Phone Fluid Mechanics

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It is said that during the Song dynasty in China, a skilled craftsman named Huai Bing monk. The famous story of The WTI Monk was found. One year in Shaanxi, two iron cows washed away by a float bridge. The iron ox weighed tens of thousands of pounds and was hard to salvage. Monk Huai Bing made two major ships, filled with sand, connected to iron cattle, and then discarded the sand in the ship. As the ship floated above, iron cattle were picked up.

 

Coincidentally, in the West more than one thousand years ago in the Song Dynasty, one person was able to solve the problem with buoyancy. This person is the Greek Archimedes (Archimedes ). In around 260 BC, the king of Syria (now in the West Island of Italy) gave Archimedes a difficult problem. Originally, the king invited people to build a pure gold crown with gold, but the King suspected that the Crown was fake, so ask Archimedes to identify. Archimedes put the crown at one end of the balance, and then put the heavy gold block on the other end of the balance, and finally dipped the balance into the water. As a result, the balance is no longer maintained after the balance enters the water.

Both stories use buoyancy to solve the problem, but there is a difference. The ancients used experience to solve the problem, and did not conduct systematic research and induction. Archimedes started with mathematics and made quantitative analysis and summarized the "Archimedes principle ".

The "Archimedes principle" puts forward that the object in the fluid is subject to upward buoyancy, and its buoyancy size is equal to the weight of the discharged fluid. The "fluid" can be liquid or gas. "Weight" is the force caused by gravity. This is the earliest law related to fluid mechanics that can be used for quantitative calculation.

During the renaissance of the 15th century, da Vinci (Art Master) continued to study the characteristics of fluid. He carefully considered the flying points of birds, observed the pattern of Ripple propagation on the water surface, and focused his interest on Eddy. Under normal conditions, the fluid is stable, but when the speed of the fluid reaches a certain size, the stability will be damaged to form a "Turbulence ". "Turbulence" is composed of vortex shards.

Da Vinci's more interesting analysis is about the principle of volume-flow conservation of liquids: In the same pipeline, the volume of fluid flowing across different areas for equal time is the same. This conclusion seems simple and implies an extremely important assumption: the fluid cannot be compressed.

 

Around 1589, the Italian scientist Galileo completed the world's first complete physics textbook: fluid mechanics. This textbook is very characteristic. Galileo used the method of ironic comedy-Like storytelling to describe his ideas, which made readers curious and influenced to some extent.

In 1641, Italian scientist Torricelli designed a famous experiment to determine the gas pressure. He found that the height of the mercury in the closed vacuum mercury tube had been 76 cm because the air was heavy and the atmospheric pressure put the mercury at 76 cm points.

The Tori split was heavily influenced by Galileo. As early as 1628, Mr. cardley, Tori's mentor, wrote a book about fluid mechanics, in which he immediately pointed out an important mistake. Casdeli is a Galileo student. Cardeli believes that if a hole is drilled at the side of a water tank, the water flow speed at the hole is proportional to the distance between the hole and the water surface. The Tori splitting experiment shows that it is proportional to the square root of the distance. The theory of Tori splitting has had a profound impact, leading to the separation of fluid mechanics from mechanics and becoming an independent discipline. Interestingly, the conclusions of Galileo and Tori split up many Archimedes's theory into bankruptcy.

Pascal, a French mathematician, undertook Tori's work. Pascal believes that the pressure of any part of the fluid in a closed container is transferred to all directions of the fluid, and the size is equal. The mathematical description of Pascal's law is: F1/A1 = F2/A2. F1, F2 is the force applied on the fluid, a1, a2 is the area of force applied. Pascal's law is of great significance in hydraulic transmission. Pascal has also become an international pressure unit.

At the same time, Boyle, an Irish scientist, began studying the "Elasticity" of the air ". He found a goat's bladder and pulled gas from the air. As a result, the bladder became smaller. Fill in the gas and increase the bladder. In 1662, he carefully studied the experimental data and proposed the "wave ear Law": the product of the gas pressure and volume is a constant.

In 1687, Newton published the mathematical principles of natural philosophy. Newton and others think that the old fluid mechanics is far from the engineering practice and decide to increase some coefficients. Newton introduced the friction between objects into the fluid and thought that there was also a "viscosity force" similar to the friction force in the fluid ". Rond d'alembert, a French physicist, made an experiment using a ship in Water Based on Newton's idea, proving that the viscous resistance in the fluid has a square relationship with the moving speed of the object.

