CHAPTER 9
CONVECTION IN TURBULENT CHANNEL FLOW
9.1 Introduction
 We will begin this subject with the criteria for fully developed velocity and
temperature profiles.
 Will focus most of our attention on analyzing fully developed flows.
 As in Chapter 6, our analysis is limited to general boundary conditions:
(i) uniform surface temperature, and (ii) uniform heat flux.
9.2 Entry Length
 Common rules of thumb:
Lh Le

 10
De De
(6.7)
o De is the hydraulic or equivalent diameter
De 
4 Af
P
o Bejan [1] recommends (6.7) particularly to Pr = 1 fluids.
1
 White [2] recommends the following approximation:
Lh
 4.4 Re1/6
De
De
(9.1)
Lh
 0.623 Re1/4
De
De
(9.2)
 Latzko (see Reference 3) suggests:
 Thermal entry length doesn’t lend itself to a simple, universallyapplicable equation, since the flow is influenced so much by fluid
properties and boundary conditions.
 The hydrodynamic entry length is much shorter for turbulent flow than
for laminar. In fact, the hydrodynamic entrance region is sometimes
neglected in the analysis of turbulent flow.
9.3 Governing Equations
 Consider flow through a circular pipe.
 Assume 2D, axisymmetric,
incompressible flow.
2
9.3.1 Conservation Equations
 Conservation of Mass:
u 1 

 rvr   0
x r r
(9.3)
 x-momentum equation reduces to:
u
v
u
1 dp 1  
u 
 vr r  

r




M

x
r
 dx r r 
r 
(9.4)
 Conservation of Energy:
T
T 1 
u
 vr

x
r r r

T 
 r    H  r 


(9.5)
9.3.2 Apparent Shear Stress and Heat Flux
 Similar to that of the flat plate:
 app
u
    M 

r
(9.6)
T
     H 
cp
r
(9.7)

qapp
3
9.3.3 Mean Velocity and Temperature
Mean Velocity
 Calculating by evaluating the mass flow rate in the duct:
ro
m   um A    u  2 r  dr
0
 Assuming constant density:
ro
ro
1
2
um 
u  2 r  dr  2  urdr
2 
 ro 0
ro 0
(9.8)
Bulk Temperature
 Evaluating by integrating the total energy of the flow:
ro
Tm 
 Turdr
0
ro
 urdr
0
 Can be simplified by substituting the mean velocity, equation (9.8),
4
ro
2
Tm 
Turdr
2 
um ro 0
(9.9)
9.4 Universal Velocity Profile
9.4.1 Results from Flat Plate Flow
 Already seen that the universal velocity profile in a pipe is very similar
to that of flow over a flat plate at zero or favorable pressure gradient.
 We even adapted a pipe flow friction factor model to analyze flow over a
flat plate using the momentum integral method.
 It is apparent, then, that the characteristics of the flow near the wall of
a pipe are not influenced greatly by the curvature of the wall of the radius
of the pipe.
 Therefore a reasonable start to modeling pipe flow is to invoke the
two-layer model that we used to model flow over a flat plate:
Viscous Sublayer:
u  y 
(8.54)
Law of the Wall:

u 
1

ln y   B
 We also have continuous wall law models by Spalding (8.63) and
Reichardt (8.64) that have been applied to pipe flow.
(8.58)
5
Wall Coordinates for Internal Flow
 Note that for pipe flow, the wall coordinates are a little different than for
flat-plate flow.
 First, the y-coordinate for pipe flow is
y  ro  r
(9.10)
 So the wall coordinate y+ is
*
r

r
u


y  r   r   o

o
(9.11)
 The velocity wall coordinate is the same as before,
u
u  *
u

(8.49)
 and the friction velocity is the same,
u*   o / 
(8.46)
 The friction factor is based on the mean flow velocity instead of the
free-stream velocity:
o
Cf 
(9.12)
2
 1 / 2   um
6
 So the friction velocity can be expressed as:
u*  um C f / 2
9.4.2 Development in Cylindrical Coordinates
 The velocity profile data for pipe flow matches that of flat plate flow
o This fact allowed us to develop expressions for universal velocity
profiles solely from flat plate (Cartesian) coordinates.
 Would we have achieved the same results if we had started from the
governing equations for pipe flow (i.e., cylindrical coordinates)?
 Assume fully developed flow. x-momentum reduces to
1   r

