6.55 and 6.56, approximate profiles for tangential velocity and temperature across the boundary layer must be defined.. For example, a cubic parabola profile of the typeT = a + by + cy2+ dy
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C2 = −C1 =2c p (T0 − TAW)
U2 The solution forθAW(ηB ) is obtained numerically by solving eq (6.51) From this
θAW(0) ≡ r c = b(Pr) Pr1/2 (forgases)
wherer c is called the recovery factor Using this, the adiabatic wall temperature is
calculated as
TAW = T∞+ r c U2
2c p
Forlow velocities this can be approximated as
TAW = T∞
The temperature profile for various choices ofT0is shown in Fig 6.9 from Gebhart (1971)
The heat flux can be written as
q
0 = −k ∂T ∂y
y=0 = −k U2
2c p
U
νx
1/2
[θ
AW(0) + C1φ(0)]
= k(T0− TAW)
0.332
U
νx
1/2
· Pr1/3
(6.54)
Figure 6.9 Boundary layer temperature profiles with viscous dissipation (From Gebhart, 1971.)
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and with the definition
q
0 = h x (T0 − TAW)
the Nusselt numberbecomes the well-known Pohlhausen (1921) solution:
Nu= 0.332Re1/2
x · Pr1/3
where the properties can be evaluated at the reference temperature recommended by Eckert:
T∗ = T∞ + (T W − T∞ ) + 0.22(TAW − T∞ )
6.4.7 Integral Solutions for a Flat Plate Boundary Layer with Unheated Starting Length
Consider the configuration illustrated in Fig 6.10 The solution for this configuration can be used as a building block for an arbitrarily varying surface temperature where
a similarity solution does not exist Assuming steady flow at constant properties and
no viscous dissipation, the boundary layer momentum and energy equations can be integrated across the respective boundary layers to yield
d
dx 0
u(U − u) dy = ν ∂u ∂y
y=0
(6.55)
d dx
T
0
u(T∞− T ) dy = α ∂T
∂y
y=0
(6.56)
To integrate eqs (6.55) and (6.56), approximate profiles for tangential velocity and temperature across the boundary layer must be defined For example, a cubic parabola profile of the typeT = a + by + cy2+ dy3can be employed, with the conditions
␦T
x
T0
x0
Figure 6.10 Hydrodynamic and thermal boundary layer development along a flat plate with
an unheated starting length in a uniform stream
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∂T
∂y
T =T∞
By evaluating the energy equation at the surface, an additional condition can be developed:
∂2T
∂y2
y=0= 0
Using these boundary conditions, the temperature profile can be determined as
T − T0
T∞− T0
= 3
2
y
δT −1
2
y
δT
3
(6.58)
Polynomials of higherordercan be selected, with the additional conditions deter-mined by prescribing additional higher-order derivatives set to zero at the edge of the boundary layer In a similar manner, a cubic velocity profile can be determined by prescribing
u(y = 0) = v(y = 0) = 0 u(y = δ) = u∞
∂u
∂y
y=δ= 0
and determining
∂2u
∂y2
y=0= 0
This results in
u
U =
3 2
y
δ −
1 2
y
δ
3
(6.59)
These profiles are next substituted into the integral energy equation and the inte-gration carried out Assuming that Pr> 1, the upperlimit needs only to be extended
toy = δ T, because beyond this,θ = T0− T = T − T∞= θ∞and the integrand is zero In addition, defining,r = δ T /δ, it noted that
δT
0
(θ∞− θ)u dy = θ∞Uδ
3
20r2− 3
280r4
(6.60)
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Becauser < 1, the second term may be neglected in comparison to the first Now r
is a function ofx and the integral of eq (6.60) may be put into the integral energy
equation of eq (6.56), yielding
2r2δ2dr
dx + r3δ
dδ
dx =
10α
From the integral momentum equation,δ(x) can be determined as
δ(x) = 4.64 νx
U
1/2
This yields
r3+ 4r2x dr
dx =
13 14Pr which can be solved to yield
1.026Pr1/3
1−x0
x
3/41/3
wherex0is the unheated starting length The heat transfer coefficient and the Nusselt numberare then
h = −k(∂T /∂y)| T0 − T y=0
∞ =θk
∞
∂θ
∂y
y=0= 3
2
k
δT =
3 2
k rδ
or
h = 0.332k Pr1/3
[1− (x0/x) 3/4]1/3
U
νx
1/2
(6.62) and
Nu= 0.332Pr1/2· Re1x /2
[1− (x0/x) 3/4]1/3 (6.63)
generalized for any surface temperature variation of the type
T0 = T∞ + A +
∞
n=1
Fora single step,
T0 − T
T0 − T∞ = θ(x0, x, y)
