Application of adjusted Youngs-VOF method for shallow water problems

Application of adjusted Youngs-VOF method for shallow water problems | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Application of adjusted Youngs-VOF method for shallow water problems Dongmiao Zhao, Jian Chen, Qi Liu, Yi Liu, Yang Zhou, Yi Liu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4324162/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Shallow water problems such as tidal bore, dam-break wave, shallow water wave etc. are important for design, construction, and operation of water conservancy projects. In this study, a 2-D vertical flow model having a strong adaptability for the nonlinear free surface and describing normally for bed source term is established by combining the nonlinear turbulence model and the Youngs-VOF method. The turbulent equations are discreted with implicit scheme in the vertical coordinate and explicit scheme in the horizontal coordinate based on the power-law scheme. The model is applied for a dam break wave problem and transcritical flow problem. The results show the adjusted model have a good performance in predicting xxx on the shallow water problems with strong nonlinearity, such as tidal bores. Our adjusted model and findings may benefit the design and management of water conservancy infrastructures in practice. VOF shallow water free surface dam break wave transcritical flow Figures Figure 1 Figure 2 Figure 3 1 Introduction Tidal bore, dam-break wave, shallow water wave etc. of typical shallow flow phenomena observed in nature directly influence the design, construction, and operation of water conservancy projects, such as the structural safety, failure risk assessment of levee, reservoir, and wharve. Shallow water flow are characterized by a smaller scale in water depth compared to the plane, strong nonlinearity at the free surface, and minimal lateral variation in hydraulic properties, which falls under non-homogeneous nonlinear hyperbolic partial differential problems. As a result, it is difficult to obtain an analytical solution for the discontinuous surface and complex riverbed boundary conditions. The accurate numerical solution for the shallow water problem relies on effectively capturing flow discontinuity and appropriately addressing variations in riverbed. Currently, most studies developing based on one-dimensional or plane two-dimensional coordinate system mainly utilizes high-resolution schemes with a special processing method for bed source terms such as Godunov, TVD, Runge–Kutta, WENO in order to manage a complex discontinuous flow in shallow water and ensure accurate numerical solution while preventing the dispersion of false flow [ 1 – 5 ]. The hydraulic characteristics vary along the water depth should be considered in the design and study of water conservancy projects for the economy, reliability and security, and base on the characteristics of shallow water flow, the shallow water problems can be reducible to a 2-D vertical flow problems. In this paper, a numerical model is presented that is suitable for the change in the bed and possesses robust capability in capturing free surfaces by using the normal grid division method to process the bed source terms, improving the Young’s method [ 6 ] which is is a widely used free surface treatment method with a strong universal applicability for nonlinear free surface problems by a more accurate ELVIELRA method [ 7 ] for free surface tracking, and incorporating the turbulent model. The model that cloud simulate the varying in the vertical direction of velocity and pressure is tested quite well in dam break wave and transcritical flow problems, and it may be used as the foundation for the study of sediment transport under the action of tidal bore. 2 Materials and methods 2.1 Flow model The flow is modeled by means of the following Reynolds-averaged equations: $$\frac{{\partial u}}{{\partial x}}+\frac{{\partial v}}{{\partial z}}=0$$ 1 $$\frac{{\partial u}}{{\partial t}}+u\frac{{\partial u}}{{\partial x}}+v\frac{{\partial u}}{{\partial z}}= - \frac{1}{\rho }\frac{{\partial p}}{{\partial x}}+{g_x}+\left( {\upsilon +{\upsilon _t}} \right)\left( {\frac{{{\partial ^2}u}}{{\partial {x^2}}}+\frac{{{\partial ^2}u}}{{\partial {z^2}}}} \right)+2\frac{{\partial {\upsilon _t}}}{{\partial x}}\frac{{\partial u}}{{\partial x}}+\frac{{\partial {\upsilon _t}}}{{\partial z}}\left( {\frac{{\partial u}}{{\partial z}}+\frac{{\partial v}}{{\partial x}}} \right) - \frac{2}{3}\frac{{\partial k}}{{\partial x}}$$ 2 $$\frac{{\partial v}}{{\partial t}}+u\frac{{\partial v}}{{\partial x}}+v\frac{{\partial v}}{{\partial z}}= - \frac{1}{\rho }\frac{{\partial p}}{{\partial z}}+{g_z}+\left( {\upsilon +{\upsilon _t}} \right)\left( {\frac{{{\partial ^2}v}}{{\partial {x^2}}}+\frac{{{\partial ^2}v}}{{\partial {z^2}}}} \right)+2\frac{{\partial {\upsilon _t}}}{{\partial z}}\frac{{\partial v}}{{\partial z}}+\frac{{\partial {\upsilon _t}}}{{\partial x}}\left( {\frac{{\partial u}}{{\partial z}}+\frac{{\partial v}}{{\partial x}}} \right) - \frac{2}{3}\frac{{\partial k}}{{\partial z}}$$ 3 where u and v are velocity components; p is pressure; \({g_x}\) and \({g_z}\) are body force; ρ is fluid density; \(\upsilon\) is kinematic viscosity; \({\upsilon _t}\) \(={c_\mu }{{{k^2}} \mathord{\left/ {\vphantom {{{k^2}} \varepsilon }} \right. \kern-0pt} \varepsilon }\) , is eddy viscosity. The turbulent kinetic energy k and its dissipation rate ε are obtained from the k-ε nonlinear turbulence model: $$\frac{{\partial k}}{{\partial t}}+u\frac{{\partial k}}{{\partial x}}+v\frac{{\partial k}}{{\partial z}}=\frac{\partial }{{\partial x}}\left[ {\left( {\upsilon +\frac{{{\upsilon _t}}}{{{\sigma _k}}}} \right)\frac{{\partial k}}{{\partial x}}} \right]+\frac{\partial }{{\partial z}}\left[ {\left( {\upsilon +\frac{{{\upsilon _t}}}{{{\sigma _k}}}} \right)\frac{{\partial k}}{{\partial z}}} \right]+{G_k} - \varepsilon$$ 4 $$\frac{{\partial \varepsilon }}{{\partial t}}+u\frac{{\partial \varepsilon }}{{\partial x}}+v\frac{{\partial \varepsilon }}{{\partial z}}=\frac{\partial }{{\partial x}}\left[ {\left( {\upsilon +\frac{{{\upsilon _t}}}{{{\sigma _\varepsilon }}}} \right)\frac{{\partial \varepsilon }}{{\partial x}}} \right]+\frac{\partial }{{\partial z}}\left[ {\left( {\upsilon +\frac{{{\upsilon _t}}}{{{\sigma _\varepsilon }}}} \right)\frac{{\partial \varepsilon }}{{\partial z}}} \right]+\frac{\varepsilon }{k}\left( {{C_1}{G_k} - {C_2}\varepsilon } \right)$$ 5 where $${G_k}={\upsilon _t}\left[ {2{{\left( {\frac{{\partial u}}{{\partial x}}} \right)}^2}+2{{\left( {\frac{{\partial v}}{{\partial z}}} \right)}^2}+{{\left( {\frac{{\partial u}}{{\partial z}}+\frac{{\partial v}}{{\partial x}}} \right)}^2}} \right]$$ 6 The turbulent constants are determined as: \({c_\mu }=\) 0.09, \({c_1}=\) 1.44, \({c_2}=\) 1.92, \({\sigma _1}=\) 1.0, \({\sigma _2}=\) 1.3 [ 8 ] . 