Non-hydrostatic Pressure Distribution Research Articles

FINITE-AMPLITUDE INTERNAL SOLITARYWAVES IN A SHEAR FLOW OF A TWO-LAYER FLUID

We consider a nonlinear system of equations that describes the propagation of finite-amplitude internal waves in two-layer stratified shallow water under a cover in the Boussinesq approximation. Within the framework of this model, we study solitary waves, taking into account the nonhydrostatic pressure distribution in one or both layers, and also consider the case of weakly nonlinear waves. In the class of traveling waves, this model is reduced to an ordinary differential equation for determining the profile of a solitary wave in a shear flow. This equation admits an analytical study. The conditions for the existence of solitary waves adjacent to a given constant flow are determined by two dimensionless parameters related to the wave speed and the velocity shear of the undisturbed flow. Examples of solitary waves are given. These examples demonstrate the effects of the velocity shear on the waveform.

A hybrid 1D-2D Lagrangian solver with moving coupling to simulate dam-break flow

In dam-break problems, two-dimensional (2D) models on the vertical plane appropriately capture the non-hydrostatic pressure distribution but with a high computational cost. This study presents a hybrid Lagrangian solver of one-dimensional (1D) shallow water flow and a 2D large-eddy simulation with a sub-particle scale to simulate dam-break flows over frictional beds. The proposed 1D-2D hybrid scheme is a useful trade-off between the accuracy of the 2D models on the vertical plane and the computational efficiency of 1D shallow water equations models. The configuration of sub-domains is established so that the 1D solves the regions with dominantly hydrostatic pressure, while the 2D solver accounts for the non-hydrostaticity. A moving coupling method is introduced based on two-layer shallow water equations with a uniform velocity distribution that constructs two sub-domains, including inner and outer zones in the section coupling model (SCM), or upper and lower layers within the layer coupling model (LCM). Then, the region with high vertical accelerations is substituted by the 2D model in the SCM or mixed SCM-LCM in which the overlapping zone and boundary interface move with time, based on the 1D flow feature. Also, a new method is proposed to deal with irregular boundary interfaces in the LCM while satisfying the momentum transfer. The present hybrid model is advantageous as it meets mass conservation, and there is no need for typical inflow/outflow boundary conditions at the coupling interfaces for simulating dam-break waves. Moving particle simulation (MPS) is utilized to solve the Lagrangian equations, which is advantageous in dealing with free-surface problems with large deformation. The model validation reveals that the hybrid results of water depths are in very good agreement with observed data and those obtained from the 2D model for selected dam-break flows with the ratio of initial depths downstream to upstream up to 0.4. In this respect, the average root-mean-square-error (RMSE) is computed at 8.09 mm in the SCM, 8.16 mm in the mixed SCM-LCM, and 7.48 mm in the pure 2D model, while both hybrid models save the computational time by 55% compared to the 2D model. Moreover, the mixed SCM-LCM preserves the nonlinearity of the wavefront over long times, whereas the SCM converts the wavefront into a shockwave considering the same number of fluid particles.

Non-Hydrostatic Transitional Open-Channel Flows from a Supercritical to a Subcritical State

Abstract In this study, a depth-averaged numerical model was employed to investigate the two-dimensional flow features of transitional open-channel flows from a supercritical to a subcritical state. Compared to a shallow-water model, the proposed model incorporates supplementary terms to account for the effects of non-uniform velocity and non-hydrostatic pressure distributions. The model equation was solved numerically by means of the Adams–Bashforth–Moulton scheme. A wide variety of transitional open-channel flow problems such as hydraulic jumps was considered for assessing the suitability of the numerical model. The results of the model for the free-surface profile, pressure distribution, and characteristics of the first wave of an undular jump were compared with the experimental data, and the agreement was found to be satisfactory. Despite the effects of the three-dimensional characteristics of the flow and the bulking of the flow caused by air entrainment, the model performed reasonably well with respect to the simulations of the mean flow characteristics of the curvilinear turbulent flow problems. Furthermore, the results of this investigation confirmed that the model is more suitable for analyzing near-critical turbulent flow problems without cross-channel shock waves.

