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NASA
TECHNICAL
NOTE
•
NASA TN D-8430
!
t477-
Z
I--
18992
He,. _ o_
MF Ao I
UncJ.a-_
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\_ "_
REVISED
FORTRAN
CALCULATING
ON
THE
VELOCITIES
HUB-SHROUD
SURFACE
OF
AN
MIXED-FLOW
I - User's
Theodore
Lewis
.
FOR
AND
STREAMLINES
MIDCHANNEL
AXIAL-,
_.- _I
RADIAL-,
TURBOMACHINE
OR
STREAM
OR
ANNULAR
DUCT
Manual
Katsanis
Research
Cleveland,
PROGRAM
21299
Ohio
and
William
D.
McNally
Center
44135
NATIONAL AERONAUTICSAND SPACE ADMINISTRATION .
WASHINGTON, D. C. •
MARCH 1977
1, Report
"' 4.
No.
2.
NASA
TN D-8430
Title
Subtitle
and
REVISED
Government
FORTRAN
Accession
PROGRAM
No.
3.
Recipient's
CALCULATING
5.
Report
FOR
VELOCITIES
AND STREAMLINES
ON THE HUB-SHROUD
MIDCHANNEL
STREAM
SURFACE
OF AN AXIAL-,
RADIAL-,
OR MIXED-FLOW
TURBOMACHINE
OR ANNULAR
DUCT
I- USER'S MANUAL
6,
8.
Katsanis
and William
D.
Performing
Organization
Lewis
Research
National
and
McNally
Sponsoring
Ohio
Agency
Space
and
13.
Address
National Aeronautics
and Space
15,
D. C.
Supplementary
No
Unit
No.
Contract
or
Grant
Type
Report
No,
of
and
Period
Covered
Note
Administration
Sponsoring
Agency
Code
20546
NASA
TN
on the Hub-Shroud
D-7343
(FORTRAN
Mid-Channel
Flow
Program
Surface
for Calculating
Velocities and Streamlines
of an Axial- or Mixed-Flow
Turbomachine,
1973)
Abstract
A FORTRAN-I
V" computer
free
transonic
flow
solution
The
blade
may
be fixed
axial,
row
mixed,
supersedes
shroud,
or
velocities.
Transonic
on
D-7343.
is made
Subsonic
solutions
Meridional
stream
(Suggested
developed
and
is a revision
the
blades
Upstream
and
downstream
for
loss
angles
solutions
on the
are
obtained
stream
obtained
by
be
flow
a detailed
surface
twisted
surface
by
Distribution
STAR
leaned.
variables
and
may
well
or
shock-
as
Flow
this
vary
The
may
from
results
hub
mass
that
blade
so-
uses
information
Pages
22,
flow.
Statement
- unlimited
Category
01
flow; Radial-inflow
turbine; Centrifugal- compressor
19.
Security
Classif,
(of this
Unclassified
report)
20.
Security
Classif,
(of
this
Unclassified
page)
21.
No,
of
105
For sale by the National Technical Information Service, Springfield, Virginia 22161
to
include
stream-function
method
be
report
approximate
a finite-difference,
Unclassified
Mixed-
and
program
as
at a reduced
flow; Midchannel
subsonic
of a turbomachine.
pressure.
a velocity-gradient
18.
Transonic
may
of stagnation
solution
turbomachine;
obtains
stream
of a previous
to correct
by Author(s))
plane; Turbomachine
that
midchannel
rotating,
stream-function
surface, Axial-flow
flow turbomachine;
been
hub-shroud
program
flow
are
a finite-difference,
Words
or
and
has
the
This
streamlines,
lution.
Key
TN
provision
surface
from
program
radial.
NASA
and
velocities,
17.
Report
Notes
Supersedes
16.
Organization
Technical
14.
Washington,
Performing
Administration
44135
Name
Code
505 -04
11.
and
Organization
Work
Address
Center
Aeronautics
Cleveland,
12.
Name
1977
Performing
E -8734
10.
9,
No.
Date
Narch
7. Author(s)
Theodore
Catalog
Price"
A06
CONTENTS
Page
SUMMARY
_TRODUCTION
METHOD
BASIC
1
.......................................
....................................
OF ANALYSIS
...............................
ASSUMPTIONS
...............................
SOLUTION
BY COMBINATION
OF METHODS
SUBSONIC
STREAM-FUNCTION
SOLUTION
TRANSONIC
BLADE
VELOCITY-GRADIENT
SURFACE
OF PROGRAM
DESCRIPTION
OF INPUT
INPUT
....................
APPROXIMATE
VELOCITIES
APPLICATION
...................
SOLUTION
.........
..........................
............................
AND OUTPUT
10
.......................
11
........................................
Input
Special
Dictionary
Instructions
(a) Units
for
Preparing
of measurement
(b) Damping
(c) Hub and
factors
flow
mesh
(e) Upstream
and
blade
DNEW
channel
and
points
output
26
26
....................
geometry
flow
26
..................
27
data
flow
(j) Straight
cascade
conditions
thickness
for
(i) Incompressible
infinite
25
.....................
..............................
surface
for
and
downstream
(g) How to specify
(h) Requests
Input
...........................
FNEW
shroud
(d) Orthogonal
(f) Mean
17
..................................
spline
coordinates
curves
27
.................
28
...............
29
..................
29
..........................
31
............................
31
..........................
(k) Choosing
a value
for
REDFAC
.......................
31
(l) Choosing
a value
for
ANGROT
.......................
31
OUTPUT
31
.......................................
Printed
Output
..................................
32
Plotted
Output
..................................
52
Error
Messages
NUMERICAL
EXAMPLE
66
.................................
...............................
7O
iii
PRECEDING
PlaiCE BLANK
NOT
FILMED
APPENDIXES
A-
GOVERNING
B -
DERIVATION
OF
STREAM-FUNCTION
C o DERIVATION
OF
VELOCITY-GRADIENT
D-
LOSS
E -
DEFINING
F -
INCIDENCE
G-
BLADE
CORRECTIONS
AVERAGE
H-
SYMBOLS
REFERENCES
EQUATIONS
...........................
EQUATION
..............
EQUATION
..............................
REDUCED--MASSAND
73
DEVIATION
SURFACE
DENSITIES
VELOCITIES
FLOW
85
90
PROBLEM
CORRECTIONS
AND
............
77
................
.................
92
93
BLADE-TO--BLADE
.............................
95
....................................
98
....................................
101
iv
REVISED
FORTRAN
PROGRAM
STREAMLINES
ON THE
SURFACE
FOR CALCULATING
HUB-SHROUD
OF AN AXIAL-,
STREAM
DUCT
MANUAL*
Katsanis
Lewis
AND
OR MIXED-FLOW
OR ANNULAR
I - USER'S
by Theodore
MIDCHANNEL
RADIAL-,
TURBOMACHINE
VELOCITIES
and William
Research
D.
McNally
Center
SUMMARY
A FORTRA
N-IV computer
program
has been developed that obtains a detailed sub-
sonic or transonic flow solution on the hub-shroud
single blade row of a turbomachine.
duct without blades.
supersonic flow.
blade row may
mldchannel
stream
surface of a
A solution can also be obtained for an annular
The flow must be essentially subsonic, but there may be locally
The solution is for two-dimensional,
adiabatic shock-free flow.
be fixed or rotating, and the blades may be twisted and leaned.
flow may be axial, mixed, or radial. Upstream
and downstream
vary from hub to shroud, and provision is made
for an approximate
of stagnation pressure.
The
The
flow conditions can
correction for loss
Viscous forces are neglected along solution mesh
lines running
from hub to tip.
The present program
sedes NASA
nonaxial
any
flows
without
specified
ments
for
The
the
TN D-7343.
basic
tained
solution.
mation
that
The
part
II as
program
to use
of solution
and
includes
the
solution
manual.
program
listing
TN D-7343.
and
to allow
detailed
consists
of the
flow.
be obtained.
The
for
improve-
solution
with
report,
and
to illustrate
a detailed
part
part
of the
stream
solution
flow
at the full
mass
the equations
the
program
user's
contains
use
This
solution
mass
I,
of
the re is locally
transonic
I as the
solution
function.
When
at a reduced
It explains
example
,
NASA
This
as it is.
a numerical
to handle
blades,
numerous
subsonic
solution
in two volumes,
gives
without
stream-function
is reported
complete
Supersedes
must
a velocity-gradient
program
to make
equations
to obtain
the
ducts
function
to strictly
finite-difference
the programmer's
necessary
stream
of a finite-difference,
The
is used
and
finite-difference
a transonic
to extend
calculations.
on the
is limited
are
annular
distribution,
efficient
nonlinear,
by a combination
revisions
to handle
is based
however,
flow,
gradient
the program
primary
and
analysis
solution,
supersonic
The
loss
accurate
simultaneous,
basic
and this report super-
restriction,
streamwise
more
is a revision of a previous program
and
a velocity-
provides
involved
the
and
information
and
of the program.
procedure.
infor-
flow.
manual
all
is ob-
the
method
Part
II
INTRODUCTION
The
design
analyzing
unsteady,
Clearly,
est
of blades
such
tions
The
inviscid
are
just
require
surfaces
beginning
to obtain
are
stream
surface
dimensional
from
S1
most
a channel
many
solution
surface,
significant
variation
input
is needed
either
by the
on the
which
surface.
solves
The
tions
into
when
blade
passages
the
and
have
been
tary
or are
been
thoroughly
Research
reported
in the
of limited
tested
of stream
many
on a blade-to-blade
meridional
or midchannel
However,
(normal
and
when
three-
can
often
be obtained
to the
flow).
This
is
equation
above
1.
but
this
for
refined
For
such
the
program
as the
result
report
uses
cases,
equation
cases
on the
and
from
midchannel
with
of these
of extensive
method
subsonic
of both.
(velocity
The
described
in this
gradient)
finite-difference
methods,
both
combined
are
is very
usage
at the
method
is used
the finite-difference
in a way
to obtain
that
is
flows.
of a turbomachine
programs
herein
curved
promising
surface
solu-
a solution
encountered
most
stream
obtain
in obtaining
the
me-
same
can
to completely
reported
when
can be obtained
function
also
is limited
many
especially
or by a finite-difference
is difficulty
are
surface
is
consideration
velocity-gradient
2),
stream
Difficulties
solution
solution
in many
there
However,
The
the
is efficient
flowon
literature.
generality.
for
direction,
The
(ref.
