Ch 9. Wall Turbulence; Boundary-Layer Flows
Chapter 9 Wall Turbulence. Boundary-Layer Flows
9.1 Introduction
• Turbulence occurs most commonly in shear flows.
• Shear flow: spatial variation of the mean velocity
1) wall turbulence: along solid surface → no-slip condition at surface
2) free turbulence: at the interface between fluid zones having different velocities, and at boundaries of a jet → jet, wakes
• Turbulent motion in shear flows
- self-sustaining
- Turbulence arises as a consequence of the shear.
- Shear persists as a consequence of the turbulent fluctuations.
→ Turbulence can neither arise nor persist without shear.
9.2 Structure of a Turbulent Boundary Layer
9.2.1 Boundary layer flows
(i) Smooth boundary
Consider a fluid stream flowing past a smooth boundary.
→ A boundary-layer zone of viscous influence is developed near the boundary.
1) Re Recrit
→ The boundary-layer is initially laminar.
u u() y
9-1 Ch 9. Wall Turbulence; Boundary-Layer Flows
2) Re Recrit
→ The boundary-layer is turbulent.
u u() y
→ Turbulence reaches out into the free stream to entrain and mix more fluid.
→ thicker boundary layer: turb 4 lam
External flow
Turbulent boundary layer
xx crit , total friction = laminar
xx crit , total friction = laminar + turbulent
9-2 Ch 9. Wall Turbulence; Boundary-Layer Flows
(ii) Rough boundary
→ Turbulent boundary layer is established near the leading edge of the boundary
without a preceding stretch of laminar flow.
Turbulent boundary layer
Laminar sublayer
9.2.2 Comparison of laminar and turbulent boundary-layer profiles
For the flows of the same Reynolds number (Rex 500,000)
Ux Re x
9-3 Ch 9. Wall Turbulence; Boundary-Layer Flows
turb 3.9 lam
2) Mass displacement thickness, *
h u Eq. (8.9): * (1 )dy 0 U
* turb * 1.41 lam
3) Momentum thickness, θ
h uu Eq. (8.10): (1dy ) 0UU
turb 2.84 lam
→ Because of the higher flux of mass and momentum through the zone nearest the wall
for turbulent flow, increases of * and θ rate are not as large as δ.
9.2.3 Intermittent nature of the turbulent layer
• Outside a boundary layer
→ free-stream shearless flow (U) → potential flow (inviscid)
→ slightly turbulent flow
→ considered to be non-turbulent flow relative to higher turbulence inside a turbulent
boundary layer
9-4 Ch 9. Wall Turbulence; Boundary-Layer Flows
• Interior of the turbulent boundary layer (δ)
~ consist of regions of different types of flow (laminar, buffer, turbulent)
~ Instantaneous border between turbulent and non-turbulent fluid is irregular and changing.
~ Border consists of fingers of turbulence extending into the non-turbulent fluid and fingers of non-turbulent fluid extending deep into the turbulent region.
~ intermittent nature of the turbulent layer
• Intermittency factor, Ω
Ω = fraction of time during which the flow is turbulent
Ω = 1.0, deep in the boundary layer
= 0, in the free stream
tt 0
t t0 t
①Average position of the turbulent-nonturbulent interface = 0.78 δ
②Maximum stretch of interface = 1.2 δ
③Minimum stretch of interface = 0.4 δ
9-5 Ch 9. Wall Turbulence; Boundary-Layer Flows
9-6 Ch 9. Wall Turbulence; Boundary-Layer Flows
• Turbulent energy in a boundary layer, δ
u'''2 v 2 w 2 - Dimensionless energy 2 (9.1) u*
where u 0 = shear velocity *
U Re 73,000 Re 4 106 for turbulent layer x
u'''2 v 2 w 2 222,,, uuu'''***
- smooth wall → v'0 at wall
- rough wall → v'0 at wall
- smooth & rough wall
→ turbulent energy ≠ 0 at y =
9-7 Ch 9. Wall Turbulence; Boundary-Layer Flows
9.3 Mean-flow characteristics for turbulent boundary layer
○ Relations describing the mean-flow characteristics
→ predict velocity magnitude and relation between velocity and wall shear or pressure gradient forces
→ It is desirable that these relations should not require knowledge of the turbulence details.
○ Turbulent boundary layer
→ is composed of zones of different types of flow
→ Effective viscosity ( ) varies from wall out through the layer.
