Investigation on numerical schemes in the simulation of barotropic cavitating flows

A numerical methodology for the simulation of cavitating flows is considered. A homogeneous-flow cavitation model, accounting for thermal effects and active nuclei concentration, is considered, which leads to a barotropic state law. The continuity and momentum equations for compressible inviscid flows are discretized through a finite-volume approach, applicable to unstructured grids. The numerical fluxes are computed by shockcapturing schemes and adhoc preconditioning is used to avoid accuracy problems in the low-Mach regime. Second-order accuracy in space is obtained through MUSCL reconstruction. Time advancing is carried out by an implicit linearized scheme. Two different numerical fluxes are investigated here, viz. the Roe and the Rusanov schemes. For the Rusanov flux two different time linearizations are proposed; in the first one the upwind part of the flux function is frozen in time, while in the second one its time variation is taken into account, although in an approximated manner. The different schemes and the different linearizations are appraised for the quasi 1D-flow in a nozzle through comparison against exact solutions and for the flow around a hydrofoil mounted in a wind tunnel through comparison against experimental data. Non-cavitating and cavitating conditions are simulated. It is shown that, for cavitating conditions, the Rusanov scheme together with the more complete time linearization allows time steps much larger than for the Roe scheme to be used. Finally, the results obtained with this scheme are in good agreement with the exact solutions or the experimental data for all the considered test cases.http://deepblue.lib.umich.edu/bitstream/2027.42/84242/1/CAV2009-final42.pd

An implicit low-diffusive HLL scheme with complete time linearization: Application to cavitating barotropic flows

A numerical method for generic barotropic flows is presented, together with its application to the simulation of cavitating flows. A homogeneous-flow cavitation model is indeed considered, which leads to a barotropic state equation. The continuity and momentum equations for compressible flows are discretized through a mixed finite-element/finite-volume approach, applicable to unstructured grids. P1 finite elements are used for the viscous terms, while finite volumes for the convective ones. The numerical fluxes are computed by shock-capturing schemes and ad-hoc preconditioning is used to avoid accuracy problems in the low-Mach regime. A HLL flux function for barotropic flows is proposed, in which an anti-diffusive term is introduced to counteract accuracy problems for contact discontinuities and viscous flows typical of this class of schemes, while maintaining its simplicity. Second-order accuracy in space is obtained through MUSCL reconstruction. Time advancing is carried out by an implicit linearized scheme. For this HLL-like flux function two different time linearizations are considered; in the first one the upwind part of the flux function is frozen in time, while in the second one its time variation is taken into account. The proposed numerical ingredients are validated through the simulations of different flow configurations, viz. the Blasius boundary layer, a Riemann problem, the quasi-1D cavitating flow in a nozzle and the flow around a hydrofoil mounted in a tunnel, both in cavitating and non-cavitating conditions. The Roe flux function is also considered for comparison. It is shown that the anti-diffusive term introduced in the HLL scheme is actually effective to obtain good accuracy (similar to the one of the Roe scheme) for viscous flows and contact discontinuities. Moreover, the more complete time linearization is a key ingredient to largely improve numerical stability and efficiency in cavitating conditions. (C) 2010 Elsevier Ltd. All rights reserved

Numerical Simulation of the Flow in a Turbopump Inducer in Non-Cavitating and Cavitating Conditions

A numerical methodology for the simulation of cavitating flows in real complex geometries is presented. A homogeneous-flow cavitation model, accounting for thermal effects and active nuclei concentration, which leads to a barotropic state law is adopted. The continuity and momentum equations are discretized through a mixed finite-element/finite-volume approach, applicable to unstructured grids. A robust preconditioned low-diffusive HLL scheme is used to deal with all speed barotropic flows. Second-order accuracy in space is obtained through MUSCL reconstruction. Time advancing is carried out by a second-order implicit linearized formulation together with the Defect Correction technique. The flow in a real 3D inducer for rockets turbopumps is simulated for a wide range of conditions: different flow rates and rotating speeds as well as non-cavitating and cavitating flows are considered. The results obtained with this numerical approach are compared with experimental data

Linearized implicit time advancing and defect correction applied to sediment transport simulations

The numerical simulation of sediment transport problems is considered in this paper. The physical problem is modeled through the shallow-water equations coupled with the Exner equation to describe the time evolution of the bed profile. The spatial discretization of the governing equations is carried out by a finite-volume method and a modified Roe scheme designed for non-conservative systems. As for the time advancing, starting from an explicit method, a linearized implicit scheme is generated, in which the flux Jacobian is computed through automatic differentiation. Second-order accuracy in space is then obtained through MUSCL reconstruction and in time through a backward differentiation formula associated with a defect-correction approach. The implicit time advancing is compared in terms of accuracy and computational time with the explicit approach for one-dimensional and two-dimensional sediment transport problems, characterized by different time scales for the evolution of the bed and of the water flow. It is shown that, whenever the use of large time steps is compatible with the capture of the water flow dynamics and of the bedload evolution, the implicit scheme is far more efficient than its explicit counterpart with a CPU reduction up to more than two orders of magnitude. This makes implicit time differencing an attractive option for complex real life applications in this area

A 3D finite volume scheme for the simulation of edge plasma in Tokamaks*

A finite volume method in cylindrical coordinates for the simulation of the edge region in tokamaks is proposed. In contrast to standard methods, which require the projected form of the equations in curvilinear coordinates, a new method is proposed. This technique is based on the discretization on the vector form of the equations and then on the projection on the basis of the curvilinear system associated to each control volume. The proposed methodology has been validated on several simple test cases and applied on a 3D simulation of a reduced MHD model

A 3D finite volume scheme for the simulation of edge plasma in Tokamaks

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Implicit time advancing applied to shallow water problems coupled with different models of sediment transport

The numerical simulation of sediment transport problems is considered. The physical problem is modeled through the shallow-water equations coupled with the Exner equation to describe the time evolution of the bed profile. Different models of solid transport discharge of increasing complexity are considered. The spatial discretisation of the governing equations is carried out by a finite-volume method and a modified Roe scheme designed for non-conservative systems. Linearized implicit schemes for time advancing are built through a recently proposed strategy, based on automatic differentiation to compute the flux Jacobians and on the defect correction approach to reach second-order accuracy. Explicit schemes for time advancing are compared with implicit ones in one-dimensional sediment transport problems, characterized by different time scales for the evolution of the bed. It is shown that, independently of the model used for the solid transport discharge, for slow and intermediate speeds of interaction bewteen the bedload and the water flow, for which the use of large time steps is compatible with the capture of the bed evolution, implicit time advancing is far more efficient than explicit one with a computational cost reduction up to more than three orders of magnitude

Comparison of Explicit and Implicit Time Advancing in the Simulation of a 2D Sediment Transport Problem

The simulation of sediment transport, based on the shallow-water equations coupled with Grass model for the sediment transport equation is considered. The aim of the present paper is to investigate the behavior of implicit linearized schemes in this context. A finite-volume method is considered and second-order accuracy in space is obtained through MUSCL reconstruction. A second-order time accurate explicit version of the scheme is obtained through a two step Runge-Kutta method. Implicit linearized schemes of second-order of accuracy in time are derived thanks to a BDF method associated with a Defect Correction technique. The different time-advancing schemes are compared, using a 2D sediment transport problem, with different types of flow/bed interactions. The implicit one largely outperforms the explicit version for slow flow/bed interactions while in the case of fast flow/bed interactions, the CPU time of both time integration schemes are comparable. Thus, the implicit scheme turns out to be a good candidate to simulate flows with sediment transport in practical applications