Simulating water hammer with corrective smoothed particle method

The corrective smoothed particle method (CSPM) is used to simulate water hammer. The spatial derivatives in the water-hammer equations are approximated by a corrective kernel estimate. For the temporal derivatives, the Euler-forward time integration algorithm is employed. The CSPM results are in good agreement with solutions obtained by the method of characteristics (MOC). A parametric study gives insight in the e¿ects of particle distribution, smoothing length and kernel function. Three typical water-hammer problems are solved. CSPM will not beat MOC in classical water-hammer, but it has potential for water-hammer problems with free surfaces as seen in column separation and slug impact

Rapid filling of pipelines with the SPH particle method

The paper reports the development and application of a SPH (smoothed particle hydrodynamics) based simulation of rapid filling of pipelines, for which the rigid-column model is commonly used. In this paper the water-hammer equations with a moving boundary are used to model the pipe filling process, and a mesh-less Lagrangian particle approach is employed to solve the governing equations. To assign boundary conditions with time-dependent (upstream) and constant (downstream) pressure, the SPH pressure boundary concept proposed recently in literature is used and extended. Except for imposing boundary conditions, this concept also ensures completeness of the kernels associated with particles close to the boundaries. As a consequence, the boundary deficiency problem encountered in conventional SPH is remedied. The employed particle method with the SPH pressure boundary concept aims to predict the transients occurring during rapid pipe filling. It is validated against laboratory tests, rigid-column solutions and numerical results from literature. Results obtained with the present approach show better agreement with the test data than those from rigid-column theory and the elastic model solved by the box scheme. It is concluded that SPH is a promising tool for the simulation of rapid filling of pipelines with undulating elevation profiles. Keywords: Rapid filling of pipelines; Undulating elevation profile; SP

SPH simulation of free overfall in open channels with even and uneven bottom

The free overfall can be used as a simple and accurate device for flow measurement in open channels. In the past, the solution to this problem was found mainly through simplified theoretical expressions or on the basis of experimental data. In this paper, using the meshless smoothed particle hydrodynamics (SPH) method, the free overfall in open channels with even and uneven bottom is investigated. For the even bottom case, subcritical, critical and supercritical flows are simulated. For the uneven bottom case, supercritical flows with different Froude numbers are considered. The free surface profiles are predicted and compared with theoretical and experimental solutions in literature and good agreements are obtained. Keywords: SPH, free overfall, even and uneven bottom, subcritical flow, critical flow, supercritical flow

Momentum conserving methods that reduce particle clustering in SPH

In this paper we present two remedies for particle clustering in SPH. Since particle clustering is the consequence of a diminishing kernel gradient for small inter-particle distances, the first method uses a convex kernel with a non-zero kernel gradient at the origin. The second method is based on inter-particle collisions. They are both compared with conventional SPH in several case studies. The results show a great improvement in particle distribution, where particle clustering is strongly reduced or absent, with only a small influence on the accuracy of the computations

The momentum flux coefficient for modelling steady and unsteady head losses in turbulent pipe flow

In this paper the momentum flux coefficient (ß) is used for the modelling of steady and unsteady head losses in turbulent pipe flows. For this purpose a transport equation is derived, which describes the change of ß in space and time, due to the inertia-driven shift and the shear-driven recovery of velocity profiles. It is demonstrated that ß represents steady friction and it is made plausible thatß represents the turbulence intensity. The key assumption is that the unsteady wall shear stress may be described as a logical extension of the steady wall shear stress and as such is correlated to unsteady head losses. The governing equation for ß is added to the two classical water-hammer equations, where the quasi-steady Darcy-Weisbach friction coefficient is replaced by the dynamic momentum flux coefficient ß (x,t). Thus a convenient friction model is proposed in which steady and unsteady friction are mathematically described in the same manner. Numerical simulations show that the effect of ß as momentum correction factor is negligible, but that its use in an unsteady Darcy-Weisbach type of friction term results in significant damping and rounding of pressure waves

The momentum flux coefficient for modelling steady and unsteady head losses in turbulent pipe flow

