(THRu| i ,o.A?-(COZ · 1. 2 Basic Equations of Unsteady Motion 1.3 Initial and Boundary Conditions...
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,o.A?- _Accession No.
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S ID 66-46-I
STUDY OF LONGITUDINAL OSCILLATIONSOF PROPELLANTTANKS AND
WAVE PROPAGATIONS IN FEEDLINES
Part I--One-Dimensional Wave Propagationin a Feed Line
31March 1966 NAS8-11490
Prepared by
Henry Wing, Staff Investigator GPO PRICE $
Clement L. Tai, Principal InvestigatorCFSTI PRICE(S) $
II
Approved by Hard copy (HC)
.i---- /../.----,).. _ _ Microfiche (MF)
. _" If 653 July 65
F.C. Hung/Assistant Director, Structures and Dynamics
NORTH AMERICAN AVIATION, INC.SPACE and INFORMATION SYSTEMS DIVISION
https://ntrs.nasa.gov/search.jsp?R=19660015657 2020-07-22T20:25:02+00:00Z
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Accession No.
A
31 March 1966
i S I D 66-46-I
STUDY OF LONGITUDINAL OSCILLATIONS
OF PROPELLANT TANKS AND
WAVE PROPAGATIONS IN FEED LINES
Part i--One-Dimensional Wave Propagation
in a Feed Line
NAS8-I1490
Prepared by
Henry Wing, Staff Investigator
Clement L. Tai, Principal Investigator
Approved by
F.C. Hung/Assistant Director, Structures and Dynamics
P NORTH AMERICAN AVIATION, INC.
SPACE and INFORMATION SYSTEMS DIVISION
. • TECHNICAL REPORT INDEX/ABSTRACT
D,
ACCESSION N UM B E RL__J_ DOCUMENT SECURITY CLASSIFICATIONUnclassified IITITLE OF DOCUMENT
"A Study of Longitudinal Oscillations of Propellant Tanks and
Wave Propagations in Feed Lines"
LIBRARY USE ONLY
C. L. Tai. M. M. H. Loh, S. A......... Fun H. Win S. Uchi am___aa
CODE ORIGINATING AGENCY AND /
OTHER SOURCES DOCUMENT NUMBER
, NAA ISID 66-46 (Five Volumes)
P"_'ET'CA"_ON DATE ] CONTRACT NUMBER
31 March 1966 1 NAS8-I1490
DESCRIPTIVE TERMS
fluid dynamics
pipeline dynamics
wave propagation
longitudinal oscillations
shell dynamics
ABSTRACT
The work done under this contract is being published in five separate parts:
(SID 66-46-i) Part I One-Dimensional Wave Propagation in a Feed Line
(by H. Wing and C. L. Tai)
(SID 66-46-2) Part II Wave Propagation in an Elastic Pipe Filled with
Incompressible Viscous Fluid
(by M. M. H. Loh and C. L. Tai)
(SID 66-46-3) Part III Wave Propagation in an Elastic Pipe Filled with
Incompressible Viscous Streaming Fluid
(by S. A. Fung and C. L. Tai)
(SID 66-46-4) Part IV Longitudinal Oscillation of a Propellmlt-Filled
Flexible Hemispherical Tank
(by H. Wing and C. L. Tai)
(SID 66-46-5) Part V Longitudinal Oscillation of a Propellm_t-Filled
Flexible Oblate Spheroidal Tank
(by S. Uchiyama and C. L. Tai)
FORM 131-V REV. 6-64
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
m,
FOREWORD
This report was prepared by the Space and Information Systems
Division of North American Aviation, Inc., Downey, California, for the
George C. Marshall Space Flight Center, National Aeronautics and Space
Administration, Huntsville, Alabama, under Contract No. NAS8-11490,
"Study of Longitudinal Oscillations of Propellant Tanks and Wave Propaga-
tions in Feed Lines, " dated January 6, 1965. Dr. George F. McDonough
(Principal) and Mr. Robert S. Ryan (Alternate) of Aero-Astrodynamics
Laboratory, MSFC, are Contracting Officer Representatives. The work is
published in five separate parts:
Part I - One-Dimensional Wave Propagation in a Feed Line
Part II - Wave Propagation in an Elastic Pipe Filled With
Incompressible Viscous Fluid
Part HI - Wave Propagation in an Elastic Pipe Filled With
Incompressible Viscous Streaming Fluid
Part IV - Longitudinal Oscillation of a Propellant-Filled Flexible
Hemispherical Tank
Part V - Longitudinal Oscillation of a Propellant-Filled Flexible
Oblate Spheroidal Tank
The project was carried out by the Launch Vehicle Dynamics Group,
Structures and Dynamics Department of Research and Engineering Division,
S&ID. Dr. F.C. Hung was the Program Manager for North American
Aviation, Inc. The study was conducted by Dr. Clement L. Tai (Principal
Investigator), Dr. Michael M.H. Loh, Mr. Henry Wing, Dr. Sui-An Fung,
and Dr. Shoichi Uchiyama. Dr. James Sheng, who started the investigation
of Part IV, left in the middle of the program to teach at the University of
Wisconsin. The computer program was developed by Mr. 1%.A. Pollock,
Mr. F.W. Egeling, and Mr. S. Miyashiro.
o,°
- 111 -
SID 66-46-1
• NQRTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
b
ABSTRACT
In this report, the dynamics of elastic propellant /
feed lines with streaming fluid are analyzed using the
nonlinear one-dimensional unsteady compressible
fluid flow equations. The viscous effect of the fluid
is assumed to be expressible inthe form of a hydrau-
lic resistance corresponding to turbulent flow.
Transfer functions relating the pressures, velocities,
and the corresponding phase angles were derived
from the linearized equations for the perturbed state.
The governing equations were also converted into a
system of four first-order ordinary nonlinear differ-
ential equations by the method of characteristics.
These were then transposed into finite difference form
and solved numerically on a digital computer. Impulse,
step, and sinusoidal velocity disturbances were con-
sidered in the numerical solution for a feed line with
and without a control device. The physical param-
eters of the liquid hydrogen line on the Saturn S-II
engine are used for the numerical calculations to
illustrate the results.
- V -
SID 66-46- 1
NORTH AMERICAN AVIATION, INC. SPACE and INF_0RMATION SYSTEMS DIVISION
g
C ONT ENT S
NOMENC LAT URE
INTRODUCTION
SECTION i. EQUATIONS OF UNSTEADY FLOW IN PIPES
1. 1 Variation of Cross-Sectional Area of a Pipe as a Function
of Pressure
1. 2 Basic Equations of Unsteady Motion
1.3 Initial and Boundary Conditions
I. 3. 1 Air Chamber
1.3.2 Surge Tank
i. 3.3 Tank-Type Damper
SECTION 2. SOLUTION OF EQUATIONS OF UNSTEADY
MOTION BY LINEARIZATION
2. I Boundary Condition at X = _ With Pipe Vibration
2. 2 Physical Constants and Results of Linearized Analysis
SECTION 3.
UNST EADY
CHARACTERISTICS .
3.1
3.2
3.3
3.4
3.5
SOLUTION OF NONLINEAR EQUATIONS OF
MOTION BY THE METHOD OF
Specified Time Intervals
Selection of Mesh Ratio
Computational Problem
Interpolation Formulas
Finite Difference Approximation
SECTION 4.
4.1
4.2
4.3
COMPUTATIONAL PROCEDURE
Initial Conditions.
Boundary Condition at Left End of Line
Boundary Condition at Right End of Line
4. 3. l Velocity Impulse Disturbance
4. 3. 2 Velocity Step Disturbance
4. 3. 3 Sinusoidal Velocity Disturbance
4. 3.4 Feed Line With Pressure Regulating Device
4 ...... Modified Form of Boundary Condition
4.3. 5 Numerical Procedure
Page
ix
4
5
6
7
7
8
11
17
19
21
23
23
25
25
28
31
31
32
33
34
36
36
38
39
39
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SID 66 -46- 1
NORTH AMERICAN AVIAT!ON, INC.
SPACE and INFORMATION SYSTEMS DIVISION
SECTION 5. COMPUTATIONAL APPROACH TO SPECIFIC
PROBLEM .
SECTION 6. NUMERICAL RESULTS.
6. 1 Discussion of Results.
C ONC LUSIONS
AREAS FOR FURTHER STUDY
REFERENCES
APPENDIX A. TRANSFER FUNCTIONS
APPENDIX B. VELOCITY AND PRESSURE DATA
Page
41
43
43
47
49
51
53
67
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SID 66-46-I
NORTH AMERICAN AVIATION, INC. SPACE and 1NI_)RMATION SYSTEMS DIVISION
P,
4
a
a m
A
A d
AhA
S
A.pl
C
D
E
f
f(t)
g
g(t)
h
II , 12
J
k_, k t
K
K 1
L
NOMENC EAT URE
Velocity of pressure wave
Missile acceleration
Cross-sectional area of feed line
Cross-sectional area of tank damper
Cross-sectional area of feed line wall
Cross-sectional area of surge tank
Cross-sectional area of pump inlet
Feed line restriction coefficient
Feed line diameter
Young's modulus of line material
Darcy-Weisbach friction coefficient
Prescribed function for pressure at left end of line
Gravitational constant
Prescribed function for mass flow rate at right end of line
Wall thickness
Functions defined by Equation (48)
Square root of minus one
Laminar and turbulent resistance coefficient
Modified bulk modulus defined by Equation (13)
Bulk modulus of fluid
Length of feed line
- ix -
SID 66 -46 - I
NORTH AMERICAN AVIATION, INC.