2.2 Boundary conditions 2.2.1 Riverbed The slip boundary condition is assumed to solve the velocity near the riverbed, and the production term of kinetic energy and the turbulence kinetic energy dissipation rate are calculated by wall-function approach [ 9 ]: \({G_k} \approx {\tau _w}\frac{{{\tau _w}}}{{\kappa \rho c_{\mu }^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}}k_{p}^{{1/2}}\Delta {z_p}}}\) , \(\varepsilon =\frac{{c_{\mu }^{{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-0pt} 4}}}k_{p}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}}}}{{\kappa \Delta {z_p}}}\) (7) The resultant wall shear stress \({\tau _w}\) is related to \({u_p}\) is the velocity parallels the riverbed by log law: \(\frac{{{u_p}c_{\mu }^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}}k_{p}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}}}{{{{{\tau _w}} \mathord{\left/ {\vphantom {{{\tau _w}} \rho }} \right. \kern-0pt} \rho }}}=\left\{ {\begin{array}{*{20}{c}} {\frac{1}{\kappa }\ln \left( {E{z^*}} \right){\kern 1pt} {\kern 1pt} ,\quad {z^*}>11.225} \\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} \quad \;{\kern 1pt} {\kern 1pt} {z^*}{\kern 1pt} {\kern 1pt} {\kern 1pt} \quad {\kern 1pt} {\kern 1pt} ,\quad \,{\kern 1pt} {z^*}<11.225} \end{array}{\kern 1pt} } \right.\,\) with \({z^*}={{c_{\mu }^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}}k_{p}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}\Delta {z_p}} \mathord{\left/ {\vphantom {{c_{\mu }^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}}k_{p}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}\Delta {z_p}} \upsilon }} \right. \kern-0pt} \upsilon }\) (8) Where \(\kappa\) is van Karman constant ( \(\kappa\) = 0.42); E is roughness coefficient, and determined by: $$E=\exp \left[ {\kappa \left( {B - \Delta B} \right)} \right]$$ 9 where B is additive constant ( B = 5.2); \(\Delta B\) is roughness function, and calculated as flows [ 10 ]: \(\Delta B=\left\{ {\begin{array}{*{20}{c}} {\quad \quad \quad \quad \;\;\;\;\quad \quad 0\quad \quad \quad \quad \quad \quad \quad \quad \,\,\;\;,\,\;k_{s}^{+}<2.25} \\ {\left( {B - 8.5+\frac{1}{\kappa }\ln k_{s}^{+}} \right)\sin \left[ {0.4258\left( {\ln k_{s}^{+} - 0.811} \right)} \right]\;,\;2.25 \leqslant k_{s}^{+}<90\;} \\ {\,\quad \quad \;\quad \,B - 8.5+\frac{1}{\kappa }\ln k_{s}^{+}\quad \,\,\quad \quad \quad \quad \,,\;\,k_{s}^{+} \geqslant 90} \end{array}} \right.\) with \(k_{s}^{+}={u_*}{k_s}/\upsilon\) (10) where \({u_*}\) is bed shear velocity ( \({u_*}\) \(=\sqrt {{{{\tau _w}} \mathord{\left/ {\vphantom {{{\tau _w}} \rho }} \right. \kern-0pt} \rho }}\) ); \({k_s}\) is equivalent roughness height. 2.2.2 Free surface The velocity and turbulent kinetic energy to the normal is set to zero, and the dissipation rate is obtained from the following relation. $$\varepsilon =\frac{{{{\left( {k\sqrt {{c_\mu }} } \right)}^{1.5}}}}{{0.07\kappa h}}$$ 11 where h is water depth. 2.3 Numerical method A staggered grid is used to discretize the equations. The Reynolds-averaged equations are solved using a finite difference method, and the finite volume method based on the power-law scheme is introduced to populate the turbulence model. The generic form of the equations is as follows [ 11 ]: $${a_P}\phi _{P}^{{n+1}}={a_N}\phi _{N}^{{n+1}}+{a_S}\phi _{S}^{{n+1}}+{a_E}\phi _{E}^{n}+{a_W}\phi _{W}^{n}+{S_\phi }$$ 12 where $${a_P}={a_N}+{a_S}+\Delta F+{{\Delta V} \mathord{\left/ {\vphantom {{\Delta V} {\Delta t}}} \right. \kern-0pt} {\Delta t}} - {S_p}\Delta V$$ , $${P_i}={{{F_i}} \mathord{\left/ {\vphantom {{{F_i}} {{D_i}}}} \right. \kern-0pt} {{D_i}}},\,A\left( {\left| {{P_i}} \right|} \right)=\hbox{max} \left[ {\left( {0,\,\left( {1 - 0.1{{\left| {{P_i}} \right|}^5}} \right)} \right)} \right],\,\left( {i=n,\,s,\,e,\,w} \right)$$ , $${a_N}={D_n}A\left( {\left| {{P_n}} \right|} \right)+\left( {\left| { - {F_n}} \right|} \right),\,{F_n}{\text{=}}{\left( {v\Delta x} \right)_n},\,{D_n}={\left( {{{\Gamma \Delta x} \mathord{\left/ {\vphantom {{\Gamma \Delta x} {\Delta z}}} \right. \kern-0pt} {\Delta z}}} \right)_n}$$ , $${a_S}={D_s}A\left( {\left| {{P_s}} \right|} \right)+\left( {\left| { - {F_s}} \right|} \right),\,{F_s}{\text{=}}{\left( {v\Delta x} \right)_s},\,{D_s}={\left( {{{\Gamma \Delta x} \mathord{\left/ {\vphantom {{\Gamma \Delta x} {\Delta z}}} \right. \kern-0pt} {\Delta z}}} \right)_s}$$ , $${a_E}={D_e}A\left( {\left| {{P_e}} \right|} \right)+\left( {\left| { - {F_e}} \right|} \right),\,{F_e}{\text{=}}{\left( {v\Delta x} \right)_e},\,{D_e}={\left( {{{\Gamma \Delta x} \mathord{\left/ {\vphantom {{\Gamma \Delta x} {\Delta z}}} \right. \kern-0pt} {\Delta z}}} \right)_e}$$ , $${a_W}={D_w}A\left( {\left| {{P_w}} \right|} \right)+\left( {\left| { - {F_w}} \right|} \right),\,{F_w}{\text{=}}{\left( {v\Delta x} \right)_w},\,{D_w}={\left( {{{\Gamma \Delta x} \mathord{\left/ {\vphantom {{\Gamma \Delta x} {\Delta z}}} \right. \kern-0pt} {\Delta z}}} \right)_w}$$ , $${S_\phi }=\left( {{S_c}+{{\phi _{p}^{n}} \mathord{\left/ {\vphantom {{\phi _{p}^{n}} {\delta t}}} \right. \kern-0pt} {\delta t}}} \right)\Delta V - \left( {{a_E}+{a_W}} \right)\phi _{p}^{n}+{F_e}{\phi _E} - {F_w}{\phi _W}$$ , in which φ stands for k and ε , and Γ is the diffusion coefficient for φ ; δt is time step, and S p and S c are the source terms for the discrete equations of φ . For solving the advection equation of F , a step function (the modified Youngs-VOF) is employed. The free surface slope is calculated by the ELVIRA method which is more accurate than the more commonly used VOF interface reconstruction algorithms. 3 Results and discussions 3.1 Numerical stability analysis To verify the numerical stability of the improved Youngs-VOF method, we design the following calculation example: the length of computational domain is 12m, horizontal bottom, initial water level is 0.8m, vertical uniform flow velocity at the inlet is 0.75 m/s and sudden water level rise of 0.4m within 0 ~ 2s. The influence of gravity is not considered in the calculation process. The computational grid is ∆x = ∆z = 0.05m, and the velocity field of the entire computational domain is given based on the vertical uniform distribution without considering the turbulence effect. Figure 1 shows the shape of the free surface profile at the relevant time. As shown in this figure, the improved Youngs-VOF method is relatively stable, and no numerical oscillation or diffusion occurs in the calculation process. 