In this work, we present a one-dimensional numerical model for the hyperbolic balance laws related to the incompressible fluid flow in a pipe. The problem is made challenging by the hypothesis that the tube can be collapsible and characterized by variable mechanical and geometrical properties. In order to increase the generality of the model, we assume that the variation of the properties of the tube can be discontinuous [1]. Moreover, the nature of the adopted tube-law, describing the relation between the pressure and the cross-section, leads to the presence of non-conservative products in the balance laws. The model belongs to the family of the Runge-Kutta discontinuous Galerkin schemes [2] and is third order accurate in space and time. To cope with the presence of the non-conservative products, a path-consistent approach is followed [3] and the DOT Riemann solver is here applied [4]. The use of a linear path is not sufficient to achieve satisfactory results; therefore, the curvilinear path proposed by Müller & Toro [5] is selected. To avoid unphysical oscillations, a WENO limiter is included in the model. Interestingly, the application of standard WENO techniques are not suitable for this particular scheme and a modified approach is here proposed. The natural application of the model is the simulation of blood flow in the circulatory system, with particular emphasis on the blood flow in the veins and in vessels surgically repaired by implantation of vascular prostheses. References 1. E. F. Toro and A. Siviglia. Flow in Collapsible Tubes with Discontinuous Mechanical Properties: Mathematical Model and Exact Solutions. Communications in Computational Physics, 13(2):361–385, 2013. 2. B. Cockburn and C. W. Shu. Runge-Kutta Discontinuous Galerkin Methods for Convection-dominated Problems. Journal of Scientific Computing, 16(3):173–261, 2001. 3. M. Parés. Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework. SIAM Journal on Numerical Analysis, 44:300–321, 2006. 4. M. Dumbser and E. F. Toro. A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems. Journal of Scientific Computing, 48(1-3):70–88, 2011. 5. L. O. Müller and E. F. Toro. Well-balanced high-order solver for blood flow in networks of vessels with variable properties. International Journal for Numerical Methods in Biomedical Engineering, 29:1388–1411, 2013.
A discontinuous Galerkin scheme for the simulation of flows in collapsible tubes with discontinuous mechanical properties
CALEFFI, Valerio;
2015
Abstract
In this work, we present a one-dimensional numerical model for the hyperbolic balance laws related to the incompressible fluid flow in a pipe. The problem is made challenging by the hypothesis that the tube can be collapsible and characterized by variable mechanical and geometrical properties. In order to increase the generality of the model, we assume that the variation of the properties of the tube can be discontinuous [1]. Moreover, the nature of the adopted tube-law, describing the relation between the pressure and the cross-section, leads to the presence of non-conservative products in the balance laws. The model belongs to the family of the Runge-Kutta discontinuous Galerkin schemes [2] and is third order accurate in space and time. To cope with the presence of the non-conservative products, a path-consistent approach is followed [3] and the DOT Riemann solver is here applied [4]. The use of a linear path is not sufficient to achieve satisfactory results; therefore, the curvilinear path proposed by Müller & Toro [5] is selected. To avoid unphysical oscillations, a WENO limiter is included in the model. Interestingly, the application of standard WENO techniques are not suitable for this particular scheme and a modified approach is here proposed. The natural application of the model is the simulation of blood flow in the circulatory system, with particular emphasis on the blood flow in the veins and in vessels surgically repaired by implantation of vascular prostheses. References 1. E. F. Toro and A. Siviglia. Flow in Collapsible Tubes with Discontinuous Mechanical Properties: Mathematical Model and Exact Solutions. Communications in Computational Physics, 13(2):361–385, 2013. 2. B. Cockburn and C. W. Shu. Runge-Kutta Discontinuous Galerkin Methods for Convection-dominated Problems. Journal of Scientific Computing, 16(3):173–261, 2001. 3. M. Parés. Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework. SIAM Journal on Numerical Analysis, 44:300–321, 2006. 4. M. Dumbser and E. F. Toro. A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems. Journal of Scientific Computing, 48(1-3):70–88, 2011. 5. L. O. Müller and E. F. Toro. Well-balanced high-order solver for blood flow in networks of vessels with variable properties. International Journal for Numerical Methods in Biomedical Engineering, 29:1388–1411, 2013.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
