What is Fluid Dynamics?

Hydrodynamics is a branch of fluid mechanics. It studies the motion of fluid as a continuous medium under force and its interaction with boundaries. Broadly speaking, the research also includes the interaction of fluids and other motion patterns. The difference between fluid dynamics and hydrostatics lies in the former studying fluids in motion; the difference between fluid dynamics and fluid kinematics lies in the former considering forces acting on fluids. Fluid dynamics includes liquid dynamics and gas dynamics. Its research methods, like fluid mechanics, include theory, calculation and experiment. The three methods complement each other and promote each other.

Chinese name: Hydrodynamics
Foreign name: fluid dynamics

Attributes: Branch of fluid mechanics
Technical principle: Conservation of mass and conservation of momentum
Composition: Inviscid incompressible fluid dynamics,

Principles of Fluid Dynamics Technology

The basic axioms of fluid dynamics are conservation laws, especially conservation of mass, conservation of momentum (also known as Newton's second and third laws), and conservation of energy. These conservation laws are based on classical mechanics and have been modified in quantum mechanics and general relativity. They can be represented by Reynolds transport theorem.

In addition to the above, the fluid is also assumed to adhere to the "continuum assumption". Fluids are composed of molecules that collide with each other and with solids. However, the continuity hypothesis considers that the fluid is continuous rather than discrete. Therefore, properties such as density, pressure, temperature, and velocity are all considered to be well-defined at infinitely small points and continuously change from one point to another. The fact that fluids are made up of discrete molecules is ignored.

If the fluid is dense enough to become a continuum without ionized components and the speed is very slow relative to the speed of light, then the momentum equation of the Newtonian fluid is the "Navier-Stokes equation". It is a non-linear differential equation, which describes that the stress of a fluid flow is linearly dependent on velocity and pressure. The unsimplified Navier-Stokes equation does not have a general closed form solution, so it can only be used in computational fluid dynamics, otherwise it needs to be simplified. The equation can be simplified in many ways to make it easy to solve. Some of these methods allow closed-form solutions to suitable fluid mechanics problems.

In addition to the conservation equations for mass, momentum, and energy, there are also thermodynamic equations of state that make pressure a function of other thermodynamic variables of the fluid, and the problem can be limited.

Hydrodynamic composition

The branch of fluid mechanics that studies the laws of moving fluids and the interaction between moving fluids and boundaries. The main contents of fluid dynamics include: basic equations of fluid dynamics, inviscid incompressible fluid dynamics, viscous incompressible fluid dynamics, aerodynamics, and turbine mechanical aerodynamics.

Type of flow: steady flow, unsteady flow

Flow patterns: laminar, turbulent

Flow stability: incompressible flow, compressible flow, viscous flow, inviscid flow

Hydrodynamic research point

Hydrodynamic stress tensor

Based on the fact that the inviscid fluid has no resistance to shear deformation and the stationary fluid cannot withstand shear stress, it can be asserted that in the inviscid or stationary fluid, the shear stress is zero, and the normal stress (that is, the normal stress) p _xx = p _yy = p _zz = p. p is called the pressure function of non-viscous fluid or stationary fluid, and it characterizes the stress state of non-viscous fluid or stationary fluid at any point. In fluid dynamics, one can use p _x , p _y , p _z or a combination of nine quantities p _ij (i, j = 1, 2, 3) to fully describe the stress situation at one point. The second-order tensor composed of p _ij is called stress tensor.

Relationship between hydrodynamic stress tensor and deformation rate tensor

Newton's law of viscosity

Applies only to shear flows (see Newtonian fluids). For general flow, suppose: (1) The stress tensor of the moving fluid tends to the stress tensor of the stationary fluid after the motion stops, so

Where p _ij is the stress tensor; p is the pressure; _ij is the Kroneck symbol;

Is the stress tensor (2) Each component of the stress tensor is the velocity gradient tensor

Linear homogeneous function of each component (this assumption is a logical generalization of the Newtonian viscosity formula); (3) the fluid is isotropic. From this, the relationship between the stress tensor and the deformation rate tensor s _ij can be deduced:

