View source for Physics:Law of the wall

{{Short description|Relation of flow speed to wall distance}}
[[File:Law of the wall (English).svg|thumb|upright=2|law of the wall, horizontal velocity near the wall with mixing length model]]
In [[Physics:Fluid dynamics|fluid dynamics]], the '''law of the wall''' (also known as the '''logarithmic law of the wall''') states that the average [[Physics:Velocity|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 [[Physics:Fluid|fluid]] region. This law of the wall was first published in 1930 by Hungarian-American [[Biography:Mathematician|mathematician]], [[Engineering:Aerospace engineering|aerospace engineer]], and [[Physicist|physicist]] [[Biography:Theodore von Kármán|Theodore von Kármán]].<ref>{{citation | last=von Kármán | first=Th. | year=1930 | title=Mechanische Ähnlichkeit und Turbulenz | journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Fachgruppe 1 (Mathematik) | volume=5 | pages=58–76 }} (also as: [https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930094805.pdf “Mechanical Similitude and Turbulence”], Tech. Mem. NACA, no. 611, 1931).</ref> 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.<ref name="mohrig">{{cite web|url=http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-110-sedimentary-geology-fall-2004/lecture-notes/l2_mass_mom_cons.pdf|title=Conservation of Mass and Momentum|last=Mohrig|first=David|year=2004|work=12.110: Sedimentary Geology, Fall 2004|publisher=MIT OCW|access-date=2009-03-27}}</ref>

==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 number]]s — in an overlap region with approximately constant [[Physics:Shear stress|shear stress]] and far enough from the wall for (direct) [[Physics:Viscosity|viscous]] effects to be negligible:<ref name=Schlichting_Gersten>Schlichting & Gersten (2000) pp. 522–524.</ref>

:<math>u^+ = \frac{1}{\kappa} \ln\, y^+ + C^+,</math> {{pad|5em}} with {{pad|1em}} <math>y^+ = \frac{y\, u_\tau}{\nu},</math> {{pad|2em}} <math>u_\tau=\sqrt{\frac{\tau_w}{\rho}}</math> {{pad|1em}} and {{pad|1em}} <math>u^+ = \frac{u}{u_\tau}</math>

where 
:{| border="0"
|-
|<math>y^+</math>|| is the wall coordinate: the distance ''y'' to the wall, made [[Physics:Dimensionless|dimensionless]] with the friction velocity ''u<sub>τ</sub>'' and [[Physics:Kinematic viscosity|kinematic viscosity]] ''ν'',
|-
|<math>u^+</math>|| is the dimensionless velocity: the velocity ''u'' parallel to the wall as a function of ''y'' (distance from the wall), divided by the [[Physics:Shear velocity|friction velocity]] ''u<sub>τ</sub>'',
|-
|<math>\tau_w</math>|| is the wall shear stress,
|-
|<math>\rho</math>|| is the fluid [[Physics:Density|density]],
|-
|<math>u_\tau</math>|| is called the friction velocity or [[Physics:Shear velocity|shear velocity]],
|-
|<math>\kappa</math>|| is the [[Physics:Von Kármán constant|Von Kármán constant]],
|-
|<math>C^+</math>|| is a constant, and
|-
|<math>\ln</math>|| is the [[Natural logarithm|natural logarithm]].
|}
From experiments, the von Kármán constant is found to be <math>\kappa\approx0.41</math> and <math>C^+\approx5.0</math> for a smooth wall.<ref name=Schlichting_Gersten/>

With dimensions, the logarithmic law of the wall can be written as:<ref>Schlichting & Gersten (2000) p. 530.</ref>

:<math>{u} = \frac{u_\tau}{\kappa} \ln\, \frac{y}{y_0}\ </math>

where ''y<sub>0</sub>'' 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 [[Physics:Laminar sublayer|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 <math>\delta_\nu</math> and a characteristic roughness length-scale <math>k_s</math>,<ref name="mohrig"/>

