In Physics, What Is a Boundary Layer?

The boundary layer is a thin flow layer that cannot be ignored due to the viscous force that is close to the object surface in a high Reynolds number flow. This concept was first proposed by Ludwig Prandtl, German, in 1904, the founder of modern fluid mechanics. Since then, boundary layer research has become an important subject and field in fluid mechanics.

The boundary layer is a thin flow layer that cannot be ignored due to the viscous force that is close to the object surface in a high Reynolds number flow. This concept was first proposed by Ludwig Prandtl, German, in 1904, the founder of modern fluid mechanics. Since then, boundary layer research has become an important subject and field in fluid mechanics.

Chinese name

Boundary layer

Foreign name

boundary layer

Other name

Flow boundary layer, surface layer

Presenter

Ludwig Prandtl

Presentation time

1904

Introduction to Boundary Layer

If a fluid with very low viscosity (such as water, air, etc.) is in contact with an object and has relative movement at a large Reynolds number, the thin fluid layer near the object surface will be reduced in velocity due to viscous shear stress; the fluid close to the object surface Adhesion to the surface, relative speed to the surface

Schematic diagram of parallel flow without angle of attack along the boundary layer of a slab.

Equal to zero; with the object facing up, the latitude of each layer gradually increases until it is equal to the free flow velocity. L-Plante referred to this thin layer of fluid deceleration from the object side as the boundary layer. Figure 1 is a schematic diagram of the boundary layer of a parallel flow without an angle of attack along a flat plate.

From the surface of the object, the fluid velocity rapidly increases to the local free-flow velocity, that is, the velocity corresponding to the ideal bypass flow.

Temperature boundary layer

Generally the same magnitude as the incoming flow speed. Therefore, the normal velocity of the boundary layer in the direction perpendicular to the surface has a large gradient. Even if the viscosity of the fluid is not large, such as air, water, etc., the viscous force is still large relative to the inertial force, which plays a significant role, so it is viscous flow. Outside the boundary layer, the velocity gradient is small, the viscous force can be ignored, and the flow can be regarded as non-viscous or ideal flow. At high Reynolds numbers, the boundary layer is very thin and its thickness is much smaller than the length along the flow direction. According to the magnitude comparison of scale and velocity change rate, the Navier-Stokes equation can be simplified into a boundary layer equation. When solving the high Reynolds number flow problem, the flow can be divided into two parts: the viscous flow in the boundary layer and the ideal flow outside the boundary layer. The boundary layer includes laminar, turbulent, mixed flow, low speed (incompressible), high speed (compressible), and two-dimensional and three-dimensional. Because viscosity is closely related to heat conduction, in addition to the velocity boundary layer, there is also a temperature boundary layer in high-speed flows. (The picture shows the relationship between the boundary layer in water and the friction resistance)

A brief history of the boundary layer

At the end of the nineteenth century, the science of fluid mechanics began to develop in two directions, and these two directions have virtually nothing in common. One direction is theoretical fluid dynamics, which is from Euler, a frictionless, viscous fluid. The equation of motion was developed and reached a high degree of perfection. However, due to the obvious contradiction between the results of this so-called classical fluid dynamics and experimental results-especially the very important issue of pressure loss in pipes and channels and the resistance of moving objects in fluids-this is up to Lambert was paradoxical. Because of this, practical engineers have developed a highly empirical discipline, that is, hydraulics, in order to solve important problems that arise in the rapid development of technology. Hydraulics is based on a large amount of experimental data, and it is very different from theoretical fluid dynamics in terms of methods and research objects.

At the beginning of the twentieth century, L. Prandtl was famous for solving how to unify these two branches of fluid dynamics that run counter to each other. He established a close connection between theory and experiment, and paved the way for the exceptionally successful development of fluid mechanics. Even before Prandtl, people have realized that in many cases, the results of classical fluid dynamics are inconsistent with experimental results because the theory ignores fluid friction. Moreover, the complete equation of motion (Navier-Stokes equation) with frictional flow has long been known. However, because solving these equations is mathematically difficult (except for a few special cases), the path to viscous fluid motion is theoretically impeded. In addition, in the two most important fluids, water and air, due to their low viscosity, in general, the forces produced by viscous friction are much smaller than other forces (gravity and pressure). Because of this, it is difficult to understand how the friction force, which is ignored by classical theory, can affect the motion of fluids to such a great extent.

