A few years ago I and my colleague Peter Wall wrote a short summary for Switch Function and and Complementarity Problem based formulations in a chapter which is available Open Access:
Andreas Almqvist and Peter Wall (October 26th 2016). Modelling Cavitation in (Elasto)Hydrodynamic Lubrication, Advances in Tribology, Pranav H. Darji, IntechOpen, DOI: 10.5772/63533. Available from: https://www.intechopen.com/books/advances-in-tribology/modelling-cavitation-in-elasto-hydrodynamic-lubrication
Since there are many more models for cavitation in lubrication flows it would be interested to hear from the community which ones you think are the most suitable ones and in which situations they should/could be applied.
Cheers, Andreas
This is an interesting topic. My preferred approach for complementarity problems in general is by recasting the inequality constraints into a non-smooth equality. We applied this approach in our following publication for modeling the case where the lubricant film is detached from one of the two surfaces:
Article Finite element method for starved hydrodynamic lubrication w...
In the method presented in the paper we combined a free surface flow simulation, governed by surface tension, in the starved region, with hydrodynamic flow simulation in the fully flooded region. A quite advanced case of a complementarity problem in lubrication, not seen often in the literature.
In the introduction of the paper we also list other works that might be interesting for this discussion. Just to highlight one:
Article Finite deformation effects in soft elastohydrodynamic lubric...
Great question Andreas Almqvist! One which I've debated a lot!
The complementarity problem is an excellent and effective way of solving the problem of hydrodynamic cavitation. Have you (or someone else) applied this method to highly-loaded contacts (GPa pressures) such a line and point contact geometries out of interest? Great to see answers already from authors adapting the approach for their specific problems here too.
I see the expected pressure magnitude being one of the main drivers behind the choice of model for this phenomena. I have reviewed (a little) the modelling approaches to hydrodynamic cavitation with the late, great Duncan Dowson as part of this publication: Article An Arbitrary Lagrangian-Eulerian Formulation for Modelling C...
. In this article we developed a 'moving mesh' type of approach for modelling the onset of cavitation but did not consider film reformation. This method can be applied to low pressure (MPa) and high pressure (GPa) problems without any difference in the implementation.For lower pressure environments (MPa) the Elrod-Adams algorithm is still widely used as the baseline method for simulating cavitation including papers published by my group this year: Article Two-scale homogenization of hydrodynamic lubrication in a me...
. There are several publications optimising and adapting this method with varying degrees of success in the literature, too many to list here.More recently this method has been combined with the complementarity condition to formulate the problem as described by Andreas and others. This also links to recent improvements in the solution approach by Woloszynski and the use of the Fischer-Burmeister equation to formulate the complementarity problem (see very recent work by Ron van Ostayen at TU Delft).
The complementarity method is a more stable than Elrod-Adams approach numerically because of the lack of switching functions and more robust mathematical formulation, but this does require a LCP solver which may be prohibitive to some. This is likely why Elrod-Adams remains widely used given it can be used with a standard PDE solver and setup readily with commercial codes, albeit with switching functions included making the problem numerically stiff.
There is also the issue of the change of variable approach from pressure to a derived field used explicitly for the cavitation problem, which then conflates the solution to the lubrication flow in terms of one variable. Therefore several authors use a pressure-only type approach such as that which I explored with Andreas here: Article On the Two-Scale Modelling of Elastohydrodynamic Lubrication...
. This article uses the Söderfjäll model to create a blending function reducing the density and viscosity of the lubricant rapidly to almost zero in the presence of negative pressure. This simulates the effect of cavitation solely on the lubricant properties and produces the required pressure distribution efficiently with rupture and reformation included. A similar but different approach was used by Gao et al in this paper:Article The role of micro cavitation on EHL: A study using a multisc...
. The approach is very useful for where there are many regions of cavitation occurring within the domain.Note that all the applications discussed so far are conventionally developed for low pressure (MPa) environments, and that in the classic high pressure (GPa) contact problems cavitation is managed with less rigour of the pressure profile shape/value. This is because of the large separation in magnitude between the max pressure and the vapour pressure in such contacts, with the requirement only being to resolve the pressure and pressure gradient to zero at the location of the onset of cavitation (JFO conditions).
For GPa problems the JFO conditions are conventionally achieved either with the use of a free boundary problem (see the classic multilevel solutions of Venner) or with the penalty approach of Wu (see more recent solutions by Habchi). Neither of these approaches fully resolve the problem of cavitation and this has led to several two-phase type models being developed which require very fine grids to resolve the flow. Multi-phase CFD simulations have also been developed recently by Imperial College London to investigate cavitation in much more depth for highly-loaded contacts and resolving all the flow (see Hartinger et al. and Hajishafiee et al.). These two- or multi- phase models also allow variance in the film thickness direction and are therefore not necessarily based on the Reynolds equation, making them more complex descriptions of the flow that are harder to solve numerically.
So the question comes down to whether resolving the full pressure shape/value is of importance to the outcome required, which itself is connected to the magnitude of the expected pressure. There is also the question of what type of solver you have readily available, whether a change of variable doesn't conflate the problem formulation, and if you're interested in very fine detail about the flow features.
Personally I would be in favour of the ALE or complementarity approaches for any problem/geometry but this certainly needs more development, hence my question at the start. I also think a review paper on the topic is well overdue!
Hi Andreas Almqvist
I think this article might useful for your study: Article Hydrodynamic studies on liquid-liquid two phase flow separat...
Wish you all the best!
KHOA
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