3.2 Dam break wave Furthermore, there exist shock waves in the dam-break process, and the ability of numerical methods to capture shock waves can be tested based on this type of problem. Ying et al. [ 12 ], Wang et al. [ 13 ] and Mohapatra et al. [ 14 ] employed similar examples to test their respective algorithms. Given the computational domain length of 40 m and bottom elevation b = 0, the initial conditions are as follows: \(\left( {{h_L}=1.0{\text{m}},{\kern 1pt} \,{u_L}=0.0} \right)\left| x \right.<20,{\kern 1pt} \,\left( {{h_R}=0.2{\text{m}},{\kern 1pt} \,{u_R}=0.0} \right)\left| x \right. \geqslant 20\) m The computational grid is ∆x = 0.20m, ∆z = 0.10m. Figure 2 compares the numerical solution and exact solution (obtained from the one-dimensional homogeneous shallow water equation at the given time. As shown in Fig. 2 , the overall trend of the numerical solution is consistent with the exact solution. Since the upwind scheme is used, certain numerical diffusion occurs in the numerical solution, which is more obvious at the changes in the free surface. 3.3 Transcritical flow Due to different initial boundary conditions given, subcritical flow, shock wave and supercritical flow may exist simultaneously in the computational domain, which leads to certain difficulties in numerical simulation. The harmony and adaptability of numerical methods are generally tested based on this type of problem. Such as Vazquez et al. [ 15 ], Zhou et al. [ 16 ] and Wang et al. [ 3 ] used similar examples to test the shallow water models they established. Given the computational domain length of 25m and initial water level of 0.4m, the bottom elevation is as follows. \(b\left( x \right)=\left\{ \begin{gathered} 0.2 - 0.05{\left( {x - 10} \right)^2},\quad \quad 8 \leqslant x \leqslant 12 \hfill \\ \quad \quad \quad 0.0\quad \quad \quad \,\,,\quad \quad \;x12\;\quad \quad \hfill \\ \end{gathered} \right.\) (1) Shock-free transcritical flow problem Given discharge per unit width q at the inlet of 1.53 m 2 /s and computational grid ∆x = 0.04m, a non-uniform grid is used in the vertical direction, and a vertically uniform flow velocity distribution is given at the inlet. Figure 3 a compares the numerical solution and exact solution (obtained from the one-dimensional shallow water equation without considering the time partial differential terms) when the calculation is stable. (2) Shock transcritical flow problem The discharge per unit width q at the inlet is 0.17 m 2 /s, and the computational grid is ∆x = ∆z = 0.01m. A vertically uniform flow velocity distribution at the inlet is given. Figure 3 b compares the numerical solution and exact solution when the calculation is stable. The numerical and exact solutions of the water depth in the stable flow zone on the left side of the shock-free transcritical flow are 1.0057 and 1.0147m respectively which are 0.4040 and 0.4m respectively on the right side. The numerical and exact solutions of the water depth in the stable flow zone on the left side of the shock transcritical flow are 0.4052 and 0.4062m respectively which are 0.3999 and 0.4m respectively on the right side, and the numerical and exact solutions of the minimum water depth at the shock wave are 0.3004 and 0.2996m respectively. It can be seen that the improved Youngs-VOF method has a good harmony and adaptability in numerical solutions. 4 Conclusions A 2-D vertical flow model is developed that the Youngs-VOF method is improved by a more accurate ELVIELRA method for free surface tracking in its normal direction. After numerical stability analysis, and verification analysis on the dam-break wave and transcritical flow problems, the proposed numerical model has good stability, strong ability and high accuracy in capturing the free surface. Compared with the traditional water level-bottom topography formulation (Pan et al. 2003), surface gradient method (Zhou et al. 2001), feature decomposition method (Wang et al. 2005) etc. for bed-slope source terms, a conventional grid division method is used to process the bed-slope source terms, which can provide a basis for studying shallow water problems with strong free surface nonlinearity. Declarations Author Contribution Dongmiao Zhao, Qi Liu and Jian Chen wrote the main manuscript text, Yi Liu (Wenzhou Water Conservancy Survey and Design Institute Co., Ltd.) and Yang Zhou prepared figures 1-3 and references, and Yi Liu (Earth, Ocean and Atmospheric Sciences (EOAS) Thrust, Function Hub, The Hong Kong University of Science and Technology) checked the manuscript. All authors reviewed the manuscript. Data Availability Data generated during the current study are available from authors on reasonable request. References Pan CH, Lin BY, Mao XZ. A Godunov-type scheme for 1-D shallow water flow with uneven bottom. Advances in Water Science. 2003;14(4):430-436(In Chinese). https://doi.org/10.3321/j.issn:1001-6791.2003.04.008. Crnjaric-Zic N, Vukovic S, Sopta L. Extension of ENO and WENO schemes to one-dimensional sediment transport equations. Computers & Fluids. 2004;33:31-56. https://doi.org/10.1016/S0045-7930(03)00032-X. Wang ZL, Geng YF, JIN S. Numerical modeling of 2-D shallow water flow with complicated geometry and topography. Journal of Hydraulic Engineering. 2005;35(4):1-9(In Chinese). https://doi.org/10.3321/j.issn:0559-9350.2005.04.010. Zhao ZY, Zhang QH, Li SS. Application of Runge-Kutta discontinuous Galerkin scheme for one-dimensional shallow water equations. Advances in Water Science. 2012;23(5):695-701(In Chinese). https://doi.org/10.14042/j.cnki.32.1309.2012.05.010. Hu WY, Li XG, Deng YX. A new third order WENO scheme for solving shallow water equations. Yellow River. 2023;45(12):31-36(In Chinese). https://doi.org/10.3969/j.issn.1000-1379.2023.006. Youngs DL. Time-dependent multi-material flow with large fluid distortion. Numerical methods for fluid dynamics. 1982;273-285. Pilliod JE, Puckett EG. Second-order accurate volume-of-fluid algorithms for tracking material interfaces. Journal of Computational physics. 2004;199(2):465-502. https://doi.org/10.1016/j.jcp.2003.12.023. Launder BE, Spalding DB. The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering. 