The fact that no complete theory for existence and regularity of the Navier-Stokes is available does not mean that they are not physically sound. The lack of a complete mathematical justification is at least partly compensated by the understanding of the shortcomings of this physical model (no serious scientist assumes that individual NS solutions are meaningful at high Reynolds numbers and the interpretation of the results is always based on statistical hypotheses). This is a rather common situation in theoretical physics. There is no complete understanding of the Dirac equation either, but you do get PET scans and they work and the underlying physics is understood (to the extent that it is) thanks to the same equation. I recall that, as a student, I asked a former CERN director, who was teaching a course on quantum field theory I sat in, about the rigorous definition of all the field derivatives he was writing on the blackboard. His answer was more or less: take them to be limits in the weakest topology you know. As a mathematician I was horrified and I still agree today that a better understanding of the mathematical foundations of many important theories is needed. On the other hand, in many situations a more heuristic and less formal approach has helped going beyond the limitations of existing mathematical models without having to stop doing research because we still do not have a mathematically perfect theory.

The fact that no complete theory for existence and regularity of the Navier-Stokes is available does not mean that they are not physically sound. The lack of a complete mathematical justification is at least partly compensated by the understanding of the shortcomings of this physical model (no serious scientist assumes that individual NS solutions are meaningful at high Reynolds numbers and the interpretation of the results is always based on statistical hypotheses). This is a rather common situation in theoretical physics. There is no complete understanding of the Dirac equation either, but you do get PET scans and they work and the underlying physics is understood (to the extent that it is) thanks to the same equation. I recall that, as a student, I asked a former CERN director, who was teaching a course on quantum field theory I sat in, about the rigorous definition of all the field derivatives he was writing on the blackboard. His answer was more or less: take them to be limits in the weakest topology you know. As a mathematician I was horrified and I still agree today that a better understanding of the mathematical foundations of many important theories is needed. On the other hand, in many situations a more heuristic and less formal approach has helped going beyond the limitations of existing mathematical models without having to stop doing research because we still do not have a mathematically perfect theory.

Luca addressed the point of view as a mathematician and I see his opinion.

As an engineer I can add that all we study is just a result of a specific model of the real world, wherein we make some physical assumptions that can be also weak or not still fully demonstrated.

A bird flies, he does't care if we model its lift by using either Bernoulli equation or a more sophysticated 3d unsteady NSE formulation.

The present situation is that we can use some mathematical formulations that predict the generation of lift but we are still looking for a full physical description of that.

Great. But that argument can be used for any unproven theory for flight. Why push NS equations when we don't even know that they work? Thanks.

From my perspective, if a given theory can explain experiments and also predict the outcome of experiments, it is valid within the limits of the underlying assumptions.

Maybe, at some point something better comes along with fewer and less restrictive assumptions, but as long as good experiments do not disprove the theory, it is valid.

Maybe I misunderstand the question but, solving the Navier-Stokes equations actually gives, in most of the practical occasions, very accurate predictions of the forces that a fluid and an object exchange when the former flows around the latter (or the other way around, depending from the reference frame).

As an engineer, I know that the limitations of the N.S. equations are on the Newton Laws and/or in the constitutive equations, so well within the production tolerances of the manufacts we are interested in or any other error we introduce in our computations. As a matter of fact, I don't need to understand the principles behind the N.S. equations, nor any of their limitations, to actually use them fruitfully (as long as I use them within the above mentioned limits). Actually, as an engineer, I am mainly concerned about the results, so I prefer to not bond to any specific theory or explanation, but just the scientific method and how to use it in order to bound the error on my calculations. From this point of view, I ensure you, the only thing I need is to be able to repeat an experiment (as long as what I'm interested is observed to follow deterministic rules).

Still, what do we mean by understanding here? How do you even prove the understanding? Are we talking about a cause-effect relationship? Is there a reference language, beyond mathematics itself, with which we formalize theories? Or said otherwise, do you know a principle by which every phisical process or mechanism should be describable in layman terms? Does this principle also have bounds on the allowed accuracy of the description?

