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<!DOCTYPE html>
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<head>
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<meta name="description" content="Learn about the Reynolds Number">
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<title>Reynolds Number</title>
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<h1>Fluid Fluency</h1>
<h2>An Online Repository for Fluid Physics Notes</h2>
<h2>by Rónán Gissler</h2>
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Intermediate
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</div>
<h2>Reynolds Number</h2>
<div id="beginner_content">
<section>
<h3>Mathematical Definition</h3>
<div class="equation_line">
<span class="equation">
<span>
\[ Re = \frac{\rho U L}{\eta} = \frac{U L}{\nu} \]
</span>
</span>
</div>
<p>  where \(\rho\) is the fluid's density, \(U\) is some characteristic
speed such as the average speed of the bulk flow, \(L\) is some characteristic
length (diameter of a cylinder in the flow, chordlength of an airfoil, etc.),
\(\eta\) is the fluid's dynamic viscosity, and \(\nu\) is the fluid's kinematic
viscosity.</p>
</section>
<section>
<h3>Selecting a Characteristic Lengthscale and Velocity</h3>
<p>  </p>
</section>
<section>
<h3>Physical Interpretations</h3>
<dl>
<dt>Ratio of Inertial to Viscous Effects</dt>
<dd><p>  We can obtain the Reynolds number from the differential form
of the Navier-Stokes Equation by using dimensional analysis for the
inertial and viscous terms.</p>
<div class="equation_line">
<span class="equation">
<span>
\( \dfrac{d\vec{\mathbf{v}}}{dt} \)
</span>
<span>\( + \)</span>
<span style="color: orangered">
\( (\vec{\mathbf{v}} \cdot \nabla) \vec{\mathbf{v}} \)
</span>
<span>\( = \)</span>
<span>\( - \dfrac{1}{\rho} \nabla P \)</span>
<span>\( + \)</span>
<span style="color: orangered">
\( \nu \nabla^2 \vec{\mathbf{v}} \)
</span>
<span>\( + \)</span>
<span>\( \vec{\mathbf{f}} \)</span>
</span>
</div>
<div class="equation_line">
<span class="equation">
<span>
\( (\vec{\mathbf{v}} \cdot \nabla) \vec{\mathbf{v}}
= \left[ \dfrac{U^2}{L} \right] \)
</span>
<span>\( \qquad \)</span>
<span>
\( \nu \nabla^2 \vec{\mathbf{v}} = \left[ \dfrac{\nu U}{L^2} \right] \)
</span>
</span>
</div>
<div class="equation_line">
<span class="equation">
<span>
\( Re = \dfrac{(\vec{\mathbf{v}} \cdot \nabla) \vec{\mathbf{v}}}{\nu \nabla^2 \vec{\mathbf{v}}} =
\dfrac{U^{\cancel{2}}/\cancel{L}}{(\nu \cancel{U})/L^{\cancel{2}}}
= \dfrac{U L}{\nu} \)
</span>
</span>
</div>
</dd>
<dt>Ratio of Inertial to Vsicous Forces</dt>
<dd>
<p>  Alternatively, we could obtain the Reynolds number from the
ratio of inertial to viscous forces. The inertial force can be imagined as
a jet impinging on a surface. The jet exerts a pressure on the surface
proportional to \(\rho U^2\) (where U is the velocity of the jet) and the area
of that surface is proportional to \(L^2\) (where L is some lengthscale of the
impact surface). The viscous force can be imagined as a sheet of viscous fluid
rubbing against a surface. The fluid exerts a shear stress on the surface
proportional to \(\frac{\eta U}{L}\) and the area of that surface is again
proportional to some \(L^2\).</p>
<div class="equation_line">
<span class="equation">
<span>
\( F_{inertial} = \rho U^2 L^2 \)
</span>
<span>\( \qquad \)</span>
<span>
\( F_{viscous} = \eta U L \)
</span>
</span>
</div>
<div class="equation_line">
<span class="equation">
<span>
\( Re = \dfrac{F_{inertial}}{f_{viscous}}
= \dfrac{\rho U^{\cancel{2}} L^{\cancel{2}}}{\eta \cancel{U} \cancel{L}}
= \dfrac{\rho U L}{\eta} \)
</span>
</span>
</div>
</dd>
<dt>Ratio of Diffusive to Convective Timescales</dt>
<dd>
<p>  </p>
<div class="equation_line">
<span class="equation">
<span>
\( t_{diffusive} = \dfrac{L^2}{\nu} \)
</span>
<span>\( \qquad \)</span>
<span>
\( t_{convective} = \dfrac{L}{U} \)
</span>
</span>
</div>
<div class="equation_line">
<span class="equation">
<span>
\( Re = \dfrac{t_{diffusive}}{t_{convective}}
= \dfrac{L^{\cancel{2}} / \nu}{\cancel{L} / U}
= \dfrac{U L}{\nu} \)
</span>
</span>
</div>
</dd>
</ul>
</section>
<section>
<h3>Common Misconception</h3>
<p>  Engineers often use the Reynolds number to evaluate
whether a flow is laminar or turbulent. Turbulent and laminar flows
exhibit significantly different behaviors so it is useful to have
a metric by which we can estimate whether the flow will be laminar
or turbulent. You may have heard the following rules of thumb:
</p>
<br>
<p>
External Flows: Laminar - Re < \(10^3\), Turbulent - Re > \(10^5\)<br>
Internal Flows: Laminar - Re < \(10^3\), Turbulent - Re > \(10^5\)
</p>
<br>
<p>  These rules of thumb are generally useful, but of
course in practice it's not that simple. Whether the flow is
turbulent or not depends on several factors besides the Reynolds
number including surface roughness of solids interacting with the fluid,
whether or not the fluid is non-Newtonian as the Reynolds number
implicitly assumes, and initial levels of velocity fluctuations in the flow.
Still skeptical that the Reynolds number isn't all you need?
Take a quick look at all the academic papers published on
turbulence at low Reynolds number.
</p>
</section>
</div>
<div id="intermediate_content">
<section>
<h3></h3>
</section>
</div>
<div id="advanced_content">
<section>
<h3></h3>
</section>
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</article>
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