FULLY DEVELOPED & PERIODIC FLOWS
UNICAMP
Source of Information on Phoenics Lecture: fully developed
flows
Includes:
1.
2.
3.
4.
5.
Introduction,
Single Slab fully developed flows,
Activation terms,
Periodic flows and
References.
UNICAMP
FULLY DEVELOPED & PERIODIC FLOWS
Advantages of fully developed regime
•
The case of developed flow in a duct or passage of constant
cross-section will be referred to as single-slab fully-developed
flow, since such flows can be simulated by performing
calculations on a single-slab of computational cells.
•
This is useful when it is desired to compute a fully-developed
pipe-, plane-walled-channel- or Couette-flow situations,
without, as was formerly necessary, simulating the flow in a
long duct, either elliptically or parabolically.
•
In other words, the Friction factor and Nusselt numbers can
be determined without having to solve the entrance region
problem.
•
The problem of hydrodynamically and thermally developed
flow inside constant cross-section ducts is of interest in
numerous practical applications. Examples are flows in pipes,
canals, sewers, and ventilation systems.
UNICAMP
•
•
FULLY DEVELOPED & PERIODIC FLOWS
Hydrodynamic Definition
For constant property flow in
a duct of constant cross
section, the velocity
distribution becomes
independent of the
streamwise coordinate at
sufficiently large distances
from the inlet.
Such unchanging velocity
distribution is said to fully
developed.
Le  0,06(d)Re
- Laminar
Le  4,40(d)Re(1/6) - Turbulent
UNICAMP
FULLY DEVELOPED & PERIODIC FLOWS
Thermal Definition
•
For the temperature field, the fully developed regime is
not as easily characterized as that for the velocity.
•
In fact, the temperature does not become independent of
the streamwise coordinate.
•
The usual definition of thermally developed regime is tha
the SHAPES of the temperature distributions at sucessive
streamwise locations are the same, so that the sucessive
distributions can be brought together by a suitable scale.
•
In such thermally developed flow, the heat transfer
coefficient is independent of the streamwise coordinate.
UNICAMP
•
FULLY DEVELOPED & PERIODIC FLOWS
Thermal Definition
The dimensionless form of the temperature is given as
function of the wall and the bulk temperatures:

•
Tw  T
Tw  Tb
In such thermally developed flow, the heat transfer
coefficient is independent of the streamwise coordinate.
UNICAMP
FULLY DEVELOPED & PERIODIC FLOWS
Equations set
•
The fluid flow conservation equations (constant properties)
are greatly simplified for the fully developed regime.
•
Without the streamwise convection terms the mass,
momentum and energy equations reduces to:
Mass

0
u v

x y
  2w  2w 
 uw   vw 
dP

Momentum Z 


 

 x 2 y 2 
x
y
dz


  2u  2u 
 uu   vu 
P

Momentum X 


 

 x 2 y 2 
x
y
dx


  2v  2v 
 vu   vv 
P

Momentum Y 


 

2
2
 x
x
y
dy
y 

Energy H


 C p Tu
x
  CpTv 
y
  2T  2T 
q "z

  k

2
2
 x
z
y 

UNICAMP
•
FULLY DEVELOPED & PERIODIC FLOWS
Equations set
The mass equation does not have dw/dz because the flow is
fully developed!
u v
0

x y
•
The Z momentum, the pressure gradient has to be specified
(external source), i.e., it is not deduced from the equations set.
•
The convection terms retain terms relative to the secondary
flow components only.
  2w  2w 
uw  vw 
dP



 

 x 2 y 2 
x
y
dz


UNICAMP
FULLY DEVELOPED & PERIODIC FLOWS
Advantages of fully developed regime
•
The u and v momentum equations represent the secondary
flow.
•
Notice they are uncoupled from the Z momentum.
•
The secondary flow is evaluated with the transverse
pressure gradient simultaneously.
•
The calculations are performed only in one Z slab!
  2u  2u 
uu   vu 
P


   

 x 2 y 2 
x
y
dx


  2v  2v 
vu  vv 
P


   

 x 2 y 2 
x
y
dy


FULLY DEVELOPED & PERIODIC FLOWS
Equations set
UNICAMP
•
Likewise Z momentum, the energy equation retains:
•
the energy convective transport terms due to the secondary
flow,

 Cp uT
x
  Cp vT 
y
  2T  2T 
q "z

  k

 x 2 y 2 
z


•
The energy diffusive transport terms along the x and y
directions,
•
And the axial energy flux has to be specified (external source),
i.e., it is not deduced from the equations set.
q z  C p wT  k
 2T
z 2
UNICAMP
•
•
FULLY DEVELOPED & PERIODIC FLOWS
Equations set
The fluid temperature is:
For uniform wall heat flux :
T  Tb  Tw   Tw
dTw dTb q "z  P