Dear Nicholas Landell-Mills

You can also approach a study that is not based on the continuum but the final results are similar to those you get using NSE.

Similar topics on the physics that generates lift are addressed on RG and you will not find any final conclusion.

Just a search on google:

https://www.google.com/search?client=firefox-b-d&sxsrf=ALeKk01fcyNUbjMZN_9KI-tX4CRdhL0yCw:1614099113622&q=researchgate+question+aerodynamics+generation+lift+physical+explanation&sa=X&ved=2ahUKEwiawrLiu4DvAhXN_KQKHaxUCNMQ7xYoAHoECAUQNQ&biw=1545&bih=859

"Academics, engineers, aircraft manufacturers, pilots, aviation

authorities, and other pundits (e.g. NASA) promote over twelve

diverse theories of lift. New theories of lift are occasionally

proposed. This is surprising given airplanes have been flying for

over a hundred years."

Apparently this author has a deeply rooted disrespect for engineers and scientists, calling them "pundits". Just because you can look at the creation of lift in twelve different ways does not invalidate any one theory. The Navier-Stokes equations are simply too complex to summarize all fluid phenomena in a single theory, but the experimental and computational (CFD) proof exists and explains very well why things fly. Just look at the surface pressure contours of an aircraft in a virtual wind tunnel and it becomes obvious.

If you want respect surely the best strategy is to work out how airplanes fly? Then use the knowledge to do something positive, like design an affordable and efficient flying car. As it is aeronautical news flow is dominated by airplane crashes and Navier-Stokes remains unsolved.

This whole argument is nonsense in my opinion. Airplanes don't crash because of a lack of understanding in fluid mechanics or the Navier-Stokes equations, which can be solved for any flow problem down to the smallest turbulence scale with big enough computers. As an analogy, you can explain the motion of falling bodies with Newtonian or relativistic mechanics and nobody would argue that either theory has not been proven or that they invalidate each other.

Nicholas Landell-Mills

what do you mean for "NSE remains unsolved"? There is a mathematical approach to assess existance and unicity of the solution for a certain class of BCs but we have also a lot of exact particula solutions of NSE. As well as we are ablo to numerically solve and compare to experiments in several flow problems.

"Airplanes do not crash until they are airplanes" (Cit. of one my former teachers)

To be precise. There's no conclusive experimental evidence that NS Equations work, showing that they correctly explain and predict what happens in the physical world. There are unproven hypothesis based on observations. Maybe NS equations are correct, but there's no proof.

NS equations claim to describe the physical world. But there's no proof NS equations can explain observations. For example, NS equations cannot explain why lift quadruples if aircraft velocity doubles.

Quanta magazine have written numerous articles about the long list of unresolved theoretical weakness and logical issues of NS equations. see : https://www.quantamagazine.org/tag/navier-stokes-equations

For example, NS equations can't explain turbulence. The Clay Math Institute still has a $1 m award offered for anyone who can solve the existence and smoothness problem. NS equations can't explain why a stall starts at the trailing edge of a wing, nor why a supersonic shock wave starts at the leading edge of the wing. NS equations can produce multiple outputs for the same inputs. There are different versions of NS equations. It's impossible to see from the NS equation whether lift is due to low air pressure, air viscosity or friction in a particular situation. NS equations are used for stable flight, but inverted, banking, ..... or other flight manoeuvres. NS equations provide no clear answer to simple questions, e.g. if the AOA varies by 2°, how does this impact the factors of low air pressure, viscosity and friction which allegedly cause lift?

And yet NS equations are the prevailing methods used by academics, engineers and physicists to analyse lift on an airplane wing. This reliance on NS equations doesn't make sense given all the unresolved practical and theoretical issues. It also doesn't make sense that this is not a priority to fix. No conferences, little discussion, ...... which is why I wrote this post. The academics and engineers involved don't seem to care much and some even post agressive denials.