→ Theoretical solution is not practical for the general nonuniform boundary layer
→ use semiempirical procedure
9.3.1 Universal velocity and friction laws: smooth walls
(1) Velocity-profile regions
logarithmic
linear
9-8 Ch 9. Wall Turbulence; Boundary-Layer Flows
uy 1) laminar sublayer: 04*
du ~ linear dy
u u y → * (9.6) u*
~ Mean shear stress is controlled by the dynamic molecular viscosity .
→ Reynolds stress is negligible. → Mean flow is laminar.
~ energy of velocity fluctuation ≈ 0
uy 2) buffer zone: 4* 30 ~ 70
~ Viscous and Reynolds stress are of the same order.
→ Both laminar flow and turbulence flow exist.
~ Sharp peak in the turbulent energy occurs (Fig. 9.4).
uy 3) turbulent zone - inner region: * 30 ~ 70 ,and y 0.15
~ fully turbulent flow
~ inner law zone/logarithmic law
~ Intensity of turbulence decreases.
~ velocity equation: logarithmic function
u u* y 5.6 log C2 u*
9-9 Ch 9. Wall Turbulence; Boundary-Layer Flows
4) turbulent zone-outer region: 0.15y 0.4
~ outer law, velocity-defect law
5) intermittent zone: 0.4y 1.2
~ Flow is intermittently turbulent and non-turbulent.
6) non-turbulent zone
~ external flow zone
~ potential flow
[Cf] Rough wall:
→ Laminar sublayer is destroyed by the roughness elements.
9-10 Ch 9. Wall Turbulence; Boundary-Layer Flows
z
x
9-11 Ch 9. Wall Turbulence; Boundary-Layer Flows
(2) Wall law and velocity-defect law
wall law → inner region
velocity-defect law → outer region
uy 1) Law of wall = inner law; * 30 ~ 70 ;y 0.15
∼close to smooth boundaries (molecular viscosity dominant)
∼ Law of wall assumes that the relation between wall shear stress and velocity u at distance y from the wall depends only on fluid density and viscosity;
f( u , u* , y , , ) 0
Dimensional analysis yields
u u* y f (9.3) u*
i) Laminar sublayer
- mean velocity, uu
uu - velcocity gradient, ~ constant yy
uu - shear stress, 0 (9.5) yyy0
u - shear velocity, uu 2 ** y
u u y * (9.6) u*
9-12 Ch 9. Wall Turbulence; Boundary-Layer Flows
■ Thickness of laminar sublayer define thickness of laminar sublayer ( ' ) as the value of y which makes
uy * 4
4 4 4 4 ' 1 (9.7) u* 0 2 2 Ucf /2 cUf /2
u2 where c;. c local shear stress coeff 0 ff2
9-13 Ch 9. Wall Turbulence; Boundary-Layer Flows ii) Turbulent zone – inner region
Start with Prandtl's mixing length theory
du du l 2 (1) 0 dy dy
Near wall, ly (2)
2 22du y (3) dy
Rearrange (3)
du1 u * dy y y
u 1 du * dy (4) y
Integrate (4)
u u*ln y C 1
u 1 ln yC1 (9.10) u*
Substitute BC [u 0 at yy ' ] into (9.10)
1 0 lnyC ' (5) 1
9-14 Ch 9. Wall Turbulence; Boundary-Layer Flows
1 Cy ln ' 1
(/)ms2 Assume y'' y C u**(/) m s u
Then (5) becomes
1 1 1 C12 ln y ' ln C C ln (9.11) uu**
Substitute (9.11) into (9.10)
u1 1 1 u* y lny C22 ln ln C uu**
Changing the logarithm to base 10 yields
u2.3 u* y log10C 2 (9.12) u*
Empirical values of and C2 for inner region of the boundary layer
0.41;C2 4.9
u u** y u y y 5.6 log 4.9, 30 ~ 70 ,and 0.15 (9.13) u*
→ Prandtl's velocity distribution law; inner law; wall law
9-15 Ch 9. Wall Turbulence; Boundary-Layer Flows
2) Velocity-defect law
∼outer law
∼outer reaches of the turbulent boundary layer for both smooth and rough walls
→ Reynolds stresses dominate the viscous stresses.
