effects of tail geometries on the performance and …...shapes by digitizing the real caudal fins of...

1 Copyright © 2016 by ASME

Proceedings of the ASME 2016 Fluids Engineering Division Summer Meeting

FEDSM2016 July 10-14, 2016, Washington, DC, USA

FEDSM2016-7691

EFFECTS OF TAIL GEOMETRIES ON THE PERFORMANCE AND WAKE PATTERN IN FLAPPING PROPULSION

Geng Liu, Haibo Dong

Department of Aerospace and Mechanical Engineering,

University of Virginia,

Charlottesville, VA, 22903, USA

ABSTRACT Swimming fishes exhibit remarkable diversities of the

caudal fin geometries. In this work, a computational study is

conducted to investigate the effects of the caudal fin shape on the

hydrodynamic performance and wake patterns in flapping

propulsion. We construct the propulsor models in different

shapes by digitizing the real caudal fins of fish across a wide

range of species spanning homocercal tails with low aspect ratio

(square shape used by bluegill sunfish, rainbow trout, etc.) or

high aspect ratio (lunate shape adopted by tuna, swordfish, etc.),

and even heterocercal caudal fin adopted by sharks. Those fin

models perform the same flapping motion in a uniform flow to

mimic fish’s forward swimming. We then simulate the flow

around the flapping fins by an in-house immersed-boundary-

method based flow solver. According to the analysis of the

hydrodynamic performance, we have found that the lunate shape

model (high aspect-ratio) always generates a larger thrust

compared to other models. The comparison of the propulsive

efficiency shows that the large aspect ratio fins (tuna and shark)

have a higher efficiency when the Strouhal number (St) is in the

range of steady swimming (0.2<St<0.4), while the lower aspect

ratio caudal fins (catfish, trout, etc.) are more efficient when

St>0.4, in which the fish is accelerating or maneuvering. Finally,

the 3D wake patterns of those propulsors are analyzed in detail.

NOMENCLATURE

hA : amplitude of heaving motion

wA : amplitude coefficient of wavy motion

A : mean amplitude

AR: aspect ratio

c : chord length

c : mean chord length

TC : thrust coefficient

PC : power coefficient

f : flapping frequency

L : span length

Re : Reynolds number

S : fin area

St : Strouhal number

T : flapping period

hy : lateral displacement of heaving motion

wy : lateral displacement of wavy motion

: phase difference between the heaving and

wavy motion

w : wave length

INTRODUCTION Fishes in nature have a substantial diversity of the

geometry of their caudal fins. It is thought that different

geometric configurations may serve for different hydrodynamic

purposes. For example, large aspect-ratio lunate caudal fins are

usually adopted by fast swimmers which requires a high

propulsive efficiency for their long-distance cruising. Low

aspect ratio caudal fins are usually used by smaller fishes which

desire a high maneuverability to escape from predators.

Biologists have contributed a large amount of literatures in

quantifying the characteristics of the fin shape and their flapping

kinematics [1-4]. People also studied the hydrodynamics of the

flapping propulsors with different shapes separately [5-10].


There were little literatures systematically studied the effects of

shape in flapping propulsion. Several basic parameters were used

to characterize the shape, such as the aspect ratio (AR) and area

moments. Dong et al. [11] studied the wake topology and

hydrodynamic performance of pitching-plunging plate with

different aspect ratio. Raspa et al. [12] examined the role of

aspect ratio in self-propelled undulatory plates. Yeh and Alexeev

[13] studied the effect of the aspect-ratio in plunging flexible

plates. All those studies found that the varying of aspect ratio

will significantly change the propulsive performance and wake

pattern. Li and Lu [14] numerically investigated the dynamic

performance and the wake patterns of canonical plates with

different shapes and found that the thrust behavior is related to

the area moment of the plate.

In the flapping propulsion, the change of the shape not only

alters the geometric characteristics, but also changes the

kinematic features. For example, the flapping amplitude along

the trailing edge of the propuslor will vary as the trailing edge

shape changes even though the peak-to-peak amplitude of the tip

is kept the same. So in the selection of the basic control

parameters, we should consider the change of the dynamic

features with the geometry. In this paper, considering the most

important dynamic control parameters in flapping propulsion are

the Reynolds number and the Strouhal number, we first

coordinate the shape effect with the definition of these two

parameters. We select seven typical caudal fins, spanning

homocercal tails with low aspect ratio (square shape used by

bluegill sunfish, rainbow trout, etc.) or large aspect ratio (lunate

shape adopted by tuna, swordfish, etc.), and even heterocercal

caudal fin adopted by sharks. The fins are modeled as thin plates

with the thickness of only 3% of their chord length. To simplify

the problem and make a fair comparison of the different shapes,

the model fins are performing the same flapping kinematics,

which is digitized from the steady swimming of a typical

subcarangiform swimmer (bluegill sunfish). The fin models are

put in a uniform flow to mimic the fish’s forward swimming. The

three-dimensional incompressible Navier-Stokes equations are

solved with an immersed-boundary-method based flow solver.

