The UMIST-N Near-Wall Treatment Applied to Periodic Channel Flow

CHAPTER 2. TURBULENCE MODELS 27
The logarithmic law of the wall assumes a logarithmic relationship between
velocity and displacement away from a solid boundary. These quantities are
expressed nondimensionally as U + and y + respectively. (See the discussion
in Chapter 3 on nondimensionalisation.) This relationship is due to von
Kármán [28]. The relationship is
U + = 1
κ ln Ey +
(2.39)
where κ is von Kármán’s constant. E is a function of wall roughness, and the
smooth-wall value is used in this thesis. Log-law constants appear in Table
2.3. The wall roughness constant, E is not to be confused with the E term
in the ˜ε transport equation.
Table 2.3: Log-law constants [83]
κ 0.4187
E 9.793
Table 2.4 offers a qualitative appraisal of the behaviour of flow with respect
to the nondimensional wall distance, y + . This highlights the qualitatively
different behaviours exhibited by a fluid as a solid boundary is approached.
The log law applies approximately where y + > 30. Where y + < 5, flow is
characterised by Prandtl’s [54] hypothesis, that U + = y + very near the wall.
The buffer layer is a region (5 < y + < 30) where neither of these assumptions
holds. In specifying a low-Reynolds-number solution, it is important that the
gradients nearest the wall fall within the viscous sublayer (y + < 5). In a high-
Reynolds-number CFD treatment, it is important that the log-law boundary
conditions are applied at a location where y + > 30.
When used as a boundary condition on a high-Reynolds-number k-ε treat-