In the paper "Fluid Motion with Small Friction" presented at the Heidelberg Mathematical Symposium in 1904, L. Prandtl pointed out that it is possible to accurately analyze viscous flows that occur in some very important practical problems. With the help of theoretical research and a few simple experiments, he proved that the flow around a solid can be divided into two regions: one is a thin layer (boundary layer) near the object, where friction plays a major role; the other is the layer Outside the rest of the area, friction here is negligible. Based on this assumption, Prandtl successfully gave a physically thorough explanation of the significance of viscous flows, while simplifying the corresponding mathematical difficulties to the greatest extent. Even at the time, these theoretical arguments were supported by some simple experiments performed in water holes built by Prandtl himself. So he took the first step in reunifying theory and practice. Boundary layer theory has proved extremely effective in providing an effective tool for the development of fluid dynamics. Since the 20th century, with the development of the newly developed aerodynamics discipline, the boundary layer theory has been rapidly developed. In a very short time, it has become one of the cornerstones of modern fluid mechanics, along with other very important advances (wing theory and aerodynamics).

Boundary layer boundary layer thickness

Boundary layer velocity boundary layer thickness

The distance in the boundary layer starting from the object plane (local velocity is zero) and along the normal direction to a position where the velocity is equal to the local free flow velocity U (strictly equal to 0.990 or 0.995U) is recorded as .

The thickness of the boundary layer is related to the Reynolds number of the flow, the state of free flow, the roughness of the object surface, the shape of the object surface, and the extension range. Starting from the head (front edge) of the flow object, the thickness of the boundary layer gradually increases from zero along the flow direction. When the Reynolds number of the air flow is Re _x = 10, the thickness of the laminar boundary layer on the plate is 3.5 mm at a distance of 1 m from the leading edge. Thickness of the laminar boundary layer on a smooth plate

( Re _x = Ux / v , where v is the viscosity coefficient of fluid motion); the constant value when writing the equation varies with the speed percentage at the thickness of the selected boundary layer (such as 0.90, 0.99 or 0.995), generally 3.46 to 5.64 . Thickness of turbulent boundary layer on a flat plate

Its proportionality constant is about 0.37. It can be seen that because the thickness of the boundary layer is arbitrary, it is too rough to calculate the friction resistance, so in actual applications, other thicknesses are defined. For example, at low speeds, the displacement thickness ₁ (or *), the momentum (loss) thickness ₂ (or ), and a dimensionless thickness ratio are called shape factors.

Boundary layer displacement thickness

The meaning of the displacement thickness is that the fluid in the boundary layer is blocked, so the flow passing through it is reduced, which is equivalent to the external flow in the ideal flow moving a distance from the object surface, and the shape of the flowing object becomes the original geometry Add displacement thickness.

The distance that the boundary layer formed due to the fluid viscosity retardation pushes the outer mainstream away from the wall (Figure 2) can be determined by the following formula:

When parallel flow flows through the plate, ₁ / 3 in the laminar boundary layer and ₁ / 8 in the turbulent boundary layer.

() Boundary layer momentum (loss) thickness

Due to viscous retardation, the momentum lost in the boundary layer is equivalent to the thickness occupied by this momentum when calculated according to the mainstream velocity U outside the layer, that is,

When parallel flow flows through the plate, _{2 of the} laminar boundary layer is 0.13, and _{2 of the} turbulent boundary layer is 7 / 72 = 0.097, so ₁ > ₂ .

Boundary layer shape factor

The dimensionless parameter composed of the above two thickness ratios is called the shape factor, and is usually expressed as: ₁ / ₂ = H ₁₂ (also written as H at low speed). Because ₁ > ₂ , H> 1. In the laminar boundary layer, the value of H ranges from 2.0 near the stagnation point to 3.5 at the separation point. In the turbulent boundary layer, its value is uncertain. It is about 1.2 to 2.5.

