Hydrostatic Pressure Assumption Research Articles

A semi‐implicit finite volume scheme for a simplified hydrostatic model for fluid‐structure interaction

AbstractSimulating fluid‐structure interaction problems usually requires a considerable computational effort. In this article, a novel semi‐implicit finite volume scheme is developed for the coupled solution of free surface shallow water flow and the movement of one or more floating rigid structures. The model is well‐suited for geophysical flows, as it is based on the hydrostatic pressure assumption and the shallow water equations. The coupling is achieved via a nonlinear volume function in the mass conservation equation that depends on the coordinates of the floating structures. Furthermore, the nonlinear volume function allows for the simultaneous existence of wet, dry and pressurized cells in the computational domain. The resulting mildly nonlinear pressure system is solved using a nested Newton method. The accuracy of the volume computation is improved by using a subgrid, and time accuracy is increased via the application of the theta method. Additionally, mass is always conserved to machine precision. At each time step, the volume function is updated in each cell according to the position of the floating objects, whose dynamics is computed by solving a set of ordinary differential equations for their six degrees of freedom. The simulated moving objects may for example represent ships, and the forces considered here are simply gravity and the hydrostatic pressure on the hull. For a set of test cases, the model has been applied and compared with available exact solutions to verify the correctness and accuracy of the proposed algorithm. The model is able to treat fluid‐structure interaction in the context of hydrostatic geophysical free surface flows in an efficient and flexible way, and the employed nested Newton method rapidly converges to a solution. The proposed algorithm may be useful for hydraulic engineering, such as for the simulation of ships moving in inland waterways and coastal regions.

A bed pressure correction of the friction term for depth-averaged granular flow models

Depth-averaged models, such as the Savage-Hutter model with Coulomb or Pouliquen friction laws, do not in some cases preserve the physical threshold of motion. In particular, the simulated granular mass can start to flow (or stay at rest) even if the slope angle of its free surface is lower (or higher) than the repose angle of the granular material involved. The problem is related to the hydrostatic pressure assumption, associated with the direction of integration, which is orthogonal to a reference plane or a reference bottom. We propose here an initial method to correct this misleading behavior. Firstly, we define a correction of the friction term that accounts for the Jacobian of a change of coordinates, making it possible to reproduce the physical threshold of motion and thus the solutions at rest. Secondly, we observe that the 3D model presented in [F. Bouchut, I. Ionescu, and A. Mangeney. An analytic approach for the evolution of the static-flowing interface in viscoplastic granular flows. Commun, Math. Sci., 14(8):2101–2126, 2016] verifies the physical thresholds of motion because it is based on a second order correction of the pressure valid for slow granular flows. The correction proposed here ensures that the model preserves, up to the second order, the physical threshold of motion defined by the repose angle of the material. Several numerical tests are presented to illustrate certain problems related to classical depth averaged models and the remedial effect of the proposed correction, in particular through comparisons with experimental data. We finally show that this correction is not exact far from the starting and stopping phases of the granular avalanche and should be improved by adding other second order terms in the pressure approximation.

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.

Comparison between hydrostatic and total pressure simulations of dam-break flows

ABSTRACT Shallow water models are commonly used to simulate dam-break flow; however, shallow water equations usually assume a hydrostatic pressure assumption that can limit model applications. Dam-break flow can also be solved by total pressure formulation with Navier–Stokes equations, that are computationally more expensive but can deal with a wider range of problems. We propose a multiphase flow code with second-order precision, called SuLi-LS. This code uses the Level Set methodology and can consider total pressure or hydrostatic pressure assumption in Navier–Stokes solution. To analyse the differences between hydrostatic and total pressure, we compared SuLi-LS results with experimental cases. The comparison shows that total pressure consideration provides accurate results for the dam-break flow problems, and it is a considerably better approach than considering hydrostatic pressure assumption. We found that hydrostatic pressure assumption can be valid for dam-break flows in which there is an underlying water bed.

Influence on Type of Cement on The SCC Formwork Pressure During and After Casting

Formworks for SCC are usually designed under the assumption of full hydrostatic pressure. Nevertheless, current research is attempting to explain the mechanism of this phenomenon as observed pressure usually is at the lower lever than expected. This causes formworks for SCC are usually overdesigned. It was noticed the rheological properties of fresh concrete might be a key to predict the SCC formwork pressure. Therefore, knowing the influence of fresh concrete properties on formwork pressure will enable to design formworks more efficiently. This paper presents the influence of type of cement on formwork pressure caused by SCC. Mixes were design under the assumption of equal dispersion ratio. Three types of cement were investigated: portland, blast furnace and composite cement with a different w/c ratio (0.30, 0.40) and in presence of carboxyl ethers superplasticizer. Formwork pressure was determined on the element imitating a column with dimensions of 0.20x0.20m and a height of 1.20 m with casting speed of 7 m/h. Results show the formwork pressure was registered at the lower than hydrostatic level. Rheological properties had an influence on formwork pressure. It was noticed the different cement types had an influence on rheological properties. Lateral pressure reduction over time was observed with the intensity depending on the cement.

