**3. Calculation of force**

In the following section, we describe calculations carried out when *U*=*ρ*=*a*=1 unless specified otherwise.

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

According to Eq.(5), the movement in the *y*-direction of the plate with a circulation Γ<sup>1</sup> gives rise to the force *Xb* normal to it,

$$X\_b = -\rho \Gamma\_1 \dot{\mathfrak{y}}\_0. \tag{18}$$

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 *y*-direction at the initial stage *t* ≈0. Vortices rotating in the positive direction appear under the lower place near the trailing edge. Similarly, in the *ζ*-plane, the mirror images of the vortices rotating in the anticlockwise appear in a circle with radius *a*. In these, circumstances the circulation Γ around the circle is positive. In this case the force acts in the negative *x*-direction, because the sign of *Xb* is negative from Eq.(18).

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 *t*, many vortices of the same number of free vortices exist inside the circle.

P When *<sup>n</sup>* vortices are released, the intensity of the vortices is expressed as a sum *<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*κi*. At the same time, the sum of vortices within the circle of the radius *<sup>a</sup>* determines the circulation Γ of the bound vortex. The circulation of the bound vortex is 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*

**Figure 6.**

*A simple case where two free vortices κ*<sup>1</sup> *and κ*<sup>2</sup> *are released is illustrated in the ζ-plane. Two vortices of the opposite signs exist in the circle in the ζ-plane. The circulation* Γ *around the circle is the sum of two vorticities,* 2*π κ*ð <sup>1</sup>*+κ*2Þ*.*

$$
\Gamma = \sum\_{i=1}^{n} 2\pi\kappa\_i. \tag{19}
$$

Using this circulation, we try to evaluate the force generated by the heaving motion. In Eq.(18) by changing Γ<sup>1</sup> by Γ, we have the force,

$$X\_b(t) = -\rho \Gamma \frac{\mathrm{d} \mathbf{y}\_0}{\mathrm{d}t},\tag{20}$$

The variation of *Xb* calculated by using the above equation is shown in **Figure 7** as a function of nondimensional time *τ* ¼ *t=*ð Þ 2*a=U* . The variation of the heaving velocity *wh*ð Þ*τ* of the plate (Eq.(17) is also plotted there). Let us consider the initial stage when the plate moves upward. Vortices generated by the upward movement are rotating in the clockwise direction, as shown in **Figures 4** or **5**. At the initial stage, mirror images inside the circle of the radius *a* rotate in the anticlockwise direction. In other words, the plate has a positive circulation. Because the velocity *wh* is initially positive, the force *Xb* is negative from Eq.(20). The force acts against the main flow, i.e., the plate is pulled by the fluid in the negative *x*-direction. Usually, the upward motion connects with positive circulation, and hence the force becomes almost negative. On the contrary, negative circulation occurs when the motion is downwards. As a result, the sign of *Xb* has a negative value in the mean. The average value *Xb av* ð Þ is �0.240. The index (av) stands for the mean over two periods of oscillation. It should be noted that when the absolute velocity of the plate ∣*wh*∣ is the maximum, the force becomes maximum. This means that during the generation of strong vortices, the pressure at the edge becomes large. Strictly speaking, slight time delay is also observed. This may be because of the effect of the convection due to other free vortices.

The right-hand side of Eq.(20) expresses the differentiation of the virtual momentum *ρ*Γ*y*<sup>0</sup> with respect to time, if the circulation could be independent of time. It might be the incorrect estimation of the force. The right-hand side of Eq. (19) expresses the summation of all vortices and each vorticity is independent of time. However, because the number of vortices changes with time, the circulation Γ should be considered to be time-dependent. The dependence of time on the

#### **Figure 7.**

*The variation of force by the movement of the plate with circulation* Γ. *Denoting the force as Xb, the variation divided by* 2*aρU is plotted as a function of τ*ð Þ ¼ *t=*2*a=U* .

circulation must be taken into account. From this point of view, there is a room for reconsidering the results.