I
SPACE and INFORMATION SYSTEMS DIVISICrN
m I , m 2
P
P
P
Q
Ql(t)
r
R
s
t
u
u
v(tl
V
x
z
Z
(1
A
e
Roots of Equation (46)
Fluid pressure
Reference pressure
Laplace transform of fluid pressure
Mass flow rate
Mass discharge rate
Internal radius of line
Reynold's number
Laplace transform variable
Time
Average fluid velocity over line cross section
Reference fluid velocity
Fluid discharge velocity
Volume
Spatial coordinate along axis of feed line
Axial displacement component
Laplace transform of axial displacement component
Density of line material
Incremental change
Damping coefficient of bellows
Liquid elevation in surge tank from equilibrium level
Angle between axisoffeed line and normal to flight path of
missile
4
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SID 66-46-I
NO RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
KK,
I)
¢o
Subscripts
A, B, C,
R,S
Constant characterizing pressure regulating device
Viscosity of fluid
Poisson's ratio of line material
Density of fluid
Function defined by either Equations (3) or (4)
Frequency
Pertain to points according to Figure 2
Designate value at steady-state condition or value at x = 0
Designate value at x = L
Superscripts
o Initial guess value
i Value of first iteration
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SID 66-46-1
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
INT R OD UC T ION
Pressure wave propagation in a pipe filled with fluid has been investi-
gated extensively, from the early studies of surging phenomena to the recent
work on unsteady flow in blood vessels and hydraulic control systems
(References 1 to 17). Various simplifying assumptions were employed in
each particular problem to facilitate a mathematical solution. In general,
the formulation is not applicable to any problem with conditions other than
those appropriate to the particular problem.
The classical theory of surging phenomena or "water hammer," formu-
lated by Joukowsky (Reference l), is based on the linearized one-dimensional
Navier-Stokes equation for an inviscid fluid. It assumes that the pressure is
uniform across any section of the pipe and that the deflection of the pipe is
equal to the static deflection due to the instantaneous pressure in the fluid.
This theory has since been extended to two-dimensional space (References 12
and 13), with some of the simplifications made in the Joukowsky theory
relaxed (Reference 14). As far as the relations between fluid velocity,
pressure, and hoop stress are concerned, however, the improved theory
gave essentially the same results as Joukowsky's theory, which had been
c onfirmed experimentally.
Recently, the method of characteristics was applied successfully to
the numerical solutions of one-dimensional "water hammer" analysis
(References 16 and 17), including the nonlinear terms for the convective
acceleration and fluid friction. Similar nonlinear differential equations
describing the unsteady motion of a viscous compressible fluid in tubes in
terms of a hydraulic resistance were also applied to the self-induced oscil-
lations of heavy fluids (Reference 18). In this reference, the fluid density
term is assumed to be large, and the pressure gradient term is neglected,
thus making it possible to reduce the problem to the solution of first-order
quasilinear partial differential equations with nonperiodic boundary
c onditions.
The investigation of the pressure wave in the blood vessel was con-
cerned primarily with the influence of the elasticity of the walls on the
propagation of sound through the fluid medium in the vessel. Since the
physiologists considered the distensibility of the tube to be of far greater
importance than the compressibility of the fluid, the authors assumed an
incompressible and viscous fluid (References 6 to 8). The stresses and
shear are expressed in terms of the displacements, but the inertia terms
- 1 -
SID 66-46-I
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
are neglected. The significance of the effect of viscosity and internal
damping on the wave propagation is somewhat obscured in the authors' over-
simplification of the complicated solution.
The approaches used in the hydraulic and pneumatic control systems
are generally straightforward. Since pipe diameters are small, the
elasticity of the pipe may be neglected; and, because the flow rate is usually
low in the control system, the use of linearized equations with the laminar
friction loss (References 9 to ii) is justified. The results of the theoretical
frequency-response analysis compare quite well with the experimental data
from the test.
In the present investigation, it is assumed that the propellant line
connects the propellant tank and propellant pump. The diameter of the line
is much larger than that of hydraulic control lines or blood vessels, but
much smaller than the diameter of penstock in the lower plant. The source
of disturbance may be either the fuel sloshing in the propellant tank or the
variation of flow in the propellant pump. Another possibility is that the
changes in the natural frequency of the missile structure at different flight
times may also create unsteady flow in the fuel line.
The aim of this study is to determine analytically the general dynamic
behavior of the unsteady motion and to construct a reasonable model that
can be applied directly to the overall system analysis of longitudinal
oscillations of large missiles.
-2-
SID 66 -46-I
NbRTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
Pb-
SECTION i. EQUATIONS OF UNSTEADY FLOW IN PIPES
If we consider the motion of the fluid in a pipe as a whole and
introduce the _average values of velocity u, density p, and pressure p over
the cross-section A, the basic equations describing the unsteady motion of
a viscous compressible fluid in an elastic tube in terms of a hydraulic
resistance are the law of conservation of mass
a a
a--x(puA). = - -_ (pA) (i)
and the law of momentum
a a ap(puA) + _-x (puZA) = -A _x - A_(u) (Z)at
The function _ (u) is taken to be linear for laminar flow
_(u) =_u, for R< Z000 (3)
and quadratic for turbulent flow
f Z
_(u) =_--_ pu , for R > Z000 (4)
where _ is the viscosity, D the inside diameter of the pipe, R the Reynolds
number, and f the Darcy-Weisbach resistance coefficient.
To the above equations must be added the state equation for compres-
sible flow
dp = -_I dp (5)
where K 1 is the bulk modulus of the fluid which is defined as the change in
pressure per unit change in specific volume.
-3-
SID 66-46-I
NORTH AMERICAN AVIATION, INC. SPACE end INFORMATION SYSTEMS DIVISION
If the pressure variation is small, Hooke's law can be considered
valid for a liquid, i.e.,
p-P_KI ))P=Po +(6)
Po
dp = -K-_Idp(7)
where the subscript "o" designates the variables at steady state.
I. 1 VARIATION OF CROSS-SECTIONAL AREA OF A PIPE AS A FUNCTION
OF PRESSURE
It will be assumed that the pipe has a thin elastic wall and is made of
a material for which Hooke's law holds true. Let the initial pressure in the
pipe be p and the radius be r. Then the tangential and axial forces per unit
width are given by
Ne No
N o = 0-@h = pr
N =0-h=Cx x ipr
(8)
Figure i.
and the tangential strain by
1 (_8 pr i) (9)_0 =_E - VO-x) = Eh (1 - vC
where C I = 1/2 for a pipe fixed at one end and free to move throughout the
rest of its length, C 1 = v for a pipe anchored against longitudinal movement
along its length, and Cl = 0 for a pipe having expansion joints between
anchors throughout the length of the pipe.
-4-
SID 66 -46 - I
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
or
When the internal pressure is varied, the tangential strain varies by
5r 1
5_ 0 = m = N (1 - vC1) (pSr + rSp)r Eh
_r =
r25p (1 - vC1)
Eh - rp (1 - vC1)
Multiplying the above equation by Z_r and denoting C = 1 vC1, it becomes
5A- 2rACSpEh - Crp (10)
From Equation (9), Eh =(C/e0)Pr. Since c 0 << 1, the term Crp is
negligible in comparison with the term Eh, and hence
CDA
dA- Eh dp (11)
1.2 BASIC EQUATIONS OF UNSTEADY MOTION
Combining (i), (5) and (11) yields
[1 +CD] 8p Ou I_l+ CE] Dp (IZ)
Define
: v + = p (13)a
where a is the velocity of pressure wave propagation in a fluid medium con-
tained in an elastic pipe. Since the velocity of sound in the fluid is
a2 = dp/dp = K1/p, the term CD/Ehis a correction of K1, accounting for
the elasticity of the pipe wall. Equation (12) now becomes
From (I) and (2), we have
ap o___Ep = z O___u8--_+ u 8x - pa 8x
(14)
Ou 8u 1 @p 1 _(u)o-[ -;Ox-;
(15)
_
SID 66 -46 -1
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
Equations (14) and (15) are the basic equations of unsteady motion in
an elastic pipe. For small pressure variations, the value of p in the above
equations may be considered to be equal to Po of the steady state.
In the case where the effect of missile acceleration a m and the angle, O,
which the pipe makes with the horizontal are to be taken into consideration,
the term a m sin 0 has to be added to Equation (15); i.e. ,
0u 8u _ 1 8p 1 _(u) + a m sin e (15a)7_- + u Ox p Ox p
I. 3 INITIAL AND BOUNDARY CONDITIONS
It will be assumed, in this treatment of unsteady fluid motion in pipes,
that up to the moment t = 0 the fluid is in steady motion; i.e. ,
o) = u0
p(x, o) = Po
(16)
The boundary conditions depend on the nature of the disturbance of flow
at the boundaries. The following boundary conditions are expected to be
encountered in the present problem: (I) at one end of the pipe the pressure
is given as a function of time (as a particular case, this pressure may be
constant); (2) at the other end of the pipe a device is connected which varies
the flow rate of the fluid with time according to some known law--this may
be a pump, turbine, etc. This device may be connected either directly to
the pipe or through a chamber serving to regulate the flow rate or to
diminish the pressure oscillations. The boundary conditions in such cases
as these can be written functionally as
x = o p(o,t) = Po + f(t) (17a)
8x = £ K _Q(_,t) + Q(£,t) = g(t) (17b)
where f(t) and g(t) are given functions, with
f(t) = g(t) = 0 at t < 0
Q is the mass flow rate and K is a positive constant characterizing the type
of chamber in those cases where a chamber is present.
