Waves on a stream of finite depth which has a velocity defect near the free surface

1970 ◽  
Vol 41 (3) ◽  
pp. 509-521 ◽  
Author(s):  
P. L. Betts
Keyword(s):  
Free Surface ◽  
High Speed ◽  
Surface Profile ◽  
Finite Depth ◽  
Simplified Model ◽  
Stationary Waves ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Velocity Defect ◽  
A Current

The conditions under which stationary waves may exist on a stream of water of finite depth are investigated theoretically for the case of a current which is uniform except for a constant defect in velocity in a region near the free surface. The analysis is extended to provide a two-dimensional theory for the surface profile induced by a simplified model of a hovering craft. The relevance of this work to the use of high speed flumes is discussed, and an example demonstrates the importance of the velocity distribution near the free surface.

scholarly journals On bounds and non-existence in the problem of steady waves with vorticity

2015 ◽  
Vol 765 ◽  
Author(s):  
V. Kozlov ◽  
N. Kuznetsov ◽  
E. Lokharu
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Surface Profile ◽  
Finite Depth ◽  
Two Dimensional ◽  
Vorticity Distribution ◽  
Unidirectional Flow ◽  
Wave Solutions ◽  
Free Surface Profile ◽  
Parallel Shear Flow

AbstractFor the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoulli’s constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli’s constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

A Two-Dimensional Surface Profile Imaging Technique Based on Heterodyne Interferometer

2005 ◽  
Vol 295-296 ◽  
pp. 477-482
Author(s):  
K.W. Wang ◽  
Z.J. Cai ◽  
Li Jiang Zeng
Keyword(s):  
High Speed ◽  
Imaging Technique ◽  
Surface Profile ◽  
Ccd Camera ◽  
Measurement Range ◽  
Interference Signal ◽  
Two Dimensional ◽  
Step Method ◽  
Heterodyne Interferometer ◽  
Dimensional Surface

A two-dimensional surface profile imaging technique based on heterodyne interferometer is proposed. A piezo translator vibrated grating is used to generate a heterodyne signal. A high speed CCD camera is used to extract the interference signal using a five step method. The uncertainty in the displacement measurement is approximately 0.035 µm within a measurement range of 1.7 µm, confirming the two dimensional heterodyne interferometer is valid for measuring the surface profile. The method is also available for low coherence heterodyne interferometer due to the optical frequency shifts caused by the vibration of grating independent on the wavelength.

A Linearized Two-Dimensional Theory for High-Speed Hydrofoils Near the Free Surface

1966 ◽  
Vol 10 (01) ◽  
pp. 25-48
Author(s):  
Richard P. Bernicker
Keyword(s):  
Direct Problem ◽  
High Speed ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Pressure Loading ◽  
Dimensional Theory ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Sectional Shape ◽  
Infinite Set

A linearized two-dimensional theory is presented for high-speed hydrofoils near the free surface. The "direct" problem (hydrofoil shape specified) is attacked by replacing the actual foil with vortex and source sheets. The resulting integral equation for the strength of the singularity distribution is recast into an infinite set of linear algebraic equations relating the unknown constants in a Glauert-type vorticity expansion to the boundary condition on the foil. The solution is achieved using a matrix inversion technique and it is found that the matrix relating the known and unknown constants is a function of depth of submergence alone. Inversion of this matrix at each depth allows the vorticity constants to be calculated for any arbitrary foil section by matrix multiplication. The inverted matrices have been calculated for several depth-to-chord ratios and are presented herein. Several examples for specific camber and thickness distributions are given, and results indicate significant effects in the force characteristics at depths less than one chord. In particular, thickness effects cause a loss of lift at shallow submergences which may be an appreciable percentage of the total design lift. The second part treats the "indirect" problem of designing a hydrofoil sectional shape at a given depth to achieve a specified pressure loading. Similar to the "direct" problem treated in the first part, integral equations are derived for the camber and thickness functions by replacing the actual foil by vortex and source sheets. The solution is obtained by recasting these equations into an infinite set of linear algebraic equations relating the constants in a series expansion of the foil geometry to the known pressure boundary conditions. The matrix relating the known and unknown constants is, again, a function of the depth of submergence alone, and inversion techniques allow the sectional shape to be determined for arbitrary design pressure distributions. Several examples indicate the procedure and results are presented for the change in sectional shape for a given pressure loading as the depth of submergence of the foil is decreased.

