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1.
J. J. Westerink 《Advances in water resources》1987,10(4)
A finite element method formulation for solving the harmonic shallow water equations in their primitive or unmodified form is developed and analysed. The scheme, referred to as the Primitive Pseudo Wave Equation Formulation (PPWE), involves developing a weak weighted residual form of the continuity equation and furthermore forming a pseudo wave equation by substituting the discretized form of the momentum equation into the discretized form of the continuity equation. The final set of equations to be solved, the pseudo wave equation and the primitive momentum equations, deceptively resemble the discretized equations of the wave equation formulation of Lynch and Gray. Despite this resemblance, Fourier analysis indicates that the PPWE scheme is still fundamentally primitive.However, application of the PPWE scheme to a set of stringent test problems results in very good solutions with well controlled nodal oscillations. It is shown that this low degree of spurious oscillations is due to the treatment of the boundary conditions such that elevation is taken as an essential condition and normal flux is taken as a natural condition. This particular boundary condition treatment is suggested by the formation of the pseudo wave equation. Furthermore, even though the equation re-arrangement does not in itself effect the solutions, it does make the scheme much more efficient. 相似文献
2.
Waves, circulation and vertical dependence 总被引：1，自引：1，他引：0
George Mellor 《Ocean Dynamics》2013,63(4):447-457
Longuet-Higgins and Stewart (J Fluid Mech 13:481–504, 1962; Deep-Sea Res 11:529–562, 1964) and later Phillips (1977) introduced the problem of waves incident on a beach, from deep to shallow water. From the wave energy equation and the vertically integrated continuity equation, they inferred velocities to be Stokes drift plus a return current so that the vertical integral of the combined velocities was nil. As a consequence, it can be shown that velocities of the order of Stokes drift rendered the advective term in the momentum equation negligible resulting in a simple balance between the horizontal gradients of the vertically integrated elevation and wave radiation stress terms; the latter was first derived by Longuet-Higgins and Stewart. Mellor (J Phys Oceanogr 33:1978–1989, 2003a), noting that vertically integrated continuity and momentum equations were not able to deal with three-dimensional numerical or analytical ocean models, derived a vertically dependent theory of wave–circulation interaction. It has since been partially revised and the revisions are reviewed here. The theory is comprised of the conventional, three-dimensional, continuity and momentum equations plus a vertically distributed, wave radiation stress term. When applied to the problem of waves incident on a beach with essentially zero turbulence momentum mixing, velocities are very large and the simple balance between elevation and radiation stress gradients no longer prevails. However, when turbulence mixing is reinstated, the vertically dependent radiation stresses produce vertical velocity gradients which then produce turbulent mixing; as a consequence, velocities are reduced, but are still larger by an order of magnitude compared to Stokes drift. Nevertheless, the velocity reduction is sufficient so that elevation set-down obtained from a balance between elevation gradient and radiation stress gradients is nearly coincident with that obtained by the aforementioned papers. This paper includes four appendices. The first appendix demonstrates the numerical process by which Stokes drift is excluded from the turbulence stress parameterization in the momentum equation. A second appendix determines a bottom slope criterion for the application of linear wave relations to the derivation of the wave radiation stress. The third appendix explores the possibility of generalizing results by non-dimensionalization. The final appendix applies the basic theory to a problem introduced by Bennis and Ardhuin (J Phys Oceanogr 41:2008–2012, 2011). 相似文献
3.
IINTRODUCTIONNumericalmethodsasatooltosimulateflowsandpollutanttransportareincreasinglyimportantinhydraulicandenvironmentalengineering.AveryusefulapplicationofthenumericalmethodologyinengineeringproblemswouldbetosolvethesystemofZDdepth-integratedshallowwaterequations.ManysolutionsofthegoverningequationsarederivedusingtraditionalfinitedifferencemethodonCartesianregulargrids.ThedisadvantageofthismethodseemstobetheinflexibilityofCartesiangridstocomplywithirregularorcurvedperimeterswhichsur… 相似文献
4.