Boundary layer boundary layer equation

Boundary layer equations are mathematical expressions of the physical laws followed by fluid motion in the boundary layer, including boundary layer differential equations and boundary layer momentum integral equations.

Boundary layer boundary layer differential equation

Because y and the boundary layer thickness << x (length in the direction of the object plane) are of the same order, and at the same time

Based on the Navier-Stokes equation in 1904, Prandt wrote out the magnitudes of the terms in the equations and compared them with each other. Finally, the terms with magnitudes above ² were omitted to obtain the boundary layer equation. For example, the laminar boundary layer equations of a two-dimensional incompressible flow can be written as:

Boundary conditions y = 0, u = v = 0, y = , u = U (x, t) , where u and v are velocity components in x and y directions; p is pressure; is fluid density. The original momentum equation in the y direction is simplified to

It indicates that the pressure in the boundary layer along the direction perpendicular to the wall surface remains constant, that is, the pressure p at a point on the wall surface is equal to the p value calculated at this point for inviscid outflow. Therefore, in the boundary layer flow calculation, p Is a known physical quantity.

If the object surface is a curved surface, you can choose the surface coordinate system. The direction along the object surface is x , and the direction perpendicular to the object surface is y . Also draws

The increase in the y direction is also on the order of and can be ignored.

Regarding the turbulent boundary layer equation, the flow changes with time and space, and the situation is very complicated. Therefore, the physical mechanism of turbulence has not been clarified through experiments, and a recognized model has been obtained. So for many years, people have put forward various semi-empirical theories and assumptions to find the average solution for different situations.

In the general case of a turbulent boundary, the instantaneous velocity of a fluid micelle can be expressed as the sum of the average velocity and the pulsating velocity (such as the x direction, etc.). The additional turbulent stress (also known as Reynolds stress) in the turbulent boundary layer due to momentum exchange between pulsating velocities is:

It is a tensor. In a two-dimensional case, the Reynolds stress _t can be written (see turbulence theory):

In the formula, _is called the vortex viscosity coefficient, and the upper horizontal line represents the average value. The differential equations of the two-dimensional incompressible turbulent boundary layer are:

Boundary layer boundary layer momentum integral equation

This is T. von Carmen proposed in 1921, also known as Carmen integral relation, is an approximate method commonly used in engineering. It can be used for both constant and two-dimensional incompressible laminar and turbulent (using average velocity components) boundary layers. This equation is obtained by taking a control element in the boundary layer and using the momentum theorem to make the total momentum increase rate in the x direction equal to the difference between the outflow momentum and the inflow momentum per unit time. Because the momentum is calculated from the wall y = o to y = quadrature, the average is obtained, which is the approximation method. This integral relationship is:

In the formula, ₀ is the shear stress on the object surface. Substituting the displacement thickness ₁ and the momentum thickness ₂ can be written as:

or

Where

Boundary layer laminar boundary layer

When a fluid flows around an object, the boundary layer at the front or upstream of the object is usually a laminar boundary layer. Laminar boundary layer along the surface. The speed of outflow is different from the flat plate, but the velocity distribution is similar. The velocity gradient close to the object surface is large, so the shear stress is also large. The shear stress on the surface is:

Where is the hydrodynamic viscosity coefficient. By calculating ₀ , the frictional resistance coefficient and frictional resistance of the object surface can be obtained. But these calculations can only be used before the separation point.

Boundary layer differential equation of laminar boundary layer

The Binavi-Stokes equation is simple, but it is still a nonlinear partial differential equation. The early solution of the two-dimensional laminar boundary layer equation was to find the dimensionless combination of independent variables, substitute the equation into an ordinary differential equation, and then use the series method to find the friction coefficient or directly find its numerical solution. This method is called "similar solution". "Similar solutions" are available for flow around a flat plate, flow around a wedge, contracted pipe flow, and symmetrical flow around a cylinder. Due to the development of electronic computers, numerical solutions of nonlinear partial differential equations can be directly obtained using finite difference methods or finite element methods.