S2
under
proprie-
general
NASA
and has
Lewis
Center.
orthogonal
2
a
a quasi-three-
either
information
solves
surface
blades.
and
are
of flow on the meridional
when the turbomachine
However,
programs
The program
tage
regime.
solution,
Finite-difference
they
involve
to analyze
to obtaining
surface).
which
method
low-hub-tip-ratio
the finite-difference
families
surfaces
in the hub-shroud
stream
quasi-orthogonal
are
usually
but
solution.
calculations.
method,
meridional
ratios
analyses
to
solu-
of computers,
on intersecting
surface
equations
is chosen
a finite-difference
aspect
and
inviscid
generation
inviscid
significant
in blade-to-blade
transonic
Three-dimensional
1) or on the
S2
in flow properties
use
flows
and fast-
1).
quasi-orthogonal
hub to shroud
ihod,
for
ref.
(Wu's
In this report
a solution
to the
carried
out.
This solution
surface
has
steady
solutions
cross-sectional
(fig.
only
approaches
important,
on a passage
is to analyze
at present,
for
a turbomachine.
largest
of two-dimensional
are
through
on the
the present
two-dimensional
methods
even
solutions
two blades
are
flow
a quasi-three-dimensional
there
(Wu's
effects
So,
choices
Most
between
a solution
called
several
them,
of revolution
with
requires
time,
solutions.
two-dimensional
solution.
surface
viscous
is called
viscous
at present
time.
ideally
at the present
to be obtained
what
of combining
dimensional
from
turbines
turbulent
approach
computer
there
and
impossible
usual
of several
Since
ways
are
solutions
excessive
combination
compressors
three-dimensional,
solutions
computers.
separate
for
takes
and
maximum
a subsonic-flow
the
quasiadvan-
solution.
Blade-,
Flow
\
]
,- Midchannel
surface S2
Orthogonal
channel
Blade
surface -,
L Blade-to-blade
surface S1
CD-I1362-01
Figure 1. -Two-dimensionalanalysissurfaces in a turbomachine.
The
velocity-gradient
into
the
method
transonic
This
blade
row,
stream
either
shock-free
and
rotating.
The
gram
for
conditions
arbitrary
a more
solution
for
this
TSONIC
or for
if necessary
or turbine,
vary
from
been
to extend
hub
the blade
The blades
may
row
written
the
range
to perform
mixed-,
or for
or
of solutions
The
duct.
solution
An approximate
is provided.
be twisted
and
leaned
these
radial-flow
an annular
to shroud.
flow.
through
turbomachine
Upstream
is for
correction
The blade
calcula-
can have
the
information
down-
compressible,
for
row
and
and
may
high
loss
of
be either
aspect
ratio
necessary
for
distribution.
obtained
purpose
has
for an axial-,
incompressible
blade-shape
by this
program
program
analysis
is TSONIC
is calculated
MERIDL
MERIDL
written
can
thickness
detailed
The
was
pressure
or
called
a compressor
flow
stagnation
fixed
program
program
flow
used
regime.
A computer
tions.
is then
reported
provides
on blade-to-blade
(ref.
and printed
also
3).
Information
surfaces
needed
(fig.
1).
to prepare
A useful
all
the
proinput
by MERIDL.
herein
is a revision
of the
program
described
in
references 4 and 5.
capability
prove
to handle
cases
the accuracy
tended
and
to handle
original
changes
are
radial
program
and
II):
been
extended
turbines
and
centrifugal
to handle
extended
to handle
(3) The
program
has
been
extended
to permit
of loss
within
for
hub-shroud
cisely
program
by a set
has
distribution.
been
modified
so that
of tangential
(circumferential)
gram
required
specification
of thickness
blade
section.
This
thickness
parallel
blade
surfaces
(5) If desired
specified
and
the
(6) Output
components
This
quantities
and
was
to give
MERIDL
flow
has
been
ex-
for
the
program.
The
(additional
internal
configurations,
the
added
duct
to specify
thickness
including
to the original
can be specified
The
camber
by blade
line
lean,
accurately
for
some
of some
blade
to station-line
output
to give
as absolute
and
messages
have
been
been
provided.
relative
total
pro-
on an input
and non-
blade
shapes.
tangent
specification
pre-
original
camber,
mean-camber-line
as well
blades.
an arbitrary
coordinates.
to the mean
influenced
without
is in addition
the blade
trailing-edge,
been
user
This
to specify
simplifies
static
the
thickness
was
and
have
form
to im-
satisfactory
to the program
an annular
normal
difficult
leading-
as input.
revisions
input
was
revised
nonaxial
the blades.
loss
normal
the
that
program's
compressors.
been
distribution
the
of the
and second,
input
made
has
(4) The
sity,
were
program
provision
be
for
(2) The
streamwise
extension
Although
any
satisfactory
revisions
first,
offered;
required,
in part
has
made:
of the program.
is still
extensions
were
originally
input where
documented
(1) The
those
efficiency
program
major
of changes
beyond
additional
MERIDL
following
both
Two types
angles
can
shapes.
absolute
velocity
temperature,
den-
and pressure.
(7) Several
informational
(8) Additional
error
(9) Upstream
both
the
and
convergence
(10) Interpolation
been
improved
(11)
have
downstream
boundary
conditions
and
the quality
of the
and
calculation
procedures
to give
Numerous
messages
better
small
convergence
changes
have
added
solution
and
been
to the
have
near
near
been
changed
these
boundaries.
the leading
and
smoother
made
output.
solutions
to improve
to improve
trailing
edges
in these
the
have
regions.
accuracy
and
efficiency
of the program.
The
MERIDL
IBM- TSS/360variables
ble
could
Storage
compiler
The
has
been
For
60 000 words
be easily
requirements
being
IBM-TSS/360-
4
67 computer.
required
storage
mesh.
program
used.
the numerical
for
reduced
for
Run times
implemented
on the
example
a 21 × 41 grid
for
the program
code
Lewis
of this
report,
of 861 points.
The
by equivalencing
the program
NASA
of variables
depend
range
the
storage
amount
or by using
on the
from
time-sharing
computer
of
of variaa coarser
system
3 to 15 minutes
and
on
67 equipment.
MERIDL
program
is reported
in two volumes,
with
part
I as the
user's
manual
and part II (ref. 6) as the programmer's manual. This report, part I, contains all the
information necessary to use the program. It explains the method of solution, describes the input and output, gives a numerical example to illustrate the use of the program, and derives the equationsused (in appendixes A to G.) Symbols are defined in
appendix H. Part II includes a complete program listing, detailed program procedure,
andappendixesthat derive the special numerical techniques used.
METHOD
OF ANALYSIS
BA SIC A SSUMP TIONS
It is desired
cascade
fying
to determine
of blades
are
used
relative
flow
(2) The
(3) The
fluid is a perfect
only forces
along
(5) The
trary
correction
surface,
is made
(7) The
relative
stagnation
(8) The
upstream
and
and
rotating
following
in obtaining
simpli-
a solution:
heat
mesh
Cp.
line are
those
due
to
surface
near
the
linearly
the
that
leading
and
free-stream
between
pressure
downstream
the
same
trailing
shape
edges,
as the
where
an arbi-
flow.
blade
loss
has
surfaces.
is known
boundaries
through
the
of the solution
blade
row.
region
are
orthogonal
streamlines.
The
flow
may
be axial,
tion temperature
may
vary
row
row.
The
blade
blades;
or
there
may
omitted
from
the
line
in the
are
with
neglected,
streamwise
forces
are
mixed,
from
may
or radial.
hub
to shroud,
be either
be no blades
Whirl,
fixed
at all.
both
stagnation
upstream
and
or rotating,
Within
pressure,
the
with
given
and
downstream
leaned
and
assumptions,
stagnaof the
twisted
no terms
are
equations.
In connection
viscous
is a stream
to match
varies
mesh
equations
The
or
is steady.
except
velocity
blade
surface.
gas with constant
specific
a hub-shroud
orthogonal
surface
(6) The
to the
stream
a stationary
transfer.
midchannel
mean-camber
the
through
gradient.
is no heat
blade
hub-shroud
to the blade
and pressure
distribution
in deriving
(1) The
(4) There
flow
on a midchannel
assumptions
momentum
ces
the
assumption
since
direction
considered
3,
the
these
are
forces
much
indirectly
viscous
forces
are
usually
larger.
The
by specifying
along
very
effects
a hub-shroud
small.
The
of these
a streamwise
orthogonal
viscous
for-
streamwise
total-pressure-
lo s s dis tribution.
5
SOLUTIONBY COMBINATIONOF METHODS
A flow analysis on the meridional stream surface can be obtained either by the
velocity-gradient methodor by the finite-difference method. The finite-difference
method is limited to subsonic flow; the velocity-gradient method is limited to relatively
low-aspect-ratio blades. The most accurate solution is obtainedby the finitedifference technique, so that this method is usedwhere possible (i. e., for subsonic
flow). With locally supersonic flow, the finite-difference solution is first obtained at
a reduced mass flow (appendix E) for which the flow field is completely subsonic. The
streamline curvatures andflow angles throughout the passagethat are obtained from
this solution provide the information necessary to obtain an approximate velocitygradient solution at full mass flow, regardless of aspect ratio.
SUBSONIC
The
stream-function
tion on a midchannel
known
(the
momentum
stream
equation
S TREAM-
equation
hub-shroud
function)
(eq.
(eq.
(A1),
stream
ref.
SOLUTION
appendix
surface
as a function
(96),
F UNCTION
A) is a partial
(assumption
of two variables
1) on what
he calls
5, p.
and
is derived
S2
surface.
an
_on
region
_"
is on this surface
Section A-A
c-Solution
Sh roud-_,
region
/
Flow
_
A
,- Downstream
boundary
U
boun
"_
Hub
Figure 2. - Solution
region.
differential
5).
equa-
It is in one
from
Equation
un-
Wu's
(A1) is
nonlinear
but
completely
can be solved
far
(A1).
(as indicated
the blade
for
can be
the
region
It is assumed
from
elliptic
On the
normal
that
upstream
the
sumption
upstream
and
solution
must
chosen
is an orthogonal
be used
the
shroud
curves
grid
then
(fig.
3).
This
technique
nary
differential
The
mesh
is not
flow
the
flow
is
(A1) is
subsonic,
function
the
sufficiently
Equation
on the
is zero
to be zero.
are
of equaare
solution.
stream
boundaries,
boundaries
solution
is entirely
function
is assumed
the
specified
of the
equation
entire
boundary
or its
normal
on the hub and
derivative
This
of the
to the
(A1)
of
de-
1 at the
stream
is equivalent
orthogonal
mesh
that
that
equal
fit through
the
to assuming
streamlines
(as-
orthogonals
equations,
8).
is in the
3.
mesh
used
increments
are
the
given
points
which
is known
distances
determined,
in part
II of this
as a coordinate
several
by
also
The
space
the
the
mesh
t-distance
the
(ref.
Note
orthogonals
mesh on solution
region.
the hub and
Spline
orthogonals
for
solving
method
are
is normal
6).
to
technique.
lines
finite--difference
was
used
lines.
Euler
_- Streamwise or "horizontal"
fi nite-difference
between
method
system.
