The present paper describes the verification and validation of a quasi one-dimensional pressure based finite volume algorithm, implemented in Generalized Fluid System Simulation Program (GFSSP), for predicting compressible flow with friction, heat transfer and area change. The numerical predictions were compared with two classical solutions of compressible flow, i.e. Fanno and Rayleigh flow. Fanno flow provides an analytical solution of compressible flow in a long slender pipe where incoming subsonic flow can be choked due to friction. On the other hand, Raleigh flow provides analytical solution of frictionless compressible flow with heat transfer where incoming subsonic flow can be choked at the outlet boundary with heat addition to the control volume. Nonuniform grid distribution improves the accuracy of numerical prediction. A benchmark numerical solution of compressible flow in a converging-diverging nozzle with friction and heat transfer has been developed to verify GFSSP's numerical predictions. The numerical predictions compare favorably in all cases

Bandyopadhyay, Alak

Majumdar, Alok

NASA Technical Reports Server

Paper submitted to Thermal Fluid Analysis Workshop (TFAWS) to be hosted by NASA Glenn Research
Center and Corporate College, Cleveland, September 10-14, 2007
Modeling of Compressible Flow with Friction and Heat Transfer using
the Generalized Fluid System Simulation Program (GFSSP)
Alak Bandyopadhyay
Alabama A&M University
Huntsville, Alabama 3 5762
& Alok Majumdar
NASA/Marshall Space Flight Center
Huntsville, Alabama 3 5812
Abstract
The present paper describes the verification and validation of a quasi one-dimensional
pressure based finite volume algorithm, implemented in Generalized Fluid System
Simulation Program (GFSSP), for predicting compressible flow with friction, heat
transfer and area change. The numerical predictions were compared with two classical
solutions of compressible flow, i.e. Fanno and Rayleigh flow. Fanno flow provides an
analytical solution of compressible flow in a long slender pipe where incoming subsonic
flow can be choked due to friction. On the other hand, Raleigh flow provides analytical
solution of frictionless compressible flow with heat transfer where incoming subsonic
flow can be choked at the outlet boundary with heat addition to the control volume. Non
uniform grid distribution improves the accuracy of numerical prediction. A benchmark
numerical solution of compressible flow in a converging-diverging nozzle with friction
and heat transfer has been developed to verify GFSSP's numerical predictions. The
numerical predictions compare favorably in all cases.
Introduction
Most commercial network flow analysis codes lack the capability to simulate one-
dimensional flow in a rocket engine nozzle. Simulation of one-dimensional flow in
rocket nozzle requires a numerical algorithm capable of modeling compressible flow w[th
friction, heat transfer, variable cross-sectional area and chemical reaction. One of the
primary requirements of compressible flow simulation is to accurately model the inertia
force which is often neglected in many network flow analysis codes. NASA Marshall
Space Flight Center has developed a general purpose finite volume based network flow
analysis code: Generalized Fluid System Simulation Program (GFSSP) [1] which is
widely used for the design of Main Propulsion System of Launch Vehicle and secondary
flow analysis of turbopump. The purpose of the present paper is to verify and validate
GFSSP's numerical predictions with several benchmark solutions described in the
following section.
Problem Description:
In this study, mainly two types of geometries are considered - (a) a straight pipe of
constant diameter, and (b) a converging-diverging nozzle of linearly varying diameter.
https://ntrs.nasa.gov/search.jsp?R=20070036728 2019-08-30T02:05:11+00:00Z
Theeffectof friction andheattransferonthepressure,temperatureandMachno.are
studiedusingthesetwo geometries.Theproblemsaredividedinto five differentcases,
namely:
Case No. Description
.
2.
3.
4.
°
Fanno Flow - flow with friction in an adiabatic constant area pipe.
Rayleigh Flow - flow with heat transfer in a frictionless constant area pipe.
Combined Friction and Heat Transfer in a constant area pipe.
Effect of friction and area change using an adiabatic converging-diverging
nozzle.
Combined Friction and Heat Transfer in the converging-diverging nozzle.
For each of these problems the flow is assumed to be one dimensional and pressure,
temperature and Mach number is evaluated using analytical and numerical (Generalized
Fluid System Simulation Program) methods.
