3.10. BOUNDARY-LAYER FLOW AND TURBULENCE

3.10A. Boundary-Layer Flow

In Sections 3.8 and 3.9, the Navier-Stokes equations were used to find relations that described laminar flow between flat plates and inside circular tubes, flow of ideal fluids, and creeping flow. In this section the flow of fluids around objects will be considered in more detail, with particular attention being given to the region close to the solid surface, called the boundary layer.

In the boundary-layer region near the solid, the fluid motion is greatly affected by this solid surface. In the bulk of the fluid away from the boundary layer, the flow can often be adequately described by the theory of ideal fluids with zero viscosity. However, in the thin boundary layer, viscosity is important. Since the region is thin, simplified solutions can be obtained for the boundary-layer region. Prandtl originally suggested this division of the problem into two parts, which has been used extensively in fluid dynamics.

In order to help explain boundary layers, an example of boundary-layer formation in the steady-state flow of a fluid past a flat plate is given in Fig. 3.10-1. The velocity of the fluid upstream of the leading edge at x = 0 of the plate is uniform across the entire fluid stream and has the value v. The velocity of the fluid at the interface is zero and the velocity vx in the x direction increases as one goes farther from the plate. The velocity vx approaches asymptotically the velocity v of the bulk of the stream.

Figure 3.10-1. Boundary layer for flow past a flat plate.


The dashed line L is drawn so that the velocity at that point is 99% of the bulk velocity v. The layer or zone between the plate and the dashed line constitutes the boundary layer. When the flow is laminar, the thickness δ of the boundary layer increases with as we move in the x direction. The Reynolds number is defined as NRe,x = xvρ/μ, where x is the distance downstream from the leading edge. When the Reynolds number is less than 2 × 105, the flow is laminar, as shown in Fig. 3.10-1.

The transition from laminar to turbulent flow on a smooth plate occurs in the Reynolds-number range 2 × 105 to 3 × 106, as shown in Fig. 3.10-1. When the boundary layer is turbulent, a thin, viscous sublayer persists next to the plate. The drag caused by the viscous shear in the boundary layers is called skin friction and is the only drag present for flow past a flat plate.

The type of drag occurring when fluid flows by a bluff or blunt shape such as a sphere or cylinder, which is mostly caused by a pressure difference, is termed form drag. This drag predominates in flow past such objects at all except low values of the Reynolds number, and often a wake is present. Skin friction and form drag both occur in flow past a bluff shape, and the total drag is the sum of the skin friction and the form drag. (See also Section 3.1A.)

3.10B. Boundary-Layer Separation and Formation of Wakes

We discussed the growth of the boundary layer at the leading edge of a plate as shown in Fig. 3.10-2. However, some important phenomena also occur at the trailing edge of this plate and other objects. At the trailing edge or rear edge of the flat plate, the boundary layers are present at the top and bottom sides of the plate. On leaving the plate, the boundary layers gradually intermingle and disappear.

Figure 3.10-2. Flow perpendicular to a flat plate and boundarylayer separation.


If the direction of flow is at right angles to the plate, as shown in Fig. 3.10-2, a boundary layer forms as before in the fluid that is flowing over the upstream face. Once at the edge of the plate, however, the momentum in the fluid prevents it from making the abrupt turn around the edge of the plate, and it separates from the plate. A zone of decelerated fluid is present behind the plate and large eddies (vortices), called the wake, are formed in this area. The eddies consume large amounts of mechanical energy. This separation of boundary layers occurs when the change in velocity of the fluid flowing past an object is too large in direction or magnitude for the fluid to adhere to the surface.

Since formation of a wake causes large losses in mechanical energy, it is often necessary to minimize or prevent boundary-layer separation by streamlining the objects or by other means. This is also discussed in Section 3.1A for flow past immersed objects.

3.10C. Laminar Flow and Boundary-Layer Theory

1. Boundary-layer equations

When laminar flow is occurring in a boundary layer, certain terms in the Navier-Stokes equations become negligible and can be neglected. The thickness of the boundary layer δ is arbitrarily taken as the distance away from the surface where the velocity reaches 99% of the free stream velocity. The concept of a relatively thin boundary layer leads to some important simplifications of the Navier-Stokes equations.