Fiqu re 3. - Orthogonal
method
streamwise
improved
method.
that
a predictor-corrector
as the
report
of grid
hub-shroud
orthogonal
and
mesh
type
Rungo-Kutta
along
direction
The
to obtain
obtained
second-order
The
7.
along
to the
streamwise
With
are
The
finite-difference
by the program.
in reference
resulting
by the
equations.
is generated
reported
into
(ref.
(A1) is obtained
finite-difference
is analogous
in figure
orthogonal
values
of equation
the
follows
normal
method
s-distance
cated
for
is divided
are
Heun's
when
boundaries
on the
are
stream
downstream
boundary
the
conditions
the
for
downstream
effect
when
The
downstream
A grid
the
method
5).
numerical
generate
are
and
and
a negligible
boundaries.
to the
8, p.
The
upstream
boundary
conditions
four
function
the
2) is considered
Therefore,
proper
These
on all
shroud.
flow.
when
region.
that
in fig.
so as to have
subsonic
solved
rivative
by the finite-difference
subsonic.
A finite
tion
iteratively
s
equations
that
the
or
and
to this,
ordi-
t.
The
as indion the
orthogonal
The
finite-difference
linear.
sity
These
equations
is assumed;
omitted
for
ized.
tion
for
are
then
solved.
with
again
a final
each
step
The
method
an optimum
two levels
density
solved
and by iteration
For
equations
of the
used
iteration
to the
is internal
outer
iteration.
iteration.
After
the
cal partial
The
details
II (ref.
stream
tained
with
to the full
solved
there
region
from
only
the
velocity
function
and
and
by using
we have
and
The
is the
the
is found
equations
inner
user
it is always
technique
9)
iteration,"
to the
distribution
be
(ref.
iteration."
report,
programming
must
method,
"inner
apparent
in this
is subsonic.
an iterative
in the
iteration
is repeated,
overrelaxation
in the "outer
is mentioned
is locally
supersonic
supersonic
by using
and
flow.
outer
by numeri-
(A5) and
(A6).
described
in
are
the
with
there
The
SOLUTION
equation
region
that
flow.
However,
solution
flow,
in the
means
to subsonic
a reduced-flow
flow
APPROXIMATE
but is hyperbolic
conditions
supersonic
velocity-gradient
(A1) is no longer
of supersonic
will
flow
probably
method.
method
solution
the finite-difference
10).
cannot
can be ob-
method
This
(ref.
be shock
finite-difference
an approximate
velocity-gradient
equations
as an initial-value
varying
so the
stream
flow
equa-
technique
and
extending
is described
3.
any given
8
entire
locally
in reference
(eq.
where
by getting
The
made
VELOCITY-GRADIENT
the boundary
in going
be used
this
case
changes
losses
are
to obtain
6).
in the
This
is also
linear-
The
process
are
solu-
is used
equations
is successive
this
entirely
terms.
finite--difference
is performed
is obtained,
nonlinear
den-
terms
approximate
that
if the
overrelaxation
procedure
other
an initial
are
first
can be obtained
Since
terms
the
(A1) is non-
nonlinear
equations
This
equations
factor.
of the
remaining
solution.
the
equation
iteration
information
an improved
solution
original
to obtain
of the
to solve
function
The
provides
the linearized
of the numerical
the
elliptic
an estimate
an iteration
TRANSONIC
For
and
to the program,
differentiation
part
solution
nonlinear
When
terms.
iteration
The
the
On the first
solved
This
overrelaxation
the corrections
then
to obtain
of this
since
the finite--difference
are
converged
of iteration.
of the
so that
function.
nonlinear
iteratively.
some
iteration
stream
are
solved
linearizes
first
estimate
tions
can be
linearized
the
better
this
the
These
equations
vertical
values
(A12))
will
problem,
mesh
of
W
are
line
running
at the hub,
be found.
When
equations
(A7) to (All).
where
the velocity
from
hub to tip.
a solution
equation
satisfying
(A7) has
been
Equation
W is specified
By finding
the
several
specified
solved,
mass
subject
(A7) is
at the
hub for
solutions
for
flow
to giving
the
a
correct mass flow, for every hub-shroud mesh line in the region, the entire velocity
distribution at full mass flow has been obtained.
BLADE SURFACEVELOCITIES
The solution that is obtained by either the finite--difference or velocity-gradient
method is for the midchannel surface betweenthe blades. Whenthere are blades, the
blade surface velocities are of greater interest. These can be estimated since the
blade loading is dependenton the rate of changeof whirl. By assuming a linear variation of velocity betweenblade surfaces, equation (A13) can be derived for calculating
the blade surface velocities.
APPLICATION
The
for
program
design,
other
reference
and
programs
11 describes
thicknesses
using
be made
changes
file,
may
inlet
flow
more
than
outlet
whirl
distribution,
stream
on various
TSONIC
is calculated
ence
For
both
useful.
from
loss
solution
in blade
more-detailed
3).
Most
and printed
how
shape
compressors,
flow
for
depends
coordinates
can be
in detail.
checked
Usually,
distribution.
example,
distribution
stream
of the
to obtain
have
by
changes
These
hub and
may
on the
shroud
pro-
to be changed.
accuracy
MERIDL
surface
and
use
of the bound-
on incidence
analysis
program
to compute
12 of the
from
MERIDL.
be considered
at this
output
and deviation
for
on the
can be obtained
Item
this
may
A useful
required
by MERIDL.
flow
velocities
surfaces.
information
distribution
information
by the
blade
directly
or whirl
VII) gives
is achieved
blade--to-blade
(ref.
50) explains
12 (ch.
off-design
flow pattern
surface,
is TSONIC
(p.
MERIDL
design
distribution
shape;
axial
it is used
used.
purpose
changes
of the
and
For
blade
a desirable
the blade
When
mean-camber-line
This
flow
tool.
program.
blade
blade.
the
as a design
this
give
to achieve
just
and
with
will
to analyze
involve
analyses
Output
that
design
a reasonable
midcharmel
be used
blade
conditions
analysis
compressor
program
the accuracy
When
for
should
an axial
to the
and
Of course,
both
a program
for
the MERIDL
must
ary
can be used
OF PROGRAM
good
for
this
input
section
by
for
Printed
Further
time.
designs
Referand for
conditions.
cases
where
blade
The
the flow
to blade
CHANEL
and
program
is well
from
hub
obtains
guided
in the
to shroud,
a solution
channel
the
but has
CHANEL
on a channel
large
program
cross-section
variations,
(ref.
13) is
surface.
9
The CHANEL program is particularly useful for calculating choking mass flow through
a blade row.
DESCRIPTION
The
the blade
shown
principal
row
coordinates
of input
to be analyzed.
in figure
shroud.
block
Each
and
4.
blade
The
blade
section
two sets
OF INPUT
required
The
for
geometry
shape
is given
is described
of blade-shape
AND OUTPUT
the program
is given
in cylindrical
as a series
by a set
coordinates
of
is a geometric
z
coordinates,
of blade
and
on any
r
sections
general,
w;
w2.W m+
10
of
as
from
hub to
mean-camber-line
z
Figure 4. - Cylindrical
description
coordinate system and velocity components.
smooth
surface
of
revolution.
metric
Other
description
such
as mass
tion
inputs
of hub
flow
mesh.
There
Ouptut
is given
points
include
Output
row,
or
consists
At station
speed,
are
also
variables
input
at any or
more
and
principal
mesh,
and
output
gas
conditions,
a geo-
operating
conditions
constants,
of the finite-difference
the
desired
geometric
(2) along
blocks
lines
from
and
including
hub
to shroud
data.
(1) on all mesh
streamlines
velocity
density,
solu-
of output
locations:
user-designated
r-coordinates
is given,
flow
a description
user-designated
of z-
downstream
to indicate
all of three
solution
(3) along
and
appropriate
rotational
mainly
lines
shroud,
and
of the orthogonal
the blade
and
upstream
through
(station
components
lines}.
and
temperature,
angles.
and pressure.
INPUT
Figure
5 shows
input
data
card
may
be put
in the
All
numbers
the
integers
The
stream
flow
Next,
tangential
velocity
total
pressure
Finally,
a loss
distribution
the
Input
bles
are
is given
and
in the
are
for
punched
problem
on the
data
cards.
identification.
The
first
Any information
card.
three
input
cards
point)
in a five-column
on all other
beginning
field
data
with
(fig.
cards
MBI,
5}.
are
LSFR,
These
real
must
and IMESH
be all
numbers
right
(punch
are
ad-
decimal
fields.
that
several
First,
options
the
user
function
{SFIN
the
user
can
specify
(VTHIN
and
VTHOUT}.
(PROP}
for
specifying
can specify
conditions
or as a function
either
whirl
At the
loss
within
is not given,
these
upstream
the
and
downstream
the loss
input
may
as a func-
(LOSOUT}
is assumed
to vary
and
or absolute
station
be specified.
down-
(RADIN
LAMOUT}
pressure
blade
either
of radius
(LAMIN
of total
and
either
may
If the
linearly
abso-
be given.
loss
between
edges.
are
both
in figures
section
(PERLOS}
the blade
trailing
exist
and SFOUT)
or fractional
distribution
variables
shown
serves
of this
variables
through
leading
which
as they
on the
conditions.
stream
variables
a title,
5 indicates
of the
RADOUT).
lute
input
input
80 columns
in 10-column
Figure
tion
is for
(no decimal
justed.
point}
the
geometric
6 to 10.
Special
and nongeometric.
Further
Instructions
information
for
Preparing
The
geometric
concerning
Input
(pp.
the
input
input
varia-
variables
25 to 31).
11
12
REPRODUCIBILITY
ORIGINAL
PAGE
OF
THE
IS POOR
Z
o
._="
Z
0
m
m
I
8
o
A
.-J
J
Z
Z
0
o
w
Q_
Z
o
Z
Z
o
A
._J
0
Z
v
i
I.--
o
o
0
.-.I
.--J
Z
Z
Z
o
o
A
m
m
_
m
_
m
/
A
_
I_
m
m
I---
Z
0
o
Z
0
Z
0
Z
I-V'I
I-I_
Z
Q
I--0
_
A
_
wA
1
_
_
_
0
z
Z
0
Z
Q
_A
....1
N
0
._J
I:_
0
._J
C.)
N
I
saP_lq 0u aJe a JeLl} _! spJe:_ asaq} l!tuO
13
[i] Hub or shroud spline point
0 Hub point where orthogonal
NBLPL = 6 -_
E3
mesh begins,
_
i
L
I
RI"_P
-
_
I
I....
\ ,'
-1'
L
_Jz_
Figure 6. - Input variables -'hub,
..,-
ends, or
changes spacing
ONTIP=7
ZTOUT
section
input
t
'see fig" lO)
shroud,
_Zo#oHuUTB=ll
and blade sections.
1
OUT = 5
-, T
SFo[_RADINI
T';' _u,m[
?
