Near-critical hydraulic flows in two-layer fluids

2007 ◽  
Vol 575 ◽  
pp. 187-219 ◽  
Author(s):  
ALFRED KLUWICK ◽  
STEFAN SCHEICHL ◽  
EDWARD A. COX
Keyword(s):  
Wave Speed ◽  
Evolution Equations ◽  
Fluid Layer ◽  
Bottom Topography ◽  
Propagation Speed ◽  
Internal Layer ◽  
Hydraulic Jumps ◽  
Weakly Nonlinear ◽  
Mixed Nonlinearity ◽  
Density Ratios

This paper deals with the propagation of nearly resonant gravity waves in two-layer flows over a bottom topography assuming that both fluids are incompressible and inviscid. Evolution equations are derived for weakly nonlinear surface-layer and internal-layer waves in the hydraulic limit of infinite wavelength. Special emphasis is placed on the flow regime where the quadratic nonlinear parameter associated with internal-layer waves is small or vanishes. For example, this is the case for all possible density ratios if the velocities in both layers are equal and if the interface height is close to one-half the total fluid-layer height. The waves then exhibit so-called mixed nonlinearity leading in turn to the formation of positive and negative hydraulic jumps. Considerations based on a model equation for the internal dissipative–dispersive structure of hydraulic jumps indicate that the admissibility of discontinuities in this regime depends strongly on the relative magnitudes of dispersion and dissipation. Surprisingly, these admissible hydraulic jumps may violate the wave-speed-ordering relationship which requires that the upstream wave speed does not exceed the propagation speed of the discontinuity. An important example is provided by the inviscid hydraulic jump, which has been known for some time, although its non-classical nature, in that it transmits rather than absorbs waves, has apparently not been recognized before.

scholarly journals Properties of large-amplitude internal waves

1999 ◽  
Vol 380 ◽  
pp. 257-278 ◽  
Author(s):  
JOHN GRUE ◽  
ATLE JENSEN ◽  
PER-OLAV RUSÅS ◽  
J. KRISTIAN SVEEN
Keyword(s):  
Fresh Water ◽  
Solitary Waves ◽  
Particle Tracking Velocimetry ◽  
Wave Speed ◽  
Propagation Speed ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Nonlinear Interface ◽  
The Waves ◽  
Theoretical Results

Properties of solitary waves propagating in a two-layer fluid are investigated comparing experiments and theory. In the experiments the velocity field induced by the waves, the propagation speed and the wave shape are quite accurately measured using particle tracking velocimetry (PTV) and image analysis. The experiments are calibrated with a layer of fresh water above a layer of brine. The depth of the brine is 4.13 times the depth of the fresh water. Theoretical results are given for this depth ratio, and, in addition, in a few examples for larger ratios, up to 100[ratio ]1. The wave amplitudes in the experiments range from a small value up to almost maximal amplitude. The thickness of the pycnocline is in the range of approximately 0.13–0.26 times the depth of the thinner layer. Solitary waves are generated by releasing a volume of fresh water trapped behind a gate. By careful adjustment of the length and depth of the initial volume we always generate a single solitary wave, even for very large volumes. The experiments are very repeatable and the recording technique is very accurate. The error in the measured velocities non-dimensionalized by the linear long wave speed is less than about 7–8% in all cases. The experiments are compared with a fully nonlinear interface model and weakly nonlinear Korteweg–de Vries (KdV) theory. The fully nonlinear model compares excellently with the experiments for all quantities measured. This is true for the whole amplitude range, even for a pycnocline which is not very sharp. The KdV theory is relevant for small wave amplitude but exhibit a systematic deviation from the experiments and the fully nonlinear theory for wave amplitudes exceeding about 0.4 times the depth of the thinner layer. In the experiments with the largest waves, rolls develop behind the maximal displacement of the wave due to the Kelvin–Helmholtz instability. The recordings enable evaluation of the local Richardson number due to the flow in the pycnocline. We find that stability or instability of the flow occurs in approximate agreement with the theorem of Miles and Howard.