For two-dimensional laminar flow in the x and y directions of a fluid having a constant density, Eqs. (3.7-36) and (3.7-37) become as follows for flow at steady state as shown in Figure 3.10-1 when we neglect the body forces gx and gy:

Equation 3.10-1


Equation 3.10-2


The continuity equation for two-dimensional flow becomes

Equation 3.10-3


In Eq. (3.10-1), the term (μ/ρ)(2νx/∂x2) is negligible in comparison with the other terms in the equation. Also, it can be shown that all the terms containing νy and its derivatives are small. Hence, the final two boundary-layer equations to be solved are Eqs. (3.10-3) and (3.10-4):

Equation 3.10-4


2. Solution for laminar boundary layer on a flat plate

An important case in which an analytical solution has been obtained for the boundary-layer equations is for the laminar boundary layer on a flat plate in steady flow, as shown in Fig. 3.10-1. A further simplification can be made in Eq. (3.10-4) in that dp/dx is zero since v is constant.

The final boundary-layer equations reduce to the equation of motion for the x direction and the continuity equation as follows:

Equation 3.10-5


Equation 3.10-3


The boundary conditions are νx = νy = 0 at y = 0 (y is distance from plate), and νx = v at y = ∞.

The solution of this problem for laminar flow over a flat plate giving νx and νy as a function of x and y was first obtained by Blasius and later elaborated by Howarth (B1, B2, S3). The mathematical details of the solution are quite tedious and complex and will not be given here. The general procedure will be outlined. Blasius reduced the two equations to a single ordinary differential equation which is nonlinear. The equation could not be solved to give a closed form, but a series solution was obtained.

The results of the work by Blasius are given as follows. The boundary-layer thickness δ, where νx 0.99v, is given approximately by

Equation 3.10-6


where NRe,x = ρ/μ. Hence, the thickness δ varies as

The drag in flow past a flat plate consists only of skin friction and is calculated from the shear stress at the surface at y = 0 for any x as follows:

Equation 3.10-7


From the relation of νx as a function of x and y obtained from the series solution, Eq. (3.10-7) becomes

Equation 3.10-8


The total drag is given by the following for a plate of length L and width b:

Equation 3.10-9


Substituting Eq. (3.10-8) into (3.10-9) and integrating,

Equation 3.10-10


The drag coefficient CD related to the total drag on one side of the plate having an area A = bL is defined as

Equation 3.10-11


Substituting the value for A and Eq. (3.10-10) into (3.10-11),

Equation 3.10-12


where NRe,L = Lvρ/μ. A form of Eq. (3.10-11) is used in Section 14.3 for particle movement through a fluid. The definition of CD in Eq. (3.10-12) is similar to the Fanning friction factor f for pipes.

The equation derived for CD applies to the laminar boundary layer only for NRe,L less than about 5 × 105. Also, the results are valid only for positions where x is sufficiently far from the leading edge so that x or L is much greater than δ. Experimental results on the drag coefficient to a flat plate confirm the validity of Eq. (3.10-12). Boundary-layer flow past many other shapes has been successfully analyzed using similar methods.

3.10D. Nature and Intensity of Turbulence

1. Nature of turbulence

Since turbulent flow is important in many areas of engineering, the nature of turbulence has been extensively investigated. Measurements of the velocity fluctuations of the eddies in turbulent flow have helped explain turbulence.

For turbulent flow there are no exact solutions of flow problems as there are in laminar flow, since the approximate equations used depend on many assumptions. However, useful relations have been obtained by combining experimental data and theory. Some of these relations will be discussed.

Turbulence can be generated by contact between two layers of fluid moving at different velocities or by a flowing stream in contact with a solid boundary, such as a wall or sphere. When a jet of fluid from an orifice flows into a mass of fluid, turbulence can arise. In turbulent flow at a given place and time, large eddies are continually being formed which break down into smaller eddies and finally disappear. Eddies can be as small as about 0.1-1 mm and as large as the smallest dimension of the turbulent stream. Flow inside an eddy is laminar because of its large size.

