Parabolic Flow
Parabolic flows are characterized by a dominant flow direction such that flow
quantities are determined by the upstream and cross-stream conditions. No influence is
coming from the downstream of the dominant flow direction. There are many parabolic
flows in engineering applications such as boundary layer flows, duct flows, jet and
mixing layers. Since the predominant flow direction behaves somewhat like time
coordinate, 2-dimensional steady parabolic flows can be treated as 1-dimensional
transient convection diffusion problem and 3-dimensional steady parabolic flows as 2dimensional transient convection-diffusion problem.
1. Two-dimensional parabolic flow
The mass conservation equation in 2-dimensional steady flow in Cartesian
coordinate is
u v
(1)

0
x
y
The x-momentum equation is
 
u   
u 
p
(2)
 uu      vu       S
x 
x  y 
y 
x
For the convection dominated flow, diffusion in the x-direction is small compared to
convection, uu   u , and the second term in the first bracket can be neglected.
x
Since u  v, y-direction momentum equation is not needed and the pressure gradient
along the y-direction is zero. Thus the x-momentum equation reduces to

uu     vu   u    dp  S
(3)
x
y 
y 
dx
1.1 Finite Volume Representation
Finite volume representation of parabolic flow is obtained by integrating Eqs(1) and
(3) over shaded control volume as shown in Fig. 1. All flow properties, density, pressure
and x-velocity are defined at the center of control volume surface (circle) and y-velocity
is defined at the corner of control volume (diamond). All flow properties are known at x
and flow properties at x  x is to be determined.
Integrating Eq.(1), we get
u
P
where

 u P y  Fn  Fs  0
0
(4)
Fn , s  v n , s x
1
Superscript 0 implies known flow properties at x. It is assumed that Fn , s evaluated at
x  x prevails over x . It is like fully implicit scheme used in time dependent
formulation.
N
n
y  y
y n
y
P
0
P
y s
y
s
S
x
x
x  x
Fig. 1 Control volume for 2-D parabolic flow in x-direction
Integrating the x-momentum equation over the control volume, dv  dxdydz , and
taking an unit depth in z-direction, we have
uu 
P

 dp 
0
 uu P y  J n  J s     xy  S c  S P u P xy (5)
 dx  P
where

u 
J n,s   vu    x
y  n, s

and source term is linearized. The pressure gradient term is excluded from the source
term to emphasize its importance.
Now multiplying Eq. (4) by u P and subtract it from Eq. (5) and introducing
J n  Fn u P  a N u P  u N 
J s  Fs u P  a S u S  u P 
the following finite volume representation of x-momentum equation results
aP u P  a N u N  aS u S  b
where
a N  Dn A Pn    Fn ,0
(6)
(6a)
2
aS  Ds A Ps   Fs ,0
(6b)
a P  a N  a S  a P0  S P xy
(6c)
a P0  u P y
0
(6d)
 dp 
(6e)
b    xy  S C xy  a P0 u P0
 dx  P
Diffusion conductance and convection strength are evaluated as before and power law
scheme is used for the interface flux approximation. For example, at the north interface,
 x
Dn  n
y n
Fn  v n x
P n  Dn / Fn
A Pn   0, 1  0.5 Pn
(6.f)

5
We observe a strong similarity between the 2- D parabolic flow and 1-D transient
convection-diffusion problem in y-direction. Table 1 shows differences between two
formulations.
Table 1
a
0
P
Dn , s
Fn , s
2-D Parabolic
u 0P xy / x
Transient 1-D
 P0 y(1x1) / t
 n , s x / y n , s
v n,s x
 n, s (1x1) / y n, s
v n,s (1x1)
Once velocity u is found by solving Eq.(6) by TDMA, y-direction velocity is calculated
by using the continuity, i.e., Eq. (4).
v n  v s  u P  u 0P  y
x
Eq.(7) provides v at all y starting with the known boundary value of v at y=0.
(7)
Unlike time dependent convection-diffusion formulation, there is no need to
iterate within a x step as long as it is not too large. The pressure gradient in the xdirection must be prescribed in the source term.
1.2 External Boundary Layer
3
In the external boundary layer, pressure gradient along the x-direction can be
determined by the free stream velocity.
d
 dp 
(8)
  
u2 P
 dx  P dx
where u (x) is the free stream velocity.