:{| border="0"
|-
| <math>k_s < \delta_\nu\,</math> || : hydraulically smooth flow,
|-
| <math>k_s \approx \delta_\nu\,</math>  || : transitional flow,
|-
| <math>k_s > \delta_\nu\,</math>        || : 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, <math>Re_w</math>, where

:<math>Re_w=\frac{u_\tau k_s}{\nu}.\ </math>

The flow is hydraulically smooth for <math>Re_w<3</math>, hydraulically rough for <math>Re_w>100</math>, and transitional for intermediate values.<ref name="mohrig"/>

Values for <math>y_0</math> are given by:<ref name="mohrig"/><ref name="whipple">{{cite web|url=http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-163-surface-processes-and-landscape-evolution-fall-2004/labs/roughnes_handout.pdf|title=Hydraulic Roughness|last=Whipple|first=Kelin|year=2004|work=12.163: Surface processes and landscape evolution|publisher=MIT OCW|access-date=2009-03-27}}</ref>

:{| border="0"
|-
| <math>y_0=\frac{\nu}{9 u_\tau}</math> &nbsp; || for hydraulically smooth flow
|-
| <math>y_0=\frac{k_s}{30}</math>              || for hydraulically rough flow.
|}

Intermediate values are generally given by the empirically derived Nikuradse diagram,<ref name="mohrig"/> though analytical methods for solving for this range have also been proposed.<ref>{{citation | journal=Sedimentary Geology | volume=163 | issue=3–4 | year=2004 | pages=311–321 | doi=10.1016/j.sedgeo.2003.07.005 | title=An integrated law of the wall for hydrodynamically transitional flow over plane beds | first=J.P. | last=Le Roux |bibcode = 2004SedG..163..311L }}</ref>

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

:<math>k_s \approx 3.5 D_{84},\ </math>

where <math>D_{84}</math> is the average diameter of the 84th largest percentile of the grains of the bed material.<ref>{{cite web | url=http://www.et.byu.edu/~fange/Research/Articles/Thesis/Equivalent%20sand%20roughness%20of%20Nikuradse.doc | title=Equivalent sand roughness of Nikuradse (ks) | last=Haws | first=Benjamin | access-date=2009-03-27  }}{{Dead link |date=July 2014}}</ref>

