Most secondary school textbooks, in their chapter about fluid dynamics, seem to suggest that "steady flow" and "laminar flow" are synonyms.

Though I never received any fluid mechanics course when I was at the university, it's pretty obvious to me that flows can be laminar but non-steady. But what about the converse? Can a steady flow be non-laminar?

If I skim through more advanced textbooks and lecture notes, I can't find any direct reference of a strict relation between the two concepts, neither positive nor negative. Yet "between the lines" most of the texts seem to take as a fact that steady implies laminar.

Is that true? Is a steady non-laminar flow something theoretically possible in some context (eg. inviscid flow in a purely continuum-mechanical model of a fluid) but physically unobtainable in any actual fluid? Is the implication blatantly false?

My imagination has apparently no problem at visualizing some sort of weird self-intersecting (and consequently non-laminar?) flow which doesn't vary over time. Am I missing something? I definitely guess that I am.

  • $\begingroup$ Since the discussion isn't directly addressing the point I'm more concerned about, I'll try to make my request more concrete (by expliciting some - I understand, questionable - definitions): Let's assume that "steady" means that for any point the (partial) time derivative of the velocity field is zero. And let's say that "laminar" reduces to asking that streamlines be well-defined and not self-intersecting. If my definition of fluid permits situations where two or more currents pass through each other, can I have a steady, non-laminar flow? If need be setting viscosity to 0? $\endgroup$ – wago Dec 21 '15 at 18:11
  • $\begingroup$ Keep in mind that real turbulent flows are inherently unsteady flows, i.e. their flow patterns are constantly changing. It is however possible to define a statistically steady state where the averaged flow field doesn't change in time $\endgroup$ – nluigi Dec 22 '15 at 16:19
  • $\begingroup$ 'Steady' by your definition (which I think is standard) seems to me to directly imply there is no turbulence. Turbulent flows necessarily involve changes in the velocity field over time. I think a lot of people are taking 'steady' to mean 'statistically stable', which turbulent flows can be. (you could, however, have laminar flows that are not steady, I think). $\endgroup$ – AGML Aug 19 '16 at 19:10
  • $\begingroup$ Yes, you could have laminar flows that aren't steady-state. If a slow laminar flow is accelerated slowly, without increasing the $Re$ number too much, it will remain laminar during that transience. No problemo. $\endgroup$ – Gert Aug 19 '16 at 20:27

The answer depends on your definition of "steady". A flow is called turbulent when small oscillations are no longer damped, but instead excited. Therefore when looking at the fluid on a microscopic scale, a turbulent flow is not steady.

However, turbulence can be modeled on a macroscopic scale (cf. https://en.wikipedia.org/wiki/Turbulence_modeling), effectively encapsulating the local non-steadiness. Thus on a macroscopic scale, turbulent flow can be steady.

An example for a (macroscopic) partially turbulent steady flow is the flow around an airfoil. At the nose, the flow is laminar. At some point (the transition point) at both upper and lower surface, the flow becomes turbulent. (The location of the transition point for a dedicated airfoils is depending on flow velocity and the angle of attack.)


Most secondary school textbooks, in their chapter about fluid dynamics, seem to suggest that "steady flow" and "laminar flow" are synonyms.

You'll have to point to specific textbooks because mine don't say that, primarily because it is incorrect.

By steady flow simply understand constant flow rate. Understood that way, steady flow can be laminar, turbulent (or in that grey area between the two). Constant (volumetric) flow rate says nothing about the Reynolds Number at all.


Illustrating laminar (top) and turbulent (bottom) flow, with associated radial flow speed gradients:

Laminar and turbulent flow/

For laminar flow the radial speed distribution is given by:


Suitable integration between $0$ and $R$ gives the volumetric flow rate $F$:


With $A$ the cross-section.

For turbulent flow (velocity profile not very well drawn), the exponent $2$ becomes much higher and the velocity gradient $v(r)$ much flatter near the centre. Here too, volumetric flow rate can be obtained by integration over $0$ to $R$. At very high Reynolds Numbers, then $\frac{dv(r)}{dt} \approx 0$.

  • $\begingroup$ Eh? Steady flow should mean that the fluid variables (in particular, the speed distribution) are time-independent. This seems obviously incompatible with turbulence. $\endgroup$ – AGML Aug 19 '16 at 19:07
  • $\begingroup$ No, it's not, A well established turbulent flow would have a velocity macro-distribution that's time invariant. Not so at the micro-level, obviously. $\endgroup$ – Gert Aug 19 '16 at 20:17
  • $\begingroup$ Well, ok, upvoting then. $\endgroup$ – AGML Aug 19 '16 at 23:04

Yes, a steady flow is always laminar (but not conversely as you already understood). Turbulent flows are by definition time-dependent (and thus unsteady) flows and therefore not laminar.

Turbulent flows can however be statistically stationary. This means that average quantities (such as mean velocity, turbulent kinetic energy and even higher moments), do not change in time (if you take a different time range).

  • $\begingroup$ "Turbulent flows are by definition transient flows". Not true at all. Turbulent flows can exist forever. $\endgroup$ – Gert Dec 21 '15 at 22:51
  • $\begingroup$ @Gert Transient means time-dependent. $\endgroup$ – Bernhard Dec 22 '15 at 6:58
  • $\begingroup$ Yes, I speak English too. You're still wrong. That 'different time range' you talk about is the normal time range. In REAL applications. $\endgroup$ – Gert Dec 22 '15 at 13:02
  • $\begingroup$ @Gert I don't get your point. "Turbulent flows can exist forever". I agree, but where do I say something else? I thought that was your interpretation of "transient" (i.e. non-eternal). I don't get what you mean with "normal time range" in "real application"? What I mean was, that averaging over $[t_1,t_2]$ and $[t_3,t_4]$, results in the same statistics. Can you maybe expand on your comment, so that I understand my confusion? I will then improve my answer. $\endgroup$ – Bernhard Dec 22 '15 at 13:14
  • $\begingroup$ Firstly, read the question again: it's about steady flow in secondary school text books. You think they mean the RT instability? After a while flow settles to laminar, turbulent or poorly defined, forever. The initial transiency isn't highly relevant to the question, IMHO. $\endgroup$ – Gert Dec 22 '15 at 13:41

No not necessarily. Steady flow is defined for either fully developed laminar flow, or turbulent flow. The term 'steady' applies to a non-transient nature of the flow being described within some region of space and boundaries, and statistically at a macroscopic level.

Pulsatile flows are non-steady flows and laminar flows that are not fully developed are non-steady. Although turbulent flows at a microscopic level do not have a well define velocity vector, statistical measure at a macroscopic level can, and as long as that velocity is constant, the flow can be considered steady.


According to my concepts, Laminar flow flows in the form of sheets or lamina. So magnitude is not something that we should concern, only the direction will do the whole understanding. Steady flow is that the flow parameters are invariant of time for the fixed space. Now for the flow to be Laminar the particles in the one sheet must have the some direction (successors's direction of velocity is the same as predecessor's ) so the magnitude of the particles velocity at some fixed space( in one sheet) may or may not be same at all time i.e. the flow can be unsteady as well. For the reverse it is true as well that the steady flow can be turbulent because laminar flow is strictly direction dependent.


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