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Let's imagine a viscous fluid flowing horizontally through a cylinder with inner diameter $d$. The Reynolds number for that setting is described as:

$$Re=\frac{u\cdot d}{\nu}.$$

The fluid is first flowing in a laminar manner.

After some point $x$ I increase the velocity from $u$ to $u'$, the Reynolds number increases and when $Re$ is higher than approximately $3000$ the flow will become turbulent.

After some point further down, say $y$, I decrease the velocity back from $u'$ to $u$.

Will the transition from turbulent to laminar flow in this case be the same as when going from laminar to turbulent? Will it take the same amount of time? Do I have to invest the same amount of energy to transfer it back to laminar as much as I had to invest to increase the velocity to turn it turbulent in the first place? Do I have to decrease the velocity further down than the initial $u$ to go back to laminar of will it turn laminar as soon the velocity is $u$?

Can I imagine this to be similar to the phase transition going from liquid to gas, which is theoretically a reversible process or is this kind of a transition fundamentally different when going one way than the other?

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    $\begingroup$ There is an uncertain region in the Moody diagram that implies that the change from laminar to turbulent or the change from turbulent to laminar, occurs somewhat randomly. For more info, see en.wikipedia.org/wiki/Moody_chart $\endgroup$ Commented May 17 at 16:49

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Disclaimer!!!

  • Fluid dynamics may be counterintuitive, results are likely to be case-dependent, so that results for a flow can be hard to generalize to other flows;
  • focusing on low-speed fluid dynamics, the dynamics of the flow is governed by a constrained (due to incompressibility) transport equations; so, along with the incompressiblility constraints to be satisfied thanks to pressure gradient in the flow, the main physical process is likely to be transport/convection providing a direction of propagation of the information/disturbances; as an example, the direction of propagation of the information/disturbances may be the average velocity: what happens upstream strongly influences what happens downstream, while the opposite influence is likely to be lower;
  • beside pressure gradient enforcing incompressibility, and convection, viscosity may play major role both in triggering instabilities (lots of flows are stable in the inviscid limit, and viscosity is required for the instability to occur) and damping small turbulence scales; it's role is not trivial too;
  • as I won't try to find exotic similarities between different flows, if there is no good reason, I won't try to find similarities - even formal - with other physical processes: the similarity you're looking for with reversible/irreversible process makes no sense to me (I don't know any model/reference about that, my knowledge is finite, but it's full of people working on fluid dynamics. If you have any reference, please share it).
  • beside classical, simple flows, usually with low Reynolds number (for viscous flows), or with very high Reynolds number (where viscosity and vorticity is lumped in very thin regions - wakes and vortices), it's hard to solve fluid dynamics problem analytically and numerical simulation is likely to be the way.

Actual question - what we know. As suggested in the comment by Davide White, Moody's diagram could be useful as a starting point: in Moody's diagram you can find some characteristic of a statistical steady flow in a pipe with constant section (thus the axial direction is homogeneous). This diagram links the friction coefficient with the Reynolds number of the pipe flow $Re = \frac{\rho U D}{\mu} = \frac{4 Q}{\pi D \mu} = \frac{1}{\mu}\sqrt{\frac{4\rho U Q}{\pi}}$ (being $Q$ the mass flow, $Q = \frac{\pi}{4}\rho U D^2$) and the non-dimensional roughness of the surface $\frac{\epsilon}{D}$.

Actual question - differences from Moody's setup and remarks.

  • You're aiming at reducing velocity to get laminar flow, reducing the Reynolds number. Looking at the expression of the Reynolds number as a function of $U$, with constant massflow $Q$, this could be achieved increasing the section of the pipe: this is a first difference with the system summarize in Moody's diagram

  • The way you increase this section (sharp angles? blended divergent?) may strongly influence the transition between flow regimes. Anyway, in general, you should got a laminar regime far enough from this transition region of the section: viscosity needs some space to damp the structures that are convected downstream

  • To help viscosity, you could add some nets/honeycombs in the pipe to condition the flow, breaking large/medium structures in the flow into smaller structures that could be more easily damped by viscosity. Adding these elements generates a lumped pressure drop, and increases the required power to push the fluid in the pipe

  • I won't draw any general conclusion about time/length for turbulent to laminar and laminar to turbulent regimes, since they're case dependent

  • I won't think about energy to put in the system to make laminar turbulent or to make it laminar. To make a fluid flows into a pipe you always need to provide power to the system (by means of pumps, usually), for both laminar and turbulent regimes; if you want to extract mechanical power from the flow, you need some moving elements (like turbines) If you want to think in terms of energy/power, I'd suggest to look for hydraulics and work in term of pressure and total pressure drops, usually treating variable-section pipes and systems with semi-empirical relations:

    • for constant-area pipe sections, you may assume distributed pressure drop with different expressions (the same that are summarized in the Moody's diagram) for laminar or turbulent regimes. Assuming homogeneity along each pipe section $$\Delta P_i = L_i \frac{d P}{d x}(Re_i, \tilde{\varepsilon}_i)$$

    • for variable section elements, inflows/outflows, nets/honeycombs,... a lumped pressure drop usually occurs with expression

      $$\Delta P_k = \alpha_k \frac{1}{2} \rho U_k^2 \ ,$$

      being $U_k$ a characteristic velocity of the fluid through $k^{th}$ element and $\alpha_k$ a non-dimensional coefficient typical of the element.

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