_
LAMIN I 71
or
I
1
I
I
/
_FSFOUT
I
)
I
J
I_
_
RADIN
R_IN I
|
_-'-zoMom/
_
RTOUT
RADOUTI
ZOMIN-,,./
"-_ZHIN
=
RADOUT
-_-(_
_PROP or LOSOUT
' _LLAMOUT
or VTHOUT
/
l] I
"
or
,.OUT
ZHOUT--
-_
,
'
Z
Figure 7. - Input variables - upstream and downstream
14
flow variables.
0 Hub point where orthogonal mesh begins.
ends. or changes spacing
MHT = 9
J
f
8
,- Downstream
boundary of
7
-4-
orthogonal
mesh
Upstream boundary
of ortho, )nal mesh--,
I
MM = 20
3
ROMOUT
IMBI = 5
ROMBO
-_--- 70MI N-_
-.,,
-_ZOMB
,._
ROMB! ZOMBO
t
÷
ZOMOUT_
I=Z
ROMIN_ _
Figure 8. - Input variables - orthogonal
=
NSL = 8
mesh.
ZTST
FLFR(8I = 1.0
I
I
FLFR(T)
II
I
FLFR(6)
I
1
I
I
1
I
L
E
I
I
t _--I-_
I
I
I
I
FLFR 131
I
I
FLFR(2)_
I
RTST
I
FLFR(1) - O. "__RHST
=
ZHST
=
Figure 9. - Input variables - locating streamlines
and station
lines for output.
15
E
u
%
u
#
A
E
A
E
Z
?
E
N
v
|
m
i
A
u
i
c:l_
ql
16
.g
_4
Input Dictionary
The input variables are described in terms of a consistent set of SI units: newtons,
kilograms, meters, joules, kelvins, and seconds. The program, however, will run
with input in any consistent set of units.
The input variables, in the order that they appear in figure 5, are the following:
GAM
Specific-heat ratio, y
AR
Gas constant, J/(kg) (K)
MSFL
Total mass flow through entire circumferential annulusof blade
row, kg/sec
OMEGA
Rotational speed, w,
is in opposite
REDFAC
Factor
VELTOL
direction
by which
assure
mass
subsonic
left
blank,
tion
(k),
p.
flow
throughout
tolerance
w
flow
a value
is negative
in fig.
must
if rotation
4.
be reduced
passage.
of 1.0
on maximum
all mesh
may
in order
REDFAC
will be used.
change
for
reduced-mass-flow
blank,
and
a value
be left
Whatever
mum
of FNEW
and
used
by the program.
0.01
velocity
points,
the program.
tolerance,
shown
(MSFL)
case
that
to
may
See
be
sec-
31.
over
VELTOL
Note
of that
flow
in which
Convergence
ation,
rad/sec.
value
DNEW
before
A value
is a medium
will
be used
is multiplied
it is printed
and
of 0. 001 for
tolerance,
by the
0.1
by
mini-
subsequently
VELTOL
and
iter-
solution.
of 0.01
is given
in each
is a tight
is a loose
toler-
ance.
FNEW
DNEW
Damping
factor
A value
of 0.5
in which
case
tion
p.
(b),
Damping
will
FNEW.
Number
mesh
the
for
program
Ft
from
FNEW.
will
use
iteration
to iteration.
FNEW
a value
may
of 0.5.
be left
See
blank,
sec-
26.
on calculation
to iteration.
use
of
is suggested
factor
iteration
MBI
on calculation
a value
See
of 0.5.
section
of vertical
(ZOMIN)
MBI = 0 if there
of
DNEW
(b),
mesh
to point
are
DNEW
p.
a(rV0)/at
may
within
be left
does
blank,
not have
blade
and
row
from
the program
to be equal
to
26.
lines
of first
no blades.
from
left
mesh-size
See
fig.
boundary
change
8 and
of orthogonal
(ZOMBI).
section
(d),
Use
p.
27.
17
MBO
Total number
of vertical mesh
lines from
nal mesh
(ZOMIN)
Use
= 0 if there are no blades.
MBO
to point of second
left boundary
mesh-size
of orthogo-
change
See fig. 8 and
(ZOMBO).
section
(d),
p. 27.
MM
Total number
of vertical mesh
of orthogonal
mesh
fig. 8 and section
MHT
Total number
thogonal
NBL
Number
NBL
N-HUB
Number
maximum
of 50.
Number
hub
to shroud
of blade
and
row.
TIP,
PRIP,
RIqUB
See
of or-
Use
and
RTIP
arrays,
maximum
(c), p. 26.
LAMIN,
PROP,
arrays
of flow properties
VTHIN),
maximum
of blade planes
arrays
LOSOUT,
See fig. 7 and
of 50.
maximum
of 50.
on which
blade
See fig. 6 and
of data points per blade
maximum
VTHOUT),
(e), p. 27.
or blade sections
mean
of flow proper-
LAMOUT,
section
etc. ) are given to describe
etc., arrays,
maxi-
(e), p. 27.
RADOUT,
of 50.
arrays,
(c), p. 26.
of data points given in downstream
thickness,
RBL,
of I00.
See fig. 8 and section (d), p. 27.
See fig. 6 and section
RADIN,
maximum
Number
from
of data points given in upstream
ties (SFOUT,
NPPP
of 100.
See fig. 6 and section
Number
RBL,
spaces
of spline points given in ZTIP
Number
Number
maximum
in total circumference
See fig. 7 and section
NBLPL
mesh
of spline points given in ZHUB
(SFIN,
NOUT
to ZOMOUT),
= 1 if there are no blades.
of 50.
NIN
mesh,
left to right boundaries
(d), p. 27.
of horizontal
of blades
mum
NTIP
(ZOMIN
lines from
data (ZBL,
shape
section
and blade
(f),p. 28.
section or blade plane
of 50.
in ZBL,
See fig. 10 and sec-
tion (f),p. 28.
NOS TA T
Number
ZHST
of stations from
and
ZTST)
at which
See fig. 9 and section
which
case
18
Number
NOSTAT
maximum
may
should be included
in
of 50.
be left blank,
for ZHST,
in
ZTST,
arrays.
of streamlines
in FLFR)
at which
NSL
be left blank,
may
(located by coordinates
output is desired,
(h), p. 29.
no input cards
RI-IST, and RTST
NSL
hub to shroud
from
hub to shroud
output is desired,
in which
(designated
maximum
case no cards
of 50.
by values
See fig. 9.
should be included
for
FLFR
array.
If NSL is left
equal
to 11 and
print
lines
that
by 10 percent
vary
blank,
requested
theprogram
streamline
of total
will
output
flow
set
it
on 11 stream-
(i. e.,
0,
10,
20,
...,
100 percent).
NLOSS
Number
of points
maximum
use
fractional
of 50.
If NLOSS
loss
distribution
a linear
case,
at which
no input
cards
is zero
loss
(PERLOS)
or left
blank,
in the
should
the program
streamwise
be included
for
is specified,
will
direction.
PERCRD
In this
or PERLOS
arrays.
LSFR
Integer
(0 or 1) indicating
ditions
LTPL
are
Integer
given
Integer
loss
(0 or
Integer
is necessary
tion
LBLAD
(/),
Integer
p.
(0,
are
given
and
normal
mean
fig.
0 (THBL)
given.
and
lower
blade
section
ZOMBI
angle
z-Coordinate
6 to 8 and
(ANGROT)
fit curves.
m.
change
section
no blades
blade
given.
tangential
= 2, upper
for
02
See
sec-
coordinates
shape
O (THBL)
If LBLAD
= 1,
0-thickness
blade
t0/r
surface
(TH2BL)
leading-
all
if LROT
and
is (0).
See figs.
6 and
p.
spacing
27.
Set
See
boundary
of vertical
in mesh
(d),
streamwise
of left
of intersection
first
and
whether
of intersection
z-Coordinate
for
whirl
are
01
given.
See
28.
used
card
hub profile,
where
downstream
trailing-edge
mean
file and /_te (BETALE
and BETATE)
fig. 10 and section
(f), p. 28.
of axis
this
mean
are
surface
(f), p.
1) indicating
Omit
with
(0)
as input.
two blade--shape
= 0,
If LBLAD
blade--shape
angles
specified
(1).
See
ZOMIN
(1).
pressure
rotation
spline
which
t n (TNBL)
shape
(0 or
deg.
and
coordinate
If LBLAD
thickness
are
Rotation
total
radius
as input.
shroud
1, or 2) indicating
10 and
Integer
whether
hub and
as input.
(TH1BL)
A NGROT
(1) is given
(0) or
(1) is given
upstream
flow con-
31.
blade
(TTBL)
LETEAN
(1) for
pressure
whether
(0 or 1) indicating
function
downstream
of stagnation
velocity
and downstream
of stream
whether
1) Indicating
(0) or tangential
LROT
upstream
as a function
(0 or 1) indicating
or fractional
LAMVT
whether
8 and
spline
section
ZOMBI
fit curves,
(l),
P.
of orthogonal
section
mesh
occurs
are
line
(MBI),
and
mesh
(d),
with
m.
ZOMBO
31.
p.
27.
hub profile
See
figs.
to zero
case.
19
ZOMBO
z-Coordinate of intersection of vertical mesh line with hubprofile where secondchangein mesh spacing occurs (MBO), m.
Seefigs. 6 and8 and section (d), p. 27.
ZOMOUT
z-Coordinate of intersection of right boundary of orthogonal mesh
(MM)with hub profile, m. Seefigs. 6 and8 and section(d), p. 27.
ItOMIN, ROMBI,
IIOMBO, and
ROMOUT
r-Coordinates corresponding to ZOMIN, ZOMBI, ZOMBO, and
ZOMOUT, m. Leave these spacesblank if LROT = 0.
ZI[ [,'13
Array
of z-coordinates
boundary
Illt UB
Array
of flow
Array
of flow
Array
ZttIN
of flow
z-Coordinate
stream
blank
(e),
SFIN
blank
(LSFR
= 1),
SFIN
RADIN
p.
spaces
Array
TIP
Array
from
given,
2O
26.
section
(c),
shroud
or
section
(c),
this
p.
26.
top
p.
26.
on which
entire
up-
card
andMBI¢0.
See fig.
7 and
with
shroud
of line
on which
are
profile
corresponding
to ZHIN
and
ZTIN,
if LROT
If RADIN
= 0.
and
RTIN
of stream
line
when
of absolute
cannot
function
on which
when
on which
K.
Leave
See
LSFR
of r-coordinates
hub
p.
or top
of line
m.
RHIN
is given
is given
m.
(c),
fig.
7 and sec-
27.
along
shroud
6 and
given,
of values
shroud
given,
LAMVT=0,
conditions
r-Coordinates
Array
are
26.
shroud
defining
fig.
p.