Airbus and Boeing seem content with CFD analysis showing smooth airflow over the wings, and unconcerned that they still cannot explain how this airflow creates lift.

I've met plenty pilots who find it very puzzling that engineers still don't know how airplanes fly. Pilots are also concerned that wings are designed primarily by intuition, trial and error. This creates some distrust of aero engineers.

I think you are making some fundamental confusion about what we are talking about. No one says that NSE exactly describes the fluid dynamics for any situation! NSE are the mathematical equations corresponding to a chosen model for the continuum. Even an assessed demonstration of existence and unicity of a solution of NSE does not change the fact that it is a simplified model of the reality. NSE describes turbulence, differently from what you stated, at least for some specific problems. But turbulence does not care if you want to use NSE, particle description, discrete vortex methods or anything else. It exists and mankind inteprets turbulence, as well as any other problem assuming a model for the study.

This is a physical basis in every theory, general relativity is a model, too. Any physical description followed by a mathematical set of equations is a model of the real world.

It is well know that NSE fails to describe some specific phenomena. But I can say for sure that the stall condition does not change the mathematical character of the flow problem, NSE can describe that, provided that you are adequately educated to solve this problem.

Ok, we gotta be honest here, when an aerospace engineer graduates he/she is given a certain amount of fairy dust. It is limited and totally up to him to decide how to spend it in his whole lifetime. That's how we make things fly, not trough MODELS that, WITHIN OUR CAPABILITY OF PERFORMING AND REPEATING EXPERIMENTS, are accurate enough to serve the purpose. Luckily, all the other branches of classical mechanics are instead well understood, and you don't have to worry about them.

And no, the fairy dust is not available on request.

What appears from this post, is also the confusion in considering the NSE like some exotic model that strongly differs from classical newtonian physics while, actually, the set of NSE are deduced from the classical law F=ma.

Fliippo - Yes. But the NS Equations propose a force that pulls (sucks) a wing up, with no reactive force. This is the reverse of Newtonian physics which proposes a pushing force, which then creates a reactive force in the desired direction. So is it acceptable to say the end result of NS Equations are not in anyway Newtonian?

Why? You can consider the law F=ma (written in a material volume), assuming a steady condition in the whole domain (air+airfoil) to get the sum of the acting forces to be zero. And yes, air reacts to the presence of a moving body. The airfoil in air is like a totally submerged second WW U-boat in sea :-). The resulting total force F must not produce an acceleration.

Again, you stated a result about the lift that does not depend on the specific NSEs, you can also use other simplified mathematical models . Conversely, turbulence can be described by the set of NSE.

.. and just as a practical rude experience, anyone has tried to put his hand out of a car window and we experience the force acting on the hand. We are moving the particle of fluid, "compressing" the frontal part of air so that a pressure difference is generated. We must exert a counterforce to take the hand in equilibrium. Thus, yes, we can experience lift that need a reacting force to mantain the steady state.

"But the NS Equations propose a force that pulls (sucks) a wing up, with no reactive force."

Not true! - You have equal and opposite forces in any flow. The fluid pushes or pulls on a body and the body pulls or pushes back with the same force. This is true for the entire body or any subsurface of it.

"For example, NS equations can't explain turbulence."

Also not true! - The NS equations can capture even the smallest scales of turbulence provided the flow is sufficiently resolved. This has been done for a number of problems using a CFD method called DNS (Direct Numerical Simulations) and has been validated with physical experiments. While we still do not have computers with enough processing power to run DNS on full size aircraft, some day we will.

Turbulence develops from infinitesimal disturbances in any flow whether physical or numerical, and these disturbances stay bounded (i.e. do not blow up) if you employ proper CFD methods like TVD (Total Variation Diminishing) in your discretization. Please see my publication on the evolution of vortex instability in flow over a cylinder; it starts with machine zero disturbances.

