Fluid Fluency

Mathematical Definition

  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.

Selecting a Characteristic Lengthscale and Velocity

Physical Interpretations

Ratio of Inertial to Viscous Effects

  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.

\( \dfrac{d\vec{\mathbf{v}}}{dt} \) \( + \) \( (\vec{\mathbf{v}} \cdot \nabla) \vec{\mathbf{v}} \) \( = \) \( - \dfrac{1}{\rho} \nabla P \) \( + \) \( \nu \nabla^2 \vec{\mathbf{v}} \) \( + \) \( \vec{\mathbf{f}} \)

\( (\vec{\mathbf{v}} \cdot \nabla) \vec{\mathbf{v}} = \left[ \dfrac{U^2}{L} \right] \) \( \qquad \) \( \nu \nabla^2 \vec{\mathbf{v}} = \left[ \dfrac{\nu U}{L^2} \right] \)

\( Re = \dfrac{(\vec{\mathbf{v}} \cdot \nabla) \vec{\mathbf{v}}}{\nu \nabla^2 \vec{\mathbf{v}}} = \dfrac{U^{\cancel{2}}/\cancel{L}}{(\nu \cancel{U})/L^{\cancel{2}}} = \dfrac{U L}{\nu} \)

Ratio of Inertial to Vsicous Forces

  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 \(L^2\).

\( F_{inertial} = \rho U^2 L^2 \) \( \qquad \) \( F_{viscous} = \eta U L \)

\( Re = \dfrac{F_{inertial}}{f_{viscous}} = \dfrac{\rho U^{\cancel{2}} L^{\cancel{2}}}{\eta \cancel{U} \cancel{L}} = \dfrac{\rho U L}{\eta} \)

Ratio of Diffusive to Convective Timescales

  

\( t_{diffusive} = \dfrac{L^2}{\nu} \) \( \qquad \) \( t_{convective} = \dfrac{L}{U} \)

\( Re = \dfrac{t_{diffusive}}{t_{convective}} = \dfrac{L^{\cancel{2}} / \nu}{\cancel{L} / U} = \dfrac{U L}{\nu} \)

Common Misconception

  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:

External Flows: Laminar - Re < \(10^3\), Turbulent - Re > \(10^5\)
Internal Flows: Laminar - Re < \(10^3\), Turbulent - Re > \(10^5\)

  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.