Momentum integral of boundary layer laminar boundary layer

It is much simpler than the numerical solution of partial differential equations, but it cannot provide detailed changes in flow characteristics (such as velocity distribution) in the boundary layer. Therefore, it is a practical engineering method if only the change of the characteristic physical quantities of the boundary layer (such as displacement thickness, wall shear stress, etc.) along the object plane is required. For some complex flow problems, such as the interaction between viscous and non-viscous flows, it is often used to calculate the characteristics of the boundary layer.

Calculation of Boundary Layer 3D Laminar Boundary Layer

If it is a rotating symmetric body, it can be converted into a two-dimensional form through a conversion formula [such as a Mangerer conversion formula], and the existing two-dimensional solution can be used. Three-dimensional calculations around arbitrary objects are much more complicated than two-dimensional ones, so they can only rely on numerical solutions.

Boundary layer transition and stability of laminar boundary layer

Since O. Reynold's experiments on circular tube flow proved that the flow in the tube was laminar and then transitioned to turbulent flow. He used a dimensionless ratio (that is, Reynolds number) as the flow parameter. For each specific shape, there is a critical Reynolds number. For example, the critical Reynolds number of a circular tube is 2000. Above this value, the laminar flow transitions to turbulent flow (see laminar flow). There is a similar concept of the critical Reynolds number in the boundary layer, but the Reynolds number of the boundary layer is usually written as

The critical Reynolds number Re _cr can be obtained experimentally. The transition from laminar to turbulent flow is not only related to the Reynolds number, but also affected by many other parameters, such as turbulence of external flow, back pressure gradient, fluid injection, centrifugal force flowing through concave surface, buoyancy in non-uniform flow, surface Roughness, heat exchange between the fluid and the object surface, etc., will increase instability factors and easily cause the transition of the laminar boundary layer.

Boundary layer laminar boundary layer stability theory

In terms of theory, the small disturbance stabilization theory is commonly used, that is, assuming that laminar flow is composed of average flow (which can be regarded as steady flow) plus small disturbance sinusoidal flow. If the small disturbance increases with time, it is unstable Yes, it is possible to transition into turbulence. The so-called Orr-Sommerfeld equation is generally the equation of the small perturbation theory (see Fluid Motion Stability).

When discussing the stability of parallel flow boundary layers, Tolmin-Schlichting stability theory is commonly used. Its basic idea is that when the laminar boundary layer flows through the object surface, it always receives some small disturbances (such as the tip, rough plate surface, etc.), so the laminar boundary layer contains many velocity pulsations with very small amplitude. , Its frequency range is very wide. In some cases, if the pulsation at one frequency is strengthened and the other frequencies are weakened. Then the former rapidly increases the amplitude at this frequency (this wave in the boundary layer is called the Tolmin-Schlichting wave), which makes the laminar flow unstable and leads to the formation of turbulence. Conversely, if the amplitude of all frequencies of the pulsation is reduced, the laminar flow is stable.

Boundary layer transition from laminar boundary layer to turbulent boundary layer

The stability theory of laminar flow cannot explain all the physical phenomena of the transition from laminar flow to turbulent flow. Over

Fig. 3.Laminar boundary layer transition process on the plate.

Crossing is a very complicated flow process until now. People don't know it clearly yet. From a simple flow around the plate, if the turbulence of the outflow is low, the transition from the laminar boundary layer to the turbulent boundary layer passes through the following stages when flowing through a smooth plate (Figure 3): Near the front of the plate is a stable layer The boundary layer of the flow, after the critical Reynolds number. There are unstable two-dimensional Tolmin-Schlichting waves. The unstable three-dimensional laminar wave continues to develop and forms small vortices. At very strong local shear stresses, the vortex ruptures and generates three-dimensional turbulent pulsations. In places where the turbulence velocity fluctuates greatly, many turbulent shift spots are generated; Many turbulent spots are united to develop into a fully developed turbulent boundary layer.

In most cases, when the turbulent spots develop to complete turbulence, many separate gases are formed simultaneously

Fig. 4 Relationship between boundary layer resistance coefficient Cf and Reynolds number Re

bubble. In the above process, only , , and can be analyzed theoretically. Other processes need to be further explored in the future. The instability of the centrifugal flow along the curved surface is different from the above. For example, laminar flow between two inner and outer cylinders that can rotate concentrically. When the inner cylinder rotates and the outer cylinder does not move, a Taylor vortex is generated (see fluid motion stability). When Jane does not move, the laminar flow is stable; when the two cylinders rotate in opposite directions, they are unstable. Another example is laminar flow along a concave surface. The Gertler disaster, which is perpendicular to the flow direction, also causes instability.

