Jan 15, 2009 - where ReL is the Reynolds number of the plate based on the length of the plate in the streamwise direction. Even though the variation of cf with ...
Turbulent Boundary Layer
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9.5 Turbulent Boundary Layer Understanding the mechanics of the turbulent boundary layer is important because in the majority of practical problems it is the turbulent boundary layer that is primarily responsible for surface shear force, or surface resistance. In this section the velocity distribution in the turbulent boundary layer on a flat plate oriented parallel to the flow is presented. The correlations for boundary-layer thickness and shear stress are also included.
Velocity Distribution The velocity distribution in the turbulent boundary layer is more complicated than the laminar boundary layer. The turbulent boundary has three zones of flow that require different equations for the velocity distribution in each zone, as opposed to the single relationship of the laminar boundary layer. Figure 9.6 shows a portion of a turbulent boundary layer in which the three different zones of flow are identified. The zone adjacent to the wall is the viscous sublayer; the zone immediately above the viscous sublayer is the logarithmic region; and, finally, beyond that region is the velocity defect region. Each of these velocity zones will be discussed separately.
Figure 9.6 Sketch of zones in turbulent boundary layer.
Viscous Sublayer The zone immediately adjacent to the wall is a layer of fluid that is essentially laminar because the presence of the wall dampens the cross-stream mixing and turbulent fluctuations. This very thin layer is called the viscous sublayer. This thin layer behaves as a Couette flow introduced in Section 9.1. In the viscous sublayer, τ is virtually constant and equal to the shear stress at the wall, τ0. Thus du/dy = τ0/µ, which on integration yields (9.19) Dividing the numerator and denominator by ρ gives
(9.20)
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The combination of variables has the dimensions of velocity and recurs again and again in derivations involving boundary-layer theory. It has been given the special name Shear velocity. The shear velocity (which is also sometimes called Friction velocity is symbolized as u* Thus, by definition, (9.21) Now, substituting u* for sublayer:
in Eq. (9.20), yields the nondimensional velocity distribution in the viscous
(9.22) Experimental results show that the limit of viscous sublayer occurs when yu*/ν is approximately 5. Consequently, the thickness of the viscous sublayer, identified by δ′, is given as (9.23) The thickness of the viscous sublayer is very small (typically less than one-tenth the thickness of a dime). The thickness of the viscous sublayer increases as the wall shear stress decreases in the downstream direction.
The Logarithmic Velocity Distribution The flow zone outside the viscous sublayer is turbulent; therefore, a completely different type of flow is involved. The mixing action of turbulence causes small fluid masses to be swept back and forth in a direction transverse to the mean flow direction. A small mass of fluid swept from a low-velocity zone next to the viscous sublayer into a higher-velocity zone farther out in the stream has a retarding effect on the higher-velocity stream. Similarly, a small mass of fluid that originates farther out in the boundary layer in a high-velocity flow zone and is swept into a region of low velocity has the effect of accelerating the lower-velocity fluid. Although the process just described is primarily a momentum exchange phenomenon, it has the same effect as applying a shear stress to the fluid; thus in turbulent flow these “stresses” are termed apparent shear stresses, or Reynolds stresses after the British scientist-engineer who first did extensive research in turbulent flow in the late l800s. The mixing action of turbulence causes the velocities at a given point in a flow to fluctuate with time. If one places a velocity-sensing device, such as a hot-wire anemometer, in a turbulent flow, one can measure a fluctuating velocity, as illustrated in Fig. 9.7. It is convenient to think of the velocity as composed of two parts: a mean value, , plus a fluctuating part, u′. The fluctuating part of the velocity is responsible for the mixing action and the momentum exchange, which manifests itself as an apparent shear stress as noted previously. In fact, the apparent shear stress is related to the fluctuating part of the velocity by (9.24) where u′ and ν′ refer to the x and y components of the velocity fluctuations, respectively, and the bar over these terms denotes the product of u′ ν′ averaged over a period of time.* The expression for apparent shear stress is not very useful in this form, so Prandtl developed a theory to relate the apparent shear stress to the temporal mean velocity distribution.
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Figure 9.7 Velocity fluctuations in turbulent flow. The theory developed by Prandtl is analogous to the idea of molecular transport creating shear stress presented in Chapter 2. In the turbulent boundary layer, the principal flow is parallel to the boundary. However, because of turbulent eddies, there are fluctuating components transverse to the principal flow direction. These fluctuating velocity components are associated with small masses of fluid, as shown in Fig. 9.8, that move across the boundary layer. As the mass moves from the lower-velocity region to the higher-velocity region it tends to retain its original velocity. The difference in velocity between the surrounding fluid and the transported mass is identified as the fluctuating velocity component u′. For the mass shown in Fig. 9.8, u′ would be negative and approximated by†
where du /dy is the mean velocity gradient and ℓ is the distance the small fluid mass travels in the transverse direction. Prandtl identified this distance as the “mixing length.” Prandtl assumed that the magnitude of the transverse fluctuating velocity component is proportional to the magnitude of the fluctuating component in the |u′| which seems to be a reasonable assumption because both components arise principal flow direction: |ν′| from the same set of eddies. Also, it should be noted that a positive ν′ will be associated with a negative u′ so the product will be negative. Thus the apparent shear stress can be expressed as (9.25) A more general form of Eq. (9.25) is
which ensures that the sign for the apparent shear stress is correct.
