Flat plate boundary layer and pipe flow Next we will examine the characteristics of two turbulent flows near solid

3.4.2 Flat plate boundary layer and pipe flow Next we will examine the characteristics of two turbulent flows near solid

walls. Due to the presence of the solid boundary, the flow behaviour and turbulence structure are considerably different from free turbulent flows. Dimensional analysis has greatly assisted in correlating the experimental data. In turbulent thin shear layer flows a Reynolds number based on a length scale L in the flow direction (or pipe radius) Re

2 L (e.g. U is always very large = 1 m/s, L = 0.1 m and ν = 10 m /s gives Re = 10 5

). This implies that the inertia forces are overwhelmingly larger than the viscous forces at these scales.

If we form a Reynolds number based on a distance y away from the wall (Re y = Uy/ ν) we see that if the value of y is of the order of L the above argument holds. Inertia forces dominate in the flow far away from the wall. As y is decreased to zero, however, a Reynolds number based on y will also decrease to zero. Just before y reaches zero there will be a range of values of y for which Re y is of the order of 1. At this distance from the wall and closer the viscous forces will be equal in order of magnitude to inertia forces or larger. To sum up, in flows along solid boundaries there is usually a substantial region of inertia-dominated flow far away from the wall and a thin layer within which viscous effects are important.

Close to the wall the flow is influenced by viscous effects and does not depend on free stream parameters. The mean flow velocity only depends on the distance y from the wall, fluid density ρ and viscosity µ and the wall shear stress τ w . So

U = f( y, ρ, µ, τ w ) Dimensional analysis shows that

Formula (3.16) is called the law of the wall and contains the definitions of two important dimensionless groups, u + and y + . Note that the appropriate velocity scale is u τ = τ w / ρ, the so-called friction velocity.

Far away from the wall we expect the velocity at a point to be influenced by the retarding effect of the wall through the value of the wall shear stress, but not by the viscosity itself. The length scale appropriate to this region is the boundary layer thickness δ. In this region we have

U = g(y, δ, ρ, τ w )

58 CHAPTER 3 TURBULENCE AND ITS MODELLING

Dimensional analysis yields U

=g u

C δ F The most useful form emerges if we view the wall shear stress as the cause of a velocity deficit U max − U which decreases the closer we get to the edge of the boundary layer or the pipe centreline. Thus

U max −U

=g u

B E (3.17)

This formula is called the velocity-defect law. Linear or viscous sub-layer --- the fluid layer in contact with a

smooth wall At the solid surface the fluid is stationary. Turbulent eddying motions must also

stop very close to the wall and the behaviour of the fluid closest to the wall is dominated by viscous effects. This viscous sub-layer is in practice extremely thin ( y + < 5) and we may assume that the shear stress is approximately con- stant and equal to the wall shear stress τ w throughout the layer. Thus

∂U τ( y) = µ

≅ τ w ∂y

After integration with respect to y and application of boundary condition U = 0 if y = 0, we obtain a linear relationship between the mean velocity and the distance from the wall

After some simple algebra and making use of the definitions of u + and y + this leads to

u + =y + (3.18) Because of the linear relationship between velocity and distance from the wall

the fluid layer adjacent to the wall is also known as the linear sub-layer. Log-law layer --- the turbulent region close to a smooth wall

Outside the viscous sublayer (30 <y + < 500) a region exists where viscous and turbulent effects are both important. The shear stress τ varies slowly with distance from the wall, and within this inner region it is assumed to be constant and equal to the wall shear stress. One further assumption regard- ing the length scale of turbulence (mixing length ᐉ m = κy, see section 3.7.1 and Schlichting, 1979) allows us to derive a functional relationship between u + and y + that is dimensionally correct:

u + = ln( y + ) +B= ln(Ey + ) (3.19) κ

3.4 CHARACTERISTICS OF SIMPLE TURBULENT FLOWS 59

Numerical values for the constants are found from measurements. We find von Karman’s constant κ ≈ 0.4 and the additive constant B ≈ 5.5 (or E ≈ 9.8) for smooth walls; wall roughness causes a decrease in the value of B. The values of κ and B are universal constants valid for all turbulent flows past smooth walls at high Reynolds number. Because of the logarithmic relation- ship between u + and y + , formula (3.18) is often called the log-law, and the

layer where y + takes values between 30 and 500 the log-law layer. Outer layer --- the inertia-dominated region far from the wall