Assume velocity defect (reduction) at y wall shear stress
U u y g (9.14) u*
Substituting BC [ u U at y ] into Eq. (9.12) leads to
Uu2.3 * ' logC2 (9.15) u*
Subtract (9.12) from (9.15)
Uu2.3 * logC2 ' u*
u2.3 u* y log C2 u*
U u2.3 u** u y log log CC22 ' u*
2.3 u* logC3 uy* 2.3 y log C3 where and C3 are empirical constants
9-16 Ch 9. Wall Turbulence; Boundary-Layer Flows
y i) Inner region; 0.15
→ 0.41 ,C3 2.5
U u y 5.6log 2.5 (9.16) u*
y ii) Outer region; 0.15
→ = 0.267, C3 = 0
U u y 8.6 log (9.17) u*
→ Eqs. (9.16) & (9.17 ) apply to both smooth and rough surfaces.
→ Eq. (9.16) = Eq. (9.13)
Fig. 9.8 → velocity-defect law is applicable for both smooth and rough walls
9-17 Ch 9. Wall Turbulence; Boundary-Layer Flows
Table 9-1
■ For smooth walls
u u y a) laminar sublayer: * u*
→ viscous effect dominates. b) buffer zone
u u y c) logarithmic zone: 5.6log* 4.9 u*
→ turbulence effect dominates. d) inner law (law of wall) region
9-18 Ch 9. Wall Turbulence; Boundary-Layer Flows
■ For both smooth and rough walls
U u y e) inner region: 5.6 log 2.5 u*
U u y f) outer region: 8.6 log u* g) outer law (velocity - defect law) region
9-19 Ch 9. Wall Turbulence; Boundary-Layer Flows
(3) Surface-resistance formulas
1) local shear-stress coefficient on smooth walls velocity profile ↔ shear-stress equations
c f U 2 uU* / (9.18) 2 uc* f
where c f = local shear - stress coeff.
1 c [Re] c U22 0 f U 0 22f
c uU0 f * 2
i) Apply logarithmic law
Substituting into @ y , u U into Eq. (9.12) yields
Uu2.3 * logC4 (A) u*
Substitute (9.18) into (A)
2 2.3 U c logf C (9.19) c 2 4 f
~ c f is not given explicitly.
9-20 Ch 9. Wall Turbulence; Boundary-Layer Flows ii) For explicit expression, use displacement thickness * and momentum thickness θ instead of δ
1 Clauser: 3.96 log Re 3.04 (9.20) * c f
1 Squire and Young: 4.17 log Re 2.54 (9.21) c f
UU* where Re ; Re *
Ux Re , Ref (Re ),Re * xx
iii) Karman's relation
~ assume turbulence boundary layer all the way from the leading edge
(i.e., no preceding stretch of laminar boundary layer)
1 4.15log(Rexfc ) 1.7 (9.23) c f
9-21 Ch 9. Wall Turbulence; Boundary-Layer Flows iv) Schultz-Grunow (1940)
0.370 c f 2.58 (9.24) (logRex )
Comparison of (9.23) and (9.24)
2) Average shear-stress coefficient on smooth walls
Consider average shear-stress coefficient over a distance l along a flat plate of a width b
1 total drag() D bl C U2 bl 2 f
D C f bl U 2 /2
U
b
l
9-22 Ch 9. Wall Turbulence; Boundary-Layer Flows i) Schoenherr (1932)
1 4.13log(RelfC ) (9.26) C f
→ implicit
Ul where Re l ii) Schultz-Grunow
0.427 29 C fl2.64 , 10 Re 10 (9.27) (log Rel 0.407)
Comparison of (9.26) and (9.27)
9-23 Ch 9. Wall Turbulence; Boundary-Layer Flows
3) Transition formula
■ Boundary layer developing on a smooth flat plate
∼ At upstream end (leading edge), laminar boundary layer develops because viscous effects dominate due to inhibiting effect of the smooth wall on the development of turbulence.
∼ Thus, when there is a significant stretch of laminar boundary layer preceding the turbulent layer, total friction is the laminar portion up to xcrit plus the turbulent portion from xcrit to l.
∼ Therefore, average shear-stress coefficient is lower than the prediction by Eqs. (9.26) or (9.27).
→ Use transition formula
0.427 A C f 2.64 (9.28) (log Rell 0.407) Re
9-24 Ch 9. Wall Turbulence; Boundary-Layer Flows
Ux where A/ Re = correction term = f (Re ), Re crit l crit crit
→ A = 1,060~3,340 (Table 9.2, p. 240)
Eq. (9.28) falls between the laminar and turbulent curves.
