The question brings me back to my master thesis, about 53 years ago.

I was checking if viscoelastic fluids delayed the onset of turbulence in the flow through round pipes. But let us go to your question, which I assume relates to newtonian fluids.

I was repeating the Newton's experiment also with water (newtonian fluid) and a carefully designed bellmouth at the pipe entrance.

In my apparatus, with very low disturbances, my critical Reynolds number (transition from laminar to turbulent flow) turned out to be about 14,000. And this was found also by changing the pipe diameter.

I was really surprised because also I knew the magic value of 2300. But looking at the literature, I found that, by carefully avoiding all possible disturbances, people had reached Reynolds number as high as 1,000,000 still having laminar flow through the pipe. Sorry, but I do nit remember the reference.

But I found also that, if I was running my test, e.g. at a Reynolds number of 8,000 with laminar flow in the pipe, and a big truck passed by my lab, the flow suddendly turned from laminar to turbulent. This further testifies that disturbances are important.

The conclusion is that 2300 is generally valid for industrial situations because transition occurs depending on your experimental facility.

Best regards and good luck for your work

These values are obtained from experiment (observing the streamlines in the flow, for example). The values are approximate, depending on the entrance conditions. Transition to the turbulent flow can be delayed to a much higher value of Re, if proper care is taken at the entrance.

Thank you, Dmitry.

These values have been determined experimentally by Reynolds.

I want to know: how are these values calculated?

* Reynolds = velocity of the fluid in the tube x diameter of the tube / viscosity of the fluid that flows in the tube with a fixed temperature.

the viscosity is determined by a table in function of the temperature of the fluid

- the diameter of the tube is well known

- speed is unknown

for calculating speed we have two methods

Q the flow = speed x the section of the tube = the volume of a quantity of the fluid / the time. the easiest is the second. a cronometer and a graduated tube and you have the Flow. if you have the flow and the section then you can have the speed.

Now you are replacing the speed in Reynolds * and you will have a number that will tell you the nature of your flow. Laminar or turbulent.

If you find it difficult to understand my answer. I can be clearer

The concept was introduced by Sir George Stokes in 1851, but the Reynolds number was named by Arnold Sommerfeld in 1908 after Osborne Reynolds (1842–1912), who popularized its use in 1883

https://en.wikipedia.org/wiki/Reynolds_number

The Reynolds number is a measure of the ratio between inertial and viscous forces in the flow. Many experimental observations of laminar and turbulent were successfully correlated using the Reynolds number, thus its general use in the literature whenever the flow phenomena is assumed to be determined by the balance between inertial and viscous forces. Eugene T. Adiutori published "The New Heat Transfer", 1989 considered heretical in most engineering circles because he advocated abandoning non-dimensional numbers such as the Reynolds number. ET Audiutori was probably kooky, but one must admire his courage to publish those ideas.

I recommend reading his book to appreciate the wealth of data we inherit from past researchers courtesy of their use of Reynolds and other non-dimensional numbers.

I recommend you to have a portion of test section prepared with a high speed camera to capture the flow, along with the basic devices for a convection flow such as a flow, temperature & pressure measurements. Connect them to a Data Acquisition system. Start taking readings by varying the flow rate & at the same time have the camera capturing the flow. If you find the image of the flow as smooth, its laminar. Once you find the image of the flow as violent, it is turbulent. But at certain flow rate, you would be finding that there would be an irregular pattern of flow, which is transition. Just love doing it. And take over the measurement readings and start doing your calculations to find the Laminar, Turbulent & Transition Reynolds number for your experiments.

CAUTION IS THAT YOU NEED TO CALIBRATE EVERY GADGET BEFORE YOU START YOUR EXPERIMENTS.

Hope this helps you for better understanding.

As have been said, transition to turbulent flow was investigated experimentially and in common practise the Reynold's number could be well applied. Who is interested in more details, have a look on "Hof et al. Experimental observations of nonlinear traveling waves in turbulent pipe flow. Science 2004, vol. 305, pp. 1594-1597"

It is an empirical value for different flow condition.

As Prof.Mohsen said, it is different for different flow conditions & hence you would need a good visualisation technique to adopt a range of "Re" for your experimental conditions, as previously said.

See the Reynolds' experiments.

The question brings me back to my master thesis, about 53 years ago.

I was checking if viscoelastic fluids delayed the onset of turbulence in the flow through round pipes. But let us go to your question, which I assume relates to newtonian fluids.

I was repeating the Newton's experiment also with water (newtonian fluid) and a carefully designed bellmouth at the pipe entrance.

In my apparatus, with very low disturbances, my critical Reynolds number (transition from laminar to turbulent flow) turned out to be about 14,000. And this was found also by changing the pipe diameter.

I was really surprised because also I knew the magic value of 2300. But looking at the literature, I found that, by carefully avoiding all possible disturbances, people had reached Reynolds number as high as 1,000,000 still having laminar flow through the pipe. Sorry, but I do nit remember the reference.

