scholarly journals Some explicit solutions of the three-dimensional Euler equations with a free surface

2021 ◽  
Author(s):  
Calin I. Martin
Keyword(s):  
Functional Equation ◽  
Free Surface ◽  
Euler Equations ◽  
Vertical Structure ◽  
Three Dimensional ◽  
Finite Depth ◽  
Radial Solutions ◽  
Pressure Function ◽  
Constant Vorticity ◽  
Vorticity Vector

AbstractWe present a family of radial solutions (given in Eulerian coordinates) to the three-dimensional Euler equations in a fluid domain with a free surface and having finite depth. The solutions that we find exhibit vertical structure and a non-constant vorticity vector. Moreover, the flows described by these solutions display a density that depends on the depth. While the velocity field and the pressure function corresponding to these solutions are given explicitly through (relatively) simple formulas, the free surface defining function is specified (in general) implicitly by a functional equation which is analysed by functional analytic methods. The elaborate nature of the latter functional equation becomes simpler when the density function has a particular form leading to an explicit formula of the free surface. We subject these solutions to a stability analysis by means of a Wentzel–Kramers–Brillouin (WKB) ansatz.

On constant vorticity water flows in the -plane approximation

2019 ◽  
Vol 865 ◽  
pp. 762-774 ◽  
Author(s):  
Calin Iulian Martin
Keyword(s):  
Free Surface ◽  
Velocity Field ◽  
Three Dimensional ◽  
General Setting ◽  
Stationary Flow ◽  
Water Flows ◽  
Constant Vorticity ◽  
Vorticity Vector

We consider here three-dimensional water flows in the $\unicode[STIX]{x1D6FD}$-plane approximation. In a quite general setting we show that the only flow exhibiting a constant vorticity vector is the stationary flow with vanishing velocity field and flat free surface.

The translation of two bodies under the free surface of a heavy fluid

1950 ◽  
Vol 46 (3) ◽  
pp. 453-468 ◽  
Author(s):  
A. Coombs
Keyword(s):  
Free Surface ◽  
Large Range ◽  
Three Dimensional ◽  
Finite Depth ◽  
Arbitrary Cross Section ◽  
Wave Resistance ◽  
Vertical Force ◽  
First Approximation ◽  
Special Case ◽  
Two Bodies

1. Many investigations have been made to determine the wave resistance acting on a body moving horizontally and uniformly in a heavy, perfect fluid. Lamb obtained a first approximation for the wave resistance on a long circular cylinder, and this was later confirmed to be quite sufficient over a large range. In 1926 and 1928, Havelock (4, 5) obtained a second approximation for the wave resistance and a first approximation for the vertical force or lift. Later, in 1936(6), he gave a complete analytical solution to this problem, in which the forces were expressed in the form of infinite series in powers of the ratio of the radius of the cylinder to the depth of the centre below the free surface of the fluid. General expressions for the wave resistance and lift of a cylinder of arbitrary cross-section were found by Kotchin (7) using integral equations, and the special case of a flat plate was evaluated. He continued with a discussion of the motion of a three-dimensional body. More recently, Haskind (3) has examined the same problem when the stream has a finite depth.

Bow and stern flows with constant vorticity

1999 ◽  
Vol 399 ◽  
pp. 277-300 ◽  
Author(s):  
SCOTT W. McCUE ◽  
LAWRENCE K. FORBES
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Radiation Condition ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Boundary Integral ◽  
The Body ◽  
Boundary Integral Method ◽  
Infinite Body ◽  
Constant Vorticity

Free surface flows of a rotational fluid past a two-dimensional semi-infinite body are considered. The fluid is assumed to be inviscid, incompressible, and of finite depth. A boundary integral method is used to solve the problem for the case where the free surface meets the body at a stagnation point. Supercritical solutions which satisfy the radiation condition are found for various values of the Froude number and the dimensionless vorticity. Subcritical solutions are also found; however these solutions violate the radiation condition and are characterized by a train of waves upstream. It is shown numerically that the amplitude of these waves increases as each of the Froude number, vorticity and height of the body above the bottom increases.

scholarly journals Smoothly attaching bow flows with constant vorticity

2001 ◽  
Vol 42 (3) ◽  
pp. 354-371
Author(s):  
S. W. McCue ◽  
L. K. Forbes
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Finite Depth ◽  
Surface Flow ◽  
Boundary Integral ◽  
The Body ◽  
Boundary Integral Method ◽  
Front Face ◽  
Infinite Body ◽  
Constant Vorticity

AbstractThe free surface flow of a finite depth fluid past a semi-infinite body is considered. The fluid is assumed to have constant vorticity throughout and the free surface is assumed to attach smoothly to the front face of the body. Numerical solutions are found using a boundary integral method in the physical plane and it is shown that solutions exist for all supercritical Froude numbers. The related problem of the cusp-like flow due to a submerged sink in a comer is also considered. Vorticity is included in the flow and it is shown that the behaviour of the solutions is qualitatively the same as that found in the problem described above.

