Prandtl’s mixing length model is an example of this class where the eddy diffusivities and H are modeled in terms of the gradients of the mean flow: = 2 ∂ ¯u ∂y and H = C2 ∂ ¯u ∂y 6.90 w
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τ = τm+ τT = ρ(ν + ) ∂ ¯u
q= q
m + q
T = −ρc p (α + H ) ∂ ¯T ∂y (6.87) These can be integrated to yield
¯u = y
0
τ
T0 − ¯T = y
0
q
6.4.12 Algebraic Turbulence Models
The simplest class of turbulence closure models is the zero-equation model or first-order closure model Prandtl’s mixing length model is an example of this class where
the eddy diffusivities and H are modeled in terms of the gradients of the mean flow:
= 2
∂ ¯u ∂y and H = C2
∂ ¯u ∂y (6.90) where is the mixing length Here C is found from Pr T:
C = H
=
1
PrT where PrT is the turbulent Prandtl number
6.4.13 Near-Wall Region in Turbulent Flow
The mean momentum equation can be written near the wall as
ρ ¯u ∂ ¯u ∂x + ρ¯v ∂ ¯u ∂y −∂τ ∂y +d ¯p dx = 0 (6.91)
In a region close to the wall, the approximations
ρ ¯u ∂ ¯u
and the normal velocity at the wall,
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can be made With these, the momentum equation becomes
ρv0
∂ ¯u
∂y −
∂τ
∂y +
d ¯p
dx = 0
which can be integrated usingτ = τ0and¯u = 0 at y = 0:
τ
τ0
= 1 + ρv0¯u
τ0
+d ¯p
dx
y
τ0
(6.93) Nearthe wall, a friction velocityv∗can be defined as
τ0
ρ = C f
U2
2 = (v∗)2 which yields
v∗=
τ
0
ρ
1/2
(6.94) Moreover, normalized variables can be defined near the wall:
u+= v ¯u∗ = ¯u/U
v+
0 = v0
p+= µ(d ¯p/dx)
ρ1/2τ3/2
0
(6.95d)
and substitution of eqs (6.95) into eq (6.93) yields
τ
In the absence of pressure gradient and transpiration(p+= 0, v0+= 0),
τ
τ0 = 1 or τ0= ρ(ν + ) d ¯u dy (6.97)
In a region very close to the wall (called the viscous sublayer),ν and eq (6.97)
can be integrated from the wall to a nearby location (say, 0≤ y+≤ 5) in the flow
¯u
0
d ¯u = τ0
µ
y
0
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In a region farther out from the wall, turbulence effects become important However, the near-wall region still has effectively constant total shear stress and total heat flux:
τ = τ0+ ρ(ν + ) ∂ ¯u
∂y = ρ
ν + l2∂ ¯u
∂y
∂ ¯u
∂y = ρ
ν + κ2y2∂ ¯u
∂y
∂ ¯u
∂y (6.99)
where the mixing length near the wall is taken as = κy, with κ = 0.40 being the von K´arm´an constant In the fully turbulent outer region,ν , resulting in
τ0= ρκ2y2
∂ ¯u
∂y
2
or κy ∂ ¯u
∂y =
τ
0
ρ
1/2
= v∗ (6.100) This can be expressed as
κy+∂u+
∂y+ = 1
which can be integrated to yield the law of the wall:
u+= 1
Withκ = 0.4 and C = 5.5, this describes the data reasonably well for y+ > 30,
as seen in Fig 6.14 Minorvariations in theκ and C values are found in the literature,
as indicated in Fig 6.14 In the range 5 < y+ < 30, the buffer region, both ν
Figure 6.14 Velocity profiles in turbulent flow past a flat surface in the wall coordinates
(Data from the literature, as reported by Kays and Crawford, 1993.)
