Discussion: Reason of Turbulence!!
Hi, friends:
Please forgive me if my question is too stupid. I'm wondering of the reason of turbulence for a long time. Three options came to my mind: 1)Nonlinearity 2)Molecular/microscale motions 3)Diffusion First of all, all the Reynolds Stress components come out of Convection term in the NS equation, so without Nonlinearity should have no turbulence??? But, if we rewrite the NS equation in Lagragian form, there is no nonlinearity at all, and I'm wondering what's the point? You may say, okay no nonlinearity in the equation, but there are nonlinearity on boundary conditions, then is turbulence always generated on boundary? Should be not true, if you read the following: Secondly, you may agree with me that Molecular motions always exist. And a high speed flow will have higher michanical energy. This energy tends to transfer(don't know whether is transfer is to seek minimum energy in the system)to heat which means that the molecular movement will be enhanced and the noise is then amplified. A high speed flow will easily get unstable, but instability is not the precise definition of turbulence. (How should we define turbulence?) Thirdly, can turbulence be free of vorticies? If not, then the reason for generating vorticies will be a reason of turbulence. While having a look at the vorticity equation, for fluid with p=p(rho), the only reason to generate vorticity is the viscosity term, the convection terms only deform the vorticity(vorticity tilting and stretching). In my understanding viscosity is closely related to diffusion. And does this mean that diffusion is the reason of turbulence? If this argument is denied, then why people always use eddyviscosity/diffusion concept to simulate turbulence? I doubt the justification of eddyviscosity approach on turbulence modeling for I doubt the turbulence is generated by diffusion, and I doubt whether turbulence will definitely/solely behave like momentum diffusion. One of the success of eddy viscoity, is it can give a logrithmic boundary layer solution. Isn't this somehow too a posteriori? Since people first found the logrithmic profile from experiments, and then try to fit it by eddy viscosity. And, interestingly, people find more and more complicated eddyviscosity to fit different turbulence structure(like space varying/time varying/discontinuous eddy viscosity functions), which is beyond physical meaning, we are going for mathematical games,not physics!!! I admit there are other ways like Prandtl Mixing length, is it essencially different? Some may say, diffusion is the macroscale behavior of thermomotion, then is diffusion the only macroscale behavior when we apply the continuity assumption? If not, did we miss anything in NS eqn? I was reading Perter Nielsen et al 2002 Coastal Engineering Vol 45 paper "Vertical fluxes of sediment in oscillatory sheet flow" (http://www.sciencedirect.com/science?_ob=IssueURL&_tockey=%23TOC%235966%232002% 23999549998%23284119%23FLA%23Volume_45,_Issue_1,_P ages_169_(March_2002)&_auth=y&_acct=C000015498&_version= 1&_urlVersion=0&_userid=260508&md5=77f7ca689fd0812 a9431b44ebe443f06). On page 63, a interesting point madethe variation of suspended sediment concentration is in contrast to diffusion models. And he indicated lake of convection/nonlinearity in the model for sediment concentration. And the data call for negative/"exotic" diffusivity(page 66, fig5), which means pure diffusivity(for concentration)eddy viscosity(for momentum of fluid) is a not good candidate for fitting the data!!! We are doing parameter recoginition from turbulence and the parameter(eddy viscoty) is has to be so complicated to fit the data!!! Aren't we projecting the turbulence question on to a wrong parameter space? Nowadays, people resort to computers to simulate turbulence, but I don't see any breakthrough on physics(may be somebody knows, pls pardon my ignorant mind here and kindly tell me) Another point is that Reynold Number means the nonlinearity v.s. diffusivity, if it's higher, then turbulence usually occur. But if there is no diffusivity, Reynolds number is undefined. What's the point again? Means diffusivity not most important and there is something else behind the scene? People also came up with "Geophysical Turbulence" concept which is of very large scale, and the eddyviscosity they got is several orders of magnatude away from common "Turbulence", is Geophysical Turbulence really turbulence? Then what is turbulence? Finally, I have inquries on turbulence closure for Raynolds Eqn. Look how Raynolds stresses are "made up": They are new variable out of averaging process. Problem being the averaging is not precisely defined. So Raynolds Stresses in my poor mind are made up mathematically, just like spliting a equation x=5 into Xbar+x'=5, and don't know how Xbar is defined, and ask for solution of both Xbar and x', which is no way. It's hopeless to work on closure problem like this. People can argue that the Raynolds Eqn+ somehow problematic closures solved many many real problems in engineering. Yes, they did, but we are putting some unresolved phase information of x' into a term called Reynolds stress,and make some parameterization for this unkown term to fit real data. So we are still relying on experimental data to define x' which again means to define the averaging process Xbar!! Isn't there any way out of this circle? But for a good point of the physical cause of turbulence, we won't be able to solve turbulence problem. All comments are welcome! Sincerely Wen Long 
Dear Wen,
that is a funny and interesting post you wrote. If you are aware that by introducing Reynoldsaveraging on the momentum equation you do not, of course, solve the problem, but introduce additional unknowns, i.e., the second order statistical moments of the fluctuating velocity (Reynolds stresses), everything is still clearly defined. Coming up with additional transport equations for these stresses you will find third order statistical moments. So the process actually never ends and you have to make assumptions or introduce models based on physical meaning to close the problem. However, the terms in these new transport equations like diffusion, cross diffusion, pressurestrain correlation and so on can be identified or interpreted with physical meaning. That is not a mathematics game at all. So there is no 'reason for turbulence'. Turbulence is just a name we chose for a natural phenomenon or for what happens in nature or for what we observe.And as far as I see it, the assumptions are rather obvious and starting with ReynoldsStress transport models you can derive explicit algebraic stress models, nonlinear eddyviscosity models and even oneequation models or simple eddyviscosity concepts. Doing this one clearly recognizes what the faults in different modelingconcepts are and why most effects in 3D unsteady flows cannot be captured by simple twoequation models. Best regards, Ulf. 
Seems some subjects are never too old to answer ;)

You are right, that it is what I thought. However, I hope a serious answer might help ;)

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