4
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SID 66 -46-I
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
The boundary conditions of three cases at x = % when the pipe is
connected to a chamber designed to reduce the pressure oscillations will
be discussed in the following paragraphs.
I. 3. I Air Chamber
Let p andVbe the pressure and volume of the air in the vessel, and
let Po and Vo be their steady flow values. If the compression of the air in
the chamber is considered as isothermal, then
pV = PoVo (185
During unsteady flow the increase in the quantity of liquid in the vessel will
be
d(V - V)o
Po dt - (puASx= % - Ql(t) (19)
where (puA)x=_ = Q is the mass flow rate of the liquid out of the pipe and
into the air vessel, and Ql(t) is the mass discharge rate of the liquid from
the vessel.
For small pressure variations, it is permissible to use the linearized
continuity equation. From (18), (14), and (135 it follows:
d___y= _.P°V° 8p Vo K 8Q
dt 2 0t Po 8xp PoAo
(20)
The substitution of (20) into (195 gives the following boundary conditions
atx= _:
VKo 8Q
Po Ao OxQ = Ql(t) (21)
1.3.2 Surge Tank
If the same notation is used, then the rate of increase of the liquid
volume in the surge tank is
dV
Po-_ = Qx=_ - Q1 (t) (22)
-7 -
SID 66 -46 - i
NORTH AMERICAN AVIATION, INC.SPACE and INFORMATION SYSTEMS DIVISION
where
V= qAs
rl = the increase of liquid level in the surge tank
A = cross-sectional area of the tanks
The increase in pressure of the liquid in the tank will be
V
P - Po = Po gq = Po gs
dvA AIK o __ _ s Op _ s 'Pdt pog at pog _ _-Ao x]o
Hence the boundary condition at x = £ will have the form
As K aQ
A pog axo
+ Q = Ql(t)(23)
1.3.3 Tank-Type Damper
The variation of the mass rate of the liquid contained in the damping
tank is
dpVd--t-- = (pAu)x=£ - Q1 (t) (24)
where V is the volume of the tank, Qx=# = (pAu)x=,_. is the mass of liquid
passing from the pipe into the tank, and Ql(t) is the mass discharge rate.
If the tank volume is V = AdL with the variation of tank length being
assumed small, then
dAd(25)dpV dd__td---_-=V + pL dt
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SIX) 66 -46 - I
NORTH AMERICAN AVIAT!ON, INC. SPACE and INFORMATION SYSTEMS DIVISION
From (ZO),
then
(7), and (11)
0p_ K cgQ cgp Po 0p
at Po Ao 0x at K 1 at at
aAd CdDdAd 0p
Eh at
dpV -KL% 1 8Q
where Ad,
restraint constant of the damper, respectively.
D d and C d are the cross-sectional area, diameter, and the end
The boundary condition for x = £ will be
(26)
Ad /_ll OQCdDd_
KL_- ° + _-_dT_-x + Q: Ql(t)(Z7)
-9-
SID 66 -46-I
N'O RT H AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
SEC TION 2. SOLUTION OF EOUATIONS OF UNSTEADY
MOTION BY LINEARIZATION
For small pressure variations, the equations of unsteady motion in an
elastic pipe with turbulent friction losses from (14) and (15) are
Op ap z o____u (z8)8-_ + u _xx = -Po a 8x
8u 8u I 8p f 2
8-T"+ u -_x - Po 8x 2D u (29)
Let
P = Po +AP
u = u +AuO
or written simply as
or written simply as
P=Po+P
U=U +U
O
(30)
Equations (28) and (29) yield the following sets of equations for the steady
and the perturbed motions.
8Po 2 8UoU _" -o --_-x Po a Ox (31)
uo
8u fo I 8Po 2
-- = U
8x Po 8x 2D o(3Z)
_- + (u° + u)\ ox +_x = -_oa \_ +(33)
+ u) _ + - \axP o
+ u)2 (34)
-I1-
SID 66 -46 -I
NORTH AMERICAN AVIAT!ON, INC. SPACE and INFORMATION SYSTEMS DIVISION
Considering the steady-state values of Po and u o to be known and their
variations along the x-axis small, then the perturbed equations after neg-
lecting terms of higher order will be
ap ap Z au (35)a-l-+ u ....o 8x Po a ax
Ou 1 ap8u + u ...... ku (36)
at o ax Po ax
where
fuo
k =--=kD t
o
for turbulent friction loss and
32Vk - -----_ - k%
p Do o
for laminar friction loss
Using Laplace transform and the assumption of zero initial conditions
p(x,O) = O, u(x, O) = 0 (37)
Equations (35) and (36) become
ap 2 8U (38)sP+uo _x Po a 8x
8U 1 8P (39)(s + k)U + u .....o ax Po 8x
or upon differentiation,
82P aP 2 82U
u _ + s 8x Po a 8x 2°Sx
(4O)
uo
8U 1 8ZP
--+ (s + k) _xx = - p% 8x 2
(41)
- 12-
SID 66-46-i
N'O RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
After some algebraic manipulation between Equations (38) through (41), we
have two decoupled differential equations, one for U(x, s) and one for P(x, s).
(az -uZ) --aZU-uo(ZS +k)_%U- s(s +k) U = 0o 8x 2
(42)
(az - U Z) 82P 8P
o 8x Z Uo(ZS +k)_x - s(s +k) P = 0 (43)
Let
U(x, s) = A(s) emx
mxP(x, s) = B(s) e
where m is determined by the characteristic equation
(44)
(a2 2- u 2)m - u (2s + k)m - s(s + k) = 0 (45)o o
/22Uo(ZS +k) +VUo k + 4aZs(s +k)
m -- (46)2(a 2 - u 2)
O
Hence
= m2xU A 1 em Ix + A 2 e
P = B 1 emlx + B 2 em2 x
(47)
The relations between A and B can be determined by putting (47) into
(38) and (39)
I I xIA2 oa2 Im°xAlPoaZml+Bluoml+Bl s eml + mz+Bzuomz+B2 s e _ --0
x m,_ xluoml+_ml+Al(S+k) e mt + 2Uom2+_m2+A2(s+k e -
Po Po=0
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SID 66 -46 -I
NORTH AMERICAN AVIAT!ON, INC. SPACE and INFORMATION SYSTEMS DIVISION
Since
mlxe #0,m 2 x
e #0
we have
p°aZml 1
B I =-AI Luoml +sj
[Po(Uoml + s +k)]B 1 = _ A 1[ ml
Hence
2
po a m
u m+so
- PoUo
s+k
+ Po m
Poa2m2 ]
BZ = -AZ [Uomz + sj
p oluomz + s + kJB 2 = _ A 2 m 2
_[I1,1 for m = m 1
[i2, for rn m 2(48)
-4
and
U(x, s) = A 1 e mix + A 2 e m2x [
!mlx m2xt
P(x, s) = -AII 1 e A212 e J
(49)
Let the boundary condition at x = 0 be
U(o, s) = Uo (s) P(o, s) = Po (s)
When applied to (49), it yields
U =A I +A 2o
P = -Allo 1
A 1 =
P + Uol 2o
12 - II
P + Uol Io
A 2 = ii I2
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Hence
Atx=£
U(x,s) =
P(x,s) =
P + Uol 2 P + Uol 1o mlx oe
I2 - I1 IZ - I1
rn2xe
II(P +UoI 2) I2(P + UoI I)o mix oe +
12 - I 1 12 - I 1
e m2x }(5O)
[i2 ii i [ 1U_ U _ eml£ __ em2_ 1 (eml£= o I2 I1 - Iz - I1 + Po I2 - I1 - em2"_)
[iii2 [i2 iI ]P£ = Uo 12 -T 1 (em2_- eml + Po 12 -I 1 em2 - 12 - I1 eml _
Let G(s) be the transfer matrix defined by
(51)
G(s) =
/ I2 mi£ I1 m2£e e
Iz - I 1 12 I 1
I 112
IZ - II
(em2"6 _ eml
then (50) can be written as
I (eml £ _ m2_ )
12 _ ii e
Iz em2_ Ii eml
Iz - II Iz - II
(5Z)
U£
P£
= G(s)
U o
Po
(53)
which represents the dynamic characteristics of the unsteady friction flowin an elastic line.
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From (53), we have the following relations:
Uo(S ) I2 + 1 em2_ - I1 + 1
(54)
P_(s)
Po(S)
(Iz - Ii) e (ml+m2)i
ml_ ZII + I2 e - I2 + II e m2
(55)
U£ e(ml+m2) _
Uz(s) _ (I2 - Ii)
)Uo(S) UZ UZI2 + 1 em2'Z - I1 + e ml
(56)
where m I, m 2, II and I2 are functions of s as defined in Equations (46)
and (48).
