A boundary layer is a thin layer of viscous fluid close to the solid surface of a wall in contact with a moving stream in which (within its thickness δ) the flow velocity varies from zero at the wall (where the flow “sticks” to the wall because of its viscosity) up to U_{e} at the boundary, which approximately (within 1% error) corresponds to the free stream velocity (see Figure 1). Strictly speaking, the value of δ is an arbitrary value because the friction force, depending on the molecular interaction between fluid and the solid body, decreases with the distance from the wall and becomes equal to zero at infinity.
The fundamental concept of the boundary layer was suggested by L. Prandtl (1904), it defines the boundary layer as a layer of fluid developing in flows with very high Reynolds Numbers Re, that is with relatively low viscosity as compared with inertia forces. This is observed when bodies are exposed to high velocity air stream or when bodies are very large and the air stream velocity is moderate. In this case, in a relatively thin boundary layer, friction Shear Stress (viscous shearing force): τ = η[∂u/∂y] (where η is the dynamic viscosity; u = u(y) – “profile” of the boundary layer longitudinal velocity component, see Figure 1) may be very large; in particular, at the wall where u = 0 and τ_{w} = η[∂u/∂y]_{w} although the viscosity itself may be rather small.
It is possible to ignore friction forces outside the boundary layer (as compared with inertia forces), and on the basis of Prandtl’s concept, to consider two flow regions: the boundary layer where friction effects are large and the almost Inviscid Flow core. On the premises that the boundary layer is a very thin layer (δ << L, where L is the characteristic linear dimension of the body over which the flow occurs or the channel containing the flow, its thickness decreasing with growth of Re, Figure 1), one can estimate the order of magnitude of the boundary layer thickness from the following relationship:
For example, when an airplane flies at U_{e} = 400 km/hr, the boundary layer thickness at the wing trailing edge with 1 meter chord (profile length) is m. As was experimentally established, a laminar boundary layer develops at the inlet section of the body. Gradually, under the influence of some destabilizing factors, the boundary layer becomes unstable and transition of boundary layer to a Turbulent Flow regime takes place. Special experimental investigations have established the existence of a transition region between the turbulent and laminar regions. In some cases (for example, at high turbulence level of the external flow), the boundary layer becomes turbulent immediately downstream of the stagnation point of the flow. Under some conditions, such as a severe pressure drop, an inverse phenomenon takes place in accelerating turbulent flows, namely flow relaminarization.
In spite of its relative thinness, the boundary layer is very important for initiating processes of dynamic interaction between the flow and the body. The boundary layer determines the aerodynamic drag and lift of the flying vehicle, or the energy loss for fluid flow in channels (in this case, a hydrodynamic boundary layer because there is also a thermal boundary layer which determines the thermodynamic interaction of Heat Transfer).
Computation of the boundary layer parameters is based on the solution of equations obtained from the Navier–Stokes equations for viscous fluid motion, which are first considerably simplified taking into account the thinness of the boundary layer.
The solution suggested by L. Prandtl is essentially the first term of power series expansion of the Navier–Stokes equation, the series expansion being performed for powers of dimensionless parameter (δ/L). The smaller parameter in this term is in zero power so that the boundary layer equation is the zero approximation in an Asymptotic Expansion (at large Re) of the boundary layer equation (asymptotic solution).
A transformation of the Navier–Stokes equation into the boundary layer equations can be demonstrated by deriving the Prandtl equation for laminar boundary layer in a two-dimensional incompressible flow without body forces.
In this case, the system of Navier–Stokes equations will be:
After evaluating the order of magnitude of some terms of Eq. (2) and ignoring small terms the system of Prandtl equations for laminar boundary layer becomes:
in which x, y are longitudinal and lateral coordinates (Figure 1); v is the velocity component along “y” axis; p, pressure; t, time; and n the kinematic viscosity.