1974;3(2):269-289. https://doi.org/10.1016/0045-7825(74)90029-2. Versteeg HK, Malalasekera W. An introduction to computational fluid dynamics: the finite volume method. Prentice Hall. 2007. Cebeci T, Bradshaw P. Momentum Transfer in Boundary Layers . New York: Hemisphere Publishing Corporation. 1977. WANG FJ. Computational fluid dynamics analysis-theory of application of CFD software. Beijing: Tsinghua university press. 2004(In Chinese). Ying XY, Khan AA, Wang SSY. Upwind conservative scheme for the saint venant equations. Journal of Hydraulic Engineering. 2004;130(10):977-987. https://doi.org/10.1061/(ASCE)0733-9429(2004)130:10(977). Wang L, Zheng SP. Solving shallow water wave equation based on moving grid entropy stable scheme. Chinese journal of hydrodynamics. 2020;35(2):188-193(in Chinese). https://doi.org/10.16076/j.cnki.cjhd.2020.02.006. Mohapatra PK, Eswaran V, Bhallamudi MS. Two-dimensional analysis of dam break flow in vertical plane. Journal of Hydraulic Engineering. 1999;125(2):183-192. https://doi.org/10.1061/(ASCE)0733-9429(1999)125:2(183). Vazquez-Cendon ME. Improved Treatment of Source Terms in upwind schemes for the shallow water equations in channels with irregular geometry. Journal of Computational Physics. 1999; 148(2):497-526. https://doi.org/10.1006/jcph.1998.6127. Zhou JG, Causon DM, Mingham CG, et al. Ingram. The surface gradient method for the treatment of source terms in the shallow-water equations. Journal of Computational Physics. 2001;168 (1):1-25. https://doi.org/10.1006/jcph.2000.6670. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4324162","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":300322292,"identity":"4ba24a86-d84c-48ab-bb91-2ca5a340eea6","order_by":0,"name":"Dongmiao Zhao","email":"","orcid":"","institution":"Wenzhou Water Conservancy Survey and Design Institute Co., Ltd.","correspondingAuthor":false,"prefix":"","firstName":"Dongmiao","middleName":"","lastName":"Zhao","suffix":""},{"id":300322297,"identity":"282cdcb3-7496-42d4-bdc2-a445be22a809","order_by":1,"name":"Jian 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Technology","correspondingAuthor":false,"prefix":"","firstName":"Yi","middleName":"","lastName":"Liu","suffix":""}],"badges":[],"createdAt":"2024-04-25 12:33:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4324162/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4324162/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":56238757,"identity":"09fc2d05-56e2-475a-b4d7-9e2e27c2949d","added_by":"auto","created_at":"2024-05-10 09:25:23","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":47194,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4324162/v1/deea4c3b23057ee5fda01457.png"},{"id":56238756,"identity":"b29e9b9c-98ff-46b0-a0fd-d875f1543c3b","added_by":"auto","created_at":"2024-05-10 09:25:16","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":113015,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4324162/v1/233ea8168e45a9564e6a2e2f.png"},{"id":56238754,"identity":"f41c0c5f-c8ff-4c67-8777-53edde80be46","added_by":"auto","created_at":"2024-05-10 09:25:14","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":69653,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4324162/v1/9bab5352fe12bdd71524c1f1.png"},{"id":57201600,"identity":"55b7f449-ef51-4411-b523-2df6975f8e6c","added_by":"auto","created_at":"2024-05-27 10:03:10","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":549335,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4324162/v1/9b1794cd-5871-4623-9e71-a108e6ff6455.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Application of adjusted Youngs-VOF method for shallow water problems","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eTidal bore, dam-break wave, shallow water wave etc. of typical shallow flow phenomena observed in nature directly influence the design, construction, and operation of water conservancy projects, such as the structural safety, failure risk assessment of levee, reservoir, and wharve. Shallow water flow are characterized by a smaller scale in water depth compared to the plane, strong nonlinearity at the free surface, and minimal lateral variation in hydraulic properties, which falls under non-homogeneous nonlinear hyperbolic partial differential problems. As a result, it is difficult to obtain an analytical solution for the discontinuous surface and complex riverbed boundary conditions.\u003c/p\u003e \u003cp\u003eThe accurate numerical solution for the shallow water problem relies on effectively capturing flow discontinuity and appropriately addressing variations in riverbed. Currently, most studies developing based on one-dimensional or plane two-dimensional coordinate system mainly utilizes high-resolution schemes with a special processing method for bed source terms such as Godunov, TVD, Runge\u0026ndash;Kutta, WENO in order to manage a complex discontinuous flow in shallow water and ensure accurate numerical solution while preventing the dispersion of false flow [\u003cspan additionalcitationids=\"CR2 CR3 CR4\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. The hydraulic characteristics vary along the water depth should be considered in the design and study of water conservancy projects for the economy, reliability and security, and base on the characteristics of shallow water flow, the shallow water problems can be reducible to a 2-D vertical flow problems. In this paper, a numerical model is presented that is suitable for the change in the bed and possesses robust capability in capturing free surfaces by using the normal grid division method to process the bed source terms, improving the Young\u0026rsquo;s method [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] which is is a widely used free surface treatment method with a strong universal applicability for nonlinear free surface problems by a more accurate ELVIELRA method [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] for free surface tracking, and incorporating the \u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003eturbulent model. The model that cloud simulate the varying in the vertical direction of velocity and pressure is tested quite well in dam break wave and transcritical flow problems, and it may be used as the foundation for the study of sediment transport under the action of tidal bore.