In this paper a fuzzy dynamic wave routing model (FDWRM) for unsteady flow simulation in open channels is presented. The continuity equation of the dynamic wave routing model is preserved in its original form while the momentum equation is replaced by a fuzzy rule based model which is developed on the principle that during unsteady flow the disturbances in the form of discontinuities in the gradient of the physical parameters will propagate along the characteristics with a velocity equal to that of velocity of the shallow water wave. The model gets rid off the assumptions associated with the momentum equation by replacing it with the fuzzy rule based model. It overcomes the necessity of calculating friction slope (Sf) in flow routing and hence the associated uncertainties are eliminated. The robustness of the fuzzy rule based model enables the FDWRM to march the solution even in regions where the aforementioned assumptions are violated. Also the model can be used for flow routing in curved channels. When the model is applied to hypothetical flood routing problems in a river it is observed that the results are comparable to those of an implicit numerical model (INM) which solves the dynamic wave equations using an implicit numerical scheme. The model is also applied to a real case of flow routing in a field canal. The results match well with the measured data and the model performs better than the INM. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
5.
Shaohua Marko HSU Associate Professor Dr. Department of Hydraulic Engineering Feng Chia University Taichung Taiwan China Wei-Sheng YU Environment Section Agricultural Engineering Research Center Chungli Taiwan China Tzu-Ming LIU Graduate stu 《国际泥沙研究》1997,(3)
1.IN~DUCTIONTurbiditycurrentisoneclassofflowsnameddensitycurrentorgravitycurrent(therHunterRouse(Yih(1980)),whichdePictstheintmsionofheaVyfluidintoalighterone.Usually,thedensitydifferencebetWeentWonuidisrelativelysmallandmixingacrosstheimerfaceoccurs.ThedrivingforceofdensitycurrentsisnotdensitydifferenceitselfbutthedifferenceinspeCmcweights.Turbiditycurrentisnamedwhenthedensitydifferenceisespeciallycausedbysuspendedfinesedimentparticles.Sincesediment-ladenflowcaninteraCtwiththelowerbou… 相似文献
6.
《Advances in water resources》1998,21(5):327-337
A p finite element scheme and parallel iterative solver are introduced for a modified form of the shallow water equations. The governing equations are the three-dimensional shallow water equations. After a harmonic decomposition in time and rearrangement, the resulting equations are a complex Helmholz problem for surface elevation, and a complex momentum equation for the horizontal velocity. Both equations are nonlinear and the resulting system is solved using the Picard iteration combined with a preconditioned biconjugate gradient (PBCG) method for the linearized subproblems. A subdomain-based parallel preconditioner is developed which uses incomplete LU factorization with thresholding (ILUT) methods within subdomains, overlapping ILUT factorizations for subdomain boundaries and under-relaxed iteration for the resulting block system. The method builds on techniques successfully applied to linear elements by introducing ordering and condensation techniques to handle uniform p refinement. The combined methods show good performance for a range of p (element order), h (element size), and N (number of processors). Performance and scalability results are presented for a field scale problem where up to 512 processors are used. 相似文献
7.
高精度频率域弹性波方程有限差分方法及波场模拟 总被引：18，自引：4，他引：14
有限差分方法是波场数值模拟的一个重要方法,但常规的有限差分法本身存在着数值频散问题,会降低波场模拟的精度与分辨率,为了克服常规差分算子的数值频散,本文采用25点优化差分算子,再根据最优化理论求取的优化系数,建立了频率空间域中弹性波波动方程的差分格式;为了消除边界反射,引入最佳匹配层,构造了各向同性介质中弹性波方程在不同边界和角点处的边界条件. 最后由弹性波波动方程和边界条件,通过频率域有限差分法,分别利用不同震源对弹性波在均匀各向同性介质、层状介质及凹陷模型中的传播过程进行了数值正演模拟,得到了单频波波场、时间切片和共炮点道集,为下一步的研究工作（如成像、反演）提供了研究基础. 相似文献
8.