  Cp
dz
dz
m
dTw dTb

dz
dz
•
For constant wall temp :
•
Notice, for uniform wall heat flux the bulk and wall temperature
are related to the wall heat flux, but for constant temperature
the axial gradient of both temperatures are equal but there is no
explicit relationship with the wall heat flux.
UNICAMP
•
•
•
FULLY DEVELOPED & PERIODIC FLOWS
Equations set
The axial energy gradient, dqz/dz, is expressed as a
volumetric source S as a function of the wall and bulk
temperatures, which for
dT
d
S  WC p
  WC p Tb  Tw   Tw 
dz
dz
uniform wall heat flux is:

dTw
dTb
S  WCp
  WCp
   WCp
dz
dz
•
•
 q "z  P 


  Cp 
 m


Where P is the pipe perimeter and qz is the known wall heat
flux (W/m2)
constant wall temp is:
dTb
S  WC p

dz
Single-slab fully-developed flows
UNICAMP
•
The analysis of such problems is now possible in PHOENICS
by:
1. performing the calculation on a single IZ slab of computational
cells;
2. deactivating the axial convection terms for all solved-for
variables;
3. and deactivating the built-in pressure-gradient term in the W1
momentum equation.
• If the duct wall is circular or an infinite plane, the flow is onedimensional and then the pressure P1 and cross-stream
velocity (U1 or V1) need not be solved. Consequently, all
convection terms can be de-activated. For any other duct
cross-section, the flow in the slab will be two-dimensional and
so the pressure P1 must be solved and only the axialconvection terms can be deactivated.
Given Pressure Gradient
UNICAMP
• Since the user must render the W1 pressure-gradient term
inactive, it is then usually necessary to prescribe a mean
axial pressure gradient in order to create finite axial
velocities, so that the mass flow rate is an outcome of the
PHOENICS calculation.
Given the bulk or wall temperature gradient
UNICAMP
• When it is the mass flow rate that is known and the user
requires the pressure gradient, this procedure requires a
number of separate calculations to be made where the
pressure gradient is prescribed and then adjusted from
calculation to calculation until the computed mass flow rate
matches the specified value.
• Therefore, a more economical alternative has been
provided whereby PHOENICS automatically calculates the
pressure gradient from overall continuity for a given mass
flow rate. The pressure adjustment procedure is based on
that proposed by Patankar and Spalding [1972] for parabolic
flows, and the PHOENICS implementation is described by
Madhav [1992].
Given energy rate (J/sec)
UNICAMP
• Under certain heating and cooling conditions, such as
constant heat flux or uniform temperature at the duct walls,
a fully developed temperature profile can exist in the sense
that the dimensionless temperature profile remains invariant
with distance along the duct.
•There are a large number of possible thermal boundary
conditions for a rectangular duct, since any of the four walls
may be adiabatic, of constant heat flux or constant
temperature, or thermal conduction in the surrounding wall
may be significant. The only cases presently supported by
PHOENICS are surface heat flux constant and surface
temperature constant in the peripheral direction.
Flow Activation Settings
UNICAMP
•
•
•
•
Use VR to set fluid properties, domain , objects and
boundary conditions, etc. But do not use VR (3.5.1) to
activate model ‘fully developed’.
Make sure that your 2D (xy) grid has NZ=1
Go to Q1 GROUP 7 and write:
1. SOLVE (P1,U1,V1,W1) if your domain is 2D (XY)
2. FDSOLV(FLOW, mass flow) to prescribe the mass flow in
kg/sec going thru Z slab OR choose step 3 instead
3. FDSOLV(DPDZ, press grad) to prescribe the pressure
gradient in Pa/m thru Z slab
Go to Q1 GROUP 19 and write:
1. PARSOL=F
2. FDFSOL=T
Heat Activation Settings
UNICAMP
•
•
•
•
•
Several tests were performed with PHOENICS v 3.5.1 but
they failed to predicted the Nusselt number for a laminar
flow with constant temperature in a square duct.
The Nu expected value is of 2.98 while the Nu PHOENICS
value is of 3.5. Certainly there is some problem with the
built in routine which will be fixed in the future.
Another bug is that the routine works only for enthalpy
equation. If one try temperature (Tem1) equation the
results are ‘more’ non-sense.
Tests for h1 equation and constant wall heat flux also
were not successful.
Building the thermal fully develop routine using In-Form
is a good exercise, try it !
