Definition of fluid
If you want to study the properties of fluids, you must first define the fluid, that is, you must first understand what the fluid is. The most common fluids that can be observed and perceived in real life are water and air, which belong to liquids and gases. From the perspective of perceptual observation and analysis, the most significant feature of our common fluid is that it has a strong "inclusiveness", the ancients used "sea and rivers" to describe the vastness of the sea, and the air is everywhere. In contrast, solids always have a fixed shape in our general impression, and to change the shape of a solid, some "hard operations" are usually required, such as cutting. To sum up, the main difference between fluids and solids is that solids always have a certain shape, and it is not easy to change; Fluids, on the other hand, appear in various shapes, and pouring water into a cup is the shape of a cup, and pouring it on a table is a pool. At the same time, solids are often not easily inclusive, and it is almost impossible to place objects that do not belong to themselves in solids without damaging them.
After understanding the gap between solids and fluids in the popular sense, it is also necessary to describe them in scientific language, that is, write a scientific definition, here directly give the following definition of fluid:
An object that can undergo continuous deformation by a small tangential force (shear force) is a fluid
If you want to limit the discussion of the definition of fluid to this, it is too superficial, looking at the essence through the phenomenon, what kind of essential gap gives the fluid the ability to continuously deform when subjected to tangential force, while the solid can only be destroyed after a certain degree of force?
This provides a basic idea of understanding the phenomena of the macroscopic world from a microscopic perspective. Here it is pointed out that a significant difference between fluids and solids is pointed out: whether it is a liquid or a gas, the distance between the elementary particles that make them is much greater than that of a solid, that is, the combined relationship between the particles of the solid is closer The flow capacity of the fluid, that is, the ability to be sheared and deformed, comes from this microscopic property. This method of microscopic analysis has many applications in the study of compressible fluids and viscous fluids.
Classification of flows
Continuum versus Free Molecue Flow
Imagine a fluid flowing across the surface of an object, and in the vast majority of cases, the distance between the elementary particles in the fluid, i.e. the free path, is too small compared to the macroscopic size of the object. In other words, the object cannot distinguish the impact of different particles in the fluid, and the fluid is regarded as a continuous substance without interruption, and the flow at this time is called continuous flow.
Corresponding to continuous flow, if the distance between elementary particles is too large, the object can already perceive different impacts produced by different particles, and the flow is called free particle flow.
The vast majority of fluids in practical applications are considered to flow continuously, also known as the continuum hypothesis, and there are only very few cases where particle flow needs to be studied. At the same time, a flow in between the two has the characteristics of both, which is called a low-density flow, which will not be detailed here.
Inviscid vs. Viscous Flow
Basic physics tells us that the basic particles that make up matter are always in constant irregular thermal motion, and at the same time, particles will collide due to motion, and collisions will inevitably be accompanied by the exchange of mass, momentum, and energy, and fluids are no exception.
It is these microscopic exchange phenomena that cause mass diffusion, friction, and thermal conduction, respectively. In actual research, the fluid that cannot be ignored by such phenomena is called viscous flow, and in order to facilitate the study of fluids that "exchange phenomena" do not have too much impact on the research, we usually ignore them, which is the non-viscod flow.
Compressible versus Incompressibel flows
In flow, if the density of the fluid is considered constant, the flow is incompressible. Conversely, the flow of density as a variable is compressible. A detailed definition of compressibility is described in a later section.
In real life, any fluid is compressible to a certain extent, and there is no real completely incompressible fluid; However, for the convenience of research, fluids whose compressibility has little effect on their related properties are generally considered incompressible. Generally speaking, due to the small intermolecular spacing of liquids, its compressibility is generally much smaller than that of gases; Therefore, there is also a statement that liquids are incompressible and gases are compressible, which is obviously imprecise, but it is also true to a certain extent.
Mach Number Regimes
Flow has been classified by continuity, compressibility and viscosity, and although they all play an important role and importance in fluid mechanics, the well-known classification is defined around the fundamental physics of velocity.
Definition of flow rate
We can easily define the velocity of solids, because most solids do not deform in motion, and each part of the solid has the same velocity, and it is convenient to use the concept of particles to abstract the velocity of the solid as a whole into the concept of particles. However, in the face of fluids, which are easily deformed in flow, each part of the fluid may have a different flow rate. For example, the wind speed at the center of a tornado and the edge of the vortex is very different, and it is a world away from the largely stationary air in the distance, but they belong to the same continuous flow field. Therefore, how to define the speed of the flowing fluid is an important problem. Referring to the method of abstracting particles from solids, the concept of "point" is also used to define the flow rate; However, the point at this time no longer originates from the object, because the fluid acting on the object will change at this time; Taking the specific and unchanging spatial points in the space as the basis for defining the flow velocity, we define the flow velocity as follows:
For a fixed point in space, the velocity of the fluid microelement flowing through that point is the velocity of that point
Definition of Mach number
Among the many dimensionless parameters in fluid mechanics, the Mach number is the most well-known and indeed has a wide range of applications.