Efficient two-layer non-hydrostatic wave model with accurate dispersive behaviour

A 2-layer non-hydrostatic model with improved dispersive behaviour is presented. Due to the assumption of a constant non-hydrostatic pressure distribution in the lower layer, the dispersive behaviour is improved without much additional computational time. A comparison with linear wave theory showed that this 2-layer model gives a better result for the dispersion relation and shoaling of waves in intermediate water. This means that the 2-layer model is applicable in shallow and intermediate water depths (up to relative depths kh equals 4), whereas the 1-layer model is only applicable in shallow water depths (kh smaller than 1). Three laboratory experiments, including a fringing reef and a barred beach, were used to validate the presented mode for different hydrodynamic conditions. Based on these results, it can be concluded that the 2-layer model can be applied to accurately simulate the bulk wave height and spectral properties. The low frequency wave height, the setup and in particular the second order statistics contain more scatter, but the model accurately captured the general trend. Furthermore, the model showed good results for complex bathymetries in shallow to intermediate water.

An adoption of the Spalart–Allmaras turbulence model for two- and three-dimensional free surface environmental flows

In this paper, the Spalart–Allmaras turbulence model, very popular for compressible aerospace applications, is adapted for incompressible free surface environmental flows. The Spalart–Allmaras model is a Reynolds-averaged Navier–Stokes model which solves for one transport equation of a viscosity-like variable. Besides its efficiency and reasonable computational cost, the model can take into account all sources of turbulence production and destruction. The Spalart–Allmaras model was implemented in two- and three-dimensional cases within a massively parallel software. For two-dimensional applications, the Spalart–Allmaras model was used together with the shallow water equations. Although dealing with turbulence with depth-averaged shallow water equations may be perplexing, horizontal turbulent structures could be adequately retrieved. For three-dimensional applications, the Spalart–Allmaras model was introduced with the incompressible Navier–Stokes equations. Numerical resolution is based on a fractional step algorithm that, unlike in a majority of published studies, considers a real free surface, a σ-transform moving mesh and a non-hydrostatic pressure distribution. Numerical resolution uses a mixed finite volume-finite element algorithm. The advection part could be handled, among other options, using a characteristics-based scheme. The Spalart–Allmaras model was validated against experimental laboratory observations and compared to other turbulence models. Results confirmed the potential of the adapted SA turbulence model, as both averaged and fluctuating fields (i.e. kinetic energy and velocity) showed quality comparable to, or even better, than those obtained by the well-known k-ϵ turbulence model.

Hyperbolic model of internal solitary waves in a three-layer stratified fluid

We derive a new hyperbolic model describing the propagation of internal waves in a stratified shallow water with a non-hydrostatic pressure distribution. The construction of the hyperbolic model is based on the use of additional instantaneous variables. This allows one to reduce the dispersive multi-layer Green–Naghdi model to a first-order system of evolution equations. The main attention is paid to the study of three-layer flows over an uneven bottom in the Boussinesq approximation with the additional assumption of hydrostatic pressure in the intermediate layer. The hyperbolicity conditions of the obtained equations of three-layer flows are formulated, and solutions in the class of travelling waves are studied. Based on the proposed hyperbolic and dispersive models, numerical calculations of the generation and propagation of internal solitary waves are carried out and their comparison with experimental data is given. Within the framework of the proposed three-layer hyperbolic model, a numerical study of the propagation and interaction of symmetric and non-symmetric soliton-like waves is performed.

Numerical and Experimental Study of Classical Hydraulic Jump

The present work is an effort to simulate numerically a classical hydraulic jump in a horizontal open channel with a rectangular cross-section, as far as the jump location and free surface elevation is concerned, and compare the results to experiments with Froude numbers in the range 2.44 to 5.38. The governing equations describing the unsteady one-dimensional rapidly varied flow have been solved with the assumption of non-hydrostatic pressure distribution. Two finite difference schemes were used for the discretization of the mass and momentum conservation equations, along with the appropriate initial and boundary conditions. The method of specified intervals has been employed for the calculation of the velocity at the downstream boundary node. Artificial viscosity was required for damping the oscillations near the steep gradients of the jump. An iterative algorithm was used to minimize the difference of flow depth between two successive iterations that must be less than a threshold value, for achieving steady state solution. The time interval varied in each iteration as a function of the Courant number for stability reasons. Comparison of the numerical results with experiments showed the validity of the computations. The numerical codes have been implemented in house using a Matlab® environment.