r r  
 1 p

  x
(9.13)
 Rearranging and integrating, we obtain an expression for the shear stress
anywhere in the flow:
r p
 (r ) 
C
2 x
(9.14)
 The constant C is zero, since we would expect the velocity gradient (and
hence the shear stress) to zero at r = 0.
7
 Evaluating (9.14) at r and ro and taking the ratio of the two gives:
 (r ) r

o
ro
(9.15)
u  o

 constant
r 
(9.16)
 This result shows that the local shear is a linear function of radial location.
 But the Couette Flow assumption meant that τ is
approximately constant in the direction normal to the
wall! How do we reconcile this?
o Remember that the near-wall region over which we
make the Couette flow assumption covers a very
small distance.
o Therefore we could assume that, in that small
region vary close to the wall of the pipe, the shear
is nearly constant, τ = τo
o Thus the Couette assumption approximates the behavior near the
pipe wall as
   M 
 Exp. data suggest that the near-wall behavior is not influenced by the
outer flow, or even the curvature of the wall.
8
9.4.3 Velocity Profile for the Entire Pipe
 The velocity gradient (and the shear stress) is supposed to be zero at the
centerline of the pipe.
 Unfortunately, none of the universal velocity profiles we’ve developed so
far behave this way
 Reichardt attempted to account for the entire region of the pipe. He
suggested a model for eddy viscosity:
2

r 
M  y 
r

 1   1  2   

6 
ro  
 ro  

(9.17)

1.5  1  r / ro  

B
u  ln  y
2
  1  2  1  r / ro  


(9.18)

 Which leads to the following expression for the velocity profile:

1
 Reichardt used κ= 0.40 and B = 5.5.
 The profile does not account for the viscous sublayer, but as r →ro,
equation (9.18) does reduce to the original Law of the Wall form,
equation (8.58).
9
9.5 Friction Factor for Pipe Flow
9.5.1 Blasius Correlation for Smooth Pipe
 Based on dimensional analysis and experimental data, Blasius developed
a purely empirical correlation for flow through a smooth circular pipe:
C f  0.0791ReD1/4 (4000<ReD<105)
(9.19)
o The friction factor is based on the mean flow velocity,
Cf 
o
 1 / 2   um 2
(9.12)
 Later correlations have proven to be more accurate and versatile, but this
correlation led to the development of the 1/7th Power Law velocity profile.
9.5.2 The 1/7th Power Law Velocity Profile
 Discovered independently by Prandtl [7] and von Kármán [8].
 Begin with the Blasius correlation, which can be recast in terms of wall
1/4
shear stress:

2
r
u


o
or
1 / 2   um 2
 0.0791  o
 
m


 o  0.03326  um7/4 ro1/4 1/4
(a)
10
 Assume a power law velocity profile:
u  y
 
uCL  ro 
q
(b)
 Assume that the mean velocity in the flow can be related to the centerline
velocity as:
uCL   constant  u
(c)
 Substituting (b) and (c) for the mean velocity in (a) yields :
  y
 o  (const )   u  
  ro 
 Simplifies to:
1/ q



7/4
ro1/4 1/4
 o  (const )  u 7/4 y ( 7/4 q ) ro(7/4 q 1/4) 1/4
(d)
 Both Prandtl and von Kármán argued that the wall shear stress is not a
function of the size of the pipe.
 Then the exponent on ro should be equal to zero.
 Setting the exponent to zero, the value of q must be equal to 1/7, leading to
the classic 1/7th power law velocity profile,
11
1/7
u  y
 