Substituting s = jco in (54) to (56), we can compute the magnitude and
phase angle ofU (jo_)
0
(a) the ratio of velocity to pressure deviations, p (j¢0)' at x = 0,versus frequency o
(b) the ratio of pressure deviation at x = £ to pressure deviation at
P_(jo_)
x = 0 p (jco)' versus frequency.O
(c) the ratio of velocity deviation at x = £ to velocity deviation at
u_(j_)
x = 0, U (J_• ), versus frequency.O
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The boundary condition at x = % is
a a 7.(_, t)AIE _xZ(£, t) + _ _-_ = p£(A-Api ) (62)
where A is the cross-sectional area of pipe, AI, the cross-sectional area of
pipe wall, Api the cross-sectional area of pump inlet and _ the coefficient of
damping of the bellow• The Laplace transform of (6Z) is
a
AIE _x Z(x, s) Ix=£ + _sZ(x, s) Ix=£ = p_(s)[A-Api ] (63)
Sub stituting
Z(x, s)Ix= _ = B(s) sinh_-_sxlx=_ (64)
into the above equation and solving for B(s), we have
B(s) :P_(s) [A - Api ]
V_ v_-_s s_nn_- sAlE s cosh £ +_ s " -%/'_-S
(65)
Hence, (64) becomes
z(t,s)=P_(s) [A - Api ]
S [AIE_//-_"_ cot h%/_ s_ + _]
(66)
The transfer function relating the velocity at x=% to the pressure at
x=¢, is
A - As Z(_,s) _ pi
P£(s) AIE%/_ cot h%/'_ s£ +{
This factor is to be added to the term U£/P_, in Equations (54), (55), and
(56). The deviations in magnitude and phase which occur near the natural
frequency of the longitudinal vibrations of the pipe can be seen from the
computational results shown in Appendix A.
(67)
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2.Z PHYSICAL CONSTANTS AND RESULTS OF LINEARIZED ANALYSIS
The following physical constants are based mostly on information
applicable to Saturn S-II Stage.
a
Constant
Table I
internal radius of pipe (D = 2r)
length of pipe between tank and
P
K1
h
E
(1
A
A h
Api
C
v
V
uo
Values for LH 2 Line
0. 333 ft.
20.83 ft. (outboard)
pump
velocity of pressure wave
1a =
cD+ E---/-
density of fluid at steady state
bulk modulus of fluid
thickness of pipe wall
Young' s modulus of pipe wall
density of pipe wall
cross-sectional area of pipe
31.9Z ft. (center
3700 ft/sec
0. 136 Ib-secZ/ft Z
2. 15 x 106 ib/ft Z
-31.83 x l0 ft
4.32 x l09 ib/ft 2
14.8 ib-secZ/ft 4
0. 348 ftZ
cross-sectional area of pipe wall
cross-sectional area of pump inlet
restriction factor of pipe
Poisson' s ratio of pipe wall
damping coefficient of bellow
viscosity of fluid
steady-state fluid velocity
0. 00383 ft2
0. Z56 ft2
0.85
0.30
0 (assumed)
-50. 029 x 10
51.5 ft/sec
Ib - se c /ftg
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NO RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISIQN
U_
P
k t
Constant
Table I (Gont)
steady-state velocity-pressure
ratio (determined from the steady-
state pressure versus flow curve)
laminar resistance coefficient
kf = 3Z_/po D2
Darcy-Weisback resistance coeff.
Values for LH 2 Line
-41.55 x 10 per sec
-41.55 x 10 per sec
turbulent resistance coefficient
fu0
kt - D
O. 0775 (outboard}
O. 059 (center}
5.98 per sec (outboard)
4.54 per sec (center)
With the above information the frequency response equations (54),
(55), and (56) were programmed on the IBM 7094 for both LOX and LH 2 lines.
The numerical results for the outboard LH 2 line of S-If stage are illustrated
in Appendix A, Figures A-I through A-12. The magnitude and phase angle
of Uo(jco)/Po(j_) versus 0_ are shown in Figures A-I through A-4 according to
the type of hydraulic resistances and with or without the effect of line
vibrations. Similarly, the magnitude and phase angle of P_(jc0)/Po(j_0)
versus _0 are shown in Figures A-5 through A-8. The magnitude and phase
angle of U_(j_)/Uo(j¢o ) versus ¢0 are shown in Figures A-9 through A-I2.
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SECTION 3. SOLUTION OF NONLINEAR EQUATIONS OF UNSTEADY
MOTION BY THE METHOD OF CHARACTERISTICS
The transient pressure of unsteady flow with a turbulent resistance in
an elastic pipe undergoing an acceleration am requires the solution of the two
nonlinear partial differentialEquations (14) and (15a}.
310 + 3p 3uat U_x = -paZ a-_" (68)
au + ua___u= lop_ f_L.u 2 + am sin8 {69)3t 3x p 3x 2D
The governing equations cannot be solved analytically. However, these two
equations can be transformed into a set of ordinary differential equations by
the method of characteristics and then solved numerically.
Define a characteristic curve of Equations (68) and (69) to be a curve
along which a solution of Equations (68) and (69) may have discontinuities in
the derivatives of the velocity and pressure. Suppose that such a curve is
given by
x = x(s), t = t(s) (70)
and that u and p are prescribed at each point of this curve. Then u and p
become functions of s and we have a system of four equations in four unknown
partial derivatives Ux, ut' px and Pt" These four equations are
3__u+ uOU + 1 3p _ f u 2 + a m sin O8 t 3x p 3x 2D
ZOU _p+ 3ppa _xx + at U_xx : 0
(continued on next page)
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8u dt + 8ua-i- = du
8_p_pdt + 8pat _x dx = dp
(7l)
The discontinuity relation requires that there be at least two distinct
sets of values of these derivatives which satisfy Equations (71). The con-
ditions under which Equations (T1)will admit of more than one solution
require that
1
i u o ?-
0 pa 2 1 u
dt dx 0 0
0 0 dt dx
=0 (72.)
and
1 u 0 - fu2 + am si'neZD
0 pa z 1 0
dt dx 0 du
0 0 dt dp
= 0 (73)
Expanding these determinants (72) gives
dx-- - U =-l-adt
(74)
and (73) gives
du f u2 1 (dx _tp)d--i-+ _-_ -amsinO +----_ _-- upa
=0 (75)
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Putting the relation (74) into (75), it becomes
du _D i dpd--t-+ ug - a m sin O ±_-_ d-_ = 0 (76)
In these equations where the ± sign appears, the plus sign is associated with
the characteristic C + and the minus sign with the characteristic C-.
Equations (74) define the slope of the characteristics C + and C- as
a function of velocity of flow, u, and the velocity of pressure wave, a.
Equations (76) are the compatibility equation relating the velocity u and
pressure p along the characteristic lines C + and C-.
Equations (76) are two separate, total differential equations with t as
independent variable and u and p as dependent variables. If u = u(x,t) and
p = p(x,t) satisfies Equations (68) and (69) then Equations (74) determine
two families of characteristic curves C + and C" in the x,t plane belonging
to the solution u(x,t) and p(x,t). These four Equations (74) and (76) can be
solved by finite difference approximations with appropriate initial and bound-
ary conditions.
3. 1 SPECIFIED TIME INTERVALS
There are two ways in which the characteristic curves C + and C- may
be used to obtain an approximate numerical solution to the original partial
differential equations. One method involves the use of a grid of character-
istics and the other makes use of specified time intervals. In using the
former method, it would be difficult to arrange the computations so that the
points of intersection of the characteristics occurred at values of x and t
on a rectangular grid. The use of specified time intervals has one main
advantage over the use of a grid of characteristics in that it provides results
of the flow field at different times. Hence, from a programming viewpoint,
the ordering of the computations would appear to be easier by this method
than by the use of a characteristic grid, since the values of the dependent
variables are known at predetermined points and the additional information
of the corresponding x and t coordinates does not have to be recorded.
For this reason, this is the approach which will be used in the present
analy s is.
3. Z SELECTING MESH RATIO
The first step in using the method of specified time intervals requires
selecting a mesh ratio_t/Ax for the finite difference approximation. The
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selection of an "admissible" mesh ratio plays a vital part in the success of
the numerical process. It insures that the difference equations actually do
approximate the differential equations so that anumerical solution of the
former should be a good approximation to the exact solution.
The region bounded by the two characteristics C + and C-, and a third
non-characteristic curve such as PRCS shown in Figure g is a region in
which a unique solution to the original partial differential equations would
exist provided conditions along RCS are prescribed. Thus, it is only logical
to select a mesh ratio such that the solution will lie within this "zone of
influence. " This condition will be satisfied if the mesh ratio is less than the
slope of the characteristic curves as given by Equation {74), since geometri-
cally the mesh ratio plays the role of "finite difference characteristics. "
Under this conditon, the finite difference scheme should be a valid represen-
tation of the differential Equations (14) and (15a).
If the variation in the velocity of wave propagation is assumed small
so that it may be taken equal to its steady-state value a o, then for
At i-<- (77)
Ax u±a0
we would expect the solution of the difference equations to tend to that of the
differential equations as At --- 0. Geometrically, this means that R must
lie within AC and S within BC in Figure 2. The condition given by Equa-
tion (77) also insures the stability of the numerical process. Since the
_Lh
C
\C ÷
x
Figure 2.
