scholarly journals Bubbles rising in an inclined two-dimensional tube and jets falling along a wall

1998 ◽  
Vol 39 (3) ◽  
pp. 332-349 ◽  
Author(s):  
J. Lee ◽  
J.-M. Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Froude Number ◽  
Numerical Solutions ◽  
Surface Condition ◽  
Two Dimensional ◽  
Left Wall ◽  
Free Surface Condition ◽  
Inclined Tube ◽  
The Right

AbstractThe motion of a two-dimensional bubble rising at a constant velocity U in an inclined tube of width H is considered. The bubble extends downwards without limit, and is bounded on the right by a wall of the tube, and on the left by a free surface. The same flow configuration describes also a jet emerging from a nozzle and falling down along an inclined wall. The acceleration of gravity g and the surface tension T are included in the free surface condition. The problem is characterized by the Froude number the angle β between the left wall and the horizontal, and the angle γ between the free surface and the right wall at the separation point. Numerical solutions are obtained via series truncation for all values of 0 < β < π. The results extend previous calculations of Vanden-Broeck [12–14] for β = π/2 and of Couët and Strumolo [3] for 0 < β < π/2. It is found that the behavior of the solutions depends on whether 0 < β 2π/3 or 2π/3 ≤ β < π. When T = 0, it is shown that there is a critical value F of Froude number for each 0 < β 2π/3 such that solutions with γ = 0, π/3 and π - β occur for F > Fc F = Fc and F < Fc respectively, and that all solutions are characterized by γ = 0 for 2π/3 ≤ β < π. When a small amount of surface tension T is included in the free surface condition, it is found that for each 0 < β < π there exists an infinite discrete set of values of F for which γ = π - β. A particular value F* of the Froude number for which T = 0 and γ = π - β is selected by taking the limit as T approaches zero. The numerical values of F* and the corresponding free surface profiles are found to be in good agreement with experimental data for bubbles rising in an inclined tube when 0 < β < π/2.

Waves and wave resistance of thin bodies moving at low speed: the free-surface nonlinear effect

1975 ◽  
Vol 69 (2) ◽  
pp. 405-416 ◽  
Author(s):  
G. Dagan
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Surface Condition ◽  
Perturbation Expansion ◽  
Wave Resistance ◽  
The Body ◽  
Thin Body ◽  
Two Dimensional ◽  
First Order ◽  
Flow Past

The linearized theory of free-surface gravity flow past submerged or floating bodies is based on a perturbation expansion of the velocity potential in the slenderness parameter e with the Froude number F kept fixed. It is shown that, although the free-wave amplitude and the associated wave resistance tend to zero as F → 0, the linearized solution is not uniform in this limit: the ratio between the second- and first-order terms becomes unbounded as F → 0 with ε fixed. This non-uniformity (called ‘the second Froude number paradox’ in previous work) is related to the nonlinearity of the free-surface condition. Criteria for uniformity of the thin-body expansion, combining ε and F, are derived for two-dimensional flows. These criteria depend on the shape of the leading (and trailing) edge: as the shape becomes finer the linearized solution becomes valid for smaller F.Uniform first-order approximations for two-dimensional flow past submerged bodies are derived with the aid of the method of co-ordinate straining. The straining leads to an apparent displacement of the most singular points of the body contour (the leading and trailing edges for a smooth shape) and, therefore, to an apparent change in the effective Froude number.

An Investigation on Two-Dimensional Nonlinear Sloshing in Rectangular Tank

2015 ◽  
Author(s):  
Dongya Zhao ◽  
Zhiqiang Hu ◽  
Gang Chen
Keyword(s):  
Numerical Simulation ◽  
Free Surface ◽  
Liquid Surface ◽  
Surface Condition ◽  
Excitation Frequency ◽  
Surface Elevation ◽  
Two Dimensional ◽  
Free Surface Condition ◽  
Rectangular Tank ◽  
Oscillation Movement

