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Linearized NS euqations: how to solve them?(problem with Matrix operations..)

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Old   November 18, 2009, 07:58
Default Linearized NS euqations: how to solve them?(problem with Matrix operations..)
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matteo lombardi
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Hello,
I am trying to solve the steady linearized navier-Stokes equation (which may arise from Newton Formulation of the NS or adjoint problems or other..).

Let's be more general and let's say that i I have the following system:
1)Au+Bp=F
2)Cu=0
where u is the velocity vector, p is the pressure, A,B and C are the corrispective Matrix and F is the know-term of the momentum equations (sources+explicit terms+...)

Possible classical startegies:
a)Can I solve the problem with the classical Pressure Matrix formulation?
that would mean getting u from equation 1) thus u=A^-1F - A^-1Bp
thus i get the eqaution for p is: CA^-1Bp=CA^-1F
Once p known, I can get U. (Of course I need good preconditioner and so on but let's remain simple at the moment..)

To implement this method, I would need to be able to create those matrices, and be able to do matrices operations and solve linear problems passing to OF directly the matrices... Is it possible in openFoam?

From what i know OF can solve only segregated problems (u and p segregated, and even u is solved segregated between components, i.e. solve first Ux, then Uy, then Uz). Am I right?
Thus probably solvng A^1, where A is a vector matrix, is not possible, right?

Am I correct about my assumptions about OF capabilities?

b)if I had a stokes problem, i.e. no convective terms, I could apply the divergence operator to the momentum equations, get a laplacian equation for the pressure, and then correct u. Unfortunately I have some convective terms, thus some terms on the momentum equations don't go away.. thus i cannot "segregate" u and p with this approach..
Am i correct? any other ideas?

c)I could try to adapt a strategy like the simple method to my problems, using some sorts of predictor/corrector strategy. Thus i solve the momentum euqations with the old U, then get the new P, correct U, get new P, correct U and so on until a certain convergence..
Does this make any sense?(I have tried to implement thsi strategy but so far quite unsuccessfully..)

Could anyone point me to any article,book, notes about those problem is usually solved with Finite Volumes and segregated approach?
(Mainly I want to solve the Oseen equation with finite Volume in a segregated way.)

thank you very much,
regards,
Matteo
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