27.
flow
(e),
See
6 and
with hub profile
of intersection
upstream
IttlI N, RTIN
p.
z-Coordinate
tion
m.
conditions
if LSFR=0,
section
ZTIN
channel,
section
defining
points
(c),
hub or bottom
6 and
See fig.
or bottom
section
defining
points
of input
hub
6 and
See fig.
m.
of intersection
flow
points
m.
channel,
defining
See fig.
of input
of r-coordinates
boundary
m.
channel,
of flow
points
of input
of z-coordinates
boundary
tlTIP
channel,
of r-coordinates
boundary
ZTIP
of input
upstream
= 1.
total
to shroud
See fig.
points
flow
See
flow
fig.
7 and
temperatures
on which
7 and
upstream
section
(e),
points
other.
from
conditions
along
line
are
section
section
27
given.
(e), p. 27.
from
hub to
given,
(e),
hub to
are
m.
p.
T!1 at input points
flow conditions
p.
these
as input
7 and
conditions
Leave
to each
input
See fig.
of input
LSFR
for
upstream
= 0.
is given
be equal
m.
RADIN
27.
along
are
line
PRIP
Array
of absolute
from
hub to shroud
N/m 2.
LAMIN
total
Array
See
fig.
of values
pressures
p_
on which
7 and
upstream
section
of absolute
at input
(e),
whirl
from
hub
to shroud
m2/sec.
tion
VTHIN
LAMIN
(e),
Array
on which
p.
is given
flow
p.
conditions
at input
v/
line
are
given,
points
along
line
i
upstream
when
along
27.
(rV,_
\
points
flow
LAMVT
conditions
= 0.
are
See fig.
given,
7 and
sec-
at input
i
upstream
flow
con-
27.
of values
of absolute
tangential
velocity
(V_]
\t,/
points
along
line
are
given,
ditions
See fig.
ZHOUT
7 and
z-Coordinate
blank
and
section
SFOUT
RADOUT
Array
= 1),
LOSOUT
Leave
this
m.
m.
corresponding
to ZHOUT
and
blank
= 0.
entire
See
fig.
7
27.
conditions
are
if LROT
RHOUT
and
RTOUT
of stream
is given
section
given,
are
given,
SFOUT
given,
on which
profile
along
Array
from
of line
shroud
of values
line
of line
See
fig.
when
on which
of input
when
for
= 0.
downstream
is given
cannot
on which
7 and
LSFR=
See fig.
points
flow
1.
points
flow
along
sec-
Leave
as input
to each
from
conditions
7 and
line
conditions
See fig.
m.
is given
be equal
input
downstream
LSFR
ZTOUT,
If RADOUT
function
on which
of r-coordinates
RADOUT
= 1.
hub profile
with
shroud
Array
with
andMBI_0.
p.
LAMVT
27.
spaces
shroud
PROP
p.
when
27.
conditions
flow
r-Coordinates
(LSFR
p.
of intersection
(e),
these
(e),
is given
LAMVT=0,
(e),
z-Coordinate
RTOUT
section
on which
VTHIN
if LSFR=0,
downstream
RHOUT,
to shroud
m/sec.
flow
card
tion
hub
of intersection
downstream
ZTOUT
from
are
7 and
other.
hub
to
are
given.
section
from
hub
(e),
p.
27.
to
given,
m.
section
(e),
p.
27.
of absolute
total pressures
p_ at input points
along line
hub to shroud
on which downstream
flow conditions
are
N/m 2.
(e),
p.
PROP
is given
when
LTPL
= 0.
See
fig.
7 and
27.
Array of fraction of absolute totalpressure
loss (p'
o, id - p_)/
p'
at
input
points
along
line
from
hub
to
shroud
on which
o, id
downstream
flow conditions
are given.
LOSOUT
is given when
LTPL
= 1.
See fig.
7 and
section
(e),
p.
27.
21
For the case without blades, omit input cards for the variables LAMOUT through
BETATE, inclusive.
LAMOUT
Array of values of absolute whirl (r)V0 o atinputpointsalongline
from hub to shroud on which downstream flow conditions are
given, m2/sec. LAMOUT is given when LAMVT = 0. See fig.
and
VTHOUT
section
Array
(e),
of values
points
along
conditions
LAMVT
ZBL
p.
are
m/sec.
See fig.
7 and
first,
followed
by successive
array
(f),
p.
line,
when
array
blade
blade
is given
shroud.
corresponding
m.
See
to ZBL,
fig.
6 and
TTBL
face
coordinates,
section
(LBLAD
sec-
or the
in
of
normal
to mean
coordinates,
surface
m.
TNBL
suction
(LBLAD=0).
tangential
and pressure
(LBLAD
and
it is feasible
array
the
cam-
When
curvatures,
surfaces,
either
and TH2BL
10 and
1.
for
to give
fig.
circumference.
blade
thicknesses
= 2).
thickblade
See fig.
28.
Two-dimensional
array
thicknesses,
corresponding
is the blade
See
origin
the
blade
and pressure
= 1),
The
RBL
to ZBL,
0 is positive
thicknesses
small
THIBL
(f), p.
4).
is 0 or
to ZBL,
normal
rad.
around
LBLAD
lean,
corresponding
coordinate
(fig.
it is recommended
ness,
TTBL
(f),
2 to 50) of
section
up to the
surface,
of blade
suction
Otherwise
22
rotation
only
near-parallel
and
blade
tangential
corresponding
the
section
(from
hub
surface,
can be anywhere
is little
to use
The
of positive
is given
there
mean
28.
Two-dimensional
ber
The
of 0-coordinates,
describing
0-coordinates
THBL
blade
describing
10 and
by a series
sections
mean
27.
28.
Two-dimensional
direction
6 and
flow
when
of points
of r-coordinates,
describing
section
p.
at input
downstream
is given
(e),
hub to shroud.
array
p.
section
is described
from
(f),
VTHOUT
See figs.
sections
of points
TTBL
m.
(V0),,
on which
of z-coordinates
surface
Two-dimensional
velocity
hub to shroud
given,
surface,
This
tangential
blade
tion
TNBL
from
array
blade
of points
THBL
line
= 1.
28.
27.
of absolute
Two-dimensional
mean
RBL
p.
7
of blade
tangential
tangential
to ZBL,
thickness
(circumferential)
RBL
coordinates,
in meters,
rad.
divided
by
sur10(a)
RBL.
TTBL
section
THIBL
(f), p.
array
only
when
array
corresponding
given
only when
Array
BETATE
Array
0.
ZHST
Array
See
RttST
Array
ZTST
Array
with
See
RTST
Array
FLFR
fig.
9 and
array
Array
have
there
leading
NSL = 0),
values
of streamwise
0.
and
are
1.
for
blades,
and
Omit
trailing
p.
= 0.
Omit
section
this
(h), p. 29.
output
card
sta-
if NOSTAT
9and
this
card
If no cards
are
0.0,
will
= 0.
this
card
the
first
and
0.1,
section
if NSL = 0.
See
for
0.9,
When
no blades,
between
the
are
upstream
1.0.
should
respectively.
distance
and
the
to the
PERCRD
is the fractional
there
FLFR
assign
corresponding
values,
(h), p. 29.
along
...,
= 0.
this
streamlines
given
0.2,
if NLOSS
last
Omit
automatically
distances
Omit
array.
designating
program
to FLFR:
distance
this
Seefig.
Omit
29.
the
edges.
array.
9and
to ZTST
function
PERCRD
sta-
p. 29.
fractional
array.
output
of hub-shroud
if LROT=0.
(h),
if
if NOSTAT
to ZHST
corresponding
0or
card
28.
card
See fig.
m.
is to be printed.
is the fractional
input
profile,
(h),
deg.
29.
of intersections
of stream
(i. e.,
PERLOS
p.
if
angles,
this
(f), p.
this
deg.
card
of hub-shroud
corresponding
section
followingll
PERCRD
(h),
section
output
= 1; omit
Omit
28.
28.
tangency
section
if LROT=0.
NOSTAT=
which
m.
this
is
(f), p.
angle,
(f), p.
of intersections
shroud
of values
and
TH2BL
section
= 1; omit
28.
(fig.
rad.
and
is
(f), p.
coordinates
tangency
section
TH1BL
and section
10(c)
LETEAN
10(d)
of r-coordinates
Array
fig.
and
only when
section
9 and
See
10(d)
of z-coordinates
cardif
and
(fig.
rad.
coordinates,
LETEAN
hub profile,
with
RBL
mean-camber-line
NOSTAT=0or
fig.
surface
when
of r-coordinates
tion lines
blade
See fig.
9 and
cardif
of lower
only
of z-coordinates
fig.
10(c)
See fig.
0.
tion lines
10(b)
coordinates
coordinates,
mean-camber-line
is given
LETEAN=
See fig.
surface
See fig.
= 2.
of trailing-edge,
BETATE
= 1.
= 2.
LBLAD
is given
LETEAN=
RBL
to ZBL,
of leading-edge,
BETALE
blade
to ZBL,
LBLAD
Two-dimensional
(10(c))
LBLAD
of upper
corresponding
given
BETALE
only whe_
28.
Two-dimensional
(10(c))
TH2BL
is given
When
between
the
PERCRD
downstream
stations.
23
PERLOS
The
Array of fractional loss distribution in the streamwise direction
within the blade, or between upstream and downstream input stations when there are no blades. Omit this card if NLOSS= 0.
remaining
what
output
seven
the
of this
analysis
For
iterations
would
and
the
final
the
A zero
give
to the
on the
the final
is really
the output
in any of these
called
print
for
A large
input
are
mass
to 1.,
used
although
flow by
REDFAC
each
and
The
a transonic
after
to indicate
analysis.
to obtain
integer
results
analysis
by
reduced-mass-flow
given
associated
indicates
iteration.
integer
will
should
the
iter-
mesh
point
the
obviously
not to call
associated
will
does
the
for
first
each
in
not give
more
with
for
cause
solution,
only
that
A 3,
integer
transonic
give
with
iteration.
however,
of
to be
associated
on every
after
multiple
variables
output
ITSON,
output
the
Any nonzero
be used
gives
the
the
the output
and
indicates
with
that
iterations.
Care
list
the
iterations
at other
following
IMESH,
a stream-function
third
last
iteration.
The
the
or plot
on every
iteration.
converged
useful.
wishes
and
0.1
solution.
ISUPER,
output
with
flow
except
on the first
first
mass
from
1.
can be obtained
A 1 will
the
output
full
values
reach
with
transonic
user
not
reduces
Output
variables
to be given
addition
output
after
with
increasing
beginning
iteratively
used
is to be omitted.
example,
output
then
all these
need
variables,
methods.
also
have
The program
are
or plotted.
variable
value
final
problem
at which
printed
the
integer
resulting
velocity-gradient
and
should
is desired.
solves
ation
PERLOS
iteration
output
than
of these
variables:
IMESH
Major
output
point
at every
indices
and
components
ISLINE
coordinates,
of the
and magnitude,
critical
not requested,
output
called
for
by ISLINE
or ISTATL
along
streamlines
(indicated
where
streamlines
mesh.
