Boussinesq approximation

[anshchaube]

This issue can be closed when Boussinesq approximation has been implemented and tested in our flat plenum model.

[anshchaube]

@dshaver-ANL @katyhuff this is what I am doing for Boussinesq in Nek's userf function (explanation follows):

      g = 9.8
      dh = 0.015845536
      uav = 0.219457
      uavsq = uav*uav

      rhoe = 2692.0 !Janz 1988
      a = 0.562
      alpha = a/(rhoe-(a*temp))
      tref = 0.0

      ffx = 0.0
      ffy = 0.0
      ffz = (g*dh/uavsq)*(1-alpha*(temp-tref))

The Boussinesq term ffz doesn't have a rho value multiplied to it as Nek automatically multiplies ffz by rho (which is 1 for our non-dimensionalized case anyway).

For the non dimensionalized case, the Boussinesq term is: ffz = (gL / Vav^2) [ 1 - alpha_exp (T - Tref) where L is the characteristic length (hydraulic dia), Vav the characteristic velocity, and the coefficient of expansion alpha_exp is alpha = - (1/rho) d(rho)/dT rho_{molten-salt} = rho_experimental - (constant)*T Plugging this rho into the definition of alpha yields what I have coded up.

Does this look correct? Furthermore, is this appropriate for our case where the difference between the inlet and the outlet temperatures is about 30 K ? Just for reference, our dRho/Rho_ref over this range is ~ 0.7%.

[anshchaube]

This is blowing up, just like #10. This one has lx1=8, genmap interval = 0.1, and variable dT with target CFL of 0.2.

Step    422, t= 1.0117773E+00, DT= 3.2136566E-42, C=******* 3.9314E+03 9.3883E+00
             Solving for Hmholtz scalars
        422  Scalars done  1.0118E+00  4.9035E-02
             Solving for fluid
        422  Project PRES                  NaN          NaN          NaN   7   8
        422  PRES gmres       200          NaN          NaN   1.0000E-05   4.6727E+00   8.9877E+00    F
             L1/L2 DIV(V)           3.4499+120   6.2886+125
             L1/L2 QTL              0.0000E+00   0.0000E+00
             L1/L2 DIV(V)-QTL       3.4499+120   6.2886+125
             WARNING: DIV(V)-QTL too large!
        422  Fluid done  1.0118E+00  9.2433E+00
Step    423, t= 1.0117773E+00, DT= 2.5709253E-42, C=******* 3.9409E+03 9.4704E+00