The hydrodynamic performance (thrust, power and propulsive

efficiency) and the wake patterns are carefully analyzed and

compared. The detailed information about the methodology and

the results are presented in the following sections.

METHODOLOGY

Physical model

We digitize the caudal fin (or fluke) profiles of several

common fishes (or mammal) as shown in Figure 1. Those models

are cross a wide range of fish/mammal species, from fresh water

fish to marine animals; from low aspect-ratio (AR) to high aspect

ratio tails; from square shape to forked shape and eventually to

lunate geometry. Aspect-ratio is one of the important parameters

in quantifying the propulsor shape. It is defined as follows [11],

2LAR

S (1)

where S and L are the area and the span length of the propulsor,

respectively. We summarize AR of those propulsor models in

Table 1, in which the catfish has the lowest AR while the tuna has

the highest one.

Figure 1. Propulor models with different shapes digitized from the

real fish or aquatic mammals.

Table 1. AR of caudal fin models

species catfish trout sunfish bluefish

AR 1.218 1.951 2.137 2.802

species dolphin shark tuna

AR 2.815 3.503 6.652

To make a fair comparison among those models, we employ the

same flapping kinematics to them. Based on the high-speed

videos of a sunfish in steady swimming [15], the flapping motion

can be decomposed into two parts, i.e., a heaving motion and

wavy motion, which are expressed as follows,

h wy y y (2)

where y represents the lateral displacement; hy and wy are

the displacement due to the heaving and wavy motion,

respectively.

2

sinh h

ty A

T

(3)

sin 2w w

w

x ty A x

T

(4)

In equation (3) and (4), hA is the heaving amplitude; wA the

coefficient of the waving amplitude; w wavelength; the

phase difference between heaving and the waving; t is time; T is

the flapping period. The values of those parameter employed in

this paper are shown in Table 2.

trout bluefish tuna

catfish sunfish dolphin shark


Figure 2. Schematic of the flow past the flapping model during a

flapping cycle (Top view).

Table 2. values of kinematic parameters

hA wA w

0.133c 0.416c 6.0c -90˚

c is the chord length of the model. The top view of the fin

flapping during a full cycle is shown in Figure 2. This fin model

is put in a uniform flow with a speed of U.

Non-dimensional parameters

There are two key non-dimensional parameters in this

problem, i.e., the Reynolds number and and the Strouhal number.

The Reynolds number is determined by the reference speed,

reference length and the fluid viscosity ( ). We specify U as the

reference speed. In a flapping propulsion problem, like a

pitching-plunging motion, the chord length is usually specified

as the length scale [11, 16]. Considering that the chord length is

varying along the span, we use the mean chord length as the

length scale, which is,

S

cL

(5)

Then the Reynolds number is defined as,

ReUc

(6)

The Strouhal number is a parameter which quantifies the

characteristics of the unsteady flow. It is related to the vortex

shedding frequency and the wake width and the flow speed in

the wake. In a flapping propulsion problem, people usually

employ the flapping frequency and tip amplitude and the forward

speed of the propulsor to estimate the Strouhal number [17].

Again, as in the definition of the Reynolds number, the amplitude

along the trailing edge of those fin models should be different

even though the tip amplitude is the same. To eliminate the effect

of the shape on the flapping amplitude, we use the mean

amplitude on the trailing edge ( A ) to quantify the Strouhal

number.

fASt

U

(7)

where f is the flapping frequency f=1/T.

Numerical method

The numerical methodology of the immersed-boundary-method-

based Navier-Stokes equation solver employed in the current

study is briefly introduced here. The 3D incompressible Navier-

Stokes equations are discretized using a cell-centered, collocated

arrangement of the primitive variables, and is solved using a

finite difference-based Cartesian grid immersed boundary

method [18]. The equations are integrated in time using the

fractional step method. A second-order central difference scheme

is employed in space discretization. The Eulerian form of the

Navier-Stokes equations is discretized on a Cartesian mesh and

boundary conditions on the immersed boundary are imposed

through a ghost-cell procedure. This method was successfully

applied in many simulations of flapping propulsion [19-27].