1.2.1 Momentum boundary layer over a flat plate
As an example of external boundary layer, consider air flow over a flat plate as
shown in the figure. Air properties at 300 K are used. Since flat plate, the pressure
gradient is zero. In selecting calculation domain, x distance is limited by laminar flow
constraint and maximum y distance should be well above the momentum boundary layer.
u  6 m / s
air
Calculation
domain
x
Fig. 2 Momentum boundary layer over a flat plate
Y-direction grid is clustered near the solid wall to capture boundary layer thickness and
smaller step size x is used near x=0 to capture rapid change in flow quantities.
Numerical results in terms of local shear stress is calculated and compared with
known Blasius solution (table 2). Also non-dimensional velocity u / u  is compared with
known solutions (Fig. 3). Numerical local shear stress agrees well with the Blasius
solution except near the leading edge. This is expected since boundary layer solution is
not valid near the leading edge. A good agreement is shown for the velocity profile.
4
Table 2 Local shear stress comparison
x(m)
Blasius
Present
0.231
0.0429
0.0405
0.451
0.0307
0.0299
0.671
0.0252
0.0250
1.111
0.0196
0.0197
1.21
0.0187
0.0189
1.2
1
0.8
u/uinf
Blasius
Present
0.6
0.4
0.2
0
0
2
4
6
8
10
12
eta
Fig. 3 u / u
vs 
1.2.2 Thermal boundary layer
Using enthalpy as the dependent variable, the steady energy equation in xdirection is
uh vh    h     h 
p
p
  
  u  v    S h

 
x
y
x  c P x  y  c P y 
x
y
p
T
In the boundary layer, u  v,
. For slow speed flow
 0, and uh  
y
x
viscous dissipation   0. Thus above equation reduces to
5
uh  
 h 
dp
  u
(9)
  vh 
 Sh
x
y 
c P y 
dx
Eq.(9) is parabolic in x-direction and can be solved by parabolic flow approximation. The
first term in RHS of Eq. (9) is generally very small compared with enthalpy flux and can
be also neglected in slow speed flow. This assumption is consistent with neglecting
viscous dissipation term in Eq.(9).
By solving momentum boundary layer and thermal boundary layer
simultaneously, combined developing boundary layer flow can be analyzed. As an
example of combined boundary layer flow, consider previous problem with the solid wall
temperature is raised to 700 K. Numerical results are compared with exact solution in
terms of local Nusselt number.
Table 3 Local Nusselt number comparison
x(m)
Blasius
Present
0.231
71.56
67.16
0.451
99.97
96.45
0.671
121.97
119.03
1.111
156.94
154.80
1.21
163.79
161.78
The agreement becomes better at locations away from the leading edge.
A more general external boundary layer problem with non-zero pressure gradient
can be easily analyzed by modifying the pressure term using free stream velocity, u x  .
The control of momentum and thermal boundary layers can be achieved by blowing and
suction at the plate. This can be accomplished by specifying non-zero v at the plate
surface. Some of these applications are left as exercise problems.
1.3 Internal Flow
Another example of 2-dimensional parabolic flow is steady flow through ducts.
In this case, pressure gradient can not be determined by the free stream velocity. Instead
mass conservation is used to find the pressure gradient. Since exact pressure gradient is
not known, calculation begins with an assumed pressure gradient at x. That is
*
0
 dp 
 dp 
   