Constant Area Duct
For the first three problems, geometry is the same, a constant area pipe, as shown in the
Figure 1 as given below•
Pl = 50 psia
Ta = 80 F
M 1= 0.5
Fluid: Nz
D = 6 inch
J
4 L = 3207 inches
Figure 1 Schematic diagram for a constant area pipe.
The pipe is assumed to have a constant friction factor (f). The length of the pipe is chosen
so that the flow becomes choked at the exit of the pipe as determined from analytical
solution.
Converging-Diverging Nozzle
A schematic diagram for a converging-diverging nozzle is shown in figure 2. It is
assumed that the diameter of the nozzle is changing linearly, and it is at the lowest at the
throat. The dimensions are arbitrarily chosen and might not resemble a realistic nozzle.
The objective of the present work is to validate GFSSP with analytical solution. The
nozzleis assumedto havea constantfriction factor (f) andconstantheat flux (q). Three
different subproblemswill be considered flow with friction only (case4), flow with
heattransferonly (case5) andbothfriction andheattransfer(case6).
Ol ---
Pl = 50 psia Dexit = 12 in
T1 = 80 F
M1 = 0.25
Figure 2 Schematic of a converging-diverging nozzle
Benchmark Solutions
The generalized one dimensional compressible flow can be described mathematically
using the following conservation equations. These equations are applicable to study the
combined effect of area change, friction and heat transfer in a converging-diverging
nozzle as well as in a constant area duct.
The equations are in the differential form.
Mass Conservation:
d9 dA + dV = 0 (1)
9 A V
Momentum Conservation:
dp _ 7M 2 fdx 4_7M 2 dV= 0 (2)
p 2 D V
3
Where,M = Machno.
V V
Usingthedefinition of Machnumber,Stagnationtemperature,Equations(1) and(2) can
beexpressedasanordinarydifferentialequationof 1storder.
l+ -lM2 r+/1+taT01 1[ f dAdM "(1 - 7M 2 7M 2dx - 2T o dx - 7M2 A _xx (3)
With boundary value, M(x = O) = M1 (3b)
dT° in equation (3) can be determined from energy equation which can be expressed as:
dx
q(z_D)dx = mcpdT 0 (4)
Given the inlet conditions 3(b), the 1st order differential equation (eqn. 3) in M is solved
to find the Mach number at any x location. As this equation is a nonlinear equation in M
this equation is solved by using 4 th order Runge-Kutta method [2].
From Mach number, the static temperature and pressure can be determined from the
following equations:
T(x) _ To(x ) 1+ 7-12 (_a(_v,_,0,2
T(0)
To (0) 1 + V(M(x))2
(5)
p(x) A(0) M(0) .T((-_x)
p(0)- a(x) M-_ 1_T---_ (6)
Numerical Modeling
GFSSP employs a finite volume formulation of mass, momentum and energy
conservation equations in a network consisting of nodes and branches [3]. Energy
conservation equations are expressed in terms of entropy with entropy generation due to
viscous dissipation. Mass and energy conservation equations are solved for pressures and
entropies at the nodes while the momentum conservation equation is solved at the
branches to calculate flow rate. The friction in pipe was modeled by Darcy friction factor.
The pressure drop in a pipe is expressed as:
4
2Ap = k f m (7)
8fL
where, Ks = (7b)
2 5
Pu 71; D gc
For the numerical simulation using GFSSP, the entire domain (pipe or the nozzle) is
divided into several sectors, and each of these sectors is represented by a pipe of constant
diameter. The diameters for adjacent sectors will vary for converging-diverging nozzle.
The fluid is assumed to be Nitrogen. The numerical solutions generated by GFSSP is
presented in the results and discussion section and compared with the analytical
solutions.
Discretization
For the numerical simulation, the pipe in Figure 1 is divided into finite number of pipes
of non-uniform length as shown in Figure 3. A node is being placed at the end of each
pipe sectors including the two boundary nodes. Both uniform and non-uniform node
distributions have been tried for the simulations. It has been observed that a total of 21
nodes with clustered nodes near the inlet and exit (figure 3), is sufficient for the constant
area pipe (cases 1 through 3) to get grid independent solution.
