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- Author or Editor: Qingping Zou x
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Abstract
Second-order analytical solutions are constructed for various long waves generated by a gravity wave train propagating over finite variable depth h(x) using a multiphase Wentzel–Kramers–Brillouin (WKB) method. It is found that, along with the well-known long wave, locked to the envelope of the wave train and traveling at the group velocity Cg
, a forced long wave and free long waves are induced by the depth variation in this region. The forced long wave depends on the depth derivatives hx
and hxx
and travels at Cg
, whereas the free long waves depend on h, hx
, and hxx
and travel in the opposite directions at 
Abstract
Second-order analytical solutions are constructed for various long waves generated by a gravity wave train propagating over finite variable depth h(x) using a multiphase Wentzel–Kramers–Brillouin (WKB) method. It is found that, along with the well-known long wave, locked to the envelope of the wave train and traveling at the group velocity Cg
, a forced long wave and free long waves are induced by the depth variation in this region. The forced long wave depends on the depth derivatives hx
and hxx
and travels at Cg
, whereas the free long waves depend on h, hx
, and hxx
and travel in the opposite directions at
Abstract
Based on the spectral eddy viscosity model of bottom boundary layers, the spectral representation of bottom friction and dissipation for irregular waves is reduced to an equivalent monochromatic wave representation. The representative wave amplitude and frequency are chosen so that the bottom velocity and bottom shear stress variances of the equivalent wave model are identical to those of the spectral model. Moreover, these variances have to satisfy the same relationship as those of a monochromatic wave with the same frequency. According to the wave bottom boundary layer theory, the ratio between bottom stress spectrum and bottom velocity spectrum has a frequency dependence of ω q , where the exponent q is a positive constant. The representative wave frequency and direction are obtained based on this power law, whereas in previous studies they were derived using a Taylor expansion of the ratio about a particular frequency or were proposed heuristically. Previous equivalent wave theories are therefore valid only for narrowbanded wave spectra. The present theory, however, is applicable to a wide variety of wave spectra including broadbanded and multimodal spectra.
Abstract
Based on the spectral eddy viscosity model of bottom boundary layers, the spectral representation of bottom friction and dissipation for irregular waves is reduced to an equivalent monochromatic wave representation. The representative wave amplitude and frequency are chosen so that the bottom velocity and bottom shear stress variances of the equivalent wave model are identical to those of the spectral model. Moreover, these variances have to satisfy the same relationship as those of a monochromatic wave with the same frequency. According to the wave bottom boundary layer theory, the ratio between bottom stress spectrum and bottom velocity spectrum has a frequency dependence of ω q , where the exponent q is a positive constant. The representative wave frequency and direction are obtained based on this power law, whereas in previous studies they were derived using a Taylor expansion of the ratio about a particular frequency or were proposed heuristically. Previous equivalent wave theories are therefore valid only for narrowbanded wave spectra. The present theory, however, is applicable to a wide variety of wave spectra including broadbanded and multimodal spectra.
Abstract
To calculate the effects of turbulent relaxation on oscillatory turbulent boundary layers, a viscoelastic term is added to an eddy viscosity model. The viscoelastic term parameterizes the lag of turbulent properties in response to imposed oscillatory shear and is proportional to the ratio between the timescales of eddy dissipation and of the oscillating flow. It is found that the turbulent relaxation plays an important role in the phase variations of velocity and shear stress with elevation, and that it decreases the friction factor and the phase lead of bed shear stress over free stream velocity.
To assess the effects of turbulent diffusion in this problem, the viscoelastic model is extended by further introducing a turbulent diffusion term in the model. The comparisons between these two models indicate that turbulent diffusion significantly reduces the magnitudes of shear stress and velocity perturbation in the outer region of the boundary layer. It is also found that the effects of turbulent relaxation and diffusion increase with increasing relative roughness. As a result, the analytical solutions demonstrate an overall improvement over the eddy viscosity model in predicting the observed temporal evolution of velocity and shear stress profiles; this improvement is more distinct for rough beds than smooth beds.
Abstract
To calculate the effects of turbulent relaxation on oscillatory turbulent boundary layers, a viscoelastic term is added to an eddy viscosity model. The viscoelastic term parameterizes the lag of turbulent properties in response to imposed oscillatory shear and is proportional to the ratio between the timescales of eddy dissipation and of the oscillating flow. It is found that the turbulent relaxation plays an important role in the phase variations of velocity and shear stress with elevation, and that it decreases the friction factor and the phase lead of bed shear stress over free stream velocity.
To assess the effects of turbulent diffusion in this problem, the viscoelastic model is extended by further introducing a turbulent diffusion term in the model. The comparisons between these two models indicate that turbulent diffusion significantly reduces the magnitudes of shear stress and velocity perturbation in the outer region of the boundary layer. It is also found that the effects of turbulent relaxation and diffusion increase with increasing relative roughness. As a result, the analytical solutions demonstrate an overall improvement over the eddy viscosity model in predicting the observed temporal evolution of velocity and shear stress profiles; this improvement is more distinct for rough beds than smooth beds.