As seen in **Figure 7**, the force *Xb* varies with a period *π=ν*ð Þ ¼ *T=*2 and has a negative value on an average. However, we did not take into account the variation of Γ. The circulation Γ around the plate changes with the same period *π=ν*. In Eq. (20) we took into account the differentiation of the vitual momentum partly, and it could not give the correct force induced by virtual momentum. The *x*-component of the real virtual momentum, *ρ*Γð Þ*t y*0ð Þ*t* , has two time-dependent variables, Γ and *y*0. To estimate the correct force *Xb*, we should take into account the variation of the virtual momentum. The correct expression for the force:

$$X\_b = -\frac{\mathbf{d}}{\mathbf{d}t} \left(\rho \Gamma(t) \boldsymbol{\uprho}\_0(t)\right). \tag{21}$$

In the present situation, Γ and *y*<sup>0</sup> are both periodic function of time whose period is 2*π=ν*ð Þ ¼ *T* . The product of two periodic functions with the same period is also a periodic function. The differentiation with respect to *t* is also a periodic function whose average is zero. Finally, we conclude that the force *Xb* gives no net force, or *Xb av* ð Þ=0. Here the subscript (av) stands for the average over two periods of time, 2*T*.

#### **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 *y*-direction. This problem was first discussed and the solution was analytically given by Kármán-Sears in the linear limit [15]. Their solution corresponds to the sum of the forces *Yv* and *Yd*. The force *Yv* has already been given in Eq.(15) only when one free vortex exists. For the present aim, however, the formula should be extended to include all the vortices. In the following, according to [15] the variation of force divided by 2*aρU* is shown.

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

When *WT*=0.5 and *ν*=0.5 the variations are given in **Figure 8**. The variation of *Yv* in the VM, and that of the sum of *Yv* and *Yd* in the DI. The change of the sum *Yv*+*Yd* agrees well with the analytical result of [15]. In particular, the agreement becomes better for a lower *WT*. It is seen that the two components *Yv* and *Yd* have an importance of the same degree on the generation of force. At an initial stage, *τ* ≈0, the sum has a negative value, and the minimum value of the force occurs at the stage where the velocity in the *y*-direction becomes maximum, which corresponds to the initial instant *τ* ¼ 0. The force acts as a drag in this heaving motion. It is interesting to investigate the *y*-component of the force with respect to the virtual momentum. Because such a force in the *y*-direction is not related directly to the thrust force, hereafter we will not discuss it further.

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 *x* ¼ 40 at *τ* ≈20.

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 *κ*<sup>1</sup> as

$$X\_{\upsilon} = 2\pi\rho\kappa\_{\mathsf{U}}\dot{\mathfrak{y}}\_{1}.\tag{22}$$

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 *Xv av* ð Þ is calculated to be 0.148. This means that the force acts in the positive *x*-direction or the fluid pushes the plate to the direction of the flow. Similarly, the plate is adding the force to the flow as a reaction. In this sense, we may regard the positive *Xv av* ð Þ as a thrust. Whether this force acts as a thrust or a drag depends on the combination of the sign of *κ*<sup>1</sup> and that of the velocity *y*\_0. Possible combinations are listed in **Table 1**.

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 *t* ¼ 19*:*8 and *t* ¼ 39*:*8, it is found that

#### **Figure 8.**

*The variation of the force in the y*-*direction Yv due to the change of the virtual momentum. The variation of the sum of Yv and Yd of the DI is also plotted.*

#### **Figure 9.**

*The force generated from the variation of the virtual momentum is Xv* þ i*Yv. The variation of the x-component, Xv=*2*aρU, is plotted as a function of the dimensionless time τ when WT=0.5 and ν=0.5.*


#### **Table 1.**

*Signs of κ*1*, velocity and force. The combination of the sign of κ*<sup>1</sup> *and y*\_<sup>1</sup> *determine whether Xv acts as a thrust or a drag.*

the vortices rotating in the positive direction move upward and those rotating in the negative direction moves downward. This tendency is pronounced for the vortices existing near the trailing edge. It is noted that the force has its peak when the heaving velocity *wh* is at the maximum or the minimum. The period of the force oscillation is *T=*2. When the plate passes through *y* ¼ 0, the vortex with a strong intensity is generated. At this instant, the force reaches the maximum.

