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### 8.2 Equilibrium log-law wall functions

The following wall functions are those appropriate to a near-wall layer in local equilibrium:

U+ = ln(E*y+)/κ

k = Uτ2/√ℂμ

ε = ℂμ3/4*k3/2/(κ*δ)

ω=√k/(κ*ℂμ1/4*δ)

(8.2.1)

(8.2.2)

(8.2.3)

(8.2.4)

where:

• U+= Ur/UΤ;
• Ur is the absolute value of the resultant velocity parallel to the wall at the first grid node;
• Uτ is the resultant friction velocity ( = √(τw/ρ) );
• δ is the normal distance of the first grid point from the wall;
• y+ is the dimensionless wall distance ( = Uτ*δ/ν );
• νl is the laminar kinematic viscosity;
• μ=CμCd=0.09;
• κ=0.41 is von Karman's constant; and
• E is a roughness parameter, which is equal to 8.6 for smooth walls.

Equation (8.2.1) is the well-known logarithmic law of the wall, and this law should be applied to a point whose y+ value is in the range 30 ≤ y+ <130.

The boundary condition (8.2.2) for the turbulent kinetic energy k assumes that the turbulence is in local equilibrium, and consequently, this set of wall functions are not really suitable under separated conditions because turbulent energy diffusion towards the wall is significant, leading to appreciable departures from local equilibrium.

The wall function defined by eqn (8.2.1) is implemented in the momentum equations by way of source terms per unit wall surface area, Sm, which take the form:

Smo = ρ*s*|Ur|*(Vw-V)

(8.2.5)

where ρ denotes the fluid density, s the friction factor ( = τw/(ρ*Ur2) ), V denotes the in-cell value of velocity, and Vw denotes the value of the velocity at the wall. The friction factor is s determined from the laminar and turbulent friction factors, sl and st, as follows:

s = max(sl, st)

sl = 1./Re

st = [κ/ln(E*Re*√st)]2

Re = Ur*δ/νl

(8.2.6)

(8.2.7)

(8.2.8)

(8.2.9)

and the expression for st follows from equation (8.2.1), and its value is determined by iteration from equation (8.2.8).

For heat and mass transfer at the wall, the flux of the variable φ between the fluid and the wall is given by:

Sφ = ρ*St*|Ur|*(φw-φ)

(8.2.10)

in which φ denotes the in-cell value of the heat- or mass-transfer variable, and φw denotes the value of φ at the wall. The Stanton number St is defined by:

St ≡ -qw/(ρ*Cp*|Ur|*(φ-φw) = h/(ρ*Cp*|Ur|)

(8.2.11)

where h is the local heat transfer coefficient; qw is the wall heat flux, and Cp is the specific heat.

The Stanton number in equation (8.2.10) is computed from:

St = max(Stl, Stt)

Stl = 1./(σt*Re)

Stt = st/[σt*(1.+Pm*√st)]

Pm = 9.*(σlt-1)*(σtl)1/4

(8.2.12)

(8.2.13)

(8.2.14)

(8.2.15)

In the foregoing equations, σt and σl denote the turbulent and laminar Prandtl numbers respectively, and equation (8.2.15) defines the smooth-wall sublayer-resistance function Pm of Jayatilleke [1969], which is valid for moderate to high Prandtl number fluids, i.e. for σl ≥0.5.

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