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fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
, the law of the wall (also known as the logarithmic law of the wall) states that the average
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
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 In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shea ...
region. This law of the wall was first published in 1930 by Hungarian-American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
,
aerospace engineer Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: aeronautical engineering and astronautical engineering. Avionics engineering is s ...
, and
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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:Schlichting & Gersten (2000) pp. 522–524. :u^+ = \frac \ln\, y^+ + C^+, with y^+ = \frac, u_\tau=\sqrt and u^+ = \frac where : From experiments, the von Kármán constant is found to be \kappa\approx0.41 and C^+\approx5.0 for a smooth wall. With dimensions, the logarithmic law of the wall can be written as: : = \frac \ln\, \frac\ 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 \delta_\nu and a characteristic roughness length-scale 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, Re_w, where :Re_w=\frac.\ The flow is hydraulically smooth for Re_w<3, hydraulically rough for Re_w>100, and transitional for intermediate values. Values for 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 \approx 3.5 D_,\ where D_ is the average diameter of the 84th largest percentile of the grains of the bed material.


Power law solutions

Works by Barenblatt and others have 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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
approach. However, in 2014, Frewer et al. refuted these results.


For scalars

For scalars (most notably temperature), the self-similar logarithmic law of the wall has been theorized (first formulated by B. A. Kader) and observed in experimental and computational studies. In many cases, extensions to the original law of the wall formulation (usually through integral transformations) are generally needed to account for compressibility, variable-property and real fluid effects.


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, : 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 :u^+ \neq y^+ :u^+ \neq \frac \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 In fluid dynamics, turbulence kinetic energy (TKE) is the mean kinetic energy per unit mass associated with eddies in turbulent flow. Physically, the turbulence kinetic energy is characterised by measured root-mean-square (RMS) velocity fluct ...
\kappa ^+ function and the van Driest mixing length equation. Comparisons with
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data of fully developed turbulent channel flows for 109 showed good agreement.


Notes


References

* *


Further reading

*{{citation , journal=AIAA Journal , volume=47 , issue=3 , year=2009 , title=Evidence of nonlogarithmic behavior of turbulent channel and pipe flow , first1=Matthias H. , last1=Buschmann , first2=Mohamed , last2=Gad-el-Hak , doi=10.2514/1.37032 , pages=535 , bibcode = 2009AIAAJ..47..535B


External links


Definition from ScienceWorldFormula on CFD Online
Fluid dynamics Turbulence