L = 3207 inches
ooo • •
• • • • • • • • • • • 0000
0 0.2 0.4 0.6 0.8 1
XIL
Figure 3. Non-uniform node distribution for the constant area duct.
Results & Discussion
The analytical solution for all the cases are obtained from the generalized numerical
solution of equation (3) and these solutions have also been reproduced and verified with
Fanno and Rayleigh tables available in any standard text book [4]. Cases 1 through 3 are
for constant area ducts and cases 4 and 5 are for variable area ducts (nozzle flow).
5
Case 1: Fanno Flow:
Flow with friction, but no area change and heat transfer. No heat transfer implies the
stagnation temperature is constant and dTo/dx = O.
Before presenting the results for Fanno flow, choice of pipe length as 3207 inches. is
explained as below.
From the analytical solution for Fanno flow [4], the critical length of the pipe (L*). is
determined from the following equation:
(7)
M is the inlet Mach number. The critical length of the pipe is the length required for the
flow to choke at the exit (i.e. M = 1 at exit). With an inlet Mach no of 0.5 and a friction
factor of 0.002, and pipe diameter of 6 inches the critical length is calculated to be 3207
inches. This length is kept fixed for the cases of constant area pipes.
Figure 4 shows a plot of the pip* ratio with different types of node distribution for the
numerical solution compared with the analytical solution. A non-uniform node
distribution with a total of 21 nodes (20 control volumes) is sufficient to get a grid-
independent solution. The plot also shows that the numerical solution using GFSSP
agrees very well with that ofthe analytical solution.
Fanno Flow: Pressure with axial distance
2.25 -,-------------------------,
2.00
1.75
Co
-Co
1.50
1.25
Non Uniform Grid with 20 grids
......Theoretical
- Uniform Grids with 40 nodes
- Uniform Grids with 20 nodes
1.000.800.60OAO0.20
1.00 +------,-------,-------,-------,----------.
0.00
Figure 4. Pressure distribution for Fanno Flow with various grid distributions.
6
Fanno Flow: Temperature with Axial Distance
1.16 -,-----------------------,
--Theoretical
-NonUniform Grid with 20 nodes
r----"'=:::::=O::::;;;;;;;;;;;;......~~ Uniform Grids with 20 nodes-Uniform Grids with 40 nodes1.12
1.08
1.04
1.000.800.60OAO0.20
1.00 +-------,----,-------r-----,-------+
0.00
Figure 5. Temperature distribution for Fanno Flow with various grid distributions.
The corresponding temperature distribution is shown in figure 5.
The Mach no. is a derived quantity for the numerical solution and it is calculated from the
mass flow rate, temperature and pressure as follows.
M_V_4~~-JT
- C - n ~yg: pD2 (8)
Where, R = gas constant = 55.19 ft-Ibtl(lbmR), gc = 32.17 ft/sec2. p is in psia, Tin
Rankine and D is the diameter in inches, m is the flow rate in lbm/s. Figure 6 show a plot
of the Mach no. along the axial direction, and again the agreement with the analytical
solution is quite good. The slight difference even at the inlet is because, in GFSSP the .
mass flow rate is not prescribed, rather the pressure boundary condition is specified, and
the flow rate is computed.
Fanno Flow: Mach No. Plot
0.8
o
z
~ 06 L:::::::::::::::::::::::::::::::::::::::::::;::::::::;::::::::;:::::::;;::::;;---
0.4
0.80.60.40.2
0.2 +----,--------,------r------.-----I
o
xlL
Figure 6. Case1 - Fanno Flow. Plot of Mach number along the pipe length.
7
Case 2 - Rayleigh Flow: Flow with no friction and a uniform heat transfer
Figures 7, 8 and 9 show the distribution of temperature, pressure and Mach number for a
heat input rate of 2088 Btu/sec on the same pipe geometry that has been used for Fanno
flow. The inlet Mach number is chosen as 0.46 and it has been analytically calculated that
with this inlet Mach number, a heat input of 2088 Btu/sec makes the flow choked at the
exit (i.e. Mach number = 1). Further increase in the heat rate will make the flow
supersonic and in the present work, only subsonic flow is being considered. Again, the
numerical results show quite good agreement (within 5%) with the analytical solution.
