Modelling of an Internal Wave Gravity Current Using Eulers Equations

2002 ◽  
Author(s):  
Deborah J. Wood
Keyword(s):  
Internal Wave ◽  
Wave Speed ◽  
Gravity Current ◽  
Laboratory Scale ◽  
Navier Stokes ◽  
Linear Wave ◽  
Field Observations ◽  
Medium Density ◽  
Sloping Bottom

In nature where thermoclines exist an internal wave may form, and if a sloping bottom is also present then a gravity current may occur. In this study we use a Navier-Stokes solver to solve Eulers equations to simulate the generation and evolution of such a wave. The thermoclines used in this study are similar to those seen in nature except scaled down to the laboratory scale used by some ongoing experiments. We find that the Navier-Stokes solver generates and evolves a wave similar to experimental observations. The head of the gravity current is dominated by medium density fluid with the thermocline thickness growing and becoming thickest at the centre of the head. Maximum velocities of approximately 0.5 of the linear wave speed are found which are similar to experimental and field observations.

scholarly journals Gravity currents and internal waves in a stratified fluid

2008 ◽  
Vol 616 ◽  
pp. 327-356 ◽  
Author(s):  
BRIAN L. WHITE ◽  
KARL R. HELFRICH
Keyword(s):  
Internal Waves ◽  
Wave Speed ◽  
Numerical Solutions ◽  
Gravity Current ◽  
Gravity Currents ◽  
Stratified Fluid ◽  
Front Speed ◽  
Long Wave ◽  
Conjugate State ◽  
Energy Conserving

A steady theory is presented for gravity currents propagating with constant speed into a stratified fluid with a general density profile. Solution curves for front speed versus height have an energy-conserving upper bound (the conjugate state) and a lower bound marked by the onset of upstream influence. The conjugate state is the largest-amplitude nonlinear internal wave supported by the ambient stratification, and in the limit of weak stratification approaches Benjamin's energy-conserving gravity current solution. When the front speed becomes critical with respect to linear long waves generated above the current, steady solutions cannot be calculated, implying upstream influence. For non-uniform stratification, the critical long-wave speed exceeds the ambient long-wave speed, and the critical-Froude-number condition appropriate for uniform stratification must be generalized. The theoretical results demonstrate a clear connection between internal waves and gravity currents. The steady theory is also compared with non-hydrostatic numerical solutions of the full lock release initial-value problem. Some solutions resemble classic gravity currents with no upstream disturbance, but others show long internal waves propagating ahead of the gravity current. Wave generation generally occurs when the stratification and current speed are such that the steady gravity current theory fails. Thus the steady theory is consistent with the occurrence of either wave-generating or steady gravity solutions to the dam-break problem. When the available potential energy of the dam is large enough, the numerical simulations approach the energy-conserving conjugate state. Existing laboratory experiments for intrusions and gravity currents produced by full-depth lock exchange flows over a range of stratification profiles show excellent agreement with the conjugate state solutions.

Internal-Wave-Driven Mixing: Global Geography and Budgets

2017 ◽  
Vol 47 (6) ◽  
pp. 1325-1345 ◽  
Author(s):  
Eric Kunze
Keyword(s):  
Internal Wave ◽  
Internal Tide ◽  
Internal Tides ◽  
Tidal Mixing ◽  
Linear Wave ◽  
Power Sources ◽  
The North ◽  
Dissipation Rates ◽  
Thickness Variability ◽  
Significant Pathway

AbstractInternal-wave-driven dissipation rates ε and diapycnal diffusivities K are inferred globally using a finescale parameterization based on vertical strain applied to ~30 000 hydrographic casts. Global dissipations are 2.0 ± 0.6 TW, consistent with internal wave power sources of 2.1 ± 0.7 TW from tides and wind. Vertically integrated dissipation rates vary by three to four orders of magnitude with elevated values over abrupt topography in the western Indian and Pacific as well as midocean slow spreading ridges, consistent with internal tide sources. But dependence on bottom forcing is much weaker than linear wave generation theory, pointing to horizontal dispersion by internal waves and relatively little local dissipation when forcing is strong. Stratified turbulent bottom boundary layer thickness variability is not consistent with OGCM parameterizations of tidal mixing. Average diffusivities K = (0.3–0.4) × 10−4 m2 s−1 depend only weakly on depth, indicating that ε = KN2/γ scales as N2 such that the bulk of the dissipation is in the pycnocline and less than 0.08-TW dissipation below 2000-m depth. Average diffusivities K approach 10−4 m2 s−1 in the bottom 500 meters above bottom (mab) in height above bottom coordinates with a 2000-m e-folding scale. Average dissipation rates ε are 10−9 W kg−1 within 500 mab then diminish to background deep values of 0.15 × 10−9 W kg−1 by 1000 mab. No incontrovertible support is found for high dissipation rates in Antarctic Circumpolar Currents or parametric subharmonic instability being a significant pathway to elevated dissipation rates for semidiurnal or diurnal internal tides equatorward of 28° and 14° latitudes, respectively, although elevated K is found about 30° latitude in the North and South Pacific.

