**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 Γ moves with a constant speed d*x*0*=*d*t*, a fluid force acts perpendicular to the direction of motion. The magnitude is known to be *ρ*ð Þ d*x*0*=*d*t* Γ. It should be noted that the magnitude is the derivative of the virtual momentum *ρx*0Γ with respect to time, see [3], Art.157. Here, *ρ* is the density of fluid. This is a simplest application of a well-known law that governs the conservation of virtual momentum. In other words, this is a typical example of the second law of motion in the vortex motion. In general, the virtual momentum plays an essential role in the generation of force instead of the normal momentum. As illustrated above, in unsteady flows, the virtual momentum is important for the generation of force. We would like to illustrate the role of the virtual momentum by applying it to a heaving motion of a thin plate.

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, [4–6]). 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

#### **Figure 1.**

*Vortex street and an object in the stream. (a) an object fixed in the stream; (b) a thin aerofoil heaving vertically. Two thick arrows denote the direction of momentum increase.*

#### *Thrust Force Generated by Heaving Motion of a Plate: The Role of Vortex-Induced Force DOI: http://dx.doi.org/10.5772/intechopen.100435*

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 Γ. Even in unsteady conditions, we assume that the fluid flows smoothly at the trailing edge according to Kutta's condition. The circulation Γ is determined by this smoothness condition. The velocity around the leading edge would diverge and hence the pressure may not be finite because the edge is a mathematical singular point.

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 *z* ¼ *z*0ð Þ*t* in the complex *z*-plane or at *z*<sup>0</sup> ¼ 0. The coordinates *z* and *z*<sup>0</sup> are related by the equation

$$\mathbf{z}'(t) = \mathbf{z} - \mathbf{i}\mathbf{y}\_0(t). \tag{1}$$

Consider a uniform flow whose velocity is *U* in the *x*-direction and a bound vortex of a constant circulation Γ around the plate and no free vortices. The circulation is positive when the fluid rotates in the anticlockwise direction, while the vorticity is positive for vortices rotating in the clockwise direction. The force *X* þ i*Y* acting on the object located at *z*<sup>0</sup> ¼ *x*<sup>0</sup> þ i*y*<sup>0</sup> is given by:

$$\mathbf{X} + \mathbf{i}Y = \mathbf{i}\rho\Gamma(\dot{\mathbf{z}}\_0 - \mathcal{U}) - 2\pi\rho a^2 \ddot{\overline{\mathbf{z}}}\_0 + 2\pi\rho a^2 \ddot{\overline{\mathbf{z}}}\_0,\tag{2}$$

where the dot denotes the derivative with respect to time *t* [8]. Here, the length of the plate is 4*a* (=*L*) and located parallel to the uniform flow (see **Figure 2**).

**Figure 2.** *Coordinates system. The heaving plate is located at y* = *y*0, �2*a* ≤*x*<sup>0</sup> ≤2*a*.

Confining ourselves to the oscillation only in the *y*-direction, or *z*\_<sup>0</sup> ¼ i*y*\_0ð Þ*t* , the force can be:

$$\mathbf{i}\prime X + \mathbf{i}Y = \mathbf{i}\rho\Gamma(\mathbf{j}\dot{\mathbf{y}}\_0 - \mathcal{U}) + 4\pi\mathbf{i}\rho a^2 \ddot{\mathbf{y}}\_0. \tag{3}$$

For cases without any motion, the above equation is written simply as *Y* ¼ �*ρU*Γ, which corresponds to the lift known as the Kutta-Joukowski theorem.

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

$$m'\frac{\text{d}^2\text{y}\_0(t)}{\text{d}t^2},\tag{4}$$

where *m*<sup>0</sup> is called the virtual mass. The direction of this force is parallel to the direction of motion. Accordingly, this force which acts in the *y*-direction cannot contribute to the propulsion. The virtual mass for this thin plate is expressed as *πρ*ð Þ *<sup>L</sup>=*<sup>2</sup> <sup>2</sup> <sup>¼</sup> *<sup>m</sup>*<sup>0</sup> ð Þ (see [9], Art. 9.222, [10–12] for the general discussion). This force acting only in the *y*-direction is independent of vortex formation and shedding. The force is not related to thrust, and hence we will not discuss this force any more. Finally, from Eq.(3) the force in the *x*-direction is

$$X = -\rho \dot{\boldsymbol{\wp}}\_0 \boldsymbol{\Gamma}.\tag{5}$$

This formula corresponds to the Kutta-Joukowski theorem. When the object with the circulation Γ is located at *z* ¼ *z*1, the virtual momentum is expressed as �i*ρz*1Γ.

