In fluid dynamics, the law of the wall (also known as the logarithmic 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 in 1930 by Hungarian-American mathematician, aerospace engineer, and physicist Theodore von Kármán. 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 + = 1 κ ln y + + C + , {\displaystyle u^{+}={\frac {1}{\kappa }}\ln \,y^{+}+C^{+},} with y + = y u τ ν , {\displaystyle y^{+}={\frac {y\,u_{\tau }}{\nu }},} u τ = τ w ρ {\displaystyle u_{\tau }={\sqrt {\frac {\tau _{w}}{\rho }}}} and u + = u u τ {\displaystyle u^{+}={\frac {u}{u_{\tau }}}}
where
From experiments, the von Kármán constant is found to be κ ≈ 0.41 {\displaystyle \kappa \approx 0.41} and C + ≈ 5.0 {\displaystyle C^{+}\approx 5.0} for a smooth wall. With dimensions, the logarithmic law of the wall can be written as:
u = u τ κ ln y y 0 {\displaystyle {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 {\displaystyle \delta _{\nu }} and a characteristic roughness length-scale k s {\displaystyle k_{s}} ,
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, R e w {\displaystyle Re_{w}} , where
R e w = u τ k s ν . {\displaystyle Re_{w}={\frac {u_{\tau }k_{s}}{\nu }}.\ }
The flow is hydraulically smooth for R e w < 3 {\displaystyle Re_{w}<3} , hydraulically rough for R e w > 100 {\displaystyle Re_{w}>100} , and transitional for intermediate values. Values for y 0 {\displaystyle y_{0}} are given by:
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 ≈ 3.5 D 84 , {\displaystyle k_{s}\approx 3.5D_{84},\ }
where D 84 {\displaystyle D_{84}} is the average diameter of the 84th largest percentile of the grains of the bed material.
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