复杂水流的高分辨率数值模拟
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摘要
本文对自由面复杂水流运动问题,采用带源项的浅水方程和Navier-Stokes方程进行模拟,建立了浅水方程的高精度高分辨率非结构化网格有限体积离散模型,以及三维Navier-Stokes方程的动压结构化分层网格的半隐式有限差分离散模型。全文的总体内容如下:
第一,基于空间交错网格系统,采用半隐式格式对圣维南方程进行离散,同时利用加通量修正的迎风格式对对流项进行处理,使格式满足在初始数据光滑处达高阶精度和间断处达低阶精度的自动转换,避免了数值求解中在间断处产生非物理的数值振荡。通过实例验证了此法具有很好的守恒性、精确性以及很强的间断捕捉性能。
第二,基于非结构化三角形网格技术,建立了二维浅水方程的高精度高分辨率有限体积数值离散模型。源项采用矩阵向量散度和全隐的处理方式,保证了格式的和谐性和稳定性。模型在干湿界面处利用能确保质量守恒的有效动边界处理技术,增强了模型处理非恒定动边界问题的能力,提高了模型的实用性。采取加通量限制的空间重构方法获得空间高阶精度和防止间断处非物理振荡的产生。通过部分溃坝、斜水跃问题以及新安江模型验证了本文二维浅水模型的精度、和谐性和捕捉间断的能力。为了使模型能够应用于实际水流问题的计算,又进一步将其应用到了英那河水库的溃坝洪水研究中,体现了模型解决实际工程问题的能力。
第三,基于垂向直接分层的平面为结构化四边形网格剖分的处理技术,建立了自由面流动问题的三维Navier-Stokes方程的半隐式有限差分动压模型。在水深方向上积分满足运动学边界条件的连续性方程得到水位方程,将其与动量离散方程联立数值求解,即可实现对自由面进行捕捉。动压修正值是通过求解压力泊松方程来获得的。出流边界利用在水平动量方程右端附加人工阻尼项的海绵层处理来消减波能,并于海绵层末端进一步加入辐射边界条件,减少反射波的影响。采用κ-ε双方程紊流模型求解涡粘性系数,使三维模型达到封闭。最后通过三维线性驻波、Delft水力学中Scheldt水槽实验的周期性入射波通过淹没潜堤的传播问题以及连接丹麦和瑞典间海域的潮流场模拟等算例,从多方面验证了模型具有稳定、实用、高效、灵活等的优点,不但能对短波问题进行精确成功的模拟,而且也适合对大范围的水体进行长时段的模拟。
本文所选择的验证算例全都是具有解析解或实测数据的,通过与模拟值的比较分析表明本文所建立的数学模型具有很好的水流模拟性能。
In this paper,free surface complex flow problems,using shallow water equations of source terms and Navier-Stokes equations modeling.a high-order and high-resolution discrete model for shallow water equation based on unstructured grid finite volume technique and a hydrodynamic pressure discrete model for Navier-Stokes equation base on structured layered grid semi-implicit finite difference numerical technique are established for the free-surface practical flow problems.For one-dimensional shallow water equations(Saint-Venant equations),in based on a space staggered grid system,a semi-implicit discrete scheme that has strong capacity to capture discontinuous is applied,simultaneity,the flux limiter method, is implemented to control the momentum conservative of the proposed scheme.For bed slope source term and friction slope source terms in two-dimensional shallow water,we present a fully implicit method consists of treating the bed slope source term as the divergence of a proper matrix.This method guarantees harmoniousness and stability of model.Based on Eulerian-Lagrangian method and k-εequations turbulence model,a semi-implicit discrete model used the structured grid for three-dimensional Navier-Stokes equations with stability hydrodynamic pressure is established.Applying object oriented programming language C# based on Visual studio.net2005 compiling environment,the writing,debugging and verification of the program is done:
First,based on space staggered grid system,using semi-implicit scheme Saint-Venant equations discretized.In order to improve the accuracy,the flux limiter method has been used. This high-order resolution method switches between the second-order approximation when the data are sufficiently smooth and the first-order approximation near a discontinuity.This treatment avoids nonphysical numerical oscillations near a discontinuity.A few computational examples on classical test cases are given to illustrate the properties of the present method in terms of balance,accuracy and efficiency.
