Great Country Academician

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

".Referring to the 'Thermal Conductivity Compressible Navier-Stokes Equation Paper' by Professors Pan Ronghua and Zhang Weizhe, and on this basis, relax the initial value conditions."

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

"There exists some normal constant C and no η \u003e 0 such that for any (x, t) ∈ (0, 1) (0, ∞)."

"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 spanning three centuries, and it is beyond imagination to solve it.

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

However, there are only a handful of people who have made major breakthroughs in it.

In the current mathematics world, the greatest progress in the NS equation is still a phased result that he and Fefferman advanced together when he was in Princeton.

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

But now, Xu Chuan wants to push it further, so as to give a finite boundary domain and the condition of having a 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 of plasma turbulence in the chamber of a controllable nuclear fusion reactor and use a supercomputer to perform control calculations.

For Xu Chuan, he is not looking forward to solving the NS equation at present, and 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 didn't even get close to the foot of the mountain, and people couldn't see its top, they could only look at it from a distance through the mist.

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 "existence of smooth solutions of 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 hadn't gone out for more than a week.

His advancement of the NS equation was smooth at the beginning. Partial differential equations were one of his research fields in his previous life. In addition, he took mathematics as his major in this life. In this area, he has successfully surpassed the previous life. Go out for a longer distance.

But this does not allow him to go on smoothly on the NS equation. Two days ago, he fell into a bottleneck, and he 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 the coefficient. It is derived from Chen Zhida's deformation tensor S+R decomposition theory for wall flow with zero pressure gradient, and the deformation in the theoretical equation of the velocity profile U(y) is obtained. here.

From this equation, it can be concluded that with the increase of the wall distance, the scale of turbulent flow evolves from a tiny scale with ultra-high wavenumbers to a super-large scale with zero wavenumbers.

In general, it can almost replace Euler's equation for all turbulent flows, and obtain a universally valid equation system.

Furthermore, for this equation, it has been confirmed that Prandtl's logarithmic law of 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 operating state calculated by the mathematical formula is exactly the same as the actual operating state.

If this can be done, it can be used to establish a mathematical model to realize the prediction and control of turbulent flow.

However, it has a fatal problem!

That is, the turbulent region is the region where cosA can not be approximated to 1 to close to 0, and a generally effective analytical solution is difficult to obtain.

This is the most deadly point for the strangely shaped controllable fusion reactor chamber.

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

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

Therefore, although the micro-element body acceleration derived by S+R is consistent with the Lie derivative in essence, they are very different in mechanical (physical) interpretation.

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

Consequently, there is little supporting literature in the theoretical community for the wall flow equations presented here, as well as for the general equations for turbulence.

That is to say, even if Xu Chuan wanted to consult and learn from previous literature and papers, he couldn't do it.

This is an almost completely empty field.

In the study room, after crumpling the manuscript paper in his hand and throwing it into the trash can aside, Xu Chuan let out a long sigh of relief staring at the brand new A4 paper.

Since the derivation entered the bottleneck, he has been stuck on this problem for almost ten days, but found nothing.

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

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

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

Maybe he needs a little help.

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

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

The NS equation is the same as the Yang-Mills gauge field existence and mass interval assumptions, both of which are not only difficult problems in mathematics, they are also difficult problems in physics.

Perhaps, he can think of a way from a physical point of view.

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 speeding plane to a bathtub full of water.

Its essence lies in the infusion of energy from the largest scale to the smallest through the formation, interaction and demise of vortices.

In simple terms, orderly fluid flow creates individual vortices, and these vortices interact, break up into smaller eddies, and then the smaller eddies continue to interact, and so on...

But this chaos has puzzled scientists for centuries.

There is currently no mechanistic framework that resolves how the interactions between vortices drive such an energy cascade.

And for physicists, facing a difficult problem, there is a solution that physicists often use!

That is to put these things together and completely "crush" them!

For example, in order to understand the basic components of the universe, theoretical physicists have built large-scale strong particle colliders to accelerate microscopic particles and let them collide to obtain data.

This time, in order to reveal the basic mechanism of turbulent flow and find a solution to the NS equation, Xu Chuan decided to let the vortex collide with the vortex, and see its structure and movement from the microscopic level with his own eyes.

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

Regarding Xu Chuan's request, this Dean Yu agreed without thinking about it.

In the physics experiment building, Xu Chuan called two of his students and asked them to help. Under Yu Yongwang's arrangement, Nanjing University also called in two doctoral students to help.

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

Various sea creatures can create vortex rings underwater with air and fast-moving water.

This is because when the circular air bubble moves forward, it will be squeezed by the frontal water and the water surface friction force from the side to the rear, which will cause the original circular air bubble to be flattened, and the edge will be compressed by the backward pressure. The force will disturb the air at the edge to rotate, thus forming a vortex at the edge, and gradually being separated in the middle, forming a vortex ring.

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

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

In the laboratory, Gu Bing reported loudly to complete the work in his 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 erection and debugging of the ultra-high resolution camera.

"Professor report, the ultra-high resolution camera is ready and ready to record."

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, and under precise control, the vortex ring manufacturing instruments located on both sides of the water tank fired a bubble forward at the same time. Under the high-speed movement, the bubbles evolved into a vortex ring, and then collided together at the very 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 in a piece of dye.

But for Xu Chuan, this was enough.

In this laboratory, Xu Chuan specially found a powerful scanning laser, which is synchronized 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 sent it to the computer.

All that remains is to reconstruct the collision process using a 3D visualization program.

"Professor, are you done with this experiment?"

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

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

"Can I ask what this is studying? Eddy flow? Turbulent flow?"

Called over in a hurry, Amelia and Gu Bing were a little curious about what their mentor was doing after he disappeared for half a month.

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

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

Xu Chuan said with a smile: "Of course, the NS equation is originally for the study of fluid mechanics, and vortex is also a part of fluid mechanics."

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

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

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

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

She majored in mathematical physics in college, and she is also very interested in NS equations. If she joins Xu Chuan's research, even if she can't help much, she can definitely learn a lot.

On the side, Gu Bing also cast a look of expectation.

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

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 guide the students much, but this year is different. At the beginning of the year, he personally deployed a Hodge-like math problem and handed it over to them.

It is estimated that this difficult problem can consume all their daily time.

If they can solve it, they will not be far from graduation.

After several days of tossing, the 3D visualization reconstruction of the vortex collision was finally completed.

Nantah University immediately sent the reconstructed data over.

After receiving the data, Xu Chuan made some green tea and turned on the computer.

Since Qiu Chengtong got inspiration from the tea mist before, he has also started to make tea and drink tea now, 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 has also become 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 reconstruction of the 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 be formed at their edges.

The crests of these waves develop finger-like filaments that grow along the core perpendicular to where the collision occurred.

These "fingers" then rotate in the opposite direction to the neighboring "fingers," creating a new array of micro-vortices that interact for a few milliseconds.

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

But it also brings a kind of vague inspiration to Xu Chuan.

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

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

He always felt that these things gave him an inexplicable 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 brighten up!

He remembered where he had seen this familiar thing, and he knew how to advance the NS equation!

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