Mir Aamir Abbas

No, it is not the velocity magnitude to determine turbulence but an onset which is intrinsic in the quadratic non linearity of the convective term in the momentum equation.

Apart some classical cases (one is the O. Reynolds experiment in pipe), the Reynolds number value cannot be always used ad a threshold to assess transition to turbulence. That depends on the lenght scale you use for the Re number, the value can be small at certain characteristic scale of turbulence and high at the integral scale.

If fluid flows fast, particles have high kinetic energy which is "too much" for viscosity of the fluid to damp. So their path would not be "parallel" to each other (laminar) and they start to have chaotic path. The ratio of inertia forces to viscose forces (Reynolds number) is a good indicator for outset of turbulence. However, there is no theorem that directly relating the Reynolds number to turbulence. You can think from statistical thermodynamic view point to turbulence: it is form of entropy of energy (tendency to distribute high energy towards the lowest energy zones of the system) When flow is turbulent, particles exhibit additional transverse/random motions that increases the rate of energy and momentum exchange between them thus increasing the energy dispersion in the system.

Well, the nature of flow whether it's laminar or turbulent is described by non-dimensional Reynolds number (Re) which is the ratio of inertia forces to viscous forces. The inertia forces are responsible for kinetic energy as well. If flow is laminar then Re < 2100 i.e. the gain in kinetic energy is bounded and this make all the fluid particles to obey same change in pressure. But when Re>2100, which is the case of non-laminar flow (transitional or turbulence region), the increase is kinetic energy is not bounded and after a threshold value (depends on the domain and nature of problem) this bring non-uniform pressure changes in all fluid particles. This is completely chaotic and due to same the fluid particles flow randomly. Because of the same, the fluid velocity is also chaotic. Thus unbounded gain in kinetic energy cause the turbulence.

The additional kinetic energy gain in fluid particles is due to some external forces or domain of problem.

Bharat Soni

I think that the common understanding of this topic relating turbulence and a Reynolds number value is somehow misleading. What is more, turbulence is not chaos in a random meaning.

Let me use the simplest model, the Burgulence, that is the 1D Burgers equation du/dt+u*du/dx=mu*d2u/dx^2. I don't introduce any Re number.

1) First assume mu=0 (inviscid fluid) and an initial condition representing a smooth field like sin(2*pi*x/L), that is with the only wavenumber k=1. The quadratic non linearity will generate immediately the k=2 component and so on progressing in time will be produced a range of Fourier components in the solution. Without viscosity, this cascade will never terminate (actually, for the inviscid Burgers equation the solution will be singular at a breaking time).

2) Now assume a finite value for mu (real fluid) and the same initial condition. Depending now on the rate between the diffusive flux (mu*du/dx) and the convective one (u^2) the cascade generated by the quadratic non linearity will be terminated at some wavenumber, that is at some characteristic lenght where viscosity acts. That is not defined explicitly by a unique lenght but rather by the range of lenghts where du/dx is sufficiently relevant to produce mu*du/dx of the same ordere of magnitude of the convective term.

In conclusion, the Reynolds number is somehow misleading. In turbulence we don't have a single characteristic lenght but a range ofcharacteristic lenght scales, each one producing a value for the local Re number. Just think about the classical theory of the wall turbulence. A laminar region is still existent at a small lenght scale for which Re is small. Laminar, transitional and turbulent regions generally coexist. The classical experiment in pipes by O. Reynolds is a specific case but can be misleading

Kaveh Zamani's explanation seems acceptable to me.

Kaveh Zamani

your statements are generally correct, however there is a fundamental difference between turbulence and thermodynamic concept of kinetic energy. In the former, there is not the scale separation we can adopt, conversely, for the thermodynamic concept. In turbulence you can also accept a backward energy cascade.

Hi Filippo Maria Denaro, Kaveh Zamani, Bharat Soni and Mladen Bošnjaković, I can see from the answers that the excess kinetic energy causes the fluid particles to cross layers in a turbulent flow. But why can't it let the fluid particles just flow in layers like in laminar flow. Is there something we neglect in laminar flow that becomes non-negligible in a turbulent flow?

