Signs of

## Abstract

To understand the force acting on birds, insects, and fish, we take heaving motion as a simple example. This motion might deviate from the real one. However, since the mechanism of force generation is the vortex shedding due to the motion of an object, the heaving motion is important for understanding the force generated by unsteady motion. The vortices released from the object are closely related to the motion characteristics. To understand the force acting on an object, information about momentum change is necessary. However, in vortex systems, it is impossible to estimate the usual momentum. Instead of the momentum, the “virtual momentum,” or the impulse, is needed to generate the force. For calculating the virtual momentum, we traced all vortices over a whole period, which was carried out by using the vortex-element method. The force was then calculated based on the information on the vortices. We derived the thrust coefficient as a function of the ratio of the heaving to travelling velocity.

### Keywords

- heaving motion
- virtual momentum
- unsteady effect
- extended Blasius formula
- vortex street

## 1. Introduction

Motion of insects or birds is inherently unsteady. The creatures utilise the unsteadiness efficiently. For example, a coherent structure called the leading edge vortex (LEV) plays an essential role in the generation of unsteady force. Many authors have published studies on the topic and hilighted its importance, experimentally and numerically. The magnitude of the unsteady force cannot be explained by a steady-state approach. In many cases, the unsteadiness generates greater forces more efficiently than that in the steady state [1, 2]. Experiments have been conducted in three-dimensional space and numerical analyses have been carried out to understand the mechanism of force generation. These studies explained several aspects of unsteady phenomenon, but the role of vortices generated close to the object is still unclear. How does the behaviour of vortices affect the generation of force? In particular, how does momentum change depend on the force? We are not sure how to estimate the momentum of a vortex system, because the usual momentum has no definite value. Our aim is to establish a rule that governs the force generation by the momentum change. Characteristics such as the magnitude, the rotation direction, and the position are key to determining the momentum. Unless we determine their properties, the evaluation of force cannot be made quantitatively.

When an object of a constant circulation

A lot of attention has been paid to the dependence of parameters characterising the unsteadiness known as the reduced frequency or the Strouhal number of the propulsive motion of insects, fish and humans (for example, [4, 5, 6]). Here, we also discuss the dependence of the reduced frequency on the thrust.

The heaving motion of a thin plate is the simplest and most suitable example of the analysis of unsteady phenomena. In addition, the heaving motion is solved in the limit as the heaving amplitude becomes smaller. For investigating the unsteady phenomenon, the vortex motion is a key concept. The analytical tool used here is not specific and can be extended to wider problems.

## 2. Direct effect of a heaving plate

First, we have a look at the relation between the force acting on a body fixed in a stream and the free vortices flowing behind it. It is known that a drag acts on a still body set in the stream. We can see two vortex rows here, called the Kármán vortex street (see Figure 1(a)).

We can also notice another similar vortex street behind the flying birds and the swimming fish. However, the direction of rotation of the vortices is inverse. In the case of the Kármán street, a momentum defect is observed while the momentum seems to increase behind the birds and fish. In the latter case, a thrust acts on the object to move forward due to the increase in momentum. As an example, we show the vortex street appearing in heaving motion (see Figure 1(b)). In pitching motion, a similar street can be observed (see example, [7]). In general, those cases where backward momentum increases generate thrust acting against the flow. In the figure, the thick arrows denote the direction of the increased momentum.

To understand the mechanism of thrust generation we study the heaving motion of a thin plate in a uniform flow. We assume that the plate has a constant circulation

To evaluate the force acting on an object, we usually integrate the pressure on the surface of the object. However, because a simple plate has two singular points at the leading and trailing edges. In particular, the estimation of the pressure at the leading edge is almost impossible when Kutta’s condition is applied at the trailing edge. Instead of the integration of pressure, we apply Newton’s second law of motion, which states that the force is a result of the momentum change. However, it is known that the estimation of momentum is almost impossible, and hence virtual momentum has to be used instead.

### 2.1 Effect of bound vortex

The coordinates system is shown in Figure 2. A thin aerofoil is located at

Consider a uniform flow whose velocity is

where the dot denotes the derivative with respect to time

For cases without any motion, the above equation is written simply as

The second term on the right-hand side indicates the drag defined as

where

This formula corresponds to the Kutta-Joukowski theorem. When the object with the circulation

Eq. (5) can be derived easily by considering the virtual momentum. For an object with a constant circulation

### 2.2 Effect of free vortex

Next, we proceed to discuss about the effect of free vortices on the force. The general rule for estimating the force, when the viscosity is negligible, is the Blasius formula, see [10]. Since the formula is valid only for steady flow conditions, it has to be extended to include the unsteady effect. The extended formula for the force

where

where

First, we consider Eq.(10). This force is dependent on the object form. To integrate it we map a plate in the

When a vortex is located at

where

and the convection velocity,

It is easy to see that the right-hand side of Eq.(12) is pure imaginary, because the right-hand side expresses the sum of a complex and its complex conjugate. This means that the force has only a

From Eq.(10), we have

where

Formulas (5) and (15) are the main targets for the calculation of thrust.

