Depth-averaged Horizontal Velocity Research Articles

Effect of water depth on Kelvin–Helmholtz instability in a shallow water flow

It is well known that the interface between two regions of an incompressible ideal fluid flow moving in a relative motion is necessarily destabilized, regardless of the velocity difference’s strength. This phenomenon is the so-called Kelvin–Helmholtz instability. However, a large number of works demonstrated a surprising result that the instability is suppressed for shallow water flows; the interface is stabilized if the Froude number, defined by the velocity difference’s ratio to the gravity wave’s speed, is sufficiently large. In a limited way, these authors used the shallow water equations without the higher-order effect of the dispersive terms. Thus, this investigation aims at examining these higher-order dispersive effects to analyze the interface stability problem of tangential-velocity discontinuity in shallow water flows. In particular, we use the Green–Naghdi equations to introduce the dispersive terms related to the depth and the depth-averaged horizontal velocities of the fluid. We show that the interface stability depends on the Froude number (i.e., the velocity difference’s strength) and the water depth. A critical value of the Froude number to stabilize the interface is smaller than the case of no dispersive terms, and the flow in a deeper region is more stable than in a shallower one. We also consider the distribution of kinetic and potential energy to clarify a feature characteristic of a large class of instabilities in shallow water flow. The instability of flows is caused by the decrease in the kinetic energy during the perturbation of waves. This phenomenon is known as negative energy modes and plays a vital role in applying the model to industrial equipment. A conclusion is that the equipartition of energies occurs if and only if the velocity difference is zero and the water depth is shallow enough to ignore the dispersive terms.

Shock-Capturing Boussinesq Modelling of Broken Wave Characteristics Near a Vertical Seawall

Broken wave characteristics in front of a vertical seawall were modeled and studied using a shock-capturing Boussinesq wave model FUNWAVE-TVD. Validation with the experimental data confirmed the capability of FUNWAVE-TVD in predicting the wave characteristics via the shock-capturing method. Compared to the results obtained from the Boussinesq model coupled with an empirical breaking model, the advantage of the present shock-capturing model for the broken waves near a vertical seawall was clearly revealed. A preliminary investigation of the effects of the key parameters, such as the incident wave height, water level at the seawall, and seabed slope, on the wave kinematics (i.e., the root mean square of the surface fluctuations and depth-averaged horizontal velocity) near the seawall was then conducted through a series of numerical experiments. The numerical results indicate the incident wave height and the water depth at the seawall are the important parameters in determining the magnitude of the wave kinematics, while the effect of the seabed slope seems to be insignificant. The role of the breaking point locations is also highlighted in this study, in which case further breaking can reduce the wave kinematics significantly for the coastal structures predominately subjected to broken waves.

Should tsunami simulations include a nonzero initial horizontal velocity?

Tsunami propagation in the open ocean is most commonly modeled by solving the shallow water wave equations. These equations require initial conditions on sea surface height and depth-averaged horizontal particle velocity or, equivalently, horizontal momentum. While most modelers assume that initial velocity is zero, Y.T. Song and collaborators have argued for nonzero initial velocity, claiming that horizontal displacement of a sloping seafloor imparts significant horizontal momentum to the ocean. They show examples in which this effect increases the resulting tsunami height by a factor of two or more relative to models in which initial velocity is zero. We test this claim with a “full-physics” integrated dynamic rupture and tsunami model that couples the elastic response of the Earth to the linearized acoustic-gravitational response of a compressible ocean with gravity; the model self-consistently accounts for seismic waves in the solid Earth, acoustic waves in the ocean, and tsunamis (with dispersion at short wavelengths). Full-physics simulations of subduction zone megathrust ruptures and tsunamis in geometries with a sloping seafloor confirm that substantial horizontal momentum is imparted to the ocean. However, almost all of that initial momentum is carried away by ocean acoustic waves, with negligible momentum imparted to the tsunami. We also compare tsunami propagation in each simulation to that predicted by an equivalent shallow water wave simulation with varying assumptions regarding initial velocity. We find that the initial horizontal velocity conditions proposed by Song and collaborators consistently overestimate the tsunami amplitude and predict an inconsistent wave profile. Finally, we determine tsunami initial conditions that are rigorously consistent with our full-physics simulations by isolating the tsunami waves from ocean acoustic and seismic waves at some final time, and backpropagating the tsunami waves to their initial state by solving the adjoint problem. The resulting initial conditions have negligible horizontal velocity.Graphical abstractGRAPHICAL ABSTRACT CAPTION