The particularities of near-critical flows formation in open channels

This article considers the current problems of near-critical flows. It provides short characteristics of each phenomenon and describes the cases of near-critical flow formation during the operation of the different hydraulic structures. Each of the considered phenomena has a number of characteristic features which distinguish them from the usual subcritical and supercritical flows with smooth or slowly varied movement. Such properties include the wave-like or roller nature of free-surface curves, the presence of a streamline inclination and curvature, and also a non-hydrostatic pressure distribution in depth mainly in the vertical cross-section of these phenomena. Therefore, during mathematical and numerical simulations at the designing stage of hydraulic structures it is necessary to take into account the additional parameters which characterise the particularities of near-critical flows. In cases in which these moments are neglected, there are many cases of accidents and damage being caused to structures which are operated in conditions of near-critical flow formation. An objective of this work is to provide a detailed analysis of the particularities of near-critical flows and show their negative consequences on hydraulic structures. The article presents the results of the mathematical and hydraulic simulation of wavelike near-critical flows and a comparison of the full-scale measurement and a mathematical model of translation waves.

On Steady Two-dimensional Free-surface Flows with Spatially-varied Discharges

Abstract In an open-channel, the transition of a flow from a subcritical to a supercritical state may occur as a result of a lateral inflow or outflow that produces a streamwise discharge variation. Apparently, such a transition cannot be modeled accurately by a conventional hydrostatic pressure approach. In this study, a depth-averaged model that accounts for the effects of a spatially-varied discharge and a non-hydrostatic pressure distribution was developed and applied to simulate the transcritical flow in a lateral-spillway channel and the subcritical flow in a main channel fitted with side weirs. The model results for the axial free-surface profile and variation of discharge in the main channel were compared with the results of a shallow-flow model and experimental data, thereby resulting in a closer match to the measurements than the shallow-flow model. Overall, the investigation results confirmed the efficiency and validity of the non-hydrostatic depth-averaged model in simulating the mean flow characteristics of the subcritical and transcritical free-surface flows with spatially increasing or decreasing discharges, thus demonstrating its potential to be used as a numerical tool in engineering practice.

Simulation of free surface flows with non-hydrostatic pressure distribution

In this work, a free surface flow simulator is developed in which Navier–Stokes equations using Marker and Cell (MAC) method in the framework of finite difference methodology have been solved. The free surface is tracked by the volume of fluid (VOF) method. The numerical code without free-surface is validated against the flow past a square cylinder. Three different free surface flows, i.e., dam break flow, two-dimensional cavity filling and undular bore, are studied to demonstrate the efficacy of the developed numerical model to simulate free surface flow. The numerical model used in the present work involves tracking of free surface of a single fluid in a two-fluid system. The parameters which can affect the interface orientation of the fluid is given as boundary condition at the interface. The inherent advantage of such numerical models is its ability to track free surface for high density and viscosity ratio fluids like air–water. The numerical model used in the present work is capable of solving such high density ratio two-fluid systems for which the effects due to surface tension are negligible. Results from all the problems are compared with earlier results available in literature.

Two-dimensional Boussinesq equations applied to channel flows: deducing and applying the equations

ABSTRACT A basic hypothesis adopted for theoretical formulation of fluid flows is the hydrostatic pressure distribution. However, many researchers have pointed out that this simplification can lead to errors, in cases such as dam break flow. Discrepancy between computational solution and the experiment is attributed to the pressure distribution. These findings are not new, but it is not presented any formulation in the literature that considers the non-hydrostatic pressure distribution in 2D flow. This article deduces the Boussinesq Equations as an evolution of the Shallow Water Equations with the hypothesis of non-hydrostatic pressure distribution in the vertical direction. XYZ Orthogonal Cartesian System is used, considering the influence of channel bed slope and head losses of flow. It is presented the non-hydrostatical correction in the Boussinesq equation in two dimension using Fourier series. The solution uses Runge-Kutta Discontinuous Galerkin Method and the formulation is applied to a cylindrical dam-break.

On the steady two-dimensional open channel flow

This note concerns the open channel flow problem. The flow is assumed to be steady (with constant discharge), two-dimensional (with non-hydrostatic pressure distribution), without both wave breaking and sediment transport. The flow problem consists in finding the location of the free surface, and the velocity and pressure fields. The problem solution is obtained by a hybrid model which incorporates the dissipation in potential flows. In order to set up the model, this note proposes a new Picard iteration for the potential velocity field. By coupling approximate expressions for the velocity field to dynamic equations, the flow problem is reduced to one of solving a nonlinear ordinary differential equation. To illustrate the model validation, numerical simulations are performed for open channel flow over curved bottom, and the free overfall problem. In terms of free surface location and bottom pressure profile, the obtained results are in good agreement with data in literature.