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velocity u < < a o, the criteria used in the present study for selecting the
mesh ratio was
At 1
Ax a0
The accuracy of the finite approximations deteriorates if the mesh ratio is
much less than 1/a o although their solution will be stable. Optimum accuracy
is obtained when the finite difference characteristics coincide with the con-
tinuous characteristics of the governing differential equations. For a system
whose true characteristics are curved such as in the present case, the
optimum accuracy cannot be obtained using a fixed rectangular grid since
the finite difference characteristics must necessarily have smaller slopes
than the true characteristics to ensure stability of the numerical process.
3.3 COMPUTATIONAL PROBLEM
The computational problem which must be dealt with is illustrated in
Figure Z. On the rectangular grid in the x,t plane as shown here, the
velocity and pressure are known at each mesh point, such as A, B, and C
for time tl, either as given initial conditions or as a result of a previous
stage of the calculations. The problem is to determine the conditions
(velocity and pressure) at a typical point P corresponding to time ti + I.
The coordinates of point P in the x,t plane are therefore known. Through
this known point, we pass the two characteristic curves C + and C-, inter-
secting the line t = ti at points R and S respectively. Both the velocity
and pressure are unknown at the three points P, R, and S, and the space
variable is unknown at the latter two points, giving a total of eight unknowns.
3.4 INTERPOLATION FORMULAS
In the analysis to follow, the various variables associated with a given
point will have that point's designation used as a subscript, such as uA,
PA, XA ..... etc.
Since the characteristics C + and C- pass through points R and S
respectively, expressions are needed for the velocity and pressure at these
two points in order to compute the condition at P using these two curves.
This is done by fitting a parabola through the known values at A, C, and B,
and then interpolating. For example, let us consider the case of calculating
the velocity uR at point R. (See Figure 3.}
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uC
uR
u A
//
XA x R x C
Figure 3.
Using the second order equation
u(x) --a + b(x - Xc) + c(x - Xc)2 (78)
and applying it to the three points x A, x B, and x C where the velocities are
known, we obtain three algebraic equations for determining the coefficients
a, b, and c. Applying Equation (78)to x = x C shows immediately that
a=u C(79)
Application to the remaining two points yields
uA = a + b(x A - Xc) + c(x A - xc) Z (80)
u B = a + b(x B - Xc) + c(x B - xc)Z (81)
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Since successive mesh points along the x axis are _x units apart, we can
express xA and xB in terms of x C as
x A = x C -_x
x B = x C + Ax
On substituting these two relations into Equations (80) and (81), they become
uA = a -b • _x + c(_x) z (82_)
uB = a + b • Ax + c(_x) Z (83)
Making use of the result given by Equation (79) and solving Equations (8Z)
and (83) simultaneously, we find that
1
b - aax (UB - UA) (84)
C - 2 (UA + UB - ZUc)(85)
With the three coefficients a, b, and c determined, we apply Equation (78)
to x --x R and obtain
]
UR : Uc + Z_x (UB - UA)(XR - Xc) "_z(m×)
Z (UA + UB - 2Uc)(xR - Xc)2
(86)
The same functional form given by Equation (78) is also assumed for
the pressure variation between the three adjacent points xA, XB, and x C.
Under this assumption, the coefficients given by Equations {79), (84), and
(85) are valid provided that the velocities uA, UB, and u C are replaced
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respectively by the corresponding pressures PA' PB' and PC" Thus, the
following quadratic interpolation relationships can be written down
immediately:
i i
PR = PC + 2-_--_x(PB - PA){XR - xC) +2(a×)
Z {PA + PB - ZPc)(XR - xc )Z(87)
1 1Us --uc + z-N-£x(uB " UA)(xs - xc) +
Z(Ax)Z (UA + uB- ZUc)(X S - xc)Z
(88)
I
PS = PC + _ {PB - pA)(X S - Xc) + 1 (PA + PB - 2Pc)(Xs - Xc )22(Ax)z (89)
These four equations form a necessary part in the computational scheme.
The other four equations needed to obtain a solution are given by the charac-
teristic Equations (74) and (76).
3.5 FINITE DIFFERENCE APPROXIMATION
Let us confine our immediate attention to the two characteristic equa-
tions associated with the characteristic C+; i.e., we consider the plus sign
in Equations {74) and (76). These two equations are transposed into finite
difference form by replacing the differential coefficient by a divided difference
over an interval and replacing the other variables by the arithmetic mean
over the interval. This finite difference approximation is one of second
order, and hence it is compatible with the interpolation formulas given in
Equations {86) through {89).
On applying the foregoing procedure to Equations (74) and {76), and
transposing, we obtain
uR + Up o)XR = Xc - 2 "+ a At (90)
f Up)g PP - PRUp - UR+-_-_(UR+ _t-a sin 8.At + = 0 (91)m Po ao
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It is assumed that the friction coefficient is a constant, and furthermore,
that the variations in density and velocity of wave propagation are sufficiently
small so that they may be taken equal to their respective value at steady-
state conditions. The subscript "o" refers to steady-state values.
Similarly, the corresponding equations associated with the character-istic C- are
_(u S + Up o)XS = XC 2 a At (92)
PP - PSUp) 2Up - u S + (uS + At- a sin O • At - O (93)m Po ao
The above four equations, together with the four interpolation formulas,
Equations (86) through (89), form a set of eight nonlinear simultaneous
algebraic equations for the determination of the eight unknowns XR, UR,
PR' xs, us, PS, up and pp. The problem now is to organize a computational
tional procedure suitable for solving these eight equations on a digital
computer. Without resorting to a simultaneous solution of these eight equa-
tions, the most practical way would be by iteration.
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SECTION 4. COMPUTATIONAL PROCEDURE
One method of obtaining the conditions at point P by iteration using the
nonlinear algebraic equations (86) through (93) is the following:
1. Make an estimate of the conditions at R, S, and P by assuming
that they are the same as those existing at C. In other words,
o o o
u R = uS = Up = u C
o o o
PR = PS = PP = PC
The superscript "o" refers to the initial guess, or zeroth
approximation.
Z. Using these estimates, calculate x_ and x ° from Equations (90)
and (92), respectively.
1 1 13. Making use of the results in step 2, calculate u ' PR' Us' and PSfrom Equations (86) through (89) inclusive, respectively.
4. With the values obtained thus far for the first iteration, solve
Equations (91) and (93) simultaneously for u 1 and p_.
The foregoing process is continued until successive values for both
velocity and pressure at point P are within the prescribed tolerance.
4. 1 INITIAL CONDITIONS
In order to start the computational procedure, as outlined above, we
must have the initial conditions corresponding to time t = 0. The initial
conditions in the propellant line are taken as those corresponding to the
steady-state condition. Due to friction losses along the length of the pro-
pellant line, the pressure at each mesh point on the x-axis must necessarily
be different from one mesh point to another. The friction coefficient can be
calculated from the experimental data that gives pipe resistance versus flow
rate. It is tacitly assumed that this friction coefficient obtained under
steady-state conditions is applicable to transient phenomena and also remains
a constant throughout the line.
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Within the framework of one-dimensional flow, the velocity u in the
line must be a constant in the case of steady flow. This is immediately
apparent from the continuity Equation (1) when one invokes the condition of
steady flow whereby the flow field is independent of time, and hence, reducing
it to
_-x-x(pUa ) = 0
Since we are considering propellant lines of constant cross-sections and
ignoring any variation in p, it is readily seen that u must be a constant.
Under this condition, the equation of motion given by Equation (15a) reduces
to
Po
2U
dp + _..f. o-- -a sin Odx D g m
= 0
On integrating this equation and applying the result to two consecutive mesh
points, we have
Pl
U o
- P2 = Po _ _--a sinm(94)
where the subscript "1" designates the upstream mesh point and the subscript
"2" the downstream mesh point. In the absence of the acceleration term
a m sin e, Equation {94) is equivalent to the well-known formula used in
hydraulics to calculate the head loss in a pipe.
4. Z BOUNDARY CONDITION AT LEFT END OF LINE
For the boundary condition at the left end of the line, we assume that
the pressure in the tank remains a constant. This condition is very nearly
fulfilled since the behavior of the system is of interest only during the first
few seconds following the initiation of the disturbance. The value of this
constant pressure is taken equal to the steady-state value just prior to the
introduction of the disturbance in the line.
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A C B
Figure 4.
To calculate the velocity corresponding to this given pressure, we may
proceed in the following manner (refer to Figure 4):
1. Assume initially that the velocities Up and u S are equal to u A ,
i.e., u_ = u_ = uA .
2. Using these assumed values, calculate x_ from Equation (92) with
x C replaced by xA = 0 .
1 13. For this value of
x S, compute u_ and p_ from Equations (88) and(89), respectively.
4. Solve for uI from Equation (93) using the results of Step 3.
This process is repeated until two consecutive values of up agree within the
specified accuracy.
4. 3 BOUNDARY CONDITION AT RIGHT END OF LINE
It is assumed that the disturbance in the system originates at the right
end of the line where the pump is located. Some of the various boundary
conditions which may be considered have been already discussed in Section
1. 3. Before considering these boundary conditions where some sort of
regulating device is present in this preliminary investigation, we will merely
specify a particular disturbance. Since any sort of disturbance will affect
the flow rate and hence the fluid velocity, we will consider the disturbance
as a change in velocity. The disturbance is taken in such a manner that it
causes a decrease in velocity at the point of origin, and hence in turn pro-
duces an increase in pressure.
