When is the Reynolds number rho U d/mu versus rho U L/mu
PK Bot’s answer:
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ENGINEERING CONTEXT: When evaluating fluid flow in different devices, it is essential to choose a length scale reflective of the geometry; for example, in a pipe flow the characteristic length is the diameter d (i.e. \mathit{Re} = \rho U d/\mu), whereas in external flows or boundary layers the characteristic length is a relevant dimension such as the chord length or plate length L (i.e. \mathit{Re} = \rho U L/\mu). The engineering outcome is the ability to apply dynamic similarity for testing or designing devices such as pipelines or aerodynamic surfaces.
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PHYSICS: The Reynolds number compares inertial forces, proportional to \rho U^2, to viscous forces, proportional to \mu U/\ell, where the best length scale \ell reflects the distance over which velocity gradients operate to make viscous stresses. In practice, the length scale is often chosen prioritizing convenience over physical relevance, as long as the length scale used can be related to the most physically relevant length scale.
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ASSUMPTIONS: Forming a Reynolds number does not assume anything; it is simply a rearrangement of relevant parameters. For the Reynolds number to be useful, however, those parameters need to be representative of the forces that are dominating fluid behavior.
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SOLUTION TECHNIQUE:
– In fully developed pipe flow, we use the pipe diameter d so that
\mathit{Re}_d = \frac{\rho U d}{\mu}
– In flows over flat plates or airfoils, we use the length L so that
\mathit{Re}_L = \frac{\rho U L}{\mu}
d exemplifies a case where we can easily pick a length scale that is representative of the distance over which velocities change so that \mu U/d is a good approximation of the local viscous stress. In a pipe, the maximum speed is at the centerline, the minimum is at the wall. The length over which the speed goes from minimum to maximum back to minimum is d. We could have used r as well; the reason we use d is simply convention.
L exemplifies a case where the length scale that makes \mu U/L a good approximation of the local viscous stress is not immediately known. For flow over a flat plate or over an airfoil, that length scale would be the thickness of the boundary layer \delta, except you don’t know \delta when you just look at an airfoil. However, the distance along a surface x or its total length L are related to the boundary layer thickness, because boundary layer thickness increases monotonically as we move along the plate or airfoil. So, given that it is inconvenient to define \mathrm{Re} by use of \delta, we do the next best thing, we define it by use of x or L.
Because \mathrm{Re}_d is using a length scale that means something different from \mathrm{Re}_L, similar phenomena occur at different values of \mathrm{Re}. For example, flow through a pipe goes turbulent somewhere between \mathrm{Re}_d= 1000-4000, whereas a flat plate boundary layer goes turbulent around \mathrm{Re}_L= 5e5. If we calculated \mathrm{Re}_\delta= for a flat plate, the transition to turbulence would be closer to the pipe value for \mathrm{Re}_d.
- REFLECTION: The selection of d versus L is a geometric consideration that is meant to be representative of viscous effects. Selecting it properly ensures that the Reynolds number represents the correct balance of inertial and viscous effects. Selecting it in a way that matches convention lets you compare your results to standard references.
To formulate this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Appendix C.4: (HIGHLY RELEVANT)
- Appendix C Fundamentals: Reynolds Number (HIGHLY RELEVANT)
- Chapter 10 Fundamentals: Reynolds Number (HIGHLY RELEVANT)
- Chapter 6 Fundamentals: Reynolds Number (HIGHLY RELEVANT)
- Appendix C.2: Dimensional Analysis to Generalize Experiments (HIGHLY RELEVANT)
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