On the initial values of kinematic and dynamic free-surface boundary conditions for water wave problems
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On the initial values of kinematic and dynamic free-surface boundary conditions for water wave problems * Dong-qiang Lu Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, China Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai 200072, China (Received May 28, 2020, Revised July 22, 2020, Accepted July 24, 2020, Published online September 3, 2020) ©China Ship Scientific Research Center 2020 Abstract: The kinematic and dynamic boundary conditions on the free surface of a fluid should be posed for water wave problems. In the framework of potential theory for an inviscid and incompressible fluid with an irrotational motion, the combined boundary condition, which involves the velocity potential only, is often used by eliminating the elevation terms mathematically. Such a combination is correct for the solutions in the frequency domain, and is not feasible for an initial-boundary-value problem in the time domain since it leads to a totally different physical formulation. The correct initial conditions for pure gravity waves and hydroelastic waves are presented. Key words: Initial values, boundary conditions, water wave, hydroelasticity
Water wave problem, an interesting topic in fluid mechanics, is traditionally studied in the framework of the irrotational motion in an inviscid, incompressible and homogenous fluid. For an incompressible fluid, the conservation law of mass is represented by u = 0 , where u is the velocity vector. Thus u can be expressed in terms of a potential function ( x t ) as u = under the assumption of an irrotational motion, where x = (x y z ) in the Cartesian coordinates and t the time. Furthermore, for an inviscid fluid, the conservation law of momentum yields the Euler equation without the viscosity term as follows
P 1 2 + + e + g = B(t ) t 2
(1)
where Pe ( x t ) the total pressure, the density, g the gravitational acceleration, ( z t ) the surface elevation, B (t ) the Bernoulli constant, and * Project supported by the National Natural Science Foundation of China (Grant No. 11872239). Biography: Dong-qiang Lu (1972-), Male, Ph. D., Professor Corresponding author: Dong-qiang Lu, E-mail: [email protected]
z = (x y ) . B (t ) can usually be set as zero by re-defining the potential without affecting the velocity vector. By introducing t
= B ( )d
(2)
0
we have
B(t ) , = t t = = u ,
t =0
= t =0 .
Therefore, ( x t ) is used as the velocity potential hereinafter. For a moving surface, Eq. (1) reformulated in terms of is employed as the dynamic boundary condition on z = . The kinematic boundary condition on z = reads + + = t x x y y z
(3)
Equation (3) indicates that the fluid particles on the surface move only tangentially. Another approach for the boundary conditions is a combination of Eqs. (1) and (3) by eliminating
mathematically all the elevation terms with . One can take the total der
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