The Vertical Mode Decomposition of Surface and Internal Tides in the Presence of a Free Surface and Arbitrary Topography

AbstractThe method of decomposing surface and internal tides determines the expression for internal tide energy, energy flux, and energy conversion. The de facto standard is to define surface tides as depth-averaged pressure and horizontal velocity and internal tides as the residuals. This decomposition, which is equivalent to projecting motion onto vertical modes that obey a rigid lid, is known to produce spurious energy conversion CS through movement of the free surface. Here, motion is instead projected onto modes that obey a linear, free-surface boundary condition. The free-surface modes are shown to obey a more complicated orthogonality condition than rigid-lid modes but are still straightforward to calculate numerically. The resulting decomposition (i) completely eliminates spurious energy conversion CS and (ii) leads to a more precise expression for topographic internal tide generation C, which now depends on horizontal gradients in the vertical structure of the surface tide. Numerical simulations and rough global estimates indicate that corrections to C are a maximum of a few percent. However, CS produces spurious energy flux divergences/convergences in the open ocean, which are the same order of magnitude [O(1–10) mW m−2] as open-ocean internal tide energy dissipation.

Observations of pockmark flow structure in Belfast Bay, Maine, Part 3: implications for sediment transport

Current observations and sediment characteristics acquired within and along the rim of two pockmarks in Belfast Bay, Maine, were used to characterize periods of sediment transport and to investigate conditions favorable to the settling of suspended sediment. Hourly averaged Shields parameters determined from horizontal current velocity profiles within the center of each pockmark never exceed the critical value (approximated with the theoretical model of Dade et al. 1992). However, Shields parameters estimated at the pockmark rims periodically exceed the critical value, consistent with conditions that support the onset of sediment transport and suspension. Below the rim in the near-center of each pockmark, depth-averaged vertical velocities were less than zero (downward) 60% and 55% of the time in the northern and southern pockmarks, and were often comparable to depth-averaged horizontal velocities. Along the rim, depth-averaged vertical velocities over the lower 8 m of the water column were primarily downward but much less than depth-averaged horizontal velocities indicating that suspended sediment may be moved to distant locations. Maximum grain sizes capable of remaining in suspension under terminal settling flow conditions (ranging 10–170 μm) were typically much greater than the observed median grain diameter (about 7 μm) at the bed. During upwelling flow within the pockmarks, and in the absence of flocculation, suspended sediment would not settle. The greater frequency of predicted periods of sediment transport along the rim of the southern pockmark is consistent with pockmark morphology in Belfast Bay, which transitions from more spherical to more elongated toward the south, suggesting near-bed sediment transport may contribute to post-formation pockmark evolution during typical conditions in Belfast Bay.

A solution of the conservation law form of the Serre equations

The nonlinear and weakly dispersive Serre equations contain higher-order dispersive terms. These include mixed spatial and temporal derivative flux terms which are difficult to handle numerically. These terms can be replaced by an alternative combination of equivalent temporal and spatial terms, so that the Serre equations can be written in conservation law form. The water depth and new conserved quantities are evolved using a second-order finite-volume scheme. The remaining primitive variable, the depth-averaged horizontal velocity, is obtained by solving a second-order elliptic equation using simple finite differences. Using an analytical solution and simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including flows with steep gradients. It is only slightly more computationally expensive than solving the shallow water wave equations. doi:10.1017/S144618111600002X

A SOLUTION OF THE CONSERVATION LAW FORM OF THE SERRE EQUATIONS

The nonlinear and weakly dispersive Serre equations contain higher-order dispersive terms. These include mixed spatial and temporal derivative flux terms which are difficult to handle numerically. These terms can be replaced by an alternative combination of equivalent temporal and spatial terms, so that the Serre equations can be written in conservation law form. The water depth and new conserved quantities are evolved using a second-order finite-volume scheme. The remaining primitive variable, the depth-averaged horizontal velocity, is obtained by solving a second-order elliptic equation using simple finite differences. Using an analytical solution and simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including flows with steep gradients. It is only slightly more computationally expensive than solving the shallow water wave equations.