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4.3. 1 Velocity Impulse Disturbance
Art
Figure 5.
If the righthand end of the feed line is provided with a valve and this
valve is suddenly closed a certain amount and then reopened instantaneously,
we would have an impulse disturbance. The magnitude of this disturbance
will depend upon the amount of closure before reopening takes place. The
maximum impulse will occur, of course, when the valve is closed completely
before it is reopened. (See Figure 5.)
For a velocity change that takes the form of an impulse, the conditions
at the right end must be recalculated before proceeding with the solution in
the x,t plane. The corresponding pressure change Ap due to a Au change
in velocity can be found from momentum considerations. The result is
Ap = Po ao Au (95)
Therefore, the conditions which we have at the right end at time t = 0 due
to the impulse are
U ----UO
P = PO
(96)
where u and Po are the steady-state velocity and pressure, respectively,• O
prtor to the introduction of the impulse.
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A C B
C +/
Figure 6.
In calculating the two unknowns at point P when t > 0, another relation-
ship is required besides the equation for the characteristic curve C +. This
additional relation is obtained from the condition that the discharge at anytime must obey the orifice law. Hence,
1/2Ip_\
Up = u (-_) (97)
where the quantities with a bar over them represent reference values.
They may be thought of as the "equilibrium conditions" about which the
transients caused by the disturbance oscillate. In the case of an impulse,the steady-state values are used as reference.
The calculation of Up and pp may proceed as follows:
1. Assume that initially the velocities at R and P are equal to that
at B (refer to Figure 6); i. e. ,
o o
u R = Up = u B
2. Calculate x_ from Equation (90) with x C replaced by x B = • .
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1
4.
ICompute uI and PR from Equations (86) and (87), respectively.
1 1
Solve Equations (91) and (97) simultaneously for Up and pp .
As before, this procedure is repeated until two successive values of both
Up and pp are within the specified tolerance.
4.3.2 Velocity Step Disturbance
We define a step disturbance as that caused by an instantaneous partial
closure of the valve with the valve held at this new position. Thus, the step
disturbance in this sense does not mean that the velocity is maintained at
this new value for all time, but the position of the valve itself represents
a step function. The position of the valve creates a new "equilibrium
position" which the flow will eventually attain after the transients due to the
disturbance have subsided.
As in the case of a velocity impulse, we must first calculate the new
conditions at the right end due to the step disturbance before proceeding with
the solution. These new conditions are computed in exactly the same manner
as was done in the case of the impulse. The pressure change is given by
Equation (95), and the new conditions by Equation (96). The only difference
between the two cases is that for the step disturbance we use the new con-
ditions given by Equation (96) for reference values in Equation (97) instead
of the steady-state values. With this one change, the computational proce-
dure for calculating the velocity and pressure at point B on the boundary
for t > 0 is identical to that outlined for the case of an impulse.
4.3.3 Sinusoidal Velocity Disturbance
The procedure given in Section 4.3. Z for handling a step disturbance
may be extended to handle any other type of disturbance or pulse shapes
simply by replacing the given disturbance by an equivalent series of step
functions. By equivalent, it is meant that both the composite step function
representation of the given disturbance and the disturbance itself have the
same area under the curve.
Suppose that in the absence of propagation effects, the disturbance
under consideration took the form of a sine function OAB as shown in
Figure 7. We first divide the half-time period tA into suitable time incre-
ments At. From the computational standpoint, it is convenient to choose
AT equal to either At, the time interval used in the difference equations, or
a multiple thereof. Within each interval, the ordinate for the "step" is
selected by either visual inspection or other means such that the area under
the "step" is equal to that under the actual curve. In this way we replace
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4
A B
Figure 7.
the sine curve by the "staircase"approximation 01234 A...B, and this
function is the one which willbe dealt with in our subsequent discussion.
Corresponding to the instantaneous velocity change AU a for the
first step "012" of the staircase approximation, the pressure change is
obtained from Equation (95) as before. Since it is assumed that the dis-
turbance takes place only after steady-state conditions have prevailed in
the line, the new conditions at the righthand end after the disturbance has
taken place is given by Equation (96). The calculation for the pressure
and velocity at the right end for 0 < t < AT now proceed in the manner
given in Section 4.3. l where the new conditions existing there are used
as reference values. This iteration process is continued for as many
steps as necessary until the specified tolerance on the dependent variables
have been met. We now arrive at point "2" on our staircase approxi-
mation so that step "234" comes under consideration. The response of
the system due to this step input during the time AT < t < 2 AT is
handled in the identical fashion as for the previous step in which
the reference values to be used now in Equation (97) are furnished by the
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conditions existing at time t = AT with the effects of AU b included. This
same computational procedure is continued for each succeeding "step. "
4.3.4 Feed Line With Pressure Regulating Device
In Section i. 3, the boundary condition at the right end was derived for
the case of a pressure regulating device, such as a surge tank, air chamber,
or tank damper, being present in the line. In each of these three cases, it
was shown that the boundary condition can be expressed as a first order
partial differential equation in the form
OQ +K-_- Q g(t) at x = L (98)
where Q is the mass flow rate entering the regulating device, and g(t), a
prescribed function, is the mass flow rate discharging from the device. The
constant g is dependent upon the type of regulating device under considera-
tion, and the value it assumes for the various devices is given in Section 1.3.
It would appear that one approach of incorporating Equation (98) as a
boundary condition into the righthand end of the feed line would be to inte-
grate it and obtain a relation between Q(t) and g(t) . For any prescribed
disturbance function g(t) then, Q(t) will be known and hence also the velocity
u(g,t) and the discharge velocity are known. Therefore the computational
procedure given in Sections 4.3. I, 4.3.2, or 4.3.3 may be employed,
depending on the type of disturbance under consideration. Since Equation(98)
is a boundary condition, what we are seeking is a particular solution.
For any given disturbance, there are a number of particular solutions which
would satisfy Equation (98). Without knowing the general form of the closed
form solution for the pressure and velocity along the feed line itself, it is
impossible to tell which one of the particular solutions is the correct one.
Therefore the only other alternative is to express this equation in finite
difference form and solve it simultaneously with the other appropriate
equations at the right end, numerically.
The derivative 8Q/Ox appearing in Equation (98) can be expressed by
a divided difference over an interval Ax. However this "average" slope of
Q over the interval may be quite different from the slope 8Q/0x at x = g.
Although this difficulty may be alleviated somewhat by dividing the last inter-
val into smaller intervals, the numerical procedure would be complicated.
The reason for our difficulty in obtaining a representative finite
approximation for Equation (98) is that although it dictates the condition
which must prevail for all values of time t , it involves the derivative aQ/ax
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which in the x, t plane is in a direction normal to the t axis. Thus if the
boundary condition can be rewritten so that any derivatives present are
with respect to time instead of the spatial variable x , then it can be put in
finite difference form solely in terms of the variables at x = g. Having the
equation in this form should permit a more accurate numerical treatment
of the boundary condition at the right end of the feed line than would be
pos sible otherwis e.
4.3.4. 1 Modified Form of Boundary Condition
The boundary condition given by Equation (17b) stems from the con-
tinuity in mass flow which must exist at the junction of the device and the
feed line. For each of the three pressure regulating devices considered,
this boundary condition can be rewritten in the form
A K @--_-P+ Q = g(t) (99)o at
by utilizing the relation between 0Q/0x and 0p/at given in Section I. 3.
The constant K characterizes the damping device and is equal to K divided
by the modified bulk modulus K defined in Equation (13), and A o is thecross-sectional area of the feed line.
If our attention is focussed on feed lines of constant cross sections
such as in the present case, and the variation in the fluid density is assumed
sufficiently small so that it is essentially equal to its steady-state value, it
is advantageous from a computational point of view to rewrite the quantity Q
in Equation (99) in terms of the fluid velocity u. Since Q = 9oAou ,
Equation (99) takes the form
_P
_T + u = v(t) (100)
The quantity v(t) will be equal to the discharge velocity, provided the dis-
charge cross-sectional area is the same as that of the feed line; otherwise
it will be simply proportional to it. The proportionality constant in this
case is equal to the ratio of the discharge cross-sectional area to the feed
line cross-sectional area.
4. 3. 5 Numerical Procedure
Equation (I00) may be transposed into a finite difference equation
using the technique given in Section 3.5. Employing the same notation
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as shown in Figure 6, the resulting equation takes the form
PP - PB 1 1
B) = (Vp+V B) (101)
The three quantities with the subscript B are known from the calculation for
the previous time interval, and since v is a prescribed function, Vp is also
known. Thus pp and Up are the two unknowns in Equation (101). The com-
putational procedure to determine these two unknowns may proceed in the
following manner.
I. Assume initially that u_ = u_ = u B
O
2. Calculate xR from Equation (90) with x C replaced by L .
3. Compute u_ and p_ from Equations (86) and (87) respectively.
4. Solve Equations (91) and (101) simultaneously for up and p_ .
This process is continued until successive values for both Up and pp are
in agreement within the specified limits.
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SECTION 5. COMPUTATIONAL APPROACH TO SPECIFIC PROBLEM
We are now in a position to handle a specific problem under the bound-
ary conditions described previously. Referring to Figure 8, suppose that we
have decided to divide the propellant line into five equal incremental lengths
so that the left and right boundaries coincide with A0 and A5 respectively.
One way in which the computation can be carried out would be the following.
