Depth-averaged Momentum Equations Research Articles

Subtidal to Supertidal Variability of Reynolds Stresses in a Midlatitude Stratified Inner Shelf

AbstractWe describe the spatiotemporal variability and vertical structure of turbulent Reynolds stresses (RSs) in a stratified inner shelf with an energetic internal wave climate. The RSs are estimated from direct measurements of velocity variance derived from bottom-mounted acoustic Doppler current profilers. We link the RSs to different physical processes, namely, internal bores, midwater shear instabilities within vertical shear events related to wind-driven subtidal along-shelf currents, and nonturbulent stresses related to incoming nonlinear internal wave (NLIW) trains. The typical RS magnitudes are O(0.01) Pa for background conditions, with diurnal pulses of O(0.1–1) Pa, and O(1) Pa for the NLIW stresses. A NLIW train is observed to produce a depth-averaged vertical stress divergence sufficient to accelerate water 20 cm s−1 in 1 h, suggesting NLIWs may also be important contributors to the depth-averaged momentum budget. The subtidal stresses show significant periodic variability and are O(0.1) Pa. Conditionally averaged velocity and RS profiles for northward/southward flow provide evidence for downgradient turbulent momentum fluxes, but also indicate departures from this expected regime. Estimates of the terms in the depth-averaged momentum equation suggest that the vertical divergence of the RSs are important terms in both the cross-shelf and along-shelf directions, with geostrophy also present at leading-order in the cross-shelf momentum balance. Among other conclusions, the results highlight that internal bores and shoaling NLIWs may also be important dynamical players in other inner shelves with energetic internal waves.

The structure of incipient coastal counter currents in South Portugal as indicator of their forcing agents

The alongshore subtidal water circulation along the South Portugal inner shelf is characterized by the temporal alternation of equatorward (i.e., broadly eastward) flows related to coastal upwelling processes and poleward (i.e., broadly westward) Coastal Counter Currents (CCCs). The objective of this study is to get insights about the main drivers of CCCs based on kinematic parameters describing the structure of the flow at the moment it changes direction. The parameters are derived from an extensive bottom-mounted ADCP dataset (16 deployments; 34,121 hourly records) collected at a single mooring (23 m water depth). Results show that the so-called incipient flows present contrasted general patterns whether they turn from equatorward to poleward or the opposite. Complementary observations at a nearby station indicate that these characteristics are spatially consistent along the studied area. Although 70% of CCCs are generated under favourable wind conditions (Levanter), these flows generally develop through the bed layer, in particular in summer. Hence, the Levanter wind - expected to promote flow setup through the surface layer - is not the main driver of CCCs in most cases. The general structure of incipient CCCs strongly suggests that the dominant force competing with the wind stress is an alongshore pressure gradient (APG). Furthermore, the maximum equatorward flow magnitude before CCCs setup is significantly correlated with the following (poleward) acceleration of incipient CCCs near the bed. Such relation is consistent with the development of CCCs due to the unbalance of an APG (produced during active upwelling) when wind relaxes. This process is further supported by an analysis of the depth-averaged momentum equation which suggests that the coastal circulation is mainly driven by linear dynamics in the region.

On the dynamics of quasi-steady gravity currents flowing up a slope

Quasi-steady gravity currents propagating first on a horizontal and then up a sloping boundary are investigated by means of theoretical analysis and laboratory experiments. The bottom slope ranged from 0.18 to 1 and full- and partial-depth configurations were considered. The developed theoretical model, using the depth averaged momentum equation, provides new physical insight into the importance of the different forces that act on the current and accounts for the gravity component along the slope, whose effect increases with both the slope angle and the ratio of current to ambient fluid depths. The height of the current decreases linearly with up-slope distance and the spatial rate of decrease, expressed by the current shape parameter is determined from the theory, using the measured up slope distance at which the current stops. This current shape parameter is found to depend on the slope only and it is not dependent on the current to ambient fluid depths. It can then be used to calculate the current velocity and the up-slope distance reached by the current. It is shown that the front velocity of all performed experiments is predicted by the theory indicating that the theory remains valid up to a slope equal to 1.