Output
velocity
and
are
includes
components
streamline
region,
z,
and
curvature.
an estimate
except
crossed
r,
magnitude,
Where
of suction-
for
m
ratio,
debugging
the
relative
flow
velocity
purposes,
lines
streamline
cases.
at each
point
of the orthogonal
ratio,
passes
and pressure--surface
This
since
coordinates,
velocity
mesh
angles.
in FLFR)
streamline
critical
is,
in most
by values
mesh
that
and
is preferable
by vertical
and
value,
velocity
is usually
output
mesh,
stream-function
output
Major
orthogonal
relative
flow
within
angles,
the blade
velocities
is also
at locations
specified
printed.
ISTATL
Major
output
by ZHST
FLFR
tion
24
and
along
ZTST
array.
that
absolute
station
arrays
Output
lines
from
and at values
corresponds
velocity
hub to shroud,
components,
of stream
to that
given
as well
function
for
ISLINE
as density,
specified
with
the
pressure,
by
addiand
temperature for static conditions and for absolute and relative stagnation
conditions are printed. ISTATL should be zero if NOSTAT is zero.
1-PLOT
Plotting indicator requesting output to be plotted on microfilm. Any nonzero value in IPLOT will causethe input data and generatedorthogonal
mesh to be plotted. Also, at each iteration that is a multiple of IPLOT,
streamlines will be plotted and meridional and surface velocities will be
plotted for each streamline value. These will also be plotted after the
final transonic solution.
ISUPER
Integer (0 or 1) indicating which solution (subsonicor supersonic) of the
velocity-gradient equation is desired. If ISUPER= 0, only the subsonic
solution
will
solutions
ITSON
Integer
for
the
when
TSONIC
given
for
put.
Usually,
this
TSONIC.
user
0, no extra
on the
mesh,
after
and
LAMVT.
arrays
with
plotted
input
the
arrays
downstream
should
cubic
for
for
Also
supersonic
value
use
in calculating
input
= 0, no information
TSONIC
will
be listed
data
after
will
with
ITSON
(i. e.,
> 20)
debug
output
is desired.
be
other
subsonic
for
is printed.
mesh
as well
each
If IDEBUG
and
printed.
Fifteen
iteration
the
out--
converwill
achieve
of the
of debug
of these
If
> 0, blade
coefficients
as 21 arrays
Instructions
no errors
should
following
geometry
of the finiteoutput
arrays
on the
or-
and
are
change
reduced-mass--flow
be smooth
enough
curves
(see
thoroughly
solution
that
re-is a
into
that
section
by the program
form
the hub and
the hub,
(g),
after
p.
spline
units;
specified
shroud
fitting
any
improper
for
and blade
input
those
by LSFR,
arrays
These
time
new
it is submitted.
bounds
shroud,
29).
first
before
the input
of the
Input
the
inconsistent
with
not being
Preparing
to MERIDL
reasons:
not agreeing
input
for
in input
be checked
input
on microfilm
output
are
geometric
spline
and
of IDEBUG.
to have
Therefore,
stream
If ITSON
additional
alternate
equations,
Special
made
for
information
a high
whether
information
multiple
input
3).
only wants
Using
IDEBUG=
printed
or whirl;
(ref.
is desired
Otherwise.
the
indicating
thogonal
commonly
subsonic
result.
difference
run.
R = 1, both
information
program
is reached.
Integer
It is unusual
If ISUPE
will be printed.
indicating
gence
IDEBUG
be printed.
and
the
set
Errors
sign
arrays;
LTPL,
on
is
are
w,
and
V 0,
up-
and
blade-geometry
sections
will
geometric
is completed;
data
arrays
the
be fit well
are
all
microfilm
25
output will indicate whether or
especially
that
from
(a) Units
throughout
on the
this
system
are
report.
input
The
tent
gas
constant
gives
of units
can
is mass
per
in the
be employed,
factors
on the
Ft
a(rV0)/0t
large
that,
Using
values
are
be used.
O(rV0)/_t
reduce
not made
DNEW
of FNEW
not converge,
rate
of convergence
Limited
variables
to obtain
experience
are
For
ing is required.
most
The
user
iteration.
The
for
maximum
gradually
approach
user
learn
effects
can
the
and
subsonic
flows,
DNEW
this
in the
have
range
with
FNEW
and
are
would
of these
DNEW
in
FNEW
the
case
are
and
most
(Note
that
Ft
or
will
DNEW
are
is auto-
and
are
to
F t or
this
1.0.
the gradients
with
so
if
and
Ft
changes
However,
and
each
diverge.
change
FNEW
are
for
often
VELTOL
are
in velocity
before
and
printed
at any
convergence.
in FNEW
DNEW
Maximum
(DVTHDR)
change
for
of changes
DNEW
During
Therefore,
used.
consis-
values
solution
that
time.
of flow
DNEW
some
damp-
commonly
FNEW
and
other.)
0(rV0)/at
VELTOL
(A1).
terms
of 0.5
been
any
and
new
where
is not
unit
units.
a portion
can be obtained
x
per
FNEW
the
be used
of convergence.
relative
any
of convergence,
however,
experience
rates
Ft
it should
and
to each
to gain
to maximize
change
flows,
with
FNEW
× Acceler-
Since
be reduced.
as when
solutions
time.
variables.
should
as low as 0.05
to be equal
will have
effectively
the maximum
values
for
unit
of the predicted
should
rate
and
of equation
only
of these
accuracy
that
of FNEW
at times
do not have
shown
allows
caution
temperature,
is mass
to these
fraction
reduced
same
practical
1.0
be used
x Length)/(Mass
flow
iteration,
depend
may
of the program,
on each
or DNEW
of the
the
terms
that
= Mass
variables
changes
is the
so that
per
14) is used
of units
Force
mass
input
portion
values
converged
Values
although
DNEW
from
of (Force
0(rV0)/0t
than
FNEW
has
not large,
to 1.0.
or
Because
reduced
- The
accepted
less
length,
and
calculated
to the previous
too small.
matically
DNEW
and
does
the
used,
was
to be added
(rV0)/0t
equal
value
The
change
of FNEW
The
and
be checked,
constants
if force,
is not labeled
DNEW.
any
(ref.
set
of length
output
should
of Units
consistent
units
volume,
units
output
not use
obtained
the
or finite-difference
calculated.
if the full
are
unit
the
as damping
subsonic
does
any
have
chosen
and
and
each
then
FNEW
of the
System
example,
units
All
if it is reasonable.
program
mass
factors
iteration
to see
For
R must
velocity
smooth.
Therefore,
(b) Dampin_
used
them
the
used.
was
International
the program.
Density
then
set
being
for
set,
- The
independently,
Temperature).
Output
data
However,
of units
chosen
ation.
input
of measurement.
in preparing
time
a new
not the input
minimum
with
point
By observing
and ]:)NEW
for
he can
use
values
theoutput
is also
these
different
runs
and
on
printed,
values,
the
of the pro-
gram.
tc) Hub and
geometry
must
26
have
shroud
is specified
the
same
flow-channel
in the
z-origin
ZHUB,
_eometry.
RHUB
(typically
and
- The
ZTIP,
the blade
hub
RTIP
leading
and shroud
arrays.
edge
flow-channel
Both
at the hub).
of these
These
curves
two
arrays must extendfar enoughupstream and downstream to
downstream
stream
they
boundaries
input
will
stations
are
N-HUB and
NTIP)
If the
user
effect.
situation
/d) Ortho_onal
MBO,
MM,
MHT;
ZOMOUT.
ROMBO,
and
ROMOUT
must
ZOMBO
hub,
use
although
spacing
by
locating
lines
MHT
follow
usually
pass
spaces
adequately
given
lines
of length
NIN.
(or LOSOUT),
(fig.
tion
7).
and
SFIN
and
Ordinarily,
stream
along
function
streamlines
specified
ZHOUT,
(LAMVT
ZTOUT
region
range
for
fits
of
RADIN),
TIP,
the
upstream
(SFIN
and
upstream
= 0) and
inputs
are
conditions.
are
portions
of the
downstream
In this
are
superfluous,
in the
instance,
distance
is determined
of the
by
vertical
edges
vertical
mesh
are
mesh
required
not
lines
The
will
suggested
to cover
the
are
solution
flow
case
are
all
used,
blade
which
of length
along
are
input
blade.
these
are
are
all
PROP
NOUT
with
the
assump-
the
flow
field
region:
conditions
these
arrays,
to establish
(MBI _ 0), the
so that
conditions
(or RADOUT),
streamlines,
of the
blades
flow
SFOUT
which
conditions
or downstream
there
at the
the four
(or VTHIN)
in the
arrays,
flow
SFOUT).
blade
For
on
and
between
hub to shroud.
LAMIN
given
along
and
ZOMBI
of the
the blade
are
located
Mesh-size
none
- Upstream
and
momentum
downstream
from
all
8).
direction
that
ROMBI,
requested.
So some
ZOMBI,
geometry.
(or VTHOUT)
downstream
(fig.
vertical
ZOMBO
PRIP,
conditions
of angular
and
and
flow
locations
be located
progress
on blade
edges
relation
by MBI,
ZOMIN,
The
hub will
8).
blockage
ROMIN,
input.
since
and
condition.
z-coordinates,
Usually,
(fig.
this
by
= 1),
spaces
lines
shroud
as they
depending
and
in the
the hub
is specified
trailing
of mesh
distance.
ZOMBI
LAMOUT
Upstream
the
four
by the
or trailing-edge
Downstream
upstream
far,
spline
flow
lines
(LROT
and
the
along
is specified
to these
spacing
downstream
(or
of conservation
in the
this
Relatively
accurate
realistic
geometric
leading
along
hub--shroud
between
is 15 to 30,
in the
Mesh
to the hub or
(e) Upstream
to the
spaces
leading-
in or out of the blade
other
the number
mesh
in the
the blade
of mesh
extend
and
down-
result.
to include
mesh
The
is established
and
- ZOMBI.
a more
is specified
to correspond
etc.)
spaced
orthogonal
number
close
arrays
of the mesh
as all
direction
(ZOMIN,
ZOMBO
do not
may
blockage
shroud
be specified.
z-origin
do not have
MBO - MBI evenly
given
and
smooth,
of orthogonal
ANGROT
also
usually
horizontal
z-coordinates
number
When
must
are
they
in the
upstream
(2 to 10 is a typical
calculate
represent
the positioning
the same
locations
the
upstream
channel
boundary-layer
will
- The
and
and
hub,
cover
the
If they
flow
surfaces
the hub and
the output
ZOMBO,
the
of the
revise
mesh.
and
as
given.
smooth
the program
the amount
he should
In this
are
an incorrect
these
to have
as well
6).
knows
profiles,
data
and
to describe
in order
(fig.
mesh,
streamflow
extrapolated
needed
surfaces
shroud
where
be linearly
few points
these
of the orthogonal
given
values
If,
at all
in addition,
ZHIN,
variables
apply
as a function
ZTIN
need
of
points
the whirl
is
and
not be specified.
