Convective term unsteady 2D NS equations

General Discussion
1

Hi everyone, I am implementing unsteady 2D Navier-Stokes equations but I am having troubles with the convective term.
Looking at the FreeFem documentation, I have found the built-in function convect that works pretty well for my kind of problem. The code that implements the convective term in this way is the following one:

varf navierstokes([ux, uy, p], [vx, vy, q])  = int2d(Th)(
      rho/dt* [ux, uy]'* [vx, vy]
    + mu* (Grad(u):Grad(v))
    - p* div(v)
    - q* div(u)
    - 1e-10 *p*q    
	)
	+ int2d(Th) (
      [fx, fy]' * [vx, vy]
    + rho/dt* [convect([upx, upy], -dt, upx), convect([upx, upy], -dt, upy)]'* [vx, vy]
    )
  + on(noslip, ux=0, uy=0)
  + on(inflow, ux=uin, uy=0)
 ;

where up variables are the ones related to the previous time-step.
Thus, I have tried to directly implement the weak formulation of my problem, but it does not work. The related code is the following one:

varf navierstokes([ux, uy, p], [vx, vy, q])  = int2d(Th)(
      rho/dt*  [ux, uy]'* [vx, vy] 
    + mu* (Grad(u):Grad(v))
    - p* div(v)
    - q* div(u)
    - 1e-10 *p*q    
	)
	+ int2d(Th) (
      [fx, fy]' * [vx, vy]
    - rho/dt* [upx,upy]'*[vx, vy] 
	)
	+int2d(Th)(
	[upx,upy]'*Grad(u)*[vx, vy]
	)
  + on(noslip, ux=0, uy=0)
  + on(inflow, ux=uin, uy=0)
 ;

Have someone ever had the same problem? Can you help me?

Thanks in advance

2

Without the whole script it is impossible to sort out the error.

Is your problem is small and 2D, you can use the following approach:

This code uses implicit time stepping and LU factorization, and solves the perturbation for of the equations (stationary base flow and solves for the fluctuations). It worked for me nicely.

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