But I found also that, if I was running my test, e.g. at a Reynolds number of 8,000 with laminar flow in the pipe, and a big truck passed by my lab, the flow suddendly turned from laminar to turbulent. This further testifies that disturbances are important.

The conclusion is that 2300 is generally valid for industrial situations because transition occurs depending on your experimental facility.

Best regards and good luck for your work

Mr. professor Maria Carlomagno. Thank you for the answer. I should remind you that, fluid dynamics is based on a loose theoretical foundation. The Navier-Stokes equations are mathematically difficult to be analytically solved, and with too many ignored terms and simplifying assumption, it is not a reliable analytical tool to validate the experiments. Fluid dynamics is accompanied by too many contradictions, as d'Alembert paradox, and also too many other unexplained phenomena as vorticity and eddy generation and so on...

Wind-tunnel and boundary-layer are only a contraption to remedy the problem, but actually they exacerbate the mess. However, I have spotted on faultlines of the Navier-Stokes equations, as it is the subject of one of my projects in ResearchGate: https://www.researchgate.net/project/Analytical-fluid-dynamics-through-modified-Navier-Stokes-equations

Using the coupled-strain theory, I am trying to truly analytically solve the modified Navier-Stokes equations for 2D laminar and turbulent flow, and I've made some breakthrough in the way just recently. Regards.

The Orr–Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow. The solution to the Navier–Stokes equations for a parallel, laminar flow can become unstable if certain conditions on the flow are satisfied, and the Orr–Sommerfeld equation determines precisely what the conditions for hydrodynamic stability are. And then critical Reynolds number can be found from stability curve. This critical number is a criteria showing start of transition in a particular flow provided amplitude amplification of the perturbation occurs.

Above statements reference is: en.m.wikipedia.org/wiki/Orr–Sommerfeld_equation

Thank you Mr. professor Jahanmiri.

Greeting Mr Chomachar,

Greeting Professor Carlomagno,

It is a pity that we could not meet after your excellent conference in Sorrento!

I am continuing to organize small conferences and the next one in going to be hold at early December at IPMech RAS (http://ipmnet.ru/files/conf/2018waves_school/Ann_SCHOOL_2018_en.pdf) and pleased to invite everybody to come.

Concerning O. Reynolds experiment:

The facility still exists in the Manchester University and everybody can define “its own critical number” which value depends on a number of factors: quality of flow tubes geometry, the intensity of traffic on the adjacent street and even on the weight and dexterity of the experimenter. Vagn Ekman could increase the "critical number" on the Reynolds facility to 20,000 and wanted to increase it further, but the grant was not extended...

So there are no laminar and turbulent flows. There are comparisons of different incompatible models derived by using non-identical transformation of the basic equations set.

Every new equations set (including reduced, constitutive -turbulence of stability equations) define new physical properties of a fluid flow, which cannot be compared between themselves or observed in experiment with the accuracy control.

The set of fundamental equations is unique that is invariant on spatial or temporary scales and can be applied at any values of flows parameters.

This thought was clearly understood by G.G. Stokes, who wrote in his “Report on Prof. O. Reynolds paper” on April 19, 1883:

“IN THE ONE PART OF LANGUAGE SEEMS TO IMPLY WHICH WAS NOT PERHAPS INTENDED THAT HE DISCOVERED NEW DIMENSIONAL PROPERTIES OF FLUIDS, AND MIGHT EVEN LEAD TO THE SUPERPOSITON THAT HE HAD SHOWN THAT ANOTHER CONSTANT BEYOND THOSE RECOGTNAISED WAS NECESSARY ON ORDER TO DEFINE A FLUID MECHANICALLY. THIS CERTAINLY IS NOT THE CASE; THE DIMENSIONAL PROPERTIES ARE ALREADY INVOLVED OBVIOUSLY IN THE EQUATION OF MOTION; AND THERE IS ABSOLUTELY NOTHING TO PROVE THAT HE HAS DISCOVERED THE NECESSITY OF AN ADDITIONAL CONSTANT TO DEFINE A FLUID”.

The problem is that the approximation of a homogeneous fluid transforms the well posed complete system with equation of state and energy (or at least with variable stratified density) into the overdetermined system.

Sincerely,

Yuli D Chashechkin

Dear prof. Chashechkin ,

the conference in Sorrento was exactly twenty years ago and I am glad you enjoyed it. Thanks for your comments which I reccomended.

Concerning your conference at the end of this year, I would very glad to come back to Moscow (last time I was at TSAGI for their 90th annibersary) and participate, but I have to admit that, even if being still scientifically active, I retired as Emeritus Professor from my University. Therefore, I do not have any institution paying for my travel expenses, so, unless someone kindly takes care of them, I cannot participate to any meeting

Best regards and good luck for your work

Giovanni Maria Carlomagno

Please refer my book:

Turbulent-Nonturbulent Interface Detection methods in Transitional flows.