scholarly journals Free-surface flows emerging from beneath a semi-infinite plate with constant vorticity

2002 ◽  
Vol 461 ◽  
pp. 387-407 ◽  
Author(s):  
SCOTT W. McCUE ◽  
LAWRENCE K. FORBES
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Infinite Plate ◽  
Free Surface Flows ◽  
Surface Flow ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Constant Vorticity ◽  
Uniform Shear ◽  
Uniform Shear Flow

The free-surface flow past a semi-infinite horizontal plate in a finite-depth fluid is considered. It is assumed that the fluid is incompressible and inviscid and that the flow approaches a uniform shear flow downstream. Exact relations are derived using conservation of mass and momentum for the case where the downstream free surface is flat. The complete nonlinear problem is solved numerically using a boundary-integral method and these waveless solutions are shown to exist only when the height of the plate above the bottom is greater than the height of the uniform shear flow. Interesting results are found for various values of the constant vorticity. Solutions with downstream surface waves are also considered, and nonlinear results of this type are compared with linear results found previously. These solutions can be used to model the flow near the stern of a (two-dimensional) ship.

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.

Properties of Finite-Difference Operators for the Steady-Wave Problem

1993 ◽  
Vol 37 (01) ◽  
pp. 1-7
Author(s):  
John S. Letcher
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Radiation Condition ◽  
Free Surface Flows ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Difference Operators ◽  
Surface Boundary ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

A feature of most implementations of Dawson's boundary-integral method for steady free-surface flows is the use of upstream finite-difference operators for the streamwise derivative occurring in the linearized free-surface boundary condition. An algebraic analysis of a family of candidate operators reveals their essential damping and dispersion error characteristics, which correlate well with their observed performance in two-dimensional example flows. Some new operators are found which perform somewhat better than Dawson's, but the general outlook for accurate results using difference operators is nevertheless bleak. It is shown that the calculation necessarily diverges as panel size is reduced, and a breakdown at higher speeds is also inevitable. More promise appears to lie in satisfying the radiation condition by several alternative ways, which are briefly discussed.

The effect of surfactant on bursting gas bubbles

1995 ◽  
Vol 302 ◽  
pp. 231-257 ◽  
Author(s):  
Jeremy M. Boulton-Stone
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Stress Condition ◽  
Numerical Technique ◽  
Gas Bubbles ◽  
Free Surface Flows ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Surface Shear ◽  
Surface Compression

A numerical technique, based on the boundary integral method, is developed to allow the modelling of unsteady free-surface flows at large Reynolds numbers in cases where the surface is contaminated by some surface-active compound. This requires the method to take account of the tangential stress condition at the interface and is achieved through a boundary-layer analysis. The constitutive relation that forms the surface stress condition is assumed to be of the Boussinesq type and allows the incorporation of surface shear and dilatational viscous forces as well as Marangoni effects due to gradients in surface tension. Sorption kinetics can be included in the model, allowing calculations for both soluble and insolube surfactants. Application of the numerical model to the problem of bursting gas bubbles at a free surface shows the greatest effect to be due to surface dilatational viscosity which drastically reduces the amount of surface compression and can slow and even prevent the information of a liquid jet. Surface tension gradients give dilatational elasticity to the surface and thus also significantly prevent surface compression. Surface shear viscosity has a smaller effect on the interface motion but results in initially increased surface concentrations due to the sweeping up of surface particles ahead of the inward-moving surface wave.

Development of the boundary layer at a free surface from a uniform shear flow

1966 ◽  
Vol 25 (1) ◽  
pp. 87-95 ◽  
Author(s):  
Simon L. Goren
Keyword(s):  
Boundary Layer ◽  
Mass Transfer ◽  
Free Surface ◽  
Shear Flow ◽  
Surface Velocity ◽  
Cube Root ◽  
Two Dimensional ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
Capillary Jets

The development of the boundary layer accompanying the formation of a free surface at y′ = 0, from the two-dimensional uniform shear flow u′ = ωyω, is discussed. The analysis shows that the surface velocity and surface position vary as the cube root of the distance downstream, while the mass-transfer coefficient varies inversely as the cube root of this distance. It is shown how these may be applied to the formation of capillary jets.

Steep solitary waves in water of finite depth with constant vorticity

1994 ◽  
Vol 274 ◽  
pp. 339-348 ◽  
Author(s):  
J.-M. Vanden-Broeck
Keyword(s):  
Integral Equation ◽  
Solitary Waves ◽  
Numerical Scheme ◽  
Integral Equation Method ◽  
Finite Depth ◽  
Boundary Integral ◽  
Constant Vorticity ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
Shear Current

Solitary waves with constant vorticity in water of finite depth are calculated numerically by a boundary integral equation method. Previous calculations are confirmed and extended. It is shown that there are branches of solutions which bifurcate from a uniform shear current. Some of these branches are characterized by a limiting configuration with a 120° angle at the crest of the wave. Other branches extend for arbitrary large values of the amplitude of the wave. The corresponding solutions ultimately approach closed regions of constant vorticity in contact with the bottom of the channel. A numerical scheme is presented to calculate directly these closed regions of constant vorticity. In addition, it is shown that there are branches of solutions which do not bifurcate from a uniform shear flow.

scholarly journals Three-dimensional free-surface flow over arbitrary bottom topography

2018 ◽  
Vol 846 ◽  
pp. 166-189 ◽  
Author(s):  
Nicholas R. Buttle ◽  
Ravindra Pethiyagoda ◽  
Timothy J. Moroney ◽  
Scott W. McCue
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Bottom Topography ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Three Dimensions ◽  
Computational Time ◽  
Algebraic Equations

We consider steady nonlinear free surface flow past an arbitrary bottom topography in three dimensions, concentrating on the shape of the wave pattern that forms on the surface of the fluid. Assuming ideal fluid flow, the problem is formulated using a boundary integral method and discretised to produce a nonlinear system of algebraic equations. The Jacobian of this system is dense due to integrals being evaluated over the entire free surface. To overcome the computational difficulty and large memory requirements, a Jacobian-free Newton–Krylov (JFNK) method is utilised. Using a block-banded approximation of the Jacobian from the linearised system as a preconditioner for the JFNK scheme, we find significant reductions in computational time and memory required for generating numerical solutions. These improvements also allow for a larger number of mesh points over the free surface and the bottom topography. We present a range of numerical solutions for both subcritical and supercritical regimes, and for a variety of bottom configurations. We discuss nonlinear features of the wave patterns as well as their relationship to ship wakes.

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