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and are important and the experimental data from various investigators can be
approximated by
u+ = 5 ln y+− 3.05
6.4.14 Analogy Solutions for Boundary Layer Flow
Consider the apparent shear stress and heat flux in turbulent flat plate boundary layer flow:
τ = ρ(ν + ) ∂ ¯u
∂y and
∂ ¯u
∂y =
τ
q= −ρc p (α + H ) ∂ ¯T
∂y and
∂ ¯T
∂y = −
q
ρc p (α + H ) (6.103)
These expressions can be integrated from the wall to a location in the free stream,
u =
0
τ
T0 − T =
0
q
Underthe assumptions that
Pr= PrT = 1 and q q
0
=ττ
0 then
T0 − T =
0
q
ρc p (ν + ) dy =
q
0
τ0
0
τ
ρc p (ν + ) dy (6.106)
Upon division by the velocity expression
T0 − T
u = q0
c pτ0 then
q
0
T0 − T
x
k =
τ0
ρu2
ρu1x
µ
µc p
C f
2 Rex· Pr(6.107)
Formoderate Rex
C f =0.058
Re0x .2
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and forPr= PrT = 1 and no pressure gradient,
Nux = 0.0296Re0.80
undergo transition, and become turbulent along the plate length The average heat transfer coefficient in such a case can be determined by assuming an abrupt transition
at a locationx T:
¯h = L1 x T
0
h L dx + L
x T
h T dx
(6.109)
whereh L andh T are the local heat transfer coefficient variations for the laminar
and turbulent boundary layers, respectively A critical Reynolds number ReT can be selected and eq (6.109) can be normalized to obtain the average Nusselt number
NuL = 0.664Re0.50
T + 0.036Re0L .8− Re0.8
T
(6.110)
stress and heat flux in eqs (6.102) and (6.103), division and introduction of the turbulent scales leads to
τ
τ0 =1+ν ∂u+
q
q
0
=
1
Pr+ /ν
PrT
∂T+
The turbulent boundary layer may be considered to be divided into an inner region,
a bufferregion, and an outerregion Forthe innerregion(0 ≤ y+ < 5), = 0 and
q= q
0, which results in
1= 1
Pr
∂T+
At the outeredge of the innerlayer,T+
s = 5Pr
Forthe bufferregion(5 ≤ y+ < 30),
τ = τ0 q= q
0 u+= 5 + 5 lny+
or
∂u+
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Substituting eq (6.113a) into eqs (6.111) provides
1=1+νy5+ (6.114a) or
ν =
y+
With eq (6.114b) put into eq (6.112),
1=
1
Pr +y+− 5
5PrT
∂T+
Then integration across the buffer region gives
T+− T+
s = y
+
5
dy+ 1/Pr + (y+− 5)/5Pr T (6.116)
or
T+− T+
s = 5PrT ln
1+ Pr
PrT
y+
5 − 1
(6.117) This may be applied across the entire buffer region:
T+
b − T+
s = 5PrT ln
1+ Pr
PrT
30
5 − 1
(6.118)
Forthe outerregion(y+> 30), where ν and H ν,
τ
τ0
= ν
∂u+
q
q
0
= /ν
Pr
∂T+
Similar distributions of shear stress and heat transfer rate in the outer region can be assumed so that
τ
τb =
q
q
b
(6.121) Becauseτ and qare constant across the inner two layers,τb = τ0andq = q
0 Hence
τ
τ0
= q
q
0
(6.122)
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With eqs (6.121) and (6.122) in eq (6.120),
ν
∂u+
∂y+ = νPrT
∂T+
∂y+ and
∂T+
∂y+ = PrT ∂u+
Integrating across the outer layer yields
T+
1 − T+
b = PrT (u+
1 − u+
whereT+
1 andu+
1 are in the free stream and
u+
b = 5 + 5 ln 6
so that
T+
1 − T+
b = PrT u+
Addition of the temperature profiles across the three layers gives
T+
1 = 5Pr + 5PrT lnPrT+ 5Pr
6PrT + PrTu+
1 − 5 (6.126) and because
T+= ρc p (T0 q− ¯T )
0
v∗= ρc p (T0 q− ¯T )
0
τ
0
ρ
1/2
= Pr· Rex
Nux
C
f
2
1/2
(6.127) the Nusselt numberbecomes
Nux = Rex (C f /2)
1/2
5+ 5PrT
Pr ln
PrT + 5Pr
6PrT
+PrT
Pr (u+
1 − 5)
(6.128)
This yields the final expression with St= Nux /Re x· Pr:
(C f /2)1/2{5Pr + 5PrT ln[(PrT + 5Pr)/6Pr T]− 5PrT} + PrT (6.129)
For
C f = 0.058
Re0.20 x
eq (6.129) can be written as
Nux = 0.029Re0.8
where the parameterG is
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(0.029/Re .20
x )1/2 {5Pr + 5 ln[(1 + 5Pr)/6] − 5} + 1 (6.131)
As pointed out by Oosthuizen and Naylor(1999), eq (6.131) is based on PrT = 1
but does apply forvarious Pr
inte-gral forms of the momentum and energy equations can also be developed for turbulent flow by integrating the mean flow equations across the boundary layer from the sur-face to a locationH outside the boundary layer The integral momentum equation
can be written as
d dx
H
0
¯u( ¯u − U)dy = −τ0
ρ = −U2
dδ2
dx = −(v∗)2 (6.132)
where the momentum thicknessδ2is
δ2= H
0
¯u
U
1−U ¯u