Internal Wave Propagation from Pulsating Sources in a Two-Layer Fluid of Finite Depth

2012 ◽  
Vol 201-202 ◽  
pp. 503-507
Author(s):  
David O. Manyanga ◽  
Wen Yang Duan
Keyword(s):  
Free Surface ◽  
Internal Wave ◽  
Internal Waves ◽  
Wave Length ◽  
Three Dimensional ◽  
Vertical Direction ◽  
Finite Depth ◽  
Green Functions ◽  
Design And Optimization ◽  
Underwater Structures

The influence of internal waves is very important in the Engineering Analysis, Design and Optimization. To study the internal wave properties, we model a two-layer fluid and generalize to multiple layers. In a two-layer fluid with the upper layer having a free surface, there exist two modes of waves propagating due to the free surface and the interface. This is due to the density difference in the vertical direction of the water, due to the variation in salinity and temperature where waves from underwater structures are of importance. In this case the fluid is assumed to be non viscous, incompressible and the flow is non rotational. On the other hand, there is need for appropriate Green functions to analyze these properties. In this paper, we use the three dimensional Green functions for a stationary oscillating source to study the internal wave characteristics. Some of the behavior studied in this work includes effects of internal waves on the surface and internal wave amplitudes. Further, an investigation of the influence of internal waves on the wave length, frequency and period is made.

THREE-DIMENSIONAL WAVE-FREE POTENTIALS IN THE THEORY OF WATER WAVES

2013 ◽  
Vol 55 (2) ◽  
pp. 175-195 ◽  
Author(s):  
HARPREET DHILLON ◽  
B. N. MANDAL
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Surface Condition ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Finite Depth ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Thin Elastic Plate ◽  
Linearized Theory

AbstractProblems of wave interaction with a body with arbitrary shape floating or submerged in water are of immense importance in the literature on the linearized theory of water waves. Wave-free potentials are used to construct solutions to these problems involving bodies with circular geometry, such as a submerged or half-immersed long horizontal circular cylinder (in two dimensions) or sphere (in three dimensions). These are singular solutions of Laplace’s equation satisfying the free surface condition and decaying rapidly away from the point of singularity. Wave-free potentials in two and three dimensions for infinitely deep water as well as water of uniform finite depth with a free surface are known in the literature. The method of constructing wave-free potentials in three dimensions is presented here in a systematic manner, neglecting or taking into account the effect of surface tension at the free surface or for water with an ice cover modelled as a thin elastic plate floating on the water. The forms of the wave motion at the upper surface (free surface or ice-covered surface) related to these wave-free potentials are depicted graphically in a number of figures for all the cases considered.

Prediction of Ship Motions Via a Three-Dimensional Time-Domain Method Following a Quad-Tree Adaptive Mesh Technique

2020 ◽  
Vol 27 (1) ◽  
pp. 29-38
Author(s):  
Teng Zhang ◽  
Junsheng Ren ◽  
Lu Liu
Keyword(s):  
Free Surface ◽  
Incident Wave ◽  
Time Domain ◽  
Three Dimensional ◽  
Adaptive Mesh ◽  
Wave Profile ◽  
Time Domain Method ◽  
Ship Motions ◽  
Quad Tree ◽  
Mesh Technique

AbstractA three-dimensional (3D) time-domain method is developed to predict ship motions in waves. To evaluate the Froude-Krylov (F-K) forces and hydrostatic forces under the instantaneous incident wave profile, an adaptive mesh technique based on a quad-tree subdivision is adopted to generate instantaneous wet meshes for ship. For quadrilateral panels under both mean free surface and instantaneous incident wave profiles, Froude-Krylov forces and hydrostatic forces are computed by analytical exact pressure integration expressions, allowing for considerably coarse meshes without loss of accuracy. And for quadrilateral panels interacting with the wave profile, F-K and hydrostatic forces are evaluated following a quad-tree subdivision. The transient free surface Green function (TFSGF) is essential to evaluate radiation and diffraction forces based on linear theory. To reduce the numerical error due to unclear partition, a precise integration method is applied to solve the TFSGF in the partition computation time domain. Computations are carried out for a Wigley hull form and S175 container ship, and the results show good agreement with both experimental results and published results.

Analysis of three-dimensional aerospace configurations using the Euler equations

1989 ◽  
Author(s):  
N. KROLL ◽  
C. ROSSOW ◽  
S. SCHERR ◽  
J. SCHOENE ◽  
G. WICHMANN
Keyword(s):  
Euler Equations ◽  
Three Dimensional
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