" NS equations can't explain why a stall starts at the trailing edge of a wing, nor why a supersonic shock wave starts at the leading edge of the wing."

Once again, not true! - The NS equations perfectly predict these phenomena in a properly set up CFD simulation. I typically use an impulsive acceleration from rest as the initial condition and then march the problem in time, but one can also accelerate more slowly. I have seen shock waves form at the leading edge and boundary layers separate at the trailing edge many times in my aerodynamic (CFD) simulations.

"NS equations can produce multiple outputs for the same inputs."

Not really true! - When you run the NS equations through a CFD simulation with identical initial and boundary conditions you always get the same result. But there are some flow problems that will develop an alternate solution with only minor modifications of the initial condition, i.e. the way you start your flow simulation. One such example is an unstarted supersonic spike inlet.

Trying to be more constructive, the allegory of a wing being sucked upward instead of pushed is just a semantic construction. What is sucking if not being pushed?

The wing is definitely pushed upward because the pressure beneath is larger than the one above. It just happens that, with respect to the undisturbed conditions, the pressure below it is not as much increased as the one above is decreased.

But isn't this what exactly happens when you drink a coke with a straw?

Dear Mr. Mills,

It was a great pleasure to read your energetic essay, whose extensive illustrations highlight the incompleteness and inconsistency of modern aero- and hydrodynamics. However, not all publications on the topic were included in your impressive list of references. Moreover, some of the missed groups of papers are directly related to the essence of the issue under discussion.

Concerning the content of a current problem of drag and lift on a body in a flow.

Since Newton, and in fact, in pre-Newtonian times (Galileo, Descartes), the description of nature and the anthropogenic (technogenic) contribution to its changes developed not only on the basis of the axiomatic principles of a material point motion (Newton's Three Laws) but also on the development of the concept of conservation laws for matter (substance), momentum and energy. In classical solid body mechanics, these laws include the principle of conservation of angular momentum as well.

As applied to the mechanics of fluids and gases, the conservation laws are presented in the form of the fundamental equations of matter, momentum, energy, or temperature transfer (so on continuity, Navier-Stokes, Fourier, Fick equations indicated firstly at 1944 [1-3] and the empirical equation of state and supported in [2, 3]. In recent years, the content of the equation of state has been redefined, and the internal energy the derivatives of which determine the density, pressure, temperature, and other thermodynamic quantities is included in it more completely [4]. Internal energy can be quickly converted into thin flow components by localized processes of direct atomic-molecular interactions [5].

The fundamental system of equations of the mechanics of inhomogeneous fluids is closed, self-consistent, and resolvable [6]. Its complete solutions, constructed taking into account the compatibility condition, include waves of various types, ligaments (thin extended elements – interfaces, envelopes, fibers, bundles, ...), as well as various products of their nonlinear interaction that are vortices, jets, wakes, ... which can be distinguished in the structure of flows. Ligaments link processes at the atomic-molecular level with macroscopic flow components [7].

The approximation of an actually homogeneous fluid, in which the density is identically constant and, as a coefficient, can be omitted in all equations, transforms a solvable fundamental system into a system of Navier-Stokes equations (NSE). As noted by Mase, Serrin [8, 9], and other scientists, NSE does not include basic parameters that determine the properties of fluids (density), and in fact, the system represents some kind of special continuous transformation of metric space into itself (parameter of transformation is time). The NSE system with constant density is degenerate. Some of the singularly perturbed solutions in the low-viscosity approximation equal identically [4]. In this regard, NSE is solvable only when considering problems in a two-dimensional formulation, which is used when comparing the results of calculating the flows around a body using various models of the fluid (actual or potentially homogeneous, strongly or weakly stratified [10].