### 2.3 Determination of positions and velocities of a vortex

Now, we discuss how to generate a vortex under our boundary condition. What determines the vorticity and its position? Consider a flat plate set parallel to the flow (see Figure 2). Even in unsteady motion, the flow is subject to the condition that the fluid flows smoothly at the trailing edge. In other words, Kutta’s condition at the edge must be satisfied at all times. We consider the heaving motion whose velocity, perpendicular to the plate is expressed as

In the above equation,

Because the plate has a velocity in the

We proceed to the next step to discuss the problem of movement of vortices. A vortex moves by the other free vortices including the bound vortex and the uniform velocity. The induced velocity

Actual calculations were done in the

### 2.4 Calculation results

In the calculations, we determine the physical variables by choosing

In Figure 5, the distribution of vortices

## 3. Calculation of force

In the following section, we describe calculations carried out when

### 3.1 Direct force by movement of a plate with a circulation

According to Eq.(5), the movement in the

We investigate the thrust generation due to the movement of a thin flat plate in more detail. When the motion is subjected to Eq.(17), we consider the force in the

The case where two free vortices are outside the circle is shown in Figure 6. For more than a vortex in the flow field there must be mirror images whose sign is opposite to the free vortices. In general, at time

When

Using this circulation, we try to evaluate the force generated by the heaving motion. In Eq.(18) by changing

The variation of

The right-hand side of Eq.(20) expresses the differentiation of the virtual momentum

As seen in Figure 7, the force

In the present situation,

### 3.2 Effect of moving vortices

In this subsection, we discuss the force resulting from the movement of free vortices. First, we show the result of the force in the

At the initial stage, it is seen from Figures 4 and 5 that vortices of positive vorticity appear. These vortices travel to the position near

Next we consider the thrust component of the force generated by the change of virtual momentum. From Eq.(15) the force component is expressed for a vortex

Taking into account all the vortices existing in the flow field, we can get a complete set of the component for the present problem. The variation is shown in Figure 9. It seems to oscillate sinusoidally except for the initial stage and has a positive value in the mean. For this example, the mean value

^{a}(Direction of rotation) | ^{b} (Direction of force) | Thrust or drag | ||
---|---|---|---|---|

I | thrust | |||

II | drag | |||

III | drag | |||

IV | thrust |

Behind the heaving plate there appear two vortex streets, as shown in Figures 3 and 4. The upper street consists of vortices rotating in the positive direction, and the lower one consists of vortices rotating in the negative one. By inspecting the distributions of vortices at two different times

Table 1 suggests that the sign of the force

By comparing Figures 7 and 9 it is clear that the force component

### 3.3 Effect of heaving amplitude on the force

It seems that the thrust force is generated due to the motion of the plate against the fluid. To understand the role of the heaving amplitude

Next, we show the variation of the thrust

### 3.4 Effect of heaving frequency on the force

From the previous subsection, it can be seen that

We summarise the thrust coefficient in the nondimensional form,

where

## 4. Concluding remarks

Thrust force can be generated by a simple heaving motion of a plate. The force is perpendicular to the direction of oscillation. A pair of rows of vortices plays an important role in the generation of the force. The two vortex streets give rise to an increase in momentum in the direction normal to the direction of oscillation. The word “momentum” here does not mean the usual momentum but the virtual one, because the usual momentum cannot be determined in such a vortex system. The direct integration of the pressure around the surface of a body is not a correct way to know the thrust generation. Application of the virtual momentum to the generation of force made the estimation of the force possible.

In general, the most important parameter characterising the unsteady flow is the reduced frequency

Our result is for the coefficient of thrust

The proportional constant is nondimensional and does not depend on the parameter

Although our analysis is confined to the heaving motion of a thin plate, we summarise that the force due to the vortex movement can be expressed as a function of nondimensional quantity in a simple form. It is expected that our analysis could apply to more complex movement of an aerofoil.

## Acknowledgments

The author thanks Professor Hidenobu Shoji of Tsukuba University for many useful discussions and important information regarding vortex element methods.

## Nomenclature

a | Radius of circle in mapped plane |

CT | Thrust coefficient |

k (=2aρν/U) | Reduced frequency |

L | Chord length (=4a) |

T=2π/ν | Period of oscillation |

U | Uniform velocity |

WT | Amplitude of heaving velocity |

X+iY | Complex force |

x+iy | Coordinates in complex plane |

w=u−iv | Complex velocity |

ν | Radian frequency of heaving |

Γ | Circulation along the curve in the anticlockwise direction |

κ | Vorticity (positive for clockwise, negative for anticlockwise) |

ρ | Density of fluid |

ζ | Plane mapped from real z-plane |

VM | Virtual momentum component |

DI | Direct interaction component |

v | Virtual component |

d | Direct interaction component |

b | Bound vortex |

av. | Average |

LEV | Leading edge vortex |