3(a), Lm/D = 2,while in figure 3(b), Lm/D � 3.8. For the case in figure 3(c), the piston was passingthrough the position L/D � 8 at the time the picture was taken. The piston motionwas only stopped later at Lm/D = 14.5. In all cases, vortices were generated withsimilar impulsive piston motion to that depicted in figure 2.

One striking feature in these pictures is the existence of a trailing jet of fluid behindthe leading vortex ring in figure 3(c) and lack of it in figures 3(a) and 3(b). It appearsthat in figures 3(a) and 3(b) almost all of the discharged fluid has been entrainedinto the vortex ring. However, for the case in figure 3(c), the vortex ring shows aclear separation from the active trailing jet-like region behind it. It is apparent thatthe formation of the vortex ring has been completed and the vorticity is no longerentrained from the shear layer region of the trailing jet. It is interesting to note that thesize of the leading vortex ring in figure 3(c) is approximately the same as that of thevortex ring in figure 3(b) and is larger than that depicted in figure 3(a). Consideringthat the pictures are taken at the same downstream position of X � 9D, this variation

• Is there an “optimum” vortex ring for maneuvering?

• What is the best tail shape and flexibility, and best stroke kinematics to make an “optimum” vortex ring for maneuvering?

Gharib M, Rambod E, Shariff K (1998) A universal time scale for vortex ring formation. J Fluid Mech 360:121–140

Monday, May 13, 13

vortex ring formation

Taylor GI (1952) Formation of a vortex ring by giving an impulse to a circular disk and then dissolving it away. J. Applied Phys. v.24, no.1, pp.104-5

Ushear layer

Gharib M, Rambod E, Shariff K (1998) A universal time scale for vortex ring formation. J Fluid Mech 360:121–140

shear layer

discpiston-cylinder

U

L

D

� = 12UL

L � 4D� = 2

� UD

L � ??

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swimming

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swimming (slow motion)

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vortex chain

Nauen and Lauder, JEB 2002

w

U

!

!

laser plane

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kinematics

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body wave

C = � · f

U = ⇥ · f

phase speed

swimming speed

for thrust:

therefore � > 1.2 · ⌅

C > 1.2 · U

M. J.  Lighthill (1960) “Note on the swimming of slender fish” Journal of Fluid Mechanics, vol 9, no. 2, pp 305-317

Monday, May 13, 13

RoboFish design

Figure 5-2: Cable torque transmission.

5.1.1 Actuation Mechanism

The required actuation can be implemented using rotational or linear actuators. For

the design examples presented in this chapter, RC (radio control) servo motors are

used. Figures 5-2 and 5-3 display different views of two transmission designs used in

the prototypes. A transmission consists of a servo, a servo support, a rigid plate, and

a mechanisms to transmit the servo forces to the rigid plate. A chosen transmission

mechanism is then embedded inside the fish-like compliant body. During actuation,

the transmission rigid plate applies the forces from the servo to the compliant body.

Figure 5-2 shows a transmission mechanism that uses two cables attached to a servo

motor to transmit torque to the rigid plate. Figure 5-3 shows an alternative mecha-

nism that uses a flexure linkage connected to the rigid plate to transmit the required

torque to a section inside the body. In this configuration the servo arm acts as the

coupler link and the rigid plate is the ground link. The cable mechanism is simpler

and more easily scalable. However, during a cycle the cable mechanism only transmits

Viscoelastic material: Silicone/polyurethane gel:E ~ 103-105 Pa, μ ~ 1-100 Pa•s

Modified Bernoulli-Euler beam equiation:

Desired tail kinematics:

Ref.: Pablo Valdivia y Alvarado, PhD Thesis, MIT, 2007.

⇤ = E� + µ�

⇤M � EIh�� � µIh��

⌅��=

�⇥A + ⇥w

�4 b2

⇥h

h(x = ⌅, t) = Hsin(�t)

Monday, May 13, 13

complexity

Figure 5-2: Cable torque transmission.

5.1.1 Actuation Mechanism

The required actuation can be implemented using rotational or linear actuators. For

the design examples presented in this chapter, RC (radio control) servo motors are

used. Figures 5-2 and 5-3 display different views of two transmission designs used in

the prototypes. A transmission consists of a servo, a servo support, a rigid plate, and

a mechanisms to transmit the servo forces to the rigid plate. A chosen transmission

mechanism is then embedded inside the fish-like compliant body. During actuation,

the transmission rigid plate applies the forces from the servo to the compliant body.