uCL  ro 
(9.20)
 Experimental data show that this profile adequately models the velocity
profile through a large portion of the pipe, and is frequently used in
models for momentum and heat transfer.
 Limitations:
o Accurate for only a narrow range of Reynolds numbers (roughly,
104 to 106).
o Yields an infinite velocity gradient at the wall
o Does not yield a gradient of zero at the centerline
Nikuradse’s Improvement to the 1/7th Power Law
 Another student of Prandtl, Nikuradse [10] measured velocity profiles in
smooth pipe over a wide range of Reynolds numbers, and reported that
the exponent varied with Reynolds number,
 y
u
 
uCL  ro 
n
(9.21)
12
 Also correlated pipe friction factor of the form
C
Cf 
Re1/D m
(9.22)
 As one might expect, Nikuradse’s results show that the velocity profile
becomes fuller as the mean velocity increases.
9.5.3 Prandtl’s Law for Smooth Pipe
 Whereas the Blasius correlation is purely empirical, we can develop a
more theoretical model for friction factor by employing the universal
velocity profile.
13
 Begin with the Law of the Wall, equation. Substitute the wall coordinates
u+ and y+, as well as the friction velocity u*   o /   um C f / 2 :
u
um
2
1  yum
 ln 
C f   
Cf 
 B
2 
(9.23)
2
1  Re D
 ln 
C f   2
Cf 
 B
2 
(9.24)
 If we assume that the equation holds at any value of y, we could
evaluate the expression at the centerline of the duct, y = ro=D/2, where
u  uCL :
uCL
um
 We now have a functional relationship for the friction factor.
 However, the ratio uCL/um is still unknown.
Evaluating the Mean Velocity for Prandtl’s Law
 Goal is to integrate the Law of the Wall velocity profile (8.58) across
the pipe.
 Start with expression for the mean velocity, equation (9.10). Using the
variable substitution y = ro – r , (9.10) becomes
14
r
r
1 o
2 o
um 
u(2 r )dr  2  u( ro  y )dy
2 
 ro 0
ro 0
(9.25)
 Then, substitute the Law of the Wall for u .
 Performing the integration, it can be shown that the mean velocity
becomes
*
 1  ro u 
3 
um  u  ln 

 B


2





*
(9.26)
 Or, making substitutions again for u*
um  um
C f  1  ReD
 ln 
2    2

Cf 
3 

 B
2 
2 

(9.27)
 Warning: the term we were trying to evaluate, um, cancels out of the
expression! However, uCL doesn’t not appear either.
 We can use the above expression directly to find an expression for Cf.
Rearranging, and substituting the values κ= 0.41 and B = 5.0 gives
1
Cf / 2


 2.44ln ReD C f / 2  0.349
15
 This development ignores the presence of a viscous sublayer or a wake
region.
 Empirically, a better fit to experimental data is
1
Cf / 2


 2.46ln C f / 2 D C f / 2  0.29
(ReD  4000)
(9.28)
 This is called Prandtl’s universal law of friction for smooth pipes.
o Sometimes referred to as the Kármán-Nikuradse equation.
 Note that, despite the empiricism of using a curve fit to obtain the
constants in (9.28), using a more theoretical basis to develop the function
has given the result a wider range of applicability than Blasius’s
correlation.
 Equation (9.28) must be solved iteratively for Cf. A simpler, empirical
relation that closely matches Prandtl’s is:
Cf
2
 0.023 ReD1/5
(3  104 <ReD  106 )
(9.29)
 This correlation is also suitable for non-circular ducts, with the
Reynolds number calculated using the hydraulic diameter.
16
9.5.4 Effect of Surface Roughness
 From our discussion of turbulent flow over a rough flat plate, we saw
that roughness shifts the universal velocity profile downward.
 We could write the velocity profile in the logarithmic layer as:

u 
1

ln y   B  B
ΔB is the shift in the curve, which increases with wall roughness k+
 The behavior of the velocity profile also depends on the geometry of the
roughness, like rivets to random structures like sandblasted metal.
 The following model is based on equivalent sand grain roughness [2],
1
f
1/ 2