In 1755, the Swiss scientist Euler assumed an ideal fluid (no viscosity, not compressed) and listed the basic equations of fluid mechanics. In this model, only the pressure of the fluid movement is determined. The three vector fields defined by the model are velocity field U, pressure field P, and density field rock. They are functions of position s and time t.

This equation seems complicated. It is actually a replica of Newton's second law. Newton's second law said, "the acceleration of an object is proportional to the external force of the object, and is inversely proportional to the object quality ". The mathematical expression is a = F/M. The Euler's formula is also very similar. The two items on the left are the acceleration of the particle motion, and the right is the sum of the pressure changes of particles per mass.

The first item on the left indicates the time change rate of the particle velocity in Space S. The second item on the left indicates that the quality point at the speed of U has reached another location after slight time changes. The new location is different from the original location. The partial derivative of u to S is the rate of change of speed at different positions. multiplied by u, a new change of speed at the position is obtained. We only learned the first item in the middle school physics because we have different definitions for the solid (solid) and particles. The second is about the changes brought about by Fluid Particle Motion.

If the three-dimensional coordinates are U, V, W, and the gravity direction are considered, the following Euler's equations can be deduced.

The Euler's equation shows that the internal pressure of the liquid can simulate the movement of the liquid particles, and the internal pressure can be solved by the speed. The Euler's equation is a non-linear equation, that is, it is difficult to use today's computers to solve it. In Europe more than two hundred years ago, people tried to simplify this equation, one of which is most famous as the "bernuoli equation.

D. Bernoulli, a Swiss scholar, believes that one-dimensional scenarios of fluid with constant density and constant flow (the flow field does not change with time, and the flow field is just a function of spatial coordinates) can be summarized as follows:

The bernuoli equation can be obtained by the Euler's equation after the above simplification. From the kinetic energy theorem we have learned, we know that the second side of velocity is actually the kinetic energy of an object. Therefore, the bernuoli equation tells us that the pressure has caused the change of kinetic energy. Because the viscosity is not considered, the "constant" on the right is not accurate.

In the 18th century, the French scientist lagrmann proposed a method for studying fluid mechanics. It focused on the movement of a single particle, analyzed its trajectory, and obtained the movement of the entire fluid. The resulting semi-Laplace method only studies the point of the destination point on the spatial node. In comparison, Euler's method focuses on space points and divides the space into a grid. Each node on the grid has the same speed, mass, and density. The variation of these quantities reflects the variation of the fluid. He is also known as the "point-of-origin" method, and also known as "space law ".

As mentioned above, even though Euler and bernauli use the mechanics law and calculus created by Newton, they did not consider the fluid viscosity proposed by Newton. The theoretical results are far from the experimental results. However, an overly fine equation cannot be solved. Fluid Mechanics is divided into two schools: the fluid theory school that supports continuous theoretical derivation and the water power school that uses semi-theoretical and semi-practical measurements. The two factions had to argue with each other over one hundred years.

The key to achieving breakthroughs in pure theoretical research is to establish a suitable viscosity model. In 1822, Navy first used a differential method to establish a equations for uncompressed viscous fluids. In 1845, stochastes further improved these equations and introduced the viscosity coefficient. These equations are often referred to as the "equations of the NS ).

The NS equations tell us how viscous a liquid is. It has several basic assumptions: 1. The fluid must be continuous (there is no gap inside, such as dissolved bubbles); 2. All vector fields (velocity, pressure, density ...) All can be classified. It is extremely difficult to solve NS equations. Most questions in engineering can only obtain approximate answers. Currently, scientists have only obtained more than 70 precise special solutions.

The NS equation is extremely complex. If we discuss unzipped fluids and assume that the viscosity coefficient is a constant, we can simplify the NS equations into the following simple form:

(1)

(2)

(1) The left side of the formula is the acceleration of the Euler's equation. The first item on the right is the pressure factor of the Euler's equation. The third item F on the right is the external force attached to the liquid. The second item on the right is the sticky item introduced by stochastes. In other words, the Euler's equation is a special case where the viscosity coefficient of uncompressed fluids is 0. If you carefully observe the second item, you will find that it is very similar to the diffusion mode of hot motion.

(2) formula represents the non-compression of the fluid, which is the momentum conservation equation.