**Table 1** suggests that the sign of the force *Xv* is positive. In fact, it is seen from **Figure 9** that the average of the force *Xv* is calculated to be positive.

By comparing **Figures 7** and **9** it is clear that the force component *Xv* is small compared to *Xb*. The heaving motion has an influence more effectively on the generation of force in the *y*-direction As far as the thrust force is concerned, however, the force *Xb* produced directly by the heaving motion has no effect. Therefore, the force we should take into account is the force *Xv* only as a thrust.

#### **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 *WT*, we plotted *Xv av* ð Þ 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*

**Figure 10.** *Thrust variation as a function of WT when U=1 and ν=0.5. Thrust increases proportional to W*<sup>2</sup> *T.*

a function of *WT* in **Figure 10**. As mentioned earlier, the subscript (av) means the average over two periods. It is easy to see that the thrust is proportional to *W*<sup>2</sup> *T* except when the *WT* value is lage. In this case, the proportional constant is estimated to be 0.592. In addition to *Xv av* ð Þ, the variation of *Xb av* ð Þ is also plotted for comparison.

Next, we show the variation of the thrust *Xv av* ð Þ*=*2*aρU* as a function of *U* in **Figure 11**. The curve seems to be inversely proportional to *U* except for large values of *U*. This means that the thrust *Xv* does not depend on the velocity *U*. When *WT*=0.5, the constant of proportionality is estimated as 0.148.

#### **3.4 Effect of heaving frequency on the force**

From the previous subsection, it can be seen that *Xv* is proportional to *W*<sup>2</sup> *<sup>T</sup>* and does not depend on *U*. To confirm it, we have plotted the nondimensional variable *Xv av* ð Þ*=*ð Þ *<sup>L</sup>=*<sup>2</sup> *<sup>=</sup>ρU*<sup>2</sup> as a function of ð Þ *WT=<sup>U</sup>* <sup>2</sup> in **Figure 12** for three different *WT*'s, i.e., 0.3, 0.5 and 0.7. The coefficient *Xv av* ð Þ*=*2*aρU*<sup>2</sup> is called the thrust coefficient denoted as *CL*. The coefficient has almost linear relation to the velocity ratio *WT=U*. The constant of proportionality must be nondimensional. In such unsteady locomotions, the most important dimensionless parameter is the reduced frequency *k*ð Þ ¼ 2*aρν=U* or the Froude number. However, **Figure 12** gives no defined dependence of the reduced frequency on the coefficient *CT*. The plotted data include various values of *k* between 0*:*2≤*k*≤ 2*:*5. We can draw our conclusion that the reduced frequency *k* does not affect the thrust coefficient in this heaving motion.

We summarise the thrust coefficient in the nondimensional form,

$$\mathbf{C}\_{T} = \frac{X\_{v(av)}}{L\frac{1}{2}\rho l\ell^{2}} \propto \left(\frac{\mathcal{W}\_{T}}{\mathcal{U}}\right)^{2},\tag{23}$$

where *L* is the chord length being equal to 4*a*.

**Figure 11.**

*Thrust variation as a function of U for WT* ¼ 0*:*5*. The thrust is inversely proportional to U. The proportional constant is estimated as 0.148.*

**Figure 12.**

*Thrust coefficient CT as a function of W*ð Þ *<sup>T</sup>=<sup>U</sup>* <sup>2</sup> *. For three different WT the data are almost decomposed into a unique line.*

### **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

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

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 *k* or Froude number. How the parameter plays a part in the generation of force has been a main concern of many people. Many researchers have tried to address this problem experimentally. However, the task was difficult to address, and only few researchers have answered analytically.

Our result is for the coefficient of thrust *CT*,

$$\mathbf{C}\_{T}\mathbf{x}\left(\frac{\mathbf{W}\_{T}}{U}\right)^{2}.\tag{24}$$

The proportional constant is nondimensional and does not depend on the parameter *k* expressing the unsteadiness of flow. The thrust force *Xv av* ð Þ is independent of uniform velocity *<sup>U</sup>*, and therefore the coefficient *CT* is proportional to ð Þ *WT=<sup>U</sup>* <sup>2</sup> .

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.

### **Acknowledgements**

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