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and for an arbitrary variation,
T0 − T∞= x
0 [1− θ(x0, x, y)]∂T
∂x0 dx0+
k
i=1
[1− θ(x0i , x, y)] ∆T0 ,i (6.65)
q
0 = k x
0
∂θ(x0, x, 0)
∂y
dT0 dx0 dx0+
k
i=1
∂θ(x0 i , x, 0)
This procedure using the integral momentum equation can also be generalized to
a turbulent boundary layer with an arbitrary surface temperature variation
6.4.8 Two-Dimensional Nonsimilar Flows
When similarity conditions do not apply, as in the case of an unheated starting length plate, two classes of approaches exist for solution of the governing equations The integral method results in an ordinary differential equation with the downstream coordinatex as the independent variable and parameters associated with the body
profile shape and the various boundary layer thicknesses as the dependent variable
Generally, such solutions result in correlation relationships that have a limited range
of applicability These are typically much faster to compute
Differential methods solve the partial differential equations describing numerically
the conservation of mass, force momentum balance, and conservation of energy
These equations are discretized over a number of control volumes, resulting in a set
of algebraic equations that are solved simultaneously using numerical techniques
These solutions provide the detailed velocity, pressure, temperature, and density fields for compressible flows The heat transfer rates from various surfaces can also be determined from these results In the 1980s and 1990s, the computational hardware capabilities expanded dramatically and the differential methods have become the most commonly used methods for various complex and realistic geometries
6.4.9 Smith–Spalding Integral Method
The heat transfer in a constant-property laminar boundary layer with variable veloc-ityU(x) but uniform surface temperature can be obtained via the Smith–Spalding
integral method (1958), as described by Cebeci and Bradshaw (1984) A conduction thickness is defined as
δc= k(T0 − T∞ )
q
0
= − T0 − T∞
(∂T /∂y) y=0 (6.67)
This is expressed in nondimensional form as
U(x)
ν
dδ2
c
dx = f
δ2
c
ν
dU
dx , Pr
= A − Bδ2c
ν
dU
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whereA and B are Prandtl number–dependent constants Forsimilarlaminarflows,
the Nusselt numberis given as
Nu= k(T0 q − Tx
∞) = −
x(∂T /∂y)0 T0 − T∞ = f (Pr,m,0)Re1x /2 (6.69)
and using the definition of the conduction thickness yields
δ2
whereC = f (Pr, m, 0) The parameters in eq (6.68) can be expressed as
U(x)
ν
dδ2
c
dx =
1− m
δ2
c
ν
dU
dx =
m
Equation (6.69) is a first-order ordinary differential equation that can be integrated as
δ2
c= νA
x
0
U B−1 dx
U B + δ2
c U B
i
U B
where the subscripti denotes initial conditions The normalized heat transfer
coeffi-cient in the form of a local Stanton number (St= Nu/Re x· Pr) is
ρc p U(T0 − T∞ ) =
k
ρc p Uδ =
c1(U∗) c2
x∗
0
(U∗) c3dx∗1/2
1
ReL
1/2
(6.72)
Here
c1= Pr−1· A −1/2 , c2= B
2, c3 = B − 1
and provided in Table 6.1 and
U∗= U(x) U
∞ x∗ =L x ReL= U∞νL
whereU∞is a reference velocity, typically the uniform upstream velocity, andL is a
length scale characteristic of the object
As an example of the use of the Smith–Spalding approach, consider the heat transfer in cross flow past a cylinder of radiusr0 heated at a uniform temperature
The velocity distribution outside the boundary layer is given by
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TABLE 6.1 Parametersc1, c2 , an dc3 as Functions of the Prandtl Number
Source: Cebeci and Bradshaw (1984).
U(x) = 2U∞sinθ = 2U∞sin x
wherex is measured around the circumference, beginning with the front stagnation
point Forsmall values ofθ, sin θ ≈ θ, and the free stream velocity approaches that
for a stagnation point where similarity exists The computed results for Nux /Re1/2
x
are shown in Fig 6.11 for three values of Pr
0 0.2 0.4 0.6 0.8 1 1.2
Pr = 0.7
Pr = 5
Pr = 10
180 (deg)
x
r0
Figure 6.11 Local heat transfer results for crossflow past a heated circular cylinder at uniform temperature for various Pr values, using the integral method (From Cebeci and Bradshaw, 1984.)
