Definition of fluidsTo study the properties of a fluid, one must first define the fluid, that is, first understand what the fluid is. The most common fluids that can be observed and perceived in real life are water and air, which are classified as liquids and gases

Definition of fluids

To study the properties of a fluid, one must first define the fluid, that is, first understand what the fluid is. The most common fluids that can be observed and perceived in real life are water and air, which are classified as liquids and gases. From the perspective of perceptual observation and analysis, the most prominent feature of our common fluids is their strong "inclusiveness". The ancients used the phrase "the sea embraces all rivers" to describe the vastness of the sea, while air is ubiquitous. On the contrary, solids always have a fixed shape in our general impression, and changing the shape of solids usually requires some "hard operations" such as cutting. In summary, the main difference between fluids and solids is that solids always have a certain shape and are not easy to change; However, fluids appear in various shapes. Pouring water into a cup is the shape of the cup, and pouring it onto a table is a puddle. Meanwhile, solids are often not easily inclusive, and it is almost impossible to place objects that do not belong to them within the solid without causing damage.

aboutDefinition of fluids

An object that can undergo continuous deformation under a small tangential force (shear force) is a fluid

If we want to limit the discussion of fluid definition to this, it would be too shallow. Looking at the essence through phenomena, what kind of essential difference gives fluids the ability to undergo continuous deformation when subjected to tangential force, while solids can only fail after reaching a certain degree of force?

Here is a basic approach to understanding the phenomena of the macro world from a micro perspective. Here is a significant difference between fluid and solid:Whether it is liquid or gas, the distance between the elementary particle that make up them is far greater than that of the solid, that is, the combination and connection between the particles of the solid should be closerThe flow ability of fluids, that is, the ability to undergo shear deformation, comes from this microscopic characteristic. This method of analyzing from a microscopic perspective has many applications in the study of compressible and viscous fluids.

Classification of Flows

ContinuumVersusFreeMolecueFlow

Imagine that the fluid flows through the surface of an object. In most cases, due to the distance between the elementary particle in the fluid, namelyFree pathCompared to the macroscopic size of objects, the order of magnitude is too small. Or rather, objects cannot distinguish the collisions caused by different particles in the fluid, and consider the fluid as an uninterrupted and continuous substance. The flow at this time is calledContinuous flow

Continuous flowFree particle flow

Continuous flowContinuum assumptionIn very few cases, it is necessary to study particle flow. At the same time, a flow between the two has both characteristics, and is calledLow density flowWe won't go into detail here.

InviscidVersusViscousFlow

Basic physical knowledge tells us that the elementary particle that make up matter are always moving irregularly in the process of heat, and at the same time, particles will collide due to movement, and the collision will inevitably be accompanied by the exchange of mass, momentum, and energy, and fluid is no exception.

It is precisely these micro level exchange phenomena that have caused the macro levelMass diffusion, viscous friction, and thermal conduction In practical research, fluids that cannot be ignored for such phenomena are calledViscousflowIn order to facilitate the study of fluids that do not have a significant impact on the "exchange phenomenon", we usually ignore them, which isInviscidflow

Compressible Versus Incompressible flows

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In real life, any fluid is to some extent compressible, and there is no truly incompressible fluid; But for the convenience of research, the fluid whose compressibility has little effect on its related properties is usually regarded as incompressible fluid. Generally speaking, the compressibility of liquid is generally much smaller than that of gas due to the small spacing between molecules; Therefore, there are also claims that liquids are incompressible while gases are compressible, which is obviously not rigorous but to some extent true.

MachNumberRegimes

Passed on topContinuity, compressibility and viscosityFlows are classified. Although they all play an important role in fluid mechanics, the well-known classification method focuses onVelocityThis basic physical property is defined.

Definition of flow rate

We can easily define the velocity of a solid because the vast majority of solids do not deform during motion, and each part of the solid has the same velocity. It is convenient to use the concept of particles to abstract the velocity of the solid as a whole into the concept at this point. But in the face of fluids, which are prone to deformation during flow, each part of the fluid may have different flow velocities. There is a significant difference in wind speed between the center of a tornado and the edge of a vortex, which is vastly different from the basically stationary air in the distance. However, they all belong to a continuous systemFlowfield It can be seen that how to define the velocity of a flowing fluid is a very important issue. Referring to the method of abstracting particles from solids, the concept of "point" is also used to define flow velocity; However, at this point, the point no longer originates from the object, as the fluid acting on the object will change; Using specific and unchanging spatial points in space as the basis for defining flow velocity, we define the flow velocity as follows:

For a certain point in space, the velocity of the fluid element passing through this point is the velocity at that point

Definition ofMach number

Among the many dimensionless parameters in fluid mechanics,Mach number is the most widely known and indeed has a wide range of applications.