A coupled model is developed to investigate the dynamic interaction between an offshore pile, a porous seabed and seawater when subjected to the pseudo-Stoneley wave along the seabed and the seawater interface. The pile and the seabed are treated as the porous medium governed by Biot's theory, while the seawater is considered as an acoustic medium and is described by the conventional Helmholtz equation. The free field solution of the incident pseudo-Stoneley wave is obtained using Biot's theory and the potential method. Based on the boundary element method (BEM) for the porous medium and the acoustic medium, three BEM formulations are constructed for the pile, the seabed and the seawater, respectively, which are combined together using the continuity conditions between the pile, the seabed and the seawater to formulate the coupled model for the system. As shown in numerical examples, when the system is subjected to the pseudo-Stoneley wave, the maximum pore pressure of the seabed usually occurs at the region near the interfaces between the seabed and the seawater. 相似文献
9.
《Advances in water resources》1987,10(3):138-148
The amplitude and phase of 11 tidal constituents for the English Channel and southern North Sea are calculated using a frequency domain, finite element model. The governing equations — the shallow water equations — are modifed such that sea level is calculated using an elliptic equation of the Helmholz type followed by a back-calculation of velocity using the primitive momentum equations. Triangular elements with linear basis functions are used. The modified form of the governing equations provides stable solutions with little numerical noise. In this field-scale test problem, the model was able to produce the details of the structure of 11 tidal constituents including O1, K1, M2, S2, N2, K2, M4, MS4, MN4, M6, and 2MS6. 相似文献
10.
《Advances in water resources》2002,25(8-12):1105-1117
Macroscopic differential equations of mass and momentum balance for two immiscible fluids in a deformable porous medium are derived in an Eulerian framework using the continuum theory of mixtures. After inclusion of constitutive relationships, the resulting momentum balance equations feature terms characterizing the coupling among the fluid phases and the solid matrix caused by their relative accelerations. These terms, which imply a number of interesting phenomena, do not appear in current hydrologic models of subsurface multiphase flow. Our equations of momentum balance are shown to reduce to the Berryman–Thigpen–Chen model of bulk elastic wave propagation through unsaturated porous media after simplification (e.g., isothermal conditions, neglect of gravity, etc.) and under the assumption of constant volume fractions and material densities. When specialized to the case of a porous medium containing a single fluid and an elastic solid, our momentum balance equations reduce to the well-known Biot model of poroelasticity. We also show that mass balance alone is sufficient to derive the Biot model stress–strain relations, provided that a closure condition for porosity change suggested by de la Cruz and Spanos is invoked. Finally, a relation between elastic parameters and inertial coupling coefficients is derived that permits the partial differential equations of the Biot model to be decoupled into a telegraph equation and a wave equation whose respective dependent variables are two different linear combinations of the dilatations of the solid and the fluid. 相似文献
11.
《Advances in water resources》1999,22(5):461-478
Fractional flow formulations of the multi-phase flow equations exhibit several attractive attributes for numerical simulations. The governing equations are a saturation equation having an advection diffusion form, for which characteristic methods are suited, and a global pressure equation whose form is elliptic. The fractional flow approach to the governing equations is compared with other approaches and the implication of equation form for numerical methods discussed. The fractional flow equations are solved with a modified method of characteristics for the saturation equation and a finite element method for the pressure equation. An iterative algorithm for determination of the general boundary conditions is implemented. Comparisons are made with a numerical method based on the two-pressure formulation of the governing equations. While the fractional flow approach is attractive for model problems, the performance of numerical methods based on these equations is relatively poor when the method is applied to general boundary conditions. We expect similar difficulties with the fractional flow approach for more general problems involving heterogenous material properties and multiple spatial dimensions. 相似文献
12.