Three-dimensional modeling of flow characteristics and the influence of non-hydrostatic pressure in a 193° sharp open channel bend

A numerical model of three-dimensional free surface flows has been developed to simulate the flow pattern and bed shear stress in a sharp channel bend with 193° curved reach. The numerical model solved 3D Reynolds-averaged Navier-Stokes equations with standard k – ε turbulence model to compute the flow field and the finite-volume method were used as the discretization scheme. The governing equations were solved by employed the explicit projection method with two steps. The numerical predictions were verified with experiment results and showed that the model enables to resolve the complex three-dimensional flow problems and capture the main flow features in sharp channel bends. The influence of non-hydrostatic pressure was revealed by the results with and without hydrostatic pressure assumption. It concluded that the non-hydrostatic model almost has no noticeable influence on flow pattern in this sharp channel bend with highly three-dimensional flow characteristics.

Effects of capillary fringe and truncation factor on pore-water pressure and water table variation in dewatering and watering sand column experiments

Abstract Spatiotemporal variations of the pore-water pressure and water table response to small amounts of water exchange were measured in a sand column. In the dewatering experiment, the pressure head variation can be divided into three stages: the initial pressure head declining phase when water table is above the sand surface, a sudden drop in accordance with the capillary fringe when water level passes through the sand surface, and a slow declining phase with respect to the specific yield of the sand. As for the watering experiment of the shallow groundwater with different initial truncation factors (TF), the addition of a minute amount of water results in a rapid rise of the pressure head initially, then a gradually increase to its equilibrium. When the upper extent of the fully saturated capillary fringe coincides with the sand surface (TF ≤ 1), a thin film of water addition produces nearly instantaneous increase in pore-water pressure, corresponding to a rapid rise of groundwater table to the sand surface. Pressure head increment increases linearly with the TF value and reaches its maximum when TF value approaches unity. Considering the vertical distribution of the pressure head, its gradient becomes larger right after the water addition, then decreases to its initial value of hydrostatic condition in the end. In the case of TF > 1, there is an unsaturated layer above the groundwater table, the pressure head increases after the water addition, whose magnitude however decreases as the increase of TF value. Nevertheless, the final position of the groundwater table is still below the sand surface, and the gradient of the pressure head vertical distribution dose not vary significantly throughout the entire process. In consideration of hydrostatic pressure assumption, temporal variation of the water table was also quantitatively obtained based on the measured instantaneous pore-water pressure data, which is consistent with time-varying process of the pressure head during both the dewatering and watering experiments.

Hydraulic analogy and visualisation of two-dimensional compressible fluid flows: part 1: theoretical aspects

The principles of the well-known, hydraulic analogy are explained (see Courant and Friedricks, 1948; Loh, 1969). The governing equations of two-dimensional compressible fluid two-dimensional flows in non-dimensional form, based on conservation laws are first discussed. Then, the isentropic flow condition is introduced to produce the isentropic Euler equations. On the other hand, the equations governing surface waves on thin water layers over a flat surface are derived in non-dimensional form, using the assumption of hydrostatic pressure across the water layer, hence the analogy between the two problems is established. The normalised density of the compressible flow corresponds to the normalised height of the thin water layer and the speed of sound corresponds to the speed of surface waves in water, hence, the Mach number corresponds to Froude number. Finally, it is shown that the analogy can be used to visualise supersonic and transonic two-dimensional flow patterns, including shock waves and expansion fans around airfoils and in convergent/divergent nozzles. Also, nonlinear water waves of finite amplitude, in dispersive media, are discussed. In part 2 of this study, water table experiments are presented together with qualitative and quantitative measurement techniques.