Mir Aamir Abbas

laminar flows are not only when a fluid flows along steady layers...For example the laminar vortex shedding behind a bluf body is a case where the flow is separated. The starting point is that the laminar flow can be unstable in such a way to develop an unsteady behavior and a fully 3D flow. The instability is intrinsic in the non linear character of the governing equations. You cannot just observe a value for the Reynolds number. For example, the laminar solution in a pipe (Hagen-Poiseulle solution) is valid for any value of the Reynolds number. But the problem is the stability of the solution to perturbations.

Thus, your question is much more related to the instability topic.

It all comes down to the action of different forces ...

If the forces of inertia can be significantly different in some parts of the fluid flow section then a "collision" of the individual masses and changes in the direction of motion occur (because pressure field is changed localy) ...

Maybe this is cause of instability ...

Mladen Bošnjaković

It is a complex framework, for example experiments in pipes still do not agree about a value of the transitional Re number. Pressure perturbations can onset the instability to turbulence, as well as just some roughness of the pipe or in the device.

What should be clear is the fact that turbulence is generally everywhere, the laminar regime is just a specific case.

Filippo Maria Denaro , thank you prof. for your comments on Burger's equation and turbulence.

I agree with your comments that only Re is not sufficient to categorize flow as turbulent. There you need some other factors as well, and hence I tried to explain the things in terms of kinetic energy.

The examples which you described related to Burger's equation will either lead to contact wave or shock wave. I think it's concept is different to turbulence. Although shock wave also propagates to change pressure and velocity suddenly.

Also, as you mentioned " Turbulence is not chaotic in random meaning" - It depends on physics and domain of problem again. Please correct me if I am wrong.

Mir Aamir Abbas , dear Prof. , we are not missing any phenomena while defining the laminar flow where all the fluid particles follow parallel layers. We are only obeying the basic definition of tangential shear applied to fluid. As you know that when a tangential shear is applied to fluid, it deforms fluid packet continuously. and the momentum is transferred layer by layer in lateral direction. The tangential shear is applied by either initial relative velocity of fluid with respect to the boundary.

Bharat Soni

" The examples which you described related to Burger's equation will either lead to contact wave or shock wave. I think it's concept is different to turbulence. "

No. The original work,

Burgers, J. M. (1948). A mathematical model illustrating the theory of turbulence,

is a prototypal model for turbulence, the viscous equation is parabolic and the solution is always regular. Clearly the model cannot describe the vorticity dynamics but it is realted to turbulence.

And turbulence is never chaos in the sense of randomness. Just have a look to the spectra of a white rumour and compare to a turbulence spectra.

Turbulence is nothing but access of energy. Amount of bulk energy that could not be easily transferred to shape the fluid motion. Because the nature prefers equality, some sort chaotic process that leads to disruption abruptly I guess, tries to dessipate the excess energy. Well it is definitely related to kinetics and pressure.

Flow will turbulent only when the initial disturbance get amplified. It may be either low velocity or high velocity. If the disturbance is not amplify then the flow will no more turbulence even if the velocity is very high in the flow..

High shear stress caused by the high velocity gradient across the flow is the one of the reasons of emerging turbulence in some of the flows besides high Reynolds number.

Hi Mohsen Jahanmiri. Thanks for the answer. But I still am not able to understand what is it that high shear stress can do which low shear stress cannot do.

It simply pumps high energy into the flow which causes turbulence to persist.

Mohsen Jahanmiri

Do you agree that turbulence is caused by amplification of disturbance and that in absence of disturbance, the flow will remain laminar at even high speed flows.

Mir Aamir Abbas

It is not exactly the amplification of the magnitude of disturbance but the creation of a wide spectrum of components to produce turbulence. The aspect of the onset of transition from laminar to turbulent regimes is complex. The flow can remain laminar even if a perturbation is present (for example the laminar vortex shedding). Conversely, a turbulent flow can also relaminarize. In practice, it is unlikely that high speed flows remains laminar, disturbance are present also in the small sound waves intensity.