\u003c/p\u003e"},{"header":"2 Materials and methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Flow model\u003c/h2\u003e \u003cp\u003eThe flow is modeled by means of the following Reynolds-averaged equations:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\frac{{\\partial u}}{{\\partial x}}+\\frac{{\\partial v}}{{\\partial z}}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\frac{{\\partial u}}{{\\partial t}}+u\\frac{{\\partial u}}{{\\partial x}}+v\\frac{{\\partial u}}{{\\partial z}}= - \\frac{1}{\\rho }\\frac{{\\partial p}}{{\\partial x}}+{g_x}+\\left( {\\upsilon +{\\upsilon _t}} \\right)\\left( {\\frac{{{\\partial ^2}u}}{{\\partial {x^2}}}+\\frac{{{\\partial ^2}u}}{{\\partial {z^2}}}} \\right)+2\\frac{{\\partial {\\upsilon _t}}}{{\\partial x}}\\frac{{\\partial u}}{{\\partial x}}+\\frac{{\\partial {\\upsilon _t}}}{{\\partial z}}\\left( {\\frac{{\\partial u}}{{\\partial z}}+\\frac{{\\partial v}}{{\\partial x}}} \\right) - \\frac{2}{3}\\frac{{\\partial k}}{{\\partial x}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\frac{{\\partial v}}{{\\partial t}}+u\\frac{{\\partial v}}{{\\partial x}}+v\\frac{{\\partial v}}{{\\partial z}}= - \\frac{1}{\\rho }\\frac{{\\partial p}}{{\\partial z}}+{g_z}+\\left( {\\upsilon +{\\upsilon _t}} \\right)\\left( {\\frac{{{\\partial ^2}v}}{{\\partial {x^2}}}+\\frac{{{\\partial ^2}v}}{{\\partial {z^2}}}} \\right)+2\\frac{{\\partial {\\upsilon _t}}}{{\\partial z}}\\frac{{\\partial v}}{{\\partial z}}+\\frac{{\\partial {\\upsilon _t}}}{{\\partial x}}\\left( {\\frac{{\\partial u}}{{\\partial z}}+\\frac{{\\partial v}}{{\\partial x}}} \\right) - \\frac{2}{3}\\frac{{\\partial k}}{{\\partial z}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eu\u003c/em\u003e and \u003cem\u003ev\u003c/em\u003e are velocity components; \u003cem\u003ep\u003c/em\u003e is pressure;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({g_x}\\)\u003c/span\u003e\u003c/span\u003eand\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({g_z}\\)\u003c/span\u003e\u003c/span\u003eare body force; \u003cem\u003eρ\u003c/em\u003e is fluid density; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\upsilon\\)\u003c/span\u003e\u003c/span\u003eis kinematic viscosity; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\upsilon _t}\\)\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(={c_\\mu }{{{k^2}} \\mathord{\\left/ {\\vphantom {{{k^2}} \\varepsilon }} \\right. \\kern-0pt} \\varepsilon }\\)\u003c/span\u003e\u003c/span\u003e, is eddy viscosity. The turbulent kinetic energy \u003cem\u003ek\u003c/em\u003e and its dissipation rate \u003cem\u003eε\u003c/em\u003e are obtained from the \u003cem\u003ek-ε\u003c/em\u003e nonlinear turbulence model:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\frac{{\\partial k}}{{\\partial t}}+u\\frac{{\\partial k}}{{\\partial x}}+v\\frac{{\\partial k}}{{\\partial z}}=\\frac{\\partial }{{\\partial x}}\\left[ {\\left( {\\upsilon +\\frac{{{\\upsilon _t}}}{{{\\sigma _k}}}} \\right)\\frac{{\\partial k}}{{\\partial x}}} \\right]+\\frac{\\partial }{{\\partial z}}\\left[ {\\left( {\\upsilon +\\frac{{{\\upsilon _t}}}{{{\\sigma _k}}}} \\right)\\frac{{\\partial k}}{{\\partial z}}} \\right]+{G_k} - \\varepsilon$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\frac{{\\partial \\varepsilon }}{{\\partial t}}+u\\frac{{\\partial \\varepsilon }}{{\\partial x}}+v\\frac{{\\partial \\varepsilon }}{{\\partial z}}=\\frac{\\partial }{{\\partial x}}\\left[ {\\left( {\\upsilon +\\frac{{{\\upsilon _t}}}{{{\\sigma _\\varepsilon }}}} \\right)\\frac{{\\partial \\varepsilon }}{{\\partial x}}} \\right]+\\frac{\\partial }{{\\partial z}}\\left[ {\\left( {\\upsilon +\\frac{{{\\upsilon _t}}}{{{\\sigma _\\varepsilon }}}} \\right)\\frac{{\\partial \\varepsilon }}{{\\partial z}}} \\right]+\\frac{\\varepsilon }{k}\\left( {{C_1}{G_k} - {C_2}\\varepsilon } \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${G_k}={\\upsilon _t}\\left[ {2{{\\left( {\\frac{{\\partial u}}{{\\partial x}}} \\right)}^2}+2{{\\left( {\\frac{{\\partial v}}{{\\partial z}}} \\right)}^2}+{{\\left( {\\frac{{\\partial u}}{{\\partial z}}+\\frac{{\\partial v}}{{\\partial x}}} \\right)}^2}} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe turbulent constants are determined as: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({c_\\mu }=\\)\u003c/span\u003e\u003c/span\u003e0.09,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({c_1}=\\)\u003c/span\u003e\u003c/span\u003e1.44,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({c_2}=\\)\u003c/span\u003e\u003c/span\u003e1.92,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma _1}=\\)\u003c/span\u003e\u003c/span\u003e1.0,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma _2}=\\)\u003c/span\u003e\u003c/span\u003e1.3 [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] .\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Boundary conditions\u003c/h2\u003e \u003cdiv id=\"Sec5\" class=\"Section3\"\u003e \u003ch2\u003e2.2.1 Riverbed\u003c/h2\u003e \u003cp\u003eThe slip boundary condition is assumed to solve the velocity near the riverbed, and the production term of kinetic energy and the turbulence kinetic energy dissipation rate are calculated by wall-function approach [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({G_k} \\approx {\\tau _w}\\frac{{{\\tau _w}}}{{\\kappa \\rho c_{\\mu }^{{{1 \\mathord{\\left/ {\\vphantom {1 4}} \\right. \\kern-0pt} 4}}}k_{p}^{{1/2}}\\Delta {z_p}}}\\)\u003c/span\u003e \u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varepsilon =\\frac{{c_{\\mu }^{{{3 \\mathord{\\left/ {\\vphantom {3 4}} \\right. \\kern-0pt} 4}}}k_{p}^{{{3 \\mathord{\\left/ {\\vphantom {3 2}} \\right. \\kern-0pt} 2}}}}}{{\\kappa \\Delta {z_p}}}\\)\u003c/span\u003e\u003c/span\u003e (7)\u003c/p\u003e \u003cp\u003eThe resultant wall shear stress \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\tau _w}\\)\u003c/span\u003e\u003c/span\u003e is related to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u_p}\\)\u003c/span\u003e\u003c/span\u003e is the velocity parallels the riverbed by log law:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\frac{{{u_p}c_{\\mu }^{{{1 \\mathord{\\left/ {\\vphantom {1 4}} \\right. \\kern-0pt} 4}}}k_{p}^{{{1 \\mathord{\\left/ {\\vphantom {1 2}} \\right. \\kern-0pt} 2}}}}}{{{{{\\tau _w}} \\mathord{\\left/ {\\vphantom {{{\\tau _w}} \\rho }} \\right. \\kern-0pt} \\rho }}}=\\left\\{ {\\begin{array}{*{20}{c}} {\\frac{1}{\\kappa }\\ln \\left( {E{z^*}} \\right){\\kern 1pt} {\\kern 1pt} ,\\quad {z^*}\u0026gt;11.225} \\\\ {{\\kern 1pt} {\\kern 1pt} {\\kern 1pt} \\quad \\;{\\kern 1pt} {\\kern 1pt} {z^*}{\\kern 1pt} {\\kern 1pt} {\\kern 1pt} \\quad {\\kern 1pt} {\\kern 1pt} ,\\quad \\,{\\kern 1pt} {z^*}\u0026lt;11.225} \\end{array}{\\kern 1pt} } \\right.