Withdrawal of a stratified fluid from a vertical two-dimensional duct

1975 ◽  
Vol 70 (2) ◽  
pp. 321-332 ◽  
Author(s):  
Jorg Imberger ◽  
Chris Fandry
Keyword(s):  
Number Theory ◽  
Free Surface ◽  
Froude Number ◽  
Stratified Fluid ◽  
Two Dimensional ◽  
Tank Bottom ◽  
Vertical Duct

On the basis of a small Froude number theory it is shown that previous horizontalduct models of the withdrawal of fluid from a stratified tank with a free surface and a fixed bottom are inappropriate at large times and that the flow should be modelled by that from an infinite vertical duct. The falling horizontal free surface in the tank is replaced by a vertically moving column of stratified fluid and the tank bottom is modelled by a stagnant pool of fluid below. This model is analysed and a solution uniformly valid in time is presented. The properties of the solution are then compared with existing theories and experiments.

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.

scholarly journals Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid

2007 ◽  
Vol 570 ◽  
pp. 297-305 ◽  
Author(s):  
AXEL DELONCLE ◽  
JEAN-MARC CHOMAZ ◽  
PAUL BILLANT
Keyword(s):  
Froude Number ◽  
Dimensional Stability ◽  
Numerical Study ◽  
Three Dimensional ◽  
Stratified Fluid ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Homogeneous Fluid ◽  
Sheared Flow ◽  
Vertical Scales

This paper investigates the three-dimensional stability of a horizontal flow sheared horizontally, the hyperbolic tangent velocity profile, in a stably stratified fluid. In an homogeneous fluid, the Squire theorem states that the most unstable perturbation is two-dimensional. When the flow is stably stratified, this theorem does not apply and we have performed a numerical study to investigate the three-dimensional stability characteristics of the flow. When the Froude number, Fh, is varied from ∞ to 0.05, the most unstable mode remains two-dimensional. However, the range of unstable vertical wavenumbers widens proportionally to the inverse of the Froude number for Fh ≪ 1. This means that the stronger the stratification, the smaller the vertical scales that can be destabilized. This loss of selectivity of the two-dimensional mode in horizontal shear flows stratified vertically may explain the layering observed numerically and experimentally.

scholarly journals The Froude number for solitary water waves with vorticity

2015 ◽  
Vol 768 ◽  
pp. 91-112 ◽  
Author(s):  
Miles H. Wheeler
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Froude Number ◽  
Simple Proof ◽  
Upper Bound ◽  
Water Waves ◽  
Wave Speed ◽  
Critical Value ◽  
Arbitrary Distribution ◽  
Two Dimensional

We consider two-dimensional solitary water waves on a shear flow with an arbitrary distribution of vorticity. Assuming that the horizontal velocity in the fluid never exceeds the wave speed and that the free surface lies everywhere above its asymptotic level, we give a very simple proof that a suitably defined Froude number $F$ must be strictly greater than the critical value $F=1$. We also prove a related upper bound on $F$, and hence on the amplitude, under more restrictive assumptions on the vorticity.

Drag on a sphere moving horizontally in a stratified fluid

2000 ◽  
Vol 418 ◽  
pp. 339-350 ◽  
Author(s):  
M. D. GREENSLADE
Keyword(s):  
Number Theory ◽  
Froude Number ◽  
Translational Motion ◽  
Laboratory Data ◽  
Wave Theory ◽  
Stratified Fluid ◽  
Wave Generation ◽  
Rectangular Section ◽  
Size And Shape ◽  
Linear Gravity

The steady translational motion of a sphere in a Boussinesq stratified fluid, where the motion is parallel to the stratification surfaces, is studied. A model based on existing linear gravity wave theory for large Froude numbers and on new theory for small Froude numbers is presented. The small-Froude-number theory describes wave generation and the presence of a rectangular-section attached wake whose size and shape is controlled by the size and shape of the wave generation regions. Existing laboratory data are used to evaluate the model's prediction for the drag coefficient of the sphere as a function of Froude number.