In this paper the momentum flux coefficient (ß) is used for the modelling of steady and unsteady head losses in turbulent pipe flows. For this purpose a transport equation is derived, which describes the change of ß in space and time, due to the inertia-driven shift and the shear-driven recovery of velocity profiles. It is demonstrated that ß represents steady friction and it is made plausible thatß represents the turbulence intensity. The key assumption is that the unsteady wall shear stress may be described as a logical extension of the steady wall shear stress and as such is correlated to unsteady head losses. The governing equation for ß is added to the two classical water-hammer equations, where the quasi-steady Darcy-Weisbach friction coefficient is replaced by the dynamic momentum flux coefficient ß (x,t). Thus a convenient friction model is proposed in which steady and unsteady friction are mathematically described in the same manner. Numerical simulations show that the effect of ß as momentum correction factor is negligible, but that its use in an unsteady Darcy-Weisbach type of friction term results in significant damping and rounding of pressure waves

The momentum flux coefficient for modelling steady and unsteady head losses in turbulent pipe flow

In this paper the momentum flux coefficient (ß) is used for the modelling of steady and unsteady head losses in turbulent pipe flows. For this purpose a transport equation is derived, which describes the change of ß in space and time, due to the inertia-driven shift and the shear-driven recovery of velocity profiles. It is demonstrated that ß represents steady friction and it is made plausible thatß represents the turbulence intensity. The key assumption is that the unsteady wall shear stress may be described as a logical extension of the steady wall shear stress and as such is correlated to unsteady head losses. The governing equation for ß is added to the two classical water-hammer equations, where the quasi-steady Darcy-Weisbach friction coefficient is replaced by the dynamic momentum flux coefficient ß (x,t). Thus a convenient friction model is proposed in which steady and unsteady friction are mathematically described in the same manner. Numerical simulations show that the effect of ß as momentum correction factor is negligible, but that its use in an unsteady Darcy-Weisbach type of friction term results in significant damping and rounding of pressure waves

Rapid filling of pipelines with the SPH particle method

The paper reports the development and application of a SPH (smoothed particle hydrodynamics) based simulation of rapid filling of pipelines, for which the rigid-column model is commonly used. In this paper the water-hammer equations with a moving boundary are used to model the pipe filling process, and a mesh-less Lagrangian particle approach is employed to solve the governing equations. To assign boundary conditions with time-dependent (upstream) and constant (downstream) pressure, the SPH pressure boundary concept proposed recently in literature is used and extended. Except for imposing boundary conditions, this concept also ensures completeness of the kernels associated with particles close to the boundaries. As a consequence, the boundary deficiency problem encountered in conventional SPH is remedied. The employed particle method with the SPH pressure boundary concept aims to predict the transients occurring during rapid pipe filling. It is validated against laboratory tests, rigid-column solutions and numerical results from literature. Results obtained with the present approach show better agreement with the test data than those from rigid-column theory and the elastic model solved by the box scheme. It is concluded that SPH is a promising tool for the simulation of rapid filling of pipelines with undulating elevation profiles. Keywords: Rapid filling of pipelines; Undulating elevation profile; SP

Simulating water hammer with corrective smoothed particle method

The corrective smoothed particle method (CSPM) is used to simulate water hammer. The spatial derivatives in the water-hammer equations are approximated by a corrective kernel estimate. For the temporal derivatives, the Euler-forward time integration algorithm is employed. The CSPM results are in good agreement with solutions obtained by the method of characteristics (MOC). A parametric study gives insight in the e¿ects of particle distribution, smoothing length and kernel function. Three typical water-hammer problems are solved. CSPM will not beat MOC in classical water-hammer, but it has potential for water-hammer problems with free surfaces as seen in column separation and slug impact

Simulating water hammer with corrective smoothed particle method

The corrective smoothed particle method (CSPM) is used to simulate water hammer. The spatial derivatives in the water-hammer equations are approximated by a corrective kernel estimate. For the temporal derivatives, the Euler-forward time integration algorithm is employed. The CSPM results are in good agreement with solutions obtained by the method of characteristics (MOC). A parametric study gives insight in the e¿ects of particle distribution, smoothing length and kernel function. Three typical water-hammer problems are solved. CSPM will not beat MOC in classical water-hammer, but it has potential for water-hammer problems with free surfaces as seen in column separation and slug impact