Hydrostatic versus nonhydrostatic hydrodynamic modelling of secondary flow in a tortuously meandering river: Application of Delft3D

AbstractGiven the importance of pressure gradients in driving secondary flow, it is worth studying how the modelled flow structures in a natural river bend can be impacted by the assumption of hydrodynamic pressure. In this paper, the performance of hydrostatic versus nonhydrostatic pressure assumption in the three‐dimensional (3D) hydrodynamic modelling of a tortuously meandering river is studied. Both hydrostatic and nonhydrostatic numerical models were developed using Delft3D‐Flow to predict the 3D flow field in a reach of Stillwater Creek in Ottawa, Canada. An acoustic Doppler velocimeter was employed to measure the 3D flow field at a section in a sharp bend of the simulated river at two flow stages. The results of the Delft3D hydrostatic model agreed well with the acoustic Doppler velocimeter measurements: The hydrostatic model predicted reasonably accurately both the streamwise velocity distribution across the section and the magnitude and location of the primary secondary flow cell. The results of the Delft3D nonhydrostatic approximation showed that the model was not conservative and could not accurately generate either the secondary flow or the streamwise velocity distribution. This study illustrated the superior performance of the hydrostatic over nonhydrostatic 3D modelling of the secondary flow using Delft3D. Several possible reasons for unfavourable performance of the nonhydrostatic version of Delft3D are discussed, including the pressure correction technique employed in Delft3D. Considering the uncertainties that may arise in both modelling and field measurements, the 3D hydrostatic Delft3D model was capable of reasonably predicting the river bend flow structures in the studied meandering creek.

Refinement on non-hydrostatic shallow granular flow model in a global Cartesian coordinate system

Current shallow granular flow models suited to arbitrary topography can be divided into two types, those formulated in bed-fitted curvilinear coordinates, and those formulated in global Cartesian coordinates. The shallow granular flow model of Denlinger and Iverson \cite{Denlinger2004} and the Boussinesq-type shallow granular flow theory of Castro-Orgaz \emph{et al}. \cite{Castro2014} are formulated in a Cartesian coordinate system (with $z$ vertical), and both account for the effect of nonzero vertical acceleration on depth-averaged momentum fluxes and stress states. In this paper, we first reformulate the vertical normal stress of Castro-Orgaz \emph{et al}. \cite{Castro2014} in a quadratic polynomial in the relative elevation $\eta$. This form allows for analytical depth integration of the vertical normal stress. We then calculate the basal normal stress based on the basal friction law and scaling analysis. These calculations, plus certain constitutive relations, lead to a refined full non-hydrostatic shallow granular flow model, which is further rewritten in a form of Boussinesq-type water wave equations for future numerical studies. In the present numerical study, we apply the open-source code TITAN2D to numerical solution of a low-order version of the full model involving only a mean vertical acceleration correction term. To cure the numerical instability associated with discretization of the enhanced gravity, we propose an approximate formula for the enhanced gravity by utilizing the hydrostatic pressure assumption in the bed normal direction. Numerical calculations are conducted for several test cases involving steep slopes. Comparison with a bed-fitted model shows that even the simplified non-hydrostatic Cartesian model can be used to simulate shallow granular flows over arbitrary topography.

Influence of the Rheological Properties of SCC on the Formwork Pressure

Formworks for self-compacting concrete (SCC) are commonly designed under the assumption of full hydrostatic pressure. However, current research shows that SCC's design pressure could be reduced, if the concrete's rheological properties were taken, into account. Knowing the relationship between the properties and the pressure, we can prevent formwork overdesign. This research was based, on the assumption that fresh concrete can be described as a Bingham fluid. This paper presents the correlations between static and dynamic yield stress, and lateral formwork pressure. Measured rheological parameters were compared to standard technical concrete tests. Formwork pressure were determined on the element imitating a column with dimensions of 0.20 x 0.20 m and a height of 1.20 m with two casting speeds 1 and 7 m/h. Three types of cement, 2 superplasticizers, and 2 w/c ratios were used. A correlation between rheological parameters and pressure reduction over time was observed. Based on our results we propose a methodology and testing sequence applicable in practice.

Three-dimensional shallow water system: A relaxation approach

We study a three-dimensional shallow water system, which is obtained from the three-dimensional Navier–Stokes equations after Reynolds averaging and under the simplifying hydrostatic pressure assumption. Since the three-dimensional shallow water system is generically not hyperbolic, it cannot be numerically solved using hyperbolic shock capturing schemes. At the same time, existing simple finite-difference and finite-volume methods may fail in simulations of unsteady flows with sharp gradients, such as dam-break and flood flows. To overcome this limitation, we propose a novel numerical method, which is based on a relaxation approach utilized to “hyperbolize” the three-dimensional shallow water system. The extended relaxation system is hyperbolic and we develop a second-order semi-discrete central-upwind scheme for it. The proposed numerical method can preserve “lake at rest” steady states and positivity of water depth over irregular bottom topography. The accuracy, stability and robustness of the developed numerical method is verified on five numerical experiments.