Rayleigh Flow: Temperature Plot (M = 1 at exit)
1.05 -,---------------------,
0.95
'1:: 0.9
1-0.85
0.8
0.75
0.80.60.40.2
0.7 +----,------,------r---~r__-----I
o
x/L
Figure 7: Temperature distribution for Rayleigh Flow (Q = 2088 Btu/s)
Rayleigh Flow: Pressure Distribution (M = 1 at exit)
0.80.4 0.6
xlL
Analytic
0.2
1+----,------,------r----r--------li
o
1.6
1.
1.
·p/p
1.4
Figure 8. Pressure distribution for Rayleigh Flow (Q = 2088 Btu/s).
8
Rayleigh Flow: Mach No. Plot
0.8
~ 0.6 L~s::::!1'===~::!!==~~~
s:. Analytical
~ 0.4
==
0.2
0.80.60.40.2
0+----,-----,--------,-------,------1
o
xlL
Figure 9. Mach no. distribution for Rayleigh Flow (Q = 2088 Btu/s).
Case 3: Combined Friction and Heat Transfer in a constant area Pipe
There is no standard table available for the combined effect of friction and heat transfer,
and analytical solution can only be obtained by solving the differential equation in Mach
number (eqn. 3). Figure 10, 11 and 12 show the combined effect of friction and heat
transfer on the temperature, pressure and Mach no. respectively. The inlet Mach no. is
0.45. All three plots correspond to f = 0.002, Q = 555 Btu/sec. The pressure and
temperature are plotted as dimensional quantities, with inlet pressure as 50 psia and inlet
temperature of 80 F.
Pressure Plot for Combined Friction and Heat Transfer
(f =0.002 and Q =555 Btu/s)
50 r-;;;:::;;=-------;=:::.:===~--~
45
~ 40
E:
~ 35
j
til
til
~ 30
Co
25
0.80.6040.2
20 +-----,-------r-------,----,------I
o
x/L
Figure 10. Effect of friction and heat transfer on the Temperature.
9
Temperature Plot: Combined Friction and Heat Transfer
(f =0.002, Q =555 Btuts)
120
LL
- 100f!
:::l
...
l.'!!
CIl 80
Co
E
~ 60
0.80.60.40.2
40 +-__--,--__--,-__-,--__----,-__-----J
o
x1L
Figure 11. Effect of friction and heat transfer on the Pressure.
Mach No. Plot for Combined Friction and Heat Transfer
0.8
0.2
0.80.60.40.2
o+-------,------,--------,------,-----~
o
x1L
Figure 12. Effect of friction and heat transfer on the Mach number.
Case 4 and Case 5: Nozzle Flow with Friction Only, and both Friction with Heat
The nozzle dimensions and operating conditions are given in figure 2. The wall friction
factor is taken as 0.05. The nozzle is assumed to be adiabatic. For the numerical
simulation the pressure and temperature at the inlet are specified as given and exit
pressure is specified. The exit pressure is calculated from the analytical solution to ensure
that the inlet Mach number is 0.25.
Figure 13 shows the effect of node distribution for the numerical simulation and it is
observed that about 64 nodes, mostly uniform, except clustered near the inlet and throat,
produces grid independent solution. The numerical solution agrees well with the
analytical except near the throat.
10
Plot of Pressure for friction factor = 0.05
50
Uniform Grid: 25 nodes
Uniform Grid: 50 nodes
45
-
.!!! 40tfj
c.
-(1) 35I-
~
tfj MixedTypetfj
(1) 30 with 64l-
n.
Analytical
25
20
0 50 100 150 200 250
X(inches)
Figure 13. Grid study on the pressure distribution in a converging-diverging nozzle.
Figure 14 and 15 show the effect of friction factor and both friction factor and heat
transfer on the Mach number distribution and the pressure distribution respectively.
Nozzle Flow: Mach No. Plot
0.9
0.8
0.7
0
Z 0.6
.c
(,) 0.5
nl
~ 0.4
0.3
0.2
0.1
0
0 50 100 150
Axial Distance, X (inches)
- ..