Eq. (5) can be derived easily by considering the virtual momentum. For an object with a constant circulation Γ<sup>1</sup> located at the position *z*1, the momentum, or more precisely the virtual momentum, *P*, of the flow is expressed as �i*ρz*1Γ1. When the vortex moves at the speed *z*\_ 1, the force *F* acts on it as a result of momentum change, i.e.,

$$F = -\frac{\mathbf{d}P}{\mathbf{d}t} = \mathbf{i}\rho \dot{\mathbf{z}}\_1 \Gamma\_1. \tag{6}$$

#### **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 ð Þ *X*, *Y* , as seen in, for example, [13], is given as

$$\mathbf{X} - \mathbf{i}Y = \frac{\mathbf{i}\rho}{2} \oint\_{B} \left(\frac{\mathbf{d}f}{\mathbf{d}z}\right)^{2} \mathbf{d}z + \mathbf{i}\rho \oint\_{B} \frac{\mathbf{d}\overline{f}}{\mathbf{d}t} \,\mathrm{d}\overline{\mathbf{z}},\tag{7}$$

where *B* denotes the path along the surface of an object in the anticlockwise direction. In the above equation, *f z*ð Þ is the complex potential defined by *f z*ð Þ¼ *ϕ*ð Þþ *x*, *y* i*ψ*ð Þ *x*, *y* . Here *ϕ* and *ψ* are the velocity potential and the stream function, respectively. The bar denotes the complex conjugate. Thisin **Figure 10** formula expresses two typical types of forces. One is the virtual momentum (VM) component, and the other the direct-interaction (DI) component. VM acts due to the change in momentum and DI is the direct interaction of the vortices with the body, which becomes important when the vortex is near the object. We denote the two forces *Fv* for VM and *Fd* for DI to distinguish between them. Before we discuss the general case, we consider a simple one where one free vortex *κ*<sup>1</sup> exists at *z* ¼ *z*1. The forces for VM and DI are expressed as

*Thrust Force Generated by Heaving Motion of a Plate: The Role of Vortex-Induced Force DOI: http://dx.doi.org/10.5772/intechopen.100435*

$$
\overline{F}\_v = \frac{\mathbf{i}\rho}{2} \boldsymbol{\upPhi}\_B \left(\frac{\mathbf{d}f}{\mathbf{d}z}\right)^2 \mathbf{d}z,\tag{8}
$$

$$=2\oint\_{x\_1} \frac{\mathbf{i}\kappa\_1}{z - z\_1} \left[\frac{\mathbf{d}f}{\mathbf{d}z}\right]\_c \mathbf{d}z,\tag{9}$$

where d½ � *f =*d*z <sup>c</sup>* means the convection velocity at *z* ¼ *z*<sup>1</sup> by the vortex *κ*1. On the other hand, the force for DI is estimated from

$$\overline{F}\_d = \mathrm{i}\rho \overline{\oint\_B \frac{\partial f}{\partial t}} \mathrm{d}z. \tag{10}$$

First, we consider Eq.(10). This force is dependent on the object form. To integrate it we map a plate in the *z*-plane to a circle of radius *a* in the *ζ*-plane as

$$z = G(\zeta) = \zeta + \frac{a^2}{\zeta},\tag{11}$$

When a vortex is located at *z* ¼ *z*1, the integration can be carried out to give

$$\overline{F}\_d = 2\pi \dot{\rho} \kappa\_1 a^2 \left( \frac{1}{\zeta\_1^2 - a^2} \frac{d\overline{z}\_1}{dt} + \frac{1}{\overline{\zeta}\_1^2 - a^2} \frac{d\overline{z}\_1}{dt} \right),\tag{12}$$

where

$$z\_1 = G(\zeta\_1),\tag{13}$$

and the convection velocity,

$$\frac{\mathbf{dz}\_1}{\mathbf{d}t} = \begin{bmatrix} \mathbf{d}f \\ \mathbf{d}z \end{bmatrix}\_{x\_1} = \left( \mathbf{1} - \frac{a^2}{\zeta\_1^2} \right) \dot{\zeta}\_1. \tag{14}$$

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 *y*-component. Therefore, the component *Fd* is not related to the thrust. Hence, we will not discuss *Fd* anymore. Only the VM would contribute to the thrust force.

From Eq.(10), we have

$$\overline{F}\_v = X\_v - \mathrm{i}Y\_v = 2\pi\mathrm{i}\rho\kappa\_1\dot{\overline{z}}\_1,\tag{15}$$

where *z*\_ <sup>1</sup> is the covection velocity of vortex *κ*1. The above equation is to Eq.(6), because 2*πκ*1=�Γ1. To determine the convection velocity *z*\_ 1, we apply the conformal mapping Eq.(13) and trace the vortex in the *ζ*-plane and then calculate the velocity in the *z*-plane. The moving speed of vortex *κ*<sup>1</sup> in the *z*-plane is already given by Eq.(13). Hence, we have

$$\overline{F}\_v = 2\pi \text{i}\rho \kappa\_1 \left(1 - \frac{a^2}{\overline{\zeta}\_1^2}\right) \dot{\overline{\zeta}}\_1 \tag{16}$$

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

$$w\_h(t) = \mathcal{W}\_T \mathbf{e}^{\dot{\mathbf{u}}t}.\tag{17}$$

In the above equation, *ν* is the radian frequency of the heaving motion, and *WT* is the amplitude. Denoting the period of the oscillation as *T*, *T* ¼ 2*π=ν*.

