Mixing length model On dimensional grounds we assume the kinematic turbulent viscosity ν t ,

3.7.1 Mixing length model On dimensional grounds we assume the kinematic turbulent viscosity ν t ,

which has dimensions m 2 /s, can be expressed as a product of a turbulent velocity scale ϑ (m/s) and a turbulent length scale ᐉ (m). If one velocity scale and one length scale suffice to describe the effects of turbulence, dimensional analysis yields

ν t =C ϑᐉ (3.36) where C is a dimensionless constant of proportionality. Of course the

dynamic turbulent viscosity is given by µ t =C ρϑᐉ Most of the kinetic energy of turbulence is contained in the largest eddies,

and turbulence length scale ᐉ is therefore characteristic of these eddies which interact with the mean flow. If we accept that there is a strong connection

70 CHAPTER 3 TURBULENCE AND ITS MODELLING

between the mean flow and the behaviour of the largest eddies we can attempt to link the characteristic velocity scale of the eddies with the mean flow properties. This has been found to work well in simple two-dimensional turbulent flows where the only significant Reynolds stress is τ xy = τ yx =− ρ u ′v′ and the only significant mean velocity gradient is ∂U/∂y. For such flows it is

at least dimensionally correct to state that, if the eddy length scale is ᐉ,

where c is a dimensionless constant. The absolute value is taken to ensure that the velocity scale is always a positive quantity irrespective of the sign of the velocity gradient.

Combining (3.36) and (3.37) and absorbing the two constants C and c into

a new length scale ᐉ m we obtain ∂U

This is Prandtl’s mixing length model. Using formula (3.33) and noting that ∂U/∂y is the only significant mean velocity gradient, the turbulent Reynolds stress is described by

∂U ∂U

τ xy = τ yx =− ρ u ′v′ = ρᐉ m 2 (3.39)

∂y ∂y

Turbulence is a function of the flow, and if the turbulence changes it is necessary to account for this within the mixing length model by varying ᐉ m . For a substantial category of simple turbulent flows, which includes the free turbulent flows and wall boundary layers discussed in section 3.4, the turbulence structure is sufficiently simple that ᐉ m can be described by means of simple algebraic formulae. Some examples are given in Table 3.2 (Rodi, 1980).

Table 3.2 Mixing lengths for two-dimensional turbulent flows

Flow

L Mixing layer

Mixing length ᐉ m

Layer width Jet

0.07L

Jet half width Wake

0.09L

Wake half width Axisymmetric jet

0.16L

Jet half width Boundary layer ( ∂p/∂x = 0) viscous sub-layer and

0.075L

κy [ 1 − exp(− y + /26)]

log-law layer ( y/L ≤ 0.22) Boundary layer outer layer ( y/L ≥ 0.22)

thickness Pipes and channels

0.09L

Pipe radius or (fully developed flow)

0.14 – 0.08(1 − y/L) 2 − 0.06(1 − y/L ) 4 ]

channel half width

3.7 RANS EQUATIONS AND TURBULENCE MODELS

The mixing length model can also be used to predict turbulent transport of scalar quantities. The only turbulent transport term which matters in the two-dimensional flows for which the mixing length is useful is modelled as follows:

where Γ t = µ t / σ t and µ t = ρν t where ν t is found from (3.38). Rodi (1980) recommended values for σ t of 0.9 in near-wall flows, 0.5 for jets and mixing

layers and 0.7 in axisymmetric jets.

In the formulae in Table 3.2 y represents the distance from the wall and κ = 0.41 is von Karman’s constant. The expressions give very good agree- ment between computed results and experiments for mean velocity distribu- tions, wall friction coefficients and other flow properties such as heat transfer coefficients in simple two-dimensional flows. Results for two flows from Schlichting (1979) are given below in Figures 3.14a–b.

Figure 3.14 Results of calculations using mixing length model for (a) planar jet and (b) wake behind a long, slender, circular cylinder Source: Schlichting, H. (1979)

Boundary Layer Theory, 7th edn, reproduced with permission of The McGraw-Hill Companies

The mixing length has been found to be very useful in simple two- dimensional flows with slow changes in the flow direction. In these cases the production of turbulence is in balance with its dissipation throughout

72 CHAPTER 3 TURBULENCE AND ITS MODELLING

the flow, and turbulence properties develop in proportion with a mean flow length scale L. This means that in such flows the mixing length ᐉ m is pro-

portional to L and can be described as a function of position by means of

a simple algebraic formula. The majority of practically important flows, however, involve additional contributions to the budgets of turbulence properties due to transport, i.e. convection and diffusion. Moreover, the production and destruction processes may be modified by the flow itself. Consequently, the mixing length model is not used on its own in general- purpose CFD, but we will find it embedded in many of the more sophisti- cated turbulence models to describe near-wall flow behaviour as part of the treatment of wall boundary conditions.

An overall assessment of the mixing length model is given in Table 3.3.

Table 3.3 Mixing length model assessment Advantages:

• easy to implement and cheap in terms of computing resources • good predictions for thin shear layers: jets, mixing layers, wakes and

boundary layers • well established

Disadvantages: • completely incapable of describing flows with separation and recirculation • only calculates mean flow properties and turbulent shear stress