Lateral Shear Layer and Its Velocity Distribution of Flow in Rectangular Open Channels

The lateral velocity distribution of flow in the shear layer of open channel is required to many problems in river and eco-environment engineering, e.g. distribution of pollutant dispersion, sediment transport and bank erosion, and aquatic habitat. It is not well understood about how the velocity varies laterally in the wall boundary layer. This paper gives an analytical solution of lateral velocity distribution in a rectangular open channel based on the depth-averaged momentum equation proposed by Shiono & Knight. The obtained lateral velocity distributions in the wall shear layer are related to the two hydraulic parameters of lateral eddy viscosity (λ) and depth-averaged secondary flow (Γ) for given roughened channels. Preliminary relationships between the above two parameters and the aspect ratio of channel, B/H, are obtained from two sets of experimental data. The lateral width (δ) of the shear layer was investigated and found to relate to the λ and the bed friction factor (f), as described by Equation (26). This study indicates that the lateral shear layer near the wall can be very wide (δ/H = 14.6) for the extreme case (λ = 0.6 and f = 0.01).

Depth-averaged momentum equation for gravity currents with varying density: coefficient in pressure term

ABSTRACTGravity currents are often modelled by means of shallow water equations (SWEs). In these models, simplifications such as the consideration of a constant layer-averaged density are common. This note presents the complete and general derivation of a 2D depth-averaged momentum equation for gravity currents with density and velocity varying in the bed-normal direction. Special attention is given to the pressure term which is evaluated for constant, linear and exponential density profile. The shape of the density profile has implications for the momentum balance: the assumption of constant density leads to an overestimation of the driving force due to pressure gradient by a factor of 33% for linear density profile and up to 50% for an exponential profile. It also leads to an overestimation of celerity in numerical models based on traditional SWEs by factor of 22% and around 40% for linear end exponential density profiles respectively.

The Vertical Structure of Low-Frequency Motions in the Nearshore. Part II: Theory

AbstractField observations from a vertical stack of two-component current meters obtained from the 1994 Duck94 nearshore field experiment (presented in a companion paper by Lippmann, et al.) show significant vertical structure in energy, phase, and rotation of motions at low frequencies around 0.005 Hz. Low-frequency motions are typically modeled in the surfzone with the shallow-water (depth averaged) momentum equations that do not allow for any vertical structure. Following work from the shelf tidal community (Prandle), this study shows that the observations are consistent with the depth-varying momentum equations including shear stresses induced by a bottom boundary layer described by a constant eddy viscosity νt and bottom friction given by a constant drag coefficient and depth-averaged velocity . The bidirectional flow field is solved over arbitrary depth profiles varying only in the cross-shore direction h(x) in the presence of a vertically uniform mean alongshore current with cross-shore shear structure V(x). Analytic solutions are found to depend on νt, cd, h, ∂V/∂x, and the parameter , where σ and k are the radian frequency and alongshore wavenumber of the oscillating motion. Model behavior is explored by plotting solutions for a given parameter space as functions of the nondimensional depth H = λh and dimensionless friction parameter that combines the effects of bottom drag and vertical mixing. The behavioral changes in amplitude, phase shift, and rotational structure over the water column are qualitatively similar to those observed in the field.

A High-Order Numerical Method to Study Three-Dimensional Hydrodynamics in a Natural River

AbstractA high-order numerical method for three-dimensional hydrodynamics is presented. The present method applies high-order compact schemes in space and a Runge-Kutta scheme in time to solve the Reynolds-averaged Navier-Stokes equations with the k-ε turbulence model in an orthogonal curvilinear coordinate system. In addition, a two-dimensional equation is derived from the depth-averaged momentum equations to predict the water level. The proposed method is first validated by its application to simulate flow in a 180° curved laboratory flume. It is found that the simulated results agree with measurements and are better than those from SIMPLEC algorithm. Then the method is applied to study three-dimensional hydrodynamics in a natural river, and the simulated results are in accordance with measurements.