. . -
200 250
Figure 14. Plot of Mach number for nozzle flow - (a) Effect of friction and heat
transfer - analytical solution (b) Effect of friction and heat transfer -
numerical (GFSSP) solution (c) Effect of friction only - Analytical and (d)
Effect of friction only - numerical simulation.
11
Effect of both Friction and Heat Transfer on Pressure
50 -,.",;;:-----------------------,
~ 40III
Q.
-Q)
...
:::s
III
IIIQ) 30...
a.
a
20 +----,----------,-----,------,--------1
o 50 100 150 200 250
X (inches)
Figure 15. Pressure distribution for nozzle flow with (a) Both friction and heat transfer-
Analytical (b) Both friction and heat transfer - Numerical (GFSSP) (c) Friction only-
Analytical and (d) Friction only - Numerical (GFSSP).
Both of these plots show very good agreement for most of the locations between the
numerical and analytical. Figure 16 shows the corresponding temperature plot.
160 -,----------------------
140
120
LL
-CI> 100
...
::::I
-f!! 80 -,-~.,;;:
CI>
c.
E 60
CI>
I-
40
20
Nozzle Flow: Temperature Plot
c
..............~.....
d
25050 100 150 200
Axial Distance, X (inches)
0+----,-------,------,----------,------1
o
Figure 16. Temperature distribution for nozzle flow with (a) Both friction and heat
transfer - Analytical (b) Both friction and heat transfer - Numerical (GFSSP) (c) Friction
only - Analytical and (d) Friction only - Numerical (GFSSP).
All ofthese plots show a very good agreement between analytical and numerical results.
12
Conclusions
The paper presents a numerical study of the effect of friction, heat transfer and area
change in subsonic compressible flow to verify the accuracy of pressure based finite
volume algorithm implemented in Generalized Fluid System Simulation Program. The
numerical solutions of pressure, temperature and Mach number have been compared with
benchmark solution for five different cases representing the effect of friction, heat
transfer and area change. Generally there is a good agreement between the numerical
solution and benchmark solution. It has been observed that non-uniform grid improves
the accuracy of numerical solution. The observed discrepancy at the throat of the
converging-diverging nozzle is due to sharp discontinuity at the nozzle throat which has
not been accounted for in the numerical model where the nozzle has been discretized by a
series of pipes with varying cross-sectional area. The modeling of supersonic flow is
being planned for future investigations.
Acknowledgement
This work has been performed under NASA/Marshall Space Flight Center Summer
Faculty Research Program (2007) in the Thermal and Combustion Analysis Branch
(ER43) of Propulsion System Department in Engineering Directorate. The authors would
like to thank Mr. Bruce Tiller for his support and encouragement.
References
1. Majumdar, Alok, Steadman, Todd and Moore, Ric, "Generalized Fluid System
Simulation Program (GFSSP) Version 5.0." George C. Marshall Space Flight Center,
Engineering Directorate Technical Report (To be published as NASA Technical
Memorandum), February, 2007
2. Press W.H. et al., "Numerical Recipes in Fortran: The Art of Scientific Computing",
2 nd Ed.,1992, Cambridge University Press, pp 704-716.
3. Majumdar, A.K., "A Second Law Based Unstructured Finite Volume Procedure for
Generalized Flow Simulation", 37 th AIAA Aerospace Meeting Conference and Exhibit,
January 11-14, 1999, Reno, NV, AIAA 99-0934.
4. Saad, Michael A, "Compressible Fluid Flow", 2 nd Edn., 1993, Prentice Hall
Publications.
Nomenclature
Alphabets:
A
C
Area of the pipe, or local area of the nozzle at any axial location.
speed of sound
13
Cp
D
f
L
M
m
P
Q
q
R
T
V
Specific heat at constant pressure.
local diameter
Darcy Friction Factor
Length of the Pipe, Nozzle
Mach number
mass flow rate
Pressure
Total Heat Transfer Rate
Heat Flux
Gas Constant
Temperature
Velocity
Greek Symbols:
p Density
7 Specific Heat Ratio
Subscripts:
1 Inlet
0 Stagnation Properties
t at the throat of the nozzle.
Superscripts:
* Choked Properties (Corresponding to M - 1)
14


Modeling of Compressible Flow with Friction and Heat Transfer Using the Generalized Fluid System Simulation Program (GFSSP)

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