Upstream influence and the form of standing hydraulic jumps in liquid-layer flows on favourable slopes

1995 ◽  
Vol 284 ◽  
pp. 63-96 ◽  
Author(s):  
Robert I. Bowles
Keyword(s):  
Free Surface ◽  
Liquid Layer ◽  
Numerical Solutions ◽  
Free Surface Flow ◽  
Quantitative Agreement ◽  
Surface Flow ◽  
Hydraulic Jumps ◽  
Sloping Surface ◽  
Free Interaction ◽  
Limiting Case

Steady planar flow of a liquid layer over an obstacle is studied for favourable slopes. First, half-plane Poiseuille flow is found to be a non-unique solution on a uniformly sloping surface since eigensolutions exist which are initially exponentially small far upstream. These have their origin in a viscous–inviscid interaction between the retarding action of viscosity and the hydrostatic pressure from the free surface. The cross-stream pressure gradient caused by the curvature of the streamlines also comes into play as the slope increases. As the interaction becomes nonlinear, separation of the liquid layer can occur, of a breakaway type if the slope is sufficiently large. The breakaway represents a hydraulic jump in the sense of a localized relatively short-scaled increase in layer thickness, e.g. far upstream of a large obstacle. The solution properties give predictions for the shape and structure of hydraulic jumps on various slopes. Secondly, the possibility of standing waves downstream of the jump is addressed for various slope magnitudes. A limiting case of small gradient, governed by lubrication theory, allows the downstream boundary condition to be included explicitly. Numerical solutions showing the free-surface flow over an obstacle confirm the analytical conclusions. In addition the predictions are compared with the experimental and computational results of Pritchardet al.(1992), yielding good qualitative and quantitative agreement. The effects of surface tension on the jump are also discussed and in particular the free interaction on small slopes is examined for large Bond numbers.

scholarly journals Early-time free-surface flow driven by a deforming boundary

2015 ◽  
Vol 767 ◽  
pp. 811-841 ◽  
Author(s):  
C. Frederik Brasz ◽  
Craig B. Arnold ◽  
Howard A. Stone ◽  
John R. Lister
Keyword(s):  
Free Surface ◽  
Numerical Simulations ◽  
Liquid Layer ◽  
Early Time ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Solid Boundary ◽  
Jet Formation ◽  
Domain Perturbation ◽  
Quiescent Liquid

AbstractWhen a solid boundary deforms rapidly into a quiescent liquid layer, a flow is induced that can lead to jet formation. An asymptotic analytical solution is presented for this flow, driven by a solid boundary deforming with dimensionless vertical velocity $V_{b}(x,t)={\it\epsilon}(1+\cos x)\,f(t)$, where the amplitude ${\it\epsilon}$ is small relative to the wavelength and the time dependence $f(t)$ approaches 0 for large $t$. Initially, the flow is directed outwards from the crest of the deformation and slows with the slowing of the boundary motion. A domain-perturbation method is used to reveal that, when the boundary stops moving, nonlinear interactions with the free surface leave a remnant momentum directed back towards the crest, and this momentum can be a precursor to jet formation. This scenario arises in a laser-induced printing technique in which an expanding blister imparts momentum into a liquid film to form a jet. The analysis provides insight into the physics underlying the interaction between the deforming boundary and free surface, in particular, the dependence of the remnant flow on the thickness of the liquid layer and the deformation amplitude and wavelength. Numerical simulations are used to show the range of validity of the analytical results, and the domain-perturbation solution is extended to an axisymmetric domain with a Gaussian boundary deformation to compare with previous numerical simulations of blister-actuated laser-induced forward transfer.

scholarly journals Free surface flow past topography: A beyond-all-orders approach

2012 ◽  
Vol 23 (4) ◽  
pp. 441-467 ◽  
Author(s):  
CHRISTOPHER J. LUSTRI ◽  
SCOTT W. MCCUE ◽  
BENJAMIN J. BINDER
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Numerical Solutions ◽  
Power Series Expansion ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Asymptotic Results ◽  
Stagnation Points ◽  
Flow Past ◽  
Stokes Lines

The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck, (2006) Exponential asymptotics and gravity waves. J. Fluid Mech.567, 299–326, the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of inclination. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.

On Marangoni drying: nonlinear kinematic waves in a thin film

1993 ◽  
Vol 254 ◽  
pp. 649-670 ◽  
Author(s):  
S. B. G. M. O'Brien
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Order Approximation ◽  
Nonlinear Partial Differential Equation ◽  
Free Surface Flow ◽  
Wave Theory ◽  
Alcohol Concentration ◽  
Surface Flow ◽  
Singular Perturbation Theory ◽  
Higher Derivative

In the field of industrial drying, a recent innovation has exploited the occurrence of Marangoni effects in such a way that the resultant free-surface flow enhances the drying process. To this end, alcohol vapour, soluble in water, is introduced above a drying film and as a result of diffusion through the air and water phases a favourable concentration gradient gives rise to the required shear flow. We consider here a simple process driven by this mechanism, and by means of asymptotic simplification and the concepts of singular perturbation theory a leading-order approximation is obtained in which the alcohol concentration in the water is a specified function of space and time. The evolution of the free surface thus reduces to a single nonlinear partial differential equation of a similar form to the Korteweg–de Vries and Burgers equations, higher-derivative terms corresponding to surface tension and gravity effects. Numerical solutions of this equation are obtained and are compared to the application of first order nonlinear kinematic wave theory with corresponding shock solutions.