The strong coupling of applied stress and pore fluid pressure, known as poroelasticity, is relevant to a number of applied problems arising in hydrogeology and reservoir engineering. The standard theory of poroelastic behavior in a homogeneous, isotropic, elastic porous medium saturated by a viscous, compressible fluid is due to Biot, who derived a pair of coupled partial differential equations that accurately predict the existence of two independent dilatational (compressional) wave motions, corresponding to in-phase and out-of-phase displacements of the solid and fluid phases, respectively. The Biot equations can be decoupled exactly after Fourier transformation to the frequency domain, but the resulting pair of Helmholtz equations cannot be converted to partial differential equations in the time domain and, therefore, closed-form analytical solutions of these equations in space and time variables cannot be obtained. In this paper we show that the decoupled Helmholtz equations can in fact be transformed to two independent partial differential equations in the time domain if the wave excitation frequency is very small as compared to a critical frequency equal to the kinematic viscosity of the pore fluid divided by the permeability of the porous medium. The partial differential equations found are a propagating wave equation and a dissipative wave equation, for which closed-form solutions are known under a variety of initial and boundary conditions. Numerical calculations indicate that the magnitude of the critical frequency for representative sedimentary materials containing either water or a nonaqueous phase liquid is in the kHz–MHz range, which is generally above the seismic band of frequencies. Therefore, the two partial differential equations obtained should be accurate for modeling elastic wave phenomena in fluid-saturated porous media under typical low-frequency conditions applicable to hydrogeological problems. 相似文献
13.
Herman W. J. Kernkamp Henri A. H. Petit Herman Gerritsen Erik D. de Goede 《Ocean Dynamics》2005,55(3-4):351-369
In this paper, the formulations of the primitive equations for shallow water flow in various horizontal co-ordinate systems
and the associated finite difference grid options used in shallow water flow modelling are reviewed. It is observed that horizontal
co-ordinate transformations do not affect the chosen co-ordinate system and representation in the vertical, and are the same
for the three- and two-dimensional cases. A systematic derivation of the equations in tensor notation is presented, resulting
in a unified formulation for the shallow water equations that covers all orthogonal horizontal grid types of practical interest.
This includes spherical curvilinear orthogonal co-ordinate systems on the globe. Computational efficiency can be achieved
in a single computer code. Furthermore, a single numerical algorithmic code implementation satisfies. All co-ordinate system
specific metrics are determined as part of a computer-aided model grid design, which supports all four orthogonal grid types.
Existing intuitive grid design and visual interpretation is conserved by appropriate conformal mappings, which conserve spherical
orthogonality in planar representation. A spherical curvilinear co-ordinate solution of wind driven steady channel flow applying
a strongly distorted grid is shown to give good agreement with a regular spherical co-ordinate model approach and the solution
based on a β-plane approximation. Especially designed spherical curvilinear boundary fitted model grids are shown for typhoon
surge propagation in the South China Sea and for ocean-driven flows through Malacca Straits. By using spherical curvilinear
grids the number of grid points in these single model grid applications is reduced by a factor of 50–100 in comparison with
regular spherical grids that have the same horizontal resolution in the area of interest. The spherical curvilinear approach
combines the advantages of the various grid approaches, while the overall computational effort remains acceptable for very
large model domains. 相似文献
14.
基于大气边界层动量、感热和水汽通量的基本方程,定量地计算了波动海面的动量、感热和水汽通量。首先,应用Prandtl的混合长概念,推导出贴海面大气层中风速、位温和比湿的涡动交换率及其贴海面层厚度,并且证明了波面上位温或比湿贴海面层厚度与速度贴海面层厚度的比值,和平面上的相应比值完全相等。随后,利用空气动力学理论讨论了贴海面动量、感热和水汽输送的参数化问题。最后,对现有五种理论模式进行了比较说明。 相似文献
15.
