Conservative Godunov-Type Solutions for Water Hammer Flows

Simulation and analysis of water hammer flows have great significance in a wide range of industrial and including power plants, sewage pipelines, etc. So far, many numerical approaches, such as the Method of Characteristics (MOC), Finite Differences (FD), and Finite Volume (FV), have been introduced to realize the solutions of the water hammer equations.

Guinot (2000) and Zhao and Ghidaoui (2004) introduced first- and second-order explicit finite volume method (FVM) Godunov-type schemes for water-hammer problems. The latter demonstrated that the first-order Godunov scheme and the MOC scheme give identical results. Leon (2007) proposed a conservative form FVM Godunov-Type Schemes, which use a second-order accurate formulation at the internal cells and a second-order accurate boundary method. This method has been applied to water hammer flows and the results show that Leon’s scheme is more accurate than either the MOC scheme or the Godunov-type scheme of Zhao and Ghidaoui (2004). However, Leon (2007) pointed out that his scheme does not consider the influences of convective term in the governing equations.

In this work, based on the Euler equations of conservative form, first- and second-order finite volume (FV) Godunov-type schemes on water hammer problems are formulated. Based on the theory of Riemann invariant, the proposed scheme adopts virtual cells and this scheme can compute the flux of boundaries and internal cells together for increasing computational efficiency. To achieve second-order accuracy at the internal cells in the proposed approach, MUSCL -Hancock method is used. The scheme considers the influences of convective term with special method. Comparisons of the conservative form FVM Godunov-Type Scheme of Leon (2007), and the proposed two different boundary-treatments schemes in this work, are conducted. The computations are executed on various Courant numbers and the energy norm is used to evaluate the numerical accuracy of the schemes. The computation results and theoretical analysis show that Courant numbers have impact on the numerical dissipation (numerical error) and convective term have certain degree impact on the result especially at high flow rate and low sound speed. Also,the results show the influences of boundary conditions with virtual cells. The reasons of differences between conservation and non-conservation Godunov-Type solutions are analyzed on account of theoretical analysis and simulation results.