 dx  P  dx  P
The x-momentum equation with the assumed pressure gradient is
*
 dp 
(10)
a P u P*  a N u *N  a S u S*     xy  b '
 dx  P
where * represent velocity obtained by using the assumed pressure gradient. In Eq.(10)
source term is modified from that in Eq.(6) by taking out the pressure gradient term. By
solving Eq.(10 ), we obtain u P* .
The exact velocity and pressure gradient are
6
u P  u P*  u P'
*
'
 dp 
 dp 
 dp 
     
 dx  P  dx  P  dx  P
(11)
where prime denotes velocity deviation and pressure gradient deviation.
Substituting Eq.(11) into Eq.(6) and using Eq.(10), we get
'
 dp 
a P u  a N u  a S u    xy
 dx  P
'
P
'
N
'
S
Now we assume that
xy  dp 
u 
 
a P  dx 
Eq.(12) relates pressure correction to the velocity correction.
'
'
P
(12)
The mass conservation requires that
0
m   u P y   u P y
Substitute
xy  dp 
uP  u 
 
a P  dx 
into the last equation and solve for the pressure gradient correction term, noting that it is
independent of y, we have
'
*
P
'
 dp 
  
 dx  P
m    P u P* y
xyy
 P a
P
(13)
Substituting Eq.(13) into Eq.(12),
 m   u * y
 P P
 y  
'
u P    
 a P     yy
P

aP



(14)


The first bracket term in RHS of Eq.(14) is dependent on y while the second bracket term
is not.
By adding velocity and pressure corrections given by Eqs.(13) and (14)
respectively, we get corrected velocity and pressure at x  x  x . The y-velocity is
obtained as in the external parabolic flow.
1.3.1 Entrance flow between two parallel plates
Consider air entering parallel plates as shown in Fig. 4. Air velocity is 0.5 m/s
and its temperature is 300 K while the plate temperature is at 700 K. The energy equation,
7
Eq.(9) is also applicable to this situation. Appropriate modifications were added to the
program used for the external flow to account for the pressure change. The local skin
friction coefficient and local Nusselt number are compared with the known analytic
solutions.
u e  0.2 m / s
y
Te=300 K
D=0.05
m
x
Tw  700 K
x0
x=3 m
Fig. 4 Developing flow between two parallel plates
The local skin friction coefficient is defined
 x 
C f ,x  w 2
0.5u e
1
Sparrow showed by using approximate integral method that
8 uc
C f , x Re D 
3 ue
 ue
1 
 uc



(15)
1
(16)
u e D
and u c (x) is the centerline velocity. Numerical skin friction is

estimated by using Eq.(15) where wall shear stress is obtained from numerical velocity
profile. The centerline velocity is expressed as an implicit function of distance x in
analytical solution. Instead, in the present calculation, numerically obtained centerline
velocity is used in Eq.(16). Fig. 5 shows a good agreement in the local skin friction
coefficient. Fig. 6 shows local Nusselt number comparison for thermally developing flow
with constant wall temperature. Agreement with a reference 2 is not as good in the
entrance region as the fully developed region. This is partially due to the approximation
involved in analytical solution using boundary layer approximation.
where Re D 
E. M. Sparrow, “Analysis of laminar forced convection heat transfer in the entrance region of flat
rectangular ducts”,NACA TN3331,1955.
2
R. K. Shah and A. L. London, Laminar forced convection in ducts, Supplement 1 to Advances in Heat
Transfer, Academic Press, New York, 1978.
1
8
CfxRed
100
Present
Sparrow
10
-0.01
0.01
0.03
0.05
0.07
x/D/Red
Fig. 5 Local Skin friction between two parallel plates
18
Nux
16
14
Nux(Num)
12
Nux (ref)
10
8
6
4
2
0
0
0.05
0.1
0.15
0.2
(x/D/Red/Pr)^0.5
Fig. 6 Local Nusselt number between parallel plates (constant wall temperature)
9
1.3.2. Entrance flow in a circular duct
It can be shown that mass, momentum and energy conservation equations
parabolic flow through a circular ducts are
u 1  ( yv)

