Irregular frequency removal methods: theory and applications in hydrodynamics

Yujie Liu; Jeffrey M. Falzarano

doi:10.1007/s40868-017-0023-5

Marine Systems & Ocean Technology

June 2017, Volume 12, Issue 2, pp 49–64 | Cite as

Irregular frequency removal methods: theory and applications in hydrodynamics

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Yujie LiuEmail author
Jeffrey M. Falzarano

Article

First Online: 10 April 2017

Received: 02 November 2016

Accepted: 17 March 2017

129 Downloads
2 Citations

Abstract

In the problem of multi-body hydrodynamics, it is of growing interest to understand the gap resonance. One important issue is to remove the irregular frequency effect, which can be confused with the physical resonance. In this paper, an extensive survey of the previous approaches to the irregular frequency problem has been undertaken. The matrix formulated in the boundary integral equations will become nearly singular for some frequencies. The existence of numerical round-off errors makes the matrix still solvable by a direct solver, however results in unreasonably large values in some aspects of the solution, namely the irregular frequency effect. This paper will mainly discuss the lid method on the internal free surface. To reach a higher accuracy, the singularity resulting from the Green function needs special care. Finally, results with and without irregular frequency removal are shown to demonstrate the effectiveness of our proposed method. The validation cases in this paper include a single mini-boxbarge and two mini-boxbarges side by side. The major contribution of this paper is that we have investigated the difference in jump condition caused by basic settings, supplemented Kleinman’s uniqueness proof with a more detailed reasoning process, provided a different equation set for irregular frequency removal, proved the jump condition in a rigorous way, and further discussed Newman’s question about the singular terms in the jump conditions.

Keywords

Irregular frequency Green function Log singularity Hydrodynamics

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Appendix 1: Discussion on jump conditions

Problem description

In the notes attached to the WAMIT theory manual [38], it is mentioned one controversial problem in the jump condition adopted by Kleinman [21]. It is explained when the source and image source are both located at the free surface (i.e., $\zeta = 0$), the derivative would become:

$$\begin{aligned} \frac{\partial }{\partial \zeta }\left( \frac{1}{r} + \frac{1}{r'}\right) = 0 \quad \hbox {on} \ \zeta =0 \end{aligned}$$

(37)

This derivative would not lead to a delta function with area $4\pi$, which contradicts the jump condition in Kleinman [21], written as below:

$$\begin{aligned}&\lim \limits _{p \rightarrow C_\mathrm{w}^-} \int _{C_\mathrm{w}} u(q) \frac{\partial G(p, q)}{\partial n_q} \nonumber \\&\quad = 4\pi u(p) + \int _{C_\mathrm{w}} u(q)\frac{\partial G(p,q)}{\partial n_q} \hbox {d}S_q \end{aligned}$$

(38)

The jump condition above is not identical with the exact equation in Kleinman [21]. We have used notation rules mentioned in Newman [39] to derive the jump condition. For the convenience of the readers, the differences in coordinate and Green functions are listed below (Fig. 18).

Fig. 18
Coordinate setting in Newman [39]

In Newman [39] (Fig. 19):

$$\begin{aligned} G(\varvec{x};\varvec{\xi }) = \frac{1}{r} + \frac{1}{r'} + g(\varvec{x};\varvec{\xi }) \end{aligned}$$

(39)

In Kleinman [21]:

$$\begin{aligned} G(\varvec{x};\varvec{\xi }) = -\frac{1}{2\pi r} - \frac{1}{2\pi r'} + g(\varvec{x};\varvec{\xi }) \end{aligned}$$

(40)

In implementing the irregular frequency method, Newman [39] still adopted the expression in Kleinman [21]. We have also tested the program when we delete the term $4\pi u(p)$ in the final expression. In that case, we will get the wrong results. Then, the question becomes: Why the correct mathematical derivation will produce wrong results while the seemingly “wrong” expression can lead to the correct one?