A fuzzy dynamic flood routing model (FDFRM) for natural channels is presented, wherein the flood wave can be approximated to a monoclinal wave. This study is based on modification of an earlier published work by the same authors, where the nature of the wave was of gravity type. Momentum equation of the dynamic wave model is replaced by a fuzzy rule based model, while retaining the continuity equation in its complete form. Hence, the FDFRM gets rid of the assumptions associated with the momentum equation. Also, it overcomes the necessity of calculating friction slope (Sf) in flood routing and hence the associated uncertainties are eliminated. The fuzzy rule based model is developed on an equation for wave velocity, which is obtained in terms of discontinuities in the gradient of flow parameters. The channel reach is divided into a number of approximately uniform sub‐reaches. Training set required for development of the fuzzy rule based model for each sub‐reach is obtained from discharge‐area relationship at its mean section. For highly heterogeneous sub‐reaches, optimized fuzzy rule based models are obtained by means of a neuro‐fuzzy algorithm. For demonstration, the FDFRM is applied to flood routing problems in a fictitious channel with single uniform reach, in a fictitious channel with two uniform sub‐reaches and also in a natural channel with a number of approximately uniform sub‐reaches. It is observed that in cases of the fictitious channels, the FDFRM outputs match well with those of an implicit numerical model (INM), which solves the dynamic wave equations using an implicit numerical scheme. For the natural channel, the FDFRM outputs are comparable to those of the HEC‐RAS model. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
16.
In this study, a theoretical approach is used to investigate the scattering problem of circular holes under a scalene triangle on the surface. The wave displacement function is obtained by solving the Helmholtz equation that meets the zero-stress boundary conditions by adopting the method of separation of variables. Based on the complex function,multi-polar coordinate method, and region-matching technique, algebraic equations are established at auxiliary boundaries and free boundaries conditions... 相似文献
17.
18.
《Advances in water resources》2005,28(4):345-358
Nearly all generalized wave continuity (GWC)-based models utilize the velocity-based, non-conservative form of the momentum equation to obtain the depth-averaged changes in velocity. It has been hypothesized that a flux-based, conservative form of the momentum equation may improve accuracy and stability. Herein, we study the impact of the choice of dependent variable and form of the momentum equation in a GWC-based finite element shallow water model. The impact of this change on mass balance, stability, and accuracy (spatial and temporal) is rigorously assessed, first for 1D barotropic flows and then for 2D barotropic flows in a variety of basins. Both 1D and 2D results indicate that the conservative form improves mass balance on both global and local scales, with the most significant gains found in local mass balance in areas with steep bathymetry gradients. This is also the region where the conservative form shows an increase in local spatial accuracy. Taylor series analysis and numerical simulations indicate a strong correlation between local spatial truncation errors and local mass balance errors. Stability, temporal accuracy and global spatial accuracy do not show statistically significant changes between the two algorithms in both 1D and 2D studies. 相似文献
19.
The weakly nonlinear dynamics of packets of equatorial Kelvin waves is studied using singular perturbation theory applied to the shallow water wave equations. Within the limits of the perturbation theory, which is formally restricted to weak mean shear and weak nonlinearity, we derive a Nonlinear Schroedinger equation to describe the envelope of the wave packet. We find that nonlinearity has a defocusing effect so that coherent wave packets must owe their existence entirely to the generation mechanism rather than to nonlinear focusing of a broad initial disturbance. 相似文献
20.
V. P. Singh 《水文研究》1994,8(4):311-326
Error equations for the kinematic wave and diffusion wave approximations with lateral inflow neglected in the momentum equation are derived under simplified conditions for space-independent flows. These equations specify error as a function of time in the flow hydrograph. The kinematic wave, diffusion wave and dynamic wave solutions are parameterized through a dimensionless parameter γ which is dependent on the initial conditions. This parameter reflects the effect of initial flow depth, channel-bed slope, lateral inflow and channel roughness when the initial condition is non-vanishing; and it reflects the effect of bed slope, channel roughness and acceleration due to gravity when the initial condition is vanishing. The error equations are found to be the Riccati equation. The structure of the error equations in the case when the momentum equation neglects lateral inflow is different from that when the lateral inflow is included. 相似文献