More details about this method can be found in references [11,

18]. Validations about this solver can be found in our previous

works [20, 28].

The simulations were carried out on a non-uniform Cartesian

grid. The computational domain size was chosen as

15 10 10c c c with 400 160 192 (about 12.3 millions) grid

points in total. High-resolution uniform grids around the fin

models with the spacing of about 0.015c is designed to resolve

the near-field vortex structures. At the left-hand boundary, we

provide a constant inflow velocity boundary condition. The

right-hand boundary is the outflow boundary, allowing the

vortices to convect out of this boundary without significant

reflections. The zero gradient boundary condition is provided at

all lateral boundaries. A homogeneous Neumann boundary

condition is used for the pressure at all boundaries.

RESULTS

In this section, we will first compare the results

(hydrodynamic performance and the flow structures) of two

typical fin models, i.e., trout and tuna. And then we show the

results of all the models examined in this paper.

Trout .vs. tuna

In this sub-section, we compare the hydrodynamic

performance and wake patterns between trout and tuna caudal fin

models at St=0.35 and Re=2500. We simulate the cases for six

flapping cycles so that both the hydrodynamic forces and the

flow structure exhibit a good periodicities. The results shown in

the following are in the last flapping cycle.

Table 3. Hydrodynamic performance of trout and tuna caudal fin

models

trout tuna

TC 0.071 0.211

PC 0.435 0.836

0.164 0.252

U

x

y

o


Figure 3. Comparison of Thrust and power coefficients in trout and

tuna caudal models.

Figure 4. (a) Top view and (b) side view of the flow structures (

2 ) of trout caudal fin.

The force acting on the fin surface were computed through

direct integration of the surface pressure and shear. The thrust

force (T

F , opposite to x-direction shown in Figure 2) are

presented here as non-dimensional coefficients, which are

defined as 2

0.5

T

T

FC

U S , where

TC is the thrust coefficient.

We use the same way to non-dimensionalize the hydrodynamic

power as,3

0.5

h

P

PC

U S , where

hP and

PC are the

hydrodynamic power and its coefficient, respectively. And the

propulsive efficiency is defined as T PC C , where

TC

and PC are the cycle averaged thrust coefficient and power

coefficient, respectively.

Figure 5. (a) Top view and (b) side view of the flow structures (

2 ) of tuna caudal fin.

Figure 3 shows the comparison of time variation of T

C and P

C

between trout and tuna caudal fins and the cycle averaged values

of them are shown in Table 1. From Figure 3, both the force and

power coefficients exhibit half-cycle periodicity because the

flapping motion (left stroke and right stroke) is symmetric in our

model. There are two peaks and two troughs of thrust coefficient

during one flapping cycle for both trout and tuna. The trough

values in these two models are close to each other, however, the

peak values of tuna is much higher than those of trout. This leads

to a 190% increase of the mean thrust coefficient of tuna

compared to that of trout (see Table 3). The mean power

coefficient in tuna caudal fin model is nearly twice as much as

that in trout. The propulsive efficiency of the tuna caudal fin is

about 0.252, which is about 54% higher than that of trout.

The 3D vortex structures are visualized by the isosurface of

the imaginary part of the complex eigenvalue (λ) derived from

the instantaneous velocity gradient tensor, which identifies flow

regions where rotation dominates over strain [29-31]. The flow

structures of trout and tuna caudal fin models are shown in

Figure 4 and Figure 5, respectively. The overall feature of the

(a)

(b)

18°

(a)

(b)

12°

(a)

(b)


flow structures in these two models are similar to each other.

Both figures show that, at each stroke (left stroke or right stroke),

there is a vortex ring shed from the fin and the vortex rings are

inter-connected with each other. The backward flow induced by

these vortex rings is responsible for the thrust production. This

is in line with previous studies on the three-dimensional vortex

structures of fish tail [32, 33].

The difference of the 3D flow structures between trout and

tuna is obvious. From the top view (Figure 4 (a) and Figure 5

(a)), we can see that the vortex loop is expanding as they

propagate backward in both trout and tuna models, but the

expanding angle in the tuna (12˚) is less than that of trout (18˚). The lower expanding angle may lead to a more concentrated

backward flow, which explains the higher efficiency of the tuna

model.

The other obvious difference between those two models is that,

in trout model, the vortex ring seems compressed in the vertical

direction right after the shedding, so that the vertical vortex tube

shed from the trailing edge is tilted immediately. Though the

vortex rings is somewhat compressed in tuna model, we can

observe the vertical vortex tubes right behind the trailing edge.