The boundary layer is thin and the velocity at its external edge U_{e} can be sufficiently and accurately determined as the velocity of an ideal (inviscid) fluid flow along the wall calculated up to the first approximation, without taking into account the reverse action of the boundary layer on the external flow. The longitudinal pressure gradient [∂p/∂x] = [dp/dx] (at p(y) = const) in Eq. (3) can be depicted from the Euler equation of motion of an ideal fluid. From the above, Prandtl equations in their finite form will be written as:
This is a system of parabolic, nonlinear partial differential equations of the second order which are solved with initial and boundary conditions
The system of equations (4) is written for actual values of velocity components u and v. To generalize the equations obtained for turbulent flow, the well-known relationship between actual, averaged and pulsating components of turbulent flows parameters should be used. For example, for velocity components there are relationships connecting actual u and v, average ū and and pulsating u' and v' components:
After some rearrangements, it is possible to obtain another system of equations [Eq. (6)] from system (3), in particular for steady flow:
Using the following relation for friction shear stress in the boundary layer:
and taking into account that in the laminar boundary layer u = u' and it is possible to rewrite the Prandtl equations in a form valid for both laminar and turbulent flows:
The simplest solutions have been obtained for a laminar boundary layer on a thin flat plate in a two-dimensional, parallel flow of incompressible fluid (Figure 1). In this case, the estimation of the order of magnitude of the equations terms: x ~ L, y ~ δ, δ ~ allows combining variables x and y in one relation
and to reduce the solution of Eq. (8) (at dp/dx = 0) to determining the dependencies of u and v upon the new parameter ξ. On the other hand, using well-known relations between velocity components u, v and stream function ψ
it is possible to obtain one ordinary nonlinear differential equation of the third order, instead of the system of partial differential equations (8)
Here, f(ξ) is the unknown function of ξ variable: f = ƒ =
The first numerical solution of Eq. (10) was obtained by Blasius (1908) under boundary conditions corresponding to physical conditions of the boundary layer at y = 0: u = 0, v = 0; at y → ∞; u → U_{e} (Blasius boundary layer).
Figure 2 compares the results of Blasius solution (solid line) with experimental data. Using these data, it is possible to evaluate the viscous boundary layer thickness. At ξ 2.5, (u/U_{e} 0.99) (Figure 2); consequently from Eq. (9) we obtain:
From the Blasius numerical calculations of the value of the second derivative of f(ξ) function at the wall friction shear stress, the relationship in this case is:
Friction force R, acting on both sides of the plate of L length (Figure 1), is also determined from Eq. (11):
as in the friction coefficient for flat plates:
Despite the fact that Prandtl equations are much simpler than Navier–Stokes equations, their solutions were obtained for a limited number of problems. For many practical problems, it is not necessary to determine velocity profiles in the boundary layer, only thickness and shear stress. This kind of information may be obtained by solving the integral momentum equation
The integral relationship (12) is valid both for the laminar and turbulent boundary layer.
Functions which were not known a priori but which characterize distribution of fluid parameters across the layer thickness δ are under the integral in Eq. (12). And the error of calculating the integral is less than the error in the approximately assumed integrand function ρu = ρu(y). These create conditions for developing approximate methods of calculating boundary layer parameters which are less time-consuming than the exact methods of integrating Prandtl equations. The fundamental concept was first suggested by T. von Karman, who introduced such arbitrary layer thickness δ*
and momentum displacement thickness δ**
thus, we can transform Eq. (12) for two-dimensional boundary layer of incompressible fluid to:
There are three unknown functions in Eq. (15), namely, δ* = δ*(x), δ** = δ**(x) and τ_{w} = τ_{w}(x) [functions of U_{e}(x) and correspondingly which are known from computations of flow in the inviscid flow core].
The solution of an ordinary differential equation like Eq. (15) usually requires assumption (or representation) of velocity distribution (velocity profile) across the boundary layer thickness as the function of some characteristic parameters (form-parameters), and it also requires the use of empirical data about the relationship between friction coefficient C_{f} = 2τ_{w}/(ρU^{2}_{e}) and the arbitrary thickness of the boundary layer (friction law).
Some definite physical explanations can be given as far as the values of δ* and δ** are concerned. The integrand function in Eq. (13) contains after rearrangement, a term (U_{e} – u) which characterizes the velocity decrease. The integral in Eq. (14) can thus be considered as a measure of decreasing the flow rate across the boundary layer, as compared with the perfect fluid flow at the velocity U_{e}. On the other hand, the value of δ* can be considered as the measure of deviation along a normal to the wall (along “y” axis) of the external flow stream line under the influence of friction forces. From this consideration of the integral structure of Eq. (14), it is possible to conclude that δ** characterizes momentum decrease in the boundary layer under the influence of friction.