Figure 5-2 shows a transmission mechanism that uses two cables attached to a servo

motor to transmit torque to the rigid plate. Figure 5-3 shows an alternative mecha-

nism that uses a flexure linkage connected to the rigid plate to transmit the required

torque to a section inside the body. In this configuration the servo arm acts as the

coupler link and the rigid plate is the ground link. The cable mechanism is simpler

and more easily scalable. However, during a cycle the cable mechanism only transmits

!1.0 !0.$% !0.%0 !0.2% 0 0.2%

'(') + 1.04

t + 0

t + .

position [L]

RoboTuna RoboFish

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particle image velocimetry

• Vector field and vorticity contours shown

• Anticlockwise vorticity in red

• Clockwise vorticity in blue

• Static vorticity due to motion of ambient fluid

• Ignore data in the fish shadow

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dye visualization

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top view (1/3 speed)

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side view (1/3-speed)

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swimming speed

0 1.35 2.7 4.050

0.1

0.2

0.3

0.4

flapping frequency [Hz]

sw

imm

ing s

peed [L

/s]

live fish RoboFish

0.15 [

L/s]/[H

z]

0.59

[L/s]

/[Hz]

low freq.nominalhigh freq.

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0 0

0.2

0.4

0.6

0.8

1.0

St

robot, StH

robot, Stw

Danio, StH

Danio, Stw

1.35 2.7 4.05

flapping frequency [Hz]

Stw ≈ 0.52

Stw ≈ 0.20

wake geometry

BP Epps, P Valdivia y Alvarado, K Youcef-Toumi, AH Techet,Flow visualization studies of the swimming performance of biomimetic compliant fish-like robots. (in preparation.)

Stw =fw

U� constant

StH =fH

U� constant

w

U

ℓ

Γ H

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related research

! ""

!

#$%&'(!))*!+,'((!-$.(/0$1/23! 41'5$623! 05'&65&'(0!4$0&23$7(-!&0$/%! 5,(!896'$5('$1/! 0,1:$/%!";!:2<(! 05'&65&'(0! 0$.&325(-! =1'! 5,(!

>(?@AAA!620(B!C51DE!-1&F3(!'1:!:2<(!25!G5?A*HI!CF1551.E!0$/%3(!'1:!:2<(!25!G5?A*J*!!

!

Re = 4000StH = 0.7

Re = 4000StH = 0.2

Borazjani I, Sotiropoulos F (2008) Numerical investigation of the hydrodynamics of carangiform swimming in the transitional and inertial flow regimes. Journal of Experimental Biology 211:1541–1558

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optimal Strouhal number

StH = 0.3Re = 10^4 - 10^6Monday, May 13, 13

Fish Exploiting Vortices Decrease Muscle ActivityJames C. Liao, et al.Science 302, 1566 (2003);

Monday, May 13, 13

From: "David Beal <dnbeal@alum.mit.edu>" Subject: " Dead fish video" Date: "April 28, 2009 3:35:18 PM EDT" To: " Brenden Epps <bepps@mit.edu>

This video shows the dead fish in slow motion moving forwards. The blue line shows the length of the line it was attached to the cylinder with. The red line shows approximately the back of the suction region behind the cylinder. The fish's interaction with the cylinder's shed vorticity is what causes it to move forwards to that point. I also attached a real-time version.

Dave

Fish Exploiting Vortices Decrease Muscle ActivityJames C. Liao, et al.Science 302, 1566 (2003);

Monday, May 13, 13

oscillating foil propulsion

J. M. ANDERSON, K. STREITLIEN, D. S. BARRETT and M. S. TRIANTAFYLLOU (1998). Oscillating foils of high propulsive efficiency. Journal of Fluid Mechanics, 360, pp 41-72

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oscillating foil propulsion

MIT Center for Ocean Engineering, 2005

3D vortex ring

Wingtip view

Planform View

Isometric view

St=0.25 Max Angle of attack α0=15°

Phase shift between heave and pitch: ϕ = 90°

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oscillating foil propulsion

MIT Center for Ocean Engineering, 2005

St = 0.18

St = 0.35

Wake Morphology Derived from CFD Experimental

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