Second,using technique of unstructured triangle grid,a high-order and high-resolution finite volume numerical discreted model is established for the 2D shallow water equations. The proposed method consists of writing the bed slope source term as the divergence of a proper matrix,related to the static force due to bottom slope,The friction terms are treated fully implicitly to prevent numerical instabilities.The interface between dry and wet region in this model is using a moving boundary that is satisfied mass conservation.This technique improves the ability of treating the moving boundary in unsteady flow problem.The technique of space reconstruction and method of flux limiter are employed to achieve high-order spatial accuracy and to prevent nonphysical oscillations.Simultaneous,using eigen decomposition method for boundary condition is processed,computed flux of boundary edge, furthermore,the flux substitutes into solution of the finite volume discreted equations. Numerical tests of two-dimensional oblique hydraulic jump,dam break and Xin Anjiang model are employed to test the this two-dimensional model's accuracy,conservation and capability of handling discontinuous solutions.To apply the model in practical flows,an example of a dam break flood of Ying Nahe reservoir is applied to prove the model's harmoniousness and capability of simulating practical flows.
Third,based on horizontal structured grid and vertical directly layered grid treatment technology,a three-dimensional Navier-Stokes equations of free-surface flow model is established in a hydrodynamic pressure semi-implicit finite difference numerical discrete scheme.In water depth direction,free-surface equation can be obtained by integral continuity equation,which is satisfied kinematic condition at the free surface.To solve this equation and momentum equation,the free surface can be captured directly.The hydrodynamic pressure correction is solved from the pressure Poisson equation(PPE).At outflow,a combination of a sponge layer technique and a radiation boundary condition is applied to minimize wave reflection.For the sponge layer,the artificial damping terms are added to the right-hand side of horizontal momentum equation,at the end of the sponge layer,the radiation condition is used further to enhance wave absorption.In order to make three-dimensional model closed, turbulence model uses k-εequations to solve eddy viscosity coefficient.By three-dimensional linear standing wave,the wave propagation over a bar of scheldt flume experiment carried at Delft hydraulics and tidal current field simulation of waters of between Denmark and Sweden etc example,the model is verified to have advantages such as stability, high efficiency etc,and proved suitable for the simulation both water body of large range and long period.
Numerical models established by this paper provide good results by comparing numerical results with analytical solution or experimental data.
第一,基于空间交错网格系统,采用半隐式格式对圣维南方程进行离散,同时利用加通量修正的迎风格式对对流项进行处理,使格式满足在初始数据光滑处达高阶精度和间断处达低阶精度的自动转换,避免了数值求解中在间断处产生非物理的数值振荡。通过实例验证了此法具有很好的守恒性、精确性以及很强的间断捕捉性能。
第二,基于非结构化三角形网格技术,建立了二维浅水方程的高精度高分辨率有限体积数值离散模型。源项采用矩阵向量散度和全隐的处理方式,保证了格式的和谐性和稳定性。模型在干湿界面处利用能确保质量守恒的有效动边界处理技术,增强了模型处理非恒定动边界问题的能力,提高了模型的实用性。采取加通量限制的空间重构方法获得空间高阶精度和防止间断处非物理振荡的产生。通过部分溃坝、斜水跃问题以及新安江模型验证了本文二维浅水模型的精度、和谐性和捕捉间断的能力。为了使模型能够应用于实际水流问题的计算,又进一步将其应用到了英那河水库的溃坝洪水研究中,体现了模型解决实际工程问题的能力。
第三,基于垂向直接分层的平面为结构化四边形网格剖分的处理技术,建立了自由面流动问题的三维Navier-Stokes方程的半隐式有限差分动压模型。在水深方向上积分满足运动学边界条件的连续性方程得到水位方程,将其与动量离散方程联立数值求解,即可实现对自由面进行捕捉。动压修正值是通过求解压力泊松方程来获得的。出流边界利用在水平动量方程右端附加人工阻尼项的海绵层处理来消减波能,并于海绵层末端进一步加入辐射边界条件,减少反射波的影响。采用κ-ε双方程紊流模型求解涡粘性系数,使三维模型达到封闭。最后通过三维线性驻波、Delft水力学中Scheldt水槽实验的周期性入射波通过淹没潜堤的传播问题以及连接丹麦和瑞典间海域的潮流场模拟等算例,从多方面验证了模型具有稳定、实用、高效、灵活等的优点,不但能对短波问题进行精确成功的模拟,而且也适合对大范围的水体进行长时段的模拟。
本文所选择的验证算例全都是具有解析解或实测数据的,通过与模拟值的比较分析表明本文所建立的数学模型具有很好的水流模拟性能。
