The initial development of a jet caused by fluid, body and free-surface interaction. Part 2. An impulsively moved plate

2007 ◽  
Vol 578 ◽  
pp. 67-84 ◽  
Author(s):  
D. J. NEEDHAM ◽  
J. BILLINGHAM ◽  
A. C. KING
Keyword(s):  
Free Surface ◽  
Surface Deformation ◽  
Boundary Value ◽  
Outer Region ◽  
Small Time ◽  
Horizontal Strip ◽  
Initial Development ◽  
Inviscid Incompressible Fluid ◽  
Fluid Body ◽  
Inner Region

The free-surface deformation and flow field caused by the impulsive horizontal motion of a rigid vertical plate into a horizontal strip of inviscid incompressible fluid, initially at rest, is studied in the small time limit using the method of matched asymptotic expansions. It is found that three different asymptotic regions are necessary to describe the flow. There is a main, O(1) sized, outer region in which the flow is singular at the point where the free surface meets the plate. This leads to an inner region, centred on the point where the free surface initially meets the plate, with size of O(-t log t). To resolve the singularities that arise in this inner region, it is necessary to analyse further the flow in an inner-inner region, with size of O(t), again centred upon the wetting point of the nascent rising jet. The solutions of the boundary value problems in the two largest regions are obtained analytically. The solution of the parameter-free nonlinear boundary value problem that arises in the inner-inner region is obtained numerically.

The initial development of a jet caused by fluid, body and free-surface interaction. Part 1. A uniformly accelerating plate

1994 ◽  
Vol 268 ◽  
pp. 89-101 ◽  
Author(s):  
A. C. King ◽  
D. J. Needham
Keyword(s):  
Free Surface ◽  
Flow Field ◽  
Finite Depth ◽  
Small Time ◽  
Restoring Force ◽  
Initial Development ◽  
Thin Region ◽  
Fluid Body ◽  
Horizontal Momentum ◽  
Important Design

The flow field induced by a vertical plate accelerating into a stationary fluid of finite depth with a free surface and a gravitational restoring force is investigated. This is a model problem for some technologically important design issues such as the bow splash of a ship moving at forward speed. Experimentally it is found that a thin jet forms on the plate and rises rapidly upwards. We investigate this jet in the small-time approximation and find an analytical solution for the flow field in which the jet emerges out of a thin region where the horizontal momentum of the main flow is converted by inertial effects into a rising jet.

Surface-tension-driven flow in fat fluid wedges and cones

1999 ◽  
Vol 397 ◽  
pp. 45-71 ◽  
Author(s):  
JOHN BILLINGHAM
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Outer Region ◽  
Integral Transforms ◽  
Capillary Waves ◽  
Governing Equations ◽  
Double Integral ◽  
Long Time ◽  
Inner Region ◽  
Short Time

We consider the evolution under the action of surface tension of wedges and cones of viscous fluid whose initial semi-angles are close to π/2. A short time after the fluid is released from rest, there is an inner region, where surface tension and viscosity dominate, and an outer region, where inertia and viscosity dominate. We also find that the velocity of the tip of the wedge or cone is singular, of O(log(1/t)), as time, t, tends to zero. After a long time, the free surface asymptotes to a similarity form where deformations are of O(t2/3), and capillary waves propagate away from the tip. However, a distance of O(t3/4) away from the tip, viscosity acts to damp out the capillary waves.We solve the linearized governing equations using double integral transforms, which we calculate numerically, and use asymptotic techniques to approximate the solutions for small and large times. We also compare the asymptotic solution for the inviscid fat wedge with a numerical solution of the nonlinear inviscid problem for wedges of arbitrary semi-angle.

scholarly journals The effects of surface tension on the initial development of a free surface adjacent to an accelerated plate

2015 ◽  
Vol 776 ◽  
pp. 37-73 ◽  
Author(s):  
J. Uddin ◽  
D. J. Needham
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Local Structure ◽  
Contact Point ◽  
Initial Development ◽  
Early Stages ◽  
Inviscid Incompressible Fluid ◽  
Stationary Layer ◽  
Weak Surface ◽  
Theoretical Results