On the theory of electrohydrodynamically driven capillary jets

1997 ◽  
Vol 335 ◽  
pp. 165-188 ◽  
Author(s):  
ALFONSO M. GAÑÁN-CALVO
Keyword(s):  
Surface Charge ◽  
Electric Fields ◽  
Gas Flow ◽  
Wave Speed ◽  
Propagation Speed ◽  
Field Equations ◽  
Steady Regime ◽  
One Dimensional ◽  
Relaxation Effects ◽  
Capillary Jets

Electrohydrodynamically (EHD) driven capillary jets are analysed in this work in the parametrical limit of negligible charge relaxation effects, i.e. when the electric relaxation time of the liquid is small compared to the hydrodynamic times. This regime can be found in the electrospraying of liquids when Taylor's charged capillary jets are formed in a steady regime. A quasi-one-dimensional EHD model comprising temporal balance equations of mass, momentum, charge, the capillary balance across the surface, and the inner and outer electric fields equations is presented. The steady forms of the temporal equations take into account surface charge convection as well as Ohmic bulk conduction, inner and outer electric field equations, momentum and pressure balances. Other existing models are also compared. The propagation speed of surface disturbances is obtained using classical techniques. It is shown here that, in contrast with previous models, surface charge convection provokes a difference between the upstream and the downstream wave speed values, the upstream wave speed, to some extent, being delayed. Subcritical, supercritical and convectively unstable regions are then identified. The supercritical nature of the microjets emitted from Taylor's cones is highlighted, and the point where the jet switches from a stable to a convectively unstable regime (i.e. where the propagation speed of perturbations become zero) is identified. The electric current carried by those jets is an eigenvalue of the problem, almost independent of the boundary conditions downstream, in an analogous way to the gas flow in convergent–divergent nozzles exiting into very low pressure. The EHD model is applied to an experiment and the relevant physical quantities of the phenomenon are obtained. The EHD hypotheses of the model are then checked and confirmed within the limits of the one-dimensional assumptions.

PATTERNS, DEFECTS, AND EVOLUTION OF BÉNARD–MARANGONI CELLS

1996 ◽  
Vol 06 (09) ◽  
pp. 1665-1671 ◽  
Author(s):  
J. BRAGARD ◽  
J. PONTES ◽  
M.G. VELARDE
Keyword(s):  
Fluid Layer ◽  
Ambient Air ◽  
Finite Geometry ◽  
Amplitude Equations ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Numerical Exploration ◽  
Ginzburg Landau Equations ◽  
Rigid Plane ◽  
Selection Of

We consider a thin fluid layer of infinite horizontal extent, confined below by a rigid plane and open above to the ambient air, with surface tension linearly depending on the temperature. The fluid is heated from below. First we obtain the weakly nonlinear amplitude equations in specific spatial directions. The procedure yields a set of generalized Ginzburg–Landau equations. Then we proceed to the numerical exploration of the solutions of these equations in finite geometry, hence to the selection of cells as a result of competition between the possible different modes of convection.

W-shape bright and several other solutions to the (3+1)-dimensional nonlinear evolution equations

2021 ◽  
pp. 2150468
Author(s):  
Youssoufa Saliou ◽  
Souleymanou Abbagari ◽  
Alphonse Houwe ◽  
M. S. Osman ◽  
Doka Serge Yamigno ◽  
...  
Keyword(s):  
Acoustic Waves ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Singular Function ◽  
Ion Acoustic Waves ◽  
Weakly Nonlinear ◽  
Quantum Electron ◽  
Ion Acoustic ◽  
Trigonometric Function Solutions

By employing the Modified Sardar Sub-Equation Method (MSEM), several solitons such as W-shape bright, dark solitons, trigonometric function solutions and singular function solutions have been obtained in two famous nonlinear evolution equations which are used to describe waves in quantum electron–positron–ion magnetoplasmas and weakly nonlinear ion-acoustic waves in a plasma. These models are the (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov (NLEQZK) equation and the (3+1)-dimensional nonlinear modified Zakharov–Kuznetsov (NLmZK) equation, respectively. Comparing the obtained results with Refs. 32–34 and Refs. 43–46, additional soliton-like solutions have been retrieved and will be useful in future to explain the interaction between lower nonlinear ion-acoustic waves and the parameters of the MSEM and the obtained figures will have more physical explanation.

Across-shore Propagation of Subthermocline Eddies in the California Current System

2021 ◽  
Keyword(s):  
Wave Speed ◽  
High Salinity ◽  
Current System ◽  
California Current ◽  
Propagation Speed ◽  
Underwater Glider ◽  
Salt Balance ◽  
Sea Level Anomaly ◽  
California Current System ◽  
Baroclinic Rossby Wave

AbstractThough subthermocline eddies (STEs) have often been observed in the world oceans, characteristics of STEs such as their patterns of generation and propagation are less understood. Here, the across-shore propagation of STEs in the California Current System (CCS) is observed and described using 13 years of sustained coastal glider measurements on three glider transect lines off central and southern California as part of the California Underwater Glider Network (CUGN). The across-shore propagation speed of anticyclonic STEs is estimated as 1.35-1.49 ± 0.33 cm s−1 over the three transects, Line 66.7, Line 80.0, and Line 90.0, close to the westward long first baroclinic Rossby wave speed in the region. Anticyclonic STEs are found with high salinity, high temperature, and low dissolved oxygen anomalies in their cores, consistent with transporting California Undercurrent water from the coast to offshore. Comparisons to satellite sea-level anomaly indicate that STEs are only weakly correlated to a sea surface height expression. The observations suggest that STEs are important for the salt balance and mixing of water masses across-shore in the CCS.

