Similarity for boundary layer flow follows from the observation that while the boundary layer thickness at each downstream locationx is different, a scaled normal distanceη can be employe
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u∗∂T∗
∂x∗ + δ
δT v∗∂T∗
∂y∗
T = 1
Pr
δ
δT
2∂2T∗
∂y∗2
T
+ 2Ec · Φ∗+ 2βT · Ec · u∗dp∗
dx∗ +ρc qL
p U ∆T (6.21)
The corresponding dimensional form of eq (6.21) is
u ∂T
∂x + v
∂T
∂y = α
∂2T
∂y2 + βT
ρc p u
dp
dx +
µ
ρc p
∂u
∂y
2
+ q
ρc p (6.22)
Equation (6.20) can be used to demonstrate that when Pr 1,
δT
δ = O
1
Pr1/2
and when Pr 1,
δT
δ = O
1
Pr1/3
6.4.2 Similarity Transformation Technique for Laminar Boundary Layer Flow
Following the simplifications of eqs (6.19)–(6.22), the two-dimensional steady-state boundary layer equations are:
∂u
∂x +
∂v
u ∂u ∂x + v ∂u ∂y = −1ρ∂p ∂x + ν∂ ∂y2u2 = UU x+ ν∂ ∂y2u2 (6.24)
u ∂T
∂x + v
∂T
∂y = α
∂T2
∂y2 + βT u ∂p
∂x +
µ
ρc p
∂u
∂y
2 + q
ρc p (6.25)
The boundary conditions for an impermeable surface are
Equations (6.23)–(6.25) constitute a set of nonlinear partial differential equations
Under certain conditions similarity solutions can be found that allow conversion of this set into ordinary differential equations The concept of similarity means that
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certain features (e.g., velocity profiles) are geometrically similar Analytically, this amounts to combining thex and y spatial dependence on a single independent
vari-ableη The velocity components u(x,y) and v(x,y) are expressed by a single
nondi-mensional stram functionf (η), and the temperature T (x,y) into a nondimensional
temperatureφ(η).
Similarity for boundary layer flow follows from the observation that while the boundary layer thickness at each downstream locationx is different, a scaled normal
distanceη can be employed as a universal length scale Presence of natural length
scales (such as a finite-length, plate, cylinder, or sphere) generally precludes the finding of similarity solutions Using the scaled distance, the similarity procedure finds the appropriate normalized stream and temperature functions that are also valid
at all locations Following Gebhart (1980), the similarity variables are defined as
f (η) = ψ(x,y)
φ(η) = T (x,y) − T T ∞(x)
where the allowable forms ofb(x) and c(x) (defined later) and
d(x) = T0(x) − T∞(x)
j (x) = T∞(x) − Tref need to be determined The transformed governing equations in terms of the foregoing normalized variables are
f(η) + 1
b(x)
dc(x)
dx f (η)f(η) −
1
b(x)
dc(x)
c(x) b(x)
db(x) dx
[f(η)]2
ρv2c(x)[(b(x)]3
dp
φ(η)
Pr +b(x)1 dc(x) dx f (η)f(η) − b(x)d(x) c(x) dd(x) dx f(η)φ(η)
− c(x)
b(x)d(x)
dj (x)
dx f(η) +
βT
ρc p
c(x) b(x)d(x)
dp
dx f(η)
+cν2
p
[c(x)b(x)]2
d(x) [f(η)]2+
1
d(x)[b(x)]2
q
kPr= 0 (6.31)
The boundary conditions are
f(0) = f (0) = 1 − φ(0) = φ(∞) = 0
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and it is also noted that
dp
dx = −ρU
dU dx
For the momentum equation to be entirely a function of the independent variableη,
1
b(x)
dc(x)
dx = C1
c(x)
[b(x)]2
db(x)
dx = C2 This results in choices for the constants,C1andC2:
c(x) = ke kx (C1= C2)
kx q (C1 2)
withC1andC2related by
C2= q − 1 q C1 For the pressure gradient term to be independent ofx, the free stream velocity
must be
U(x) ∝ x2q−1
Also known as Falkner–Skan flow, this form arises in the flow past a wedge with an
included angleβπ as seen in Fig 6.6 In this case,
U(x) = ¯Cx m
where
2− β
as indicated in potential flow theory From the similarity requirement, the exponent
q becomes
q = m + 1
2− β
Then the pressure gradient term in eq (6.30) becomes
− 1
ρν2
1
c(x)[b(x)]3
dp
dx =
(C1− C2)3m ¯C2
ν2k4
x2m−1
x4q−3 = β
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The arbitrary constantsC1andC2may be chosen, without loss of generality, as
C1= 1 and C2= β − 1
This results in
c(x) =
2
m + 1Rex
1/2
b(x) = 1x
m + 1
2 Rex
1/2
η = y
x
m + 1
2 Rex
1/2
ψ(x,y) = νf (η)
2
m + 1Rex
1/2
and forthis choice of constants, the Falkner–Skan momentum equation and boundary conditions become
f(η) + f (η)f(η) + 1− f(η) 2β = 0
f(η = 0) = f (η = 0) = 1 − f(η = ∞) = 0
The often used (and much older) Blasius (1908) variables for flow past a flat plate (β = 0) are related to the Falkner–Skan variables η and f (η) as
ηB= 21/2 η and f (η B ) = 21/2 f (η)
Forsimilarity to hold, the energy equation must satisfy the conditions
c(x) b(x)d(x)
dd(x)
ν2[c(x)b(x)]2
c p d(x) = K3
[c(x)b(x)]2
d(x) = K3C6 (6.32b)
c(x) b(x)d(x)
dj (x)
βT
ρc p
c(x) b(x)d(x)
dp
dx = K4
c(x) b(x)d(x)
dp
dx = K4C10 (6.32d) 1
d(x)[b(x)]2
1 Pr
q
k = F1(η) (6.32e)
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6.4.3 Similarity Solutions for the Flat Plate at Uniform Temperature (m = 0)
For the case of the flat plate, the similarity equations and boundary conditions reduce to
f(η) + f (η)f(η) = 0 φ(η) + Pr · f (η)φ(η) = 0 (6.33) and
1− φ(0) = φ(∞) = 0 and f (0) = f(0) = 1 − f(∞) = 0 (6.34) Both the momentum and energy equations are ordinary differential equations in the form of two-point boundary value problems The momentum equation is solved first because it is uncoupled from the energy equation The velocity field is then substituted into the energy equation to obtain the temperature field and heat transfer characteristics
The wall heat flux is obtained as
q(x) = h x (T0− T∞) = −k ∂T
∂y
y=0 = −k(T0− T∞)φ(0) ∂η
∂y
= −k(T0− T∞)φ(0)1
x · Re1x /2 (6.35)
This results in the local Nusselt number
Nux = h x x
−φ(0)
√
2 Re
1/2
x = ¯F(Pr)Re1/2
x
where ¯F (Pr) is determined numerically and near Pr ≈ 1 is well approximated by
0.332Pr1/3so that,
Nux = 0.332Re1/2
the surface-averaged heat transfer coefficient is determined as:
¯h = A1
s h dA s =L1 L
0
h x dx = 2h
x=L
(6.37)
6.4.4 Similarity Solutions for a Wedge (m = 0)
For a wedge at a uniform surface temperature, the expressions for the surface heat flux and the Nusselt numberare
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q(x) = −k(T0− T∞)φ(0)1
x
m + 1
2 Rex
1/2
(6.38)
Nux= h x x
−φ(0)
√
2 [(m + 1)Re x 1/2 = ¯F(m, Pr)Re1/2
For a wedge with a spatially varying surface temperature, eq (6.32a)
C5= c(x)
b(x)d(x)
dd(x) dx
yields
1
d(x)
dd(x)
dx = C5b(x)
c(x) = C5m + 1
2x =
n x
where
n ≡ C5m + 1
2
is a constant This yields
d(x) = [T0(x) − T∞]= Nx n
whereN is a constant arising from the integration for the surface temperature
varia-tion The energy equation is transformed to
φ(η) + Pr
f (η)φ(η) − m + 12n f(η)φ(η)
= 0
The resulting expressions for the heat flux and the local Nusselt number are
q(x) = −kφ(0)N
(m + 1) ¯C
2ν
1/2
Nux = −φ(0)
m + 1
2 Rex
1/2
= ¯F(Pr, m, n)Re1/2
In the range of 0.70 ≤ Pr ≤ 10, Zhukauskas (1972, 1987) reports the correlation for
the data computed by Eckert:
Nu
Re1/2 x
= 0.56A
where
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β = 2m
m + 1 and A = (β + 0.20)0.11· Pr0.333+0.067β−0.026β
2
For the thermal boundary layer thickness to increase withx, two special cases of
the foregoing solution corresponding to a flat plate (m = 0) are of interest (Fig 6.7).
These correspond respectively to a uniform heat flux surface and a line heat source
atx = 0 (a line plume) The heat flux can be written as
q(x) = −kφ(0)N
U ν
1/2
x (2n−1)/2
Forthe first condition (Fig 6.7a), n = 1
2 Forthe second condition (Fig 6.7b), the
total energy convected by the flow per unit length of the source is written as
q(x) = ∞
0
ρc p u(T − T∞) dy
= νρc p c(x)d(x) ∞
0
f(η)φ(η) dη ∝ x (2n+1)/2 (6.43)
For the convected energy to remain invariant withx, n must take on the value n = −1
2
Wedge Flow Limits With regard to wedge flow limits, numerical solutions to the Falkner–Skan equations have been obtained for−0.0904 ≤ m ≤ ∞, where the lower
limit is set by the onset of boundary layer separation In addition, the hydrodynamic boundary layer thickness is
Figure 6.7 Two important cases for boundary layer flow at uniform free stream velocity
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δ(x) = ηδ
2ν
(m + 1) ¯C
1/2
x (1−m)/2
whereηδ, the nondimensional thickness of the boundary layer, is bound atx = 0
only form ≤ 1 This requires that 1 ≥ m ≥ −0.0904 Additionally, the total energy
convected by the flow perunit width normal to the plane of flow is given by
q(x) = νρc p c(x)d(x) ∞
0
f(η)φ(η) dη ∝ Nx (2n+m+1)/2
This provides the condition
2n + m + 1
2 ≥ 0 or n ≥ − m + 1
2
6.4.5 Prandtl Number Effect
Considerthe case ofn = m = 0 first In the limiting cases of Pr 1 and Pr 1
(Fig 6.8), the solution of the momentum equation can be approximated in closed form Subsequently, the energy equation can be solved For Pr 1, the velocity
componentsu and U are approximately equal throughout the thermal boundary layer.
This results inf(η) ≈ 1 or f (η) ≈ η + K Foran impermeable wall, K = 0 and
the energy equation simplifies to
φ(η) + Pr · ηφ(η) = 0
This can be integrated to yield
φ(η) = e−η2+C
where
η2
1= η2· Pr
2
Figure 6.8 Laminar flat plate boundary layer flow at limiting Prandtl numbers
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and with the boundary conditions
1= φ(0) = φ(∞) = 0
another integration provides
φ(η) = 1 − erf(η1)
where
erf(η1) = √2
π
1
0
e −v2
dv
From this the local Nusselt number is determined as
Nux = −φ√(0)
2 Re
1/2
x = √2 π
Pr
2 Rex
1/2
= 0.565Re1/2
x · Pr1/2 (6.44)
and forPr 1, the nondimensional stream function near the wall is expressed as
f (η) = f (0)
0! +f(0)
1! η + f(0)
2! η2+f(0)
3! η3+ · · · (6.45) However,f (0) = f(0) = 0 and the momentum equation shows that f(0) = 0.
This results in
f (η) = f(0)
2! η2= 0.332√
upon using
f(0) =√2f
B (0) = 0.332√2 where the subscriptB refers to the Blasius variables defined in Section 6.4.2 The
energy equation then becomes
φ(η) +0.332Pr√
This can be integrated and evaluated at the surface to yield
−φ√(0)
2 = 0.339Pr1/3 (6.48a)
Nux = 0.339Re1/2
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6.4.6 Incompressible Flow Past a Flat Plate with Viscous Dissipation
Using the Blasius normalized variables for the flow and then defining the normalized temperature as
θ(η) = 2c p (T − T∞)
U2 the energy equation and thermal boundary conditions become
θ(η) +1
2Pr· f (η)θ(η) + 2Pr · f(η)2= 0 (6.49) and
θ(η = ∞) = 0 and θ(η = 0) = 2c p (T0− T∞)
U2 = θo (constant) (6.50)
The solution to eq (6.49), which is a nonhomogeneous equation, consists of the superposition of a homogeneous part:
θH = C1φ(η) − C2 and a particular solution
θP = θAW(η)
where the subscript AW refers to an adiabatic wall condition
The governing equations and boundary conditions for these are
θ
AW+1
2Pr· f θ
AW+ 2Pr(f)2= 0 (6.51)
φ+1
2Pr· f φ= 0 (6.52) and
θ
AW(0) = θ
Forthe boundary conditions forthe complete problem to be satisfied,
θ(∞) = 0 = θAW(∞) + C1φ(∞) + C2 which leads toC1= −C2and
θ(0) = 2c p (T0− T∞)
U2 = θAW(0) + C1φ(0) + C2= 2c p (TAW− T0)
or
... substituted into the energy equation to obtain the temperature field and heat transfer characteristicsThe wall heat flux is obtained as
q(x) = h x... 6.7).
These correspond respectively to a uniform heat flux surface and a line heat source
atx = (a line plume) The heat flux can be written as
q(x)...
Nux = 0.332Re1/2
the surface-averaged heat transfer coefficient is determined as:
¯h = A1
s