Two-dimensional free-surface gravity flow past blunt bodies

1972 ◽  
Vol 51 (3) ◽  
pp. 529-543 ◽  
Author(s):  
G. Dagan ◽  
M. P. Tulin
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
The Body ◽  
Surface Gravity ◽  
Gravity Flow ◽  
Breaking Wave ◽  
Infinite Length ◽  
Two Dimensional ◽  
Flow Past ◽  
Outer Expansion

Most of the wave resistance of blunt bow displacement ships is caused by the bow-breaking wave. A theoretical study of the phenomenon for the two-dimensional steady flow past a blunt body of semi-infinite length is presented. The exact equations of free-surface gravity flow are solved approximately by two perturbation expansions. The small Froude number solution, representing the flow beneath an unbroken free surface before the body, is carried out to second order. The breaking of the free surface is assumed to be related to a local Taylor instability, and the application of the stability criterion determines the value of the critical Froude number which characterizes breaking. The high Froude number solution is based on the model of a jet detaching from the bow and not returning to the flow field. The outer expansion of the equations yields the linearized gravity flow equations, which are solved by the Wiener-Hopf technique. The inner expansion gives a nonlinear gravity-free flow in the vicinity of the bow a t zero order. The matching of the inner and outer expansions provides the jet thickness as well as the associated drag.

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.

Shear instability in a stratified fluid when shear and stratification are not aligned

2011 ◽  
Vol 685 ◽  
pp. 191-201 ◽  
Author(s):  
Julien Candelier ◽  
Stéphane Le Dizès ◽  
Christophe Millet
Keyword(s):  
Froude Number ◽  
Unstable Mode ◽  
Vertical Direction ◽  
Maximum Velocity ◽  
Stratified Fluid ◽  
Shear Instability ◽  
Two Dimensional ◽  
Horizontal Length ◽  
The Stability ◽  
Vertical Jet

AbstractThe effect of an inclination angle of the shear with respect to the stratification on the linear properties of the shear instability is examined in the work. For this purpose, we consider a two-dimensional plane Bickley jet of width $L$ and maximum velocity ${U}_{0} $ in a stably stratified fluid of constant Brunt–Väisälä frequency $N$ in an inviscid and Boussinesq framework. The plane of the jet is assumed to be inclined with an angle $\theta $ with respect to the vertical direction of stratification. The stability analysis is performed using both numerical and theoretical methods for all the values of $\theta $ and Froude number $F= {U}_{0} / (LN)$. We first obtain that the most unstable mode is always a two-dimensional Kelvin–Helmholtz (KH) sinuous mode. The condition of stability based on the Richardson number $Ri\gt 1/ 4$, which reads here $F\lt 3 \sqrt{3} / 2$, is recovered for $\theta = 0$. But when $\theta \not = 0$, that is, when the directions of shear and stratification are not perfectly aligned, the Bickley jet is found to be unstable for all Froude numbers. We show that two modes are involved in the stability properties. We demonstrate that when $F$ is decreased below $3 \sqrt{3} / 2$, there is a ‘jump’ from one two-dimensional sinuous mode to another. For small Froude numbers, we show that the shear instability of the inclined jet is similar to that of a horizontal jet but with a ‘horizontal’ length scale ${L}_{h} = L/ \sin \theta $. In this regime, the characteristics (oscillation frequency, growth rate, wavenumber) of the most unstable mode are found to be proportional to $\sin \theta $. For large Froude numbers, the shear instability of the inclined jet is similar to that of a vertical jet with the same scales but with a different Froude number, ${F}_{v} = F/ \hspace *{-.1pc}\cos \theta $. It is argued that these results could be valid for any type of shear flow.

Nonlinear free-surface flow at a two-dimensional bow

1989 ◽  
Vol 209 ◽  
pp. 57-75 ◽  
Author(s):  
Mark A. Grosenbaugh ◽  
Ronald W. Yeung
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Stagnation Point ◽  
Free Surface Flow ◽  
Surface Flow ◽  
The Body ◽  
Two Dimensional ◽  
Bow Wave ◽  
Bulbous Bow ◽  
Nonlinear Free Surface

Unsteady free-surface flow at the bow of a steadily moving, two-dimensional body is solved using a modified Eulerian-Lagrangian technique. Lagrangian marker particles are distributed on both the free surface and the far-field boundary. The flow field corresponding to an inviscid, double-body solution is used for the initial condition. Solutions are obtained over a range of Froude numbers for bodies of three different shapes: a vertical step, a faired profile, and a bulbous bow. A transition Froude number exists at which the bow wave begins to overturn and break. The value of the transition Froude number depends on the bow shape. A stagnation point is observed to be present below the free surface during the initial stage of the wave formation. For flows occurring above the transition Froude number, the stagnation point remains trapped below the free surface as the wave overturns. Below the transition Froude number, the stagnation point rises to the surface as the crest of the transient bow wave moves upstream and away from the body.

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.

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