Momentum transport and bed shear stress distribution in a meandering bend: Experimental analysis in a laboratory flume

The paper concerns the mechanisms underlying the distribution of the bed shear stress in meandering bends. Literature indicates that cross-stream circulation strongly affects the redistribution of the downstream velocity, but the feedback between them is still poorly understood. The aim of this paper is to gain some insight into how the momentum transport by cross-stream circulation contributes to the bed shear stress redistribution. Experimental analysis, based on a detailed dataset collected in a large-amplitude meandering laboratory flume, is presented. From these data an evaluation is made of the terms in the depth-averaged momentum equations and the analysis is especially devoted to terms including the momentum transport by cross-sectional motion. Results confirm that these terms exert an important role in bed shear stress estimation and, thus, they have to be adequately included in the depth-averaged models. Based on measured data, an equation expressing the interrelation between the cross-sectional momentum and the downstream velocity is also introduced.

Improvement of the Lateral Distribution Method based on the mixing layer theory

The accurate prediction of depth-averaged streamwise velocity, boundary shear stress and lateral shear stress are important requisites for the estimation of the flow depth associated with flood events in compound river channels composed of main channel and floodplain. This engineering problem may be tackled through the analytical solution of the depth-averaged momentum equation. Under uniform flow, this solution relies on the calibration of three descriptors of the bottom friction, secondary currents and lateral shear stress. In this paper, the analytical solution materialized in the Lateral Distribution Method is revisited through the consideration of a new panel division. Accurate measurements of streamwise and spanwise velocities as well as of boundary shear stress are used to obtain new predictors of the coefficients describing the effects of bottom friction, secondary currents and lateral shear.The new lateral division of the compound channel into four panels is physically based on the mixing layer width, which is computed by an iterative procedure easily implemented in practical applications.

Hydrodynamic analysis of storm movement effects on runoff hydrographs and loop-rating curves of a V-shaped watershed

[1] The effects of storm movement on rainfall-runoff processes through a V-shaped watershed are investigated using a dynamic wave model based on shallow water equations. The governing equations are solved by a fourth-order accurate finite volume method to obtain accurate computational results. Once the numerical model is verified, a series of rainfall-runoff events on a V-shaped watershed are simulated. To investigate the contribution of each term in the depth-averaged momentum equation, a scale analysis, and quantitative assessment are conducted on the basis of the computational results. As a result, it is found that the speed and the direction of storm movement can play an important role in determining peak discharge and peak arrival time. In addition, the storm movement can generate loop in the stage-discharge relationship curve. Finally, it is revealed that the rating curve depends on the watershed characteristics rather than the rainfall conditions, at least in the idealized V-shaped watershed.

Velocity profile variations in granular flows with changing boundary conditions: insights from experiments

We present results of detailed velocity profile measurements in a large series of granular flow experiments in a dam-break setup. The inclination angle, bead size, and roughness of the running surface were varied. In all experiments, the downstream velocity profiles changed continuously from the head to the tail of the avalanches. On rough running surfaces, an inflection point developed in the velocity profiles. These velocity profiles cannot be modeled by the large class of constitutive laws which relate the shear stress to a power law of the strain rate. The velocity profile shape factor increased from the head to the tail of the avalanches. Its maximum value grew with increasing roughness of the running surface. We conclude that flow features such as velocity profiles are strongly influenced by the boundary condition at the running surface, which depends on the ratio of bead size to the typical roughness length of the surface. Furthermore, we show that varying velocity profile shape factors inside gravitationally driven finite-mass flows give rise to an additional term in the depth-averaged momentum equation, which is normally solved in the simulation software of hazardous geophysical flows. We therefore encourage time dependent velocity profile measurements inside hazardous geophysical flows, to learn about the importance of this “new” term in the mathematical modeling of these flows.

