Shallow Water Flow Research Articles

Shallow flow over a bed with a lateral change of roughness

River beds frequently exhibit a lateral variation of roughness. For example, in the case of an overflowing river, the main channel has a smoother topography compared to the adjacent floodplains where vegetation and land occupation yield an important hydraulic roughness. The lateral difference in roughness can induce a high lateral velocity gradient within the river cross- section that gives birth to a mixing layer. This mixing layer leads to fluid and momentum transfers between the two adjacent beds. To understand such mix- ing processes in rivers is important for predicting stage-discharge relationships and the velocity distribution within the cross-section. In order to address these issues in the context of a shallow water flow with a water depth h of the same order as the roughness elements of the bed, experiments were undertaken in a 26 m long and 1.1 m wide glass-walled open-channel flume. One half-side of the bed was covered with an array of cubes of height k arranged in a square configuration, the other side with smooth glass. Three different levels of cube submergence h/k were examined (h/k = 0.8, 1.5 and 2). The experiments and measurements were designed to yield the flow in the complete volume of the interstices across the cube array. To achieve this, 2C-3D linear-scanning PIV measurements with zero-parallax optics were developed and set up. The mea- surements revealed the complexity of the flow structure around the interface between the rough and smooth beds. The results show that the ability of the mixing layer to exchange momentum is highly dependent on the level of the cube submergence h/k.

An Implicit Scheme for Shallow Water Flow with Wet Dry Interface

A finite difference implicit scheme is presented in this paper for solution of the shallow water equations in one dimensional (1D) form. The present model has many advantages like, handling of discontinuous and complex bed topography, satisfying C-property (preservation of motionless water surface over a wet or dry bed) and capability of handling large value of temporal step etc. Another very important feature of the present model is that, no special treatment of the source vector of the governing equations is required here to deal with very less water depth. To investigate the performance of the present model in diverse situations, it is used to replicate four different problems of known analytical solution, and the model is found to be quite capable for varied situations.

Discrete conservation properties for shallow water flows using mixed mimetic spectral elements

A mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as higher moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.

Modeling shallow water flows on general terrains

A formulation of the shallow water equations adapted to general complex terrains is proposed. Its derivation starts from the observation that the typical approach of depth integrating the Navier-Stokes equations along the direction of gravity forces is not exact in the general case of a tilted curved bottom. We claim that an integration path that better adapts to the shallow water hypotheses follows the "cross-flow" surface, i.e., a surface that is normal to the velocity field at any point of the domain. Because of the implicitness of this definition, we approximate this "cross-flow" path by performing depth integration along a local direction normal to the bottom surface, and propose a rigorous derivation of this approximation and its numerical solution as an essential step for the future development of the full "cross-flow" integration procedure. We start by defining a local coordinate system, anchored on the bottom surface to derive a covariant form of the Navier-Stokes equations. Depth integration along the local normals yields a covariant version of the shallow water equations, which is characterized by flux functions and source terms that vary in space because of the surface metric coefficients and related derivatives. The proposed model is discretized with a first order FORCE-type Godunov Finite Volume scheme that allows implementation of spatially variable fluxes. We investigate the validity of our SW model and the effects of the bottom geometry by means of three synthetic test cases that exhibit non negligible slopes and surface curvatures. The results show the importance of taking into consideration bottom geometry even for relatively mild and slowly varying curvatures.

Assessment of single‐instrument techniques for removing wave bias from Reynolds stress estimates

AbstractA new single‐instrument method for removing wave bias from Reynolds stress estimates is proposed. Assuming that the wave motions and the pressure signal are spatially coherent, the method uses a linear filtration scheme to estimate the wave velocity at the instrument based on the pressure signal, and then the estimated velocity is subtracted from the measured velocity to arrive at a wave‐free estimate of the Reynolds stress. The advantage of the proposed single‐instrument technique is that it limits the financial and logistical issues associated with deploying a second instrument. The proposed method is compared to two frequency‐domain‐based single‐instrument techniques and to a two‐instrument method that uses the velocity signal at an adjacent instrument in a linear filtration scheme to estimate and remove the wave velocity. The methods are tested and compared using synthetic time series and field data collected in tidally driven shallow water flow in Wassaw Sound, Georgia and in deeper water currents in Monterey Bay, California. The method proposed in this study offers superior performance over the other single‐instrument techniques in shallow water, comparing favorably with the two‐instrument method. In deeper water, no single‐instrument technique offers clearly superior performance.

Numerical Model of Tidal Current and Sediment Applied in Huangmaohai Bay

In this paper, a 2D model for the simulation of shallow water flow by convection and diffusion over variable bottom is presented, which is based on a finite volume method over triangular unstructured grids. The format of Reo’s approximate Riemann is adopted to solve the flux terms. And the bed slope source term is treated by split in the form of the flux eigenvector. For the diffusion terms, the divergence theorem is employed to obtain the derivatives of a scalar variable on each triangular cell. Then, the process of tidal current and sediment in Huangmaohai bay is simulated, which flow pattern is similar with the actual flow. So it is present the model could be applied to simulate the complicated current structure in the water area around hydraulic construction.