1.328 Laminar flow: C f 1/2 Rel
9-25 Ch 9. Wall Turbulence; Boundary-Layer Flows
[Ex. 9.1] Turbulent boundary-layer velocity and thickness
An aircraft flies at 25,000 ft with a speed of 410 mph (600 ft/s).
Compute the following items for the boundary layer at a distance 10 ft from the leading edge of the wing of the craft.
(a) Thickness ' (laminar sublayer) at x 10 ft
Air at El. 25,000 ft: 3 1042ft / s
1.07 1033slug / ft
600(x ) Select Re 5 105 crit 3 104
xcrit 0.25 ft ~negligible comparaed to l 10 ft
Therefore, assume that turbulent boundary layer develops all the way from the leading
edge. Use Schultz-Grunow Eq., (9.24) to compute c f
Ux 600(10) Re 2 107 x (3 104 )
9-26 Ch 9. Wall Turbulence; Boundary-Layer Flows
0.370 0.370 0.370 c 0.0022 f (logRe )2.587 2.58 (7.30) 2.58 x log 2 10
1 c U2 (1.07 10 3 )(0.0022)(600) 2 0.422 lb / ft 2 0 22f
0.422 u0 19.8 ft / s * 1.07 103
4 4(3 104 ) Eq. (9.7): ' 0.61 1044ft 7.3 10 in u* 19.8
(b) Velocity u at y '
u u y Eq. (9.6): * u*
u2 ' (19.8) 2 (0.61 10 4 ) u * 79.7 ft / s 13% of U (3 104 ) [Cf ] U 600 ft / s y '
(c) Velocity u at y / 0.15
Use Eq. (9.16) – outer law
U u y 5.6 log 2.5 u*
600 u 5.6 log 0.15 2.5 19.8
u600 91.35 49.5 459.2 ft / s 76% of U
9-27 Ch 9. Wall Turbulence; Boundary-Layer Flows
y [Cf] u U 5.6 u** log 2.5 u
(d) Distance y at y / 0.15 and thickness
Use Eq. (9.13) – inner law
u u* y 5.6 log 4.9 u*
y 459 19.8y At 0.15: 5.6 log4 4.9 19.8 3 10
19.8yy 19.8 log44 3.26; 1839 3 10 3 10
y0.028' 0.33 in 0.8 cm (B)
y Substitute (B) into 0.15
y 0.186' 2.24in 5.7 cm 0.15
0.186 Atx 10': 3049 3 103 ' 0.61 104
' 0.0003
9-28 Ch 9. Wall Turbulence; Boundary-Layer Flows
[Ex. 9.2] Surface resistance on a smooth boundary given as Ex.9.1
(a) Displacement thickness *
h * u 1 dy 0 U
* h/ uy 1 dh , / 1 (A) 0 U
Neglect laminar sublayer and approximate buffer zone with Eq. (9.16)
U u y (iy ) / 0.15, 5.6log 2.5 (9.16) u*
u u y u 1 5.6** log 2.5 (B) UUU
U u y (ii ) y / 0.15, 8.6log (9.17) u*
u u* y 1 8.6 log lnxx 2.3log (C) UU
lnx dx x ln x x Substituting (B) and (C) into (A) yields
* 0.15 1.0 y u** y y u y 2.43ln 2.5 dd 3.74ln '/ UU 0.15
0.15 1 u y y y y y y y * 2.43 ln 2.5 3.74 ln U 0.0003 0.15
u (19.8) 3.74* 3.74 0.1184 U 600
9-29 Ch 9. Wall Turbulence; Boundary-Layer Flows
* 0.1184 0.1184 (0.186) 0.022 ft
* 0.1184 11.8%
(b) Local surface-resistance coeff. c f
Use Eq. (9.20) by Clauser
1 600 0.022 3.96 log Re 3.04 Re 44,000 **4 c f 3 10 3.96 log 44,000 3.04
3 c f 2.18 10 0.00218
[Cf] c f = 0.00218 by Schultz-Grunow Eq.
9-30 Ch 9. Wall Turbulence; Boundary-Layer Flows
9.3.2. Power-law formulas: Smooth walls
● Logarithmic equations for velocity profile and shear-stress coeff .