Although the NS equations are just an approximate description, it still makes a huge step forward in the theoretical fluid mechanics, which is the first peak moment in the history of fluid mechanics. The NS equations are a summary of the past fluid mechanics history and an astonishing prediction of fluid development in the future. Modern theoretical fluid mechanics research has taken NS equations as the original starting point.

Do you still remember the "Turbulence" that fascinated Da Vinci? The NS equation is only effective for stable fluids and cannot solve the "Turbulence" problem. The fluid problems solved by the NS equation are classified as the "laminar flow" of smooth flows, which corresponds to the "Turbulence. In 1883, Renault carefully studied the Motion Mechanism of the laminar flow and turbulence, deduced a number using the non-compressed NS equation, and used this number to determine whether the fluid is a laminar flow or a turbulent flow. The Renault number is the ratio of Inertia Force (constant linear motion without external force) to viscous force (fluid friction. When the number of Renault is relatively small, the viscosity force is greater than the inertia force, the flow rate of the disturbance will quickly collapse, the fluid movement is stable, is the laminar flow; on the contrary, the inertia force is greater than the viscosity force, small Disturbance will rapidly develop into a huge disturbance, forming irregular turbulence. However, today, turbulence is still difficult to portray, and the real Formation Mechanism of turbulence is still a mystery.

In 1904, German physicist Prandtl proposed the "Boundary Layer Theory ". When the number of Renault is large, the fluid should be turbulent, but he believes that it is still a laminar flow close to the edge of the fluid. By introducing the "Boundary Layer", you can simplify the NS equations. Another major contribution of mint is to unify the fluid theory school and water school. The "Boundary Layer Theory" is the product of the combination of theory and practice.

At the beginning of the 20th century, people began to study the air dynamics required by the aircraft. He started the wing Theory Based on fluid mechanics and told people why the air could send such a heavy plane to the sky. In 1911, Hungary Von Karman (von kámán) became a student at ant. He proposed the theory of "Karman vortex. When a blocking fluid is set in a fluid, a staggered scroll occurs downstream of the blocking fluid, just like street lights on both sides of the street.

Feng Karman set up the Jet Propulsion Lab at Caltech and Qian Xuesen became his student. They jointly studied the formula of the effect of compression on the surface pressure of the fluid in the subacoustic velocity flow, called the Karman-Qian Xuesen formula ". In his memoir Qian Xuesen and the Red China, Feng Karman commented: "One of the greatest geniuses in the U.S. rocket field is my outstanding door student ".

In 1955, Qian Xuesen returned to China as a result of cardinalism in the United States. Chinese writer Zhang chunru commented in Qian Xuesen: "The World's impression of Qian Xuesen is not based on his findings in the United States, but on the achievements of leading science in China after he was evicted. He is definitely a top-notch scientist, but people who have worked with him have repeatedly stressed to me that he is not qualified to be comparable to Newton or Einstein, not even as good as his mentor Feng Karman at Caltech. Despite his theoretical efforts and great value to the development of gas dynamics in the United States, he did not drive the revolution or create new fields. If he died in 1955 and never returned to China, his life would not be the first-class biography material ."

In 1961, American Meteorological scientist Lorenz cut the input data of six decimal places into three decimal places in order to save time when simulating atmospheric motion. He found that a 0.0001 input error could result in a completely opposite result. Lorenz has established a new mathematical model for climate. This model can produce infinite divergence results. He concluded that constant weather changes cannot be correctly predicted. In October 1979, lorunz delivered a "Butterfly Effect" speech, believing that a butterfly in Brazil can shake its wings and cause a tornado in Texas. This is the famous "Chaos theory ".

The development of fluid mechanics began with Archimedes and has gone through thousands of years of research. Some people say that the beginning of chaos theory is the end of classical science, but fluid mechanics is far from over. Even if modern people can use computers to perform complex operations, it is still impossible to simulate turbulence and chaos.

In 2000, the clay Institute of Mathematics announced the seven historic "Millennium Challenge" and promised to provide 1 million US dollars to anyone who can answer any question. The sixth one is the proof of the existence and smoothness of the NS equations, which is still unclaimed. Even a relatively simple Euler's equation cannot prove its existence at present.

Flying Birds, fluctuating waves, surging flames, snow and ice, humans are trying to find the ultimate answer behind them. I believe that one day, the masterpiece of nature will appear in front of us.

Phone Fluid Mechanics