# Is cigarette smoke an example of turbulent flow?

Everywhere in internet and books that introduce to fluid dynamics mention cigarette smoke as a classical example of turbulent flow.

But the velocity, in other words the Reynolds number, is small. How could be that this flow is turbulent?

• How are you defining Reynolds number? There could be many definitions, and so there isn't just one and it isn't always small. What happens to Re if you choose the length scale to be distance from the tip of the cigarette? Then is it always small? Is velocity constant or does it change? Commented Feb 20, 2017 at 20:35
• @tpg2114 The velocity changes, the fluid accelerates. The length scale is small. I have read but I do not remember where that shear free flows become turbulent for small Re and the reason was not mentioned Commented Feb 20, 2017 at 20:45

The meaning of the Reynolds number is that the equations of viscous fluid flow allow similarity: if the geometry is matched (up to a scaling factor) between two setups, and the Reynolds numbers are matched, then the equations describing fluid motion are the same in normalized variables, which means that the character of motion (laminar or turbulent) should be matched as well. This follows from an algebraic transformation of the equations, and there is nothing there that would predict that motion has to become turbulent for $Re$ above a particular value, e.g., $Re$ = 1. On the other hand, the physical interpretation of $Re$ as the ratio of inertia to dissipation implies that increasing $Re$ should eventually lead to a transition to turbulence. However the numerical value of critical $Re$ depends on the geometry. In some standard cases, e.g., flow in round pipes, it is known that transition to turbulence happens at $Re$~1000; but this does not say that in a different setup a value as small as $Re$~10, in case of a cigarette, would not be enough to make the flow turbulent. So whether in a particular setup the Reynolds number is sufficient for transition to turbulence is up to an experiment (or a detailed simulation). For a cigarette, experimental results, e.g., in the picture, clearly demonstrate that initially laminar rise of hot air plume from a cigarette transitions to disordered, turbulent, motion of air a few cm away from the tip of the cigarette.

Below are a couple of insightful videos showing turbulent plumes from a candle https://www.youtube.com/watch?v=mLp_rSBzteI and from a match https://www.youtube.com/watch?v=TQJAokQQbmI

• There are indeed rigorous stability limits (e.g. Serrin's theorem) as obtained from fully nonlinear stability analysis. Granted, these limits are very loose: Serrin's theorem results in a lower limit of $Re_{min}\approx5.71$, far below the stability limit of any real flow. Most certainly there are no flows that are unstable at a Reynolds number of 10. Notice also that at this point I am only talking about the stability limit; the Reynolds number limit for true flow turbulence will be far higher still.
– Pirx
Commented Feb 21, 2017 at 0:55
• @Pirx: Wikipedia article en.wikipedia.org/wiki/Reynolds_number states for a sphere in a fluid "...purely laminar flow only exists up to Re = 10...". But turbulence can come in different flavors, from weakly developed to fully developed. I believe the case of a cigarette smoke is closer to "weakly" developed turbulence, but to discuss this quantitatively one would need to introduce definitions of weakly and fully developed turbulence. Commented Feb 21, 2017 at 1:33
• @Pirx: I doubt some theorem can provide rigorous stability boundaries for a practical case, transition to turbulence is sensitive to such subtle things as surface quality, no way this can be quantitatively captured by some analytic math. Commented Feb 21, 2017 at 1:40
• Nonsense. The Wikipedia article is wrong, and wildly so. Flow around a sphere is steady up to roughly a Reynolds number of 100, and certainly remains laminar up to roughly $Re=300$.
– Pirx
Commented Feb 21, 2017 at 1:42
• Weird. A rising cigarette plume (and similar small plumes from candles, matches etc) clearly demonstrates stochastic character, that's an experimental fact, I linked some videos to my answer. If you indeed work in this field maybe you should publish another article, explaining your disagreement with experimental facts. Commented Feb 21, 2017 at 4:45

First of all, cigarette smoke is cigarette smoke; it's not an example, it's not a flow, and it's not turbulence. People need to keep concepts and categories straight in their heads, otherwise nothing but complete and utter confusion arises.

Now, if you are asking whether the thermal convection flow that is induced by the heat from a cigarette in still air and that is made visible by the smoke particles from the cigarette is an instance of turbulent flow, then the answer is no. The Reynolds number in this case is indeed too low for turbulence to develop, and detailed measurements of the velocity field will clearly demonstrate that the flow is fully laminar. No responsible and even halfway competent textbook would mention this flow as an example of turbulent flow.

• I can't think of a single real life example of a cigarette where the flow remains laminar. Maybe it wasn't a perfect example, but it wasn't a bad one. A big reason they probably used it is that people have seen cigarette smoke, it's a visual of the flow, and it usually transitions from laminar to turbulent, giving a pretty good visual.
– JMac
Commented Feb 21, 2017 at 11:28
• @ JMac I guess I'll give up on this one. No point trying to discuss a non-trivial subject of research with people that have no training whatsoever. I'll just note that in turbulence research we have a very specific understanding of what we mean by the term "turbulence", and the definition is not something as silly as "looks kind of messy". There are plenty of examples of flows with chaotic and/or random-looking streaklines that are not turbulent. Specifically, the picture shown in the answer most definitely does not show turbulent flow.
– Pirx
Commented Feb 21, 2017 at 12:26
• I very much want to accept your point.Discussion is always fruitful I downvote you because your answer explains nothing, but you give some information at comments where you insult other users. Commented Feb 21, 2017 at 12:43
• @Prix as far as I'm aware the real definition of turbulence involves the flows no longer being layered. I've seen many sources which give examples of smoke plumes transitioning from laminar to turbulent in a similar way to the cigarette picture linked in the other answer. I find it odd that you are always so adamant that everyone else doesn't know what they are talking about, yet have provided no literature to support your assertion that the flow will remain fully laminar. Arrogance without support is generally not well received.
– JMac
Commented Feb 21, 2017 at 13:02
• No, turbulence is most certainly not defined as "the flows no longer being layered". See Section 1.1 "The nature of turbulence" in Tennekes & Lumley's A First Course in Turbulence. Like I said, there are many, many examples of flows that are "no longer being layered" that are not turbulent.
– Pirx
Commented Feb 21, 2017 at 13:20