scholarly journals Physical and numerical analysis of the front of a gravity current on a horizontal bottom

1998 ◽  
Vol 26 ◽  
pp. 289-295
Author(s):  
Mohamed Naaim ◽  
Thierry Pellarin
Keyword(s):  
Numerical Model ◽  
Stokes Equations ◽  
Gravity Current ◽  
Processing System ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Experimental Approaches ◽  
Two Parameters ◽  
Two Stages

In this paper, numerical and experimental approaches are applied to analyse the dynamics of the front of a gravity current. This study focused on two parameters: internal density and velocity fields. The salt concentration was determined by a potentiometric process. The internal velocities were determined using an optical device and an image-processing system. The structure of the head of the gravity current was analysed. Its density was measured and two stages of evolution were observed. This analysis allows us to coufirm the existence of two important stages. Forxf<xs, where the dynamics depend on the initial condition, the flow consists of a head and body and the front density is constant. Forxf>xs, we show that the density of the front decreases and evolves towards the Hallworth and others (1993) law. From a comparison between the experiments and the numerical model, we show that the numerical model, which is based on Navier–Stokes equations and on thek−Lturbulence model (whereLis the height of the gravity current), can predict well flow in the slump regime and in the inertia–buoyancy regime with smoothed results in the transition from the head to the body of the gravity current.

Energy balances for axisymmetric gravity currents in homogeneous and linearly stratified ambients

2008 ◽  
Vol 616 ◽  
pp. 303-326 ◽  
Author(s):  
MARIUS UNGARISH ◽  
HERBERT E. HUPPERT
Keyword(s):  
High Reynolds Number ◽  
Gravity Current ◽  
Water Model ◽  
Navier Stokes ◽  
Shallow Water Model ◽  
Fluid System ◽  
Dense Fluid ◽  
High Reynolds ◽  
Energy Balances ◽  
Two Fluid

We analyse the exchange of energy for an axisymmetric gravity current, released instantaneously from a lock, propagating over a horizontal boundary at high Reynolds number. The study is relevant to flow in either a wedge or a full circular geometry. Attention is focused on effects due to a linear stratification in the ambient. The investigation uses both a one-layer shallow-water model and Navier–Stokes finite-difference simulations. There is fair agreement between these two approaches for the energy changes of the dense fluid (the current). The stratification enhances the accumulation of potential energy in the ambient and reduces the energy decay (dissipation) of the two-fluid system. The total energy of the axisymmetric current decays considerably faster with distance of propagation than for the two-dimensional counterpart.

Numerical Simulation of Flows Past Partially-Submerged Horizontal Circular Cylinders in Free Surface Waves

2013 ◽  
Author(s):  
Hans Bihs ◽  
Muk Chen Ong
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Wave Theory ◽  
Circular Cylinders ◽  
Navier Stokes ◽  
Wave Forces ◽  
Linear Wave ◽  
Numerical Wave Tank ◽  
Linear Wave Theory ◽  
Free Surface Waves

Two-dimensional (2D) numerical simulations are performed to investigate the flows past partially-submerged circular cylinders in free surface waves. The 2D simulations are carried out by solving the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations with the k-ω turbulence model. The level set method is employed to model the free-surface waves. Validation studies of a numerical wave tank have been performed by comparing the numerical results with the analytical results obtained from the linear-wave theory. Wave forces on the partially-submerged cylinders have been calculated numerically and compared with the published theoretical and experimental data under regular-wave conditions. The free-surface elevations around the cylinders have been investigated and discussed.