Because the plate has a velocity in the *y*-direction at *t* ¼ 0, Kutta's condition is not satisfied. To satisfy the condition we set a new vortex at *x* ¼ 2*a* þ Δ*x*, and we determine the vorticity *κ*<sup>1</sup> of the vortex so as to satisfy the condition. As for setting the initial position, [14] serves as a useful reference. The condition for the flow leaving the trailing edge smoothly determines *κ*<sup>1</sup> uniquely. Later at *t* ¼ Δ*t* the vortex *κ*<sup>1</sup> moves away and hence the flow does not satisfy Kutta's condition again. To avoid the undesirable flow, we set a new vortex *κ*<sup>2</sup> at the same position as the initial position of *κ*1, i.e., at *x* ¼ 2*a* þ Δ*x*. Kutta's condition fixes the value *κ*<sup>2</sup> uniquely. Similarly, the subsequent process determines sequentially *κi*ð Þ *i* ¼ 1, 2, … .

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 *w* ¼ *u* � i*v* at *z* by the vortex *κ<sup>c</sup>* located at *z* ¼ *zc* is written as:

$$
\mu - \mathbf{\dot{w}} = \frac{\mathbf{\dot{x}}\_{\varepsilon}}{z - z\_{\varepsilon}}.
$$

Actual calculations were done in the *ζ*-plane with respect to all the vortices including those of the mirror image. The calculation step was carried out at every time for the step Δ*t*. See [14] for the suitable relation between Δ*x* and Δ*t*.

#### **2.4 Calculation results**

In the calculations, we determine the physical variables by choosing *a* ¼ 1, *ρ* ¼ 1 and *U* ¼ 1. In **Figure 3**, we show their positions and the direction of rotation for the case when *ν* ¼ 0*:*5 and *WT* ¼ 0*:*5 at *t* ¼ 19*:*8. The symbol + denotes the vortices of the clockwise rotation and those of the triangle (in red) the anticlockwise one, respectively. It is seen that the vortices rotating in clockwise direction gather at some places in the negative *y*-plane, while those rotating in anticlockwise direction gather in the positive *y*-plane. **Figure 4** shows the positions at *t*=39.8. We can find three clusters of vortices of clockwise rotation at about *x* ¼7.5, 20, and 33, and three clusters of anticlockwise rotation at *x* ¼13, 27, and 37. The clusters of positive or negative vortices occur by the interaction of each vortex. At those positions, the vorticities concentrate and have a structure in a large scale. Nonlinearity is seen even for such low *WT* (=0.5). Three clusters of vortices rotating in the clockwise direction are in the area for *y*<0, while three clusters rotating in the anticlockwise direction are in the area for *y*>0. This array of two vortex streets would generate the downward flow, which suggests that the momentum is generated in the positive *x*-direction. Momentum generation in the positive *x*-direction means the generation of thrust force, as will be explained later. The deviation of arrays from the ordered

*Thrust Force Generated by Heaving Motion of a Plate: The Role of Vortex-Induced Force DOI: http://dx.doi.org/10.5772/intechopen.100435*

#### **Figure 3.**

*Positions and the direction of rotation of vortices at t* ¼ 19*:*8 *for the case of WT* ¼ 0*:*5 *and ν* ¼ 0*:*5. *Vortices of the positive sign generated in the initial stage gather near x* = 12 *and those of the negative sign near x*=*17.5. The symbol* þ *stands for vortices rotating in the clockwise direction, while the triangle in red indicates vortices rotating in the anticlockwise direction.*

**Figure 4.**

*Positions and the direction of rotation of vortices at t* ¼ 39*:*8 *for WT* ¼ 0*:*5 *and ν* ¼ 0*:*5. *Some clusters of vortices rotating in the clockwise direction and those rotating in the anticlockwise direction appear.*

ones is the result of nonlinearity. **Figure 4** also shows the deviation of sinusoidal distribution of vortices. Next, we consider the positions of vortices at initial stages near *t*=0. Those vortices generated initially, which are distributed near *x* ¼ 40, fluctuate violently and move to the positive *y*-direction.

In **Figure 5**, the distribution of vortices *κ<sup>i</sup>* determined in the manner explained earlier is depicted. This plot shows the complex distribution of vortices based on the interactions among many vortices. This may explain the reason why the clusters are generated.

**Figure 5.** *Distribution of the vorticity at t* ¼*39.8 when WT* ¼ 0*:*5 *and ν* ¼ 0*:*5.