A Numerical Study of Non-hydrostatic Shallow Flows in Open Channels

Abstract The flow field of many practical open channel flow problems, e.g. flow over natural bed forms or hydraulic structures, is characterised by curved streamlines that result in a non-hydrostatic pressure distribution. The essential vertical details of such a flow field need to be accounted for, so as to be able to treat the complex transition between hydrostatic and non-hydrostatic flow regimes. Apparently, the shallow-water equations, which assume a mild longitudinal slope and negligible vertical acceleration, are inappropriate to analyse these types of problems. Besides, most of the current Boussinesq-type models do not consider the effects of turbulence. A novel approach, stemming from the vertical integration of the Reynolds-averaged Navier-Stokes equations, is applied herein to develop a non-hydrostatic model which includes terms accounting for the effective stresses arising from the turbulent characteristics of the flow. The feasibility of the proposed model is examined by simulating flow situations that involve non-hydrostatic pressure and/or nonuniform velocity distributions. The computational results for free-surface and bed pressure profiles exhibit good correlations with experimental data, demonstrating that the present model is capable of simulating the salient features of free-surface flows over sharply-curved overflow structures and rigid-bed dunes.

A non-hydrostatic pressure distribution solver for the nonlinear shallow water equations over irregular topography

We extend a recently proposed 2D depth-integrated Finite Volume solver for the nonlinear shallow water equations with non-hydrostatic pressure distribution. The proposed model is aimed at simulating both nonlinear and dispersive shallow water processes. We split the total pressure into its hydrostatic and dynamic components and solve a hydrostatic problem and a non-hydrostatic problem sequentially, in the framework of a fractional time step procedure. The dispersive properties are achieved by incorporating the non-hydrostatic pressure component in the governing equations. The governing equations are the depth-integrated continuity equation and the depth-integrated momentum equations along the x, y and z directions. Unlike the previous non-hydrostatic shallow water solver, in the z momentum equation, we retain both the vertical local and convective acceleration terms. In the former solver, we keep only the local vertical acceleration term. In this paper, we investigate the effects of these convective terms and the possible improvements of the computed solution when these terms are not neglected in the governing equations, especially in strongly nonlinear processes. The presence of the convective terms in the vertical momentum equation leads to a numerical solution procedure, which is quite different from the one of the previous solver, in both the hydrostatic and dynamic steps. We discretize the spatial domain using unstructured triangular meshes satisfying the Generalized Delaunay property. The numerical solver is shock capturing and easily addresses wetting/drying problems, without any additional equation to solve at wet/dry interfaces. We present several numerical applications for challenging flooding processes encountered in practical aspects over irregular topography, including a new set of experiments carried out at the Hydraulics Laboratory of the University of Palermo.

Analytical Solutions of Energy Equation for Rectangular Channels: Direct Approach

AbstractAn analytical solution of nondimensional hydraulic energy equation is derived in cosine form for the flow in rectangular open channels considering nonhydrostatic pressure and nonuniform velocity distributions across flow depth and friction and turbulent losses. The new inverted energy equation is a single equation describing all the three roots of the nondimensional hydraulic energy equation. The energy losses due to turbulence and bed shear are also implicitly accounted. Only two roots out of the three roots are practically significant, which denote the two alternate depths and are distinguishable for subcritical and supercritical flow, respectively. These equations yield direct determination of alternate depths in a single step, avoiding an iterative procedure. The use of the new equations is illustrated through worked-out examples. The new single-term analytical inverted equations are computationally simple and useful for academicians, field engineers, and practitioners in directly solving in a...

