The Prandtl mixing-length model

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3.1.3 The Prandtl mixing-length model

(a) The effective viscosity

In Prandtl's (1925) mixing-length model, the effective viscosity is taken as being proportional to the square of a quantity having the dimensions of length, ie the so-called mixing length ML, multiplied by the absolute value of the local velocity gradient.

Thus: EV = const * LM**2 * abs(vel_grad)

The mixing-length LM has to be specified as a function of position.

In unbounded flows, LM is of the order of 0.1 * layer width.

Close to a wall, LM is of the order of 0.4 * distance from the wall;

BUT, in the immediate vicinity of the wall where viscous effects predominate, it diminishes more rapidly.

(b) Advantages

Calculations based on the Prandtl mixing-length model (PMLM) are easy to make, because no additional differential equation must be solved.

In unbounded flows (ie jets, wakes, plumes), the variation of the PML across the layer width is not large, so that velocity profiles can be fairly well predicted.

Although the PMLM is not useful very close to a wall, unless modified in the manner of (say) Van Driest (see below), the processes occurring there can often be handled adequately by use of an empirically-based "wall function".

For most boundary-layer flows, at least the order of magnitude of the mixing length can be correctly guessed.

(c) Disadvantages,

For flows with recirculation, or those with non-planar walls, it is impossible to estimate the distribution of mixing-length magnitudes with acceptable accuracy. There are many such flows

The PMLM implies that the local level of turbulence depends only upon the local generation and dissipation rates; but, in reality, turbulence may be carried or diffused to locations where no turbulence is actually being generated at all.

The PMLM cannot represent this.

The PMLM implies that the "turbulent viscosity" is always positive (as do one- and two-equation models also); in reality it can change sign.

(d) Activation in PHOENICS

In order to activate the PMLM in PHOENICS, the PIL command TURMOD needs to be inserted if the Q1 file with the argument MIXLEN.

Then the choice of mixing-length formula is made by the setting of further parameters such as EL1A, IENUT, etc.

If no mixing-length formula is specified, then TURMOD(MIXLEN) selects the generalised turbulent length scale LTLS, so that TURMOD(MIXLEN) is equivalent to:

ENUT=GRND2;EL1=GRND1;EL1A=0.0;EL1B=0.41;DISWAL;GENK=T

where DISWAL activates the solution of a differential equation for LTLS (see Section 3.1.2 above on the LVEL turbulence model).

TURMOD(MIXLEN-RICE) selects the PMLM of Rice and Geary for use in 2-phase bubbly flows when ONEPHS=F (see Section 7.5 below). This TURMOD command is equivalent to:

ENUT=GRND2;EL1=GRND10;EL1A=1.0;EL1E=1.0;GENK=T

In addition the user must set CFIPB=bubble diameter. CFIPB is also used for this purpose in the interphase drag law selected by CFIPS=GRND7 (see the PHENC entry 'INTERPHASE DRAG MODELS').

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