The following relations are valid:
where H is the form-parameter of the boundary layer velocity profile. For example, for linear distribution u = ky,
At present, so-called semi-empirical theories are widely used for predicting turbulent boundary layer parameters. In this case, it is assumed that total friction stress τ in a turbulent boundary layer is a sum
Here, τ_{T} is additional (turbulent or Reynolds) friction stress, in particular, in an incompressible flow see Eq. (7).
This representation is directly connected with the system of equations of motion in the boundary layer (6). In the compressible boundary layer, density pulsations can be considered to be the result of temperature pulsations
where β = (1/T) is the volumetric expansion coefficient.
Additional semi-empirical hypotheses about turbulent momentum transfer are used for determining τ_{T}. For example,
where η_{T} is the dynamic coefficient of turbulent viscosity introduced by J. Boussinesq in 1877.
On the basis of the concept of similarity of molecular and turbulent exchange (similarity theory) Prandtl introduced the mixing length (die Mischungsweg) hypothesis. The mixing length 1 is the path a finite fluid volume (“mole”) passes from one layer of average motion to another without changing its momentum. In accordance with this condition, he derived an equation which proved to be fundamental for the boundary layer theory:
For turbulent region of the near wall flow boundary layer, L. Prandtl considered the length 1 proportional to y
where κ is an empirical constant.
Close to the wall, where η_{T} << η, viscous molecular friction [the first term in Eq. (15)] is a determining factor. The thickness of this part of the boundary layer δ_{1}, which is known as laminar or viscous sublayer, is . Outside the sublayer, the value of η_{T} increases, reaching several orders of magnitude larger than η. Correspondingly, in this zone of the boundary layer known as the turbulent core τ_{T} > _{0} = η[∂ū/∂y]. Sometimes the turbulent core is subdivided into the buffer zone, where the laminar and turbulent friction are the comparable value, and the developed zone, where τ_{T} >> τ_{0}. For this region, after integrating Eq. (18) and taking into account Eq. (19), it is possible to derive an expression for logarithmic velocity profile:
If dimensionless (or universal) coordinates are used.
where is the so-called dynamic velocity (or Friction Velocity), Eq. (20) can be rewritten in the following form:
Velocity distribution representation in universal coordinates and mathematical models for turbulent viscosity coefficient are dealt with in greater detail in the section of Turbulent Flow.
One of the current versions of the semi-empirical theory of turbulent boundary layer developed by S. S. Kutateladze and A. I. Leontiev is based on the so-called asymptotic theory of turbulent boundary layers at Re → ∞ where the thickness of laminar (viscous) sublayer δ_{1} decreases at a higher rate than δ as a result of which (δ_{1}/δ) → 0.
Under these conditions, a turbulent boundary layer with “vanishing viscosity” is developing. In this layer, η → 0 but is not equal to zero and in this respect, the layer differs from perfect fluid flow. The concept of relative friction law, introduced by S. S. Kutateladze and A. I. Leontiev (1990), indicates
The law is defined as the ratio of friction coefficient C_{f} for the condition under consideration to the value of C_{f0} for “standard” conditions on a flat, impermeable plate flown around by incompressible, isothermal flow, both coefficients being obtained for Re** = U_{e}δ**/ν. It is shown that at Re → ∞; η → 0; and C_{f} → 0, the relative variation of the friction coefficient under the influence of such disturbing factors as _{pressure gradient, compressibility}, nonisothermicity, injection (suction) through a porous wall etc., has a finite value.
The equations derived for calculating the value of Ψ have one important characteristic which makes Ψ independent of empirical constants of turbulence. In accordance with the fundamental concept of the integral “approach”, the integral momentum equation is transformed into:
Here, Re_{L} = U_{e}L/ν, b = (2/C_{f0})(ρ_{w}U_{w})/(ρ_{e}U_{e}) are the permeability parameters for the case of injecting a gas at density ρ_{w} through a permeable wall at the velocity of v_{w}. For determining the function Re** = Re** , it is necessary to calculate the distribution For this purpose, the principle of superposition of disturbing factors applies
In Eq. 24, each multiplier represents the relative friction law, taking into account the effect of one of the factors, among them compressibility Ψ_{M}, temperature (or Enthalpy) head Ψ_{T}, injection Ψ_{B}, pressure gradient Ψ_{P} and others.