Dissipative effects on the resonant flow of a stratified fluid over topography

1988 ◽  
Vol 192 ◽  
pp. 287-312 ◽  
Author(s):  
N. F. Smyth
Keyword(s):  
Bottom Topography ◽  
Stratified Fluid ◽  
Flow Speed ◽  
Resonant Condition ◽  
Weakly Nonlinear ◽  
Dissipative Effects ◽  
Long Wave ◽  
Near Resonance ◽  
Wave Limit ◽  
Wave Modes

The effect of dissipation on the flow of a stratified fluid over topography is considered in the weakly nonlinear, long-wave limit for the case when the flow is near resonance, i.e. the basic flow speed is close to a linear long-wave speed for one of the long-wave modes. The two types of dissipation considered are the dissipation due to viscosity acting in boundary layers and/or interfaces and the dissipation due to viscosity acting in the fluid as a whole. The effect of changing bottom topography on the flow produced by a force moving at a resonant velocity is also considered. In this case, the resonant condition is that the force velocity is close to a linear long-wave velocity for one of the long-wave modes. It is found that in most cases, these extra effects result in the formation of a steady state, in contrast to the flow without these effects, which remains unsteady for all time. The flow resulting under the action of boundary-layer dissipation is compared with recent experimental results.

Hydraulic jumps in a channel

2009 ◽  
Vol 618 ◽  
pp. 71-87 ◽  
Author(s):  
DANIEL BONN ◽  
ANDERS ANDERSEN ◽  
TOMAS BOHR
Keyword(s):  
Boundary Layer ◽  
Flow Rate ◽  
Eddy Viscosity ◽  
Fluid Layer ◽  
Momentum Conservation ◽  
Opening Angle ◽  
Normal Velocity ◽  
Hydraulic Jumps ◽  
Mixing Length ◽  
Critical Height

We present a study of hydraulic jumps with flow predominantly in one direction, created either by confining the flow to a narrow channel with parallel walls or by providing an inflow in the form of a narrow sheet. In the channel flow, we find a linear height profile upstream of the jump as expected for a supercritical one-dimensional boundary layer flow, but we find that the surface slope is up to an order of magnitude larger than expected and independent of flow rate. We explain this as an effect of turbulent fluctuations creating an enhanced eddy viscosity, and we model the results in terms of Prandtl's mixing-length theory with a mixing length that is proportional to the height of the fluid layer. Using averaged boundary-layer equations, taking into account the friction with the channel walls and the eddy viscosity, the flow both upstream and downstream of the jump can be understood. For the downstream subcritical flow, we assume that the critical height is attained close to the channel outlet. We use mass and momentum conservation to determine the position of the jump and obtain an estimate which is in rough agreement with our experiment. We show that the averaging method with a varying velocity profile allows for computation of the flow-structure through the jump and predicts a separation vortex behind the jump, something which is not clearly seen experimentally, probably owing to turbulence. In the sheet flow, we find that the jump has the shape of a rhombus with sharply defined oblique shocks. The experiment shows that the variation of the opening angle of the rhombus with flow rate is determined by the condition that the normal velocity at the jump is constant.

scholarly journals Hamiltonian description of wave dynamics in nonequilibrium media

1994 ◽  
Vol 1 (4) ◽  
pp. 234-248 ◽  
Author(s):  
N. N. Romanova
Keyword(s):  
Nonlinear Wave ◽  
Evolution Equations ◽  
Normal Modes ◽  
Weak Nonlinearity ◽  
Hamiltonian Description ◽  
Weakly Nonlinear ◽  
Wave Dynamics ◽  
Canonical Variables ◽  
Stable Points ◽  
Stable Area

Abstract. We consider Hamiltonian description of weakly nonlinear wave dynamics in unstable and nonequilibrium media. We construct the appropriate canonical variables in the whole wavenumber space. The essentially new element is the construction of canonical variables in a vicinity of marginally stable points where two normal modes coalesce. The commonly used normal variables are not appropriate in this domain. The mater is that the approximation of weak nonlinearity breaks down when the dynamical system is written in terms of these variables. In this case we introduce the canonical variables based on the linear combination of modes belonging to the two different branches of dispersion curve. As an example of one of the possible applications of presented results the evolution equations for weakly nonlinear wave packets in the marginally stable area are derived. These equations cannot be derived if we deal with the commonly used normal variables.

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