A numerical model of wave overtopping on seadikes

This paper describes the development of a numerical model for wave overtopping on seadikes. The model is based on the flux-conservative form of the nonlinear shallow water equations (NLSW) solved with a high order total variation diminishing (TVD), Roe-type scheme. The goal is to reliably predict the hydrodynamics of wave overtopping on the dike crest and along the inner slope, necessary for the breach modelling of seadikes. Besides the mean overtopping rate, the capability of simulating individual overtopping events is also required. It is shown theoretically that the effect of wave breaking through the drastic motion of surface rollers in the surfzone is not sufficiently described by the conventional nonlinear shallow water equations, neglecting wave setup from the mean water level and thus markedly reducing the model predictive capacity for wave overtopping. This is significantly improved by including an additional source term associated with the roller energy dissipation in the depth-averaged momentum equation. The developed model has been validated against four existing laboratory datasets of wave overtopping on dikes. The first two sets are to validate the roller term performance in improving the model prediction of wave overtopping of breaking waves. The last two sets are to test the model performance under more complex but realistic hydraulic and slope geometric conditions. The results confirm the merit of the supplemented roller term and also demonstrate that the model is robust and reliable for the prediction of wave overtopping on seadikes.

Model of Nearshore Waves and Wave-Induced Currents around a Detached Breakwater

A numerical model that combines a random wave transformation and a wave-induced current model was developed in order to predict the wave and current fields around a detached breakwater. The wave field was determined using the EBED model, as reported by Mase in 2001, with a modified energy dissipation term. The surface roller associated with wave breaking was modeled based on a modification of the equations in works by Dally and Brown, and Larson and Kraus, in which the term for the roller energy flux in the alongshore direction was added to the energy balance equation. The nearshore currents and water elevation were determined from the continuity equation together with the depth-averaged momentum equations. The model was validated by three unique high-quality data sets obtained during experiments on detached breakwaters in the large-scale sediment transport facility basin at the Coastal and Hydraulics Laboratory in Vicksburg, Miss. The calculated significant wave height and longshore current were in good agreement with these measurements, whereas the cross-shore current was underestimated because undertow processes were not included in the modeling (depth-averaged equations employed). The calculated wave setup was somewhat overestimated; however, the absolute differences between the calculations and measurements were overall relatively small.

A numerical model of nearshore waves, currents, and sediment transport

A two-dimensional numerical model of nearshore waves, currents, and sediment transport was developed. The multi-directional random wave transformation model formulated by Mase [Mase, H., 2001. Multi-directional random wave transformation model based on energy balance equation. Coastal Engineering Journal 43(4), 317-337.] based on an energy balance equation was employed with an improved description of the energy dissipation due to breaking. In order to describe surface roller effects on the momentum transport, an energy balance equation for the roller was included following Dally and Brown [Dally, W.R., Brown, C.A., 1995. A modeling investigation of the breaking wave roller with application to cross-shore currents. Journal of Geophysical Research 100(C12), 24873-24883.]. Nearshore currents and mean water elevation were modeled using the continuity equation together with the depth-averaged momentum equations. Sediment transport rates in the offshore and surf zone were computed using the sediment transport formulation proposed by Camenen and Larson [Camenen, B., Larson, M., 2005. A general formula for non-cohesive bed load sediment transport. Estuarine, Coastal and Shelf Science 63, 249-260.; Camenen, B., Larson, M., 2007. A unified sediment transport formulation for coastal inlet application. Technical report ERDC/CHL CR-07-1, US Army Engineer Research and Development Center, Vicksburg, MS.; Camenen, B., Larson, M., 2008. A general formula for non-cohesive suspended sediment transport. Journal of Coastal Research 24(3), 615-627.] together with the advection–diffusion equation, whereas the swash zone transport rate was obtained from the formulas derived by Larson and Wamsley [Larson, M., Wamsley, T.V., 2007. A formula for longshore sediment transport in the swash. Proceedings Coastal Sediments '07, ASCE, New Orleans, pp. 1924–1937.]. Three high-quality data sets from the LSTF experimental facility at the Coastal and Hydraulics Laboratory in Vicksburg, USA, were used to evaluate the predictive capability of the model. Good agreement between computations and measurements was obtained with regard to the cross-shore variation in waves, currents, mean water elevation, and sediment transport in the nearshore and swash zone. The present model will form the basis for predicting morphological evolution in the nearshore due to waves and currents with special focus on coastal structures.