The fundamental concepts of the boundary layer create conditions for explaining such phenomena as flow separation from the surface under the influence of the flow inertia, deceleration of viscous flow by the wall and adverse pressure gradient acting in the upstream direction [∂p/∂x] = [dp/dx[ >] 0 or [∂u/x∂] < 0.
If pressure gradient is adverse on the surface location between sections ‘1–4’ (see Figure 3), the velocity distribution u = u(x,y) in the boundary layer changes gradually; becoming “less full,” decreasing the inclination in the fluid jets which are closer to the wall and possessing less amount of kinetic energy (see velocity profile shapes in Figure 3) which penetrate far downstream into the region of increased pressure. In some sections, for example section ‘4’, fluid particles which are on the ‘a-a’ stream line (dotted line in Figure 3) — having completely exhausted their supply of kinetic energy become decelerated (u_{a} = 0).
Static pressure and pressure gradient value do not vary across boundary layer thickness. Therefore, fluid particles which are closer to the wall than line ‘a–a’ and possessing still less amount of energy begin to move in the opposite direction under the influence of the pressure gradient in ‘4–4’ section (see Figure 3). Thus, the relationship:
In this way, at some locations of the surface, the velocity profile changes. This change is characterized by the alteration of the sign of the derivative [∂u/∂y]_{w} from positive (section 2, Figure 3) to negative (section 4). Of course, it is also possible to define the section where [∂u/∂y]_{w} = 0 (section 3, Figure 3). This is referred to as the boundary layer separation section (correspondingly point ‘S’ on the surface of this section is the separation point). It is characterized by the development of a reverse flow zone — the flow around the body is no longer smooth, the boundary layer becomes considerably thicker and the external flow stream lines deviate from the surface of the body flown around. Downstream of the separation point, the static pressure distribution across the thickness of the layer is not steady and the static pressure distribution along the surface does not correspond to the pressure distribution in the external, inviscid flow.
The separation is followed by the development of reverse flow zones and swirls, in which the kinetic energy supplied from the external flow transforms into heat under the influence of friction forces. The flow separation, accompanied by energy dissipation in the reverse flow swirl zones, results in such undesirable effects as increases in the flying vehicles’ drag or hydraulic losses in channels.
On the other hand, separated flows are used in different devices for intensive mixing of fluid (for example, to improve mixing of fuel and air in combustion chambers of engines). When viscous fluids flow in channels with a variable cross-section (alternating pressure gradient), the separation zone may be local if the diffusor section is followed by the confusor section, where the separated flow will again reattach to the surface (see Figure 4a). When the flow separates from the trailing edge of the body (for example, from the wing trailing edge), the so-called wake is formed by “linking” boundary layers (see Figure 4b).
REFERENCES
Prandtl, L. (1904) Über Flüssingkeitsbewegungbeisehr Kleiner Reibung: Verhandl. III Int. Math. Kongr. — Heidelberg.
Blasius, H. (1908) Grenzschichten in Flüssigkeiten mit Kleiner Reibung: Z. Math. Phys., 56:1–37.
Kutateladze, S. S. and Leontiev, A. I. (1990) Heat Transfer, Mass Transfer and Turbulent Boundary Layers, Hemisphere Publishing Corporation, New York, Washington, Philadelphia, London.
Использованная литература
- Prandtl, L. (1904) Ãœber FlÃ¼ssingkeitsbewegungbeisehr Kleiner Reibung: Verhandl. III Int. Math. Kongr. â€” Heidelberg.
- Blasius, H. (1908) Grenzschichten in FlÃ¼ssigkeiten mit Kleiner Reibung: Z. Math. Phys., 56:1â€“37.
- Kutateladze, S. S. and Leontiev, A. I. (1990) Heat Transfer, Mass Transfer and Turbulent Boundary Layers, Hemisphere Publishing Corporation, New York, Washington, Philadelphia, London.