In turbulent flow the velocity is fluctuating in all directions. In Fig. 3.10-3 a typical plot of the variation of the instantaneous velocity νx in the x direction at a given point in turbulent flow is shown. The velocity is the deviation of the velocity from the mean velocity in the x direction of flow of the stream. Similar relations also hold for the y and z directions:

Equation 3.10-13


Equation 3.10-14


Figure 3.10-3. Velocity fluctuations in turbulent flow.


where the mean velocity is the time-averaged velocity for time t, νx the instantaneous total velocity in the x direction, and the instantaneous deviating or fluctuating velocity in the x direction. These fluctuations can also occur in the y and z directions. The value of fluctuates about zero as an average and, hence, the time-averaged values . However, the values of , will not be zero. Similar expressions can also be written for pressure, which also fluctuates.

2. Intensity of turbulence

The time average of the fluctuating components vanishes over a time period of a few seconds. However, the time average of the mean square of the fluctuating components is a positive value. Since the fluctuations are random, the data have been analyzed by statistical methods. The level or intensity of turbulence can be related to the square root of the sum of the mean squares of the fluctuating components. This intensity of turbulence is an important parameter in the testing of models and the theory of boundary layers. The intensity of turbulence I can be defined mathematically as

Equation 3.10-15


This parameter I is quite important. Such factors as boundary-layer transition, separation, and heat- and mass-transfer coefficients depend upon the intensity of turbulence. Simulation of turbulent flows in testing of models requires that the Reynolds number and the intensity of turbulence be the same. One method used to measure intensity of turbulence is to utilize a hot-wire anemometer.

3.10E. Turbulent Shear or Reynolds Stresses

In a fluid flowing in turbulent flow, shear forces occur wherever there is a velocity gradient across a shear plane, and these are much larger than those occurring in laminar flow. The velocity fluctuations in Eq. (3.10-13) give rise to turbulent shear stresses. The equations of motion and the continuity equation are still valid for turbulent flow. For an incompressible fluid having a constant density ρ and viscosity μ, the continuity equation (3.6-24) holds:

Equation 3.6-24


Also, the x component of the equation of motion, Eq. (3.7-36), can be written as follows if Eq. (3.6-24) holds:

Equation 3.10-16


We can rewrite the continuity equation (3.6-24) and Eq. (3.10-16) by replacing νx by :

Equation 3.10-17


Equation 3.10-18


Now we use the fact that the time-averaged value of the fluctuating velocities is zero , and that the time-averaged product is not zero. Then Eqs. (3.10-17) and (3.10-18) become

Equation 3.10-19


Equation 3.10-20


By comparing these two time-smoothed equations with Eqs. (3.6-24) and (3.10-16), we see that the time-smoothed values everywhere replace the instantaneous values. However, in Eq. (3.10-20) new terms arise in the set of brackets which are related to turbulent velocity fluctuations. For convenience we use the notation

Equation 3.10-21


These are the components of the turbulent momentum flux and are called Reynolds stresses.

3.10F. Prandtl Mixing Length

The equations derived for turbulent flow must be solved to obtain velocity profiles. To do this, more simplifications must be made before the expressions for the Reynolds stresses can be evaluated. A number of semiempirical equations have been used; the eddy-diffusivity model of Boussinesq is one early attempt to evaluate these stresses. By analogy to the equation for shear stress in laminar flow, τyx = -μ(dvx/dy), the turbulent shear stress can be written as

Equation 3.10-22


where ηt is a turbulent or eddy viscosity, which is a strong function of position and flow. This equation can also be written as follows:

Equation 3.10-23


where εt = ηt/ρ is eddy diffusivity of momentum in m2/s, by analogy to the momentum diffusivity μ/ρ for laminar flow.

In his mixing-length model Prandtl developed an expression to evaluate these stresses by assuming that eddies move in a fluid in a manner similar to the movement of molecules in a gas. The eddies move a distance called the mixing length L before they lose their identity.