Hydrodynamic Momentum and Energy Equations

The momentum equation is a mathematical expression of conservation of momentum, and its vector form is:

Where v is the velocity vector; F is the mass force acting on the unit mass; p is the pressure; and are the fluid density and the dynamic viscosity coefficient, respectively. The above formula shows that the inertial force per unit volume is equal to the mass force per unit volume plus the pressure gradient and viscous stress on the unit product. The energy equation is a mathematical expression of energy conservation, which can be written as:

Where T and s are the thermodynamic temperature of the fluid and the entropy per unit mass of the fluid; k is the thermal conductivity; q is the heat transferred into the unit mass of fluid in unit time due to radiation or other reasons; is the viscosity loss function, Its expression is

Dynamics of Hydrodynamic Vortex

The dynamic properties of vortex are mainly reflected in Kelvin's theorem and Helmholtz's theorem. If the fluid is non-viscous, positive pressure (see positive pressure fluid), and the external force has potential, the vortex will not be immortal, and the vortex and vortex tube are always composed of the same fluid particle, and the strength of the vortex tube does not change with time. . Only the three factors of the fluid's viscosity, baroclinity and external force can make the vortex produce, develop, change and die.

Hydrodynamic Bernoulli and Lagrangian integrals

Non-viscous, positive-pressure fluids can be integrated under constant and non-rotating special cases under the action of potential external forces. These two first integrals of the equation of motion are called Bernoulli integrals (see Bernoulli's theorem) and Lagrange integrals. They (especially Bernoulli integrals) are very useful in the theoretical study or practical application of fluid mechanics.

Hydrodynamic momentum theorem

For most fluid mechanics problems, in order to understand the situation of the entire flow field, the basic equations of fluid mechanics in differential form need to be solved under certain initial conditions and boundary conditions. However, sometimes you only need to know some global characteristic quantities (such as the reaction force of the fluid to the object moving in it and the energy loss of the entire flow system, etc.), you can use the global theorem in the system of integral form equations- The Momentum Theorem and Momentum Moment Theorem can directly find the characteristic quantities of interest according to the given flow parameters on the boundary, without the need to solve differential equations. The above method is simple and easy to implement and has a wide range of applications in fluid dynamics.

Main types of fluid dynamics

Compressible and incompressible streams

All fluids are compressible to some extent, in other words, changes in pressure or temperature cause changes in fluid density. However, in many cases, changes in density caused by changes in pressure or temperature are quite small and can be ignored. Such fluids can be modeled with incompressible flows, otherwise they must be described using the more general compressible flow equation.

Mathematically speaking, incompressibility means that the density of a fluid remains constant as it flows, in other words: where D / Dt is a convective derivative. This condition can simplify many equations describing fluids, especially when applied to fluids of uniform density.

The Mach number is a measure of whether a gas is compressible or not. Roughly speaking, when the Mach number is less than about 0.3, it can be explained by the behavior of incompressible flow. As for liquids, whether they are more compressible or incompressible depends on the nature of the liquid itself (especially the critical pressure and temperature of the liquid) and the conditions of the fluid (whether the liquid pressure is close to the liquid critical pressure). Acoustic problems often need to introduce compressibility considerations, because acoustic waves are compressible waves, and their properties will change with the propagation medium and pressure changes.

Viscous and non-viscous flows

Hydrodynamics When the resistance in the fluid is greater, the viscosity must be considered when describing the fluid. The Reynolds number can be used to estimate the effect of the viscosity of a fluid on the description problem. The so-called Stokes flow refers to the flow with a relatively small Reynolds number. In this case, the inertia of the fluid is negligible compared to the viscosity. A large Reynolds number of a fluid indicates that the inertia is greater than the viscosity when the fluid is flowing. Therefore, when the fluid has a large Reynolds number, assuming it is a non-viscous flow, ignoring its viscosity can be regarded as an approximation. Such an approximation can get good results when the Reynolds number is large. Even in some issues that have to be considered sticky (such as boundary issues). At the boundary between the fluid and the tube wall, there is a so-called non-slip condition, and there will be a large rate of strain rate locally, which makes the viscosity effect amplified and vorticity, so the viscosity cannot be ignored. Therefore, to calculate the net force of the pipe wall on the fluid, a viscosity equation is required. As illustrated by the Darlene's fallacy, objects cannot feel force in a non-viscous stream. The Euler equation is a standard equation for describing inviscid flows. In this case, a commonly used model uses the Euler equation to describe fluids far from the boundary, and at the boundary of contact, the boundary layer equation is used. By integrating the Euler equation on a certain streamline, the Bernoulli equation can be obtained. If the fluid is vortex-free everywhere, the Bernoulli equation describes the entire flow.