\\,\\)\u003c/span\u003e \u003c/span\u003ewith \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({z^*}={{c_{\\mu }^{{{1 \\mathord{\\left/ {\\vphantom {1 4}} \\right. \\kern-0pt} 4}}}k_{p}^{{{1 \\mathord{\\left/ {\\vphantom {1 2}} \\right. \\kern-0pt} 2}}}\\Delta {z_p}} \\mathord{\\left/ {\\vphantom {{c_{\\mu }^{{{1 \\mathord{\\left/ {\\vphantom {1 4}} \\right. \\kern-0pt} 4}}}k_{p}^{{{1 \\mathord{\\left/ {\\vphantom {1 2}} \\right. \\kern-0pt} 2}}}\\Delta {z_p}} \\upsilon }} \\right. \\kern-0pt} \\upsilon }\\)\u003c/span\u003e\u003c/span\u003e (8)\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\kappa\\)\u003c/span\u003e\u003c/span\u003e is van Karman constant (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\kappa\\)\u003c/span\u003e\u003c/span\u003e=\u0026thinsp;0.42); \u003cem\u003eE\u003c/em\u003e is roughness coefficient, and determined by:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$E=\\exp \\left[ {\\kappa \\left( {B - \\Delta B} \\right)} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eB\u003c/em\u003e is additive constant (\u003cem\u003eB\u003c/em\u003e\u0026thinsp;=\u0026thinsp;5.2);\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Delta B\\)\u003c/span\u003e\u003c/span\u003eis roughness function, and calculated as flows [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\Delta B=\\left\\{ {\\begin{array}{*{20}{c}} {\\quad \\quad \\quad \\quad \\;\\;\\;\\;\\quad \\quad 0\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\,\\,\\;\\;,\\,\\;k_{s}^{+}\u0026lt;2.25} \\\\ {\\left( {B - 8.5+\\frac{1}{\\kappa }\\ln k_{s}^{+}} \\right)\\sin \\left[ {0.4258\\left( {\\ln k_{s}^{+} - 0.811} \\right)} \\right]\\;,\\;2.25 \\leqslant k_{s}^{+}\u0026lt;90\\;} \\\\ {\\,\\quad \\quad \\;\\quad \\,B - 8.5+\\frac{1}{\\kappa }\\ln k_{s}^{+}\\quad \\,\\,\\quad \\quad \\quad \\quad \\,,\\;\\,k_{s}^{+} \\geqslant 90} \\end{array}} \\right.\\)\u003c/span\u003e \u003c/span\u003ewith \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k_{s}^{+}={u_*}{k_s}/\\upsilon\\)\u003c/span\u003e\u003c/span\u003e (10)\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u_*}\\)\u003c/span\u003e\u003c/span\u003e is bed shear velocity (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u_*}\\)\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(=\\sqrt {{{{\\tau _w}} \\mathord{\\left/ {\\vphantom {{{\\tau _w}} \\rho }} \\right. \\kern-0pt} \\rho }}\\)\u003c/span\u003e\u003c/span\u003e);\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k_s}\\)\u003c/span\u003e\u003c/span\u003e is equivalent roughness height.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e2.2.2 Free surface\u003c/h2\u003e \u003cp\u003eThe velocity and turbulent kinetic energy to the normal is set to zero, and the dissipation rate is obtained from the following relation.\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\varepsilon =\\frac{{{{\\left( {k\\sqrt {{c_\\mu }} } \\right)}^{1.5}}}}{{0.07\\kappa h}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eh\u003c/em\u003e is water depth.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Numerical method\u003c/h2\u003e \u003cp\u003eA staggered grid is used to discretize the equations. The Reynolds-averaged equations are solved using a finite difference method, and the finite volume method based on the power-law scheme is introduced to populate the turbulence model. The generic form of the equations is as follows [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]:\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$${a_P}\\phi _{P}^{{n+1}}={a_N}\\phi _{N}^{{n+1}}+{a_S}\\phi _{S}^{{n+1}}+{a_E}\\phi _{E}^{n}+{a_W}\\phi _{W}^{n}+{S_\\phi }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$${a_P}={a_N}+{a_S}+\\Delta F+{{\\Delta V} \\mathord{\\left/ {\\vphantom {{\\Delta V} {\\Delta t}}} \\right. \\kern-0pt} {\\Delta t}} - {S_p}\\Delta V$$\u003c/div\u003e\u003c/div\u003e,\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$${P_i}={{{F_i}} \\mathord{\\left/ {\\vphantom {{{F_i}} {{D_i}}}} \\right. \\kern-0pt} {{D_i}}},\\,A\\left( {\\left| {{P_i}} \\right|} \\right)=\\hbox{max} \\left[ {\\left( {0,\\,\\left( {1 - 0.1{{\\left| {{P_i}} \\right|}^5}} \\right)} \\right)} \\right],\\,\\left( {i=n,\\,s,\\,e,\\,w} \\right)$$\u003c/div\u003e\u003c/div\u003e,\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$${a_N}={D_n}A\\left( {\\left| {{P_n}} \\right|} \\right)+\\left( {\\left| { - {F_n}} \\right|} \\right),\\,{F_n}{\\text{=}}{\\left( {v\\Delta x} \\right)_n},\\,{D_n}={\\left( {{{\\Gamma \\Delta x} \\mathord{\\left/ {\\vphantom {{\\Gamma \\Delta x} {\\Delta z}}} \\right. \\kern-0pt} {\\Delta z}}} \\right)_n}$$\u003c/div\u003e\u003c/div\u003e,\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$${a_S}={D_s}A\\left( {\\left| {{P_s}} \\right|} \\right)+\\left( {\\left| { - {F_s}} \\right|} \\right),\\,{F_s}{\\text{=}}{\\left( {v\\Delta x} \\right)_s},\\,{D_s}={\\left( {{{\\Gamma \\Delta x} \\mathord{\\left/ {\\vphantom {{\\Gamma \\Delta x} {\\Delta z}}} \\right. \\kern-0pt} {\\Delta z}}} \\right)_s}$$\u003c/div\u003e\u003c/div\u003e,\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$${a_E}={D_e}A\\left( {\\left| {{P_e}} \\right|} \\right)+\\left( {\\left| { - {F_e}} \\right|} \\right),\\,{F_e}{\\text{=}}{\\left( {v\\Delta x} \\right)_e},\\,{D_e}={\\left( {{{\\Gamma \\Delta x} \\mathord{\\left/ {\\vphantom {{\\Gamma \\Delta x} {\\Delta z}}} \\right. \\kern-0pt} {\\Delta z}}} \\right)_e}$$\u003c/div\u003e\u003c/div\u003e,\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equf\" name=\"EquationSource\"\u003e\n$${a_W}={D_w}A\\left( {\\left| {{P_w}} \\right|} \\right)+\\left( {\\left| { - {F_w}} \\right|} \\right),\\,{F_w}{\\text{=}}{\\left( {v\\Delta x} \\right)_w},\\,{D_w}={\\left( {{{\\Gamma \\Delta x} \\mathord{\\left/ {\\vphantom {{\\Gamma \\Delta x} {\\Delta z}}} \\right. \\kern-0pt} {\\Delta z}}} \\right)_w}$$\u003c/div\u003e\u003c/div\u003e,\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equg\" name=\"EquationSource\"\u003e\n$${S_\\phi }=\\left( {{S_c}+{{\\phi _{p}^{n}} \\mathord{\\left/ {\\vphantom {{\\phi _{p}^{n}} {\\delta t}}} \\right. \\kern-0pt} {\\delta t}}} \\right)\\Delta V - \\left( {{a_E}+{a_W}} \\right)\\phi _{p}^{n}+{F_e}{\\phi _E} - {F_w}{\\phi _W}$$\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003ein which \u003cem\u003eφ\u003c/em\u003e stands for \u003cem\u003ek\u003c/em\u003e and \u003cem\u003eε\u003c/em\u003e, and Γ is the diffusion coefficient for \u003cem\u003eφ\u003c/em\u003e; \u003cem\u003eδt\u003c/em\u003e is time step, and \u003cem\u003eS\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eS\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e are the source terms for the discrete equations of \u003cem\u003eφ\u003c/em\u003e.