Modelling of fast hydraulic transients: issues, challenges, perspectives

The paper proposes an overview of different modelling approaches used for severe transients, particularly dam-break waves. One-, two- and three-dimensional approaches are briefly recalled with some examples in order to highlight their strengths and weaknesses, especially regarding the accurate speed of information through the numerical schemes. Two examples are developed in more details in order to highlight the result sensitivity to the selected model: the classical circular dam break and its dependency on the mesh, and the behaviour of the dam-break wave in a 90° bend. Experimental data are also presented in order to show that, even in the near-field, hydrostatic pressure assumption has a limited impact on the accuracy. Some results from Smoothed Particle Hydrodynamics are also compared to common Eulerian approaches. In order to use the best of each approach, hybrid methods are also considered through the example of the 90° bend. Fast transients generally induce erosion and deposition of large amount of sediment. The challenge is now to account for the inertia of the sediments suddenly mobilised. A two-layer model is proposed, with some assumptions, not wholly in line with experimental observations, but with promising results. Several improvements are tested but closure equations remain delicate and require more fundamental investigations.

A 3-D implicit finite-volume model of shallow water flows

A three-dimensional (3-D) model has been developed to simulate shallow water flows in large water bodies, such as coastal and estuarine waters. The eddy viscosity is determined using a newly modified mixing length model that uses different mixing length functions for the horizontal and vertical shear strain rates. The 3-D shallow water flow equations with the hydrostatic pressure assumption are solved using an implicit finite-volume method based on a quadtree (telescoping) rectangular mesh on the horizontal plane and the sigma coordinate in the vertical direction. The quadtree technique can locally refine the mesh around structures or in high-gradient regions by splitting a coarse cell into four child cells. The grid nodes are numbered with a one-dimensional index system that has unstructured grid feature for better grid flexibility. All the primary variables are arranged in a non-staggered grid system. Fluxes at cell faces are determined using a Rhie and Chow-type momentum interpolation, to avoid the possible spurious checkerboard oscillations caused by linear interpolation. Each of the discretized governing equations is solved iteratively using the flexible GMRES method with ILUT preconditioning, and coupling of water level and velocity among these equations is achieved by using the SIMPLEC algorithm with under-relaxation. The model has been tested in four cases, including steady flow near a spur-dyke, tidal flows in San Francisco Bay and Gironde Estuary, and wind-induced current in a flume. The calculated water levels and velocities are in good agreement with the measured values.

Fluid overpressures and strength of the sedimentary upper crust

The classic crustal strength-depth profile based on rock mechanics predicts a brittle strength σ1−σ3=κ(ρ¯gz−Pf) that increases linearly with depth as a consequence of [1] the intrinsic brittle pressure dependence κ plus [2] an assumption of hydrostatic pore-fluid pressure, Pf = ρwgz. Many deep borehole stress data agree with a critical state of failure of this form. In contrast, fluid pressures greater than hydrostatic ρ¯gz>Pf>ρwgz are normally observed in clastic continental margins and shale-rich mountain belts. Therefore we explore the predicted shapes of strength-depth profiles using data from overpressured regions, especially those dominated by the widespread disequilibrium-compaction mechanism, in which fluid pressures are hydrostatic above the fluid-retention depth zFRD and overpressured below, increasing parallel to the lithostatic gradient ρ¯gz. Both brittle crustal strength and frictional fault strength below the zFRD must be constant with depth because effective stress (ρ¯gz−Pf) is constant, in contrast with the classic linearly increasing profile. Borehole stress and fluid-pressure measurements in several overpressured deforming continental margins agree with this constant-strength prediction, with the same pressure-dependence κ as the overlying hydrostatic strata. The role of zFRD in critical-taper wedge mechanics and jointing is illustrated. The constant-strength approximation is more appropriate for overpressured crust than classic linearly increasing models.

A finite element/volume method model of the depth‐averaged horizontally 2D shallow water equations

SUMMARYAnalysis of surface water flows is of central importance in understanding and predicting a wide range of water engineering issues. Dynamics of surface water is reasonably well described using the shallow water equations (SWEs) with the hydrostatic pressure assumption. The SWEs are nonlinear hyperbolic partial differential equations that are in general required to be solved numerically. Application of a simple and efficient numerical model is desirable for solving the SWEs in practical problems. This study develops a new numerical model of the depth‐averaged horizontally 2D SWEs referred to as 2D finite element/volume method (2D FEVM) model. The continuity equation is solved with the conforming, standard Galerkin FEM scheme and momentum equations with an upwind, cell‐centered finite volume method scheme, utilizing the water surface elevation and the line discharges as unknowns aligned in a staggered manner. The 2D FEVM model relies on neither Riemann solvers nor high‐resolution algorithms in order to serve as a simple numerical model. Water at a rest state is exactly preserved in the model. A fully explicit temporal integration is achieved in the model using an efficient approximate matrix inversion method. A series of test problems, containing three benchmark problems and three experiments of transcritical flows, are carried out to assess accuracy and versatility of the model. Copyright © 2014 John Wiley & Sons, Ltd.

