What is the difference between maximal flow and maximum flow?

Fluid Dynamics: What is the difference between chaotic flow and turbulent flow?

• I have heard that turbulent flows are chaotic but that not all chaotic flows are turbulent. Given some characteristics of a chaotic flow (eg velocity data, etc) how could you tell if it was turbulent or just chaotic? E.g. would you expect k^(-5/3) to be true for a chaotic flow?

• Answer:

  EDIT: As pointed out, a chaotic flow has to have mixing by definition. The  following answer assumes 'chaotic' to mean 'random' which is technically  incorrect. But I've decided to leave it here anyway, because I  think that's what the OP meant. However, I still think that you won't observe a k^(-5/3) spectrum in the case of chaotic advection. I'm also adding Piyush's comment as a part of this answer. Chaotic  but non-turbulent flows can have exponential mixing. There is a whole  field of chaotic advection based on this fact. In fact, you can have  exponential mixing of mass in Stokes flow, which is as far away from  turbulence as possible. This is often used to mix fluid efficiently at  micro devices (low Re), where turbulence is simply not feasible due to  energy considerations. A  chaotic flow is one in which there seems to be a high irregularity in the behavior of one/all flow variables with time/space. While a turbulent flow certainly exhibits this behavior, there are also other properties that should be present for a flow to be called turbulent, one of which is high levels of mixing, i.e., mass/momentum/heat transfer. This is a distinct (and perhaps the most useful) property of turbulent flows which is frequently exploited. When you try to mix the sugar in a cup of coffee by stirring it, you're essentially making use of this property. This effect can be clearly seen by looking at the velocity profiles of laminar and turbulent flows through a pipe. (Taken from http://www.flowcontrolnetwork.com/articles/flow-profile-theory-practice) The lines show the magnitude of horizontal/streamwise velocities with respect to height along a pipe. You can see that the turbulent flow has a much flatter velocity profile than a laminar flow, i.e., there are higher velocities close to the wall for a turbulent flow compared to a laminar flow, while there are lower velocities close to the centerline for a turbulent flow compared to a laminar flow. This shows that velocity (momentum for an incompressible/constant density flow) is transferred to a greater extent from the fluid elements close to the centerline to the fluid elements close to the walls in case of a turbulent flow. A good example of a flow which is chaotic but is not turbulent is the trail behind an aircraft. Though the flow inside the jet trail is highly chaotic, it is not turbulent because it maintains the shape (diameter) for very large distances behind the aircraft, which means that there is very low/negligible mixing with the surrounding atmosphere. (Taken from http://www.123rf.com/photo_1585204_exhaust-trail-seen-behind-a-aircraft-very-high-in-the-sky.html) There are other properties which a flow should satisfy to be called turbulent. They are Rotationality, presence of Energy Cascades etc. Also, I wouldn't expect to see a K^(-5/3) behavior in the energy spectrum of a chaotic flow, because the energy cascade is an example of a momentum transfer which need not occur in a chaotic flow. More importantly, I have doubts as to whether the concept of an energy spectrum is even meaningful for a purely chaotic flow which is not turbulent.