This feature makes the wake have a stronger concentrated

vorticity, which is helpful to the thrust production. This is much

clearer when we show the vertical vorticity on the mid-plane of

these two cases in Figure 6. The 2D wake of the tuna caudal fin

is similar to that of a 2D model fish’s propulsion [34-37]. It is

known that the 2D model can generate more thrust than that of

3D model when the kinematics is the same.

Figure 6. Contour of y on the mid-plane of (a) trout and (b) tuna

caudal fin models.

Comparison of seven caudal fin models

In this sub-section, we simulate the hydrodynamics of the

seven caudal fin models shown in Figure 1. The Reynolds

number is fixed at Re=2500 while the Strouhal number is

varying from 0.25 to 0.6.

Figure 7. (a) Thrust coefficient and (b) propulsive efficiency of seven

caudal fin models.

Figure 7(a) shows the thrust coefficient of those seven caudal

fin models. It demonstrates that the thrust is increasing as St

increases in all models. According to this figure, the thrust is

highly dependent on the aspect-ratio. The lowest AR model

(catfish) has the lowest thrust and the highest AR model (tuna)

generates the largest thrust at the same Strouhal number. The

trend of the propulsive efficiency with respect to St is different

with that of thrust coefficient. According to Figure 7(b), we

found that the larger AR propulors (tuna and shark) are more

efficient when St < 0.4, while the lower AR propulsors (catfish,

trout, etc) have higher propulsive efficiency when St > 0.4. We

know that the Strouhal number of aquatic animals’ cruising is in

the range of 0.2 ~ 0.4 [38]. The higher St usually corresponds to

the accelerating motion or maneuvering, in which the swimming

speed is slower than the steady swimming. Our results indicate

that it is better to adopt a large AR propulsor when cruising at a

(a)

(b)

(a)

(b)


high speed, while a low AR fin is more efficient when

accelerating or maneuvering. This somewhat explains why the

high speed cruising fishes in the ocean, such as tuna, shark,

sailfish, etc., usually possess large AR caudal fins, but some

smaller fresh water fish such as sunfish and trout, which need a

high maneuverability to escape from predators, more likely

employ low AR fins.

Figure 8. 3D flow structures of (a) catfish, (b) bluefish and (c) shark

caudal fins. St=0.35, Re=2500.

Flow structures of three typical caudal fins (catfish, bluefish

and shark) among those seven models at St=0.35 are selected and

shown in Figure 8. Catfish caudal fin is the lowest AR one with

a round shape trailing edge. Its flow structure (Figure 8 (a))

doesn’t differ too much with that of trout because trout tail’s AR

is close to that of catfish. Bluefish tail is forked, so the shape of

the vortex shed from the trailing edge is a little tilted compared

with that of catfish. Another difference of the flow structures

between catfish and bluefish caudal fins is the connection

between the adjacent vortex rings. The vortical structures of

shark is quite different with those of all other models examined

in this paper because of the asymmetry of the upper and lower

lobe of the tail. The vortex ring shape is highly dependent on the

trailing edge shape. The vortical structures shed from the upper

lobe is much stronger than those shed from the lower lobe. This

is because the flapping amplitude of the upper lobe is larger than

that of the lower one. Further studies on the wake pattern of

heterocercal caudal fins will be carried out in the future.

CONCLUSIONS

The hydrodynamic performance and wake structures of

flapping propulsors with different shapes have been investigated

by solving the 3D Navier–Stokes equations using an immersed

boundary method. The propulsor models are digitized from the

real fish’s caudal fins and the flapping kinematics of a

subcarangiform swimmer (sunfish) is applied to all those

models. Our simulation results demonstrate that the lunate shape

model (large aspect-ratio) always generates a larger thrust

compared to other models. The comparison of the propulsive

efficiency indicates that the large AR fins (tuna and shark) has a

higher efficiency during cruising (St < 0.4), however, low aspect

ratio caudal fins (catfish, trout, etc.) is more efficient when fish

are accelerating or maneuvering (St > 0.4). The 3D wake patterns

of those propulsors are analyzed in detail. We found that vortex

shapes are highly dependent on the fin aspect-ratio and the

trailing edge shape. This work provides a physical insight into

the understanding of hydrodynamics and flow structures for the

propulsor with different geometries and provides guidelines to

the design of propulsors for bioinspired autonomous underwater

vehicles.

ACKNOWLEDGMENTS This work is supported by ONR MURI N00014-14-1-0533

and NSF CBET-1313217.

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effects of tail geometries on the performance and …...shapes by digitizing the real caudal fins of...

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