### 2. Equilibrium log-law wall functions

The following wall functions are those appropriate to a near-wall layer in local equilibrium, which may be written as follows:

U+/Uτ = ln(E*Y+)/κ

k = Uτ2/√(CμCd)

ε = (CμCd)0.75*k1.5/(κ*δ)

(1)

(2)

(3)

where:

• 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 ),
• (CμCd) is a constant equal to 0.09 in the standard k-ε model,
• κ is the von Karman constant and
• E is a roughness parameter.

In PHOENICS the log-law constants κ and E are set in Group 1 Section 1 of GREX3 as follows: AK = 0.41 and EWAL = 8.6, which is appropriate for smooth walls.

Equation (1) is the well-known logarithmic law of the wall.

Strictly this law should be applied to a point whose Y+ value is in the range 30 < Y+ <130.

The user may elicit printout of the Y+ values in the PHOENICS RESULT file by setting YPLS=T or WALPRN=T in the Q1 file.

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

The wall function defined by equation (1) is implemented in the momentum equations by way of source terms which take the form:

Smom = ρ*s*abs(Ur)*(Vw-V)

(4)

where rho 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 determined from:

s = max (sturb, slam)
where:

slam = 1./Re

sturb = (κ/ln(E*Re*√(sturb))2

Re is the local Reynolds number, defined as
Re = Ur*δ/ν

(5)

(6)

(7)

(8)

and the expression for sturb follows from equation (1). The value of sturb is determined by iteration from equation (7).

For heat and mass transfer at the wall, the flux of the variable phi from the wall to the fluid is given by:

Sφ = ρ*St*abs(Ur)*(φw-φ)

(9)

in which φ denotes the in-cell value of the heat- or mass-transfer variable, and φw denotes the wall value of φ. St is the Stanton number ( = -Qw/(ρ*Cp*abs(Ur)*(φ-φw) ), determined from:

St = max (Stturb, Stlam)
where:

Stlam = 1./(Prt*Re)

Stturb = sturb/(Prt*(1.+Pm*SQRT(sturb)))

where the Pm is the smooth-wall sublayer-resistance function of Jayatilleke [1969]
Pm = 9.*(Prl/Prt-1)*(Prt/Prl)**0.25
Prt and Prl denote the turbulent and laminar Prandtl numbers respectively.

(10)

(11)

(12)

(13)

This Pm-function is valid for moderate to high Prandtl number fluids, i.e. for Prl => 0.5.

Elhadidy [1980] proposed a Pm-function that applies to the whole Prl range, and is given by:

• Pm = (Ym)*[b**0.72 - 1] -(1/k)*ln{[1 - k*(Ym)*b**0.72]/[1 + k*(Y+)]} +

(1/k)*ln[(1 + k*(y+)*b)/(1 + k*(y+)]

where; Ym=11.5 and b =(Prl/Prt). This has been successfully validated for low Prl fluids such as liquid metals, but has yet to be implemented in PHOENICS.