ReD f 1/ 2
 2.0log10 
 0.8
1/ 2 
 1  0.1( k / D) ReD f 
(9.30)
where it is common to use the Darcy friction factor,
f  4C f
(9.31)
Two Important Points on Surface Roughness:
1. If the relative roughness k/D is low enough, it doesn’t have much of
an effect on the equation.
17
o Scaling shows that roughness is not important if  k / D  ReD  10
2. On the other hand, if , the roughness term  k / D  ReD  1000
dominates in the denominator, and the Reynolds number cancels;
in other words, the friction factor is no longer dependent on the ReD.
Colebrook-White Equation
 Developed for commercial pipes,
1
f
1/ 2
k/D
2.51 
 2.0log10 

1/ 2 
3.7
Re
f

D

(9.32)
 This function is what appears
in the classic Moody Chart
18
Moody Chart:
19
9.6 Momentum-Heat Transfer Analogies
 Development is applied to the case of a constant heat flux boundary
condition.
 Strictly speaking, an analogy cannot be made in pipe flow for the case
of a constant surface temperature. But resulting models approximately
hold for this case as well.
Development
 x-momentum equation (9.4) becomes, for hydrodynamically fully
developed flow,
1 dp 1  
u 

r    M  

 dx r r 
r 
(9.33a)
 Energy equation reduces to,
T 1  
T 
u

r    H 

x r r 
r 
(9.33b)
 Are the left-hand sides analogous?
20
o Note that in pipe flow the pressure gradient is non-zero, although
constant with respect to x. To ensure an analogy, then, the left side
of (9.33b) must then be constant.
o For thermally fully developed flow and a constant heat flux at the
wall, the shape of the temperature profile is constant with respect to x,
leading to:
dT
 constant
dx
o So the analogy holds on the LHS.
Boundary Conditions
 Boundary conditions must match:
At r = 0:
At r = ro:
du (0) dT (0)

0
r
r
u ( ro )  0, T ( ro )  Ts ( x )
du ( ro )
T ( ro )

 o, k
 qo
dr
r
 If we normalize as follows:
(9.34a)
(9.34b)
(9.34c)
21
T  Ts
u
x
r
U
,
, X  and R 
um
Tm  Ts
L
ro
 We can show that both the governing equations and the boundary
conditions are identical in form.
9.6.1 Reynolds Analogy for Pipe Flow
 Assume ν = α (Pr = 1) and εM = εH (Prt = 1)
o Same assumptions used to develop Reynold’s analogy for a flat plate
 Then the governing equations (9.33a) and (9.33b) are identical.
 Follow exactly the same process that we followed for the original
derivation, we find that the Reynolds analogy is essentially identical for
pipe flow,
Cf
qo
St D 

 um c p (Ts  Tm ) 2
Cf
Nu D
St D 

ReD Pr
2
(Pr  1)
(9.35)
 Note that in this case the Stanton number is defined in terms of the
mean velocity and bulk temperature, as is the wall shear stress:
 o  12 C f  um2
22
9.6.2 Adapting Flat-Plate Analogies to Pipe Flow
 We saw that Reynold’s analogy is identical for flat plate and pipe flows.
 We know that the velocity profiles near the wall are similar.
 Can we adapt other flat-plate analogies to pipe flow?
Von Kármán Analogy for Pipe Flow
 Take original von Kármán analogy, replace V∞ and T ∞ with
V  uCL and T  TCL
 These substitutions also affect the friction factor, which translates to:
o
Cf  1
2

u
2
CL
 Following the development exactly as before, the result is almost identical:
qo

 uCLc p (Ts  TCL )
Cf / 2
Cf 
 5 Pr  1  
1 5
(
Pr

1)

ln




2 
 6 
(9.36)
 Problem: the LHS and the friction factor are expressed in terms of
centerline variables instead of the more common and convenient mean
quantities um and Tm. Correct this as follows:
23
 um (Ts  Tm ) 
qo


 um c p (Ts  Tm )  uCL (Ts  TCL ) 
C
f

/ 2  um / uCL 
2
Cf 
 5 Pr  1  
( Pr  1)  ln 


2 
6


 Now, Cf is again defined in terms of the mean velocity, C f   o / 12  um2
um
1 5
uCL
and the terms qo /  um c p (Ts  Tm ) are collectively the Stanton number
for pipe flow.
 Simplifying,
 Ts  Tm 
St D 