To explain the jump conditions, the authors found that the understanding of $\partial /\partial n_{\varvec{\xi }}$ can lead to different results. One understanding is:

$$\begin{aligned} \frac{\partial }{\partial n_{\varvec{\xi }}} = \mathbf {n}_{\varvec{\xi }} \cdot \nabla _{\varvec{x}} = \mathbf {n}_{\varvec{\xi }} \cdot \left( \frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}\right) \end{aligned}$$

(41)

The other is:

$$\begin{aligned} \frac{\partial }{\partial n_{\varvec{\xi }}} = \mathbf {n}_{\varvec{\xi }} \cdot \nabla _{\varvec{\xi }} = \mathbf {n}_{\varvec{\xi }} \cdot \left( \frac{\partial }{\partial \xi }, \frac{\partial }{\partial \eta }, \frac{\partial }{\partial \zeta }\right) \end{aligned}$$

(42)

To explain the jump condition in Kleinman [21], the first one should be adopted while the second one will lead to 0 as discussed by Newman [39].

In the next sections, we will collect the materials from different researchers holding the two opinions.

Discussion

In here we are more interested in the singular terms 1 / r and $1/r'$. Thus, in the following discussions, we will only focus on the two terms, omitting the wavy Green function part. To study the difference, we used the formula in Katz and Plotkin [40] as benchmark and then introduced the controversial parts.

In Katz and Plotkin [40], from their definition of normal vector, we can get:

$$\begin{aligned} \phi (\varvec{x}) = -\frac{1}{4\pi } \int _{S} \sigma \left( \frac{1}{r}\right) \hbox {d}S + \frac{1}{4\pi } \int _{S} \mu \ \varvec{n}\cdot \nabla \left( \frac{1}{r}\right) \hbox {d}S \end{aligned}$$

(43)

where $-\mu = \phi - \phi _i, -\sigma = \frac{\partial \phi }{\partial n} - \frac{\partial \phi _i}{\partial n}, r = \sqrt{(x-\xi )^2 + (y-\eta )^2 + (z-\zeta )^2}.$

From physical intuitive, if the above equation stands for the potential at $\varvec{x}$, generated by a distribution of sources and doublets located at $\varvec{\xi }$, then $\varvec{n}$ is supposed to be $\mathbf {n}_{\varvec{\xi }}$, the normal vector at the specific doublet and $\nabla$ will be $\nabla _{\varvec{x}}$. They also derived the expression of the doublet when one sink and one source are approaching, which has confirmed this conclusion.

Furthermore, this equation is derived from the subtraction of the external potential and the internal potential.

$$\begin{aligned} \phi (\varvec{x}) = \frac{1}{4\pi } \int _S \left[ \frac{1}{r} \nabla (\phi - \phi _i) - (\phi - \phi _i) \nabla \left(\frac{1}{r}\right) \right] \cdot \varvec{n} \hbox {d}S \end{aligned}$$

(44)

This equation is coming from:

$$\begin{aligned} \phi (\varvec{x})&= \frac{1}{4\pi } \int _S \left( \frac{1}{r}\nabla \phi - \phi \nabla \frac{1}{r}\right) \cdot \varvec{n} \hbox {d}S \nonumber \\&=\frac{1}{4\pi } \int _S \left[ \frac{1}{r} \frac{\partial \phi }{\partial n_{\varvec{\xi }}} - \phi \frac{\partial }{\partial n_{\varvec{\xi }}}(\frac{1}{r}) \right] \hbox {d}S_{\varvec{\xi }}\end{aligned}$$

(45)

$$\begin{aligned} 0&= \frac{1}{4\pi } \int _S \left( \frac{1}{r}\nabla \phi _i - \phi _i \nabla \frac{1}{r}\right) \cdot \varvec{n} \hbox {d}S \nonumber \\&=\frac{1}{4\pi } \int _S \left[ \frac{1}{r} \frac{\partial \phi }{\partial n_{\varvec{\xi }}} - \phi \frac{\partial }{\partial n_{\varvec{\xi }}}\left(\frac{1}{r}\right) \right] \hbox {d}S_{\varvec{\xi }} \end{aligned}$$

(46)

Therefore, the $\nabla$ and $\mathbf {n}_{\varvec{\xi }}$ will share the same definition with (43). This conclusion is also supposed to hold when the field point is located on the boundary. From this point of physical meaning, we should adopt $\frac{\partial }{\partial n_{\varvec{\xi }}} = \mathbf {n}_{\varvec{\xi }} \cdot \nabla _{\varvec{x}} = \mathbf {n}_{\varvec{\xi }} \cdot (\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z})$.