~ universal
~ applicable over almost entire range of Reynolds numbers
● Power-law equations
~ applicable over only limited range of Reynolds numbers
~ simpler
~ explicit relations for uU/ and c f
~ explicit relations for in terms of Re and distance x
(1) Assumptions of power-law formulas
1) Except very near the wall, mean velocity is closely proportional to a root of the distance y from the wall.
1 uy∝ n (A)
2) Shear stress coeff. c f is inversely proportional to a root of Re
1 U ∝ c f m , Re Re
A cf m (9.29) U where A, m = constants
9-31 Ch 9. Wall Turbulence; Boundary-Layer Flows
3.32 [Cf] Eq. (9.29) is similar to equation for laminar boundary layer, c f Re
(2) Derivation of power equation
Combine Eqs. (9.18) and (9.29)
A c c f m (9.18): uU f U * 2
m UU222
uc* f A
m m 1 U 2 2 2
u* A
m m 1 U 2 2 u 2 * u* A
m U u 2 m B * (9.30) u*
Substitute Assumption (1) into Eq. (9.30), replace with y
m u uy 2 m B * (9.31) u*
Divide (9.31) by (9.30)
m uy2 m (9.32) U
9-32 Ch 9. Wall Turbulence; Boundary-Layer Flows
1 For 3,000 < Re < 70,000; m, A 0.0466 B , 8.74 4
1 uy7 (9.33) U
1 U u 7 8.74 * (9.34) u*
1 7 u uy* 8.74 (9.35) u*
0.0466 c f 1 (9.36) Re 4
(3) Relation for
p Adopt integral-momentum eq. for steady motion with 0 x
~ integration of Prandtl's 2-D boundary-layer equations
U 2 0 U22 c c U (9.37) x ff220
where θ = momentum thickness
h uu θ = 1 dy (A) 0 UU
Substitute Eq. (9.33) into (A) and integrate
9-33 Ch 9. Wall Turbulence; Boundary-Layer Flows
1 1 7 h y 7 y dy 0 1 (9.38) 7 0.1 72
Substitute Eqs. (9.36) and (9.38) into (9.37) and integrate w.r.t. x
0.318xx 0.318 , Re 107 (9.39) 11 x 55 Ux Rex
0.059 c , (9.40) f 1 5 Rex
Integrate (9.40) over l to get average coefficient
0.074 c , Re 107 (9.41) f 1 l 5 Rel
[Re] Derivation of (9.39) and (9.40)
U 2 Uc2 x f 2
c f (B) x 2
Substitute (9.38) and (9.36) into (B)
9-34 Ch 9. Wall Turbulence; Boundary-Layer Flows
71 1 0.0466 / (Re )4 x 72 2
7 0.0233 0.0233
72 x 11 Re 44U
0.2397 x 1 U 4
Integrate once w.r.t. x
0.2397 xC 1 U 4
B.C.: 0 at x = 0
0.2397 00C → C 0 1 U 4
0.2397 x 1 U 4
550.2397 44x 1 Ux 4
9-35 Ch 9. Wall Turbulence; Boundary-Layer Flows
4 4 0.23975 5 (0.2397)5 xx4 11 Ux45 Ux
0.318 0.318 xx → Eq. (9.39) 11 55 Ux Rex
0.0466 0.0466 (9-36): c (C) f 11 Re 44U
Substitute (9.39) into (C)
0.0466 0.062 c f 1 1 1 U 40.318 41 4 4 4 5 1 Ux Ux 5 1 U 5
0.062 0.062 0.062 0.062 → 1 1 1 1 1 (9.40) 4 5 51 5 5 Ux U5 Ux Rex 1 x U 20
Integrate (9.40) over l
1l 0.062 0.062 l 1 0.062 l 1 0.076 C dx dx dx f ll01 0 1 1 0 1 1 Re5Ux 5 U 5x 5 Re 5 xll
9-36 Ch 9. Wall Turbulence; Boundary-Layer Flows
9.3.3. Laws for rough walls
(1) Effects of roughness rough walls: velocity distribution and resistance = f (Reynolds number , roughness) smooth walls: velocity distribution and resistance = f (Reynolds number)
▪ For natural roughness, k is random, and statistical quantity
→k = ks = uniform sand grain
9-37 Ch 9. Wall Turbulence; Boundary-Layer Flows
▪ Measurement of roughness effects
a) experiments with sand grains cemented to smooth surfaces
b) evaluate roughness value ≡ height ks
c) compare hydrodynamic behavior with other types and magnitude of roughness
▪ Effects of roughness
k i) s < 1 '
~ roughness has negligible effect on the wall shear
→ hydrodynamically smooth
4 ' = laminar sublayer thickness u*
k ii) s > 1 '
~ roughness effects appear
~ roughness disrupts the laminar sublayer
~ smooth-wall relations for velocity and cf no longer hold
→ hydrodynamically rough
k iii) s > 15 ~25 '
~ friction and velocity distribution depend only on roughness rather than Reynolds number
→ fully rough flow condition
9-38 Ch 9. Wall Turbulence; Boundary-Layer Flows
▪ Critical roughness, kcrit
' kcrit 44 Re x u x * Ucf /2 1 cf Rex
' If x increases, then cf decreases, and increases.