Two-dimensional liquid sloshing in rectangular tank of FLNG system is investigated both numerically and experimentally. In numerical simulation, a time-domain scheme has been developed based on potential flow theory in boundary element method. Tank movement is defined by wall boundary condition to produce a reciprocating oscillation. Nonlinear free surface condition is adopted to capture free surface elevation. Energy dissipation caused by viscous effects is considered by applying artificial damping term to the dynamic free surface condition, which is also vital to achieve a steady-state solution. For comparison, experiments of a rectangular tank filled with water subjected to specified oscillation are carried out. As coupling effects between sloshing and tank motion is not included in this research, the testing apparatus is required to produce consistent oscillation movement and not affected by the change of filling condition and sloshing load. Liquid surface elevations in several typical places of the tank were measured. Sloshing related parameters including oscillation amplitude, frequency and filling level are analyzed systematically. It’s found that numerical simulation results have good agreement with phenomenon observed under small amplitude excitation, and this nonlinear analysis method is proved to be effective in capturing liquid surface elevation. It is found that sloshing in tank is sensitive to filling level as well as excitation frequency, especially in the crucial combination cases of them. For given filling level, sloshing tends to be violent near corresponding natural frequencies, and viscous damping has limited contribution to sloshing amplitude when resonance occurs. This fundamental investigation also paves path for the study of more complicated sloshing problems.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

A note on the instabilities of a horizontal shear flow with a free surface

2000 ◽  
Vol 406 ◽  
pp. 337-346 ◽  
Author(s):  
L. ENGEVIK
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Shear Flow ◽  
Velocity Profile ◽  
Froude Number ◽  
The Other ◽  
Algebraic Equations ◽  
Horizontal Shear ◽  
Analytical Treatment ◽  
Unstable Region

The instabilities of a free surface shear flow are considered, with special emphasis on the shear flow with the velocity profile U* = U*0sech2 (by*). This velocity profile, which is found to model very well the shear flow in the wake of a hydrofoil, has been focused on in previous studies, for instance by Dimas & Triantyfallou who made a purely numerical investigation of this problem, and by Longuet-Higgins who simplified the problem by approximating the velocity profile with a piecewise-linear profile to make it amenable to an analytical treatment. However, none has so far recognized that this problem in fact has a very simple solution which can be found analytically; that is, the stability boundaries, i.e. the boundaries between the stable and the unstable regions in the wavenumber (k)–Froude number (F)-plane, are given by simple algebraic equations in k and F. This applies also when surface tension is included. With no surface tension present there exist two distinct regimes of unstable waves for all values of the Froude number F > 0. If 0 < F [Lt ] 1, then one of the regimes is given by 0 < k < (1 − F2/6), the other by F−2 < k < 9F−2, which is a very extended region on the k-axis. When F [Gt ] 1 there is one small unstable region close to k = 0, i.e. 0 < k < 9/(4F2), the other unstable region being (3/2)1/2F−1 < k < 2 + 27/(8F2). When surface tension is included there may be one, two or even three distinct regimes of unstable modes depending on the value of the Froude number. For small F there is only one instability region, for intermediate values of F there are two regimes of unstable modes, and when F is large enough there are three distinct instability regions.

Influence of the Waterline Integral on the Solution of the Frequency-Domain Forward-Speed Radiation-Diffraction Problem of a Ship Advancing in Waves

2014 ◽  
Author(s):  
D. C. Hong ◽  
S. Y. Hong ◽  
G. J. Lee ◽  
M. S. Shin
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Green Function ◽  
Frequency Domain ◽  
Surface Condition ◽  
Numerical Results ◽  
Forward Speed ◽  
Line Integral ◽  
Free Surface Condition ◽  
Surface Green Function

The radiation-diffraction potential of a ship advancing in waves is studied using the three-dimensional frequency-domain forward-speed free-surface Green function (Brard 1948) and the forward-speed Green integral equation (Hong 2000). Numerical solutions are obtained by making use of a second-order inner collocation boundary element method which makes it possible to take account of the line integral along the waterline in a rigorous manner (Hong et al. 2008). The present forward-speed Green integral equation includes not only the usual free surface condition for the potential but also the adjoint free surface condition for the forward-speed free-surface Green function as indicated by Brard (1972). Comparison of the present numerical results of the heave-heave wave damping coefficients and the experimental results for the Wigley ship models I, II and III (Journee 1992) has been presented. These coefficients are compared with those calculated without taking into account of the line integral along the waterline in order to show the forward speed effect represented by the waterline integral when it is properly included in the free-surface Green integral equation. Comparison of the present numerical results and the equivalent time-domain results (Hong et al. 2013) has also been presented.