Returning to the problem of flow around bodies, one can say that any bodies in the flow are acted upon by three components of the force (drag, lift, and lateral sliding) and three components of the angular momentum. Some of the solutions describing the flow around bodies of the simplest shape (a plate at an arbitrary angle of attack [10, 11]) are in good agreement with the data of high-resolution shadow visualization of the flow pattern [12]. It is hoped that the flow pattern around three-dimensional bodies of arbitrary shape can also be calculated numerically at any velocity of a body based on a system of fundamental equations and supported by a high-resolution experiment. For this, the resource of modern computers seems to be sufficient, but a lot of work on developing codes is required. A general analysis of the mathematics of flows taking into account the problem of accuracy in theory and errors in the experiment was carried out in [13] written in Russian.

All cited papers are either publicly available or posted on RG.

I wish you further successful work and the preparation of new articles that will facilitate the development of contacts and discussions on topical problems of physics, which, under the influence of a number of objective and subjective factors, have decreased gradually.

References

1. Landau, L. D.; Lifshitz, E. M. Fluid Mechanics. V.6. Course of theoretical physics, Pergamon Press: Oxford, UK, 1987, pp. 93,94.

2. Müller, P. The equations of oceanic motions. CUP: Cambridge, UK, 2006.

3. Vallis, G.K. Atmospheric and oceanic fluid dynamics, CUP: Cambridge, UK 2017.

4. Feistel R. Thermodynamic properties of seawater, ice and humid air: TEOS-10, before and beyond. Ocean Sci. 2018, 14, 471–502.

5. Chashechkin Yu. D. Fast superfine components and sound packets in flows induced by a drop impact on a target fluid at rest // Fluid Dynamics & Material Processing (FDMP). 2020. V. 16, No. 4, pp. 773-800. DOI:10.32604/fdmp.2020.09001

6. Chashechkin Yu.D. Singularly perturbed components of flows – linear precursors of shock waves // Math. Model. Nat. Phenom. 2018. Vol. 13. No. 2. P. 1-29. https://doi.org/10.1051/mmnp/2018020.

7. Chashechkin Yu. D., Ilinykh A. Yu. Multiple emissions of splashes upon drop impact // Doklady Physics. 2020. V. 65. No. 10. P. 384–388. DOI: 10.1134/S1028335820100067.

8. Mase G.E. Theory and problems of continuum mechanics. McGraw-Hill Book Co., 1970.

9. Serrin, J. Mathematical Principles of Classical Fluid Mechanics. Handbuch der Physik, Band VIII/1. Springer: Berlin, Germany, 1959; pp. 125–263

10. Zagumennyi Ya. V., Chashechkin Yu. D. Pattern of unsteady vortex flow around plate under a zero angle of attack (two-dimensional problem) // Fluid Dynamics, 2016, Vol. 51, No. 3, pp. 53–70. DOI: 10.7868/S0568528116030

11. Zagumennyi, Y.V., Chashechkin, Y.D. Numerical Analysis of Flows of Stratified and Homogeneous Fluids near Horizontal and Inclined Plates // Fluid Dyn. 2019. V. 54, pp. 958–969. doi:10.1134/S0015462819070152

12. Chashechkin Yu. D., Zagumennyi I. V. Visualization of stratified flows around a vertical plate: laboratory experiment and numerical simulation // Intern. J. Comp. Meth. Exp. Measure. (CMEM). 2020. V.8 (2). P. 148 -161. DOI: 10.2495/CMEM-V8-N2-148-161.

13. Chashechkin Yu. D. Mathematics of fluid and gas flows. Proceedings of the conference “Waves and vortices in complex media@, Moscow, 01-02 December 2020. P. 198-215. (in Russian) = Чашечкин Ю.Д. Математика течений жидкостей и газов. // Волны и вихри в сложных средах: 11-ая международная конференция – школа молодых ученых; 01 – 03 декабря 2020 г., Москва: Сборник материалов школы. – М.: ООО «ИСПО-принт», 2020. – 247 с. ISBN 978-5-91741-260-3.) C. 198-215.

Yuli D. Chashechkin

Professor of Fluid Mechanics

e-mail: [email protected]