https://cfd-online.com/W/index.php?title=Special:Contributions/Jmortizz&feed=atom&limit=50&target=Jmortizz&year=&month= CFD-Wiki - User contributions [en] 2022-05-18T22:49:09Z From CFD-Wiki MediaWiki 1.16.5 https://cfd-online.com/Wiki/Category:Dimensionless_parameters Category:Dimensionless parameters 2008-01-16T15:40:42Z <p>Jmortizz: </p> <hr /> <div>Dimensionless parameters that are used in fluid dynamics to characterise and classify different flows.<br /> <br /> ==References==<br /> <br /> http://www.ichmt.org/dimensionless/dimensionless.html</div> Jmortizz https://cfd-online.com/Wiki/Richardson_number Richardson number 2008-01-16T15:37:23Z <p>Jmortizz: </p> <hr /> <div>[[Category:Dimensionless parameters]]<br /> In the stability of continuously stratified parallel shear flows the ratio of (the squares of) the buoyancy frequency to the background velocity gradient is known as the (gradient) Richardson number.<br /> &lt;br&gt;<br /> :&lt;math&gt;<br /> Ri = \frac{N^2}{U_z^2} <br /> &lt;/math&gt;<br /> &lt;br&gt;<br /> :&lt;math&gt;<br /> N = \mbox{Buoyancy frequncy} = -\frac{g}{\rho_0} \frac{\partial \bar{\rho}}{\partial z} <br /> &lt;/math&gt;<br /> <br /> Here &lt;math&gt; \rho_0 &lt;/math&gt; is the reference density and &lt;math&gt; \bar{\rho} &lt;/math&gt; is the background density field.<br /> <br /> ==References==<br /> <br /> *{{reference-book |author=Hunt, J C R | year=1998 | title= Lewis Fry Richardson and his contributions to mathematics, meteorology, and models of conflict| rest =Annual Review of Fluid Mechanics, Vol. 30, 1998, pp. xiii–xxxvi}}</div> Jmortizz https://cfd-online.com/Wiki/Reynolds_number Reynolds number 2008-01-16T15:21:42Z <p>Jmortizz: </p> <hr /> <div>The Reynolds number characterises the relative importance of inertial and viscous forces in a flow. It is important in determining the state of the [[flow]], whether it is [[laminar]] or [[turbulent]]. At high Reynolds numbers flows generally tend to be [[turbulent]], which was first recognized by [[Osborne Reynolds]] in his famous [[pipe flow experiments]]. Consider the [[Navier-Stokes equations|momentum equation]] which is given below<br /> <br /> :&lt;math&gt;<br /> \frac{\partial}{\partial t}\left( \rho u_i \right) +<br /> \frac{\partial}{\partial x_j}<br /> \left[ \rho u_i u_j + p \delta_{ij} \right] = \frac{\partial}{\partial x_j} \tau_{ij}<br /> &lt;/math&gt;<br /> <br /> The terms on the right are the [[inertial forces]] and those on the left correspond to [[viscous forces]]. If &lt;math&gt;U&lt;/math&gt;, &lt;math&gt;L&lt;/math&gt;, &lt;math&gt;\rho&lt;/math&gt; and &lt;math&gt;\mu&lt;/math&gt; are the reference values for [[velocity]], [[length]], [[density]] and [[dynamic viscosity]], then<br /> <br /> inertial force ~ &lt;math&gt;\frac{\rho U^2}{L}&lt;/math&gt;<br /> <br /> viscous force ~ &lt;math&gt;\frac{\mu U}{L^2}&lt;/math&gt;<br /> <br /> Their ratio is the Reynolds number, usually denoted as &lt;math&gt;Re&lt;/math&gt;<br /> <br /> :&lt;math&gt;<br /> Re = \frac{\mbox{inertial force}}{\mbox{viscous force}} = \frac{\rho U L}{\mu}<br /> &lt;/math&gt;<br /> <br /> In terms of the kinematic viscosity<br /> <br /> :&lt;math&gt;<br /> \nu = \frac{\mu}{\rho}<br /> &lt;/math&gt;<br /> <br /> the Reynolds number is given by<br /> <br /> :&lt;math&gt;<br /> Re = \frac{U L}{\nu}<br /> &lt;/math&gt;<br /> <br /> ==Reynolds number as a ratio of time scales==<br /> <br /> Consider an impulsively started flat plate moving in its own plane with [[velocity]] &lt;math&gt;U&lt;/math&gt;. Due to the [[no-slip condition]] on the plate a [[boundary layer]] gradually develops on the plate. At time &lt;math&gt;t&lt;/math&gt;, the thickness of the [[boundary layer]] is of the order of &lt;math&gt;\sqrt{\nu t}&lt;/math&gt; (see Batchelor(1967), section 4.3). Let &lt;math&gt;L&lt;/math&gt; be the [[characteristic length scale]]. The time taken for [[viscous]] and [[convective]] effects to travel a distance &lt;math&gt;L&lt;/math&gt; is<br /> <br /> :&lt;math&gt;<br /> T_{v} = \frac{L^2}{\nu}<br /> &lt;/math&gt;<br /> <br /> and<br /> <br /> :&lt;math&gt;<br /> T_{c} = \frac{L}{U}<br /> &lt;/math&gt;<br /> <br /> The ratio of viscous to convective time scales is<br /> <br /> :&lt;math&gt;<br /> \frac{ T_{v} }{ T_{c} } = \frac{(L^2/\nu)}{(L/U)} = \frac{UL}{\nu} = Re<br /> &lt;/math&gt;<br /> <br /> Thus the Reynolds number is a measure of the viscous and convective time scales. A large Reynolds number means that viscous effects propagate slowly into the [[fluid]]. This is the reason why boundary layers are thin in high Reynolds number flows because the fluid is being convected along the [[flow]] direction at a much faster rate than the spreading of the [[boundary layer]], which is normal to the [[flow]] direction.