\u003c/p\u003e \u003cp\u003eFor solving the advection equation of \u003cem\u003eF\u003c/em\u003e, a step function (the modified Youngs-VOF) is employed. The free surface slope is calculated by the ELVIRA method which is more accurate than the more commonly used VOF interface reconstruction algorithms.\u003c/p\u003e \u003c/div\u003e"},{"header":"3 Results and discussions","content":"\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Numerical stability analysis\u003c/h2\u003e \u003cp\u003eTo verify the numerical stability of the improved Youngs-VOF method, we design the following calculation example: the length of computational domain is 12m, horizontal bottom, initial water level is 0.8m, vertical uniform flow velocity at the inlet is 0.75 m/s and sudden water level rise of 0.4m within 0\u0026thinsp;~\u0026thinsp;2s. The influence of gravity is not considered in the calculation process.\u003c/p\u003e \u003cp\u003eThe computational grid is ∆x = ∆z\u0026thinsp;=\u0026thinsp;0.05m, and the velocity field of the entire computational domain is given based on the vertical uniform distribution without considering the turbulence effect. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows the shape of the free surface profile at the relevant time. As shown in this figure, the improved Youngs-VOF method is relatively stable, and no numerical oscillation or diffusion occurs in the calculation process.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Dam break wave\u003c/h2\u003e \u003cp\u003eFurthermore, there exist shock waves in the dam-break process, and the ability of numerical methods to capture shock waves can be tested based on this type of problem. Ying et al. [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], Wang et al. [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] and Mohapatra et al. [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] employed similar examples to test their respective algorithms.\u003c/p\u003e \u003cp\u003eGiven the computational domain length of 40 m and bottom elevation \u003cem\u003eb\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0, the initial conditions are as follows:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\left( {{h_L}=1.0{\\text{m}},{\\kern 1pt} \\,{u_L}=0.0} \\right)\\left| x \\right.\u0026lt;20,{\\kern 1pt} \\,\\left( {{h_R}=0.2{\\text{m}},{\\kern 1pt} \\,{u_R}=0.0} \\right)\\left| x \\right. \\geqslant 20\\)\u003c/span\u003e \u003c/span\u003e \u003csub\u003em\u003c/sub\u003e \u003c/p\u003e \u003cp\u003eThe computational grid is ∆x\u0026thinsp;=\u0026thinsp;0.20m, ∆z\u0026thinsp;=\u0026thinsp;0.10m. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e compares the numerical solution and exact solution (obtained from the one-dimensional homogeneous shallow water equation at the given time. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, the overall trend of the numerical solution is consistent with the exact solution. Since the upwind scheme is used, certain numerical diffusion occurs in the numerical solution, which is more obvious at the changes in the free surface.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Transcritical flow\u003c/h2\u003e \u003cp\u003eDue to different initial boundary conditions given, subcritical flow, shock wave and supercritical flow may exist simultaneously in the computational domain, which leads to certain difficulties in numerical simulation. The harmony and adaptability of numerical methods are generally tested based on this type of problem. Such as Vazquez et al. [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], Zhou et al. [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] and Wang et al. [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] used similar examples to test the shallow water models they established.\u003c/p\u003e \u003cp\u003eGiven the computational domain length of 25m and initial water level of 0.4m, the bottom elevation is as follows.\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(b\\left( x \\right)=\\left\\{ \\begin{gathered} 0.2 - 0.05{\\left( {x - 10} \\right)^2},\\quad \\quad 8 \\leqslant x \\leqslant 12 \\hfill \\\\ \\quad \\quad \\quad 0.0\\quad \\quad \\quad \\,\\,,\\quad \\quad \\;x\u0026lt;8\\;or\\,x\u0026gt;12\\;\\quad \\quad \\hfill \\\\ \\end{gathered} \\right.\\)\u003c/span\u003e \u003c/span\u003e \u003c/p\u003e \u003cp\u003e(1) Shock-free transcritical flow problem\u003c/p\u003e \u003cp\u003eGiven discharge per unit width q at the inlet of 1.53 m\u003csup\u003e2\u003c/sup\u003e/s and computational grid ∆x\u0026thinsp;=\u0026thinsp;0.04m, a non-uniform grid is used in the vertical direction, and a vertically uniform flow velocity distribution is given at the inlet. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea compares the numerical solution and exact solution (obtained from the one-dimensional shallow water equation without considering the time partial differential terms) when the calculation is stable.\u003c/p\u003e \u003cp\u003e(2) Shock transcritical flow problem\u003c/p\u003e \u003cp\u003eThe discharge per unit width q at the inlet is 0.17 m\u003csup\u003e2\u003c/sup\u003e/s, and the computational grid is ∆x = ∆z\u0026thinsp;=\u0026thinsp;0.01m. A vertically uniform flow velocity distribution at the inlet is given. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb compares the numerical solution and exact solution when the calculation is stable.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe numerical and exact solutions of the water depth in the stable flow zone on the left side of the shock-free transcritical flow are 1.0057 and 1.0147m respectively which are 0.4040 and 0.4m respectively on the right side. The numerical and exact solutions of the water depth in the stable flow zone on the left side of the shock transcritical flow are 0.4052 and 0.4062m respectively which are 0.3999 and 0.4m respectively on the right side, and the numerical and exact solutions of the minimum water depth at the shock wave are 0.3004 and 0.2996m respectively. It can be seen that the improved Youngs-VOF method has a good harmony and adaptability in numerical solutions.