Modelling free surface flow with curvilinear streamlines by a non-hydrostatic model

Abstract This study addresses a particular phenomenon in open channel flows for which the basic assumption of hydrostatic pressure distribution is essentially invalid, and expands previous suggestions to flows where streamline curvature is significant. The proposed model incorporates the effects of the vertical curvature of the streamline and steep slope, in making the pressure distribution non-hydrostatic, and overcomes the accuracy problem of the Saint-Venant equations when simulating curvilinear free surface flow problems. Furthermore, the model is demonstrated to be a higher-order one-dimensional model that includes terms accounting for wave-like variations of the free surface on a constant slope channel. Test results of predicted flow surface and pressure profiles for flow in a channel transition from mild to steep slopes, transcritical flow over a short-crested weir and flow with dual free surfaces are compared with experimental data and previous numerical results. A good agreement is attained between the experimental and computed results. The overall simulation results reveal the satisfactory performance of the proposed model in simulating rapidly varied gravity-driven flows with predominant non-hydrostatic pressure distribution effects. This study suggests that a higher-order pressure equation should be used for modelling the pressure distribution of a curvilinear flow in a steeply sloping channel.

Free-Surface Flow Over River Bottom Sill: Experimental and Numerical Study

This paper reports on a laboratory study that was performed in an experimental model in the Technological Educational Institute of Thessaly. The current research work examines the water surface profile variation in abruptly altered open channel topography. The objective is to investigate the effects of river bottom sills on flow depth for river control protection against floods. A rectangular bottom sill with one edge rounded was placed on the bottom of a rectangular, horizontal, laboratory open flume and measurements were performed of free-surface flow depth over the bottom sill, under steady flow conditions, for different inflow discharges. Moreover, the HEC-RAS simulation model was used to numerically simulate the water surface variation over the rectangular bottom sill, under steady flow conditions. A mixed flow regime simulation was performed in order to validate the transition from subcritical to supercritical flow conditions. The computed flow depth variation over the rectangular bottom sill was compared with the obtained experimental measurements in order to demonstrate the accuracy of the performed measurements. The presented experimental data may be used to assist in the development of new and the verification and refinement of existing river hydraulic numerical simulation models, particularly those which use non-hydrostatic pressure distribution.

Evaluation and modification of time splitting method applied to the fully dynamic numerical solution of water wave propagation

Time splitting method has gained popularity in the numerical solution of two and three dimensional advection diffusion problems. This paper critically evaluates the suitability of applying this method to water wave propagation. The model which is used for this purpose solves Navier-Stokes equations with dynamic pressure. Finite volume method is applied to a boundary fitted staggered grid with equal number of layers in the vertical direction. Assuming non-hydrostatic pressure distribution in the upper layer, this model is able to simulate the short wave propagation using only a few vertical layers (3-5 layers). Comparison of the results of this model with experimental and analytical data indicates that amplitude and phase errors are present which seems to be caused by applying time splitting algorithm. A method already proposed to overcome the poor results of this algorithm for the advection of scalar quantities is modified and extended to vector quantities applicable to the present study. The results of this modified method exhibit a significant improvement in the model and indicate the weakness of time splitting method in the solution of non-hydrostatic nonlinear wave propagation problems.

One-dimensional modelling of curvilinear free surface flow: generalized Matthew theory

ABSTRACTThe potential flow theory is a good mathematical approximation to various problems in the fields of free surface open channel and groundwater flows. In both fields, second-order equations are commonly derived using perturbation methods to account for the non-hydrostatic pressure distribution. However, an alternative technique is Picard's iteration approach, used by Matthew for steady open channel flows. Regretfully, this technique was not expanded to unsteady potential flow, limiting its full impact. The purpose of this work is to generalize the theory of Matthew to unsteady potential free surface flow, both for open channels and groundwater. The new development is an alternative to perturbation techniques, highlighting the relevance of Matthew's work. To illustrate the widespread problems to which the theory applies, simulations are made for a dambreak wave and for flow in sloping and curved aquifers. In both cases one-dimensional results are in good agreement with two-dimensional data.

A new depth-integrated non-hydrostatic model for free surface flows

A new depth-integrated model deploying a non-hydrostatic pressure distribution is presented. With the pressure divided into hydrostatic and dynamic components, the horizontal momentum equations were obtained by integrating the Navier-Stokes equations from the bottom to the free surface. The vertical momentum equation, in which the convective and viscosity terms were omitted, was approximated by the Keller-box scheme. The model has two steps. First, the dynamic pressure gradient terms were discretized semi-implicitly and the other terms were in explicit scheme. Second, the velocities expressed as the unknown dynamic pressure were substituted into the continuity equation, resulting in a five-diagonal symmetric matrix linear system that was solved by the conjugate gradient method. The model was validated with the propagation of a solitary wave and sinusoidal wave, indicating that it can predict free surface flows well.