Available Phoenics Library Case Numbers
UNICAMP
quasi-one-dimensional (ie fully-developed 2D)
Couette flow, Reynolds number = 10
Couette flow, various Reynolds no. and other options
1D channel; fully-developed flow; k-eps model
1D Laminar Pipe Flow with Heat Transfer.
1D Laminar Pipe Flow with Mass Transfer.
1D Turbulent Pipe Flow with Heat Transfer.
1D Laminar Couette Flow with Heat Transfer.
1D Laminar MHD Channel Flow.
1D Laminar MHD Couette Flow.
951
958
605
500
501
502
505
506
507
quasi-two-dimensional (ie fully-developed 3D)
Fully-developed flow in duct of square cross-section
2D Laminar Duct Flow with Heat Transfer.
2D Turbulent Duct Flow with Heat Transfer.
896
503
504
WORKSHOP 1
UNICAMP
•
We are going to explore the fully
developed fluid flow and heat
transfer in a square channel.
1. Using VR set up the (XY) square
with 1mx1m with NX=NY=21
2. The fluid is air (0)
3. Activate the solution for velocity
(laminar regime) and pressure
4. Set sweeps = 50, and relaxation
settings for W1 = 1E+09
5. At last use four plate objects to
define the channel walls and create
a pressure relief cell.
WORKSHOP 1
UNICAMP
•
•
It is given the axial pressure
gradient, dp/dz is 4.425E-04
Pa/m. The problem is to find the
W1 velocity field corresponding
to this dp/dz
GROUP 7
1. SOLVE(P1 ,U1 ,V1 ,W1)
2. FDSOLV (DPDZ, 4.425E-04)
•
GROUP 19
1. FDFSOL=T; PARSOL=F
• Check solution, at channel center, w1
= 1.725 m/s
• P1 equation was not used, therefore
its residual is meaningless. In fact
one could set up: SOLVE(w1) only.
•
If you got trouble in
obtaining this solution
download wksh_FULLY-1
•
UNICAMP
•
WORKSHOP 2
It is the same as WKSH 1 but
now is given the mass flow rate
(10g/sec air) and you are asked
to find the pressure gradient.
GROUP 7
1. SOLVE(P1 ,U1 ,V1 ,W1)
2. FDSOLV (MASS,1)
•
GROUP 19
1. FDFSOL=T; PARSOL=F
• Change w1 to RELAX(W1 ,FALSDT,
1.000000E+03)
• Check solution, at channel center, w1
= 1.725 m/s.
• You must have got the same solution
of wksh-fully1. Check the result file
for mass flow rate and pressure
gradient.
•
If you got trouble in
obtaining this solution
download wksh_FULLY-2
•
UNICAMP
•
•
•
•
WORKSHOP 3
We start from wksh fully 1 with
dp/dz=4.425E-04 Pa/m but now
the SOUTH wall is not
stationary but moving U1 = 1
m/sec.
Consider the flow turbulent and
use the LVEL model to
determine new velocity field.
Find the proper relaxation coef such
that 500 sweeps are enough!
Noticed you just solved a 3D velocity
field using just one-slab!
Try to explain why the w1 field is not
symmetrical
•
If you got trouble in
obtaining this solution
download wksh_FULLY-3
UNICAMP
END OF THE WORKSHOP ON FULLY
DEVELOPED FLOW
PERIODIC FLOWS
UNICAMP
• A significant number of technically important duct
configurations do not admit fully developed solutions of the
type presented.
• Among such ducts are those in which the flow cross
section is not constant but varies in a periodic manner
along the flow direction.
PERIODIC
SECTION
FLOW
• Ducts with periodic area change are frequently found in
plate-fin heat exchangers
The Periodic Regime
UNICAMP
• Consider the flow in the (XY) plane with the main direction
parallel to the x axis.
• If the periodic section has length L along the x direction
then the velocity profile at the channel inlet and outlet are
the same, i.e.
u(x,y) = u(0,y)=u(L,y)
and
v(x,y) = v(0,y)=v(L,y)
• The equation above characterizes the peridic fully
developed regime for velocity field, replacing the
conventional condition dw/dz = 0.
• The variation of u and v in a single module which
encompasses the period length L is capable of representing
the changes of u and v through out the entire flow field.
The Periodic Regime
UNICAMP
• The pressure difference along the module outlet and inlet is
constant,
px , y   px  L, y   px  L, y   px  2L, y 
•And its profile along the channel is represented below
p
x
The Periodic Regime
UNICAMP
-bx
p
The outlet to inlet pressure drop is responsible for the globalx
mass flow in the positive x-direction, defining:
px, y   px  L, y  L  b
Where b is a constant. It is then reasonable to split the
pressure field into two components: the bx related to the
global mass flow and P(x,y) related to detailed local motions
inside the module
px , y   bx  Px , y 
The Periodic Regime
UNICAMP
•The conservation equations for mass and momentum for
constant property flow are:
u v