Definition ofMach number:

The ratio of the flow velocity at a certain point in the flow field to the local sound velocity is theMach number at that pointMa

Ma=V/a

aboutLocal sound speedThe definition of sound can be explained in detail in the section on compressible fluids, which can be simply and intuitively understood as the propagation speed of sound.

Mach number criterion classification

Below is the specific content of classifying flows based on Mach numbers:

oneSubsonic flow1Local sound speedstreamline It is worth noting that for flow fields containing solids (such as aircraft wings), the interference generated by the presence of solids can propagate to the entire flow field due to the flow velocity being less than the speed of sound. This is in line with our general understanding that when an object falls into water, the ripples generated always propagate in an approximately circular shape both upstream and downstream. But don't take this for granted. When the flow rate exceeds the speed of sound, the situation becomes very different.

twoTransonic flowIf there are both supersonic (Ma> 1) and subsonic (Ma< 1) parts in the flow field, the flow at this time is called transonic flow. Due to the fact that solids usually exist in the flow field in practical research, and the disturbance of solids usually causes changes in the flow velocity of different parts of the flow field, that is, fluids that were originally subsonic flow are likely to become supersonic locally when flowing through objects, and vice versa. So transonic flow is also very common in practical application research.

threeSupersonic flow1******streamline**about

fourHypersonic flowWhen the Mach number in the flow field is very high, the flow is hypersonic. The remarkable characteristics of hypersonic flow are that the distance between the shock wave and the boundary of the object becomes very small due to the high velocity, and there are a lot of viscous effects in the flow between the shock wave and the boundary of the object, and because of the high temperature, the elementary particle in the fluid begin to react chemically to produce new substances.

Forces and Moments in Aerodynamics

In the professional applications related to aerospace, the fluid contacted is mainly air, and the branch of fluid mechanics with air as the main research object is aerodynamics. In the research of aerodynamics, the most important thing is the force on the object in the flow field, which is relied on by the paper plane we folded, the design of the aircraft structure and even the internal structure of the aeroengine.

Sources of force and torque

The force analysis of aircraft flying in the sky seems to be very complex, including the complex shape of the nose, fuselage, wings, and other parts, which can cause significant interference with the incoming air, thereby forming a complex flow field around the aircraft. However, the force analysis of aircraft is quite simple from a certain perspective, because no matter how complex the flow field is, there are only two sources of force applied to objects within it:

one

two

No matter how complex the surface of an object is, the forces and moments it receives come from the above two aspects; The pressure is always perpendicular to the surface of the object, and the tangential force is always tangent to the surface of the object and is the source of friction, which is an important component of the resistance suffered by the object.

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Let's take the force acting on a two-dimensional wing as an example to discuss the force acting on an object in a flow field. Note that two-dimensional flow refers to the flow field of a fluid being two-dimensional, with only x and y directions, without considering the z direction, that is, taking a cross-section of the wing and ignoring its length for discussion.

The force diagram of a two-dimensional wing is as follows:

The pressure and tangential force on the wing surface ultimately form a combined force R that acts on a certain force point on the wing, accompanied by the corresponding moment M at that point.

Usually, we decompose the resultant force R to define some commonly used components, in order to facilitate research and intuitive analysis of physical phenomena. In most cases, the combined force is decomposed into two different sets of component systems, namelyAxial force and normal forceLift and drag

Decomposition of resultant force

Axial force and normal forceThe component force parallel to the chord length of the wing shape (the straight line connecting the head and tail of the wing shape) is expressed in axial direction as A. The normal force perpendicular to it is represented by N.

Lift and dragThe component perpendicular to the direction of the incoming flow is called lift, denoted by L. The component force perpendicular to it and parallel to the direction of the incoming flow is called resistance, denoted by D.

The angle between the chord length direction and the inflow direction is calledAngle of attack

From the above figure, it can be seen that the angle of attack is still the angle between L and N, and D and A. Through geometric relationships, the conversion relationship between the two force systems can be easily obtained:

Lift and dragAxial force and normal forceAxial force and normal force

The x-axis in the figure divides the wing shape into upper and lower parts along the chord length, with the corresponding subscript u on the upper surface and l on the lower surface, whereandRepresenting the arc length from the leading edge point of the airfoil (i.e. the coordinate origin) around the upper or lower surface to a certain point on the airfoil surface, this setting is for the convenience of integration in the future.