Assume that both Prand PrT are equal to unity and that the mean profiles for the mean velocity and temperature in the velocity (δ) and thermal (δT) boundary layers
are given by
¯u
U =
y
δ
1/7
and T0 − ¯T
t0 − T∞ =
y
δT
1/7
(6.134) Then from the integral momentum equation
τ
τ0 = 1 −y
δ
1/7
(6.135) and because the total heat flux at any point in the flow is
ν + = α + H = τ/ρ
∂ ¯u/∂y = 7ν
C f
2
δ
xRex
1−y
δ
9/7 y
δ
6/7
(6.136)
the total heat flux at any point in the flow is
q= −ρc p (α + H ) ∂ ¯T
For the assumed profiles this can be written as
q
ρc p U(T0 − T∞) =
C f
2
1−y
δ
9/7 y
δ
6/7
(6.138)
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The wall heat flux can be determined from eq (6.138) This expression is substi-tuted on the right-hand side of the integral energy equation The ¯u and ¯T profiles and
theδ as a function of x relations are substituted on the left side and δ T is solved for
as a function ofx The final result forthe Stanton numberSt is
St= C f
2
1−x0
x
9/10−1/9
(6.139) With the same Prandtl number correction as in the case of the fully heated plate,
St· Pr0.40 = 0.0287Re −0.2
x
1−x0
x
9/10−1/9
(6.140)
Just as for laminar flow, the result of eq (6.140) can be generalized to an arbitrarily varying wall temperature through superposition (Kays and Crawford, 1980)
law for thermal boundary layers on smooth surfaces with uniform specified surface heat flux, a relationship that is nearly identical to eq (6.140) for uniform surface temperature is recommended:
St· Pr0.40 = 0.03Re −0.2
For a surface with an arbitrarily specified heat flux distribution and an unheated section, the temperature rise can be determined from
T0(x) − T∞= x0=x
x0 =0 Γ(x0, x)q (x0) dx0 (6.142) where
Γ(x0,x) =
9
10Pr−0.60· Re−0.80
x
Γ1 9
Γ8 9
(0.0287k)
1−x0
x
9/10−8/9
(6.143)
foregoing analyses The turbulent Prandtl number is determined experimentally using
PrT = uv(∂ ¯T /∂y)
Kays and Crawford (1993) have pointed out that, in general, it is difficult to measure all four quantities accurately With regard to the mean velocity data, mean temperature data in the logarithmic region show straight-line behavior when plotted in the wall coordinates The slope of this line, relative to that of the velocity profile, provides
PrT Data forairand waterreveal a PrT range from 0.7 to 0.9 No noticeable effect
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due to surface roughness is noted, but at very low Pr, a Prandtl number effect on PrT
is observed This is believed to be due to the higher thermal conductivity based on existing analyses At higherPr, there is no effect on PrT The PrT values are found
to be constant in the “law of the wall” region but are higher in the sublayer, and the wall value of PrT approaches 1.09 regardless of Pr
6.4.15 Surface Roughness Effect
If the size of the roughness elements is represented by a mean length scalek s, a
roughness Reynolds number may be defined as Re k = k s (v∗/ν) Three regimes of
roughness may then be prescribed:
Smooth Rek < 5
Transitional rough 5< Re k < 70
Fully rough Rek > 70
Under the fully rough regime, the friction coefficient becomes independent of viscosity and Rek The role of the roughness is to destabilize the sublayer For Rek >
70, the sublayer disappears and the shear stress is transmitted to the wall by pressure drag on the roughness elements The mixing length near a rough surface is given as
= κ(y+δy o ), which yields a finite eddy diffusivity at y = 0 In the wall coordinates,
based on experimental data,
δy+
0 =δy0v∗
The law of the wall for a fully rough surface is
du+
κ(y++ δy+
Because no sublayer exists for fully rough surface, eq (6.146) may be integrated from
0 toy+to provide
u+=1κlny++1κln32.6
The friction coefficient is calculated by evaluatingu+
∞while including the additive effect of the outerwake The latterresults in a displacement by 2.3 The result is
u+
∞= (C 1
f /2)1/2 = 1κln848
The heat transfer down to the plane of the roughness elements is by eddy con-ductivity, but the final transfer to the surface is by molecular conduction through the almost stagnant fluid in the roughness cavities The law of the wall is written as