Actually, the moving eddy or "lump" of fluid will gradually lose its identity. However, in the definition of the Prandtl mixing length L, this small packet of fluid is assumed to retain its identity while traveling the entire length L and then lose its identity or be absorbed in the host region.

Prandtl assumed that the velocity fluctuation is due to a "lump" of fluid moving a distance L in the y direction and retaining its mean velocity. At point L, the lump of fluid will differ in mean velocity from the adjacent fluid by Then the value of is

Equation 3.10-24


The length L is small enough that the velocity difference can be written as

Equation 3.10-25


Hence,

Equation 3.10-26


Prandtl also assumed Then the time average, is

Equation 3.10-27


The minus sign and the absolute value were used to make the quantity agree with experimental data. Substituting Eq. (3.10-27) into (3.10-21),

Equation 3.10-28


Comparing with Eq. (3.10-23),

Equation 3.10-29


3.10G. Universal Velocity Distribution in Turbulent Flow

To determine the velocity distribution for turbulent flow at steady state inside a circular tube, we divide the fluid inside the pipe into two regions: a central core where the Reynolds stress approximately equals the shear stress; and a thin, viscous sublayer adjacent to the wall where the shear stress is due only to viscous shear and the turbulence effects are assumed negligible. Later we include a third region, the buffer zone, where both stresses are important.

Dropping the subscripts and superscripts on the shear stresses and velocity, and considering the thin, viscous sublayer, we can write

Equation 3.10-30


where τ0 is assumed constant in this region. On integration,

Equation 3.10-31


Defining a friction velocity as follows and substituting into Eq. (3.10-31),

Equation 3.10-32


Equation 3.10-33


The dimensionless velocity ratio on the left can be written as

Equation 3.10-34


The dimensionless number on the right can be written as

Equation 3.10-35


where y is the distance from the wall of the tube. For a tube of radius r0, y = r0 - r, where r is the distance from the center. Hence, for the viscous sublayer, the velocity distribution is

Equation 3.10-36


Next, considering the turbulent core where any viscous stresses are neglected, Eq. (3.10-28) becomes

Equation 3.10-37


where dv/dy is always positive and the absolute value sign is dropped. Prandtl assumed that the mixing length is proportional to the distance from the wall, or

Equation 3.10-38


and that τ = τ0 = constant. Equation (3.10-37) now becomes

Equation 3.10-39


Hence,

Equation 3.10-40


Upon integration,

Equation 3.10-41


where K1 is a constant. The constant K1 can be found by assuming that ν is zero at a small value of y, say y0:

Equation 3.10-42


Introducing the variable y+ by multiplying the numerator and denominator of the term y/y0 by ν*/ν, where ν = μ/ρ, we obtain

Equation 3.10-43


Equation 3.10-44


A large amount of velocity distribution data by Nikuradse and others for a range of Reynolds numbers of 4000 to 3.2 × 106 have been obtained, and the data fit Eq. (3.10-36) in the region up to y+ of 5 and also fit Eq. (3.10-44) above y+ of 30, with K and C1 being universal constants. For the region of y+ from 5 to 30, which is defined as the buffer region, an empirical equation of the form of Eq. (3.10-44) fits the data. In Fig. 3.10-4 the following relations which are valid are plotted to give a universal velocity profile for fluids flowing in smooth, circular tubes:

Equation 3.10-45


Equation 3.10-46


Equation 3.10-47


Figure 3.10-4. Universal velocity profile for turbulent flow in smooth circular tubes.


Three distinct regions are apparent in Fig. 3.10-4. The first region next to the wall is the viscous sublayer (historically called "laminar" sublayer), given by Eq. (3.10-45), where the velocity is proportional to the distance from the wall. The second region, called the buffer layer, is given by Eq. (3.10-46), and is a region of transition between the viscous sublayer with practically no eddy activity and the violent eddy activity in the turbulent core region given by Eq. (3.10-47). These equations can then be used and related to the Fanning friction factor discussed earlier in this chapter. They can also be used in solving turbulent boundary-layer problems.

