Law of the wall

law of the wall, horizontal velocity near the wall with mixing length model

In fluid dynamics, the law of the wall states that the average velocity of a turbulent flow at a certain point is proportional to the logarithm of the distance from that point to the "wall", or the boundary of the fluid region. This law of the wall was first published by Theodore von Kármán, in 1930. It is only technically applicable to parts of the flow that are close to the wall (<20% of the height of the flow), though it is a good approximation for the entire velocity profile of natural streams.

General logarithmic formulation

The logarithmic law of the wall is a self similar solution for the mean velocity parallel to the wall, and is valid for flows at high Reynolds numbers — in an overlap region with approximately constant shear stress and far enough from the wall for (direct) viscous effects to be negligible:

$u^+ = \frac{1}{\kappa} \ln\, y^+ + C^+,$   with   $y^+ = \frac{y\, u_\tau}{\nu},$   $u_\tau=\sqrt{\frac{\tau_w}{\rho}}$   and   $u^+ = \frac{u}{u_\tau}$

where

y+	is the wall coordinate: the distance y to the wall, made dimensionless with the friction velocity uτ and kinematic viscosity ν,
u+	is the dimensionless velocity: the velocity u parallel to the wall as a function of y (distance from the wall), divided by the friction velocity uτ,
τw	is the wall shear stress,
ρ	is the fluid density,
uτ	is called the friction velocity or shear velocity
κ	is the Von Kármán constant,
C+	is a constant, and
ln	is the natural logarithm.

From experiments, the Von Kármán constant is found to be κ≈0.41 and C⁺≈5.0 for a smooth wall.

With dimensions, the logarithmic law of the wall can be written as:

${u} = \frac{u_\tau}{\kappa} \ln\, \frac{y}{y_0}\$

where y₀ is the distance from the boundary at which the idealized velocity given by the law of the wall goes to zero. This is necessarily nonzero because the turbulent velocity profile defined by the law of the wall does not apply to the laminar sublayer. The distance from the wall at which it reaches zero is determined by comparing the thickness of the laminar sublayer with the roughness of the surface over which it is flowing. For a near-wall laminar sublayer of thickness δ_ν and a characteristic roughness length-scale k_s,

$k_s < \delta_\nu\,$	: hydraulically smooth flow,
$k_s \approx \delta_\nu\,$	: transitional flow,
$k_s > \delta_\nu\,$	: hydraulically rough flow.

Intuitively, this means that if the roughness elements are hidden within the laminar sublayer, they have a much different effect on the turbulent law of the wall velocity profile than if they are sticking out into the main part of the flow.

This is also often more formally formulated in terms of a boundary Reynolds number, Re_w, where

$Re_w=\frac{u_\tau k_s}{\nu}.\$

The flow is hydraulically smooth for Re_w <3, hydraulically rough for Re_w >100, and transitional for intermediate values.

Values for y₀ are given by:

$y_0=\frac{\nu}{9 u_\tau}$	for hydraulically smooth flow
$y_0=\frac{k_s}{30}$	for hydraulically rough flow.

Intermediate values are generally given by the empirically-derived Nikuradse diagram, though analytical methods for solving for this range have also been proposed.

For channels with a granular boundary, such as natural river systems,

$k_s \approx 3.5 D_{84},\$

where D₈₄ is the average diameter of the 84th largest percentile of the grains of the bed material.

Power law solutions

Work by Barenblatt and others has shown that besides the logarithmic law of the wall — the limit for infinite Reynolds numbers — there exist power-law solutions, which are dependent on the Reynolds number. In 1996, Cipra submitted experimental evidence in support of these power-law descriptions. This evidence itself has not been fully accepted by other experts.

In 2001, Oberlack claimed to have derived both the logarithmic law of the wall, as well as power laws, directly from the Reynolds-averaged Navier–Stokes equations, exploiting the symmetries in a Lie group approach. All scaling laws derived in should represent the leading order solutions of a corresponding averaged Navier-Stokes boundary value problem for sufficiently high Reynolds numbers, and hence should all be valid in spatial regimes where the influence of the boundaries can be ignored. This derivation from first principles however lacks mathematical rigor. It is based on several critical assumptions, mainly to justify the use of the pure temporal scaling symmetry of the inviscid Euler equations, a symmetry not admitted by the viscous Navier-Stokes equations, not even to leading order in any strict perturbative sense for small viscosities. The well-known challenging issues related to the vanishing viscosity limit in transiting from the viscous Navier-Stokes to the inviscid Euler equations, particularly for bounded flows, also resides in every Lie group analysis for variable transformations, in that the limit is singularly unstable for symmetry transformations and non-unique for equivalence transformations. These crucial issues were not addressed in the derivation, making this derivation from first principles thus highly questionable. The difference here between a symmetry and an equivalence transformation is that the former defines viscosity as an arbitrary but fixed parameter, while the latter contrarily defines viscosity as an own additional Lie group variable next to all coordinates and flow variables. Eventually an ultimate test for these scaling laws allegedly emerging from first principles would be to show if the proposed Lie group methodology allows for a consistent unified treatment in agreement with experiment or DNS data when higher-order statistical quantities are included, in particular when involving the quantity of dissipation with its anomalous behavior in the vanishing viscosity limit. Otherwise the assumptions made in order to derive these proposed wall scaling laws in distorts the conclusion of dealing with a rigorous derivation from first principles.

Near the Wall

Below the region where the law of the wall is applicable, there are other estimations for friction velocity.

Viscous Sublayer

In the region known as the viscous sublayer, below 5 wall units, the variation of u⁺ to y⁺ is approximately 1:1, such that:

For  $y^+<5$

$u^+ = y^+$

where

y+	is the wall coordinate: the distance y to the wall, made dimensionless with the friction velocity uτ and kinematic viscosity ν,
u+	is the dimensionless velocity: the velocity u parallel to the wall as a function of y (distance from the wall), divided by the friction velocity uτ,

This approximation can be used farther than 5 wall units, but by y⁺=12 the error is more than 25%.

Buffer Layer

In the buffer layer, between 5 wall units and 30 wall units, neither law holds, such that:

For  $5<y^+<30$

$u^+ \neq y^+$

$u^+ \neq \frac{1}{\kappa} \ln\, y^+ + C^+$

with the largest variation from either law occurring approximately where the two equations intercept, at y⁺=11. That is, before 11 wall units the linear approximation is more accurate and after 11 wall units the logarithmic approximation should be used, though neither are relatively accurate at 11 wall units.

The mean streamwise velocity profile U+ is improved for y^+ < 20 with an eddy viscosity formulation based on a near-wall turbulent kinetic energy k^+ function and the van Driest mixing length equation. Comparisons with DNS data of fully-developed turbulent channel flows for 109<Reτ<2003 showed good agreement.

References

• Chanson, H. (2009), Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows, CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages, ISBN 978-0-415-49271-3 
• Schlichting, Hermann; Gersten, K. (2000), Boundary-layer Theory (8th revised ed.), Springer, ISBN 3-540-66270-7 

Further reading

• Buschmann, Matthias H.; Gad-el-Hak, Mohamed (2009), "Evidence of nonlogarithmic behavior of turbulent channel and pipe flow", AIAA Journal 47 (3): 535, doi:10.2514/1.37032 

External links

• Definition from ScienceWorld
• Formula on CFD Online