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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:

U_{r}*√k/U_{τ}^{2} = Ln(Ê*√k*δ/ν)/ķ

where ķ=κ*ℂ_{μ}^{1/4},
Ê=E*ℂ_{μ}^{1/4}, and
ℂ_{μ}=C_{μ}C_{d}=0.09.
Thus, the non-equilibrium wall function employs √k as the characteristic
turbulent velocity scale, rather than the friction velocity U_{τ}.
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 s_{t} is now
given by:

s_{t} = ķ*√k/[ U_{r}*Ln(Ê*√k*δ/ν) ]

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 P_{k} 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:

P_{k} = U_{τ}^{2}*U_{r}/(2*δ)

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

ε = (ℂ_{μ})^{3/4}*k*^{3/2}*Ln(Ê*√k)*δ/ν)/(2*κ*δ)

Under conditions of local equilibrium, P_{k}/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 s_{t}=U_{τ};^{2}/U_{r}^{2} and
U_{τ};^{2}=√(ℂ_{μ})*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:

St_{t} = s_{t}/[ σ_{t}*(1.+P_{m}*s_{t}*U_{r}/{ℂ_{μ}^{1/4}*√k}) ]

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

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