Chapter 272: Advancing the NS Equation From a Physical Perspective! (Second Update, Please Give Me a Monthly Ticket)

After writing the title and introduction, Xu Chuan began to enter the text.

". Quote the 'compressible Navier-Stokes equation of thermal conductivity' by Professors Pan Ronghua and Zhang Weizhe, and relax the initial value conditions on this basis."

"Then (v, υ, θ)(×)∈H*H*H becomes (v, θ)∈H(0,1), υo∈H(0,1)"

"There are some positive constants C and no η>0, so that for any (x, t)∈(0,1)(0,∞)."

"It can be obtained that C≤υ(x, t)≤C, C≤θ（x, t≤C）, and ||(υ-∫υdx, υ, θ-∫υdx)(·, t)||H(0,1)≤Ceηt"

In the study, Xu Chuan began to explore the NS equation.

This is a problem that spans three centuries, and it is more difficult than imagined to solve it.

Since Saint-Venant and Stokes independently proposed the formal equation with a constant viscosity coefficient in 1845 and named it Navier-Stokes equation, there have been numerous mathematicians and physicists who have studied it for two centuries.

However, there are only a few who have made major breakthroughs in it.

The greatest progress in the mathematics community on the NS equation is the interim results that he and Fefferman have made when he was in Princeton.

It has been possible to determine the existence of a solution given an initial condition and boundary condition in the surface space.

Now, Xu Chuan wants to take it a step further and give a finite domain and a condition with Dirichlet boundary. In three-dimensional space, the Navier-Stokes equation has a real solution and the solution is smooth.

If this step can be achieved, it is almost possible to establish a mathematical model for the plasma turbulence in the chamber of a controlled nuclear fusion reactor and use a supercomputer for control calculations.

For Xu Chuan, he is not looking forward to solving the NS equation at present. That is not a good and reliable 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 have not even approached the foot of the mountain, and people cannot see its top, but can only look at it from afar through the fog.

Xu Chuan did not dare to say that he could solve the NS equation in his lifetime.

Not only because it is difficult, but also because it is a huge systematic project.

The "existence problem of smooth solutions of the N-S equation system in three-dimensional space" defined by the Clay Institute is just a prelude to the NS equation.

In the villa, Xu Chuan has not gone out for more than a week.

His progress in the NS equation was smooth at the beginning. Partial differential equations were one of his research fields in his previous life, and in this life, he majored in mathematics. In this area, he has successfully surpassed his previous life and walked a longer distance.

But this did 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 formula on the manuscript paper.

"U``=-(1/v)(1-cosA)U."

This is a very simple formula, a harmonic equation with a function as a coefficient, which is derived from the deformation tensor S+R decomposition theory of Chen Zhida for the wall flow with zero pressure gradient, and the velocity profile U(y) theoretical equation is deformed.

From this equation, it can be obtained that as the distance from the wall increases, the scale of turbulence evolves from a microscale with ultra-high wavenumber to a super-large scale with zero wavenumber.

In general, it can almost replace the Euler equation and apply to all turbulence, and obtain a universally valid set of equations.

In addition, for this equation, it has been confirmed that Prandtl's logarithmic law velocity 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 conditions, the turbulent operation state calculated by mathematical formulas is exactly the same as the actual operation.

If this can be done, it can be used to establish a mathematical model to predict and control turbulence.

However, it has a fatal problem!

That is, the turbulent region is the region where cosA evolves from not being able to approximate 1 to being close to 0, and it is difficult to obtain a generally effective analytical solution.

This is the most fatal point for the strangely shaped controlled nuclear fusion reactor chamber.

Xu Chuan wanted to find a method to supplement or replace it, but he has not been able to do so so far.

More importantly, in mathematics, the strict acceleration formula is proved by Lie derivatives.

Therefore, although the microelement acceleration derived by S+R is essentially the same as the Lie derivative, there is a big difference in the mechanical (physical) interpretation.

At present, the scientific community generally accepts the Euler equation based on the Lie derivative, or the NS equation.

Therefore, for the wall flow equation given here and the universal equation of turbulence, there is almost no supporting literature in the theoretical community.

In other words, Xu Chuan couldn't even consult and learn from previous literature and papers.

This is an almost completely blank field.

In the study, after crumpling the manuscript paper in his hand into a ball and throwing it into the trash can on the side, Xu Chuan stared at the brand new A4 paper and breathed a sigh of relief.

Since the derivation reached a bottleneck, he has been stuck on this problem for almost ten days, but has gained nothing.

Of course, this cannot be completely said. At least he has ruled out many unusable methods in the past ten days.

He shook his head and was about to continue writing, but after thinking about it, he threw the pen aside again.

Looking up at the ceiling for a while, Xu Chuan pushed the chair away and stood up.

Perhaps, he needs a little help.

He thought of his experience in solving the problem of Yang-Mills gauge field existence and mass interval hypothesis in his previous life.

At that time, just like this time, he was restricted by a bottleneck for a long time.

And the NS equations and the Yang-Mills gauge field existence and mass interval hypothesis are both not only mathematical problems, but 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.

Turbulence is everywhere. It exists in the wake of a high-speed aircraft and in a bathtub full of water.

Its essence is to inject energy from the largest scale to the smallest scale through the formation, interaction and disappearance of vortices.

Simply put, orderly fluid flow will form vortices, which will interact with each other and split into smaller vortices, and then the smaller vortices will continue to interact, and so on...

However, this chaos has troubled scientists for centuries.