0
x y
  2u  2u 
 u
u
u 
P
   u  v   bx 
   2  2   g x
x
y 
x
y 
 t
 x
 2v 2v 
 v
v
v 
P
   u  v       2  2   g y
x
y 
y
 t
 x y 
•The term bx represents the mean pressure term. It has to be
specified externally while the P(x,y) is evaluated internally.
•The terms involving d2/dx2 were retained in recognition of
the fact that large local streamwise gradients may occur in
periodically fully developed flows.
Characteristics of the Periodic Flow Built-in Solver
UNICAMP
•
•
•
Periodic fully-developed flows can be simulated in PHOENICS
by using the built-in x-cyclic (XCYCLE=T) option provided
that:
1. the calculation does not employ cylindrical polar coordinates ( for
then there is no z-cyclic option );
2. the lateral or transverse flow is not cyclic as well; or the flow does
not have complex thermal boundary conditions.
In any of these events, the user is advised to adopt the
approach of repeatedly transferring the downstream exit
values of all variables, except pressure, to the inlet plane at
IZ=1;
Whichever approach is adopted, the solution domain is
restricted to a single geometrical module which repeats itself
in identical fashion in the actual geometry.
PHOENICS Settings
UNICAMP
• Define the domain, models, fluid properties, numerics and
•grid using VR
• You may also define plates and other types of boundary
conditions BUT DO NOT DEFINE INLET AND OUTLET, leave
them blank.
• In group 6 type: XCYCLE=T
•In group 13 specify the mean flow pressure gradient: DP/L (b)
• PATCH(PRESDROP,VOLUME,1,NX,1,NY,1,NZ,1,LSTEP)
•COVAL(PRESDROP,U1,FIXFLU,DP/L)
• do not forget to create a pressure relief
WORKSHOP 4
UNICAMP
•
We are going to
explore the periodic
fully developed
fluid flow in a
rectangular
channel(1x1/2) with
periodically spaced
thin plates.
1m
Pressure
Relief
0.5 m
Thin Plates
0.25 m
1.
2.
3.
4.
5.
6.
0.2m
0.5 m
0.25 m
Using VR set up the (XY) rectangle with 1mx0.5m with NX=NY=20 (phoenics auto
mesh)
The fluid is air (0)
Activate the solution for velocity (turbulent regime use LEVEL) and pressure
Set sweeps = 5000, you may try finding good relaxation settings
At last use two plate objects to define the channel walls, and two thin plates as a
dividing walls as indicated and create a pressure relief cell.
Write on the q1 file DP/L = 10 Pa/m (or 1mmCA/m)
WORKSHOP 4
UNICAMP
•
It is given the mean axial pressure gradient, b = Dp/dz =
10 Pa/m (or 1mmCA/m). The problem is to find the U1 and
V1 velocities fields corresponding to this dp/dz
• GROUP 6
1. XCYCLE=T (Sources: cyclic b.c. -> all slabs on)
•
•
•
GROUP 13
PATCH (PRESDROP,VOLUME,1,20,1,20,1,1,1,1)
COVAL (PRESDROP,U1 , FIXFLU
, 1.000000)
•
UNICAMP
If you got trouble in
obtaining this solution
download wksh_PERIOD-1
2D - Periodic baffled channel
WORKSHOP 4
UNICAMP
• Comments:
1. Use AUTOPLOT and compare the inlet and outlet pressure fields (i.e.)
for x1 and x 20. At the domain openings they are almost identical as
they should. Notice that PHOENICS solves for P(x,y), the actual
pressure field is recovered introducing bx component.
2. Use AUTOPLOT and compare the inlet and outlet V1 velocity (i.e.) for
x1 and x20. They are almost identical as imposed by the periodic
boundary condition.
3. Finally, use AUTOPLOT and compare the inlet and outlet U1 velocity
(i.e.) for x1 and x20. You may notice that they are not coincident, as
they should. But in fact you are not comparing U1 at the same slabs.
The C.V. for U1 is staggered. Therefore U1 values at west face of the
20th C.V. should be the U1 inlet values for the east face of the 1st C.V.,
but these values are not stored. The solved values are for the west
face of the 1st C.V., that is the reason why the velocities are not
coincidental.
UNICAMP
2D - Periodic ribbed channel
2D- Infinite array of
cylinders in cross flow
UNICAMP
END