3.2 Sub-group 1.2 in which one differential equation is used

3.2.1 Prandtl energy with prescribed length scale

(a) The effective viscosity

Prandtl, in 1945, generalised his hypothesis regarding the effective viscosity of a turbulent fluid, giving it the form:

EV = const1 * LM * SQRT (KE)

where: LM is a prescribed length scale, which may vary from place to place; and KE is the kinetic energy of the turbulent motion, deducible from the velocity fluctuations in the three directions u', v' and w'

Thus: KE = 0.5 * ( u'**2 + v'**2 + w'**2 )

(b) The source of information about KE

term representing

D(KE)/Dt time-dependence & convective transport = div( const2*EV* grad(KE)) turbulent diffusion + EV*(vel_grad)**2 kinetic-energy generation by shear - const3*k**1.5/LM kinetic-energy disipation

Thus:

• The turbulent-viscosity concept was still used.
• Comparison of the formula for EV in Prandtl's mixing-length and energy models (see section 3.1.3 above) shows the connexion:
  KE =LM* vel_grad * LM
• Whereas the PMLM requires only one empirical constant, the Prandtl energy model (PEM) requires two more.

(c) Advantages and disadvantages

• The PEM does allow for convection and diffusion of turbulence into regions where there is zero local generation. It is therefore inherently capable of simulating some phenomena more realistically than can the PML model.
• On the other hand it is no more capable than is the PMLM of determining for itself what the value of LM should be; and knowledge is almost totally absent for re-circulating and 3D flows.
• Consequently it has been little used, and is considered useful only close to walls, where the length scale is fairly well known.

(d) Activation in PHOENICS

In order to activate the PEM in PHOENICS, the PIL command TURMOD needs to be inserted in the Q1 file with the argument KLMODL. TURMOD(KLMODL) is equivalent to the following PIL commands:

      SOLVE(KE)
      ENUT=GRND3;KELIN=0
      PATCH(KESOURCE,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP)
      COVAL(KESOURCE,KE,GRND4,GRND4);GENK=T

Then the choice of LM formula is made by the setting of further parameters such as EL1, EL1A, etc (see the Encyclopaedia entry EL1).

wbs