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### 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

Prandtl postulated that the turbulence energy obeyed a transport
equation of the form:
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).

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