[mufeng]

Dear all,
Recently, I am new to the hydrodynamics' linear stability analysis. And I have read a relative literature on linear stability analysis using openFoam(https://www.sciencedirect.com/scienc...00936116300024).
On the framework of openfoam, my question is that how to obtain the Jacobian matrix of the right handside of linearized Navier–Stokes equations as mentioned in equation(6).
All suggestions are appreciated.

[mAlletto]

Have a look at this. https://www.google.com/url?sa=t&sour...7Buj1hHcIWT3Ze.

There you will find the Jacobian matrix of the linear navier stokes equation

[dlahaye]

OpenFoam does *not* apply a Newton iteration to linearize the non-linear convective terms in the momentum equation.

OpenFoam instead employs a successive iteration (fixed point iteration, Picard iteration instead, see wiki for explanation of these terms). See openfoam documentation.

OpenFoam does not compute a Jacobian.

Examples of public domain codes that do maybe compute the Jacobian include Fenics, FireDrake, NGSolve, Gridap, Ferrite, WaterLilly and Trixie.

[aerosayan]

Originally Posted by mufeng

my question is that how to obtain the Jacobian matrix of the right handside of linearized Navier–Stokes equations

I have also been searching for the same thing, and someone correct me if I'm wrong, but the Jacobian is what linearizes the equations around a certain point.

Regarding how to compute the Jacboian ...

1. Try to find out if any other research paper has already published an analytical form of the Jacobian you're looking for. I have found many for compressible euler flows, and one for navier stokes. Still searching for other research papers ...

2. If you don't find anything, you have to do it yourself. Obviously the NS equations are difficult, So, if ..

U = [rho, u*rho, v*rho, E*rho]

F = convective flux vector in x direction,

G = convective flux vector in y direction,

Computing the Jacobians dF/dU, and dG/dU is obviously a complicated thing .. because it's difficult to understand what even are the Jacobian components dF/d(rho), or dF/d(u*rho)

What I've seen other researchers do though, is reformulate the equations.

If you assume A = rho, B = u*rho, C = v*rho, D = E*rho, you can actually reformulate your flux equations with A,B,C,D ...

As a result, you can now easily compute the Jacobian components as dF/dA, dF/dB, dF/dC etc...

Since the flux equations in F are reformulated in terms of A,B,C,D, you can calculate the partial derivatives with respect to A,B,C,D