On the choice of CFD codes in the design process of planing sailing yachts

2009 ◽  
Author(s):  
Jérémie Raymond ◽  
Jean-Marie Finot ◽  
Jean-Michel Kobus ◽  
Gérard Delhommeau ◽  
Patrick Queutey ◽  
...  
Keyword(s):  
Free Surface ◽  
Design Process ◽  
Finite Differences ◽  
Potential Flow ◽  
Surface Condition ◽  
Cpu Time ◽  
Free Surface Condition ◽  
Finite Differences Method ◽  
Non Linear ◽  
Ranse Solver

The discussion is based on results gathered during the first two years of a 3 years research program for the benefits of Groupe Finot-Conq, Naval Architects. The introduction presents the objectives of the program: Setting up a practical method using numerical and experimental available tools to design fast planing sailing yachts. The aim of this paper is to compare advantages and disadvantages of four different kinds of CFD codes which are linear and non-linear potential flow approach, RANSE solver using finite differences method and RANSE solver using volume of fluid method. The Fluid Mechanics Laboratory of the Ecole Centrale de Nantes (France) has developed those three approaches so those homemade codes will be used for this study. The first one is REVA, a potential flow code with a linearised free surface condition. ICARE is a RANSE solver using finite differences method with a non linear free surface condition. It is extensively used for industrial projects as for sailing yachts projects (ACC for example). ISIS-CFD is a RANSE solver using finite volume method to build the spatial discretization of the transport equations with unstructured mesh. The latter is able to compute sprays for fast planing ships but is also the slower in terms of CPU time. In addition, we had the opportunity to test FS-FLOW which is a potential flow code with a non linear free surface condition distributed by FRIENDSHIP CONSULTING. Numerical results for the four codes are compared with the other codes' results as with tank tests data. Those tank tests were made using captive model test technique on two Open60' models. Reasons of the choice of the captive model technique are explained and experimental procedures are briefly described. Comparisons between codes are mainly based on the easiness of use, the cost in CPU time and the confidence we can have in the results as a function of the boat speed. Flow visualizations, pressure maps, free surface deformation are shown and compared. Analysis of local quantities integrated or by zone is also presented. Results are analyzed focusing on the ability of each code to represent flow dynamics for every speed with a special attention to high speeds. The practical question raised is to know which kind of answers each code can bring in terms of tendencies evaluation or sensitivity to hull geometry modifications. The main goal is to be able to judge if those codes are able to make reliable and consistent comparisons of different designs. Conclusion is that none of the codes is perfect and gather all the advantages. It is still difficult to propose a definitive methodology to estimate hydrodynamic performances at every speed and at every stage of the design process. Knowing each code limitations, it appears more coherent to use each of them at different stages of the design process: the quickest and less reliable to understand the main tendencies and the longest and more precise to validate the final options.

Interpreting Vortex Interactions With a Free Surface

1994 ◽  
Vol 116 (1) ◽  
pp. 91-94 ◽  
Author(s):  
E. P. Rood
Keyword(s):  
Free Surface ◽  
Surface Condition ◽  
Viscous Forces ◽  
Vortex Interactions ◽  
Free Surface Condition ◽  
Vortex Loops ◽  
Physical Experiments ◽  
Vorticity Flux ◽  
Tangential Acceleration

An understanding of the process by which vorticity interacts with a free surface is sought by analytical examination of the free-surface condition for the vorticity flux. A novel mechanism is suggested that permits closed vortex loops to evolve into open loops terminating at the free surface. It is hypothesized that abrupt vortex “disconnection,” observed in physical experiments, arises from a smooth diffusion of vorticity through the interface, with a necessary coincident tangential acceleration of the interface attributed to viscous forces.

scholarly journals Experimental study of the wake behind a surface-piercing cylinder for a clean and contaminated free surface