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6.4.10 Axisymmetric Nonsimilar Flows
The Smith–Spalding method (1958) can be extended to axisymmetric flows by using the Mangler transformation If the two-dimensional variables are denoted by the sub-script 2 and the axisymmetric variables by the subsub-script 3 and neglecting transverse curvature, then
dx2 =r0
L
2K
dx3 θ2 =r0
L
K
whereK is the flow index used in the Mangler transformation to relate the
axisym-metric coordinatesx and θ to two-dimensional coordinates and
r2
0(δ c )2
3=
νA x3
0
U B−1 dx3
wherer0is the distance from the axis to the surface (see Fig 6.13)
The Stanton numberis given in nondimensional form by
St= c1(r0∗) K (U∗) c2
x∗
3
0
(U∗) c3(r∗
0)2K dx∗
3
1/2
1
ReL
1/2
(6.76)
whereU∗ = U(x)/U∞ , r∗
0 = r0/L, and x ∗
3 = x3/L Here the constants c1,c2, and
c3are given in Table 6.1
As an example, heat transfer from a heated sphere of radiusa at a uniform
tem-perature placed in an undisturbed flow characterized byU∞can be calculated by this method The velocity distribution in the inviscid flow is
U = 3
2U∞sinφ = 3
2U∞sinx
The variation of Nux (U∞a/ν) −1/2downstream from the front stagnation point is
seen in Fig 6.12 The result of the similarity solution for the case of axisymmetric stagnation flow is also shown As described by Cebeci and Bradshaw (1984), this similarity solution requires the transformation of the axisymmetric equations into a nearly two-dimensional form through use of the Mangler transformation
6.4.11 Heat Transfer in a Turbulent Boundary Layer
The two-dimensional boundary layer form of the time-averaged governing equa-tions is
∂ ¯u
∂x+
∂ ¯v
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ρ
¯u ∂ ¯u
∂x + ¯v
∂ ¯v
∂y
= −d ¯p
dx + µ
∂2¯u
∂y2 − ∂
∂y (ρuv)
= −d ¯p dx +∂τ ∂y m −∂y ∂ (ρuv) (6.79)
ρc p
¯u ∂ ¯T ∂x + ¯v ∂ ¯T ∂y
= k ∂2¯T
∂y2 −∂y ∂ (ρc p vT) = − ∂q ∂y m −∂y ∂ (ρc p vT) (6.80)
whereτm = µ(∂ ¯u/∂y) and q
m = −k(∂ ¯T /∂y) are the mean shear stress and heat
flux, respectively
two-dimensional flow, with the velocity component as well as all derivatives of time-averaged quantities neglected in thez direction For axisymmetric flow, such as in
a circularjet orwithin the boundary layeron a body of circularcross section, also
called a body of revolution (Fig 6.13), Cebeci and Bradshaw (1984) show that the
governing equations can be generalized:
2
1.6
1.2
0.8
0.4
0
Nux
Ua⬁
Similarity
Similarity
Pr = 1
Pr = 10
180 (deg)
x
r0
Figure 6.12 Local heat transfer behavior for flow past a heated sphere at uniform surface temperature for various Pr values, using the integral method (From Cebeci and Bradshaw, 1984.)
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Mainstream Fluid
u⬁
x = 0 Axis of Symmetry 90 z
y
r r o
Boundary Layer
Figure 6.13 Boundary layer development on a body of revolution
∂r N ¯u
∂x +
∂r N ¯v
ρ
¯u ∂ ¯u
∂x + ¯v
∂ ¯v
∂y
= −dp
dx +
1
r N
∂
∂y
r N
µ∂ ¯u
∂y − ρuv
(6.82)
ρc p
¯u ∂ ¯T ∂x + ¯v ∂ ¯T ∂y
=r1N ∂y ∂
k ∂ ¯T ∂y − ρc p vT
(6.83)
whereN = 1 in axisymmetric flow and N = 0 in two-dimensional flow The
turbulent components of the shear stress and the heat flux are defined as
τT = −ρuv= ρ∂ ¯u ∂y and q
T = −ρc p vT= −ρc pH ∂ ¯T
∂y
where and Hare, respectively, the turbulent or eddy diffusivities of momentum and heat The governing two-dimensional equations with these incorporated become
¯u ∂ ¯u
∂x + ¯v
∂ ¯v
∂y = −
1
ρ
d ¯p
dx +
∂
∂y
(ν + ) ∂ ¯u
∂y
(6.84)
and
¯u ∂ ¯T
∂x + ¯v
∂ ¯T
∂y =
∂
∂y
(α + H ) ∂ ¯T
∂y
= ν ∂
∂y
1
Pr+ ν
1
PrT
∂ ¯T
∂y
(6.85)
where PrT = / H is the turbulent Prandtl number Solution of (6.84) and (6.85) requires modeling of the turbulent shear stress and heat flux
combina-tion of the mean and turbulent components: