Chapter 272: Advancing the NS equation from a physical perspective!
After writing the title and introduction, Xu Chuan began to enter the main text.
". Quoting Professors Pan Ronghua and Zhang Weizhe's "Paper on the Compressible Navier-Stokes Equation of Thermal Conductivity", and based on this, relax the initial value conditions."
"There exists some positive constant C and no eta > 0 such that for any (x, t) ∈ (0, 1) (0, ∞)."
"It can be obtained that C≤u(x,t)≤C, C≤θ(x,t≤C), and ||(u-∫udx,u,θ-∫udx)(·,t)||H(
0,1)≤Ceeta”
In the study, Xu Chuan began to explore the NS equation.
This is a problem that has spanned three centuries, and it is difficult to solve it beyond imagination.
Since Saint-Venant and Stokes independently proposed a formal equation with a constant viscosity coefficient in 1845 and named it the Navier-Stokes equation, there have been as many mathematicians and physicists studying it as there are across the river in the past two centuries.
crucian carp.
However, there are only a handful of people who have made major breakthroughs in this area.
In the current mathematics community, the greatest progress in the NS equation is still the phased achievement he and Fefferman promoted when he was at Princeton.
It is possible to determine the existence of a solution given an initial condition and boundary condition in the surface space.
Now, Xu Chuan wants to push it further by giving a finite domain and conditions with Dirichlet boundaries. In three-dimensional space, the Navier-Stokes equation has real solutions and the solutions are smooth.
If this step can be achieved, it will almost be possible to establish a mathematical model for the plasma turbulence in the controllable nuclear fusion reactor chamber and use a supercomputer to perform control calculations.
For Xu Chuan, he currently does not expect to solve the NS equation or anything like that, which is not a reliable and good idea.
It has been nearly two hundred years since the NS equation was proposed, and it still stands tall like a peak with no end in sight.
Countless climbers don't even get close to the foot of the mountain. People can't see the top of the mountain, they can only take a look at it from a distance through the fog.
Xu Chuan did not dare to say that he would be able to solve the NS equation in his lifetime.
Not only because it is difficult, but also because it is a huge systematic project.
The "problem of the existence of smooth solutions to the N-S equations in three-dimensional space" defined by the Clay Institute is just a prelude to the NS equations.
In the villa, Xu Chuan has not left the house for more than a week.
His advancement of NS equations was relatively smooth at the beginning. Partial differential equations was one of his research fields in his previous life. In addition, he took mathematics as his major field in this life. In this area, he has successfully surpassed the previous research field.
Went a greater distance.
But this does not allow him to go smoothly on the NS equation. Two days ago, he fell into a bottleneck and is still looking for a way to solve this problem.
In the study, Xu Chuan frowned and stared at the calculations on the manuscript paper.
"U``=-(1/v)(1-cosA)U."
This is a very simple formula. It is a harmonic equation with functions as coefficients. It is derived from Chen Zhida's decomposition theory of deformation tensor S R for wall flow with zero pressure gradient, and obtains the deformation in the velocity profile U(y) theoretical equation.
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It can be seen from this equation that as the wall distance increases, the scale of turbulence evolves from a small scale with ultra-high wave number to a super-large scale that approaches zero wave number.
In general, it can almost replace Euler's equation and is applicable to all turbulent flows, resulting in a universally valid system of equations.
In addition, for this equation, it has been confirmed that Prandtl's logarithmic speed is the theoretical solution of the equation.
Therefore, it can be considered that for ideal wall flow, the theoretical solution is consistent with the experimental solution.
To put it simply, under ideal circumstances, the turbulence operating state calculated through mathematical formulas is exactly the same as the actual operation.
If this can be done, it can be used to establish mathematical models to predict and control turbulence.
However, it has a fatal problem!
That is, the turbulent region is a region where cosA evolves from being unable to be approximated to 1 to close to 0, and a universally valid analytical solution is difficult to obtain.
This is the most fatal point for the weirdly shaped controllable nuclear fusion reactor chamber.
Xu Chuan wants to find a way to supplement or replace it, but has not been able to do it so far.
More importantly, mathematically, the strict acceleration formula is proved using Lie derivatives.
Therefore, although the acceleration of micro-elements derived using S R is essentially the same as the Lie derivative, they are very different in terms of mechanical (physical) interpretation.
At present, what is generally accepted by the scientific community is the Euler equation based on Lie derivatives, or the NS equation.
Therefore, there is almost no supporting literature in the theoretical community for the wall flow equations and the general equations of turbulence given here.
In other words, Xu Chuan couldn't even look up previous literature and papers.
This is an almost completely blank field.
In the study room, after crumpling the manuscript paper in his hand and throwing it into the trash can, Xu Chuan stared at the brand-new A4 paper and let out a long sigh of relief.
Since the derivation entered the bottleneck, he has been stuck on this problem for almost ten days, but has achieved nothing.
Of course, this cannot be completely said. At least in the past ten days, he has eliminated many unusable methods.
He shook his head and was about to continue writing, but after thinking about it, he threw the pen aside.
After looking up at the ceiling for a while, Xu Chuan pushed away his chair and stood up.
Maybe, he needs a little help.
He thought of his experience in solving the difficult problems of the existence of the Yang-Mills gauge field and the hypothesis of mass separation in his previous life.
This chapter is not finished yet, please click on the next page to continue reading the exciting content! At that time, just like this time, it was restricted by a bottleneck for a long time.
The NS equation, like the existence of the Yang-Mills gauge field and the mass separation hypothesis, are not just mathematical problems, they are also physical problems.
Perhaps, he can think of a solution from a physical perspective.
Putting aside mathematical thinking, from a physical point of view, the fastest way to study a problem is to practice it.
Turbulence is everywhere, from the wake of a high-speed airplane to a bathtub filled with water.
Its essence lies in injecting energy from the largest scale to the smallest scale through the formation, interaction and demise of vortices.
To put it simply, orderly fluid flow will form vortices. These vortices will interact and split into smaller vortices, and then the smaller vortices will continue to interact, and so on...
However, this chaos has puzzled scientists for centuries.
There is currently no mechanistic framework that can analyze how interactions between vortices drive such an energy cascade.
For physicists, when faced with a difficult problem, there is a solution that physicists often use!
That is to put these things together and completely "smash" them!
For example, in order to understand the basic components of the universe, theoretical physicists have built large strong particle colliders to accelerate microscopic particles and then collide them to obtain data.
This time, in order to reveal the basic mechanism of turbulence and find a solution to the NS equation, Xu Chuan decided to let vortices collide with each other and see its structure and movement from the microscopic level with his own eyes.
At NTU, Xu Chuan went straight to the School of Physics, found Yu Yongwang, the dean of the School of Physics, and made a request to borrow the equipment of the School of Physics.
Dean Yu agreed to Xu Chuan's request without even thinking about it.
In the physics experiment building, Xu Chuan called two of his students and asked them to help. Under the arrangement of Yu Yongwang, Nanda also called two doctoral students to help.
In fact, it is not difficult to create turbulent collisions.
Various sea creatures can create vortex rings underwater using air and fast-moving water.
This is because when a round bubble moves forward, it will be subject to the squeezing force of the water on the front and the friction of the water surface on the side. This causes the originally round bubble to be flattened, and the edge is squeezed by the backward force.
If the force is applied, the air at the edge will be disturbed to rotate, thus forming a vortex at the edge, which will gradually be separated in the middle, forming a vortex ring.
The difficulty of the experiment is to use an ultra-high-resolution camera to record the entire collision of two turbulent flows, and then use a 3D visualization program to reconstruct the collision process and determine the basic mechanism of turbulent evolution.
"Professor, I have made adjustments here. The A1 vortex ring uses green material, and the A2 vortex ring uses red material."
In the laboratory, Gu Bing reported loudly to complete the work in hand.
Xu Chuan nodded and said, "Okay."
On the other hand, with the help of students majoring in photogrammetry and remote sensing, Amelia also successfully completed the installation and debugging of the ultra-high-resolution camera.
"Reporting to Professor, the ultra-high-resolution camera is ready and can be recorded at any time."
Under the command of Xu Chuan and the help of Nanda, the equipment for the vortex ring collision experiment was quickly assembled.
The experiment started. Under precise control, the vortex ring manufacturing instruments located on both sides of the water tank simultaneously launched a bubble forward. Under high-speed movement, the bubble evolved into a vortex ring and then collided together in the center.