A tsunami simulation of Hakata Bay using the viscous shallow-water equations

The tsunami generated by the Great East Japan Earthquake caused serious damage to the coastal areas of the Tohoku district. Numerical simulations are used to predict damage caused by tsunamis. Shallow-water equations are generally used in numerical simulations of tsunami propagation from the open sea to the coast. This research focuses on viscous shallow-water equations and attempts to generate a computational method using finite element techniques based on the previous investigations of Kanayama and Ohtsuka (Coast Eng Jpn 21:157–171, 1978). First, the viscous shallow-water equation system is derived from the Navier–Stokes equations, based on the assumption of hydrostatic pressure in the direction of gravity. Next a numerical scheme is shown. Finally, tsunami simulations of Hakata Bay and Tohoku-Oki are shown using the approach.

Vertical two-dimensional non-hydrostatic pressure model with single layer

The vertical two-dimensional non-hydrostatic pressure models with multiple layers can make prediction more accurate than those obtained by the hydrostatic pres-sure assumption. However, they are time-consuming and unstable, which makes them unsuitable for wider application. In this study, an efficient model with a single layer is developed. Decomposing the pressure into the hydrostatic and dynamic components and integrating the x-momentum equation from the bottom to the free surface can yield a horizontal momentum equation, in which the terms relevant to the dynamic pressure are discretized semi-implicitly. The convective terms in the vertical momentum equation are ignored, and the rest of the equation is approximated with the Keller-box scheme. The velocities expressed as the unknown dynamic pressure are substituted into the continuity equation, resulting in a tri-diagonal linear system solved by the Thomas algorithm. The validation of solitary and sinusoidal waves indicates that the present model can provide comparable results to the models with multiple layers but at much lower computation cost.

Depth-integrated, non-hydrostatic model using a new alternating direction implicit scheme

The hydrostatic pressure assumption has been widely applied in simulating rivers, lakes and reservoirs, but it has been found inappropriate in various cases where the vertical acceleration is significant. This paper presents a depth-integrated, non-hydrostatic model using a new alternating direction implicit scheme. Using the proposed scheme, each step is split into two half steps. In the first half step, the dynamic pressure and the x-direction velocity in the continuity equation and the momentum equations in the x-direction and z-direction are expressed implicitly, and the others explicitly; in the second half step, the dynamic pressure and the y-direction velocity in the continuity equation and the momentum equations in the y-direction and z-direction are discretized implicitly, and the others explicitly. The Thomas algorithm is applied to solve the tri-diagonal linear system at each half step. The model is developed and validated with several analytical solutions and laboratory experiments. The results show that the model can provide comparable results at very low computational cost.

Implementation of a direct nonhydrostatic pressure gradient discretisation into a layered ocean model

The increasing demand for the investigation of nonhydrostatic effects in ocean modelling requires crucial modifications to models applying the hydrostatic pressure assumption. Within many studies the capability of the pressure-correcting projection method for the inclusion of the missing nonhydrostatic pressure contribution into an existing hydrostatic model kernel was verified. It provides accurate results but also requires computational (and implementational) effort for solving a Poisson equation for the pressure correction.In contrast, some studies were based on an alternative approach that does not require the inversion of a Poisson equation. Within this alternative approach the nonhydrostatic pressure contribution is calculated by an explicit vertical integration of the additional nonhydrostatic terms in the balance of vertical momentum. Since ocean models are not intended to replace classical engineering tools and are usually applied to nearly hydrostatic flows, this straight-forward extension of the hydrostatic procedure is an interesting option. However, the feasibility of the alternative approach was not tested within a full 3D explicit mode-splitting model yet. Furthermore, the nonhydrostatic capability of the alternative approach has so far not been validated against well known nonhydrostatic benchmark test cases. In order to assess the potential of the alternative approach for future ocean modelling applications, in the present study these required but still missing investigations are carried out.To demonstrate the necessary modifications to an explicit mode-splitting hydrostatic model kernel, the extension of the General Estuarine Transport Model (GETM) is outlined. The simulation results of laboratory and idealised oceanic test cases are presented and compared to analytical theory, laboratory experiments and other numerical simulations. This validation indicates the nonhydrostatic capability of the extended GETM. However, stability issues limited the efficiency and the applicability of the straight-forward inclusion of nonhydrostatic dynamics.