Doppler spectra of airborne sound backscattered by the free surface of a shallow turbulent water flow.

Measurements of the Doppler spectra of airborne ultrasound backscattered by the rough dynamic surface of a shallow turbulent flow are presented in this paper. The interpretation of the observed acoustic signal behavior is provided by means of a Monte Carlo simulation based on the Kirchhoff approximation and on a linear random-phase model of the water surface elevation. Results suggest that the main scattering mechanism is from capillary waves with small amplitude. Waves that travel at the same velocity of the flow, as well as dispersive waves that travel at a range of velocities, are detected, studied, and used in the acoustic Doppler analysis. The dispersive surface waves are not observed when the flow velocity is slow compared to their characteristic velocity. Relatively wide peaks in the experimental spectra also suggest the existence of nonlinear modulations of the short capillary waves, or their propagation in a wide range of directions. The variability of the Doppler spectra with the conditions of the flow can affect the accuracy of the flow velocity estimations based on backscattering Doppler. A set of different methods to estimate this velocity accurately and remotely at different ranges of flow conditions is suggested.

Hydrochemistry and Characteristics of Groundwater: Case Study Water Contamination at Citarum River Upstream

Rancaekek and Sayang area, West Java, are the area where many industrial factories are located.Thus, the region becomes the targeted destination for industrial development.The population in the area is rising due to the growth of industries causing the regional development becomes uncontrollable. In addition, the constant increment of waste and also poor-coordinated disposal systems may result in groundwater contamination in the areas. The rapid growth of the area increase the need for groundwater as well as the need for more research about contamination at Rancaekek and Sayang. The research aims to explore the spread of groundwater contamination in the area. The research method is carried out based on the analysis of Geological Mapping, Hydrogeological Mapping and chemical characteristics of the groundwater in the area. Chemical analyses of the groundwater were conducted through laboratory test of groundwater samples at specific spots of dug wells. The lab test results were further analyzed to determine the contamination zone. The findings reveal that the distribution of contamination in the area follow the shallow ground water flow patterns, the water contamination contains heavy metal and there is degradation of soil fertility. The findings suggest the stakeholders to delineate the contaminated area, and increase the dissemination of environmental awareness.

Riemann solution for a class of morphodynamic shallow water dam-break problems

This paper investigates a family of dam-break problems over an erodible bed. The hydrodynamics is described by the shallow water equations, and the bed change by a sediment-conservation equation, coupled to the hydrodynamics by a sediment transport (bed-load) law. When the initial states $\boldsymbol{U}_{l}$ and $\boldsymbol{U}_{r}$ are sufficiently close to each other the resulting solutions are consistent with the theory proposed by Lax (Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, 1973, SIAM), that for a Riemann problem of $n$ equations there are $n$ waves associated with the $n$ characteristic families. However, for wet–dry dam-break problems over a mobile bed, there are three governing equations, but only two waves. One wave vanishes because of the presence of the dry bed. When initial left and right bed levels ($B_{l}$ and $B_{r}$) are far apart, it is shown that a semi-characteristic shock may occur, which happens because, unlike in shallow water flow on a fixed bed, the flux function is non-convex. In these circumstances it is shown that it is necessary to reconsider the usual shock conditions. Instead, we propose an implied internal shock structure the concept of which originates from the fact that the stationary shock over a fixed-bed discontinuity can be regarded as a limiting case of flow over a sloping fixed bed. The Needham & Hey (Phil. Trans. R. Soc. Lond. A, vol. 334, 1991, pp. 25–53) approximation for the ambiguous integral term $\int \!h\,\text{d}B$ in the shock condition is improved based on this internal shock structure, such that mathematically valid solutions that incorporate a morphodynamic semi-characteristic shock are arrived at.

The space–time CESE scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients

The effects of bottom topography and horizontal temperature gradients on the shallow water flows are theoretically investigated. The considered systems of partial differential equations (PDEs) are non-strictly hyperbolic and non-conservative due to the presence of non-conservative differential terms on the right hand side. The solutions of these model equations are very challenging for a numerical scheme. Thus, our primary goal is to introduce an improved numerical scheme which can handle the non-conservative differential terms efficiently and accurately. In this paper, the space–time conservation element and solution element (CESE) method is extended to approximate these model equations. The proposed scheme has capability to overcome all difficulties posed by this nonlinear system of PDEs. The performance of the scheme is analyzed by considering several case studies of practical interest and the results of suggested scheme are compared with those of central NT scheme. The accuracy of the scheme is verified qualitatively and quantitatively.

