https://www.win.tue.nl/~atijssel/pdf_files/Bergant-Simpson-Tijsseling_2006.pdf

Water hammer with column separation: A historical review

A mathematical model has been developed to calculate transient flows in viscoelastic pipes with vapour cavitation. This model takes into account both the viscoelastic behaviour of the pipe wall and the vapour cavitation phenomenon. When the liquid pressure falls below the vapour pressure, a bubbly cavitating flow of homogeneous liquid-vapour mixture occurs in some parts of the pipeline. The mixture density is expressed by means of a non-linear expression of the liquid volume fraction. The pipe wall behaviour is simulated based on the mechanical principle, by introducing additional viscoelastic term into the mass balance fluid equation. The application of mass and momentum conservation laws yields to a system of hyperbolic partial differential equations. The later is resolved by a second order finite differences scheme. The proposed model is tested for transient flows with and without vapour cavitation. The obtained numerical results are discussed and compared with experimental data.

Numerical Simulation of Transient Flows in Viscoelastic Pipes with Vapour Cavitation

Most of today’s water supply systems are based on plastic pipes. They are characterized by the retarded strain (RS) that takes place in the walls of these pipes. The occurrence of RS increases energy losses and leads to a different form of the basic equations describing the transient pipe flow. In this paper, the RS is calculated with the use of convolution integral of the local derivative of pressure and creep function that describes the viscoelastic behavior of the pipe-wall material. The main equations of a discrete bubble cavity model (DBCM) are based on a momentum equation of two-phase vaporous cavitating flow and continuity equations written initially separately for the gas and liquid phase. In transient flows, another important source of pressure damping is skin friction. Accordingly, the wall shear stress model also required necessary modifications. The final partial derivative set of equations was solved with the use of the method of characteristics (MOC), which transforms the original set of partial differential equations (PDE) into a set of ordinary differential equations (ODE). The developed numerical solutions along with the appropriate boundary conditions formed a basis to write a computer program that was used in comparison analysis. The comparisons between computed and measured results showed that the novel modified DBCM predicts pressure and velocity waveforms including cavitation and retarded strain effects with an acceptable accuracy. It was noticed that the influence of unsteady friction on damping of pressure waves was much smaller than the influence of retarded strain.

Modeling Transient Pipe Flow in Plastic Pipes with Modified Discrete Bubble Cavitation Model

A mathematical model which describes the transient flow with vapour cavitation is presented. Homogeneous bubbly cavitating two-phase flow will occur as a consequence of pressure reduction. By analogy with the gas release phenomenon, when the liquid pressure in the pipe drops below the saturation vapour pressure, vapour will be released from the liquid according to Henry's law. The distribution of the formed vapour bubbles is assumed to be uniform and the vapour within the bubbles acts isothermally. The two constitutive equations of continuity and momentum, written in conservation form, yield to a quasi-homogeneous flow model which have the same form inside as well as outside the cavitation region, but with different density expressions as pressure functions. A two-step finite differences numerical scheme is applied for the computation of the transient cavitating flow. Computation results are favourably compared with some numerical examples obtained from relevant literature. Consistent agreement has been obtained between numerical predictions and experimental measurements in prototype pipelines.

Numerical simulation of transient vaporous cavitating flow in horizontal pipelines

The numerical vapour cavitation model presented in a previous work is extended and applied to study fluid—structure interaction in transient flows in elastic pipes. The application of mass and momentum conservation laws yields to a system of two hyperbolic partial differential equations. The obtained system is resolved by a two-step finite differences numerical scheme. By analogy with the gas release phenomenon, when the liquid pressure in the pipe drops below the vapour pressure, vapour will be released from the liquid according to Henry’s law. To put in evidence, the fluid—structure interaction, a relationship between pressure and pipe section, has been introduced. Computation results permit the representation of pressures’ histories in different pipe sections, as well as the evolution of the cavitation zones in term of the elastic Young modulus.

Modelling Vapour Cavitation in Pipes with Fluid–Structure Interaction

Un modele mathematique permettant d'etudier le phenomene de degazage de l'air dissous dans les ecoulements transitoires en conduites deformables est presente. Le degazage se produit lorsque la pression devient inferieure a la pression de saturation exprimee par la loi de Henry. Dans ces conditions, la quantite de gaz degagee est proportionnelle a la difference entre les pressions de saturation et ambiante. L'air se degage sous forme de petites bulles qui evoluent selon la loi isotherme. Dans la zone de degazage, la fraction massique en gaz libre est utilisee pour exprimer la masse volumique du fluide a l'aide d'une moyenne ponderee des masses volumiques des deux phases du melange : liquide et gazeuse. L'influence de la deformation radiale des parois de la conduite sur l'evolution du regime transitoire est representee par une relation entre la pression du fluide et la section de la conduite. La formulation mathematique aboutit a un systeme de deux equations aux derivees partielles non lineaires de type hyperbolique dont la resolution est conduite par une methode des differences finies conservatives

Lili Taïeb

Papers

Écoulements transitoires dans les conduites déformables avec dégazage de l'air dissous