Lagrangian representation for incompressible flow
by Yi Zhang
Even Eulerian representation gives a field description of flow problems, it’s characterized and troubled by the convection terms. On the other hand, Lagrangian representation does not have this drawback, although the nonlinearity problem posed by convection term is replaced by nonlinearity of mapping between initial spacial cooridnates depending on particle index and spacial coordinates at some time later, as I am about to show.
Denote spacial coordinates, velocity, pressure in Eulerian representation as , , (symbol without subscript means the vector of proper dimension), one has the governing equations for inviscid incompressible flow:
For some reference time , and material coordinate (particle index) , spacial position is described by a particle trajectory mapping from Lagrangian coordinate to Eulerian coordinate:
Mapping between material coordinate and spacial coordinate is called deformation.
When material coordinate coincides with spacial coordinate at reference time , we have
in which , and are all generally in . Then the Lagrangian representation of unknowns are
Here symbols with superscript denotes the variables in Lagrangian representation. The material derivative and spacial/reference derivative, for some variable are related by
in which is the Jacobi matrix of , i.e. the deformation gradient:
A second order tensor’s reference coordinate form and material coordinate form is connected by Piola transform:
where is a second order tensor in reference configuration, and is second order tensor corresponding to in deformed configuration. The superscript means “inverse” & “transpose”. Considering the divergence operation, we have:
The divergence theorem leads to
Following theorem can be proved for the Piola transform:
and area derivatives and are related by
Apply results above to the viscosity term in momentum equation, we have
Finally, the Lagrangian formulation for momentum equation is:
here bears the physical interpretation of volume change ratio, which is constantly one for incompressible flow. Depending on whether the reference time is taken as some time during the movement or always the initial time with initial configuration, the Lagrangian description can be divided into updated Lagrangian formulation or Total Lagrangian formulation.