Dispersion Effects on Tsunami Propagation in South China Sea

As the Manila Trench is becoming the most tsunami-hazardous, it is necessary to ascertain the wave height and arrival time in the South China Sea region through numerical simulation of tsunamis generated from potential earthquake source along the Manila subduction zone. The Okada model is employed to generate tsunami. The surface elevation and depth-averaged horizontal velocity at first 5 min, coming from the simulation of shallow water equations, are then interpolated in the weakly dispersive model (FUNWAVE) to calculate tsunami propagation and far-field impact. The characteristics of tsunami wave height distribution in South China Sea are analyzed for the assessment of tsunami hazard near coasts around South China Sea due to the hypothetical earthquakes with magnitude of [Formula: see text] and the worst case scenario ([Formula: see text]). The maximum wave height distribution computed by the Boussinesq equations is compared with that by the shallow water equations to investigate the dispersion effects on propagation of tsunami in South China Sea. It is found that the dispersion effects of the tsunami waves propagating in South China Sea are not significant if the earthquake magnitude is large enough.

Modeling Tsunami in South China Sea with Boussinesq Equations

To describe the dispersion effects, the high order Boussinesq (FUNWAVE) is adopted to simulate the tsunami propagating from Manila Trench to the South China Sea region. The Okada model is employed to generate tsunami, the surface elevation and depth-averaged horizontal velocity at t=5min computed by the shallow water equations are then interpolated to serve as the initial condition of the Boussinesq model. Different tsunami propagation scenarios with magnitude of Mw=8.0 are presented, and the arrival time and tsunami wave distribution are described and analyzed for the assessment of tsunami hazard near coasts around South China Sea. It is concluded that the dispersion effects in South China Sea is not significant.

Plunging solitary wave and its interaction with a slender cylinder on a sloping beach

Laboratory experiments were performed for solitary waves breaking on a constant slope. In the second set of experiments a vertical cylinder was installed on the slope, which was subject to the impact of a shoaling solitary wave. PIV was used to record the time history of free surface elevations, and temporal and spatial velocity variations in two fields of view (FOVs). A numerical model was also employed to simulate the three-dimensional wave–structure interaction problem. Filtered Navier–Stokes equations are solved numerically. The small scale eddies were modeled by the traditional Smagorinsky LES model and RNG LES model. The data-model comparisons were performed. The agreement is very good for the free surface elevations before breaking occurs. The laboratory wave appears to break slightly sooner than the numerically simulated wave. The comparisons for the normalized horizontal velocity profiles in the water column in both FOVs show excellent agreement. However, the magnitude of depth-averaged horizontal velocities for the numerical wave is roughly 10–15% higher than that of laboratory wave. The numerical model is used to illustrate the runup process of the solitary wave on the cylinder. The time history of wave force is also calculated and correlated to the free surface evolution.

Similarity of the velocity profile in geometrically distorted flow model

A method is presented to measure the depth-averaged horizontal velocity in distorted physical models, i.e., the depth-averaged horizontal velocity can be equivalent to a point velocity measured at approximately 0.6h below the water surface (h is the water depth). Further a scale for the Chézy coefficient in distorted models is deduced, and the velocity distributions in the vertical in prototype and distorted models are analyzed. By comparing data of the vertical variation of the horizontal velocity measured in a physical distorted model (with distortion ratio of 4.0) with prototype data, inaccuracies via the one-point method, the three-point method and the five-point method are analyzed. It is concluded that it is more accurate to measure the depth-averaged velocity via the three-point method than via the one-point method in the distorted model. What's more, the point velocity at 0.6h below the water surface approximates the depth-averaged velocity in the distorted physical model and the one-point method is proposed to be used in practice.