The moment the red and yellow vortex rings collided, they formed mixed-color ripples and rings visible to the naked eye, but in just a second, these ripples and rings dissipated into a sea of dye.
But for Xu Chuan, this is enough.
In this laboratory, Xu Chuan specially found a powerful scanning laser sheet and synchronized it with a high-speed camera. The combination of the two allows it to capture hundreds of thousands of images per second.
The ultra-high-resolution high-speed camera accurately recorded the entire experimental process and transmitted it to the computer.
All that's left is to use a 3D visualization program to reconstruct the collision process.
"Professor, are you done with this experiment?"
In the laboratory, Amelia looked curiously at her classmates who were dismantling the equipment and asked Xu Chuan.
Xu Chuan nodded and said, "Well, it's done."
"Can I ask what this is studying? Eddy flow? Turbulence?"
Being hurriedly called over, Amelia and Gu Bing were both a little curious about what their mentor was doing after disappearing for more than half a month.
Xu Chuan smiled and replied: "Study the NS equation."
Amelia opened her mouth and looked at Xu Chuan with some surprise and then at the equipment being disassembled: "Is this what you're going to use?"
Xu Chuan said with a smile: "Of course, the NS equation is designed to study fluid mechanics, and vortices are also a part of fluid mechanics."
In fact, physicists have been using vortex colliders to study turbulence since the 1990s, but those previous experiments failed to slow down and reconstruct the mechanics of the moment when the collision occurred.
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The reason why Xu Chuan did this was also because of the experience brought by rebirth.
In the aerodynamics of later generations, it was a very common thing to reconstruct the chaotic system for research, so he conveniently added it.
"Professor, can I join your research?" Amelia asked expectantly.
She studied mathematical physics in college and is also very interested in NS equations. If she joins Xu Chuan's research, even if she can't help much, she will definitely learn a lot.
On the side, Gu Bing also cast an expectant look.
Noticing the eagerness of the two students, Xu Chuan smiled and said, "You should complete the tasks I gave you before."
It's not that he doesn't want two students to participate in his project, but they probably don't have enough energy and time.
Last year he didn't take much care of the students, but this year was different. At the beginning of the new year, he personally arranged a Hodge-like math problem and gave it to them.
This problem can probably consume all their daily time.
If they can solve it, they will not be far from graduation.
After several days of work, the 3D visual reconstruction of the vortex collision was finally completed.
NTU immediately sent over the reconstructed data.
After receiving the data, Xu Chuan made some tea and turned on the computer.
After getting inspiration from Qiu Chengtong through tea mist, he has now started making tea and drinking tea, hoping to continue to get inspiration and ideas from it.
Although this was of no use, Xu Chuan unexpectedly discovered that drinking tea could help him maintain a certain concentration in his daily research, so he became accustomed to making a cup of tea before doing research.
Holding the tea cup, he took a sip and opened the reconstructed vortex ring collision experiment.
This is a completely different picture from the visual one. After the reconstructed collision, the color of the vortex ring completely disappeared or became unified.
But Xu Chuan keenly noticed that when the vortex rings collide with each other, they will be stretched outward, and antisymmetric waves will be formed at their edges.
The peaks of these waves develop into finger-like filaments that grow perpendicular to the core where the collision occurs.
Then, these "fingers" rotate in the opposite direction to the adjacent "fingers", forming a new array of micro-vortices. The interaction between these micro-vortices lasts for several milliseconds.
If you don't play it extremely slowly, it can be said that it is difficult to find these.
But it also brought a vague inspiration to Xu Chuan.
With a gentle click of the mouse, he pulled the screen to the beginning and played it again.
When new vortex arrays and ripples were formed, Xu Chuan's eyes became brighter, but there was still a trace of doubt in his bright eyes.
He always felt that these things gave him an inexplicable sense of familiarity in mathematics, but for a while he couldn't remember where he had seen them.
The mouse pulled back the progress again, and he watched the video in front of him over and over again.
Suddenly, in his mind, a piece of manuscript paper appeared in his mind, making his eyes suddenly brighten!
He remembered where he had seen this familiar thing and knew how to advance the NS equation!
PS: Yesterday’s monthly ticket has been updated. I keep my promise and give me some more monthly tickets. Dear friends, there are still about 300 monthly tickets to reach 2,000, (￣︶￣*)). Let Yawei draw a prize next month.