When the laminar boundary layer transitions to the turbulent boundary, the thickness of the boundary layer increases (Figure 3), and the resistance also increases. Taking parallel flow through the plate as an example, the relationship between the resistance coefficient C _{1 f} and the Reynolds number Re is shown in FIG. 4 (using double logarithmic coordinates).

Boundary layer turbulent boundary layer

In nature and engineering, the flow on the surface of moving objects (such as airplanes, cascades, etc.) is mostly turbulent

Figure 5 Comparison of laminar and turbulent boundary layers

Flow boundary layer. Since turbulence is vortex flow and random pulsation, the flow changes with space and time. Therefore, the internal structure of the turbulent boundary layer is much more complicated than that of the laminar boundary layer. Due to the exchange of momentum in the vertical flow direction in turbulence, its velocity distribution on the vertical section of the wall is different from that of the laminar boundary layer, and its lower end is fuller (Figure 5).

From the experimental data, the turbulent boundary layer can be regarded as consisting of an inner region and an outer region. This division method is because the viscous shear stress and pressure gradient near the wall are completely different in these two regions. The inner zone includes a sticky bottom layer close to the wall. Among them, the shear stress is the largest, consisting of many small vortices, upwards is the buffer layer, and then upwards until the outer zone of the boundary layer is a turbulent layer with a large momentum exchange composed of large-scale vortices. The outer zone is from this turbulent layer to the place where the velocity is very close to the outflow. In general, the inner zone accounts for 20% of the entire boundary layer.

Boundary layer turbulent boundary layer

From the research history of turbulent boundary layer, there are two theories, which are developed and related to each other. One is statistical theory. The other is semi-empirical theory.

In statistical theory, fluid is regarded as a continuous medium, and pulsation values such as flow velocity and pressure are regarded as continuous random functions. Turbulent flow is described by the correlation function and spectral function of each pulsation value. According to the statistical average method, the pulsating structure is found out, and various average values are substituted into the Navier-Stokes equation and other equations to obtain the so-called Reynolds equation. But statistical theory is mainly used to study uniform isotropic turbulence. Not suitable for turbulent boundary layer flow.

In another semi-empirical theory, because the number of turbulent boundary layer equations is less than the number of unknown quantities . The system of equations is not closed, so some relational expressions need to be added. Some of the less rigorous approximate theories are semi-empirical theories. These theories have no strict basis, but they are useful for solving many problems in engineering. And because some of these coefficients are obtained from experiments, the results calculated by these semi-empirical theories often agree with the experiments, but their scope of application is limited. Common semi-empirical theories include: JV Bussenieske's formula for calculating Reynolds' stress using eddy viscosity coefficients in 1877, Cylont's mixed length theory (momentum transfer theory): I. Taylor's theory of vortex transfer, Carmen's similarity theory, etc. The disadvantage of these semi-empirical theories is that the internal structure of turbulence is not analyzed, and the range of use is limited.

Boundary layer turbulent boundary layer experiment

For the study of the boundary layer, experiments are a very important means. Especially turbulent boundary layer measurements. Many countries have set up groups to continue their research. General experiments are performed in a sink or wind tunnel. The flow field display methods used are hydrogen bubble method and smoke trail method. Sleeve method applied on the surface. Measurement methods In recent times, hot wires, thermal films and laser velocimetry, laser holography, etc. have been used (see turbulence experiments).

