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1

Junction Riemann problem for shallow water equations in networks

Elshobaki Mohamed Abdelrehim Selim Ibrahim

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2018

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Vol. 22. No 1

9--84

EN

In this paper, the Riemann solution of an extended Riemann problem in channel networks is presented. The Riemann problem at a junction network is well defined in the literature. However, it is limited to symmetric networks. Here, we extend the Riemann problem to non-symmetric networks such that neither the channel width equality nor the dischargee quality are assumed. The Riemann solution is given under subcritical flow conditions to ensurethe existence and uniqueness of the solution at the junction. Taking into account the massand energy conservation laws, the necessary conditions for the Riemann solution are drawn.The results are summarized in a theorem. The theorem is illustrated with a set of numerical examples.In order to perform a one-dimensional simulation in channel networks, the inner boundary conditions at the junction (i.e., the channel intersection point) are required. It has turned out that the classical models (i.e., the Equality, Gurram, Hsu models) that have been used to supplysuch a boundary suffer from many drawbacks.Thus, here we propose to use the Riemann solution at the junction networks to provideproper boundary conditions. Then, we compare all the junction models together. The junction models are validated against experimental results found in the literature for steady state flows.Generally, the Riemann model shows good results in matching the experimental data. In particular, the Riemann model shows the best results when the bottom discontinues at the junction. For the unsteady state flows, we perform prototype case studies to test the junction models in the channel networks, and the numerical solutions are compared with the analytical solutions. The Riemann model continues to show the best results that agree with the analytical solutions. However, the validation of the junction models in the unsteady state flows remains for future work due to the limited amount of real data.

2

Second order traffic flow modelIing: supply-demand analysis of the inhomogeneous Riemann problem and of boundary conditions

Lebacque J.-P. J.-P., Mammar S., Salem H. H.

Archives of Transport

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2008

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Vol. 20, iss. 1-2

47-67

EN

Recently Aw, Rascle, and Zhang introduced a second order model (ARZ) that does not exhibit the usual drawbacks of this family of models, i.e. negative velocities and/or densities. In this paper, we analyse the inhomogeneous Riemann problem for this model, which is shown to be equivalent to the inhomogeneous Riemann problem for a related first order model with modified equilibrium flow density relationship. The boundary conditions for the ARZ model are deduced. They can be expressed in terms of an upstream demand and downstream supply.

PL

Ostatnio Aw, Rascle i Zhang zdefiniowali model drugiego rzędu (ARZ), który nie wykazuje wad dotychczasowych modeli tego typu, jak ujemne prędkości i/lub gęstości. W artykule zanalizowano niejednorodny problem Riemanna dla tego modelu, który okazal się równoważny niejednorodnemu problemowi Riemanna dla odpowiedniego modelu pierwszego rzędu ze zmodyfikowaną zależnością natężenia ruchu od jego gęstości. Wyprowadzono warunki brzegowe dla modelu ARZ, które mogą być wyrażone jako popyt dopływu i podaż odpływu.

3

Second order traffic flow modelIing: Riemann problem resolution in the homogeneous case and applications to traffic timulation

Mammar S., Lebacque J.-P. J.-P., Salem H. H.

Archives of Transport

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2008

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Vol. 20, iss. 1-2

87-111

EN

This paper presents theoretical discussions, improvement and complete resolution, in the homogenous case, of the recent second order macroscopic model developments suggested by Aw and Rascle, and Zhang, respectively. The improved model strictly respects the anisotropic character of traffic flow. The resolution of the Riemann problem is described analytically and numerically. Non trivial examples are solved completely, which can be applied to the wave-tracking discretisation of the model. Numerical simulation results are reported.

PL

Artykuł przedstawia rozważania teoretyczne, udoskonalenie i pełne rozwiązanie najnowszego modelu makroskopowego drugiego rzędu sformułowanego przez Awa i Rascle'a oraz Zhanga dla przypadku jednorodnego. Ulepszony model dokładnie odzwierciedla anizotropowy charakter przepływu ruchu. Rozwiązanie problemu Riemanna jest przedstawione analitycznie i numerycznie. Nietrywialny przykład zostałw pełni rozwiązany i może być zastosowany do śledzącej falę dyskretyzacji modelu. Przedstawiono wyniki symulacji numerycznej.

4

A boundary integral equation for the 2d external potential flow

Nasser M., Murid A., Amin N.

International Journal of Applied Mechanics and Engineering

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2006

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Vol. 11, no 1

61-75

EN

Based on the recently discovered second kind Fredholm integral equation for the exterior Riemann problem, a boundary integral equation is developed in this paper for the two-dimensional, irrotational, incompressible fluid flow around an airfoil without a cusped trailing edge. The solution of the integral equation contains one arbitrary real constant, which may be determined by imposing the Kutta-Joukowski condition. Comparisons between numerical and analytical values of the pressure coefficient on the surface of the NACA 0009 and NACA 0012 airfoils with zero angle of attack show a very good agreement.

5

Zagadnienie Riemanna : metody rozwiązań numerycznych dla potrzeb hydrodynamiki

Kuczaj A.

Biuletyn Wojskowej Akademii Technicznej

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2002

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Vol. 51, nr 5

23-47

PL

W pracy przedstawiono przegląd obecnie stosowanych metod numerycznych przy rozwiązywaniu zagadnienia Riemanna dla potrzeb hydrodynamiki. Skupiono się na metodach dla równań Eulera, przedstawiając wyniki testów analizowanych metod. Dokonano porównań pod względem dokładności metod oraz ich kosztu obliczeiowego.

EN

The review is shown of numerical methods used presently in solutiona of the Riemann problem for hydrodynamics needs. The attention is paid on the methods for the Euler equations, presenting the test results of the analyzed methods. Comparisons were carried out with respect of the method accuracy and theirs calculation costs.