0
x
y y
(17a,b,c)
uu 1  
u 
dp
 y ( vu   )   

x
y y 
y 
dx
uh 1  
 h 
dp
 y( vh 

)   u
 Sh
x
y y 
c P y 
dx
In Eq.(17) y is the radial coordinate and x is the axis. Integrating Eq.(17 ) over a control
volume, dv  dxdyyd , and taking unit radian in the angular direction, we obtain a finite
volume representation of Eq.(17) similar to Cartesian case except the control volume size
and its surface area are dependent on the radial coordinate as seen in previous chapters.
Consider air entering a circular pipe at constant velocity of 0.2 m/s. The air
temperature is 300 K and the duct wall is constant at 700 K. Numerical results are
compared with known data in terms of local skin friction coefficient and local Nusselt
number. This example is similar to the previous case except the geometry.
35
0
-0.02
30
Cfx Red (num)
25
Cfx Red (ref)
-0.04
-0.06
Cfx Red
P (pa)
20
-0.08
15
-0.1
P
-0.12
10
-0.14
5
-0.16
0
-0.18
0
0.02
0.04
0.06
0.08
0.1
x/D/Red
Fig. 7 Local skin friction coefficient and pressure variation in the entrance region of a
circular tube
10
7.5
7
6.5
Nux
6
Nux (num)
Nux (ref)
5.5
5
4.5
4
3.5
3
0
0.1
0.2
0.3
0.4
(x/D/Red/Pr)^0.5
Fig. 8 Local Nusselt number in the entrance region of a circular tube (constant wall
temperature)
Numerical results are in good agreement with reference values in both skin friction and
Nusselt number. The pressure variation is nonlinear in the developing region (x/D/Red=
~0.01) and becomes linear in the hydro-dynamically fully-developed region.
2.
Three-Dimensional Parabolic Flow
Three–dimensional parabolic flows are also characterized by a dominant flow
direction. No reverse flow is allowed and flow condition is influenced by the upstream
and cross stream conditions. Examples of 3-dimensional parabolic flow are 3dimensional boundary layer flows, flow through non-circular ducts and flow through duct
with curvatures.
To extend the numerical formulation of 2-D parabolic flow to 3-D parabolic flow,
consider a 3-D steady flow through a rectangular duct.
11
y
y
x
x
x  x
z
z
Fig. 8 Three dimensional parabolic flow in a rectangular duct showing dominant flow in
x-direction and secondary flows on y-z plane
The conservation of mass, momentum in x, y, and z-directions are
u  v  w


0
x
y
z
uu   
u   
u 
dp
  vu      wu     
 Su
x
y 
y  z 
z 
dx
uv   
v   
v 
p
  vv      wv       S v
x
y 
y  z 
z 
y
uw  
w   
w 
p
w
   ww  
  vw  
 S
x
y 
y  z 
z 
z
(18)
(19)
(20)
(21)
There are two pressure fields in these equations. The pressure gradient in the xmomentum equation is the averaged pressure over the cross section at each axial
direction and is independent of y and z-coordinates. The pressure gradients in the y and zdirection are responsible for the cross flow velocity fields. This decoupling of pressure
gradient into two parts: a dominant flow direction and a cross flow direction is an
important concept in 3-dimensional parabolic flow formulation.
Integrating Eq.(19) over a control volume (see Fig. 9) , dv  dxdydz , we obtain
finite volume representation of x-momentum equation. Using a guessed dominant
12
pressure gradient, we get, u P* at the down stream location as we did in 2-D parabolic
procedure. Similarly, mass conservation requirement provides the pressure correction
term as
'
m    P u P* yz
 dp 
(22)
  