A 3D non-linear k– ε turbulent model for prediction of flow and mass transport in channel with vegetation

The results from a 3D non-linear k– ε turbulence model with vegetation are presented to investigate the flow structure, the velocity distribution and mass transport process in a straight compound open channel and a curved open channel. The 3D numerical model for calculating flow is set up in non-orthogonal curvilinear coordinates in order to calculate the complex boundary channel. The finite volume method is used to disperse the governing equations and the SIMPLEC algorithm is applied to acquire the coupling of velocity and pressure. The non-linear k– ε turbulent model has good useful value because of taking into account the anisotropy and not increasing the computational time. The water level of this model is determined from 2D Poisson equation derived from 2D depth-averaged momentum equations. For concentration simulation, an expression for dispersion through vegetation is derived in the present work for the mixing due to flow over vegetation. The simulated results are in good agreement with available experimental data, which indicates that the developed 3D model can predict the flow structure and mass transport in the open channel with vegetation.

Horizontal Sigma Coordinate System for River-Flow Models

Because of complicated channel geometry, horizontal coordinates are important in accurate and efficient computations of river flow. We propose a simplified horizontal coordinate system for river-flow simulation, referred to here as a horizontal sigma coordinate system that uses boundary-fitted grids along river channels like a generalized curvilinear coordinate system, without losing the simplicity of Cartesian or orthogonal curvilinear coordinate systems. The proposed approach is based on a sigma coordinate system. The depth-averaged continuity and momentum equations for horizontal two-dimensional fluid motion in the Cartesian and orthogonal curvilinear coordinate systems are transformed to those in the horizontal sigma coordinate system. The form of the transformed continuity and momentum equations in the proposed system is a fairly minor modification of the form of the equations for the Cartesian and orthogonal curvilinear coordinate systems. Computational results for a straight open-channel flow with ...

Three-dimensional simulation of meandering river based on 3-D RNG κ-ɛ turbulence model

A 3-D numerical model for calculating flow in non-curvilinear coordinates was established in this article. The flow was simulated by solving the full Reynolds-averaged Navier-Stokes equations with the RNG κ–ɛ turbulence model. In the horizontal x– y-plane, a boundary-fitted curvilinear co-ordinate system was adopted, while in the vertical direction, a σco-ordinate transformation was used to represent the free surface and bed topography. The water level was determined by solving the 2-D Poisson equation derived from 2-D depth averaged momentum equations. The finite-volume method was used to discretize the equations and the SIMPLEC algorithm was applied to acquire the coupling of velocity and pressure. This model was applied to simulate the meandering channels and natural rivers, and the water levels and the velocities for all sections were given. By contrasting and analyzing, the agreement with measurements is generally good. The feasibility studies of simulating flow of the natural river have been conducted to demonstrate its applicability to hydraulic engineering research.

Viscosity-excluded formulations of 2-D river flow with a new wetting and drying algorithm

An efficient method for simulating 2-D river flow is developed in which horizontal turbulent shears are omitted from the 2-D depth-averaged momentum equations. It is shown that a pseudo-viscosity can be reproduced to take into account the lost shear action, by incorporating the vertically integrated continuity equation to the momentum equations and transforming the latter into a discrete integral form. To simulate river flows with wet and dry areas, negative water depths are allowed when solving the continuity equation. The concept of negative water depth enables us to track flow boundaries with about the same accuracy but much less effort as compared with traditional numerical methods. An optimal threshold value defining dry areas is first obtained by one-dimensional theoretical analysis and then sought by trial-and-error for two-dimensional flow simulation with tolerable node-to-node spurious oscillations, while mass is best conserved. Numerical solutions using the new procedure are compared with the one-dimensional benchmark solution of the Saint Venant equations and the experimental data from a two-stage channel. Robustness of the present approach is also tested through the study of water flow in a natural river and a hypothetical channel with several bumps.

