Unsteady free surface flow above a moving circular cylinder

Abstract

A problem on non-stationary free surface flow of infinitely deep ideal fluid generated by the motion of a submerged body is considered. The water wave problem is reduced to the integral–differential system of equations for the functions defining free surface shape, normal, and tangential velocity components on the free boundary. Small-time asymptotic solution is constructed for the case of circular cylinder that moves with constant acceleration from rest. The role of non-linearity is clarified by analysis of this approximate solution which describes the formation of added mass layers, splash jets, and finite amplitude surface waves.

Appendices

Appendix A: Derivation of integral boundary equation

Derivation of integral representation (6) uses the integral Cauchy formula for the complex velocity of the fluid $F(z,t) = U-\mathrm{i}V$, which is an analytical function of the complex variable $z = x + \mathrm{i}y$

$$\begin{aligned} 2\pi \mathrm{i}F(z,t) = \int \limits _{\varGamma (t)}\frac{F(\zeta ,t)\mathrm{d}\zeta }{\zeta - z} + \int \limits _{S(t)}\frac{F(\zeta ,t)\mathrm{d}\zeta }{\zeta - z}. \end{aligned}$$

(27)

Formula (27) contains the integral over the body surface S(t), which can be eliminated by appropriate transformation of the integrand. Namely, we use the Kelvin transformation for the Cauchy kernel

$$\begin{aligned} \frac{1}{\zeta - z} = \frac{1}{z - z_{c}}\overline{\frac{z_{c} - \zeta }{\zeta - z_{*}}}, \end{aligned}$$

where z is arbitrary in the flow domain, but $\zeta$ belongs to the circle S(t), and $z_{*} = z_{c} + r^2/({\overline{z}} - {\overline{z}}_{c})$ is the inverse image of z with respect to the circle S(t) with its center in the point $z_{c}$. The bar denotes complex conjugate. Using the boundary condition on the body surface, written in complex variables

$$\begin{aligned} {Re}\, \left\{ (F - \overline{z'_{c}(t)})(\zeta - z_{c}) \right\} = 0, \quad \zeta \in S(t), \end{aligned}$$

we obtain by the residue theorem

$$\begin{aligned} \int \limits _{S(t)}\frac{F(\zeta ,t)\mathrm{d}\zeta }{\zeta - z} = \frac{2\pi \mathrm{i}r^2z'_{c}(t)}{(z - z_{c}(t))^2} - \frac{1}{z - z_{c}}\overline{\int \limits _{S(t)}\frac{\zeta - z_{c}}{\zeta - z_{*}}F(\zeta ,t)}. \end{aligned}$$

(28)

Since $z_{*}$ is located inside the circle S(t), the integration contour in the right side of the formula (28) can be transformed into the free surface contour $\varGamma (t)$ without changing the value of the integral. As a result of this transformation we come to the following relation

$$\begin{aligned} \int \limits _{S(t)}\frac{F(\zeta ,t)\mathrm{d}\zeta }{\zeta - z} = \frac{\gamma }{z - z_{c}} + \frac{2\pi \mathrm{i}r^2z'_{c}}{(z - z_{c})^2} + \frac{r^2}{(z - z_{c})^2}\overline{\int \limits _{\varGamma }\frac{F(\zeta ,t)\mathrm{d}\zeta }{\zeta - z_{*}}}, \end{aligned}$$

(29)

where real parameter

$$\begin{aligned} \gamma = {Re} \int \limits _{\varGamma }F(\zeta ,t)\mathrm{d}\zeta \end{aligned}$$

denotes velocity circulation around the cylinder. Formulae (27) and (29) together give the representation (6).

Appendix B: Calculation of terms in small-time asymptotic solution

This part of the paper contains calculation of the coefficients (19) in asymptotic expansion (11). In order to evaluate the leading-order terms $\eta _2(x)$ and $\eta _4(x)$ in the asymptotics for the free surface elevation $\eta (x,t)$, we need to find explicitly the functions $\varphi _1$ and $\varphi _3$

$$\begin{aligned} \varphi _1 = v_d^{(1)}, \qquad \varphi _3 = v_d^{(3)} + Hu_3 + r^2 \Bigg (B_r^{(0)}u_3- A^{(2)}_r v_1\Bigg )- A^{(2)}v_1, \end{aligned}$$

(30)

which appear in the Neumann series (17). The function $\varphi _1$ can be found immediately from (14)

$$\begin{aligned} \varphi _1 = 4r^2(q'(x)\sin \theta - p'(x)\cos \theta ) . \end{aligned}$$

(31)

It is easy to see from here that $\varphi _1$ is a linear combination of the functions $p'(x)$ and $q'(x)$. In contrast, calculation of the function $\varphi _3$ leads to non-linear expressions depending on the higher-order derivatives of p(x) and q(x). Namely, substitution of identity $\eta _2 = v_1/2$ into formula (14) leads to

$$\begin{aligned} v_d^{(3)}(x) = 4r^2(p''(x)\cos 2\theta - q''(x)\sin 2\theta ) + 8r^4\Bigg ( \frac{q'(x)^2 - p'(x)^2}{2}\sin 2\theta -p'(x)q'(x)\cos 2\theta \Bigg )' + O(r^6). \end{aligned}$$

(32)

Similarly, inserting $v_1$ into recursive formula (12) gives

$$\begin{aligned} u_3(x) = - \frac{2\lambda r^2}{3}(q''(x)\cos \theta - p''(x)\sin \theta ) + \frac{8}{3} r^4\Bigg (q'(x)^2\sin \theta + p'(x)^2\cos ^2\theta - p'(x)q'(x)\sin 2\theta \Bigg )_x. \end{aligned}$$