Figure 9.8 Concept of mixing length.
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The theory leading to Eq. (9.25) is called Prandtl's mixing-length theory and is used extensively in analyses involving turbulent flow.* Prandtl also made the important and clever assumption that the mixing length is proportional to the distance from the wall (ℓ = κy) for the region close to the wall. If one considers the velocity distribution in a boundary layer where du/dy is positive, as is shown in Fig. 9.8, and substitutes κy for ℓ, then Eq. (9.25) reduces to
For the zone of flow near the boundary, it is assumed that the shear stress is uniform and approximately equal to the shear stress at the wall. Thus the foregoing equation becomes (9.26) Taking the square root of each side of Eq. (9.26) and rearranging yields
Integrating the above equation and substituting u* for
gives (9.27)
Experiments on smooth boundaries indicate that the constant of integration C can be given in terms of u*, ν, and a pure number as
When this expression for C is substituted into Eq. (9.27), the result is (9.28) In Eq. (9.28), κ has sometimes been called the universal turbulence constant, or Karman's constant. Experiments show that this constant is approximately 0.41 3 for the turbulent zone next to the viscous sublayer. Introducing this value for κ into Eq. (9.28) gives the logarithmic velocity distribution (9.29) Obviously the region where this model is valid is limited because the mixing length cannot continuously increase to the boundary layer edge. This distribution is valid for values of yu*/ν ranging from approximately 30 to 500. The region between the viscous sublayer and the logarithmic velocity distribution is the buffer zone. There is no equation for the velocity distribution in this zone, although various empirical expressions have been developed 6. However, it is common practice to extrapolate the velocity profile for the viscous sublayer to larger values of yu*/ν and the logarithmic velocity profile to smaller values of yu*/ν until the velocity profiles intersect as shown in Fig. 9.9. The intersection occurs at yu*/ν = 11.84 and is regarded as the demarcation between the viscous sublayer and the logarithmic profile. The “nominal” thickness of the viscous sublayer is (9.30)
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Figure 9.9 Velocity distribution in a turbulent boundary layer. The combination of the viscous and logarithmic velocity profile for the range of yu*/ν from 0 to approximately 500 is called the law of the wall. Making a semilogarithmic plot of the velocity distribution in a turbulent boundary layer, as shown in Fig. 9.9, makes it straightforward to identify the velocity distribution in the viscous sublayer and in the region where the logarithmic equation applies. However, the logarithmic nature of this plot accentuates the nondimensional distance yu*/ν near the wall. A better perspective of the relative extent of the regions is obtained by plotting the graph on a linear scale, as shown in Fig. 9.10. From this plot one notes that the laminar sublayer and buffer zone are a very small part of the thickness of the turbulent boundary layer.
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Figure 9.10 Velocity distribution in a turbulent boundary layer—linear scales.
Velocity Defect Region For y/δ > 0.15 and yu*/ν > 500 the velocity profile corresponding to the law of the wall becomes increasingly inadequate to match experimental data, so a third zone, called the velocity defect region, is identified. The velocity in this region is represented by the velocity defect law, which for a flat plate with zero pressure gradient is simply expressed as (9.31) and the correlation with experimental data is plotted in Fig. 9.11. At the edge of the boundary layer y = δ and (U0 - u)/u* = 0, so u = U0, or the free-stream velocity. This law applies to rough as well as smooth surfaces. However, the functional relationship has to be modified for flows with free-stream pressure gradients.
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Figure 9.11 Velocity defect law for boundary layers on flat plate (zero pressure gradient). [After Rouse 6.] As shown in Fig. 9.9, the demarcation between the law of the wall and the velocity defect regions is somewhat arbitrary, so there is considerable overlap between the two regions. The three zones of the turbulent boundary layer and their range of applicability are summarized in Table 9.2. Table 9.2
ZO"ES FOR TURBULE"T BOU"DARY LAYER O" FLAT PLATE Zone
Velocity Distribution
Range
Viscous Sublayer Logarithmic Velocity Distribution Velocity Defect Law
Power-Law Formula for Velocity Distribution Analyses have shown that for a wide range of Reynolds numbers (105 < Re < 107), the velocity profile in the turbulent boundary layer on a flat plate is approximated reasonably by the power-law equation (9.32)
Comparisons with experimental results show that this formula conforms to those results very closely over about 90% of the boundary layer (0.1 < y/δ < 1). Obviously it is not valid at the surface because (du)/(dy)|y=0 → ∞, which implies infinite surface shear stress. For the inner 10% of the boundary layer, one must resort to equations for the law of the wall (see Fig. 9.9) to obtain a more precise prediction of velocity. Because Eq. (9.32) is valid over the major portion of the boundary layer, it is used to advantage in deriving the overall thickness of the
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boundary layer as well as other relations for the turbulent boundary layer. Example 9.5 illustrates the application of various equations to calculate the velocity in the turbulent boundary layer.