 Ts  TCL 
C
f

/ 2  u m / uCL 
 um  C f 
 5 Pr  1  
1  5
(
Pr

1)

ln





 6 
 uCL  2 
(9.37)
 This is the von Kármán Analogy for pipe flow.
Estimates for Mean Temperature and Velocity
 We can develop estimates for the ratios  um / uCL and Ts  Tm  / Ts  TCL 
using the definition of mean temperature, equation (9.9).
 Estimate um and Tm using the 1/7th Law profiles, which for a circular
pipe are:
24
1/7
u  y
 
uCL  ro 
(9.20)
and, similar to (8.111) for a flat plate,
1/7
 y
T  Ts
 
TCL  Ts  ro 
(9.38)
 Substituting these models into (9.8) and (9.9), we can show that:
um
 0.817
uCL
Tm  Ts
 0.833
TCL  Ts
(9.39)
(9.40)
9.6.3 Other Analogy-Based Correlations
 A simple correlation for turbulent flow in a duct is based on the
Colburn analogy.
 Beginning with the analogy, equation (8.96), and using equation (9.27)
for the friction factor, we obtain
25
St D  0.023 ReD1/5 Pr 2/ 3
or
NuD  0.023 ReD4/5 Pr 1/ 3
(9.41)
 One of the most popular correlations is the Dittus-Boelter correlation,
which is an empirical correlation based on the Colburn analogy:
NuD  0.023 ReD4/5 Pr n
(9.42)
o where n = 0.4 for heating (Ts > Tm) and n = 0.3 for cooling.
 Although still popular, the Colburn analogy and its derivative, the
Dittus-Boelter correlation have been challenged in recent years.
 Models such as those by Petukhov and Gnielinski correlation (see
Section 9.8) are preferred for their improved accuracy and range
of applicability.
 Other analogies have been developed specifically for pipe flows, instead
of adapting existing flat-plate models. Examples
o Reichardt [16]
o Boelter, Martinelli, and Jonassen [17]
o Churchill and Zajic [18] in 2002 (which the authors claim is to
date the most accurate model for the internal flow.)
26
9.7 Algebraic Method Using Universal Temperature Profile
 As we did for flow over a flat plate, we can use the universal
temperature and velocity profiles to estimate the heat transfer in a
circular duct.
 Begin again with the definition of the Nusselt number, which for flow
in a duct can be expressed as
qoD
hD
NuD 

k
(Ts  Tm )k
(9.43)
 To later invoke the universal temperature profile, we use the definition
of T+, equation 8.102, to define the mean temperature as

m
T  (Ts  Tm )
 c p u*
qo
 (Ts  Tm )
 c p um C f / 2
qo
(9.44)
 Recall that for duct flow, the friction velocity u* is defined in terms of
the mean velocity. Substituting this expression into (9.43) for qo
and invoking the definitions of the Reynolds and Prandtl numbers,
Nu D 
ReD Pr C f / 2

m
(9.45)
T
27
 Several ways to proceed with the analysis. One approach is to evaluate
Tm+ using a dimensionless version of the mean temperature expression
ro
(9.33):
2
(9.46)
Tm   2 T  u ( ro  y  )dy 
um ro

0
 Theoretically, can simply substitute appropriate universal temperature
and velocity profiles into the above and integrate. Practically, this
requires numerical integration.
 A simpler, second approach can be taken. First, we rewrite the original
Nusselt number relation (9.43) as follows,
qoD
(Ts  TCL )
NuD 
(Ts  Tm )k (Ts  TCL )
where TCL is the centerline temperature.
 Then, substitute the definition of T+ for the centerline temperature in
the denominator, we obtain
ReD Pr C f / 2 (Ts  TCL )
N uD 

TCL
(Ts  Tm )
(9.47)
28
 We can now use the universal temperature profile, equation (8.118), to
evaluate TCL+:
Prt