From the given steady-state condition at A5, the velocity at A0 through A5 isknown and the corresponding pressure at these mesh points can be calculated
from Equation(94). If the disturbance under consideration involves an instan-
taneous velocity change, such as a step function, the conditions at mesh point
point A 5 must include the effect of the velocity change before we proceed with
the calculations. The incremental pressure change corresponding to this
incremental velocity change is computed from Equation(95). Superimposing
these incremental changes in velocity and pressure according to Equation (96)
onto the steady-state values at A5, we obtain the new conditions which mustprevail there because of the disturbance.
In the case where the feed line contains a regulating device and the
disturbance is a sinusoidal function, the foregoing calculation is not neces-
sary since the disturbance was not approximated by a series of step functions
and hence did not involve any instantaneous velocity changes.
t
t 2
t IBo 81 8 2 8 3 8 4
t=O
A0 AI A2 A3 A4 A5
Figure 8
85
X
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Having the conditions for all mesh points at t=0, we now "march out" to
time tI, and proceed to calculate the conditions at these mesh points. For
the prescribed boundary condition of constant pressure at B o, the velocity
may be determined by the procedure outlined in Section 4. 2. The conditions
at mesh points B l through B 4 are found by the procedure given in Section 4. 0.
At mesh point B 5 where both the velocity u and pressure p are unknown, we
proceed in the manner described in Section 4. 3. I, 4. 3. 2, 4. 3. 3, or 4. 3.4,
depending on the type of velocity disturbance and whether a regulating device
is present or not. As long as suitable boundary conditions are prescribed at
both ends of the feed line, the solution can be "marched out" as far as
desired.
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SECTION 6. NUMERICAL RESULTS
Some numerical results obtained from the application of the foregoing
numerical procedure to the center liquid hydrogen feed line on the SaturnS-II
engine are shown in AppendixB, Figures B-1 through B-48. The numerical
values used for the various parameters in the analysis are given in Table I
(Section Z. Z). Calculations were carried out for the feed line with and without
an air chamber when subjected to all three types of velocity disturbances
discussed previously. The air chamber was selected over the other two
damping devices discussed since it combines the space economy provided by
the tank damper and, to a lesser degree, the effectivenesswithwhichthe surge
tank alleviates pressure changes. Thus it represents a device which might
well be incorporated into a propellant feed line.
Two amplitudes were considered for each disturbance expressed as a
percent of the steady-state velocity. The amplitudes chosen were 2 and 5 per-
cent since they correspond approximately to the minimum and maximum
variation expected in the pump flow of the S-II respectively. Four frequen-cies were selected for consideration in connection with the sinusoidal dis-
turbance. They were 7. 20, 9. 70, 11. 70, and 21. 11 cps, corresponding
respectively to the first four structural modes of the spring mass "POGO"
model of the Saturn S-II engine at 95 percent burntime.
6. 1 DISCUSSION OF RESULTS
Pressure and velocity time histories were obtained both at the right
end and at the middle of the line for the case when the feed line does not con-
tain a regulating device. The results corresponding to the impulse disturb-
ance and the step disturbance for amplitudes of 2 and 5 percent of steady-
state velocity are shown in Figure B-1 through B-8. In all cases, since only
partial reflections take place at the right end, both the pressure and velocity
in the line regain steady-state conditions in a very short time, approxin_ately
0. 10 second, as shown by a decrease in amplitude of each succeeding "spike"
or "step" following the initial disturbance. This is to be expected due to the
stabilizing influence of the unidirectional flow. With the exception of the
numerical values obtained, the general behavior of the feed line subjected to
either a 2 or 5 percent impulse disturbance is the same. This also applies
to the step disturbance.
The results for the sine disturbance are given in Figure B-9 to B-24
inclusive. The sine disturbance introduces an additional parameter, input
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frequency, which does not appear in either the impulse or sine disturbance.
For the liquid hydrogen line under consideration, it takes the pressure wave
caused by a disturbance at the right end 0. 03455 second to complete one
cycle, corresponding to a frequency of 28. 96 cps. The initial reflected pres-
sure wave of a sine disturbance with this input frequency will arrive at the
right end just when the disturbance is zero. Under this condition, the maxi-
mum fluid velocity is equal to the initial steady-state velocity and the trend
of velocity variation is toward this value. In the present four cases, the
input frequencies were allbelow 28. 98 cps so that the disturbance and the
reflected waves are out of phase with one another, causing the general veloc-
ity trend to increase with time. This was true both at the right end and at
the midpoint of the feed line. The pressure tends to decrease with time
but at a much lower rate than does the velocity. If the plots were carried out
for a sufficient length of time, the velocity in all four cases tends to oscillate
about some equilibrium value, as does the pressure. After the initial tran-
sients have subsided in the first cycle, the peak-to-peak variation of each
consecutive cycle for both the velocity and pressure is essentially zero.
Hence the effect of pipe friction is very small.
Figure B-25 through B-48 contain the results for all three disturbances
when an air chamber is located at the right end of the feed line. It was
assumed that the air chamber had a cross-sectional area of 3 sq ft (diameter
of approximately 2 ft) with the air occupying a 2-foot length of the chamber
initially, giving a volume V o of 6 ft3 under steady state flow. Since Po is
4,320 psf, this yield aVvalue of 4 x 10 -3 ft3/Ib. This value of _ was used
throughout the study whenever an air chamber was considered as present in
the feed line.
The results obtained for the feed line with and without an air chan_ber
cannot be compared directly to ascertain exactly how effective the air cham-
ber is in alleviating pressure surges, since in the former case the discharge
velocity is prescribed for all time while in the latter it obeys the orifice law.
This is obvious since the time history in the two cases will differ consider-
ably from one another whether the air chamber is present or not. However,
they can be compared with each other from a qualitative point of view. This
is particularly true for the pressure variation since, broadly speaking, the
general trend and maximum peak amplitudes are approximately the san_e
whether the discharge velocity is "fixed" or "free" as in the case where the
orifice law applies.
The effect of the air chamber on velocity and pressure at the right end
is illustrated in Figure B-29 and B-31 for a 2 and 5 percent step disturbance
respectively. It is seen that the pressure fluctuations which occur in the
feed line without a pressure regulating device are completely eliminated by
the air chamber and the overpressure decreases very gradually fronl its
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initial value caused by the disturbance. The pressure fluctuations actually
are much larger than for a "free" boundary since the damping influence of
the streaming fluid is much less in the fixed case. This is shown quite
clearly in Figure B-30 and B-32 by the larger pressure fluctuations in the
middle of the line. From this observation then, we see that the air chamber
is quite an effective damping device.
The results for a prescribed impulse disturbance with an air chamber
in the feed line are shown in Figure B-Z5 through B-Z8. One feature of these
results is that they are very similar to those for the step disturbance and
differ only from a quantitative point of view. This is not too surprising when
it is seen that the only difference between the two cases is that v(t); the
prescribed discharge velocity, assumes a slightly different fixed value after
the initiation of the disturbance. Since both cases have the same boundary
condition at the right end which differ only by a constant, the two results
must necessarily be similar.
Just how effective the particular air chamber is in alleviating pressure
surges is illustrated in Figures B-33 to B-48, which are the results for the
fixed sine disturbance. Considering the 5 percent sine disturbance, we
noticed that in no case does the maximum peak-to-peak pressure variation
exceed 15 psf, which is about 1.5 percent of the overpressure for the cor-
responding "undamped" case. Naturally for the Z percent case, the peak-to-
peak pressure variation is proportionately less. The air chamber also
suppresses the velocity so that in all cases it did not vary more than I/2 fps
from the steady-state velocity of 51.5 fps. The high frequency oscillations,
which appear in the velocity variation such as in Figure B-33 and in the
pressure variation as shown in Figure B-34, have no physical significance
but are mathematical in nature. As was mentioned earlier in Section 3.2, if
the time interval selected is such that the solution obtained coincides with
the true characteristics, then these oscillations will be eliminated.
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CONCLUSIONS
The principal objective of the present investigation was to develop
some mathematical models for the unsteady fluid motion in a propellant
feed line which may be incorporated in a closed loop dynamic system for the
analysis of longitudinal oscillations of a large liquid rocket booster. With
this in mind, the dynamic behavior of the feed line was looked at both from
a linear and a nonlinear viewpoint. The analysis of the linear mathematical
model resulted in the transfer functions given in Appendix A. The similarityof results for both the frictionless line and one with friction taken into
account indicate that friction loss for this particular line is very small.
Since the cross-sectional area of the line is relatively large, this result is
to be expected. The inclusion of the longitudinal natural frequency of the line
resulted in a modification of the frequency response curve which is similar
to those given in Reference 10 and 11.
Some results based on the nonlinear differential equations governing
unsteady compressible fluid flow are shown in Appendix B. Although these
results cannot be compared directly with any known published data, they do
correspond to a modification of the water hammer problem encountered in
hydraulics in which the valve is only partially closed and is a function of
time. By neglecting the friction term and setting the velocity to zero at the
right end, the solution obtained agrees with the classical waterhammer solu-
tion. Hence the method provides a useful tool for inclusion in a system
dynamic study. Various disturbances were considered under two different
boundary conditions to illustrate the flexibility of the method.