EXAMPLE 9.5 TURBULE"T BOU"DARY-LAYER PROPERTIES Water (60°F) flows with a velocity of 20 ft/s past a flat plate. The plate is oriented parallel to the flow. At a particular section downstream of the leading edge of the plate, the boundary layer is turbulent, the shear stress on the plate is 0.896 lbf/ft2, and the boundary-layer thickness is 0.0880 ft. Find the velocity of the water at a distance of 0.0088 ft from the plate as determined by (a) The logarithmic velocity distribution (b) The velocity defect law (c) The power-law formula Also, what is the nominal thickness of the viscous sublayer?
Problem Definition Situation: Water flows past a flat plate oriented parallel to the flow. At a point downstream of the leading edge of the plate, shear stress on the plate is 0.896 lbfft2 and boundary layer thickness is 0.0880 ft. Find: 1. Velocity at 0.0088 ft from plate using a. Logarithmic velocity distribution b. Velocity defect law c. Power-law formula 2. Nominal thickness of viscous sublayer Properties: From Table A.5, ρ = 1.94 slugs/ft3, ν = 1.22 × 10-5 ft2/s. Sketch:
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Plan 1. Calculate shear velocity, u*, from Eq. (9.21). 2. Calculate u using Eq. (9.29) for logarithmic profile. 3. Calculate y/δ and find (U0 - u)/u* from Fig. 9.11. 4. Calculate u from (U0 - u)/u* for velocity defect law. 5. Calculate u from Eq.(9.32) for power law. 6. Calculate δ′N from Eq. (9.30).
Solution 1. Shear velocity
2. Logarithmic velocity distribution
3. Nondimensional distance From Fig. 9.11
4. Velocity from defect law
5. Power-law formula
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6. Nominal sublayer thickness
Review Notice that the velocity obtained using logarithmic distribution and defect law are nearly the same, which indicates that the point is in the overlap region.
Boundary-Layer Thickness and Shear-Stress Correlations Unlike the laminar boundary layer, there is no analytically derived equation for the thickness of the turbulent boundary layer. There is a way to obtain an equation by using momentum principles, empirical data for the local shear stress, and by assuming the 1/7 power velocity profile 3. The result is (9.33) where x is the distance from the leading edge of the plate and Rex is U0x/ν. Many empirical expressions have been proposed for the local shear-stress distribution for the turbulent boundary layer on a flat plate. One of the simplest correlations is (9.34a) and the corresponding average shear-stress coefficient is (9.34b) where ReL is the Reynolds number of the plate based on the length of the plate in the streamwise direction. Even though the variation of cf with Reynolds number given by Eq. (9.34a) provides a reasonably good fit with experimental data for Reynolds numbers less than 107, it tends to underpredict the skin friction at higher Reynolds numbers. There are several correlations that have been proposed in the literature; see the review by Schlichting 4. A correlation proposed by White 3 that fits the data for turbulent Reynolds numbers up to 1010 is (9.35)
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The corresponding average shear-stress coefficient is (9.36) These are the correlations for shear-stress coefficients recommended here. The boundary layer on a flat plate is composed of both a laminar and turbulent part. The purpose here is to develop a correlation valid for the combined boundary layer. As noted in Section 9.3, the boundary layer on a flat plate consists first of a laminar boundary layer that grows in thickness, develops instability, and becomes turbulent. A turbulent boundary layer develops over the remainder of the plate. As discussed earlier in Section 9.4, the transition from a laminar to turbulent boundary layer is not immediate but takes place over a transition length. However for the purposes of analysis here it is assumed that transition occurs at a point corresponding to a transition Reynolds number, Retr, of about 500,000. The idea here is to take the turbulent shear force for length L, Fs, turb (L) assuming the boundary layer is turbulent from the leading edge, subtract the portion up to the transition point, Fs, turb (Ltr) and replace it with the laminar shear force up to the transition point Fs, lam (Ltr). Thus the composite shear force on the plate is Substituting in Eq. (9.18) for laminar flow and Eq. (9.36) for turbulent flow over a plate of width B gives (9.37) where Retr is the Reynolds number at the transition, ReL is the Reynolds number at the end of the plate, and Ltr is the distance from the leading edge of the plate to the transition zone. Expressing the resistance force in terms of the average shear-stress coefficient,
, gives
Here Ltr/L = Retr/ReL. Therefore,
Finally, for Retr = 500,000, the equation for average shear-stress coefficient becomes (9.38)
The variation of Cf with Reynolds number is shown by the solid line in Fig. 9.12. This curve corresponds to a boundary layer that begins as a laminar boundary layer and then changes to a turbulent boundary layer after the transition Reynolds number. This is the normal condition for a flat-plate boundary layer. Table 9.3 summarizes the equations for boundary-layer-thickness, and for local shear-stress and average shear-stress coefficients for the boundary layer on a flat plate.