Abstract
Wind and current effects on the evolution of a two-dimensional dispersive focusing wave group are investigated using a two-phase flow model. A Navier–Stokes solver is combined with the Smagorinsky subgrid-scale stress model and volume of fluid (VOF) air–water interface capturing scheme. Model predictions compare well with the experimental data with and without wind. It was found that the following and opposing winds shift the focus point downstream and upstream, respectively. The shift of focus point is mainly due to the action of wind-driven current instead of direct wind forcing. Under strong following/opposing wind forcing, there appears a slight increase/decrease of the extreme wave height at the focus point and an asymmetric/symmetric behavior in the wave focusing and defocusing processes. Under a weak following wind, however, the extreme wave height decreases with increasing wind speed because of the dominant effect of the wind-driven current over direct wind forcing. The vertical shear of the wind-driven current plays an important role in determining the location of and the extreme wave height at the focus point under wind actions. Furthermore, it was found that the thin surface layer current is a better representation of the wind-driven current for its role in wind influences on waves than the depth-uniform current used by previous studies. Airflow structure above a breaking wave group and its link to the energy flux from wind to wave as well as wind influence on breaking are also examined. The flow structure in the presence of a following wind is similar to that over a backward-facing step, while that in the presence of an opposing wind is similar to that over an airfoil at high angles of attack. Both primary and secondary vortices are observed over the breaking wave with and without wind of either direction. Airflow separates over the steep crest and causes a pressure drop in the lee of the crest. The resulting form drag may directly affect the extreme wave height. The wave breaking location and intensity are modified by the following and opposing wind in a different fashion.
Abstract
Wind and current effects on the evolution of a two-dimensional dispersive focusing wave group are investigated using a two-phase flow model. A Navier–Stokes solver is combined with the Smagorinsky subgrid-scale stress model and volume of fluid (VOF) air–water interface capturing scheme. Model predictions compare well with the experimental data with and without wind. It was found that the following and opposing winds shift the focus point downstream and upstream, respectively. The shift of focus point is mainly due to the action of wind-driven current instead of direct wind forcing. Under strong following/opposing wind forcing, there appears a slight increase/decrease of the extreme wave height at the focus point and an asymmetric/symmetric behavior in the wave focusing and defocusing processes. Under a weak following wind, however, the extreme wave height decreases with increasing wind speed because of the dominant effect of the wind-driven current over direct wind forcing. The vertical shear of the wind-driven current plays an important role in determining the location of and the extreme wave height at the focus point under wind actions. Furthermore, it was found that the thin surface layer current is a better representation of the wind-driven current for its role in wind influences on waves than the depth-uniform current used by previous studies. Airflow structure above a breaking wave group and its link to the energy flux from wind to wave as well as wind influence on breaking are also examined. The flow structure in the presence of a following wind is similar to that over a backward-facing step, while that in the presence of an opposing wind is similar to that over an airfoil at high angles of attack. Both primary and secondary vortices are observed over the breaking wave with and without wind of either direction. Airflow separates over the steep crest and causes a pressure drop in the lee of the crest. The resulting form drag may directly affect the extreme wave height. The wave breaking location and intensity are modified by the following and opposing wind in a different fashion.
Abstract
The one-dimensional reflection of a gravity wave at a discontinuity in bottom slope is calculated from a Green–Liouville (WKB) solution of the mild-slope equation.
Abstract
The one-dimensional reflection of a gravity wave at a discontinuity in bottom slope is calculated from a Green–Liouville (WKB) solution of the mild-slope equation.
Abstract
Theoretical solutions for the wave bottom boundary layer (WBL) over a sloping bed are compared with field measurements in the nearshore zone. The WBL theory is constructed using both viscoelastic–diffusion and conventional eddy viscosity turbulent closure models. The velocity solutions are then matched with those of the interior flow, given by Chu and Mei potential theory for surface gravity waves over a sloping bottom. The field measurements were obtained with a coherent Doppler profiler over a 2° bed slope. Results are presented for both flat and rippled bed conditions, the latter being characterized by low steepness, linear transition ripples. Close to the bed, the observed velocity profiles change rapidly in amplitude and phase relative to potential flow theory, indicating the presence of a wave boundary layer with a thickness of 3–6 cm. The observed velocity and shear stress profiles are in good agreement with the theory. The sloping bottom has significant effects on the vertical velocity, but not on the horizontal velocity and shear stress. Bottom roughness and friction velocity are estimated from optimizing the model–data comparisons. The friction velocities and wave friction factors are found to be consistent with values obtained from the momentum integral method and from the nearbed turbulence intensity, and with Tolman's semiempirical formulation.
Abstract
Theoretical solutions for the wave bottom boundary layer (WBL) over a sloping bed are compared with field measurements in the nearshore zone. The WBL theory is constructed using both viscoelastic–diffusion and conventional eddy viscosity turbulent closure models. The velocity solutions are then matched with those of the interior flow, given by Chu and Mei potential theory for surface gravity waves over a sloping bottom. The field measurements were obtained with a coherent Doppler profiler over a 2° bed slope. Results are presented for both flat and rippled bed conditions, the latter being characterized by low steepness, linear transition ripples. Close to the bed, the observed velocity profiles change rapidly in amplitude and phase relative to potential flow theory, indicating the presence of a wave boundary layer with a thickness of 3–6 cm. The observed velocity and shear stress profiles are in good agreement with the theory. The sloping bottom has significant effects on the vertical velocity, but not on the horizontal velocity and shear stress. Bottom roughness and friction velocity are estimated from optimizing the model–data comparisons. The friction velocities and wave friction factors are found to be consistent with values obtained from the momentum integral method and from the nearbed turbulence intensity, and with Tolman's semiempirical formulation.
Abstract
The theoretical model for group-forced infragravity (IG) waves in shallow water is not well established for nonbreaking conditions. In the present study, analytical solutions of the group-forced IG waves at O(β
1) (β
1 = h
x
/(Δkh), h
x
= bottom slope, Δk = group wavenumber, h = depth) in intermediate water and at
Abstract
The theoretical model for group-forced infragravity (IG) waves in shallow water is not well established for nonbreaking conditions. In the present study, analytical solutions of the group-forced IG waves at O(β
1) (β
1 = h
x
/(Δkh), h
x
= bottom slope, Δk = group wavenumber, h = depth) in intermediate water and at