Therefore, for a surface of uniform roughness, it is possible to be hydrodynamically rough
upstream, and hydrodynamically smooth downstream.
k ' , downstream ' k , upstream
(2) Rough-wall velocity profiles
Assume roughness height k accounts for magnitude, form, and distribution of the
roughness. Then
uy f (9.42) uk*
Make f in Eq. (9.42) be a logarithmic function to overlap the velocity-defect law, Eq.
(9.16), which is applicable for both rough and smooth boundaries.
9-39 Ch 9. Wall Turbulence; Boundary-Layer Flows
U u y y (9.16): 5.6log 2.5, 0.15 u*
i) For rough walls, in the wall region
u rough kyuy* 5.6logC5 , 50 ~ 100, 0.15 (9.43) uy*
where C 5 = const = f (size, shape, distribution of the roughness)
ii) For smooth walls, in the wall region
usmooth u** y u y y (9.13): 5.6logC2 , 30 ~ 70, 0.15 u*
where C 2 = 4.9
Subtract Eq. (9.43) from Eq. (9.13)
uu u smooth rough uk* 5.6log C6 (9.44) uu**
→ Roughness reduces the local mean velocity u in the wall region
→ where C 5 and C 6 Table 9-4
9-40 Ch 9. Wall Turbulence; Boundary-Layer Flows
(6) Surface-resistance formulas: rough walls
Combine Eqs. (9.43) and (9.16)
U u y y 5.6 log 2.5 , 0.15 u * uk 5.6 log C uy5 * U 5.6 logC (9.45) uk7 *
U 2 5.6log C7 u* cf k
1 3.96log C (9.46) k 8 cf
9-41 Ch 9. Wall Turbulence; Boundary-Layer Flows
[Ex. 9.3]
Rough wall velocity distribution and local skin friction coefficient
- Comparison of the boundary layers on a smooth plate and a plate roughened by sand grains
Water
▪ 2 Given: 0 0.485lb / ft on both plates
U = 10 ft / sec past the rough plate
ks = 0.001 ft
Water temp. = 58 F on both plates
(a) Velocity reduction u due to roughness
From Table 1-3:
1.938slug / ft 3 ; 1.25 1052ft / sec
Eq. (9.18)
0.485 u0 0.5 ft / sec * 1.938
9-42 Ch 9. Wall Turbulence; Boundary-Layer Flows
2 2 u 0.5 c 2* 2 0.005 f U 10
uk 0.5(0.001) * s 40 1.25 10 5
u uk Eq. (9.44): 5.6log* s 3.3 u*
u0.5{5.6log 40 3.3} 2.83 ft / sec
(b) Velocity u on each plate at y0.007 ft
i) For rough plate
uy Eq. (9.43): 5.6log 8.2 uk*
0.007 u0.5(5.6log 8.2) 6.47 ft / sec 0.001
ii) For smooth plate,
u uy Eq. (9.13): 5.6log* 4.9 u*
uy 0.5(0.007) * 280 1.25 10 5
u0.5{5.6log(280) 4.9} 9.3 ft / sec
Check u 9.3 6.47 2.83→ same result as (a)
9-43 Ch 9. Wall Turbulence; Boundary-Layer Flows
(c) Boundary layer thickness on the rough plate
Eq. (9.46):
1 3.96log 7.55 k cf s
11 log 7.55 1.66 ks 3.96 0.005
46 0.046ft 0.52 in 1.4 cm ks
9-44