Stationary gravity waves on non-uniform free streams: jet-like streams

1975 ◽  
Vol 77 (2) ◽  
pp. 415-438 ◽  
Author(s):  
D. H. Peregrine ◽  
Ronald Smith
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Wave Number ◽  
Explicit Solutions ◽  
Stationary Waves ◽  
Two Dimensional ◽  
Large Wave ◽  
Surface Gravity Waves ◽  
Large Wave Number

AbstractThe basic state considered in this paper is a parallel flow of a jet-like character with the centre of the jet being at or near a free surface which is horizontal. Stationary surface gravity waves may exist on such a flow, and a number of examples are looked at for small amplitude waves. Explicit solutions are given for ‘top-hat’ profile jets and for two-dimensional flows. Asymptotic solutions are developed for stationary waves of large wave-number.

Directed fluid sheets

1976 ◽  
Vol 347 (1651) ◽  
pp. 447-473 ◽  
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Steady Motion ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Inviscid Fluids ◽  
Two Dimensional Flows

This paper is concerned mainly with incompressible inviscid fluid sheets but the incompressible linearly viscous fluid sheet is also considered. Our development is based on a direct formulation using the two dimensional theory of directed media called Cosserat surfaces . The first part of the paper deals with the formulation of appropriate nonlinear equations (which may include the effects of gravity and surface tension) governing the two dimensional motion of incompressible inviscid media for two categories, namely those ( a ) for two dimensional flows confined to a plane perpendicular to a specified direction and ( b ) for propagation of fairly long waves in a stream of variable initial depth. The latter development is a generalization of an earlier direct formulation of a theory of water waves when the fixed bottom of the stream is level (Green, Laws & Naghdi 1974). In the second part of the paper, special attention is given to a demonstration of the relevance and applicability of the present direct formulation to a variety of two dimensional problems of inviscid fluid sheets. These include, among others, the steady motion of a class of two-dimensional flows in a stream of finite depth in which the bed of the stream may change from one constant level to another, the related problem of hydraulic jumps, and a class of exact solutions which characterize the main features of the time-dependent free surface flows in the three dimensional theory of incompressible inviscid fluids.

Numerical solution for two-dimensional flow under sluice gates using the natural element method

2010 ◽  
Vol 37 (12) ◽  
pp. 1550-1559 ◽  
Author(s):  
Farhang Daneshmand ◽  
S.A. Samad Javanmard ◽  
Tahereh Liaghat ◽  
Mohammad Mohsen Moshksar ◽  
Jan F. Adamowski
Keyword(s):  
Free Surface ◽  
Numerical Technique ◽  
Surface Profile ◽  
Nonlinear Boundary Conditions ◽  
Two Dimensional ◽  
Pressure Distributions ◽  
Natural Element Method ◽  
Natural Element ◽  
Surface Profiles ◽  
Element Method

Fluid loads on a variety of hydraulic structures and the free surface profile of the flow are important for design purposes. This is a difficult task because the governing equations have nonlinear boundary conditions. The main objective of this paper is to develop a procedure based on the natural element method (NEM) for computation of free surface profiles, velocity and pressure distributions, and flow rates for a two-dimensional gravity fluid flow under sluice gates. Natural element method is a numerical technique in the field of computational mechanics and can be considered as a meshless method. In this analysis, the fluid was assumed to be inviscid and incompressible. The results obtained in the paper were confirmed via a hydraulic model test. Calculation results indicate a good agreement with previous flow solutions for the water surface profiles and pressure distributions throughout the flow domain and on the gate.

Blockage with a Free Surface

1976 ◽  
Vol 20 (04) ◽  
pp. 199-203
Author(s):  
J. N. Newman
Keyword(s):  
Free Surface ◽  
Velocity Potential ◽  
Finite Depth ◽  
Wave Resistance ◽  
Steady Flows ◽  
Two Dimensional ◽  
Present Context ◽  
Two Dimensional Flow ◽  
Field Radiation ◽  
Radiation Conditions

The occurrence of blockage, or a jump in the velocity potential between the upstream and downstream infinities, is well known for steady two-dimensional flow past a body in a rigid channel. This paper considers the analogous situation where there is a free surface, as in the wave resistance problem for submerged two-dimensional bodies in a fluid of finite depth. It is shown that blockage occurs in spite of the free surface, taking values which depend not only on the dipole moment but also upon the Froude number based on depth. The occurrence of blockage, in the present context, has a bearing primarily upon the correct formulation of far-field radiation conditions for steady flows with finite depth.