Weighted interior penalty discretization of fully nonlinear and weakly dispersive free surface shallow water flows

In this paper, we further investigate the use of a fully discontinuous Finite Element discrete formulation for the study of shallow water free surface flows in the fully nonlinear and weakly dispersive flow regime. We consider a decoupling strategy in which we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects. This source term can be computed through the resolution of elliptic second-order linear sub-problems, which only involve second order partial derivatives in space. We then introduce an associated Symmetric Weighted Internal Penalty discrete bilinear form, allowing to deal with the discontinuous nature of the elliptic problem's coefficients in a stable and consistent way. Similar discrete formulations are also introduced for several recent optimized fully nonlinear and weakly dispersive models. These formulations are validated again several benchmarks involving h-convergence, p-convergence and comparisons with experimental data, showing optimal convergence properties.

Numerical Investigation of the Morphological Dynamics of a Step-and-Pool Riverbed Using DEM-MPS

In mountain streams, riverbeds with a series of coarse gravel steps are common. The sequence flow pattern with a chute and pool is usually formed on series of steps, where a transition region from supercritical to subcritical flow appears between steps. In the transition region, a discontinuous water surface with hydraulic jumps is observed and considerable water surface fluctuations occur. These fluctuations significantly affect the bed material. Accurate measurement of the flow velocities under such violent flow conditions is difficult owing to shallow water conditions, air entrainment, and flow nonlinearity. Therefore, detailed flow structures around the steps were not previously understood. To conduct dynamic investigations of formation/collapse processes of the series of steps, numerical simulations were conducted using the discrete element method (DEM) and the enhanced moving particle semi-implicit (MPS) method. Simulation results suggest that the developed DEM-MPS coupling model will be a promising tool to investigate the morphological dynamics of the step series. Furthermore, the potential relationship between water surface elevation and the collapse of steps is implied by the simulation results.

A staggered grid finite difference method for solving the gravity wave-model equations

The gravity wave-model equations are considered. The equations govern shallow water flows, where the gravity effect is significant. The equations form a system of partial differential equations with the hyperbolic type. When the gravity wave-model equations are solved using a finite difference method with staggered grids, artificial oscillations may occur in the numerical solution. In this paper, we propose a numerical treatment of a staggered grid finite difference method for solving the gravity wave-model equations. Using our proposed treatment, artificial oscillations can be eliminated, so the resulting numerical solution mimics the physics of the problem.

Formula omitted]adaptation for Shallow Water flows: conservation, well balancedness, efficiency

We investigate the potential of the so-called “relocation” mesh adaptation in terms of resolution and efficiency for the simulation of free surface flows in the near shore region. Our work is developed in three main steps. First, we consider several Arbitrary Lagrangian Eulerian (ALE) formulations of the shallow water equations on moving grids, and provide discrete analogue in the Finite Volume and Residual Distribution framework. The compliance to all the physical constraints, often in competition, is taken into account. We consider different formulations allowing to combine volume conservation (DGCL) and equilibrium (Well-Balancedness), and we clarify the relations between the so-called pre-balanced form of the equations [7]), and the classical upwinding of the bathymetry term gradients [1]. Moreover, we propose a simple remap of the bathymetry based on high accurate quadrature on the moving mesh which, while preserving an accurate representation of the initial data, also allows to retain mass conservation within an arbitrary accuracy. Second, the coupling of the resulting schemes with a mesh partial differential equation is studied. Since the flow solver is based on genuinely explicit time stepping, we investigate the efficiency of three coupling strategies in terms of cost overhead w.r.t. the flow solver. We analyze the role of the solution remap necessary to evaluate the error monitor controlling the adaptation, and propose simplified formulations allowing a reduction in computational cost. The resulting ALE algorithm is compared with the rezoning Eulerian approach with interpolation proposed e.g. in Tang and Tang [17]. An alternative cost effective Eulerian approach, still allowing a full decoupling between adaptation and flow evolution steps is also proposed. Finally, a thorough numerical evaluation of the methods discussed is performed. Numerical results on propagation, and inundation problems shows that the best compromise between accuracy and CPU time is provided by a full ALE formulation. If a loose coupling with the mesh adaptation is sought, then the cheaper Eulerian approach proposed is shown to provide results quite close to the ALE.

Numerical Modeling of Wave Run-Up in Shallow Water Flows Using Moving Wet/Dry Interfaces

Numerical Modeling of Wave Run-Up in Shallow Water Flows Using Moving Wet/Dry Interfaces

Modeling the effects of vegetation on fluid flow through an acid mine drainage passive remediation system

The fluid dynamics of acid mine drainage (AMD) flow through Wingfield Pines passive remediation system is captured using a lattice Boltzmann model of shallow water flow. The retention time of fluid through the system is critical to the efficient treatment of these polluted waters and, therefore, the retention times in all five settling ponds and the wetlands are elucidated for both the system as it was built, and for when vegetation is present. The effects of vegetation growth is predicted to initially cause channeling, potentially decreasing retention times, but over longer times to increasingly spread the fluid flow throughout the system and increase retention times.

Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel

The formation of a periodic roll-wave train in a long channel is studied for two sets of experimental parameters corresponding to Brock’s experiments [3], [4] who measured permanent wave profiles by introducing periodic perturbations at the channel inlet. In both cases, a formed free surface profile is found in good agreement with the experimental results. Qualitative properties of solutions to the model are studied in the case where the perturbation frequency is lower than the experimental one, so longer waves are generated at the channel inlet. It is observed that the corresponding roll-wave train is strongly modulated. In the case where the waves of two different lengths are generated at the channel inlet, the coarsening is observed, i.e. the process where shorter waves disappear progressively by transferring their energy to longer waves forming later a regular roll-wave train. The coarsening phenomenon is always accompanied by a strong modulation. A comparison with the Saint-Venant equations is also performed. The formation of a single wave composing a roll-wave train is also studied in a domain with periodic boundary conditions (called “periodic box”) for the same sets of experimental parameters. The free surface profile is found also in good agreement with the experimental results. This allows us to justify the use of the “periodic box” as a simple mathematical tool for a qualitative study of stability of roll-waves. In particular, we study the stability of a single steady wave by increasing its length. It is shown that the wave becomes morphologically unstable after some critical wave length: it bifurcates into a system of two waves. In the framework of a simplified multi-D model of shear shallow water flows it is also proved that a single steady wave is stable under multi-dimensional perturbations.

Two innovative pore pressure calculation methods for shallow deep-water formations

There are many geological hazards in shallow formations associated with oil and gas exploration and development in deep-water settings. Abnormal pore pressure can lead to water flow and gas and gas hydrate accumulations, which may affect drilling safety. Therefore, it is of great importance to accurately predict pore pressure in shallow deep-water formations. Experience over previous decades has shown, however, that there are not appropriate pressure calculation methods for these shallow formations. Pore pressure change is reflected closely in log data, particularly for mudstone formations. In this paper, pore pressure calculations for shallow formations are highlighted, and two concrete methods using log data are presented. The first method is modified from an E. Philips test in which a linear-exponential overburden pressure model is used. The second method is a new pore pressure method based on P-wave velocity that accounts for the effect of shallow gas and shallow water flow. Afterwards, the two methods are validated using case studies from two wells in the Yingqiong basin. Calculated results are compared with those obtained by the Eaton method, which demonstrates that the multi-regression method is more suitable for quick prediction of geological hazards in shallow layers.

A Hybrid Method to Solve Shallow Water Flows with Horizontal Density Gradients

In this work, a model for shallow water flows that accounts for the effects of horizontal density fluctuations is presented and derived. While the density is advected by the flow, a two-way feedback between the density gradients and the time evolution of the fluid is ensured through the pressure and source terms in the momentum equations. The model can be derived by vertically averaging the Euler equations while still allowing for density fluctuations in horizontal directions. The approach differs from multi-layer shallow water flows where two or more layers are considered, each of them having their own depth, velocity and constant density. A Roe-type upwind scheme is developed and the Roe matrices are computed systematically by going from the conservative to the quasi-linear form at a discrete level. Properties of the model are analyzed. The system is hyperbolic with two shock-wave families and a contact discontinuity associated to interfaces of regions with density jumps. This new field is degenerate with pressure and velocity as the corresponding Riemann invariants. We show that in some parameter regimes numerically recognizing such invariants across contact discontinuities is important to correctly compute the flow near those interfaces. We present a numerical algorithm that correctly captures all waves with a hybrid strategy. The method integrates the Riemann invariants near contact discontinuities and switches back to the conserved variables away from it to properly resolve shock waves. This strategy can be applied to any numerical scheme. Numerical solutions for a variety of tests in one and two dimensions are shown to illustrate the advantages of the strategy and the merits of the scheme.

Implicit finite volume simulation of 2D shallow water flows in flexible meshes

In this work, an implicit method for solving 2D hyperbolic systems of equations is presented, focusing on the application to the 2D shallow water equations. It is based on the first order Roe’s scheme, in the framework of finite volume methods. A conservative linearization is done for the flux terms, leading to a non-structured matrix for unstructured meshes thus requiring iterative methods for solving the system. The validation is done by comparing numerical and exact solutions in both unsteady and steady cases. In order to test the applicability of the implicit scheme to real world situations, a laboratory scale tsunami simulation is carried out and compared to the experimental data. The implicit schemes have the advantage of the unconditional stability, but a quality loss in the transient solution can appear for high CFL numbers. The properties of the scheme are well suited for the simulation of unsteady shallow water flows over irregular topography using all kind of meshes.