xy 2 z 2
 dx  P
  P a u
P
Then the velocity correction is given by
xyz  dp 
u 
 
a Pu  dx  P
'
'
P
(23)
∆x
N
W
P
y
E
x
S
z
Fig. 9 A control volume module for u-velocity
The cross flow velocities, v and w, are defined at the nominal control volume surface
defined for u, p and density and staggered control volumes are used for the integration of
cross flow momentum equation. Fig. 10 shows staggered grid arrangement for cross flow
velocities.
13
∆y
y
∆z
z
Fig. 10 Staggered grids for the cross flow velocities on the y-z plane: v (////); w(\\\\)
Integrating y-momentum equation over a staggered control volume (shown with
//// in Fig. 10), we get a finite volume representation as
a Pv v P  a Ev v E  aWv vW  a Nv v N  a Sv v S  b v  ( p S  p P )zx
(24)
where a superscript v denotes quantities associated with y-momentum equation. All
coefficients in Eq.(24) have the usual meanings as discussed before. For example,
a Ev  De A Pe    Fe ,0
De   e
xy
z e
(25)
Fe  we xy
Similarly z-momentum equation is integrated over a control volume (shown with \\\\ in
Fig. 10), we have
a Pw wP  a Ew wE  aWw wW  a Nw wN  a Sw wS  b w  ( pW  p P )yx (26)
Eqs.(24) and (25) can not be solved until the cross flow pressure fields are known.
To start the computation, we start with an assumed cross pressure field p* everywhere.
With a guessed pressure fields, Eq.(24) is
14
a Pv v P*  a Ev v E*  aWv vW*  a Nv v *N  a Sv v S*  b v  ( p S*  p P* )zx
(27)
Subtracting Eq.(27) from Eq.(24), we have
xz '
(28)
p S  p P'
v
aP
where the primed quantities are deviations from the correct values. Eq.(28) relates the
pressure correction to the y-velocity correction. Similarly for the z-velocity correction

v P' 
wP' 

xy '

pW  p P' 
w
aP
(29)
The correct velocity and pressure fields are then
v P  v P*  v P'
wP  wP*  wP'
(30)
p P  p P*  p P'
To obtain the pressure correction, we now turn to the requirement of mass conservation.
Ultimately the correct velocity fields must satisfy the mass conservation, Eq.(18).
Rewriting Eq.(18),
v  w
u
(31)


y
z
x
The RHS of Eq.(31) is a known quantity from the x-momentum equation and Eq. (31) is
a 2-dimensional continuity with a known source term. Integrating Eq.(31) over a nominal
control volume, we have
u

 u P yz  vn  vs xz  ve  vw xy  0 (32)
Substituting Eq.(30) into Eq.(32) and using Eqs.(28) and (29), we get the pressure
correction equation,
P
0
a Pp p P'  a Ep p E'  aWp pW'  a Np p N'  a Sp p S'  d Pp
where
a  e
p
E
aWp   w
a Np   n
xy 2
a Ew
xy 2
a Pw
xz 2
a Nv
(32)
(33a)
(33b)
(33c)
15
a Sp   s
xz 2
(33d)
a Pv
a Pp  a Ep  aWp  a Np  a Sp
(33e)
and
0
d Pp  u P  u P yz  (  s vs*   n vn* )xz   w ww*   e we* xy