Numerical model simulations of continental shelf flows off northern California

Abstract The three-dimensional circulation on the continental shelf off northern California in the wind events and shelf transport (WEST) experiment region during summer 2001 is studied using the primitive equation regional ocean modeling system (ROMS). The simulations are performed with realistic topography and initial stratification in a limited-area domain with a high-resolution grid. Forcing consists of measured wind-stress and heat flux values obtained from a WEST surface buoy. The general response shows a southward coastal upwelling jet of up to 1 m s - 1 and a weakening or reversal of currents inshore of the jet when upwelling winds relax. Model results are compared to WEST moored velocity and temperature measurements at five locations, to CODAR surface current observations between Pt. Reyes and Bodega Bay, and to hydrographic measurements along shipboard survey lines. The model performs reasonably well, with the highest depth-averaged velocity correlation (0.81) at the inshore mooring (40 m water depth) and lowest correlation (0.68) at the mid-depth mooring (90 m depth). The model shows generally stronger velocities than those observed, especially at the inshore moorings, and a lack in complete reversal of southward velocities observed when upwelling winds relax. The comparison of surface velocities with CODAR measurements shows good agreement of the mean and the dominant mode of variability. The hydrography compares closely at the southern and northern edges of the survey region (correlation coefficients between 0.90 and 0.97), with weaker correlations at the three interior survey lines (correlation coefficients between 0.44 and 0.76). Mean model fields over the summer upwelling period show slight coastal jet separation off Pt. Arena and significant separation off Pt. Reyes. The cape regions also experience relatively strong bottom velocities and nonlinearity in the surface flow. Across-shelf velocity sections examined along the shelf reveal a double jet structure that appears just north of Bodega Bay and shows the offshore jet strengthening to the south. We examine the dynamics during an upwelling and subsequent relaxation event in May 2001 in which the WEST measurements show evidence of a strong flow response. The alongshelf variability in the upwelling and relaxation response introduced by Pt. Reyes is evident. Analysis of term balances from the depth-averaged momentum equations helps to clarify the event dynamics in different regions over the shelf. A clear pattern in the nonlinear advection term is due to the spatial acceleration of the southward jet around the capes of Pt. Arena and Pt. Reyes during upwelling. Results from a three-dimensional Lagrangian analysis of water parcel displacement show significant southward displacement in the coastal jet region, including a strong signal from the double jet. Alongshelf variability in parcel displacements and upwelling source waters due to the presence of Pt. Arena and Pt. Reyes is also apparent from the Lagrangian fields. A cyclonic eddy-like recirculation feature offshore of Pt. Arena prior to the upwelling event causes large patches of onshore-displaced parcels. Additionally, across-shelf variability in the response of water parcels along the D line includes decreased vertical displacement and increased alongshelf displacement in the offshore direction.

Meshing Requirements for Tidal Modeling in the Western North Atlantic

This paper presents an a posteriori approach to unstructured mesh generation via a localized truncation error analysis and applies it to the Western North Atlantic Tidal (WNAT) model domain. The WNAT model domain encompasses the Gulf of Mexico, the Caribbean Sea, and the North Atlantic Ocean east to the 60°W Meridian. Herein, we pay particular attention to the area surrounding the Bahamas. A bathymetric data set with fine resolution is employed in seven separate linear, harmonic simulations of shallow water tidal flow for seven different tidal-forcing constituents. Each set of simulation results is used to perform a truncation error analysis of a linear, harmonic form of the depth-averaged momentum equations for each of the seven different tidal-forcing frequencies. Truncation error is then driven to a more uniform, domain-wide value by solving for local node spacing requirements. The process is built upon successful research aiming to produce unstructured grids for large-scale domains that can be used in the accurate and efficient modeling of shallow water flow. The methodology described herein can also be transferred to other modeling applications.