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8.3 Non-Equilibrium log-law wall functions

A generalisation of eqn (8.2.1) to non-equilibrium conditions has been proposed by Launder and Spalding [1974], the form of which is:

Ur*√k/Uτ2 = Ln(Ê*√k*δ/ν)/ķ

(8.3.1)

where ķ=κ*ℂμ1/4, Ê=E*ℂμ1/4, and ℂμ=CμCd=0.09.

Thus, the non-equilibrium wall function employs √k as the characteristic turbulent velocity scale, rather than the friction velocity Uτ. In local equilibrium, where k is given by eqn (8.2.2), it is readily shown that eqn (8.3.1) reduces to the conventional logarithmic law of eqn (8.2.1).

The wall function defined by eqn (8.3.1) is implemented in the momentum equations via eqns (8.2.5) to (8.2.9) excepting that st is now given by:

st = ķ*√k/[ Ur*Ln(Ê*√k*δ/ν) ]

(8.3.2)

The value of k at the near-wall point is calculated from its own transport equation with the diffusion of energy to the wall being set equal to zero. This transport equation contains the production rate Pk and the dissipation rate ε, and the average rates of these two terms for the near-wall cell are determined by making an analytical integration over the control volume and assuming that the shear stress and k are constant across the near-wall cell. The mean value of turbulence energy production over the near-wall cell is represented as:

Pk = Uτ2*Ur/(2*δ)

(8.3.3)

The cell-averaged dissipation rate, appearing in the sink term for the k equation is fixed to the following expression:

ε = (ℂμ)3/4*k*3/2*Ln(Ê*√k)*δ/ν)/(2*κ*δ)

(8.3.4)

Under conditions of local equilibrium, Pk/ε must equal unity, and this may be verified by dividing eqn (8.3.3) by eqn (8.3.4), using eqn (8.3.1), and noting that st=Uτ;2/Ur2 and Uτ;2=√(ℂμ)*k.

However, in the formula for the near-wall viscosity, the dissipation rate is calculated using the values at the nodal point given by eqn (8.2.3) in the previous section.

For heat and mass transfer at the wall, the flux of φ from the wall to the fluid is again given by eqns (8.2.10) to (8.2.15), excepting that the turbulent Stanton number is now calculated from:

Stt = st/[ σt*(1.+Pm*st*Ur/{ℂμ1/4*√k}) ]

(8.3.5)

This equation represents the generalisation of eqn (8.2.14) to non-equilibrium conditions.

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