3.10H. Integral Momentum Balance for Boundary-Layer Analysis

1. Introduction and derivation of integral expression

In the solution for the laminar boundary layer on a flat plate, the Blasius solution is quite restrictive, since it is only for laminar flow over a flat plate. Other, more complex systems cannot be solved by this method. An approximate method developed by von Kármán can be used when the configuration is more complicated or the flow is turbulent. This is an approximate momentum integral analysis of the boundary layer using an empirical or assumed velocity distribution.

In order to derive the basic equation for a laminar or turbulent boundary layer, a small control volume in the boundary layer on a flat plate is used, as shown in Fig. 3.10-5. The depth in the z direction is b. Flow is only through the surfaces A1 and A2 and also from the top curved surface at δ. An overall integral momentum balance using Eq. (2.8-8) and overall integral mass balance using Eq. (2.6-6) are applied to the control volume inside the boundary layer at steady state, and the final integral expression by von Kármán is (B2, S3)

Equation 3.10-48


Figure 3.10-5. Control volume for integral analysis of the boundary-layer flow.


where τ0 is the shear stress at the surface y = 0 at point x along the plate. Also, δ and τ0 are functions of x.

Equation (3.10-48) is an expression whose solution requires knowledge of the velocity νx as a function of the distance from the surface, y. The accuracy of the results will, of course, depend on how closely the assumed velocity profile approaches the actual profile.

2. Integral momentum balance for laminar boundary layer

Before we use Eq. (3.10-48) for the turbulent boundary layer, this equation will be applied to the laminar boundary layer over a flat plate so that the results can be compared with the exact Blasius solution in Eqs. (3.10-6)-(3.10-12).

In this analysis certain boundary conditions must be satisfied in the boundary layer:

Equation 3.10-49


The conditions above are fulfilled in the following simple, assumed velocity profile:

Equation 3.10-50


The shear stress τ0 at a given x can be obtained from

Equation 3.10-51


Differentiating Eq. (3.10-50) with respect to y and setting y = 0,

Equation 3.10-52


Substituting Eq. (3.10-52) into (3.10-51),

Equation 3.10-53


Substituting Eq. (3.10-50) into Eq. (3.10-48) and integrating between y = 0 and y = δ, we obtain

Equation 3.10-54


Combining Eqs. (3.10-53) and (3.10-54) and integrating between δ = 0 and δ = δ, and x = 0 and x = L,

Equation 3.10-55


where the length of plate is x = L. Proceeding in a manner similar to Eqs. (3.10-6)-(3.10-12), the drag coefficient is

Equation 3.10-56


A comparison of Eq. (3.10-6) with (3.10-55) and (3.10-12) with (3.10-56) shows the success of this method. Only the numerical constants differ slightly. This method can be used with reasonable accuracy for cases where an exact analysis is not feasible.

3. Integral momentum analysis for turbulent boundary layer

The procedures used for the integral momentum analysis for a laminar boundary layer can be applied to the turbulent boundary layer on a flat plate. A simple empirical velocity distribution for pipe flow which is valid up to a Reynolds number of 105 can be adapted for the boundary layer on a flat plate, to become

Equation 3.10-57


This is the Blasius -power law, which is often used.

Equation (3.10-57) is substituted into the integral relation equation (3.10-48):

Equation 3.10-58


The power-law equation does not hold, as y goes to zero at the wall. Another useful relation is the Blasius correlation for shear stress for pipe flow, which is consistent at the wall for the wall shear stress τ0. For boundary-layer flow over a flat plate, it becomes

Equation 3.10-59


Integrating Eq. (3.10-58), combining the result with Eq. (3.10-59), and integrating between δ = 0 and δ = δ, and x = 0 and x = L,

Equation 3.10-60


Integration of the drag force as before gives

Equation 3.10-61


In this development the turbulent boundary layer was assumed to extend to x = 0. Actually, a certain length at the front has a laminar boundary layer. Experimental data agree with Eq. (3.10-61) reasonably well from a Reynolds number of 5 × 105 to 107. More-accurate results at higher Reynolds numbers can be obtained by using a logarithmic velocity distribution, Eqs. (3.10-45)-(3.10-47).

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