(33f)
Eq.(33f) is the residual mass in a nominal control volume due to incorrect cross flow
velocities. Eq.(32) is solved repeatedly until the mass residual in each control volume
becomes small.
In summary, the procedure to obtain the cross flow velocities is:
(1) Guess cross flow pressure fields at x+∆x. The best guess could be the value at the
immediate upstream value at x.
(2) Solve w* and v* using z-momentum and y-momentum equations with guessed
pressure.
(3) Evaluate the mass residual using Eq.(33f). If it is less than an error tolerance, no
iteration is needed. If not, solve the pressure correction equation, Eq. (32) and
update the pressure and the cross flow velocities using Eqs. (28)-(30) and go to
step (2).
(4) Repeat this process until mass residual in all control volume reduces below an
error tolerance.
2.1 Entrance flow in a rectangular duct
Consider a micro-channel flow through a duct with constant rectangular cross
sectional area as shown in Fig. 11. The width (in z-direction) and the height (in ydirection) are 0.48 mm and 0.46 mm, respectively. The water entering the channel has
uniform velocity of 1.1 m/s. All fluid properties are evaluated at 300 K and remain
constants. No heat transfer is considered. The flow is laminar with Red=603. A numerical
program based on 3-D parabolic flow as discussed in previous section is used. Fig. 12
shows the numerical results in terms of the mean pressure and the local friction factor
distribution along the channel. Results agree well with known data3 at the fully developed
region.
D. Liu and S. Garimella, “Investigation of Liquid Flow in Microchannels”, AIAA20002-2776, 8th
AIAA/ASME Joint Thermophysics and Heat Transfer Conference, June 2002, St. Louis, Missouri.
3
16
y
x
z
Fig. 11 Developing flow through a rectangular micro-channel
0
180
pbar (pa)
pbar (measured)
fRed
fRed (ref)
-500
-1000
160
140
120
100
-2000
80
f Red
Pbar (Pa)
-1500
-2500
60
-3000
40
-3500
20
-4000
0
0
5
10
15
20
25
30
x (mm)
Fig. 12 Pressure and friction factor in a developing rectangular channel flow
17
Fig. 13 shows the axial velocity distribution at x=0.025 m. The maximum velocity is at
the center and is about 2.06 m/s.
Fig. 13 Axial velocity distribution at x=0.025 m. (maximum at the center)
Fig. 14 shows the cross flow pressure distribution and the cross flow velocity
vectors at x=0.025 m. The hydrodynamic entrance length for a rectangular duct2 is
x fd ,h  0.057 Re d h Dh  0.0158 m . Thus there should be no cross flow velocity
components at x=0.025 m. However, the magnitude of cross flow velocity is very small
compared with the axial velocity. The maximum cross flow speed is about 0.054
m/s which is negligible compared to the axial velocity. The magnitude of the cross flow
pressure field (p) is in the order of 0.002 Pa. This is a small pressure compared with the
averaged pressure along the axis ( p ) which is in the order of 3000 Pa.
18
Fig. 14 Cross flow pressure distribution and cross flow velocity vectors at x=0.025 m
19
Problems
1. Consider a developing laminar flow between two concentric cylinders as shown in the
figure. The flow enters the concentric tube annulus at a uniform velocity. The inside
cylinder is insulated and the outside cylinder is at a constant temperature which is
higher than the temperature of the fluid.
r
z
Di
Do
Determine the developing combined momentum and thermal entrance flow using
2dparay.for. You may choose any fluid and any dimensions for the cylinders as long as
the flow remains laminar and Di/Do=0.10.
(a) Plot local skin friction times the Reynolds number as a function of axis. Use
hydraulic diameter for the Reynolds number.
(b) Plot the pressure variation along the axis.
(c) Plot local Nusselt number, Nux, along the axis.
(d) In the hydrodynamically-fully-developed region, compare your velocity profile
with the exact solution (Eq. (6.155) in Munson, Young and Okiishi,4th Ed.).
(e) In the thermally-fully-developed region, compare your local Nusselt number with
a known data (Table 8-2, Incropera and DeWitt).
20
2. Consider developing momentum boundary layer flow over a flat plate with weak
blowing with a vertical velocities ranging from 0.001 - 0.005 m/s. Calculate the local
skin friction  w (x) and velocity profile u / u vs . Compare with the known
solutions.
3. Free stream velocity of laminar flow over a wedge is given by
u  ( x)  Cx
where the exponent m is related to the wedge angle  by
2m

1 m
and C is a constant.
y
x
u

Plot local wall shear stress as a function of x and plot non-dimensional velocity profile as
a function of similarity variable for several values of wedge angle, β. Compare with the
available data. ( Convective Heat Transfer, Burmeister, pp. 297-307)
21