# How to determine whether a particular velocity vector will result in a laminar flow?

Lets take the substance to be water. To my knowledge , the only condition imposed on a velocity profile to distinguish between laminar and turbulent force is the Reynold's number. For lets say I have been given this velocity vector :

V = (xy) i + (yz) j + (xz) k

Now how will I determine whether water, with this velocity profile , can have a laminar flow or not ?

• Welcome to engineering SE. Curious, is this a home work question? Aug 9 '15 at 0:25
• I can't tell from your question what your level of understanding is. You mention Reynolds number. Do you know how it is calculated or how that relates to laminar flow? Velocity isn't the only thing that determines the reynolds number. That fact alone should give you a hint.
– hazzey
Aug 9 '15 at 1:15
• Could you provide a sample profile? If you truly have meaningful (equal order of magnitude) velocity components in all three dimensions the problem might have a distinct three dimensional character which might influence the answer. Aug 9 '15 at 9:22
• My main aim is inquire whether is there is any other property, empirical quantity apart from Reynolds number to determine whether situation will have a laminar flow ? With velocity given in such vector forms you are tempted to think about the curl and divergence. However, I don't find them to be of any special meaning when talking about parallel streamlines which is essentially laminar flow if I am not wrong Aug 9 '15 at 19:31

I am afraid the question you are asking might not even be a valid question.

This is because of the correlation between the Reynolds-Number ($Re$) and the occurrence of turbulence. $Re$ is not a fluid mechanic switch (turning turbulence on and off) it is more a parameter which allows for describing and clustering flow phenomena.

$Re$ describes the ratio between inertial and viscous forces. This ratio has a direct influence on the stability of the flow. Flows with high viscosity (hence low $Re$) will be able to stay laminar while being subjected to small perturbations. The same perturbations will make a flow with low viscosity turbulent. High $Re$ will be able to damp out perturbations (in velocity or pressure) because they are spread out fast enough. Maybe you can think of viscosity as the ability to conduct velocity perturbations into the surrounding fluid the same way as high thermal conductivity will conduct thermal perturbations into the surrounding fluid.

As it is a re-occurring theme here at engineering SE the answer is:

• it depends
• it's a bit more complicated

Please describe the setting in more detail:

• free flow vs. internal flow
• symmetries

and maybe provide a rough idea on the velocity profile.

Finally, the most tricky part of the $Re$ is the length-scale $L$ which has to be chosen carefully to enable the similarity. The length-scale is easy to find in simple 2D-flows like a pipe or a plate. It already gets tricky when you combine both.

• My main aim is inquire whether is there is any other property, empirical quantity apart from Reynolds number to determine whether situation will have a laminar flow ? With velocity given in such vector forms you are tempted to think about the curl and divergence. However, I don't find them to be of any special meaning when talking about parallel streamlines which is essentially laminar flow if I am not wrong Aug 9 '15 at 19:31
• You are right when you are thinking of curl and divergence. Depending on the fluid's properties it might not be able to withstand those strains in a laminar manner. Since you are very secretive about your specific problem it's hard to help you here. Maybe you should look at it from the other side. You might just perform a CFD and switch on a turbulence model. Maybe the production of turbulent kinetic energy is a measure you can use. Aug 9 '15 at 19:38
• However, the Reynolds number does not determine whether you have laminar flow or not!!! This statement is only valid under specific circumstances. And, as said before experiments show a correlation between the occurrence of turbulent structures and the Reynolds number, the inversion might not be true. Aug 9 '15 at 19:41

You are correct, the Reynolds number is what determines whether a flow is laminar or turbulent. The formula for the Reynolds number is

$$Re = \frac{\rho vL}{\mu}$$

$\rho$ is the density, $v$ is the velocity of the flow, $\mu$ is the dynamic viscosity of the fluid, and $L$ is the characteristic length associated with the flow. The first three figures can be determined from the info you have in that velocity vector and by knowing the fluid. But the characteristic length depends on where your flow is. You have to know something about the pipe you're in, the plate you're flowing against, the airfoil you're going over, etc. in order to determine the Reynolds number.

Because the Reynolds number is proportional to that length, knowing the other three values really doesn't do much, unless they create a number so disproportionately large that the length would have to be impossibly small to make the flow laminar. But you also need to know something about the geometry (essentially the same thing that would give you the characteristic length) to know where the points of laminar flow and turbulent flow start with respect to the Reynolds number. The transitional values are different with the flow in a pipe and the flow over an airfoil.

So the real answer is, you can't know whether it's laminar or turbulent with the information you've provided.

• My main aim is inquire whether is there is any other property, empirical quantity apart from Reynolds number to determine whether situation will have a laminar flow ? With velocity given in such vector forms you are tempted to think about the curl and divergence. However, I don't find them to be of any special meaning when talking about parallel streamlines which is essentially laminar flow if I am not wrong Aug 9 '15 at 19:32
• @IamAM If you want an empirical quantity, you can measure the time varying velocity profile in whatever flow you're interested in. E.g. if this is an air duct, you can take continuous measurements of air velocity in the crosswise and oncoming flow directions at various points throughout the pipe radius. High crosswise flows would indicate turbulence, as would high variation in the oncoming flow velocity and a relatively "flat" velocity profile with a sharp velocity drop at the boundary. Aug 10 '15 at 5:55