However, when coming to the Green second Identity, the controversial point will arise. The Green second identity can be written as:

$$\begin{aligned} \int _S (\Phi _1 \nabla \Phi _2 - \Phi _2 \nabla \Phi _1) \cdot \varvec{n} \hbox {d}S = \nonumber \\ \int _V (\Phi _1 \nabla ^2 \Phi _2 - \Phi _2 \nabla ^2 \Phi _1) \hbox {d}V \end{aligned}$$

(47)

When substituting $\Phi _1 = 1/r$ and $\Phi _2 = \Phi$, we are assuming the integration variable should be identical with the independent variable in $\Phi _1$ and $\Phi _2$. Therefore, if the integration variable is $\varvec{\xi }, r = \sqrt{(x-\xi )^2 + (y-\eta )^2 + (z-\zeta )^2}$, then $\varvec{n} = \mathbf {n}_{\varvec{\xi }}$ and $\nabla$ will be defined as $\nabla _{\xi }$ because $\varvec{\xi }$ is the moving variable. Besides, if we adopt such definitions, we are indeed assuming the source or doublet point locates at $\varvec{x}$ and study its effect on the closed surface S, rather than the effect at $\varvec{x}$ from the source or doublet points distributed on the surface S.

The conflicts in the definition of $\nabla$ will not affect the source formulation. However, it is important in the irregular frequency removal methodology, which needs the formulation of both external potential and internal potential. Pitifully, for the time being, we do not have the code to evaluate the potential formula and cannot give the certain answer for the choices of $\nabla$. We believe the bridge between the effect from one single double and that from a distribution of doublet needs to be unveiled.

Conclusion

In conclusion, we are not able to decide $\frac{\partial }{\partial n_{\varvec{\xi }}} = \mathbf {n}_{\varvec{\xi }} \cdot \nabla _{\varvec{x}}$ or $\frac{\partial }{\partial n_{\varvec{\xi }}} = \mathbf {n}_{\varvec{\xi }} \cdot \nabla _{\varvec{\xi }}$ because either one can be valid in some sense but meanwhile induces controversial issues.

If we choose $\frac{\partial }{\partial n_{\varvec{\xi }}} = \mathbf {n}_{\varvec{\xi }} \cdot \nabla _{\varvec{\xi }}$, the proof of Kleinman [21] is valid but the jump condition on the internal free surface is questionable.

If $\frac{\partial }{\partial n_{\varvec{\xi }}} = \mathbf {n}_{\varvec{\xi }} \cdot \nabla _{\varvec{x}}$, we will expect the singular terms in the jump condition. However, the proof of Theorem 7 in Kleinman [21] will be invalid. The jump conditions in this case will be:

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_b^{\pm }} \frac{\partial }{\partial n_{\varvec{x}}} \int _{S_b} u(\varvec{\xi }) G(\varvec{x};\varvec{\xi })\hbox {d}S_{\varvec{\xi }}\\&\quad = \mp 2\pi u(\varvec{x}) + \int _{S_b} u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{x}}}\hbox {d}S_{\varvec{\xi }} \\&\lim \limits _{\varvec{x} \rightarrow S_b^{\pm }} \int _{S_b}u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} \\&\quad = \mp 2\pi u(\varvec{x}) + \int _{S_b} u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} \end{aligned}$$

where the normal vector is from inside the body to outside and the Green function is $G = \frac{1}{r} + \frac{1}{r'} + g$. Based on numerical tests, the final equations we adopted for irregular frequency module is as below:

$$\begin{aligned} 4\pi \frac{\partial \phi ({\varvec{x}})}{\partial n_{\varvec{x}}} =&-2\pi \sigma (\varvec{x}) +PV \iint _{S_b} \ \sigma (\varvec{\xi })\frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{x}}}\hbox {d}S_{\varvec{\xi }} \\&+ \iint _{S_i} \ \sigma '(\varvec{\xi })\frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{x}}}\hbox {d}S_{\varvec{\xi }} = 4\pi V_n, \qquad \varvec{x} \in S_b \\ 4\pi \frac{\partial \phi _-({\varvec{x}})}{\partial n_{\varvec{x}}} =&-4\pi \sigma '(\varvec{x}) +\iint _{S_b} \ \sigma (\varvec{\xi })\frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{x}}}\hbox {d}S_{\varvec{\xi }}\\&- PV\iint _{S_i} \ \sigma '(\varvec{\xi })\frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{x}}}\hbox {d}S_{\varvec{\xi }} = 0, \quad \varvec{x} \in S_i \end{aligned}$$

They have a sign difference with the conclusion in Kleinman [21], and the expressions are not exact the same with Newman [39], in which a numerical test is also conducted. The profound mathematical meaning needs further investigation.

Appendix 2: Cauchy principal integral in jump conditions

Kleinman [21] adopted the jump condition derived in Günter [41]. Günter applied the integral on a closed surface to prove the jump condition. The conclusion is also valid for an open surface. As a supplement, we will prove the conclusion for the integral on an open surface, which can be adopted for the integral on piecewise smooth surface. Please note that the integral equations are constructed on smooth surfaces or at least piecewise smooth surfaces.

The integral equations we are interested in are listed as below:

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_b^{\pm }} \frac{\partial }{\partial n_{\varvec{x}}} \int _{S_b} u(\varvec{\xi }) G(\varvec{x};\varvec{\xi })\hbox {d}S_{\varvec{\xi }}\nonumber \\&\quad = \mp 2\pi u(\varvec{x}) + \int _{S_b} u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{x}}}\hbox {d}S_{\varvec{\xi }} \end{aligned}$$

(48)

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_b^{\pm }} \int _{S_b}u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }}\nonumber \\&\quad = \pm 2\pi u(\varvec{x}) + \int _{S_b} u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} \end{aligned}$$

(49)

Both of the equations are in Cartesian coordinate, and the integrand contains a singularity when field point $\varvec{x} \rightarrow \varvec{\xi }$. We will evaluate the value by the Cauchy principal value integral. Since the integrand only depends on the relative distance between source point and field point, we can evaluate the integral in panel local coordinate. The origin located at panel center, z-axis is aligned long the panel normal vector and the field point located at a distance away from the center on z-axis (Fig. 20).

Jump condition I

The singular integral can be written as:

$$\begin{aligned} \lim \limits _{\varvec{x} \rightarrow S_b^{\pm }} \frac{\partial }{\partial n_{\varvec{x}}} \int _{S_b} \frac{u(\varvec{\xi })}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l - \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }} \end{aligned}$$

First, we will study the case when $\varvec{x} \rightarrow S_b^{+}$, more specifically, $z_l \rightarrow \zeta _l^+$ and assume the field point locates still infinitesimally close to the disk with radius $\epsilon \rightarrow 0$. If writing the integral in cylindrical coordinate, we will get:

$$\begin{aligned} \lim \limits _{\varvec{z_l} \rightarrow \zeta _l^+} \frac{\partial }{\partial z} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{u(\varvec{\xi })}{\sqrt{r^2 + (z-\zeta )^2}}r \hbox {d}r \hbox {d}\theta \end{aligned}$$

Assume $r = |z-\zeta | \tan \alpha , \hbox {d}r = |z-\zeta |\sec ^2\alpha \hbox {d}\alpha$, naturally $\cos \alpha = \frac{|z-\zeta |}{\sqrt{r^2 + (z-\zeta )^2}}$.

$$\begin{aligned}&\lim \limits _{z_l \rightarrow \zeta _l^+} \frac{\partial }{\partial z} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{u(\varvec{\xi })}{\sqrt{r^2 + (z-\zeta )^2}}r \hbox {d}r \hbox {d}\theta \\&\quad = \lim \limits _{z_l \rightarrow \zeta _l^+} \frac{|\cos \alpha |}{\cos \alpha } u(\varvec{x}) \frac{\partial }{\partial z} \int _{0}^{2\pi } \int _{0}^{\alpha (\epsilon )} |z-\zeta |\cdot \hbox {d}\left( \frac{1}{\cos \alpha }\right) \hbox {d}\theta \\&\quad = \lim \limits _{z_l \rightarrow \zeta _l^+} \frac{|\cos \alpha |}{\cos \alpha } u(\varvec{x}) \frac{\partial }{\partial z} 2\pi [\sqrt{\epsilon ^2 + (z-\zeta )^2} - |z-\zeta |] \\&\quad = 2\pi u(\varvec{x}) \lim \limits _{z_l \rightarrow \zeta _l^+} \frac{|\cos \alpha |}{\cos \alpha } \left[ -\frac{(z-\zeta )}{\sqrt{\epsilon ^2 + (z-\zeta )^2}} - \frac{z-\zeta }{|z-\zeta |} \right] \end{aligned}$$

When $z_l \rightarrow \zeta _l^+$, we will have:

$$\begin{aligned} \lim \limits _{z_l \rightarrow \zeta _l^+} -\frac{(z-\zeta )}{\sqrt{\epsilon ^2 + (z-\zeta )^2}} = 0 \\ \lim \limits _{z_l \rightarrow \zeta _l^+} -\frac{z-\zeta }{|z-\zeta |} = -1 \end{aligned}$$

Then,

$$\begin{aligned} \lim \limits _{z_l \rightarrow \zeta _l^+} \frac{\partial }{\partial z} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{u(\varvec{\xi })}{\sqrt{r^2 + (z-\zeta )^2}}r \hbox {d}r \hbox {d}\theta = - 2\pi u(\varvec{x}) \end{aligned}$$

(50)

Thus, the integral will become:

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_b^{+}} \frac{\partial }{\partial n_{\varvec{x}}} \int _{S_b} \frac{u(\varvec{\xi })}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l - \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }} \nonumber \\&\quad = -2\pi u(\varvec{x}) \nonumber \\&\qquad + PV \int _{S_b} \frac{\partial }{\partial n_{\varvec{x}}} \frac{u(\varvec{\xi })}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l - \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }} \end{aligned}$$

(51)

Similarly, if $\varvec{x} \rightarrow S_b^{-}$, we will have $z_l \rightarrow \zeta _l^-$. The limit will become:

$$\begin{aligned} \lim \limits _{z_l \rightarrow \zeta _l^-} -\frac{(z-\zeta )}{\sqrt{\epsilon ^2 + (z-\zeta )^2}} = 0 \nonumber \\ \lim \limits _{z_l \rightarrow \zeta _l^-} -\frac{z-\zeta }{|z-\zeta |} = 1 \end{aligned}$$

(52)

Therefore, the integral will be:

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_b^{-}} \frac{\partial }{\partial n_{\varvec{x}}} \int _{S_b} \frac{u(\varvec{\xi })}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l - \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }} \nonumber \\&\quad = 2\pi u(\varvec{x}) \nonumber \\&\qquad + PV \int _{S_b} \frac{\partial }{\partial n_{\varvec{x}}} \frac{u(\varvec{\xi })}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l - \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }} \end{aligned}$$

(53)

Up to this step, we have finished the proof of the first jump condition (48) when field pt is approaching the body surface from both sides.

When field pt is approaching the lid surface, the similar procedure will be repeated to the image source.

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_b^{\pm }} \frac{\partial }{\partial n_{\varvec{x}}} \int _{S_b} \frac{u(\varvec{\xi })}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l + \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }}\\&\lim \limits _{\varvec{z_l} \rightarrow \zeta _l^+} \frac{\partial }{\partial z} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{u(\varvec{\xi })}{\sqrt{r^2 + (z+\zeta )^2}}r \hbox {d}r \hbox {d}\theta \\&= 2\pi u(\varvec{x}) \lim \limits _{z_l \rightarrow \zeta _l^+} \frac{|\cos \alpha |}{\cos \alpha } \big [-\frac{(z+\zeta )}{\sqrt{\epsilon ^2 + (z+\zeta )^2}} - \frac{z+\zeta }{|z+\zeta |} \big ] \end{aligned}$$

Please note that at the lid surface, $\zeta = 0$.

$$\begin{aligned} \lim \limits _{z_l \rightarrow \zeta _l^-} -\frac{(z+\zeta )}{\sqrt{\epsilon ^2 + (z+\zeta )^2}} = 0 \nonumber \\ \lim \limits _{z_l \rightarrow \zeta _l^-} -\frac{z+\zeta }{|z+\zeta |} = -1 \end{aligned}$$

(54)

Then,

$$\begin{aligned} \lim \limits _{\varvec{z_l} \rightarrow \zeta _l^+} \frac{\partial }{\partial z} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{u(\varvec{\xi })}{\sqrt{r^2 + (z+\zeta )^2}}r \hbox {d}r \hbox {d}\theta = -2\pi u(\varvec{x}) \end{aligned}$$

When taking the sum of the contributions from source and image source, we will end up in the same expression in Kleinman [21].

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_i^{\pm }} \frac{\partial }{\partial n_{\varvec{x}}} \int _{S_i} u(\varvec{\xi }) G(\varvec{x};\varvec{\xi })\hbox {d}S_{\varvec{\xi }}\\&\quad = \mp 4\pi u(\varvec{x}) + \int _{S_i} u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{x}}}\hbox {d}S_{\varvec{\xi }} \end{aligned}$$

Since the point is undefined when $z>0$, thus the jump condition for the lid will be:

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_i^{-}} \frac{\partial }{\partial n_{\varvec{x}}} \int _{S_i} u(\varvec{\xi }) G(\varvec{x};\varvec{\xi })\hbox {d}S_{\varvec{\xi }}\\&\quad = 4\pi u(\varvec{x}) + \int _{S_i} u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{x}}}\hbox {d}S_{\varvec{\xi }} \end{aligned}$$

Jump condition II

If using the same coordinate setting,

$$\begin{aligned} \lim \limits _{\varvec{x} \rightarrow S_b^{\pm }} \int _{S_b} u(\varvec{\xi }) \frac{\partial }{\partial n_{\varvec{\xi }}} \frac{1}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l - \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }} \end{aligned}$$

First investigate the case when $\varvec{x} \rightarrow S_b^{+}$, more specifically, $z_l \rightarrow \zeta _l^+$ and assume the field point locates still infinitesimally close to the disk with radius $\epsilon \rightarrow 0$. If writing the integral in cylindrical coordinate and assuming $r = |z-\zeta |\tan \alpha , \hbox {d}r = |z-\zeta |\sec ^2\alpha \hbox {d}\alpha$, we will get:

$$\begin{aligned}&\lim \limits _{z_l \rightarrow \zeta _l^+} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{\partial }{\partial \zeta } \frac{1}{\sqrt{r^2 + (z-\zeta )^2}} u(\varvec{\xi })r \hbox {d}r \hbox {d}\theta \nonumber \\&\quad = \lim \limits _{z \rightarrow \zeta ^+} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{z-\zeta }{[\sqrt{r^2 + (z-\zeta )^2}]^3} u(\varvec{\xi }) r \hbox {d}r \hbox {d}\theta \nonumber \\&\quad = \lim \limits _{z \rightarrow \zeta ^+} u(\varvec{x}) \int _{0}^{2\pi } \int _{0}^{\alpha (\epsilon )} \frac{z-\zeta }{|z-\zeta |} \frac{|\cos \alpha |^3}{\cos ^3 \alpha } \sin \alpha \hbox {d}\alpha \hbox {d}\theta \nonumber \\&\quad = \lim \limits _{z \rightarrow \zeta ^+} u(\varvec{x}) \frac{z-\zeta }{|z-\zeta |} \frac{|\cos \alpha |^3}{\cos ^3 \alpha } 2 \pi \left[ -\frac{|z-\zeta |}{\sqrt{\epsilon ^2 + (z-\zeta )^2}} +1\right] \end{aligned}$$

(55)

In this case, $\epsilon \rightarrow 0$ will lead to $\alpha \rightarrow 0, \cos \alpha > 0$; from $z \rightarrow \zeta ^+$ we will have:

$$\begin{aligned} \lim \limits _{z_l \rightarrow \zeta _l^+} -\frac{(z-\zeta )}{\sqrt{\epsilon ^2 + (z-\zeta )^2}} = 0 \\ \lim \limits _{z_l \rightarrow \zeta _l^+} +\frac{z-\zeta }{|z-\zeta |} = 1 \end{aligned}$$

Then:

$$\begin{aligned}&\lim \limits _{z_l \rightarrow \zeta _l^+} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{\partial }{\partial \zeta } \frac{1}{\sqrt{r^2 + (z-\zeta )^2}} u(\varvec{\xi })r \hbox {d}r \hbox {d}\theta = 2\pi u(\varvec{x})\nonumber \\&\lim \limits _{\varvec{x} \rightarrow S_b^{+}} \int _{S_b} u(\varvec{\xi }) \frac{\partial }{\partial n_{\varvec{\xi }}} \frac{1}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l - \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }}\nonumber \\&\quad = 2\pi u(\varvec{x}) \nonumber \\&\quad + PV\int _{S_b} u(\varvec{\xi }) \frac{\partial }{\partial n_{\varvec{\xi }}} \frac{1}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l - \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }} \end{aligned}$$

(56)

Similarly, we can get:

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_b^{-}} \int _{S_b} u(\varvec{\xi }) \frac{\partial }{\partial n_{\varvec{\xi }}} \frac{1}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l - \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }}\\&\quad = -2\pi u(\varvec{x}) \\&\qquad + PV\int _{S_b} u(\varvec{\xi }) \frac{\partial }{\partial n_{\varvec{\xi }}} \frac{1}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l - \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }} \end{aligned}$$

Thus, we have proved the second equation in the jump condition:

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_b^{\pm }} \int _{S_b}u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} \nonumber \\&\quad = \pm 2\pi u(\varvec{x}) + \int _{S_b} u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} \end{aligned}$$

(57)

when the field point is approaching the lid surface $S_i$, the contribution from the source can be explained by the above equation. Herein, we will investigate the effect from image source.

First, we assume the field point is approaching lid surface from above and what will be the integral equation behave. Please note that for image source on the lid surface, we will have $\zeta = 0$. To make the derivation valid for a general case, we keep the notation $\zeta$ in the derivation.

$$\begin{aligned} \lim \limits _{\varvec{x} \rightarrow S_b^{\pm }} \int _{S_b} \frac{\partial }{\partial n_{\varvec{\xi }}} \frac{u(\varvec{\xi })}{\sqrt{(x_l - \xi _l)^2 + (y_l - \eta _l)^2 + (z_l + \zeta _l)^2}} \hbox {d}S_{\varvec{\xi }} \end{aligned}$$

Then, from similar approach, we will have:

$$\begin{aligned}&\lim \limits _{\varvec{z_l} \rightarrow \zeta _l^+} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{\partial }{\partial \zeta } \frac{u(\varvec{\xi })}{\sqrt{r^2 + (z+\zeta )^2}} r \hbox {d}r \hbox {d}\theta \nonumber \\&= 2\pi u(\varvec{x}) \lim \limits _{z_l \rightarrow \zeta _l^+} \frac{|\cos \alpha |^3}{\cos ^3\alpha } \cdot \left[ +\frac{(z+\zeta )}{\sqrt{\epsilon ^2 + (z+\zeta )^2}} - \frac{z+\zeta }{|z+\zeta |} \right] \end{aligned}$$

From $z \rightarrow \zeta ^+$ and $\zeta = 0$, we can get:

$$\begin{aligned} \cos \alpha > 0 \\ \lim \limits _{z_l \rightarrow \zeta _l^+} \frac{(z+\zeta )}{\sqrt{\epsilon ^2 + (z+\zeta )^2}} = 0 \\ \lim \limits _{z_l \rightarrow \zeta _l^+} \frac{z+\zeta }{|z+\zeta |} = 1 \end{aligned}$$

Thus,

$$\begin{aligned} \lim \limits _{\varvec{z_l} \rightarrow \zeta _l^+} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{\partial }{\partial \zeta } \frac{1}{\sqrt{r^2 + (z+\zeta )^2}} u(\varvec{\xi }) r \hbox {d}r \hbox {d}\theta = - 2\pi u(\varvec{x}) \end{aligned}$$

(58)

Considering the effect from source and its image source, the integral regarding Green function can be written as:

$$\begin{aligned} \lim \limits _{\varvec{x} \rightarrow S_i^{-}} \int _{S_i}u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} = \int _{S_i} u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} \end{aligned}$$

(59)

Following the same approach, we can get the expression when field point is approaching the lid surface from below and end up with:

$$\begin{aligned} \lim \limits _{\varvec{x} \rightarrow S_i^{\pm }} \int _{S_i}u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} = \int _{S_i} u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} \end{aligned}$$

(60)

It shows that if we adopt $\frac{\partial }{\partial n_{\varvec{\xi }}} = \mathbf {n}_{\varvec{\xi }} \cdot \nabla _{\varvec{\xi }} = \mathbf {n}_{\varvec{\xi }} \cdot (\frac{\partial }{\partial \xi }, \frac{\partial }{\partial \eta }, \frac{\partial }{\partial \zeta })$, we will not get the jump condition as that in Kleinman [21].

If we choose $\frac{\partial }{\partial n_{\varvec{\xi }}} = \mathbf {n}_{\varvec{\xi }} \cdot \nabla _{\varvec{x}} = \mathbf {n}_{\varvec{\xi }} \cdot (\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z})$, the integral equation will be:

$$\begin{aligned}&\lim \limits _{z \rightarrow \zeta ^+} \int _{B(x,\epsilon )} \frac{\partial }{\partial z} \cdot \frac{u(\varvec{\xi })}{\sqrt{(x-\xi )^2 + (y-\eta )^2 + (z-\zeta )^2}} \hbox {d}S_{\varvec{\xi }} \\&\quad = \lim \limits _{z \rightarrow \zeta ^+} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{\partial }{\partial z} \frac{1}{\sqrt{r^2 + (z-\zeta )^2}} u(\varvec{\xi }) r\hbox {d}r\hbox {d}\theta \\&\quad = \lim \limits _{z \rightarrow \zeta ^+} \int _{0}^{2\pi } \int _{0}^{\epsilon } \frac{\zeta - z}{[\sqrt{r^2 + (z-\zeta )^2}]^3} u(\varvec{\xi }) r\hbox {d}r\hbox {d} \theta \end{aligned}$$