Dear Abbas,

I think Mr. Maria answered your question. I agree with him except that according to Orr-Sommerfield equations and related stability curve beyond certain Reynolds number amplification of disturbances is inevitable which leads to turbulence.

Hi, Maria... You explanation regarding development of turbulence due to creation of a wide spectrum of component is very interesting..... But from were the spectrum become wide.... I think it is due to the disturbance which got amplified in the flow if the Reynolds number is high enough... Yes perturbation is present in the laminar flow too but it is not enough for the spectrum to become wide...

Very interesting discussion....👍

The proportion between inertia and viscosity in the simplest words is the main reason for the turbulence phenomenon. when one overtakes the other, turbulence triggers to start because that is what perturbs the laminarity of fluid particles in a flow.

Hi @Filippo Maria Denaro and @Mohsen Jahanmiri. What can be the possible sources of the initial disturbance?

Mir Aamir Abbas

it could be due to a pressure disturbance or the presence of a body or other things

Thanks for the answer Dr.Denaro. I would like to know what happens if there is no initial disturbance? Would the there still be turbulence at high speed flows?

Mir Aamir Abbas

In a real situation is impossible to avoid disturbances in a flow, in terms of both initial and boundary conditions. On the other hand, theoretically, you know that several exact solutions exist, at any value of the Reynolds number. Thus, the problem is in the stability of such solutions to the disturbances.

Dear Aamir,

Initial disturbances could be caused by the followings,

variation of pressure, velocity, temperature, viscosity or density of flow at upstream. These can be appeared as pressure waves or sound waves which could affect the initial condition.

Thanks Mohsen Jahanmiri.

@Mir Everything boilsdown to disturbances as professor @Denaro mentioned. Such disturbances are difficult to capture and are also known as Waves. They are always present and these are the source of the chaos. Juat look what happens far from diatuebances, for example in the “inviscid” region of the flow. Nothing happens at portion of the flow. That is why flow solver that region with inviscid theory. However, once you get closer to any source lf disturbance things become more complicated.

Dear Aamir,

I should add that inception of flow transition process on a flat plate occurs with appearance of TS waves which are 2D waves. If the intensity of disturbance is more than a threshold on stability curve, these waves are augmented and will be changed to 3D waves and eventually turbulent spots are formed. Finally these spots are merged together and turbulent flow prevails.

If we imagine that we may realize an experiment where any disturbace occurs, we may that maintain laminar flow for high reynolds numbers. The question is related to initial condition which is illustrated by butterfly effect. The butterfly effect is the idea that small things can have non-linear impacts on a complex system. The concept is imagined with a butterfly flapping its wings and causing a typhoon. If one may resolve exactly Navier Stokes equations with any disturbance, including numerical one, the result would be that of a laminar flow. However if this is theoritically possible, in pratice some experiments succeds to maintain laminar flows but to a certain Reynolds number. it is like the mecanical situation of instable equilibrium

Are the velocity and pressure fields of the steady state flow dependent on the initial conditions?

Dear Mir Abbas,

It depends that which approach choosing for tracking fluid motion. Eulerian or Lagrangian?

Lagrangian approach deals with individual particles and calculates the trajectory of each particle separately, whereas the Eulerian approach deals with concentration of particles and calculates the overall diffusion and convection of a number of particles.

So for Eulerian approach, initial condition doesn't affect flow field, whereas for Lagrangian it matters.

Mir Aamir Abbas Turbulent flows are by definition unsteady. They can be considered steady on Spatio-temporal averages on scales much larger than the scales of fluctuations. The same is true for the initial and limit conditions.

Mir Aamir Abbas

If you are talking about the steady RANS equations, there are no initial coditions for such formulation! You have a system of non-linear equations that must be solved iteratively. You have only a guess condition for that.

If you assume a dependence on guess conditions you are assuming that the statistical solution is not unique. For the steady NSE you can lookm at the Hopf bufurcation for multiple possible solutions.

Any obstruction or sharp corner, such as in a faucet, creates turbulence by imparting velocities perpendicular to the flow. Furthermore, high speed flows cause turbulence. Transition to turbulence may also be triggered by surface roughness in pipe and channel flows.