2000 ◽  
Vol 402 ◽  
pp. 109-136 ◽  
Author(s):  
AMY WARNCKE LANG ◽  
MORTEZA GHARIB
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Turbulent Flows ◽  
Cross Sections ◽  
Surface Deformation ◽  
Surface Condition ◽  
Free Surface Flows ◽  
Shear Stresses ◽  
Surface Contaminants ◽  
A New Technique

This experimental investigation into the nature of free-surface flows was to study the effects of surfactants on the wake of a surface-piercing cylinder. A better understanding of the process of vorticity generation and conversion at a free surface due to the absence or presence of surfactants has been gained. Surfactants, or surface contaminants, have the tendency to reduce the surface tension proportionally to the respective concentration at the free surface. Thus when surfactant concentration varies across a free surface, surface tension gradients occur and this results in shear stresses, thus altering the boundary condition at the free surface. A low Reynolds number wake behind a surface-piercing cylinder was chosen as the field of study, using digital particle image velocimetry (DPIV) to map the velocity and vorticity field for three orthogonal cross-sections of the flow. Reynolds numbers ranged from 350 to 460 and the Froude number was kept below 1.0. In addition, a new technique was used to simultaneously map the free surface deformation. Shadowgraph imaging of the free surface was also used to gain a better understanding of the flow. It was found that, depending on the surface condition, the connection of the shedding vortex filaments in the wake of the cylinder was greatly altered with the propensity for surface tension gradients to redirect the vorticity near the free surface to that of the surface-parallel component. This result has an impact on the understanding of turbulent flows in the vicinity of a free surface with varying surface conditions.

Hydrodynamic Analysis of Two Side-by-Side Moored Floating Bodies

2010 ◽  
Author(s):  
D. C. Hong ◽  
Y. Y. Kim ◽  
S. H. Han
Keyword(s):  
Free Surface ◽  
Surface Condition ◽  
Hydrodynamic Coefficients ◽  
Potential Solution ◽  
Floating Bodies ◽  
Matching Method ◽  
Free Surface Condition ◽  
Wave Elevation ◽  
Resonance Frequencies ◽  
Boundary Matching

The hydrodynamic interaction of two bodies floating in waves is studied. The two-body hydrodynamic coefficients of added mass, wave damping and exciting forces and moments are calculated using the irregular frequency free radiation-diffraction potential solution of the improved Green integral equation associated with the free surface Green function (Hong 1987) according to the conventional two-body analysis. It is well known that the conventional two-body potential solution with usual grid fineness largely overestimates the hydrodynamic coefficients at and near the resonance frequency of the free surface in the gap between two floating bodies moored side-by-side in close proximity (Huijsmans et al. 2001, Hong et al. 2005). The two-body diffraction problem has been solved by both the conventional two-body analysis without damped free surface condition and a boundary matching method with and without damped free surface condition. Numerical results of the wave exciting force coefficients of two identical caissons floating side by side obtained by the two methods have been presented and the discrepancies between them have been discussed. Particular attention is paid to the wave elevation in the gap at the resonance frequencies. Amplitudes and phases of the scattering wave elevations in the gap at the first three free surface resonance frequencies computed by the boundary matching method without damped free surface condition have been presented. It has also been shown that the unrealistic wave elevation due to the resonance of the free surface in the gap can be reduced by imposing the damped free surface condition upon the flow in the gap as used in the oscillating water column hydrodynamics (Hong et al. 2004).

scholarly journals A model for the free-surface flow due to a submerged source in water of infinite depth

1998 ◽  
Vol 39 (4) ◽  
pp. 528-538 ◽  
Author(s):  
J.-M. Vanden-Broeck
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Stagnation Point ◽  
Surface Condition ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Nonuniform Distribution ◽  
Infinite Depth ◽  
Collocation Points ◽  
Series Truncation Method

AbstractWe consider a free-surface flow due to a source submerged in a fluid of infinite depth. It is assumed that there is a stagnation point on the free surface just above the source. The free-surface condition is linearized around the rigid-lid solution, and the resulting equations are solved numerically by a series truncation method with a nonuniform distribution of collocation points. Solutions are presented for various values of the Froude number. It is shown that for sufficiently large values of the Froude number, there is a train of waves on the free surface. The wavelength of these waves decreases as the distance from the source increases.

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