Derivation of a new asymptotic viscous shallow water model with dependence on depth

A new shallow water model with viscosity and dependence on depth is presented. It is derived from the Navier–Stokes equations, with anisotropic viscosity, in a shallow domain. Asymptotic analysis has been used as in our previous works [16–21]: a small non-dimensional parameter ε related to the depth is introduced and it is studied what happens when ε becomes small. New diffusion terms are revealed thanks to the rigorous asymptotic analysis. The bidimensional shallow water model obtained in this way considers the possibility of a non-constant bottom and calculates the depth-averaged horizontal velocity, but also the three components of velocity for all z (when the vorticity is not zero). The authors perform some numerical computations that confirm the new model is able to approximate the solutions of Navier–Stokes equations with dependence on z (reobtaining the same velocities profile), whereas the classic model only computes the average velocity.

A Boussinesq-type model incorporating random wave-breaking

A recent Boussinesq-type model is herein modified to account for breaking waves in shallow water. The model is based on a system of equations in terms of surface elevation and depth-averaged horizontal velocities, in two horizontal dimensions for fully dispersive and weakly nonlinear random waves over any finite water depth. The formulation involves five terms in each momentum equation, including the classical shallow-water equation terms, and only one frequency dispersion term. This work extends the model by including depth-induced wave-breaking in one horizontal dimension, based on the eddy viscosity and surface roller criteria. The modified model was applied to simulate the propagation and wave-breaking of regular and random waves using a simple explicit finite difference scheme. The simulation results were compared with experimental data and with results from one of the most widespread commercial Boussinesq wave models, indicating good agreement in most cases.

The kinematics and stability of solitary and cnoidal wave solutions of the Serre equations

The Serre equations are a pair of strongly nonlinear, weakly dispersive, Boussinesq-type partial differential equations. They model the evolution of the surface elevation and the depth-averaged horizontal velocity of an inviscid, irrotational, incompressible, shallow fluid. They admit a three-parameter family of cnoidal wave solutions with improved kinematics when compared to KdV theory. We examine their linear stability and establish that waves with sufficiently small amplitude/steepness are stable while waves with sufficiently large amplitude/steepness are unstable.

Boussinesq Model for Weakly Nonlinear Fully Dispersive Water Waves

In the present work a new Boussinesq dispersive wave propagation model is proposed. The model is based on a system of equations expressed in terms of the free-surface elevation and the depth-averaged horizontal velocities. The approach is developed for fully dispersive and weakly nonlinear irregular waves propagating over any constant water depth in two horizontal dimensions, but it can also be applied in mildly sloping beaches with considerable accuracy. The model in its two-dimensional formulation involves in total five terms in each momentum equation, including the classical shallow water terms and only one frequency dispersion term. The latter is expressed through convolution integrals, which are estimated using appropriate impulse functions. The formulation is fully explicit in space and thus no inversion is required for the numerical solution. The model is applied to simulate the propagation of regular and irregular waves using a simple explicit scheme of finite differences. Numerical integration of a convolution integral is also required. The results of the simulations are compared with experimental data, as well as with linear and nonlinear wave theory. The comparisons show that the method is capable of simulating weakly nonlinear dispersive wave propagation over finite constant or slowly diminishing water depth in a satisfactory way.

Twodee-2: A shallow layer model for dense gas dispersion on complex topography

twodee-2 is a F ortran 90 code based on previous code ( twodee). It is designed to solve the shallow water equations for fluid depth, depth-averaged horizontal velocities and depth-averaged fluid density. The shallow layer approach used by twodee-2 is a compromise between the complexity of CFD models and the simpler integral models. It can be used for forecasting gas dispersion near the ground and/or for hazard assessment over complex terrains. The inputs to the model are topography, terrain roughness, wind measurements from meteorological stations and gas flow rate from the ground sources. Optionally the model can be coupled with the output of a meteorological processor which generates a zero-divergence wind field incorporating terrain effects. Model outputs are gas concentration, depth-averaged velocity, averaged cloud thickness and dose. The model can be a useful tool for gas hazard assessment by evaluating where and when lethal concentrations for humans and animals can be reached.