An Experimental and Numerical Study of Flow Patterns and Air Entrapment Phenomena During the Filling of a Vertical Die Cavity

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
J. J. Hernández-Ortega ◽  
R. Zamora ◽  
J. Palacios ◽  
J. López ◽  
F. Faura
Keyword(s):  
Free Surface ◽  
High Speed ◽  
Numerical Study ◽  
Die Casting ◽  
Surface Profile ◽  
Mass Conservation ◽  
Flow Patterns ◽  
Operating Conditions ◽  
Working Fluid ◽  
Air Entrapment

One of the most important problems encountered in die-casting processes is porosity due to air entrapment in the molten metal during the injection process. The aim of this work is to study experimentally and numerically the different air entrapment phenomena that may take place in the early stages of the filling of a vertical die cavity with a rectangular shape for operating conditions typically used in low and medium-pressure die-casting processes. Special attention is given to determining the influence of the gravitational forces on the flow pattern. Numerical simulation of the flow in the die cavity is carried out for the liquid phase using a commercial computational fluid dynamics (CFD) code (FLOW-3D) based on the solution algorithm-volume of fluid (SOLA-VOF) approach to solve the coupling between the momentum and mass conservation equations and to treat the free-surface, while the amount of air evacuated through vents is calculated by using an unsteady one-dimensional adiabatic model that retains friction effects. The main characteristics of the flow at the early instants of the die cavity filling are analyzed for different operating conditions, and the different flow patterns are summarized in a map as a function of the Reynolds and Froude numbers. Also, filling visualization experiments are carried out on a test bench using water as working fluid in a transparent die model and a high-speed camera. The numerical and experimental results obtained for the free-surface profile evolution are compared for different inlet velocities of the fluid and the viability of the numerical tools used to predict the final amount of trapped air in the die cavity is discussed.

scholarly journals Modified Babenko’s Equation For Periodic Gravity Waves On Water Of Finite Depth

2019 ◽  
Vol 72 (4) ◽  
pp. 415-428
Author(s):  
E Dinvay ◽  
N Kuznetsov
Keyword(s):  
Surface Tension ◽  
Gravity Waves ◽  
Water Waves ◽  
Surface Profile ◽  
Finite Depth ◽  
Parametric Representation ◽  
Two Dimensional ◽  
Free Surface Profile ◽  
The Mean ◽  
New Equation

Summary A new operator equation for periodic gravity waves on water of finite depth is derived and investigated; it is equivalent to Babenko’s equation considered in Kuznetsov and Dinvay (Water Waves, 1, 2019). Both operators in the proposed equation are nonlinear and depend on the parameter equal to the mean depth of water, whereas each solution defines a parametric representation for a symmetric free surface profile. The latter is a component of a solution of the two-dimensional, nonlinear problem describing steady waves propagating in the absence of surface tension. Bifurcation curves (including a branching one) are obtained numerically for solutions of the new equation; they are compared with known results.

scholarly journals THE SURFACE WAVE IN A TWO-DIMENSIONAL VORTEX LAYER

2011 ◽  
Vol 1 (8) ◽  
pp. 3
Author(s):  
Tokuichi Hamada
Keyword(s):  
Free Surface ◽  
Surface Wave ◽  
Order Term ◽  
Surface Condition ◽  
Surface Profile ◽  
Second Order ◽  
Primary Wave ◽  
Two Dimensional ◽  
Dynamical Equations ◽  
Vortex Layer

Progressive surface wave in a two-dimensional vortex layer is theoretically treated. Dynamical equations and free surface conditions are shown by using the two-dimensional stream functions of wave and vortex. Then the perturbation equations are given by assuming that the ratio of length scale of vortices and wave is fairly small. The first approximate solution of wave has a usual form of an irrotational progressive wave. Vortices are assumed to be steady and to have simplified Fourier- Stleltjes form. Then the interaction of this primary wave and the vortices are examined. To satisfy the free surface condition of the second order, existent waves are formed. In the second order term of the free surface elevation, these secondary waves offset the effect of the above mentioned interaction, and so the surface profile of the primary wave is not altered by the existence of inner vortices of high frequency. Some pictures of Irregular surface waves in a turbulent flow are shown to verify this property.

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