There is currently no mechanistic framework that can analyze how the interaction between vortices drives 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-scale particle colliders to accelerate microscopic particles and then let them collide to obtain data.

This time, in order to reveal the basic mechanism of turbulence and find a way to solve the NS equation, Xu Chuan decided to let vortices collide with vortices and see its structure and movement at the microscopic level with his own eyes.

At Nanjing University, Xu Chuan went straight to the School of Physics and found the dean of the School of Physics, Yu Yongwang, and asked to borrow the equipment of the School of Physics.

Dean Yu agreed to Xu Chuan's request without thinking.

In the Physics Experiment Building, Xu Chuan called two of his students to help. Under Yu Yongwang's arrangement, Nanjing University also called two doctoral students to help.

In fact, it is not difficult to create turbulent collisions.

Various marine organisms can use air and fast-moving water to create vortex rings underwater.

This is because when a round bubble moves forward, it will be squeezed by the front water and the friction of the water surface backward from the side, which will cause the original round bubble to be flattened, and the edge will disturb the air at the edge due to the backward force, thereby forming a vortex at the edge, and gradually the middle is separated to form a vortex ring.

The difficulty of the experiment is to use an ultra-high-resolution camera to record the collision of two turbulent flows throughout the process, and then use a 3D visualization program to reconstruct the collision process and determine the basic mechanism of turbulent evolution.

"Professor, I have adjusted it here. The A1 vortex ring uses green material, and the A2 vortex ring uses red material."

In the laboratory, Gu Bing reported loudly that he had completed the work in his hands.

Xu Chuan nodded and said, "Okay."

On the other side, 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.

"Report to the 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 Nanjing University, the equipment for the vortex ring collision experiment was quickly assembled.

The experiment started. Under precise control, the vortex ring maker on both sides of the water tank simultaneously launched a bubble forward. Under high-speed motion, the bubble evolved into a vortex ring and then collided in the center.

The red and yellow vortex rings formed mixed color ripples and rings visible to the naked eye at the moment of collision, but in just one second, these ripples and rings dissipated in a piece of dye.

But for Xu Chuan, this was enough.

In this lab, 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.

The rest is to use the 3D visualization program to reconstruct the collision process.

"Professor, is this experiment done?"

In the lab, Amelia looked at the students who were disassembling the equipment curiously and asked Xu Chuan.

Xu Chuan nodded and said, "Well, it's done."

"Can I ask what this is studying? Eddy current? Or turbulence?"

Being hurriedly called over, Amelia and Gu Bing were a little curious about what their mentor had been doing after disappearing for more than half a month.

Xu Chuan smiled and replied, "Studying the NS equation."

Amelia opened her mouth and looked at Xu Chuan in surprise and then at the equipment being disassembled: "Just use this?"

Xu Chuan smiled and said, "Of course, the NS equation is originally used to study fluid mechanics, and vortices are also part of fluid mechanics."

In fact, physicists have used vortex colliders to study turbulence since the 1990s, but previous experiments have failed to slow down and reconstruct the moment of chaos when the collision occurs.

The reason why Xu Chuan did this is also because of the experience brought by rebirth.

In the aerodynamics of later generations, it is a very common thing for the system to reconstruct the chaotic system for research, so he added it.

"Then professor, can I join your research?" Amelia asked expectantly.

She studied mathematical physics in college and was also very interested in the NS equation. Even if she could not help much, she would definitely learn a lot by joining Xu Chuan's research.

On the side, Gu Bing also cast an expectant look.

Noticing the eagerness of the two students, Xu Chuan smiled and said, "You should first complete the task I gave you before."

It's not that he doesn't want the two students to participate in his topic, but they probably don't have enough energy and time.

Last year, he didn't teach students much, but this year was different. At the beginning of the year, he personally deployed a Hodge-like mathematical problem and gave it to them.

This problem is estimated to consume all their daily time.

If they can solve it, they will be close to graduation.

After a few days of hard work, the 3D visualization reconstruction of the vortex collision was finally completed.

Nanjing University sent the reconstructed data as soon as possible.

After receiving the data, Xu Chuan brewed a cup of tea and turned on the computer.

Since he got inspiration from Qiu Chengtong through tea mist before, he has now started brewing and drinking tea, hoping to continue to get inspiration and ideas from it.

Although this is useless, Xu Chuan unexpectedly discovered that drinking tea can keep him focused in his daily research, so he also started to get used to brewing a cup of tea before doing research.

Holding the teacup, 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 unified.

But Xu Chuan keenly noticed that when the vortex rings collide with each other, they will be stretched outward, and antisymmetric waves will form at their edges.

The peaks of these waves will develop into finger-like filaments, growing along the core perpendicular to the collision.

Then, the rotation direction of these "fingers" is opposite to that of the adjacent "fingers", thus forming a new micro-vortex array, and the interaction between these micro-vortices will last for several milliseconds.

If it is not extremely slow, it can be said that it is difficult to find these.

But it also brought Xu Chuan a vague inspiration.

With a light click of the mouse, he pulled the screen to the beginning and played it again.

When the new vortex array and ripples formed, Xu Chuan's eyes became brighter, but there was still a hint of doubt in his bright eyes.

He always felt that these things gave him an inexplicable sense of familiarity in mathematics, but he couldn't remember where he had seen them for a while.

The mouse was pulled back to 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 he also knew how to advance the NS equation!

PS: Yesterday's monthly ticket plus update, keep your promise, please give me some more monthly tickets, dear friends, there are still 300 monthly tickets to 2,000, \(￣︶￣*\)), let Yawei draw a prize next month!