## 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.

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## References

Ogilvie F (1963) First- and second-order forces on a cylinder submerged under a free surface. J Fluid Mech 16:451–472

Tuck EO (1965) The effect of non-linearity at the free surface on flow past a submerged cylinder. J Fluid Mech 22:401–414

Tyvand PA, Miloh T (1995) Free-surface flow due to impulsive motion of a submerged circular cylinder. J Fluid Mech 286:67–101

Tyvand PA, Miloh T (1995) Free-surface flow generated by a small submerged circular cylinders starting from the rest. J Fluid Mech 286:103–116

Miloh TA (2001) Note on impulsive sphere motion beneath a free-surface. J Eng Math 41:1–11

Pyatkina EV (2003) Small-time expansion of wave motion generated by a submerged sphere. J Appl Mech Tech Phys 44(1):32–43

Tyvand PA, Miloh T (2005) Impulsive free-surface flows and small-time expansions. In: Misra JC (ed) Modern applied mathematics. Narosa Pub. House, New Delhi, pp 320–371

Haussling HJ, Coleman RM (1977) Finite-difference computations using boundary fitted coordinates for free-surface potential flows generated by submerged bodies. In: Proceedings of second international conference numerical ship hydrodynamics. University of California at Berkeley, 19–21 Sept, pp 221–233

Haussling HJ, Coleman RM (1979) Nonlinear water waves generated by an accelerated circular cylinder. J Fluid Mech 461:343–364

Terentiev AG, Afanasiev KE, Afanasieva MM (1988) Simulation of unsteady free surface flow problem by the direct boundary element method. In: Proceedings of the IUTAM symposium advanced boundary element methods, San Antonio, pp 427–434

Telste JT (1987) Inviscid flow about a cylinder rising to a free surface. J Fluid Mech 182:149–168

Moyo S, Greenhow M (2000) Free motion of a cylinder moving below and through a free-surface. Appl Ocean Res 22:31–34

Gorlov SI (1999) Unsteady nonlinear problem of the horizontal motion of a contour under the interface between two liquids. J Appl Mech Tech Phys 40(3):393–398

Gorlov SI (2000) Nonlinear problem of a circular cylinder rising vertically to an interface between liquid media. J Appl Mech Tech Phys 41(2):280–285

Zhu X, Faltinsen OM, Hu C (2007) Water entry and exit of a horizontal circular cylinder. JOMAE Trans. ASME 129(4):253–264

Greenhow M, Moyo S (1997) Water entry and exit of horizontal circular cylinder. Philos Trans R Soc Lond Ser A 355:551–563

Ovsyannikov LV, Makarenko NI, Nalimov VI, Liapidevskii VY, Plotnikov PI, Sturova IV, Bukreev VI, Vladimirov VA (1985) Nonlinear problems of the theory of surface and internal waves. Nauka, Novosibirsk (in Russian)

Makarenko NI (2003) Nonlinear interaction of submerged cylinder with free surface. JOMAE Trans ASME 125(1):72–75

Makarenko NI (2004) Nonlinear water waves in the presence of submerged elliptic cylinder. In: Proceedings of the 23rd international conference on offshore mechanics and arctic engineering, Vancouver, Canada, 20–25 June, OMAE2004-51413

Makarenko NI, Kostikov VK (2013) Unsteady motion of an elliptical cylinder under a free surface. J Appl Mech Tech Phys 53(3):367–376

Makarenko NI, Kostikov VK (2014) Non-linear water waves generated by impulsive motion of submerged obstacles. Nat Hazards Earth Syst Sci 14:751–756

Norkin MV (2012) Formation of a cavity in the initial stage of motion of a circular cylinder in a fluid with a constant acceleration. J Appl Mech Tech Phys 53(4):532–539

Kedrinsky VK (2005) Hydrodynamics of explosion: experiments and models. Springer, Berlin

Clamond D, Grue JA (2001) Fast method for fully nonlinear water wave computations. J Fluid Mech 447:337–355

Havelock TH (1949a) The wave resistance of a cylinder started from rest. Q J Mech Appl Math 2:325–334

Havelock TH (1949b) The resistance of a submerged cylinder in accelerated motion. Q J Mech Appl Math 2:419–427

Sretensky LN (1937) A theoretical study of wave resistance. Joukovsky Cent Inst Rep 319:1–55

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This work was supported by RFBR Grant No. 15-01-03942.

## 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 \(z = x + \mathrm{i}y\)

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

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

we obtain by the residue theorem

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

where real parameter

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 \(\varphi _3\)

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

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

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

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

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:

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

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

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

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).

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Kostikov, V.K., Makarenko, N.I. Unsteady free surface flow above a moving circular cylinder.
*J Eng Math* **112**, 1–16 (2018). https://doi.org/10.1007/s10665-018-9962-x

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DOI: https://doi.org/10.1007/s10665-018-9962-x