At numerical simulation of multiphase flows (water-air) almost all authors use continuity equation as "\partial ui / \partial xi = 0". Why "rho" is considered as constant? I understand that is an incompressible flow, but "rho" varies at space and time.
Dear Mr. Monteiro
In some numerical methods "rho" is constant but in some others as WCSPH method the "rho" is not constant and the pressure term is calculated with Tait equation which is function of two rhos, the principle rho and new rho.
Best regards
Dear Leonardo,
This is a very good question. And I have to state that different authors in literature would provide to you different answers. In addition, several authors do not accept a constant density even in the case of pure liquids. This is due to the fact, that the (differential) continuity equation is a (continuum) model that is based on certain assumptions. In the case of pure liquids it is in general postulated that liquids can be treated as incompressible. In fact, to compressed liquids substantially, very high pressure or kinetic energy are required due to the molecular forces acting within in the liquid. However, if extremely high pressure gradients are available, second order effects can occur that make constant density in the differential continuity equation questionable.
For multiphase flows (air-water) the most convincing "model explanation" for a constant density in the continuity equation is that not only the liquid phase but also the gaseous phase behaves incompressible if the kinetics of the flow field does not exceed certain limits. The most popular statement is that the velocity of the gas should be sufficiently small in comparison to the velocity of sound, i.e. that the Mach number Ma should be sufficient small.
Under this condition, the density of a water-air-flow can be considered not to depend on the flow field. Instead it can be considered purely as an intensive thermodynamic parameter. As other thermodynamic parameters the density depends chiefly on the temperature and the pressure. Thus, the density can change in space if a temperature or a pressure field is present. However, for small kinetic energy of the flow (small value of Ma) the density depends in a first estimation only on the temperature. Thus, for steady flows "\partial ui / \partial xi = 0" would represent a reasonable model even for water-gas-flow if, additionally, the flow field is nearly isotherm.
With compliments
Antonio
Thank you for the answers!
Prof. Antonio Delgado, Pooyan Nikeghbali and Prof. Abdul aziz Jaafar. I'm thinking about a CFD wave problem, that can be resolved by VOF or Level-Set models with an Eulerian description.
Actually, "rho" varies at space and time, near interface flow. At a cell, sometimes it is water and sometimes it is air. Because air and water are incompressible fluids (low velocity) I can simplify "rho" from my continuity equation? Or even been an incompressible flow, if "rho" varies at space and time I can't do this simplification?
Dear Leonardo,
I dealt with geophysical flows (it is not your case but there are common features). In these problems the viscosity is not constant and varies considerably. Geophysicists beleave in many cases that the density is linearly related with the viscosity and, hence, varies. It leads to the changes in the continuity equations. I attach our paper. You can look through it As for the code for calculation it is at the site of the journal Solid Earth ( as a supplementary material for our paper, free).
Best wishes,
Igor Popov
Dear Leonardo, there is a related discussion of two-phase flows going on Researchgate in parallel to this one. I do not wish to repeat myself, so let me just make one comment. The continuity equation belongs to the Newtonian fluid differential description, however surface phenomenon does not. OK, I am referring here to the differential formulation of mass-consevation equation, not the viscous shear hypothesis. In pain terms, the infinitesimal limit cannot engulf the phase dividing surface, unless it does not move. But the interface does move, so the density follows from solving quite another equation for interaction of the two flows of two phases remaining in contact. Surface tension for failry well "mixed" phases could generate significant effects, therefore Lagrangiam codes should (someday) be able to handle this situation much better. Regards, Jerzy
Thank you very much Doctor Popov and Doctor Klimkowski!
Your answers made it clear for me now!