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$$$$ PRODT    NOTICE  CHAT      11/09/12    21:17:45     7124           
                                             DATE     11/09/12

Procedure  PRODT                                 Voir aussi :

   Syntaxe :
   _________

           P = PRODT $MD (| 'TODEF' |) UN (GB TN) ;

                          | 'TOROT' |
                          | 'COMPL' |
                          | 'MIXTE' |




       P      Local measure of the velocity gradient tensor
              CHPOINT (SCAL SOMMET)

       $MD    NAVIER_STOKES model

              MMODEL

       UN     Velocity flow field
              CHPOINT (VECT SOMMET)

       GB     Buoyancy vector g*beta (standard gravity vector * coef.
              of thermal expansion)
              POINT

       TN     Temperature field
              CHPOINT (SCAL SOMMET)


   Description :
   _____________

   This procedure returns a scalar measure of the velocity gradient,
   which contributes to the eddy viscosity computation in RANS or LES
   turbulence models (k-Epsilon, Spalart-Allmaras, Smagorinsky...).



   Comments :
   __________

   In order to model the turbulent kinetic energy production, many
   authors use the expression C * Nut * P, where:

      - C is a nondimensional factor, most often a constant, or 1
      - Nut is the eddy viscosity
      - P is local measure of the velocity gradient tensor

   THE PRODT PROCEDUR COMPUTES THE P TERM ONLY, letting the dedicated
   turbulence model routines calculate the actual production.

   /!\ P scales like the inverse square of a time
       => IT MAY BE REQUIRED TO COMPUTE ITS SQUARE ROOT!


   Different measures are available for the velocity gradient tensor:

      - 'TODEF': strain rate tensor (DEFAULT)

                   P = |S|² = 2*Sij*Sij with Sij = 0.5*(Uij+Uji)
                                             Uij = dUi/dxj


      - 'TOROT': rotation rate tensor

                   P = |R|² = 2*Rij*Rij with Rij = 0.5*(Uij-Uji)
                                             Uij = dUi/dxj


      - 'COMPL': full velocity tensor

                   P = |U|² = Uij*Uij   with Uij = dUi/dxj


      - 'MIXTE': combinaison of strain rate and vorticity
                 (Dacles-Mariani et al., AIAA Journal 33(9), 1995)

                   P = { |R| + Cprod*Min(0,|S|-|R|) }²

                                        with Cprod=2



   If GB and TN parameters are given, production or destruction of
   turbulent energy due to buoyancy is taken into account:

                   G = (GB/sgt) * Grad(TN)

                           with sgt = 0.7 (Turbulent Prandtl)

   The approched which is followed here was inspired by W.Rodi's work.
   Only the stabilizing contribution of the buoyancy force is retained:

    - if Grad(T)*GB is negative (stable stratification), the total
      production is lessened as below:

                   (P + G)(1 + c3.Rf)

                          with Rf = -G/(P+G)  (Richardson number)
                               c3 = 0.8

    - otherwise G=Rf=0 so that an instable stratification does not
      affect the production term.