In this paper,free surface complex flow problems,using shallow water equations of source terms and Navier-Stokes equations modeling.a high-order and high-resolution discrete model for shallow water equation based on unstructured grid finite volume technique and a hydrodynamic pressure discrete model for Navier-Stokes equation base on structured layered grid semi-implicit finite difference numerical technique are established for the free-surface practical flow problems.For one-dimensional shallow water equations(Saint-Venant equations),in based on a space staggered grid system,a semi-implicit discrete scheme that has strong capacity to capture discontinuous is applied,simultaneity,the flux limiter method, is implemented to control the momentum conservative of the proposed scheme.For bed slope source term and friction slope source terms in two-dimensional shallow water,we present a fully implicit method consists of treating the bed slope source term as the divergence of a proper matrix.This method guarantees harmoniousness and stability of model.Based on Eulerian-Lagrangian method and k-εequations turbulence model,a semi-implicit discrete model used the structured grid for three-dimensional Navier-Stokes equations with stability hydrodynamic pressure is established.Applying object oriented programming language C# based on Visual studio.net2005 compiling environment,the writing,debugging and verification of the program is done:
First,based on space staggered grid system,using semi-implicit scheme Saint-Venant equations discretized.In order to improve the accuracy,the flux limiter method has been used. This high-order resolution method switches between the second-order approximation when the data are sufficiently smooth and the first-order approximation near a discontinuity.This treatment avoids nonphysical numerical oscillations near a discontinuity.A few computational examples on classical test cases are given to illustrate the properties of the present method in terms of balance,accuracy and efficiency.
Second,using technique of unstructured triangle grid,a high-order and high-resolution finite volume numerical discreted model is established for the 2D shallow water equations. The proposed method consists of writing the bed slope source term as the divergence of a proper matrix,related to the static force due to bottom slope,The friction terms are treated fully implicitly to prevent numerical instabilities.The interface between dry and wet region in this model is using a moving boundary that is satisfied mass conservation.This technique improves the ability of treating the moving boundary in unsteady flow problem.The technique of space reconstruction and method of flux limiter are employed to achieve high-order spatial accuracy and to prevent nonphysical oscillations.Simultaneous,using eigen decomposition method for boundary condition is processed,computed flux of boundary edge, furthermore,the flux substitutes into solution of the finite volume discreted equations. Numerical tests of two-dimensional oblique hydraulic jump,dam break and Xin Anjiang model are employed to test the this two-dimensional model's accuracy,conservation and capability of handling discontinuous solutions.To apply the model in practical flows,an example of a dam break flood of Ying Nahe reservoir is applied to prove the model's harmoniousness and capability of simulating practical flows.
Third,based on horizontal structured grid and vertical directly layered grid treatment technology,a three-dimensional Navier-Stokes equations of free-surface flow model is established in a hydrodynamic pressure semi-implicit finite difference numerical discrete scheme.In water depth direction,free-surface equation can be obtained by integral continuity equation,which is satisfied kinematic condition at the free surface.To solve this equation and momentum equation,the free surface can be captured directly.The hydrodynamic pressure correction is solved from the pressure Poisson equation(PPE).At outflow,a combination of a sponge layer technique and a radiation boundary condition is applied to minimize wave reflection.For the sponge layer,the artificial damping terms are added to the right-hand side of horizontal momentum equation,at the end of the sponge layer,the radiation condition is used further to enhance wave absorption.In order to make three-dimensional model closed, turbulence model uses k-εequations to solve eddy viscosity coefficient.By three-dimensional linear standing wave,the wave propagation over a bar of scheldt flume experiment carried at Delft hydraulics and tidal current field simulation of waters of between Denmark and Sweden etc example,the model is verified to have advantages such as stability, high efficiency etc,and proved suitable for the simulation both water body of large range and long period.
Numerical models established by this paper provide good results by comparing numerical results with analytical solution or experimental data.
引文
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