Definition of Mach number:
The ratio of the flow velocity at a point in the flow field to the local speed of sound is the Mach number Ma there
Ma=V/a
Regarding the definition of local sound velocity, leave it in the section on compressible fluids and then introduce it in detail, which is simply and intuitively understood as the propagation speed of sound.
Mach number criterion classification
Here's how flows are classified according to Mach numbers:
1. Subsonic flow: If the Mach number of any fluid in the flow field is less than 1, that is, the flow velocity is less than the local speed of sound, the flow at this time is called subsonic flow. Subsonic flow is characterized by smooth streamlines. It is worth noting that for flow fields containing solids (such as the wings of an aircraft), the disturbance to the flow caused by the presence of solids can propagate to the entire flow field because the flow velocity is less than the speed of sound. This is in line with our general perception that when an object falls into water, the resulting ripples always propagate upstream and downstream in an approximate circle. But don't take this for granted, when the flow velocity exceeds the speed of sound, the situation is very different.
2. Transonic flow: If there is a part of both supersonic (Ma>1) and subsonic (Ma<1) in the flow field, the flow at this time is called transsonic flow. Since solids are usually present in the flow field in actual research, and the disturbance of solids usually causes changes in the flow velocity of different parts of the flow field, that is, the fluid that originally flows at subsonic speed is likely to change locally to supersonic speed when flowing through the object, and vice versa. Therefore, transsonic flow is also very common in practical application research.
3. Supersonic flow: If the Mach number anywhere in the flow field is greater than 1, the flow is called supersonic flow. Unlike subsonic flow, various disturbances in supersonic fluids will produce shock waves, and the properties of the fluid will change drastically when flowing through the shock wave, and the presence of shock waves locally destroys the continuity of flow. Its **streamline** is no longer continuously smoothed. The discussion of shock waves is also discussed in compressible fluids.
4. Hypersonic flow: When the Mach number in the flow field is very large, the flow is hypersonic flow. The salient feature of hypersonic flow is that due to the high flow rate, the distance between the shock wave and the object boundary becomes extremely small, and there is a large number of viscous effects in the flow between the shock wave and the object boundary, and due to the high temperature, the elementary particles in the fluid begin to chemically react to produce new substances.
Forces and moments in aerodynamics
In aerospace-related professional applications, the fluid exposed to is mainly air, and the branch of fluid mechanics with air as the main research object is aerodynamics. In aerodynamics-related research, the most important thing is the force of the object in the flow field, from the paper airplanes we stack, to the design of the aircraft structure and even the internal structure of the aircraft engine.
The source of force and moment
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 will greatly interfere with the incoming air, and then form a complex flow field around the aircraft. However, the force analysis of the aircraft is very simple from a certain point of view, because no matter how complex the flow field is, there are only two sources of force on the objects in it:
1. Pressure distribution on the surface of the object
2. Distribution of tangential forces on the surface of the object
No matter how complex the surface of an object is, the forces and moments it receives come from the above two aspects; Among them, 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 part of the resistance of the object.
Discussion about the forces on the wing
Let's take the force of a two-dimensional wing as an example to discuss the force of an object in a flow field. Note that two-dimensional flow refers to the flow field of the fluid is two-dimensional, only two directions x and y, without considering the z direction, that is, taking a cross-section of the wing and ignoring its length for discussion.
A schematic diagram of the force on a two-dimensional airfoil is as follows:
The pressure and tangential force on the surface of the wing finally form a resultant force R acting on a certain force point on the wing, accompanied by the moment M corresponding to the point.
Typically, we decompose the resultant force R to define some common and commonly used components to facilitate research and visual analysis of physical phenomena. In most cases, the resultant force is broken down into two different component systems, namely axial and normal force, lift and drag.
Decomposition of the resultant force
Axial and normal forces: The component force parallel to the chord length of the wing (the straight line connecting the head and tail of the wing) is axial and denoted by A. Perpendicular to it is the normal force, denoted by N.
Lift and drag: The component force perpendicular to the direction of the flow is lift, expressed in L. The component force perpendicular to it parallel to the direction of the incoming flow is the resistance, denoted by D.
Among them, the angle between the direction of chord length and the direction of incoming flow is called the angle of attack.
As can be seen from the above figure, the angle of attack is still the angle between L and N, D and A, and the transformation relationship between the two component systems can be easily obtained through the geometric relationship:
It can be seen that the lift and drag of a wing can be derived from the axial and normal forces it is subjected to, as well as the angle of attack. So how do you calculate axial and normal forces? As mentioned above, the source of any force is pressure and tangential force, so establish the coordinate system shown in the figure below for the two-dimensional wing and analyze the force on its surface:
In the figure, the x-axis divides the wing along the chord length into upper and lower parts, the lower surface corresponding to the subscript is u, the lower surface is l, where and represent the arc length from the leading edge point of the wing (that is, the coordinate origin) around the upper or lower surface to a certain point on the surface of the wing, the purpose of this setting is for the convenience of the next integration.