Propagation and transformation of periodic nonlinear shallow-water waves in basins with selected breakwater systems

This paper describes the development of a Boussinesq three-equation model for simulating propagation and transformation of periodic nonlinear waves (cnoidal waves) in an arbitrary shallow-water basin. The Boussinesq equations in terms of depth-averaged horizontal velocities and free-surface elevation are solved numerically in a curvilinear coordinate system. An Euler’s predictor–corrector finite–difference algorithm is applied for numerical computation. The effects of irregular boundary, non-uniform water depth and coastal structures inside a basin are all included in the model simulation. A second-order cnoidal wave solution for the Boussinesq equations is used as an incident wave condition. A set of open boundary conditions is also applied to effectively transmit waves out of the computational domain. Model tests were conducted by simulating waves propagating past an isolated breakwater. The effect of variable depth was examined with modeling waves over an uneven bottom with convex ramp topography. The overall evolution of wave propagation, diffraction and reflection in coupled harbors with various layouts of inner and outer breakwaters was also studied. Data comparisons reveal that the simulated wave heights agree reasonably well with laboratory measurements, especially in the region of inner basin.

Turbulent boundary-layer effects on transient wave propagation in shallow water

The depth-integrated continuity and momentum equations developed by Liu & Orfila are extended to include the effects of turbulent bottom boundary layer. The eddy viscosity model is employed in the boundary layer, in which the eddy viscosity is assumed to be a power function of the vertical elevation from the bottom. The leading-order effects of the turbulent boundary layer appear as a convolution integral in the depth-integrated continuity equation because of the boundary-layer displacement. The bottom stress is also expressed as a convolution integral of the depth-averaged horizontal velocity. For simple harmonic progressive waves, the analytical expression for the phase shift between the bottom stress and the depth-averaged velocity is obtained. The analytical solutions for the solitary wave damping rate due to a turbulent boundary layer are also derived. Prandtl's one-seventh power law is described in detail as an example.

Viscous effects on transient long-wave propagation

Using a perturbation approach and the Boussinesq approximation, we derive sets of depth-integrated continuity and momentum equations for transient long-wave propagation with viscous effects included. The fluid motion is assumed to be essentially irrotational, except in the bottom boundary layer. The resulting governing equations are differential–integral equations in terms of the depth-averaged horizontal velocity (or velocity evaluated at certain depth) and the free-surface displacement, in which the viscous terms are represented by convolution integrals. We show that the present theory recovers the well-known approximate damping rates for simple harmonic progressive waves and for a solitary wave. The relationship between the bottom stress and the depth-averaged velocity is discussed.

Velocity predictions for shoaling and breaking waves with a Boussinesq-type model

In this paper, the performance of a 1-D Boussinesq model is evaluated against laboratory data for its ability to predict surf zone velocity moments. Wave evolution over a plane beach and a complex bathymetry both extending into the surf-zone is examined for six cases. For the plane beach, these comprise two cases, a spilling and a plunging cnoidal wave. For the complex bathymetry, these comprise four cases of longer and short wavelengths (spilling and plunging breakers), with regular and irregular periodicity. The model evaluation places emphasis on parameters of the wave field that could be used for the prediction of sediment transport; orbital velocity, undertow, velocity skewness, kurtosis and asymmetry. It is found that, despite an overestimation of the depth-averaged horizontal velocity in the regular waves cases, the predicted higher order velocity moments and undertow are in good agreement with the laboratory data. A bispectral analysis demonstrates that the nonlinear transfers of energy amongst the low order harmonics are well reproduced, but energy exchanges with the higher harmonics are less well predicted. As a result, the model handles velocity moments better in the shorter wave tests than in the long wave cases where triad interactions are stronger. Of the four parameters describing wave breaking, the model behaviour is most sensitive to the critical wave front slope φ B, especially with regard to velocity skewness and kurtosis predictions. It is also found that increasing the thickness of the surface roller for the case of plunging breakers improves the model's performance.