==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.<ref>{{cite web | url=http://www.lbl.gov/Science-Articles/Research-Review/Highlights/1998/v3/CS_flaw.html | title=A flaw in the law | author=Lynn Yarris | publisher=Lawrence Berkeley National Laboratory, U.S. Department of Energy | work=Berkeley Lab: Highlights 97–98 }}</ref><ref>
{{citation | first1=G.I. | last1=Barenblatt | title=Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis | journal=Journal of Fluid Mechanics | volume=248 | pages=513–520 | year=1993 | doi=10.1017/S0022112093000874 |bibcode = 1993JFM...248..513B | s2cid=123639410 }}<br />
{{citation | first1=G.I. | last1=Barenblatt | first2=V.M. | last2=Prostokishin | title=Scaling laws for fully developed turbulent shear flows. Part 2. Processing of experimental data | journal=Journal of Fluid Mechanics | volume=248 | year=1993 | pages=521–529 | doi=10.1017/S0022112093000886 |bibcode = 1993JFM...248..521B | s2cid=121328837 }}<br />
{{citation | first1=G.I. | last1=Barenblatt | first2=N. | last2=Goldenfeld | title=Does fully developed turbulence exist? Reynolds number independence versus asymptotic covariance | journal=Physics of Fluids | volume=7 | issue=12 | year=1995 | pages=3078–3084 | doi=10.1063/1.868685|arxiv = cond-mat/9507132 |bibcode = 1995PhFl....7.3078B | s2cid=15138376 }}<br />
{{citation | first1=G. I. | last1=Barenblatt | first2=A. J. | last2=Chorin | title=Scaling laws and vanishing-viscosity limits for wall-bounded shear flows and for local structure in developed turbulence | journal=Communications on Pure and Applied Mathematics | volume=50 | issue=4 | year=1998 | pages=381–398 | doi=10.1002/(SICI)1097-0312(199704)50:4<381::AID-CPA5>3.0.CO;2-6 }}</ref> 
In 1996, Cipra submitted experimental evidence in support of these power-law descriptions.<ref>{{citation | first=Barry Arthur | last=Cipra | title=A New Theory of Turbulence Causes a Stir Among Experts | journal=Science | volume=272 | page=951 | date=May 1996 |bibcode = 1996Sci...272..951C |doi = 10.1126/science.272.5264.951 | issue=5264| s2cid=117371905 }}</ref> This evidence itself has not been fully accepted by other experts.<ref>{{citation | last1=Zagarola | first1=M.V. | first2=A.E. | last2=Perry | last3=Smits | first3=A.J. | title=Log Laws or Power Laws: The Scaling in the Overlap Region | journal=Physics of Fluids | volume=9 | issue=7 | pages=2094–2100 | year=1997 | doi=10.1063/1.869328 |bibcode = 1997PhFl....9.2094Z | citeseerx=10.1.1.503.989 }}</ref> In 2001, Oberlack claimed to have derived both the logarithmic law of the wall, as well as power laws, directly from the [[Physics:Reynolds-averaged Navier–Stokes equations|Reynolds-averaged Navier–Stokes equations]], exploiting the symmetries in a [[Lie group]] approach.<ref name=Schlichting_Gersten/><ref name=Oberlack2001>{{citation | title=A unified approach for symmetries in plane parallel turbulent shear flows | first=Martin | last=Oberlack | journal=Journal of Fluid Mechanics | year=2001 | volume=427 | issue=1 | pages=299–328 | doi=10.1017/S0022112000002408 | bibcode=2001JFM...427..299O | s2cid=122979735 }}</ref> However, in 2014, Frewer et al.<ref>{{citation | title=Is the log-law a first principle result from Lie-group invariance analysis? | first1=Michael | last1=Frewer | first2=George | last2=Khujadze | first3=Holger | last3=Foysi | year=2014 | pages=1–32 | arxiv = 1412.3069 |bibcode = 2014arXiv1412.3069F }}</ref> 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<ref>{{Cite journal |last=Kader |first=B. A. |date=1981-09-01 |title=Temperature and concentration profiles in fully turbulent boundary layers |url=https://dx.doi.org/10.1016/0017-9310%2881%2990220-9 |journal=International Journal of Heat and Mass Transfer |language=en |volume=24 |issue=9 |pages=1541–1544 |doi=10.1016/0017-9310(81)90220-9 |issn=0017-9310}}</ref>) and observed in experimental and computational studies.<ref>{{Cite journal |last1=Simonich |first1=J. C. |last2=Bradshaw |first2=P. |date=1978-11-01 |title=Effect of Free-Stream Turbulence on Heat Transfer through a Turbulent Boundary Layer |url=https://doi.org/10.1115/1.3450875 |journal=Journal of Heat Transfer |volume=100 |issue=4 |pages=671–677 |doi=10.1115/1.3450875 |issn=0022-1481}}</ref><ref>{{Cite journal |last1=Patel |first1=Ashish |last2=Boersma |first2=Bendiks J. |last3=Pecnik |first3=Rene |date=2017-08-21 |title=Scalar statistics in variable property turbulent channel flows |url=https://link.aps.org/doi/10.1103/PhysRevFluids.2.084604 |journal=Physical Review Fluids |volume=2 |issue=8 |pages=084604 |doi=10.1103/PhysRevFluids.2.084604|bibcode=2017PhRvF...2h4604P }}</ref><ref>{{Cite journal |last1=Toki |first1=Takahiko |last2=Teramoto |first2=Susumu |last3=Okamoto |first3=Koji |date=2020-01-01 |title=Velocity and Temperature Profiles in Turbulent Channel Flow at Supercritical Pressure |url=https://arc.aiaa.org/doi/10.2514/1.B37381 |journal=Journal of Propulsion and Power |volume=36 |issue=1 |pages=3–13 |doi=10.2514/1.B37381|s2cid=209963353 }}</ref><ref>{{Cite journal |last1=Guo |first1=J. |last2=Yang |first2=X. I. A. |last3=Ihme |first3=M. |date=March 2022 |title=Structure of the thermal boundary layer in turbulent channel flows at transcritical conditions |journal=Journal of Fluid Mechanics |language=en |volume=934 |doi=10.1017/jfm.2021.1157 |bibcode=2022JFM...934A..45G |issn=0022-1120|doi-access=free }}</ref> 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.<ref>Turbulent Flows (2000) pp. 273–274.{{Citation | title=Turbulent Flows | first1=Stephen | last1=Pope | edition=1st revised | publisher=Cambridge University Press | year=2000 | isbn=0-521-59125-2}}</ref>