TCL



ln yCL
 13 Pr 2/ 3  7
(9.48)
 Now, just like in our analysis for flat plate flow, we can substitute the
Law of the Wall velocity profile (8.59) for ln yCL+:

uCL

1


ln yCL
B
(9.49)
 Substituting these into the Nusselt number relation,
(Ts  TCL )
(9.50)

2/ 3
 Prt ( uCL  B )  13 Pr  7  (Ts  Tm )

 We need expressions for uCL
and (Ts  TCL ) / (Ts  Tm ). For the
ReD Pr C f / 2
NuD 
centerline velocity, we can use the definition of u+ for pipe flow:

CL
u
uCL uCL
 * 
u
um
2
Cf
(9.51)
29
 It appears that, if we are to complete the analysis, we will need to
evaluate the mean velocity and temperature after all. To avoid the
complexity of the logarithmic velocity and temperature profiles, we could
estimate these quantities using the much simpler 1/7th Law profiles,
which we saw in the last section yields
um
 0.817
uCL
Tm  Ts
 0.833
TCL  Ts
(9.39)
(9.40)
 Finally, using the definition of Stanton number, St D  NuD / ( ReD Pr ) ,
selecting Prt = 0.9 and B = 5.0, we can rearrange (9.50) obtain
St D 
Cf / 2
0.92  10.8  Pr
2/3
 0.89  C f / 2
(9.52)
 Under what conditions is this equation applicable? The ultimate test
would be to compare the expression to experimental data.
 However, since we invoked the 1/7th power law, which is valid around
1×105, it might be reasonable as a first approximation to limit this model
to ReD < 1×105.
30
9.8 Other Correlations for Smooth Pipes
Petukhov’s Model
 Petukhov followed a more rigorous theoretical development, invoked
Reichardt’s model for eddy diffusivity and velocity profile (9.15, 9.16):
St D 
Cf / 2
1.07  12.7  Pr
2/3
 1
 0.5  Pr  2000 
,  4
6
C f / 2  10  Re D  5  10 
(9.53)
 Compares well to experimental data over a wide range of Prandtl and
Reynolds numbers.
 Petukhov used the following model for friction factor, which he also
developed:
Cf
2
 (2.236 ln ReD  4.639)2
(9.54)
 Note the similarity between Petukhov’s relation (9.53) and the algebraic
result, equation (9.52). It seems as if we have captured the essential
functionality even in our modest approach.
Gnielinski’s Model
 Gnielinski modified Petukhov’s model slightly, extending the model to
include lower Reynolds numbers:
31
NuD 
( Re D  1000) PrC f / 2
1  12.7  Pr
2/3
 1
0.5  Pr  2000 

, 
3
6
3

10

Re

5

10
Cf / 2 
D

(9.55)
 Use Petukhov’s friction model in (9.55) for the friction factor.
 For all models, properties should be evaluated at the film temperature.
 As was the case with the analogy-based correlations, these correlations are
reasonable for channels with constant surface temperature as well as
constant heat flux; the flows are relatively insensitive to boundary
conditions.
9.9 Heat Transfer in Rough Pipes
 We’ve discussed the effects of roughness on the heat transfer from flat
plates in Section 8.5.6, and much of the same physical intuition applies
to flow in channels. Norris [21,3] presents the following empirical
correlation for flow through circular tubes:
n
 Cf 
 Cf

Nu

 4
 , 


Nusmooth  C f , smooth 
 C f , smooth

where n  0.68 Pr 0.215
(9.56)
32
 A correlation like Colebrook’s (9.30) could be used to determine the
rough-pipe friction factor.
 The behavior of this relation reflects what we expect physically.
o The Prandtl number influences the effect of roughness, and for very
low-Pr fluids the roughness plays little role in the heat transfer.
o Regardless of Prandtl number, the influence of roughness size is
limited: Norris reports that the effect of increasing roughness
vanishes beyond (C f / C f , smooth )  4 , and so the equation reaches
a maximum.
 Although roughness enhances heat transfer, it also increases the friction.
 Neither the friction nor the heat transfer increase indefinitely with
roughness size – both reach a limiting value.
33