27
However, in most cases, legitimate values must be supplied for ZHIN, ZTIN and
ZHOUT,ZTOUT. In this case the upstream conditions are given on a straight line that
passes through the two points given by ZHIN on the hub and ZTIN on the shroud. Down-stream conditions are given on a straight line that passes through ZHOUTon the hub
and ZTOUT on the shroud. Theselines may lie anywhere in the regions from the blade
edges upstream and downstream to the boundaries of the orthogonal mesh (figs. 7 and8).
If LROT is 1, values for RHIN, RTIN, RHOUT, and RTOUT must also be given as input.
The arrays of upstream and downstream input do not necessarily have to extend all
the way from the hub to the shroud or lie on radial lines. They will be linearly extrapolated to the hub and the shroud, if necessary, by the program, should the user only
give data in a portion of the flow channel.
(t) Mean
blade
from
hub to shroud
Each
of these
these
sections.
of data
data
When
the
z-coordinates
put
arrays.
are
always
When
sections
LBLAD
tively,
the
midchannel
for
stream
surface.
let
angles
flow
from
the
The
face
the
and
to the
or proper
so,
when
array
is given
cutting
camber
The
surface
and
TTBL
array
LBLAD
This
(fig.
input,
is used
is given
for
be given
mean
z-coordinate
ZBL
origin
ZBL
for
by
(figs.
the
smoothly
for
in-
and
RBL
other
two
mean
camber
6 and
10).
Because
should
THBL
stream
tangential
(i. e.,
when
must
inlet
back
line
and
(ref.
and
may
to obtain
giving
be for
t0/r)
(fig.
10(b)).
is for
lies
on a surthe
in-
thick-
or may
not be
proper
block-
input.
the blade
Thick
F).
array
TNBL
blade
out-
(appendix
mean
surface.
(circumferential)
3),
on the blades
in general,
and
it is difficult
coordinates
the
TNBL
radii
be exercised
Alterna-
of solidity
So,
The
on the midchannel
distance
The
(b)).
solution
fairing
10(a)).
and
surface.
is a function
= 0 (fig.
10(a)
a solution
be at different
in radians
other
camber
provides
distance
may
input
to be any
The
shape,
of
section
All the
a blade--to-blade
ends
midchannel
new
etc.
by LBLAD,
(figs.
the
all
the blade
the blade
10(a)).
for
10).
in each
each
not have
data,
used
points
sections.
at an appropriate
LBLAD
caution
as input,
not the
= 1) and
surface
edges.
when
RBL
from
to the blade--section
surfaces
is given
must
obtained
surface
whose
the
of the blade
blade
does
6 and
dictionary).
array
stream
through
line
TSONIC
input
NPPP
start
blade
controlled
The program
that
There
to describe
THBL
array.
input
card.
as that
options,
with
is described
(figs.
arrays,
on adjacent
same
surface,
normal
blade
TNBL
the
trailing
lie on a curved
age
28
onto
leading
three
four
by all
shape
two dimensional
of these
followed
0-coordinates
THBL
or planes,
necessary
stream
thicknesses
normal
(when
input
the
of revolution
nesses
1,
sections
points
in the
It obtains
TNBL
put blade
are
are
each
be the
arrays
is 0 or
is for
be used
should
- The blade
of which
of a new
given,
definition
array
may
are
There
THBL
for
analogous
of ZBL
LBLAD
blade
at the beginning
Of the four
(see
coordinates.
all
data
between
given.
thickness
NBLPL
giving
points)
all the
arrays
arrays,
has
relation
for
and
by four
arrays
(NPPP
geometric
surface
thicknesses
blades,
Al-
blades with high curvature, or blades with significant lean should use TTBL input in
preference to TNBL input.
The TH1BL and TH2BL arrays are used to give blade surface 0-coordinates
(when
LBLAD
(ref.
= 2) (fig.
10(c)).
This
input
is similar
to that
used
in the
TSONIC
program
3).
To obtain
to use
a smooth
nonzero
solution
near
values
of leading-edge
to the
four
the
leading
and
and
trailing-edge
trailing
edges,
it is recommended
thicknesses,
as figure
10 indi-
cates.
In addition
dimensional
arrays
the leading-edge
angles
are
when
The
blades
on the
program
makes
use
throughout
well
as microfilm
data
for
arrays
curves
required
for
for
purposes
presses
mathematically
the
points.
Reference
points,
usually
variable
guide,
not
curvatures
enough
points
would
accurately
curve
should
be used,
inaccuracy
for
ISUPER,
program.
this
they
Use
are
The
LETEAN
not
given.
tip does
(fig.
to obtain
blade
four
blade-shape
not
within
the boundary
= 1
the
6).
data
Ex-
where
the
for
these
curves
to compute
sections,
first
given
gradients
and
as output
arrays.
second
will
that
are
derivatives,
indicate
as
whether
input
curves.
- All the
the
four
more
should
be
since
curves
or five.
points
the
the
are
so that
The
the
spline
passing
uneven
difficult
a physical
minimum
are,
a few
or highly
to fit properly.
spline
the
passing
As a
through
of points
given
of the
with
dips,
exthe
equation
places,
number
points
that
through
accurately
cubic
any other
polynomial
specified
with
fit with
or
the
more
spline
are
determining
can be
and
curve.
closer
for
Curves
specified
follow
cubic
by an idealized
used
arrays
derivatives,
is a piecewise
method
smooth
than
calculating
curve
taken
input
these
to follow
greater
the
effect
ISLINE,
ISTATL,
the
of an
in a coordinate.
(11) Requests
IPLOT,
shape
require
points
data
mean-camber-line
spline
method,
more
outside
specify
in degrees.
It can be given
necessary
of these
spline
15 describes
By this
angle
at the blade
profile.
of interpolating,
A cubic
curve.
one
two one-
arrays
1 0(d)).
= 0 when
are
acceptable.
calculation.
spline
fits
of the blade
points
the
when
These
(fig.
the last
shroud
smooth
The
were
(g) How to specify
spline
to give
the program.
these
LETEAN
there
profiles.
of spline
plots
optional.
tangency
or completely
be used
shroud
attempt
and
to the hub or
will
and
are
surfaces
at the hub or
the boundary,
should
used
given,
given
to conform
the hub
user
that
blade-section
are
or interpolation
The
BETATE)
the blade,
mean-camber-line
input
section
crossing
meet
to describe
trailing-edge
blade
region,
necessary
and
and BETATE
have
trapolation
The
and
first
necessarily
flow
(BETALE
measured
BETALE
arrays
regard.
The
output
ITSON,
optional
The
user
data.
- The
and IDEBUG
arrays
should
ZHST,
be careful
seven
all
variables
request
PJ-IST,
different
ZTST,
to request
IMESH,
portions
RTST,
only
the
and
output
of output
FLFR
are
needed.
from
also
the
used
in
Usually,
29
Shroud
Horizon_
I
I....
orthogonals-",_L__z_
-_-_'-_-
] - -l]<
-i:
- t -I -L-
....
- --I-_7_.]-.J
["-_._:'"_I
I j,
! _
Vertical_ i
II
_,_1_,_...._
l_j_
o.hogona,s. ,__ ,
,_-Station
lines
['.'_
I,._,,,oo
_ I I.'I
'
J
'l--
._]..-_
.__
Streaml
A
MenShh_nr t output - at intersections of
orthogonal mesh
Streamline output - where streamlines
cross vertical orthogonats
Station-line output - where streamlines
cross user-designated station lines
Hub
0
[]
Streamlines and station lines
Orthogonal mesh lines
.....
Figure 11. - Location of three major types of output.
only one
lines
of the
three
(ISLINE),
ISTATL
or
outputs
major
along
are
likely,
The
frequency
at which
given
to IMESH,
streamline
ISTATL
ZHST,
output
RHST,
RTST
from
output
output
should
ZTST,
and
in these
at the blade
centages
leading
of chord
and trailing
(within
Output
edges,
and trailing
edges
The array
streamline
user
does
which
not specify
put at 10-percent
3O
with
(IMESH),
is needed,
data
along
since
by
output
the
streamISLINE
and
(fig.
11).
interpolation
(ISTATL)would
significant;
if NOSTAT
given).
straight
Through
at which
this
be chosen.
is controlled
and
lines
the
RItST,
by the
of local
chord)
and
To obtain
and
that
the
lines
RTST
to the _tersection
and
connect
user
locations,
the blade
output
the
do not have
downstream
within
(and
RHST
arrays,
These
stations
ZTST,
the
hub to shroud
is given.
edge.
zero
omit
in these
upstream
to the trailing
than
= 0,
from
values
at several
in ZHST,
is greater
If LROT
his output
at several
edges,
of 1 percent
FLFR
along
along
leading
points
ISTATL.
arrays.
trailing
the
is also
arrays
be given
the values
a tolerance
and
can be requested
from
station-line
is requested
locations
and
or
are
points
to be radial.
mesh-point
RTST
corresponding
the
calculated
only
will
mesh
(ISTATL))
be requested
Output
exactly
(at the
stations
(ISLINE)
ISLINE,
cards.
can control
of output
hub-shroud
obtained
Most
values
types
at per-
at the leading
should
correspond
points
of the
leading
the hub and shroud.
need
be given
he wishes
otherwise
streamlines.
only
output
(using
if the user
(when
FLFR),
wishes
ISLINE
to specify
or ISTATL
the program
will
are
the values
used).
automatically
of
If the
give
out-
(i) Incompressible
by using
density
special
input
(RHO).
blank
cards
flow.
In this
should
an incompressible
solution,
which
infinite
ficial
or
approximate
must
REDFAC
be redundant
all the
Using
this
infinite
(see
the
flow
will
sonic
from
for
fluid
but
is required
to avoid
the
the
for
transonic
With
(RHUB,
and
the program
and,
RTIP,
This
and
RADIN,
his
the
value
a large
RADOUT,
can
and
be calculated.
a solution
simulates
for
is known,
of NBL
obtain
therefore,
radius
P
of MSFL
will
an arti-
= (2vr)/P.
value
a value
to a straight
be infinite,
pitch
NBL
this
circular,
average
blade
from
for
to apply
would
a large
since
of input
radius,
designed
be adapted
pick
input.
to shroud
REDFAC.
supersonic
the best
section
flow,
accuracy
NUMERICAL
be used
in the
a value
axial,
slope
with
axial
inlet
and
bine
rotor
with
axial
for
LROT
will
ANGROT.
p.
for
almost
a pitch
exactly
a
discharge
70).
