ABSTRACT
Conservation equations have been derived for physical variables such as velocity components, pressure, and temperature. However, a lot can be learned about the solution of these equations if those variables are normalized. Consider the two-dimensional, steady-state boundary layer equations [1]:
∂ ∂
+ ∂ ∂
=U x
V y
0. (10.1)
U U x
V U y
P x
U y
X∂ ∂
+ ∂ ∂
= − ∂ ∂
+ ∂ ∂
+1 ρ
ν ρ
2 , (10.2)
∂ ∂
=P y
Y , (10.3)
U T x
V T y
T y C
U y
g Cp p
∂ ∂
+ ∂ ∂
= ∂ ∂
+ ∂ ∂
+α
ν ρ
2
. (10.4)
If L is a characteristic length of the flow, the normalized coordinates will be defined as
x x
L y
y
L * * .= =and (10.5)
Also, if U∞ and T∞ are free stream velocity and temperature, respectively, and Ts is a constant surface temperature, the following normalized variables can be defined:
U U U
V V
U T
T T T T
P P U
* * * *= = = − −
, , , ρ ∞2
, (10.6)
X XL
U Y
YL
U g
gL
U C T Tp * * *
( = = =
−ρ ρ ρ∞ ∞ ∞ ∞ , ,
s ) . (10.7)
Substituting these normalized variables into the boundary layer equations results in the following set of normalized two-dimensional, steady-state boundary layer equations:
∂ ∂
+ ∂ ∂
=U x
V y
* *
* *
,0 (10.8)
U U x
V U y
P x U L
U y
X* * *
* * *
* *
* *
*, ∂ ∂
+ ∂ ∂
= − ∂ ∂
+ ∂ ∂
+ν ∞
∂ ∂
=P y
Y * *
*, (10.10)
U T x
V T y U L
T y
U C L T Tp s
* * *
* * *
* * (
∂ ∂
+ ∂ ∂
= ∂ ∂
+ −
α ν ∞
2 ) * *
*. ∂ ∂
+
U y
g 2
(10.11)
It is also helpful to rewrite the expressions for the local friction coefficient and the local convection heat transfer coefficient,
C
U
y
U f
y=
∂ ∂ =
µ
ρ
21 2 ∞
and h
k T y
T T
= − ∂
∂ −
, (10.12)
in terms of the normalized variables:
C U L
U yf y
= ∂ ∂ =
ν ∞
* *
(10.13)
hL k
T yf y
= ∂ ∂ =
* *
(10.14)
Four groups of parameters appear in the normalized boundary layer equations, Equations 10.8 through 10.11, and the expressions for friction coefficient, Equation 10.13, and convection heat transfer coefficient, Equation 10.14. These are ν/U∞L, α/U∞L, νU∞/CpL(Ts – T∞) and hL/kf. It can be easily verified that these groups are all dimensionless. For example, the units for ν, U∞, and L are m2/s, m/s, and m, respectively, and therefore ν/U∞L is just a number with no unit. Three of these
dimensionless groups appear as coefficients of different terms of the boundary layer equations and therefore affect the solution of these equations. The last one, hL/kf, can be considered as a dimensionless convection heat transfer coefficient. Each of these dimensionless groups has an important physical meaning that will be described in the next section.