If set $r = |z-\zeta |\tan \alpha$, then $\hbox {d}r = |z-\zeta |\sec ^2 \alpha \hbox {d}\alpha , \cos \alpha = \frac{|z-\zeta |}{\sqrt{(z-\zeta )^2 + \epsilon ^2}}$.

$$\begin{aligned} \hbox {above} = \lim \limits _{z \rightarrow \zeta ^+} \frac{|\cos \alpha |^3}{\cos ^3 \alpha } \frac{-(z-\zeta )}{|z-\zeta |} \int _{0}^{2\pi } \hbox {d}\theta \int _{0}^{\alpha (\epsilon )} \sin \alpha \hbox {d}\alpha \end{aligned}$$

where

$$\begin{aligned} \int _{0}^{2\pi } \hbox {d}\theta \int _{0}^{\alpha (\epsilon )} \sin \alpha \hbox {d}\alpha = 2\pi [-\cos \alpha (\epsilon ) + 1] \nonumber \\ = 2\pi \left[ -\frac{|z-\zeta |}{\sqrt{(z-\zeta )^2 + \epsilon ^2}}+1\right] \end{aligned}$$

(61)

Substitute into the original integral,

$$\begin{aligned}&\lim \limits _{z \rightarrow \zeta ^+} \frac{|\cos \alpha |^3}{\cos ^3 \alpha } \frac{-(z-\zeta )}{|z-\zeta |} \int _{0}^{2\pi } \hbox {d}\theta \int _{0}^{\alpha (\epsilon )} \sin \alpha \hbox {d}\alpha \nonumber \\&\quad = \lim \limits _{z \rightarrow \zeta ^+} 2\pi u(\varvec{x}) \frac{|\cos \alpha |^3}{\cos ^3 \alpha } \left[ \frac{(z-\zeta )}{\sqrt{(z-\zeta )^2+\epsilon ^2}} - \frac{z-\zeta }{|z-\zeta |} \right] \end{aligned}$$

(62)

The evaluation point approaches the surface from above. Thus, we have $\epsilon \rightarrow 0 \Rightarrow \alpha \rightarrow 0$, thus $\cos \alpha >0. z \rightarrow \zeta ^+ \Rightarrow |z-\zeta | = (z-\zeta )$. Also, $\epsilon$ needs to be one order larger than $|z-\zeta |$. Thus,

$$\begin{aligned}&\lim \limits _{z \rightarrow \zeta ^+} \frac{(z-\zeta )}{\sqrt{(z-\zeta )^2+\epsilon ^2}} = 0\end{aligned}$$

(63)

$$\begin{aligned}&\lim \limits _{z \rightarrow \zeta ^+} \frac{|\cos \alpha |^3}{\cos ^3 \alpha } \frac{-(z-\zeta )}{|z-\zeta |} \int _{0}^{2\pi } \hbox {d}\theta \int _{0}^{\alpha (\epsilon )} \sin \alpha \hbox {d}\alpha = -2\pi u(\varvec{x}) \end{aligned}$$

(64)

Back to the original integral:

$$\begin{aligned}&\lim \limits _{\varvec{x} \rightarrow S_b^{\pm }} \int _{S_b}u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} \nonumber \\&= \mp 2\pi u(\varvec{x}) + \int _{S_b} u(\varvec{\xi }) \frac{\partial G(\varvec{x};\varvec{\xi })}{\partial n_{\varvec{\xi }}}\hbox {d}S_{\varvec{\xi }} \end{aligned}$$

(65)

Herein, we have observed the singular terms’ contribution. However, the sign of the singular term is not consistent with the jump condition in Kleinman [21].

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Yujie Liu
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Jeffrey M. Falzarano
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1.Department of Ocean EngineeringTexas A&M UniversityCollege StationUSA

Irregular frequency removal methods: theory and applications in hydrodynamics

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