scholarly journals Deep-water sediment wave formation: linear stability analysis of coupled flow/bed interaction

2011 ◽  
Vol 680 ◽  
pp. 435-458 ◽  
Author(s):  
L. LESSHAFFT ◽  
B. HALL ◽  
E. MEIBURG ◽  
B. KNELLER
Keyword(s):  
Boundary Layer ◽  
Stability Analysis ◽  
Internal Wave ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Stokes Equations ◽  
Wave Mode ◽  
Bottom Current ◽  
Field Observations ◽  
Sediment Bed

A linear stability analysis is carried out for the interaction of an erodible sediment bed with a sediment-laden, stratified flow above the bed, such as a turbidity or bottom current. The fluid motion is described by the full, two-dimensional Navier–Stokes equations in the Boussinesq approximation, while erosion is modelled as a diffusive flux of particles from the bed into the fluid. The stability analysis shows the existence of both Tollmien–Schlichting and internal wave modes in the stratified boundary layer. For the internal wave mode, the stratified boundary layer acts as a wave duct, whose height can be determined analytically from the Brunt–Väisälä frequency criterion. Consistent with this criterion, distinct unstable perturbation wavenumber regimes exist for the internal wave mode, which are associated with different numbers of pressure extrema in the wall-normal direction. For representative turbidity current parameters, the analysis predicts unstable wavelengths that are consistent with field observations. As a key condition for instability to occur, the base flow velocity boundary layer needs to be thinner than the corresponding concentration boundary layer. For most of the unstable wavenumber ranges, the phase relations between the sediment bed deformation and the associated wall shear stress and concentration perturbations are such that the sediment waves migrate in the upstream direction, which again is consistent with field observations.

scholarly journals Laminar and weakly turbulent oceanic gravity currents performing inertial oscillations

2011 ◽  
Vol 8 (5) ◽  
pp. 2001-2045
Author(s):  
A. Wirth
Keyword(s):  
Mathematical Models ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Gravity Current ◽  
Boussinesq Approximation ◽  
Turbulent Fluxes ◽  
Navier Stokes ◽  
Small Scale ◽  
Mean Flow ◽  
Inertial Oscillations

Abstract. The small scale dynamics of a weakly turbulent oceanic gravity current is determined. The gravity current considered is initially at rest and adjusts by performing inertial oscillations to a geostrophic mean flow. The dynamics is explored with a hierarchy of mathematical models. The most involved are the fully 3-D Navier-Stokes equations subject to the Boussinesq approximation. A 1-D and 0-D mathematical model of the same gravity current dynamics are systematically derived. Using this hierarchy and the numerical solutions of the mathematical models, the turbulent dynamics at the bottom and the interface is explored and their interaction investigated. Three different regimes of the small scale dynamics of the gravity current are identified, they are characterised by laminar flow, coherent roll vortices and turbulent dynamics with coherent streaks and bursts. The problem of the rectification of the turbulent fluxes, that is how to average out the fluctuations and calculate their average influence on the flow is considered. It is shown that two different regimes of friction are superposed, an Ekman friction applies to the average geostrophic flow and a linear friction, not influenced by rotation, to the inertial oscillations. The combination of the two makes the bulk friction non-local in time for the 0-D model. The implications of the results for parametrisations of the Ekman dynamics and the small scale turbulent fluxes in the planetary boundary layer are discussed.

On the Transversal Vibrations of a Conveyor Belt Using a String-Like Equation

2001 ◽  
Author(s):  
Gede Suweken ◽  
W. T. van Horssen
Keyword(s):  
Wave Speed ◽  
Dynamical Behavior ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Conveyor Belt ◽  
Linear Wave ◽  
Long Time ◽  
Vertical Vibrations ◽  
Transversal Vibrations ◽  
Initial Boundary Value

Abstract In this paper an initial-boundary value problem for a linear wave (string) equation is considered. This problem can be used as a simple model to describe the vertical vibrations of a conveyor belt, for which the velocity is small with respect to the wave speed. In this paper the belt is assumed to move with varying speed. Formal asymptotic approximations of the solutions are constructed to show the complicated dynamical behavior of the conveyor belt. It also will be shown that for this problem, the truncation method is not valid on long time scales.

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