When a vertical rigid plate is uniformly accelerated horizontally from rest into an initially stationary layer of inviscid incompressible fluid, the free surface will undergo a deformation in the locality of the contact point. This deformation of the free surface will, in the early stages, cause a jet to rise up the plate. An understanding of the local structure of the free surface in the early stages of motion is vital in many situations, and has been developed in detail by King & Needham (J. Fluid Mech., vol. 268, 1994, pp. 89–101). In this work we consider the effects of introducing weak surface tension, characterized by the inverse Weber number $\mathscr{W}$, into the problem considered by King & Needham. Our approach is based upon matched asymptotic expansions as $\mathscr{W}\rightarrow 0$. It is found that four asymptotic regions are needed to describe the problem. The three largest regions have analytical solutions, whilst a numerical method based on finite differences is used to solve the time-dependent harmonic boundary value problem in the last region. Our results identify the local structure of the jet near the vicinity of the contact point, and we highlight a number of key features, including the height of this jet as well as its thickness and strength. We also present some preliminary experimental results that capture the spatial structure near the contact point, and we then show promising comparisons with the theoretical results obtained within this paper.

scholarly journals The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow

2018 ◽  
Vol 841 ◽  
pp. 109-145 ◽  
Author(s):  
M. T. Gallagher ◽  
D. J. Needham ◽  
J. Billingham
Keyword(s):  
Free Surface ◽  
Boundary Value Problem ◽  
Contact Point ◽  
Boundary Value ◽  
Complex Structure ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Contact ◽  
Initial Development ◽  
Initial Boundary Value

The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration $\unicode[STIX]{x1D70E}$) of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the exterior horizontal, as it advances ($\unicode[STIX]{x1D70E}>0$) or retreats ($\unicode[STIX]{x1D70E}<0$) from an initially stationary and horizontal strip of inviscid incompressible fluid under gravity, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating plate advancing into a fluid as studied by Needham et al. (Q. J. Mech. Appl. Maths, vol. 61 (4), 2008, pp. 581–614). Particular attention is paid to the innermost asymptotic regions encompassing the initial interaction between the plate and the free surface. We find that the structure of the solution to the governing initial boundary value problem is characterised in terms of the parameters $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D707}$ (where $\unicode[STIX]{x1D707}=1+\unicode[STIX]{x1D70E}\tan \unicode[STIX]{x1D6FC}$), with a bifurcation in structure as $\unicode[STIX]{x1D707}$ changes sign. This bifurcation in structure leads us to question the well-posedness and stability of the governing initial boundary value problem with respect to small perturbations in initial data in the innermost asymptotic regions, the discussion of which will be presented in the companion paper Gallagher et al. (J. Fluid Mech. vol. 841, 2018, pp. 146–166). In particular, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{+}$, the free surface close to the initial contact point remains monotone, and encompasses a swelling jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times [1,\infty )$ or a collapsing jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times (0,1)$. However, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{-}$, the collapsing jet develops a more complex structure, with the free surface close to the initial contact point now developing a finite number of local oscillations, with near resonance type behaviour occurring close to a countable set of critical plate angles $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{n}^{\ast }\in (0,\unicode[STIX]{x03C0}/2)$ ($n=1,2,\ldots$).

Free-surface deformation measurement

1997 ◽  
Author(s):  
H. Stahl ◽  
Kevin Stultz ◽  
H. Stahl ◽  
Kevin Stultz
Keyword(s):  
Free Surface ◽  
Surface Deformation ◽  
Deformation Measurement ◽  
Free Surface Deformation

scholarly journals Experiments on generation of surface waves by an underwater moving bottom

2015 ◽  
Vol 471 (2178) ◽  
pp. 20150069 ◽  
Author(s):  
Timothée Jamin ◽  
Leonardo Gordillo ◽  
Gerardo Ruiz-Chavarría ◽  
Michael Berhanu ◽  
Eric Falcon
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Surface Deformation ◽  
Fluid Layer ◽  
Fluid Velocity ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass ◽  
Spatio Temporal ◽  
Experimental Findings

We report laboratory experiments on surface waves generated in a uniform fluid layer whose bottom undergoes an upward motion. Simultaneous measurements of the free-surface deformation and the fluid velocity field are focused on the role of the bottom kinematics (i.e. its spatio-temporal features) in wave generation. We observe that the fluid layer transfers bottom motion to the free surface as a temporal high-pass filter coupled with a spatial low-pass filter. Both filter effects are often neglected in tsunami warning systems, particularly in real-time forecast. Our results display good agreement with a prevailing linear theory without any parameter fitting. Based on our experimental findings, we provide a simple theoretical approach for modelling the rapid kinematics limit that is applicable even for initially non-flat bottoms: this may be a key step for more realistic varying bathymetry in tsunami scenarios.

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