Observe any point on the wing surface in the figure, and the force it is subjected to is pressureorAnd tangential forceor) The force at this point is the force on the surface of the element, which is the force per unit area (length), so it is expressed in lowercase letters. All the forces represented by letters are not fixed values, and may change with different functions due to different conditions in different problems. An important task of fluid mechanics is to obtain the distribution functions of these forces in different situations through analysis and calculation.

Due to the irregularity of the wing surface and the fact that the pressure always follows the normal direction of the surface while the tangential force is always tangent to the surface, their direction will change with the geometric shape of the surface, as shown in the above figure. For the convenience of indicating the direction, dashed lines in the vertical and horizontal directions are made as reference lines for the direction, so that the angle between the pressure and tangential force and the reference direction isAt the same time, it is specified that when starting from the dashed line and reaching the corresponding force clockwisePositive value (note allBoth should be sharp angles, and positive or negative can be determined by the direction of rotation.

Lift and drag

The above figure shows a two-dimensional wing shape with a constant cross-section extending into a three-dimensional wing, that is, a wing with a constant cross-section along the z-direction. The force acting on the three-dimensional wing is now calculated through integration, where the length along the z-direction is 1,The three-dimensional surface corresponding to the unit two-dimensional curve, as long as the pressure and tangential force areBy integrating on the entire wing, the specific derivation is as follows:

For the upper surface:

For the lower surface:

The superscript of the letter representing force represents the unit span, which means the length along the z-axis is 1.

Integrating the entire wing yields:

Axial force and normal force

The torque acting on the wing

The idea of similar integration can be used to calculate the torque borne by the wing shape. From the basic knowledge of Theoretical Mechanics, we can know that the moment at different points in the object is different. Here we select the leading edge point of the wing as the force point, and specify the direction of the moment that increases the angle of attack as the positive direction, as shown in the following figure:

The same as the force analysis, the moment at the leading edge of the differential form is first written by the force on the unit arc length:

To the upper surface:

To the lower surface:

Note that the above formula is written in the Cartesian coordinate system system, so y is negative when calculating the lower surface.

Integrate and add the above two equations to obtain:

For the above formula, if the surface shape of the object is known, thenCan be expressed as a function of arc length s, as long as we findThe force and torque of the object can be calculated.

Related dimensionless parameters

For the convenience of research, dimensionless parameters are often used instead of physical quantities with units. The use of dimensionless parameters can not only eliminate the hassle of units (note that a unified unit system should be used when calculating dimensionless coefficients), but also provide a more intuitive comparison of some properties.

Firstly, define a dimensional quantity, namely the dynamic pressure of the incoming flow:

The subscript of each parameter represents the parameter of the flow from infinity. Note that the dimensions of dynamic pressure areAnd the dimension of force isThe difference between the two Therefore, the dimensionless coefficients corresponding to forces and moments in aerodynamics are defined as follows:

Lift coefficient:

Resistance coefficient:

Normal force coefficient:

Axial force coefficient:

Moment coefficient:

Among them, S and l are set for dimensionless force and torque, respectivelyFeature area and feature lengthFeature area and feature length

Note that the above discussion is all based on three-dimensional flow, and similarly, the dimensionless coefficients under two-dimensional flow can be defined:

At this point, the characteristic area S=c (1)=c

Add two more dimensionless coefficients based on pressure and tangential force:

With the above dimensionless parameters, we can also replace the previously obtained formula for calculating forces and moments with a dimensionless form for the three-dimensional wing shown in the following figure:

Bring in the dimensionless form formula:

Similarly, the conversion formula for the decomposition of two forces can also be transformed into a dimensionless form:

Determination of force point

aboutTo achieve force balance of the entire system without additional torque.

As shown in the above figure, assuming that the combined force acts in the horizontal direction from the leading edge pointThen:

The negative sign in the equation represents that the torque of the normal force shown in the figure at the leading edge point is negative, and, default to positive. In addition, the situation shown in the figure assumes that the axial force coincides with the chord length, meaning that the force system acts on the chord length. If the point of action is not on the chord length, refer to the above method for settingThat's it.
If the action point of the pseudo force is not at the actual action point, that is, to move the force system, simply add its corresponding torque.

In Figure 1 of the above figure, the force directly acts on the leading edge point, and the torque to the leading edge point is zero, so it needs to be supplementedThe torque of; Figure 2 puts the force system in placeAt, it is necessary to supplement the torque of the pressure and tangential force atMoment at, Figure 3 directly places the force at its actual point of application without the need to add any torque.

In general, a force system can act at any point by supplementing the pressure on the surface of the object and the torque exerted by the tangential force at this point.

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