Experimental measurements show that the log-law is valid in the region

0.02 < y/ δ < 0.2. For larger values of y the velocity-defect law (3.17) provides the correct form. In the overlap region the log-law and velocity- defect law have to be equal. Tennekes and Lumley (1972) show that a matched overlap is obtained by assuming the following logarithmic form:

where A is a constant. This velocity-defect law is often called the law of the

wake.

Figure 3.11 from Schlichting (1979) shows the close agreement between theoretical equations (3.18) and (3.19) in their respective areas of validity and experimental data.

Figure 3.11 Velocity distribution near a solid wall Source: Schlichting, H. (1979) Boundary Layer Theory, 7th edn,

reproduced with permission of The McGraw-Hill Companies

The turbulent boundary layer adjacent to a solid surface is composed of two regions:

• The inner region: 10 –20% of the total thickness of the wall layer; the shear stress is (almost) constant and equal to the wall shear stress τ w . Within this region there are three zones. In order of increasing distance from the wall we have: – the linear sub-layer: viscous stresses dominate the flow adjacent to

surface – the buffer layer: viscous and turbulent stresses are of similar magnitude – the log-law layer: turbulent (Reynolds) stresses dominate.

60 CHAPTER 3 TURBULENCE AND ITS MODELLING

• The outer region or law-of-the-wake layer: inertia-dominated core flow

far from wall; free from direct viscous effects. Figure 3.12 shows the mean velocity and turbulence property distribution

data for a flat plate boundary layer with a constant imposed pressure (Klebanoff, 1955).

Figure 3.12 Distribution of mean velocity and second

moments , , and u ′ 2 v ′ 2 w ′ 2 − u ′v′ for flat plate boundary layer

The mean velocity is at a maximum far away from the wall and sharply decreases in the region y/ δ ≤ 0.2 due to the no-slip condition. High values of , , and u ′ 2 v ′ 2 w ′ 2 − u ′v′ are found adjacent to the wall where the large mean velocity gradients ensure that turbulence production is high. The eddying motions and associated velocity fluctuations are, however, also subject to the no-slip condition at the wall. Therefore all turbulent stresses decrease sharply to zero in this region. The turbulence is strongly anisotropic near

the wall since the production process mainly creates component u ′ 2 . This is borne out by the fact that this is the largest of the mean-squared fluctuations in Figure 3.12.

In the case of the flat plate boundary layer the turbulence properties asymptotically tend towards zero as y/ δ increases above a value of 0.8. The r.m.s. values of all fluctuating velocities become almost equal here, indi- cating that the turbulence structure becomes more isotropic far away from the wall. In pipe flows, on the other hand, the eddying motions transport turbulence across the centreline from areas of high production. Therefore, the r.m.s. fluctuations remain comparatively large in the centre of a pipe. By symmetry the value of − u ′v′ has to go to zero and change sign at the centreline.

This multi-layer structure is a universal feature of turbulent boundary layers near solid surfaces. Monin and Yaglom (1971) plotted data from Klebanoff (1955) and Laufer (1952) in the near-wall region and found not only the universal mean velocity distribution but also that data for second

moments , , and u ′ 2 v ′ 2 w ′ 2 − u ′v′ for flat plates and pipes collapse onto a single curve if they are non-dimensionalised with the correct velocity scale u τ . Between these distinct layers there are intermediate zones which ensure

that the various distributions merge smoothly. Interested readers may find further details including formulae which cover the whole inner region and the log-law/law-of-the-wake layer in Schlichting (1979) and White (1991).

3.5 EFFECT OF FLUCTUATIONS ON THE MEAN FLOW 61