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AREAS FOR FURTHER STUDY
The propellant feed line considered in the present analysis was straight
and of constant cross-section, which is a very special case. In actual prac-
tice, feed lines may involve a step change in cross-sectional area, branch
lines, bends, or any combination of these. Therefore, in order to obtain a
more realistic mathematical model of a propellant feed line, further study
should be undertaken to incorporate these additional parameters. The method
developed for the nonlinear model can be extended to handle the step change
in cross-sectional area or a feed line with branch lines quite readily by
using it as a "building block. " What these two cases require is a modifica-
tion of the boundary conditions at either end of the line so that it can accom-
modate the continuity condition for both mass flow and pressure at the point
of discontinuity in cross-section or junction of the branch lines.
Another area for further investigation in connection with the present
problem is a parametric study of the various pressure regulating devices to
determine how each parameter influences the dynamic response. For
example, the volume of air at steady-state flow may be varied by the con-
stant_ that characterizes the air chamber. In the case where two or more
parameters are involved, such as for the tank damper, such a study would
indicate which parameter has the greatest influence on the effectiveness of
any given device to dampen pressure surges. Going a step further, this
data can then be used to make a trade-off study, say between weight and
allowable space (total volun_e) versus damping effectiveness of a particular
device. This will provide information which can be used directly to "tailor"
a damping device for a particular feed line system and a particular n_issile.
Besides relieving the overpressure by mechanical devices, there are
other means based on the high compressibility of a gas which nlerit con-
sideration. One such method is injecting a gas, say heliun_, into the feed
line. The gas bubbles "soften" the fluid column and also act as an absorp-
tion device whenever a disturbance causes a propagating pressure wave in
the line. The other idea is to insert a gas-filled plastic bag in the annulus
between two concentric lines. The inner line, which carries the propellant,
contains a series of holes, allowing contact between the propellant and the
plastic bag. This arrangement has the effect of making the feed line highly
compliant to any pressure surges.
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RE FERENCES
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5.
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Joukowski, N. Water Hammer. Proc. of Am. Water Works Asso.
(1904)
Allievi, L. Allgemeine Theorie {iber die veranderliche Bewegung des
Wassers in Leitungen. Julius Springer, Berlin, Germany (1909)
Symposium on Water Hammer, The A. S. M. E. (1933)
Parmakian, J. Waterhammer Analysis. Dover (1963)
Rich, G. R. Hydraulic Transients. McGraw-Hill (1951}
Morgan, G. W. "On The Steady Laminar Flow of a Viscous Uncom-
pressible Fluid in an Elastic Tube," Bul. Math. Bioph.,Vol. 14 (195Z)
Karreman, G. "Some Contributions to the Mathematical Biology of
Blood Circulation Reflection of Pressure Waves in the Arterial System, "
Bul. Math. Bioph., Vol. 14 (1952)
Morgan, G. W. and W. R. Ferrante. "Wave Propagation in Elastic
Tubes Filled with Streaming Liquid," J. Acou. So. of Am. (July 1955)
Blackburn, Reethof, and Shearer. Fluid Power Control. John Wiley
(1960)
D'Sonza, A. F. and R. Oldenburger.
Lines," J. Basic Eng. (Sept. 1964)
"Dynamic Response of Fluid
Regetz, Jr., J. D. An Experimental Determination of the Dynamic
Response of a Long Hydraulic Line. NASA TND-576 (Dec. 1961)
Jacobi, W. J. "Propagation of Sound Waves Along Liquid Cylinders,"
J. Acou. So. of Am., Vol. Z] (1949), pp. 120-127
Thomson, W. T. "Transmission of Pressure Waves in Liquid Filled
Tubes," Proc. of 1st U.S. Nat. Congress of App. Mech A.S.M.S. (1951)
Skalak, R. "An Extension of the Theory of Water Hammer,"
Trans, A.S.M.E. (Jan. 1956), pp. 105-116
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15.
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19.
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Sabersky, R. H. Effect of Wave Propagation in Feed Lines on Low-
Frequency Rocket Instability. Jet Propulsion (May - June 1951)
Streeter, V. L. and C. Lai. "Waterhammer Analysis Including Fluid
Friction," ASCE Trans., Vol. IZ8 (1963)
Contractor, D. N. The Reflection of Waterhammer Pressure Waves
from Minor Losses. A.S.M.E. Paper No. 64-WA/FE-16
Kochina, N. N. "On Self-lnduced Oscillations of Heavy Fluids in
Tubes," PMM, Vol 27, No. 4 (1963), pp. 609-617
Lister, M. "The Numerical Solution of Hyperbolic Partial Differential
Equations by Characteristics, " Mathematical Methods for Digital
Computers, Edited by A. Ralston and H. S. Will. New York:
John Wiley & Sons, Inc. (1960)
Hartree, D. R. Some Practical Methods of Using Characteristics in
the Calculation of Non-Steady Compressible Flow. Atomic Energy
Commission Rept. AECU Z713 (195Z)
Crandall, S. H. En_ineerin_ Analysis. New York: McGraw-Hill
(1956), pp. 355-368
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APPENDIX A. TRANSFER FUNCTIONS
Transfer functions for the outboard liquid hydrogen line on
the Saturn S-II engine
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Frequency for Outboard LHg Line, Frictionless and Without Pipe Vibration
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.oil
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Frequency for Outboard LH 2 Line, With Turbulent Resistance and
Without Pipe Vibration
- 56 -
SID 66-46-I
NORTHt
AMERICAN AVIATION, INC.__ SPACE and INFORMATION SYSTEMS DIVISION
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- 57-
SID 66-46-i
NORTH AMERICAN AVIATION, IN(:3. SPACE and INFORMATION SYSTEMS DIVISI'ON
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Frequency for Outboard LH Z Line, With Turbulent Resistance and
Including Pipe Vibration
- 58 -
SID 66-46-i
oNORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
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Figure A-5. The Ratio of Pressure Deviation at x = ,! to Pressure Deviation
at x = 0 for Outboard LH2 Line, Frictionless and Without
Pipe Vibration
- 59 -
SID 66-46-i
NORTH AMERICAN AV IATI. O N, INC. SPACE and INFORMATION SYSTEMS DIVISION
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Figure A-6. The Ratio of Pressure Deviation at x : g to Pressure Deviation
at x = 0 for Outboard LH2 Line, With Turbulent Resistance and
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- 60 -
SID 66-46-1
N(_RTH AMERICAN AVIATION, INC.SPACE and INFORMATION SYSTEMS DIVISION
I
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Pipe Vibration
-61 -
SID 66 -46-i
NORTH AMERICAN AVIATION ' INC. SPACE and INFORMATION SYSTEMS DIVISI'ON
t0
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Figure A-8. The Ratio of Pressure Deviation at x = i_ to Pressure Deviation
at x = 0 for Outboard LH 2 Line, WLth Turbulent Resistance and
Including Pipe Vibration
- 62 -
SID 66-46-!
N() RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
So0
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Figure A-9. The Ratio of Velocity Deviation at x = _ to Velocity Deviation at
x = 0 for Outboard LH2 Line, Frictionless and Without Pipe Vibration
- 63 -
SID 66-46-1
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVIS'ION
t.4
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at x = 0 for Outboard LH 2 Line, With Turbulent Resistance and
Without Pipe Vibration
- 64 -
SID 66 -46-1
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
t.O
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at x = 0 for Outboard LH2 Line, Frictionless and Including
Pipe Vibration
-65-
SID 66-46-I
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVIS'ION
4.11
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at x = 0 for Outboard LH 2 Line, With Turbulent Resistance and
Including Pipe Vibration
- 66 -
SID 66-46-I
_O'RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
APPENDIX B. VELOCITY AND PRESSURE DATA
Velocity and pressure time histories for the center liquid hydrogen
line on the Saturn S-II engine
-67 -
SID 66 -46-1
NORTH AMERICAN AVIATION, INC. SPACE and INFOI_MATION SYSTEMS DIVISION
St.l
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Impulse Disturbance 2 Percent of Steady-State Velocity
- 69 -
SID 66-46-i
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVIS'ION
St .I
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Impulse Disturbance 2 Percent of Steady-State Velocity
- 70 -
SID 66-46- 1
. N6RTH AM ERICAN AVIAT!ON, INC. _ _ SPACE and ][NFORMAT,OI_ SYSTEMS DIVISION
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Impulse Disturbance 5 Percent of Steady-State Velocity
- 71 -
SID 66-46-1
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
oUJO_
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NORTH AMERICAN AVIATION, INC.SPACE and INFORMATION SYSTEMS DIVISION
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SID 66-46- 1
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SID 66-46- 1
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SID 66-46-I
N O_R T H AMERICAN AVIAT!ON, INC. SPACE and INFOI:tMATION SYSTEMS DIVISION
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SID 66-46-I
NO_RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
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SID 66-46- 1
NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION
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SID 66- 46- I
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With Air Chamber Under Sinusoidal Disturbance, Amplitude
5 Percent of Steady-State Velocity, Frequency ll. 70 cps
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- I14-
SID 66-46-I
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With Air Chamber Under Sinusoidal Disturbance, Amplitude
5 Percent of Steady-State Velocity, Frequency 21. Ii cps
- 115-
SID 66-46- 1
NORTH AMERICAN AVIAT!ON, INC. SPACE and INFORMATION SYSTEMS DIVISION
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5 Percent of Steady-State Velocity, Frequency 21. II cps
- 116-
SID 66-46- l