Difference between revisions of "Logarithmic velocity profile"

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== logarithmic velocity profile ==
 
== logarithmic velocity profile ==
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<div class="definition"><div class="short_definition">The [[variation]] of the mean wind speed with height in the [[surface boundary layer|surface  boundary layer]] derived with the following assumptions: 1) the mean motion is one-dimensional;  2) the [[Coriolis force]] can be neglected; 3) the [[shearing stress]] and [[pressure gradient]] are independent  of height; 4) the [[pressure force]] can be neglected with respect to the [[viscous force]]; and  5) the [[mixing length]] ''l'' depends only on the fluid and the distance from the boundary, ''l'' = ''kz''.</div><br/> <div class="paragraph">Near aerodynamically smooth surfaces, the result is  <div class="display-formula"><blockquote>[[File:ams2001glos-Le29.gif|link=|center|ams2001glos-Le29]]</blockquote></div> that is, the logarithmic velocity profile, where ''u''<sub>&#x0002a;</sub> is the [[friction velocity]] and &#x003bd; the [[kinematic  viscosity]]. ''k'' &cong; 0.4 and has been called the K&#x000e1;rm&#x000e1;n constant or [[von K&#x000e1;rm&#x000e1;n's constant]]. The  equation fails for a height ''z'' sufficiently close to the surface. For aerodynamically rough flow,  [[molecular viscosity]] becomes negligible. The profile is then  <div class="display-formula"><blockquote>[[File:ams2001glos-Le30.gif|link=|center|ams2001glos-Le30]]</blockquote></div>''z''<sub>0</sub> is a constant related to the average height &#x003b5; of the surface irregularities by ''z''<sub>0</sub> = &#x003b5;/30 and is  called the [[aerodynamic roughness length]]. Another derivation of the logarithmic profile was  obtained by Rossby under the assumption that for fully rough flow the roughness affects the [[mixing length|mixing  length]] only in the region where ''z'' and ''z''<sub>0</sub> are comparable. Then ''l'' = ''k''(''z'' + ''z''<sub>0</sub>) and  <div class="display-formula"><blockquote>[[File:ams2001glos-Le31.gif|link=|center|ams2001glos-Le31]]</blockquote></div> For statically nonneutral conditions, a [[stability]] correction factor can be included (<br/>''see'' equation in  definition of [[aerodynamic roughness length]]).</div><br/> </div><div class="reference">Haugen, D. A. 1973. Workshop on Micrometeorology. Amer. Meteor. Soc., . 392 pp. </div><br/> <div class="reference">Sutton, O. G. 1953. Micrometeorology. sect. 3.9. </div><br/>
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<div class="definition"><div class="short_definition">The [[variation]] of the mean wind speed with height in the [[surface boundary layer]] derived with the following assumptions: 1) the mean motion is one-dimensional;  2) the [[Coriolis force]] can be neglected; 3) the [[shearing stress]] and [[pressure gradient]] are independent  of height; 4) the [[pressure force]] can be neglected with respect to the [[viscous force]]; and  5) the [[mixing length]] ''l'' depends only on the fluid and the distance from the boundary, ''l'' = ''kz''.</div><br/> <div class="paragraph">Near aerodynamically smooth surfaces, the result is  <div class="display-formula"><blockquote>[[File:ams2001glos-Le29.gif|link=|center|ams2001glos-Le29]]</blockquote></div> that is, the logarithmic velocity profile, where ''u''<sub>&#x0002a;</sub> is the [[friction velocity]] and &#x003bd; the [[kinematic  viscosity]]. ''k'' &cong; 0.4 and has been called the K&#x000e1;rm&#x000e1;n constant or [[Von_kármán%27s_constant|von K&#x000e1;rm&#x000e1;n's constant]]. The  equation fails for a height ''z'' sufficiently close to the surface. For aerodynamically rough flow,  [[molecular viscosity]] becomes negligible. The profile is then  <div class="display-formula"><blockquote>[[File:ams2001glos-Le30.gif|link=|center|ams2001glos-Le30]]</blockquote></div>''z''<sub>0</sub> is a constant related to the average height &#x003b5; of the surface irregularities by ''z''<sub>0</sub> = &#x003b5;/30 and is  called the [[aerodynamic roughness length]]. Another derivation of the logarithmic profile was  obtained by Rossby under the assumption that for fully rough flow the roughness affects the [[mixing length|mixing  length]] only in the region where ''z'' and ''z''<sub>0</sub> are comparable. Then ''l'' = ''k''(''z'' + ''z''<sub>0</sub>) and  <div class="display-formula"><blockquote>[[File:ams2001glos-Le31.gif|link=|center|ams2001glos-Le31]]</blockquote></div> For statically nonneutral conditions, a [[stability]] correction factor can be included (<br/>''see'' equation in  definition of [[aerodynamic roughness length]]).</div><br/> </div><div class="reference">Haugen, D. A. 1973. Workshop on Micrometeorology. Amer. Meteor. Soc., 392 pp. </div><br/> <div class="reference">Sutton, O. G. 1953. Micrometeorology. sect. 3.9. </div><br/>  
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Latest revision as of 08:43, 8 August 2016



logarithmic velocity profile

The variation of the mean wind speed with height in the surface boundary layer derived with the following assumptions: 1) the mean motion is one-dimensional; 2) the Coriolis force can be neglected; 3) the shearing stress and pressure gradient are independent of height; 4) the pressure force can be neglected with respect to the viscous force; and 5) the mixing length l depends only on the fluid and the distance from the boundary, l = kz.

Near aerodynamically smooth surfaces, the result is
ams2001glos-Le29
that is, the logarithmic velocity profile, where u* is the friction velocity and ν the kinematic viscosity. k ≅ 0.4 and has been called the Kármán constant or von Kármán's constant. The equation fails for a height z sufficiently close to the surface. For aerodynamically rough flow, molecular viscosity becomes negligible. The profile is then
ams2001glos-Le30
z0 is a constant related to the average height ε of the surface irregularities by z0 = ε/30 and is called the aerodynamic roughness length. Another derivation of the logarithmic profile was obtained by Rossby under the assumption that for fully rough flow the roughness affects the mixing length only in the region where z and z0 are comparable. Then l = k(z + z0) and
ams2001glos-Le31
For statically nonneutral conditions, a stability correction factor can be included (
see equation in definition of aerodynamic roughness length).

Haugen, D. A. 1973. Workshop on Micrometeorology. Amer. Meteor. Soc., 392 pp.

Sutton, O. G. 1953. Micrometeorology. sect. 3.9.


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