Measurements of the Free Surface Flow Structure Under an Impinging, Free Liquid Jet

1992 ◽  
Vol 114 (1) ◽  
pp. 79-84 ◽  
Author(s):  
J. Stevens ◽  
B. W. Webb
Keyword(s):  
Free Surface ◽  
Flow Structure ◽  
Liquid Layer ◽  
Free Surface Flow ◽  
Reynolds Numbers ◽  
Surface Flow ◽  
Surface Velocity ◽  
Liquid Jet ◽  
Radial Coordinate ◽  
Measured Velocity

The objective of this research was to characterize the flow structure under an impinging liquid jet striking a flat, normally oriented surface. The approach was the measurement of the free surface velocities of the jet prior to impingement and the surface velocities of the radially spreading liquid layer. A novel laser-Doppler velocimetry technique was used. The LDV system was configured such that the measurement volume would span the time-dependent fluctuations of the free surface, with the surface velocity being measured. The mean and fluctuating components of a single direction of the velocity vector were measured. It was found that the radial liquid layer data collapsed well over the range of jet Reynolds numbers 16,000 < Re < 47,000 if plotted in dimensionless coordinates, where the measured velocity was normalized by the average jet exit velocity and the radial coordinate was normalized by the nozzle diameter. Mean liquid layer depths were inferred from the velocity measurements by assuming a velocity profile across the layer, and were reported. Pre-impingement jet measurements suggest that the flow development is nearly complete two diameters from the nozzle exit.

Supercritical withdrawal from a two-layer fluid through a line sink

1995 ◽  
Vol 297 ◽  
pp. 37-47 ◽  
Author(s):  
G. C. Hocking
Keyword(s):  
Free Surface ◽  
Solution Method ◽  
Numerical Solutions ◽  
Lower Layer ◽  
Single Layer ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Line Sink ◽  
Supercritical Flows ◽  
Good Agreement

Accurate numerical solutions to the problem of finding the location of the interface between two unconfined regions of fluid of different density during the withdrawal process are presented. Supercritical flows are considered, in which the interface is drawn directly into the sink. As the flow rate is reduced, the interface enters the sink more steeply, until the solution method breaks down just before the interface enters the sink vertically from above, and becomes flow from the lower layer only. This lower bound on supercritical flow is compared with the upper bound on single-layer (free surface) flow with good agreement.

Numerical calculations of the free-surface flow under a sluice gate

1997 ◽  
Vol 330 ◽  
pp. 339-347 ◽  
Author(s):  
J.-M. VANDEN-BROECK
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Large Amplitude ◽  
Numerical Solutions ◽  
Free Surface Flow ◽  
Radiation Condition ◽  
Surface Flow ◽  
Boundary Integral ◽  
Sluice Gate ◽  
The Waves

The free-surface flow under a sluice gate is considered. The fluid is assumed to be inviscid and incompressible. The problem is solved numerically by using a boundary integral equation technique. Accurate numerical solutions are obtained when the intersection of the upstream free surface with the gate is a stagnation point. It is shown that the radiation condition is not satisfied far upstream and that there is a train of waves on the upstream free surface. For large values of the downstream Froude number F, the amplitude of the waves is so small that the upstream free surface is essentially flat. However for small values of F, the waves are of large amplitude. They ultimately approach the Stokes' limiting configuration with an angle of 120° at their crest as F is decreased.

Flotation and free surface flow in a model for subglacial drainage. Part 2. Channel flow

2012 ◽  
Vol 702 ◽  
pp. 157-187 ◽  
Author(s):  
I. J. Hewitt ◽  
C. Schoof ◽  
M. A. Werder
Keyword(s):  
Free Surface ◽  
Channel Flow ◽  
Numerical Solutions ◽  
Flow Line ◽  
Water Pressure ◽  
Free Surface Flow ◽  
High Water ◽  
Surface Flow ◽  
Overburden Pressure ◽  
Subglacial Drainage

AbstractWe present a new model of subglacial drainage incorporating flow in a network of channels and a porous sheet, with water exchange between the two determined by pressure gradients. The sheet represents the average effect of many linked cavities, whilst the channels emerge from individual cavities that enlarge due to dissipation-induced melting. The model distinguishes cases when the water pressure drops to zero, in which case it allows for the drainage space to be only partially filled with water (free surface flow), and when the pressure reaches the ice overburden pressure, in which case it allows for uplift of the ice to whatever extent is needed to accommodate the water (flotation). Numerical solutions are found for a one-dimensional flow-line version of the model. The results capture typically observed or inferred features of subglacial drainage systems, including open channel flow at the ice margin, seasonal channel evolution, and high water pressures and uplift of the ice surface driven by rapid changes in water supply.

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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