===Viscous sublayer===
In the region known as the viscous sublayer, below 5 wall units, the variation of <math>u^+</math> to <math>y^+</math> is approximately 1:1, such that:
:For {{pad|1em}}<math>y^+<5</math>
:<math>u^+ = y^+</math>

where, 
:{| border="0"
|<math>y^+</math>|| is the wall coordinate: the distance ''y'' to the wall, made [[Physics:Dimensionless|dimensionless]] with the friction velocity <math>u_\tau</math> and [[Physics:Kinematic viscosity|kinematic viscosity]] <math>\nu</math>,
|-
|<math>u^+</math>|| 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 <math>u_\tau</math>,
|}
This approximation can be used farther than 5 wall units, but by <math>y^+=12</math> 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 {{pad|1em}}<math>5<y^+<30</math>
:<math>u^+ \neq y^+</math>
:<math>u^+ \neq \frac{1}{\kappa} \ln\, y^+ + C^+</math>
with the largest variation from either law occurring approximately where the two equations intersect, at <math>y^+=11</math>.  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 <math>u^+</math> is improved for <math>y^+<20</math> with an eddy viscosity formulation based on a near-wall [[Physics:Turbulence kinetic energy|turbulent kinetic energy]] <math>\kappa ^+</math> function and the van Driest mixing length equation. Comparisons with [[Physics:Direct numerical simulation|DNS]] data of fully developed turbulent channel flows for <math>109<Re_\tau<2003</math> showed good agreement.<ref>{{citation | title=A simple eddy viscosity formulation for turbulent boundary layers near smooth walls | first=Rafik | last=Absi | journal=Comptes Rendus Mécanique | year=2009 | volume=337 | issue=3 | pages=158–165 | doi=10.1016/j.crme.2009.03.010 |bibcode = 2009CRMec.337..158A |arxiv = 1106.0985 | s2cid=40907005 }}</ref>

==Notes==
{{reflist|2}}

==References==
*{{Citation|last=Chanson|first=H.|title=Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows|publisher=CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages|year=2009|isbn=978-0-415-49271-3|url=http://espace.library.uq.edu.au/view/UQ:191112}}
*{{Citation | title=Boundary-layer Theory | first1=Hermann | last1=Schlichting | first2=K. | last2=Gersten | edition=8th revised | publisher=Springer | year=2000 | isbn=3-540-66270-7 }}

==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==
*[https://archive.today/20130209140856/http://scienceworld.wolfram.com/physics/UniversalLogarithmicWallLaw.html Definition from ScienceWorld]
*[https://archive.today/20130119072841/http://www.cfd-online.com/Wiki/Law_of_the_wall Formula on CFD Online]
*[https://web.archive.org/web/20160428024018/http://www.wolfdynamics.com/tools.html Y+ estimator]

[[Category:Fluid dynamics]]
[[Category:Turbulence]]

{{Sourceattribution|Law of the wall}}