Boundary layer boundary layer separation

As a fluid flows across a surface, its speed and pressure change. When the flow rate decreases, the pressure must be

Figure 6 Separation of the boundary layer around the wing

increase. Due to the momentum loss of the fluid micelles in the boundary layer, if the downstream pressure increases (that is, there is a reverse pressure gradient), the momentum will decrease again, until the fluid micelles can no longer advance on the object surface and will move from the object surface Separation. This phenomenon is called boundary layer separation (Figure 76). The point at which the airflow begins to leave the surface is called the separation point. Its position is from the surface

It is determined that the normal velocity gradient of the point at the object plane is zero. Fig. 7 shows the two-dimensional flow of parallel flow through a symmetrical wing profile, the boundary layer separation at the back pressure profile at the back pressure gradient, and the velocity distribution near the separation point s. Note that there is an inflection point on the velocity distribution curve at the separation point, and there is a reverse (forward) flow along the object surface in the separation region. Create a vortex and form a trailing vortex behind the object. The boundary layer on the wing is separated, which limits the lift of the wing. And increase resistance. Separation on the wing at high angles of attack can cause aircraft arrest. Air flow separation on the blades of turbine pumps, compressors, propellers, etc. reduces mechanical efficiency and can corrode wall surfaces.

There are two cases of two-dimensional separation around the object boundary layer: one is from the separation point, the main stream leaves the object surface, and

Figure 7 Two-dimensional flow of parallel flow through a symmetrical wing section

A large vortex area is formed downstream; this separation often occurs at the rear of the wing (for example) at normal angles of attack (Figures 6 and 7); the other is from the separation point S, after which the mainstream first Leave the object surface, and then attach to the object surface at point A, forming a bubble-a local reflow area (Figure 8). Separation bubbles are mostly laminar separation, and become turbulent in the middle. The bottom layer gets kinetic energy and then attaches to the object surface. When the wing is thick, the separation often occurs at the rear of the wing; when the wing is thin, the bubble separation often occurs near the leading edge. The separation of 3D boundary layer is more complicated. There is no consensus on how to define the separation point. The separation of turbulent boundary layer is the same as the separation of laminar boundary layer.

Figure 8 Separated bubbles in front of the object

Momentum exchange, for the same shape object, the separation point of the turbulent boundary layer is later than the laminar boundary layer. Under specific circumstances. One can fix the separation point artificially, and use the vortex formed by the separation of the airflow to affect the object, which has a beneficial effect. For example, the use of edge winglets in aviation is to use its leading edge to separate the vortex to increase the non-linear lifting force on the basic wing with a small aspect ratio.

In terms of experiments. The position of the separation point can be measured by oil flow method, silk method on the surface of the model, and Preston tube.

The starting point and separation point of the boundary layer in separated flows, especially in two-dimensional unsteady and three-dimensional steady flows. The research on the flow problem near the line has been paid more and more attention, and there are some approximate theories such as three-layer structure, etc. The separation criteria for two-dimensional and three-dimensional flows have also been tried.

Boundary layer boundary layer control

In applications (e.g. for aviation vehicles), the transition and separation of laminar boundary layers, the use of wings, etc.

Figure 9

Increased drag and / or reduced lift (or even stall), so people have long sought to smooth the surface of the wing and designed a "laminar flow wing profile" to maintain a laminar boundary layer. But this kind of control is limited, so people have subsequently adopted many methods to manually control the boundary layer to achieve the purpose of affecting the structure of the boundary layer, thereby avoiding the separation of air flow in the boundary layer, and reducing the resistance and increasing the lifting force. Experiments and theories lead to the following effective methods to locally accelerate the fluid: Move part of the object surface, blow out the fluid through the nozzle (slot) on the object surface to increase the energy of surface stagnation (Figure 9) Suck the stagnation through the slits on the object surface to make the boundary layer thin to suppress separation; spray with different gases to accelerate the stagnation; change the shape of the wing

Boundary layer references

1.H. Schlichting, Boundary Layer Tayer Theory, MeGraw-Hill, New York, 1979.

2. T. Cebeci and A. M. C. Smith, Analysis of Turbulent Boundary Layers, Academic Press, New York, 1974.

3.F. M. White, Wei Zhonglei, Zhen Simiao translation: (viscous fluid dynamics), Machinery Industry Press, Beijing, 1982. (F.M.White, Viscous Fluid Flow, McGraw-Hill, New York. 1974.)

4.G. V. Lachmann, Boundary Layer and Flow Control, Pergamon Press, Oxford, 1961.

5.P. Bradshaw, An Introduction to Turbulence and Its Measurement, Pergamon Press, Oxford, 1971.