Steady and unsteady flow

Flows in which the speed and pressure of a fluid change over time are called unstable flows. The speed and pressure of an unstable flow must consider not only the position but also the effect of time. A flow in which neither the speed nor the pressure of the fluid changes over time is called a steady flow.

Laminar turbulence

When flow is dominated by vortices and apparent randomness, this flow is called turbulence. When the turbulence effect is not obvious, it is called laminar flow. However, it is worth noting that the presence of vortices in the flow does not necessarily mean that the flow is turbulent-these phenomena may also exist in laminar flow. Mathematically, the turbulent flow is usually expressed by the Reynolds separation method, that is, the turbulent flow can be expressed as the sum of the stable flow and the disturbance part. The turbulence follows the Navier-Stokes equation. Direct numerical simulation (DNS), based on the Navier-Stokes equation, can be applied to incompressible flows, and Reynolds numbers can be used to simulate turbulent flows (the computer performance and the accuracy of calculation results must be capable of being loaded. Conditions). The results of this direct numerical solution method can explain the experimental data obtained.

However, most of the flows we are interested in are that the Reynolds number is much larger than the range that DNS can simulate. Even if the computer performance continues to develop in the next few decades, it is still difficult to implement simulations. Any flying vehicle should be able to carry a person (L> 3 m) at a speed of 72 km / h (20 m / s), which is far beyond the range that DNS can simulate (Reynolds number is 4 million). Flying tools like the Airbus A300 or Boeing 747 have more than 40 million Reynolds on the wings (based on the chord). In order to be able to deal with these practical problems in life, a turbulent flow model needs to be established. The Reynolds-averaged Navier-Stokes equations combine the effects of turbulence to provide a model of turbulence that expresses additional momentum transfer caused by Reynolds stress; however, turbulence It will also increase the speed of heat and mass transfer. Large eddy simulation (LES) is also a simulation method. Its appearance is similar to that of detached eddy simulation (DES). It is a combination of turbulence simulation and large eddy simulation.

Fluid dynamics applications

Applications of fluid dynamics The object of fluid dynamics research is the state and law of fluids in motion (fluid means liquid and gas). Small disciplines under fluid dynamics include aerodynamics (for studying gases) and hydrodynamics (for studying liquids)

Fluid dynamics is a sub-discipline of fluid mechanics.

Fluid dynamics has many applications in predicting the weather, calculating the forces and moments experienced by aircraft, and the flow rate of oil in oil pipelines. Some of these principles are even used in traffic engineering. Transportation itself is regarded as a continuous fluid. To solve a typical hydrodynamic problem, many characteristics of the fluid need to be calculated, including speed, pressure, density, and temperature.

Hydrodynamic References

1. terms of: Wang Wu a "Encyclopedia of China", 74 (second edition) physics entries: Fluid Mechanics: Encyclopedia of China Publishing House, 2009-07: page 263-264

2. GK Batchelor, An Introduction to Fluid Dynamics, Cambridge Univ. Press, London, 1970.

3 L. Planter et al., Guo Yonghuai, Lu Shijia, Translation: Introduction to Fluid Mechanics, Science Press, Beijing, 1981. (L. Prandtl, et al., Führer Dvrch die Strömungslehre, Friedr. Vieweg und Sohn, Braunschweig, 1969.)

4. Edited by Wu Wangyi: Fluid Mechanics, Peking University Press, Beijing, 1982.