Note that non-linear terms in (32) and (33) can be transformed to the linear combinations of p(x), q(x), and their derivatives due to the following identities

$$\begin{aligned} p'(x)q'(x) = -\frac{1}{12} \, p'''(x), \quad p'^2(x) = \frac{1}{4} p(x) + \frac{1}{4} q'(x) + \frac{1}{12} q'''(x), \quad q'^2(x) = \frac{1}{4} p(x) + \frac{1}{4} q'(x) - \frac{1}{12} q'''(x), \ldots \end{aligned}$$

Acting by this manner we can rewrite the terms $v_d^{(3)}$ in the form (i) in (18). Consequently, the term $u_3$ can be presented as follows:

$$\begin{aligned} u_3(x) = \frac{2r^4}{9} \Bigg ( q''''(x)\cos 2\theta + p''''(x)\sin 2\theta \Bigg ) + \frac{2r^4}{3}(p'(x) + q''(x))- \frac{2\lambda r^2}{3} \Bigg ( q''(x)\sin \theta - p''(x)\cos \theta \Bigg ) + O(r^6). \end{aligned}$$

(33)

Returning to formula (30) we can calculate now all integral terms with the operators H, $A_{r}^{(0)}$, $B_{r}^{(0)}$, $A_r^{(2)}$, and $A^{(2)}$. The simplest term to calculate is $Hu_3$ since we have $Hp(x) = q(x)$, $Hq(x) = -p(x)$, and the Hilbert transform H commutes with differentiation in x. Acting by this way, we come to the formula (ii) in (18).

Further, it should be noted that the operators $A_{r}^{(0)}$, $B_{r}^{(0)}$, and $A_r^{(2)}$ are included in (17) and (30) only with a multiplier $r^{2}$, and all these operators act on the functions $\varphi _3$, $u_3$, $v_1$ which are of the order $r^2$. Therefore, it is sufficient to use here truncated versions of these operators $A_0^{(0)} = A_r^{(0)}|_{r=0}$, $B_0^{(0)} = B_r^{(0)}|_{r=0}$, and $A_0^{(2)} = A_r^{(2)}|_{r=0}$ which have the form

$$\begin{aligned} A_{0}^{(0)} v(x)= & {} \frac{1}{\pi } \int \limits _{-\infty }^{+\infty }\frac{(q'(x) - sp'(x))v(s)\mathrm{d}s}{1+s^2} = \frac{1}{\pi }q'(x)\int \limits _{-\infty }^{+\infty }p(s)v(s)\mathrm{d}s - \frac{1}{\pi }p'(x)\int \limits _{-\infty }^{+\infty }q(s)v(s)\mathrm{d}s,\\ B_0^{(0)}u(x)= & {} \frac{1}{\pi }\int \limits _{-\infty }^{+\infty }\frac{(p'(x) + sq'(x))u(s)\mathrm{d}s}{1 + s^2} = \frac{1}{\pi }p'(x)\int \limits _{-\infty }^{+\infty }p(s)u(s)\mathrm{d}s + \frac{1}{\pi }q'(x)\int \limits _{-\infty }^{+\infty }q(s)u(s)\mathrm{d}s,\\ A_0^{(2)}v(x)= & {} \frac{1}{\pi }\Bigg ( p'(x)\sin \theta - q'(x)\cos \theta \Bigg ) \int \limits _{-\infty }^{+\infty }p'(s)v(s)\mathrm{d}s + \frac{1}{\pi }\Bigg ( p'(x)\cos \theta + q'(x)\sin \theta \Bigg ) \int \limits _{-\infty }^{+\infty }q'(s)v(s)\mathrm{d}s \\&- \frac{1}{\pi }\Bigg ( q''(x)\cos \theta + p''(x)\sin \theta \Bigg ) \int \limits _{-\infty }^{+\infty }p(s)v(s)\mathrm{d}s + \frac{1}{\pi }\Bigg ( p''(x)\cos \theta - q''(x)\sin \theta \Bigg ) \int \limits _{-\infty }^{+\infty }q(s)v(s)\mathrm{d}s . \end{aligned}$$

Substitution of $v = v_1$ from (31) and $u = u_3$ from (33) and calculation of the integrals give the formulae (iii) and (v) in (18). The last term to be calculated in (30) is

$$\begin{aligned} A^{(2)}v(x)\mathrm{d}x = \mathrm{v.p.} \frac{1}{\pi } \int \limits _{-\infty }^{+\infty } \frac{\eta _2(x) - \eta _2(s)}{(x-s)^2}v(s)\mathrm{d}s - \eta '_2(x)\,\mathrm{v.p.}\frac{1}{\pi } \int \limits _{-\infty }^{+\infty } \frac{v(s)\mathrm{d}s}{x-s}. \end{aligned}$$

(34)

This expression is non-linear with respect to $\eta _2$ and $v_1$, but the first term in (34) can be presented as the commutator of the Hilbert transform H with differentiation in x as follows

$$\begin{aligned} \frac{1}{\pi } \int \limits _{-\infty }^{+\infty } \frac{\eta _2(x) - \eta _2(s)}{(x-s)^2}v_1(s)\mathrm{d}s = H(\eta _2v_1)_x - \eta _2Hv_{1x}. \end{aligned}$$

The result of calculation gives formula (iv) in (18). Finally, the combination of the above calculated terms (i)–(v) from (18) results in the formula (vi) in (18).