\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Conclusions","content":"\u003cp\u003eA 2-D vertical flow model is developed that the Youngs-VOF method is improved by a more accurate ELVIELRA method for free surface tracking in its normal direction. After numerical stability analysis, and verification analysis on the dam-break wave and transcritical flow problems, the proposed numerical model has good stability, strong ability and high accuracy in capturing the free surface. Compared with the traditional water level-bottom topography formulation (Pan et al. 2003), surface gradient method (Zhou et al. 2001), feature decomposition method (Wang et al. 2005) etc. for bed-slope source terms, a conventional grid division method is used to process the bed-slope source terms, which can provide a basis for studying shallow water problems with strong free surface nonlinearity.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eDongmiao Zhao, Qi Liu and Jian Chen wrote the main manuscript text, Yi Liu (Wenzhou Water Conservancy Survey and Design Institute Co., Ltd.) and Yang Zhou prepared figures 1-3 and references, and Yi Liu (Earth, Ocean and Atmospheric Sciences (EOAS) Thrust, Function Hub, The Hong Kong University of Science and Technology) checked the manuscript. All authors reviewed the manuscript.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eData generated during the current study are available from authors on reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003ePan CH, Lin BY, Mao XZ. A Godunov-type scheme for 1-D shallow water flow with uneven bottom. Advances in Water Science. 2003;14(4):430-436(In Chinese). https://doi.org/10.3321/j.issn:1001-6791.2003.04.008.\u003c/li\u003e\n\u003cli\u003eCrnjaric-Zic N, Vukovic S, Sopta L. Extension of ENO and WENO schemes to one-dimensional sediment transport equations. Computers \u0026amp; Fluids. 2004;33:31-56. https://doi.org/10.1016/S0045-7930(03)00032-X.\u003c/li\u003e\n\u003cli\u003eWang ZL, Geng YF, JIN S. Numerical modeling of 2-D shallow water flow with complicated geometry and topography. Journal of Hydraulic Engineering. 2005;35(4):1-9(In Chinese). https://doi.org/10.3321/j.issn:0559-9350.2005.04.010.\u003c/li\u003e\n\u003cli\u003eZhao ZY, Zhang QH, Li SS. Application of Runge-Kutta discontinuous Galerkin scheme for one-dimensional shallow water equations. Advances in Water Science. 2012;23(5):695-701(In Chinese). https://doi.org/10.14042/j.cnki.32.1309.2012.05.010.\u003c/li\u003e\n\u003cli\u003eHu WY, Li XG, Deng YX. A new third order WENO scheme for solving shallow water equations. Yellow River. 2023;45(12):31-36(In Chinese). https://doi.org/10.3969/j.issn.1000-1379.2023.006.\u003c/li\u003e\n\u003cli\u003eYoungs DL. Time-dependent multi-material flow with large fluid distortion. Numerical methods for fluid dynamics. 1982;273-285.\u003c/li\u003e\n\u003cli\u003ePilliod JE, Puckett EG. Second-order accurate volume-of-fluid algorithms for tracking material interfaces. Journal of Computational physics. 2004;199(2):465-502. https://doi.org/10.1016/j.jcp.2003.12.023.\u003c/li\u003e\n\u003cli\u003eLaunder BE, Spalding DB. The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering. 1974;3(2):269-289. https://doi.org/10.1016/0045-7825(74)90029-2.\u003c/li\u003e\n\u003cli\u003eVersteeg HK, Malalasekera W. An introduction to computational fluid dynamics: the finite volume method. Prentice Hall. 2007.\u003c/li\u003e\n\u003cli\u003eCebeci T, Bradshaw P. Momentum Transfer in Boundary Layers\u003cem\u003e. \u003c/em\u003eNew York:\u003cem\u003e \u003c/em\u003eHemisphere Publishing Corporation. 1977.\u003c/li\u003e\n\u003cli\u003eWANG FJ. Computational fluid dynamics analysis-theory of application of CFD software. Beijing: Tsinghua university press. 2004(In Chinese).\u003c/li\u003e\n\u003cli\u003eYing XY, Khan AA, Wang SSY. Upwind conservative scheme for the saint venant equations. Journal of Hydraulic Engineering. 2004;130(10):977-987. https://doi.org/10.1061/(ASCE)0733-9429(2004)130:10(977).\u003c/li\u003e\n\u003cli\u003eWang L, Zheng SP. Solving shallow water wave equation based on moving grid entropy stable scheme. Chinese journal of hydrodynamics. 2020;35(2):188-193(in Chinese). https://doi.org/10.16076/j.cnki.cjhd.2020.02.006.\u003c/li\u003e\n\u003cli\u003eMohapatra PK, Eswaran V, Bhallamudi MS. Two-dimensional analysis of dam break flow in vertical plane. Journal of Hydraulic Engineering. 1999;125(2):183-192. https://doi.org/10.1061/(ASCE)0733-9429(1999)125:2(183).\u003c/li\u003e\n\u003cli\u003eVazquez-Cendon ME. Improved Treatment of Source Terms in upwind schemes for the shallow water equations in channels with irregular geometry. Journal of Computational Physics. 1999; 148(2):497-526. https://doi.org/10.1006/jcph.1998.6127.\u003c/li\u003e\n\u003cli\u003eZhou JG, Causon DM, Mingham CG, et al. Ingram. The surface gradient method for the treatment of source terms in the shallow-water equations. Journal of Computational Physics. 2001;168 (1):1-25. https://doi.org/10.1006/jcph.2000.6670.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"VOF, shallow water, free surface, dam break wave, transcritical flow","lastPublishedDoi":"10.21203/rs.3.rs-4324162/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4324162/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eShallow water problems such as tidal bore, dam-break wave, shallow water wave etc. are important for design, construction, and operation of water conservancy projects. In this study, a 2-D vertical flow model having a strong adaptability for the nonlinear free surface and describing normally for bed source term is established by combining the nonlinear turbulence model and the Youngs-VOF method. The turbulent equations are discreted with implicit scheme in the vertical coordinate and explicit scheme in the horizontal coordinate based on the power-law scheme. The model is applied for a dam break wave problem and transcritical flow problem. The results show the adjusted model have a good performance in predicting xxx on the shallow water problems with strong nonlinearity, such as tidal bores. Our adjusted model and findings may benefit the design and management of water conservancy infrastructures in practice.\u003c/p\u003e","manuscriptTitle":"Application of adjusted Youngs-VOF method for shallow water problems","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-05-10 09:23:51","doi":"10.21203/rs.3.rs-4324162/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d0973cda-834b-4cd3-a882-9b80a926bb21","owner":[],"postedDate":"May 10th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-05-27T09:55:02+00:00","versionOfRecord":[],"versionCreatedAt":"2024-05-10 09:23:51","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4324162","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4324162","identity":"rs-4324162","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}