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Figure 9.12 Average shear-stress coefficients. Table 9.3
SUMMARY OF EQUATIO"S FOR BOU"DARY LAYER O" A FLAT PLATE Laminar Flow Re x, Re L < 5 × 10
5
Turbulent Flow Re x, Re L ≥ 5 × 105
Boundary-Layer Thickness, δ Local Shear-Stress Coefficient, cf Average Shear-Stress Coefficient, Cf Example 9.6 shows the calculation of surface resistance due to a boundary layer on a flat plate.
EXAMPLE 9.6 LAMI"AR/TURBULE"T BOU"DARY LAYER O" FLAT PLATE Assume that air 20°C and normal atmospheric pressure flows over a smooth, flat plate with a velocity of 30 m/s. The initial boundary layer is laminar and then becomes turbulent at a transitional Reynolds number of 5 × 105. The plate is 3 m long and 1 m wide. What will be the average resistance coefficient Cf for the plate? Also, what is the total shearing resistance of one side of the plate, and what will be the resistance due to the turbulent part and the laminar part of the boundary layer?
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Problem Definition Situation: Air flows past a flat plate parallel to the flow with a velocity of 30 m/s. Find: 1. Average shear-stress coefficient, Cf, for the plate. 2. Total shear force (in newtons) on one side of plate. 3. Shear force (in newtons) due to laminar part. 4. Shear force (in newtons) due to turbulent part. Assumptions: The leading edge of the plate is sharp, and the boundary is not tripped on the leading edge. Sketch:
Properties: From Table A.3,
Plan 1. Calculate the Reynolds number based on plate length, ReL. 2. Calculate Cf using Eq. (9.38). 3. Calculate the total shear force on one side of plate using 4. Using value for transition Reynolds number, find transition point. 5. Use Eq. (9.18) to find average shear-stress coefficient for laminar portion. 6. Calculate shear force for laminar portion. 7. Subtract laminar portion from total shear force.
Solution 1. Reynolds number based on plate length
2. Average shear-stress coefficient
3. Total shear force
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4. Transition point
5. Laminar average shear-stress coefficient
6. Laminar shear force
7. Turbulent shear force
If the boundary layer is “tripped” by some roughness or leading-edge disturbance (such as a wire across the leading edge), the boundary layer is turbulent from the leading edge. This is shown by the dashed line in Fig. 9.12. For this condition the boundary layer thickness, local shear-stress coefficient, and average shear-stress coefficient are fit reasonably well by Eqs. (9.33), (9.34a), and (9.34b). (9.39) which are valid up to a Reynolds number of 107. For Reynolds numbers beyond 107, the average shear-stress coefficient given by Eq. (9.36) can be used. It is of interest to note that marine engineers incorporate tripping mechanisms for the boundary layer on ship models to produce a boundary layer that can be predicted more precisely than a combination of laminar and turbulent boundary layers. Example 9.7 illustrates calculating surface resistance with a tripped boundary layer.
EXAMPLE 9.7 RESISTA"CE FORCE WITH TRIPPED BOU"DARY LAYER Air at 20°C flows past a smooth, thin plate with a free-stream velocity of 20 m/s. Plate is 3 m wide and 6 m long in the direction of flow and boundary layer is tripped at the leading edge.
Problem Definition Situation: Air flows past a smooth, thin plate. Boundary layer is tripped at leading edge. Find: Total shear resistance (in newtons) on both sides of plate.
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Sketch:
Assumptions: From Table A.3,
Plan 1. Calculate the Reynolds number based on plate length. 2. Find average shear-stress coefficient from Fig. 9.12. 3. Calculate surface resistance for both sides of plate.
Solution 1. Reynolds number
Reynolds number is less than 107. 2. Average shear-stress coefficient
3. Resistance force
Even though the equations in this chapter have been developed for flat plates, they are useful for engineering estimates for some surfaces that are not truly flat plates. For example, the skin friction drag of the submerged part of the hull of a ship can be estimated with Eq. (9.38).
Copyright © 2009 John Wiley & Sons, Inc. All rights reserved.
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