Observe any point on the surface of the wing in the figure where the forces are pressure ( or ) and tangential forces or ). The force at this time is the microsurface surface, that is, the force per unit area (length), so it is expressed in lowercase letters. All these forces represented by letters are not fixed, and may change with different functions due to different conditions in different problems, and an important task of fluid mechanics is to analyze and calculate the distribution function of these forces in different situations.
Due to the irregularity of the surface of the wing, and the pressure always follows the normal direction of the surface and the tangential force is always tangent to the surface, their direction changes with the geometry of the surface, as shown in the figure above. In order to conveniently indicate the direction, make the vertical and horizontal dashed lines as the reference lines of the direction, so that the angle between the pressure and tangential force and the reference direction is, and at the same time stipulate that the corresponding force is positive when it reaches the corresponding force clockwise from the dashed line (note that all should be acute angles, and positive and negative are judged by the direction of rotation).
Now, assuming that all the forces and angles mentioned above are known, and the geometry of the wing is determined, then we are ready to calculate the axial force and normal force on the surface of the wing, and we can calculate the lift and drag by the angle of attack.
The figure above shows that the two-dimensional airfoil and other cross-sections extend to the three-dimensional wing, that is, the wing with unchanged section in the z direction, now the force of the three-dimensional wing is calculated by 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 are integrated on the entire wing, the specific derivation is as follows:
For the upper surface:
For the lower surface:
The superscript of the letter representing the force represents the unit span, i.e. the length along the z-axis is 1.
Integral for the entire wing yields:
From this, the general method for calculating axial and normal forces is derived.
The moment on which the wing is subjected
Using similar integrals to the idea of calculating the moment experienced by a wing. From the basic theoretical mechanical knowledge, we can know that the torque at different points in the object is different, here we select the leading edge point of the wing shape as the force point, and at the same time specify that the direction of the moment that increases the angle of attack is positive, as shown in the figure below:
When analyzing the force, first write the leading edge point moment in differential form by the force per unit arc length:
For the upper surface:
For the lower surface:
Note that the above equation is written in Cartesian coordinates, so y is negative when calculating the lower surface.
Integrate and add the above two equations to get:
For the above formula, if the surface shape of the object is known, it can be expressed as a function of the arc length s, and the force and moment of the object can be calculated as long as it is solved.
Relevant dimensionless parameters
For ease of study, dimensionless parameters are often used instead of physical quantities with units. Using dimensionless parameters can not only remove the trouble of units (note that a uniform unit system should be used when calculating dimensionless coefficients), but also make more intuitive comparisons of some properties.
First, define a quantity with dimensions, the dynamic pressure of the incoming flow:
The subscript ∞ of each parameter represents the parameter that flows from infinity. Note that the dimension of the dynamic pressure is , and the dimension of the force is , and the difference between the two is . Therefore, the dimensionless coefficients corresponding to forces and moments in aerodynamics are defined as follows:
Lift coefficient:
Drag coefficient:
Normal force coefficient:
Axial force coefficient:
Torque coefficient:
where S and l are the characteristic area and characteristic length set to dimensionless forces and moments, respectively. Their selection depends on the geometry of the object studied in the flow field, the specific value is not critical, they are just a tool for dimensionlessness, the key is to clarify which geometric feature of an dimensionless parameter is the characteristic area and feature length, and to maintain consistency, the important thing is which one and not how much. The following is a scheme for the characteristic area and length of the wing and cylinder, respectively, as a reference:
Note that the above discussion is all based on three-dimensional flow, and dimensionless coefficients can be similarly defined for two-dimensional flow:
The characteristic area at this time S=c(1)=c
Add two dimensionless coefficients based on pressure and tangential force:
With the above dimensionless parameters, we can also replace the previously obtained calculation formulas for forces and moments to dimensionless form, for the three-dimensional wing shown in the figure below:
Bring in the dimensionless form formula:
Similarly, it is possible to formulate the transformation of the decomposition of two forces into dimensionless forms:
Determination of the force point
So far, we have detailed the source, direction and calculation method of the force on the object studied in aerodynamics, and the final question about force analysis is, where is the point of action of the resultant force on the object? To determine the action point of the resultant force, it is necessary to firmly grasp its characteristics, that is, the moment of the resultant force acting on the leading edge point should be exactly equal to the force balance of the whole system without additional torque.
As shown in the figure above, assuming that the resultant force acts horizontally from the leading edge point, then:
Note that the negative sign in the equation indicates that the moment of the normal force shown in the figure at the leading edge point is negative, while , defaults to positive. In addition, the situation shown in the figure assumes that the axial force coincides with the chord length, that is, the force system acts on the chord length, if the action point is not on the chord length, refer to the above method to set up.
If it is assumed that the action point of the force is not at the actual action point, that is, the force system is moved, the corresponding torque can be added directly.
In Figure 1 above, the force directly acts on the leading edge point, and the torque on the leading edge point is zero, so the torque of , needs to be supplemented; Figure 2 puts the force system at , and it is necessary to supplement the moment at the moment where the pressure and tangential force are located, ; Figure 3 places the force directly on its actual point of action without any additional torque.
In general, the force system can act at any point, as long as the pressure on the surface of the object and the moment of tangential force acting at this point can be supplemented.