<br /> <br /> ==References==<br /> *{{reference-book | author=Batchelor, G K | year=1967 | title=An Introduction to Fluid Dynamics | rest=Cambridge University Press}}<br /> <br /> *{{reference-book |author=Rott, N | year=1990 | title=Note on the history of the Reynolds number| rest =Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 1–11}}<br /> <br /> [[Category: Dimensionless parameters]]</div> Jmortizz https://cfd-online.com/Wiki/Reynolds_number Reynolds number 2008-01-16T15:13:15Z <p>Jmortizz: </p> <hr /> <div>The Reynolds number characterises the relative importance of inertial and viscous forces in a flow. It is important in determining the state of the [[flow]], whether it is [[laminar]] or [[turbulent]]. At high Reynolds numbers flows generally tend to be [[turbulent]], which was first recognized by [[Osborne Reynolds]] in his famous [[pipe flow experiments]]. Consider the [[Navier-Stokes equations|momentum equation]] which is given below<br /> <br /> :&lt;math&gt;<br /> \frac{\partial}{\partial t}\left( \rho u_i \right) +<br /> \frac{\partial}{\partial x_j}<br /> \left[ \rho u_i u_j + p \delta_{ij} \right] = \frac{\partial}{\partial x_j} \tau_{ij}<br /> &lt;/math&gt;<br /> <br /> The terms on the right are the [[inertial forces]] and those on the left correspond to [[viscous forces]]. If &lt;math&gt;U&lt;/math&gt;, &lt;math&gt;L&lt;/math&gt;, &lt;math&gt;\rho&lt;/math&gt; and &lt;math&gt;\mu&lt;/math&gt; are the reference values for [[velocity]], [[length]], [[density]] and [[dynamic viscosity]], then<br /> <br /> inertial force ~ &lt;math&gt;\frac{\rho U^2}{L}&lt;/math&gt;<br /> <br /> viscous force ~ &lt;math&gt;\frac{\mu U}{L^2}&lt;/math&gt;<br /> <br /> Their ratio is the Reynolds number, usually denoted as &lt;math&gt;Re&lt;/math&gt;<br /> <br /> :&lt;math&gt;<br /> Re = \frac{\mbox{inertial force}}{\mbox{viscous force}} = \frac{\rho U L}{\mu}<br /> &lt;/math&gt;<br /> <br /> In terms of the kinematic viscosity<br /> <br /> :&lt;math&gt;<br /> \nu = \frac{\mu}{\rho}<br /> &lt;/math&gt;<br /> <br /> the Reynolds number is given by<br /> <br /> :&lt;math&gt;<br /> Re = \frac{U L}{\nu}<br /> &lt;/math&gt;<br /> <br /> ==Reynolds number as a ratio of time scales==<br /> <br /> Consider an impulsively started flat plate moving in its own plane with [[velocity]] &lt;math&gt;U&lt;/math&gt;. Due to the [[no-slip condition]] on the plate a [[boundary layer]] gradually develops on the plate. At time &lt;math&gt;t&lt;/math&gt;, the thickness of the [[boundary layer]] is of the order of &lt;math&gt;\sqrt{\nu t}&lt;/math&gt; (see Batchelor(1967), section 4.3). Let &lt;math&gt;L&lt;/math&gt; be the [[characteristic length scale]]. The time taken for [[viscous]] and [[convective]] effects to travel a distance &lt;math&gt;L&lt;/math&gt; is<br /> <br /> :&lt;math&gt;<br /> T_{v} = \frac{L^2}{\nu}<br /> &lt;/math&gt;<br /> <br /> and<br /> <br /> :&lt;math&gt;<br /> T_{c} = \frac{L}{U}<br /> &lt;/math&gt;<br /> <br /> The ratio of viscous to convective time scales is<br /> <br /> :&lt;math&gt;<br /> \frac{ T_{v} }{ T_{c} } = \frac{(L^2/\nu)}{(L/U)} = \frac{UL}{\nu} = Re<br /> &lt;/math&gt;<br /> <br /> Thus the Reynolds number is a measure of the viscous and convective time scales. A large Reynolds number means that viscous effects propagate slowly into the [[fluid]]. This is the reason why boundary layers are thin in high Reynolds number flows because the fluid is being convected along the [[flow]] direction at a much faster rate than the spreading of the [[boundary layer]], which is normal to the [[flow]] direction.<br /> <br /> ==References==<br /> *{{reference-book | author=Batchelor G K | year=1967 | title=An Introduction to Fluid Dynamics | rest=Cambridge University Press}}<br /> <br /> *[http://158.110.32.35/download/CERN07/R_ARFM_90.pdf] Rott, N., “Note on the history of the Reynolds number,” Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 1–11.<br /> <br /> [[Category: Dimensionless parameters]]</div> Jmortizz