(use
When
flow
LROT
ANGROT
from
the
should
should
than
value
for
- If the
specify
discharge
be less
the
should
1.0,
be 1.0.
usually
of REDFAC
REDFAC
stream
However,
between
should
is 1.0,
function,
0.5
be used
the
full
mass
and
no tran-
be made.
not be given
= 1 and
REDFAC
possible
solution
of the hub or shroud
radial
must
the largest
EXAMPLE,
need
use
maximum
REDFAC
calculation
ANGROT
- If possible,
finite--difference
velocity-gradient
45 ° from
input
to 1.0
a cascade
Then,
radius,
hub
can
such
should
in the
mean
at large
a value
For
axial),
the
be set
to the
by the program,
special
is primarily
input
user
arrays
AR equal
not used
can be calculated
use
set
on MERIDL
cascade.
(l) Choosing
from
input
is locally
0.95.
for
about
slightly
(k) Choosing
and
to use
are
should
for
r = 1000.
r-coordinate
artificial
very
The
of
to an integer
can be established
if there
be adopted.
and
No other
program
radius
can be executed
accurate.
but the
the
of blades
RBL)
straight
rows;
PROP
input.
less
- The
Since
neighborhood
radius,
and
cascade.
blade
number
varies
except
cases
GAM = 0.
and
that
must
be rounded
PRIP,
case,
mean
that
TIP,
Use
the
as well.
in the
AR.
with
rotating
convention
cascade,
case,
infinite
cascade
GAM and
flow
be furnished
would
(i) Straight
stationary,
for
- Incompressible
have
rotated
have
is close
= 0).
in degrees
axis.
ANGROT
ANGROT
to axial
(within
If the flow
so as
For
A radial
45 °
more
to minimize
example,
= 45 °.
about
deviates
than
the
an impeller
inflow
tur-
= -45 °.
OUTPUT
There
are
(1) Main
(2) Debug
four
output
different
- controlled
output
(3) Information
(4) Plotted
types
output
by the
- controlled
for
the
TSONIC
- controlled
of output
generated
variables
by the
IMESH,
MERIDL
ISLINE,
program:
ISTATL,
and
ISUPER
by IDEBUG
program
- controlled
by ITSON
by IPLOT
31
Most of this output is optional and is controlled by the final input card, as already described.
The output controlled by each of the variables IMESH, ISLINE, and ISTATL is essentially the same but is given at different locations for the convenienceof the user.
The IMESH output is given at the orthogonal mesh points along horizontal mesh lines,
as indicated in figure 11. The ISLINE output is given along streamlines where the
streamlines are intersected by the vertical orthogonal mesh lines (fig. 11). The ISTATL
output is given from hub to shroud along station lines (fig. 11) where these lines are
intersected by the streamlines. Also, additional ISTATL output is given beyondthat
given with IMESH or ISLINE output.
In the following sections, output is presented from the problem solved in the section
NUMERICAL EXAMPLE (p. 70). Since the complete output would be lengthy, only the
first few lines of each section of output are reproduced here. For debugoutput
(IDEBUG_ 0), output labels are simply internal variable names, which are defined in
the main dictionary of part II (ref. 6}.
The following three sections discuss the different sections of printed output, the
plotted output, and all possible error messages.
Printed Output
Table I presents the printed output from the numerical example. Each section of
this output has been numbered to correspond to the following description:
il) The first output is a listing of the input data. Variable namesare used as
labels, and the output corresponds to the input form (fig. 5). This output is listed for
every run, regardless of the values given to IMESH, etc.
(2) The second output gives information to assist the user in checking input blade
geometry. This output is listed for every run. For each input blade section from hub
to shroud the following is given: ZBL, RBL, THBL, and TTBL, all defined as in the
input dictionary; blockage at each input point (tangentialblade thickness TTBL divided
by blade--to--bladepitch in radians); SZRBL, the meridional arc length of the input blade
section; and DTHDSPand DTHDSP2, first and secondderivatives of TttBL against
SZRBL along the input blade sections. The smoothnessof DTHDSPand DTHDSP2indicates where inaccurate data or errors may have occurred in input blade geometry.
(3) The third output corresponds to IDEBUG. It has four principal sections, as the
printed output indicates. The first section gives blade geometry where the blade is intersected by the 21 × 21 alternate mesh usedfor calculating gradients of the mean
camber surface. The z, r, 0, and tangential
thickness
coordinates
are given,
followed
by the
32
gradients
of
0,
calculated
by spline
curves,
along
the
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47
nate
mesh.
This
first
of debug
output
lists
are
listed
only
also
cients
of the
mesh
giving
the
This
as the
solution.
fifth
input
output
for by the
flow
of variables.
factor
LOSOUT
array
given.
This
(this
of the
run
ORF
the
are
sixth
hub,
relative
ity.
output
ILE(J)
within
(7) The
When
the
change
occurred,
verged
points.
If there
output
change
to the
is referred
to
reduced-mass-flow
of the program.
at the points
array
along
is printed
by the
becomes
than
less
relative
change
and
FT(F_
FNEW
used
are
next
with
the
line
if PROP
is
values
each
for
iteration.
iteration,
each
the
calculated
that
line
points
and
FT,
and
change
reduced
where
the
(reduced
or full
mass
the maximum
number
and
of uncon-
FT are
of interest.
maximum
change
in DVTHDT
The
variables
DNEW
input
change
either
of these
in either
DNEW
or
the
in veloc-
of VELTOL
indices
changes
maximum
it is suggested
value
of DVTHDT
iteration.
of the
If the
and
relative
(either
in velocity,
minimum
of the mesh
DVTHDT,
maximum
the
mesh
edges.
about
are
horizontal
numbers
the input
given
jth
trailing
stream-function
the behavior
the fraction
following
the
problem,
printed
to control
and
is determined
the
line
information
Also
(2(rV0)/0t)
mesh
convergence
converged.
and
For
leading
is considered
the average
arrays.
vertical
or DNEW),
maximum
the
ITE
of FNEW
the
for
gives
and
the
closest
is a convergence
is increasing
ILE
are
Convergence
Therefore,
are
and
maximum
solution
the
and ITE(J)
seventh
minimum
flow)
lists
the blades
velocity.
by the
and
variables
DVTHDT
FNEW
to be
or FT
be de-
creased.
(8) The
function
point.
eighth
solution.
The
output
output
The
given
corresponds
output
is given
at each
of the point,
stream
W0, relative
relative
flow
velocity
W, critical
angle
fl, and mesh
48
to
equations.
iteratively
on each
follow-
output
sets
of overrelaxation
iterations"
automatically
are
The
two different
coeffi-
orthogonal
listed
of the finite-difference
of the "outer
conditions
are
quantities
the
the
of IDEBUG.
u is solved
calculated
list
and
two sections
value
section
These
output
function
value
second
constant.
of debug
contains
calculated
The
as input.
from
used
the
held
These
function
each
are
once.
stream
solution
stream
is given
lists
downstream
(6) The
that
output
the
quantities
the
only
sections
for
as called
during
that
iteration.
mesh
the
is given
fourth
overrelaxation
for
ORF
each
indicates
iteration")
(5) The
given
output
equation
The
where
solution,
and
is solved
with
successive
"inner
that
output
quantities
third
orthogonal
fourth
matrix
The
change
of the
in the
mesh
equation
changing
(4) The
be used
once.
that
Lag an iteration
of debug
orthogonal
matrix
quantities
section
function
u,
mesh
to IMESH
along
point
meridional
velocity
angle
_.
for
each
of the
horizontal
includes
velocity
ratio
one
the
mesh
following:
Wm,
W/Wcr
iterations
relative
, meridional
line
of the
at each
z- and
mesh
r-coordinates
tangential
flow
stream-
velocity
angle
_,
(9) The ninth output corresponds to ISLINE for one of the iterations of the streamfunction solution. This output is given along each streamline, corresponding to a given
stream-function value. The points along the streamline correspond to where it is intersected by the vertical mesh lines. The origin for the m-coordinate is located where
each streamline begins at the upstream boundary of the orthogonal mesh. The output
given at each streamline point includes the following: z-., r-, andm-coordinates of the
point; meridional velocity Wm; relative tangential velocity W0; relative velocity W;
critical velocity ratio W/Wcr: meridional flow angle a; relative
flow angle
/3;
streamline
curvature
suction-surface
(10) The
mass-flow
solution.
ZTST
by the
streamlines
identical
second
Output
(the
output
of output
the blade-to-blade
to ISTATL
of output
along
being
given
includes
each
is given
values
in the
for
passage,
the estimated
velocities.
it is now given
is given
arrays.
to the
row
Instead
output
ZHST,
within
corresponds
(ISLINE),
This
and
pressure--surface
10th output
or streamlines
shroud.
1/rc;
and
the
given
one of the
along
in the
other
of the
station
at each
FLFR
ISLINE,
for
array).
with
following
_Nnl )fs--
point
the
horizontal
direction,
lines
specified
these
The
output
of the
variables:
of the
mesh
along
where
addition
additional
iterations
station
given
reduced-
lines
lines
(IMESI-1)
from
hub to
by the input
lines
are
at each
stream
a repeat
point
function
of
crossed
is
u.
z-
The
and
-' I
(Wm Jbf
Figure 12. - Definition
of incidence
angles.
49
r-coordinates
absolute
of the point,
static pressure
tangential velocity
lute total temperature
V0,
(11) The
and blade
mesh
lines intersects
dence
quired
after each
These
12th output
to prepare
(The TSONIC
The
compressor
shown
are defined
to ITSON.
bf
p,
p', absop",
and
was
angles
each of the horizontal
ibf or deviation
is based
(subscript
fs
as shown
in fig. 12).
in figure
12.
It is a listing of the information
analysis
program
re-
of reference
in the future.
will be given in the updated
The
of this report.
version
in figure 13 are those computed
The
3.
definiof
run for both the hub and tip blade sections
example
on
inci-
output is given for either IMESH,
and will be reported
for TSONIC
and deviation
in fig. 12); the unblocked
blade-to-blade
used in the numerical
in the input form
where
velocity diagrams
is being updated
program
incidence
incidence
(subscript
corresponds
TSONIC
blade
total pressure
flabs' relative total pressure
iteration in which
angles
tions of all the input variables
TSONIC.)
absolute
and unblocked
The blocked
input for the TSONIC
program
V,
and trailing edges
on the free-stream
or ISTATL.
(12) The
of blocked
the blade.
output is printed
ISLINE,
consists
within the blade
ifs is based
This
flow angle
at the blade leading
the velocity diagram
velocity
T, static density
T".
11th output
angles
absolute
T', absolute
relative total temperature
p, static temperature
of the
input numbers
for use with the TSONIC
run at
the hub.
The
without
information
revision.
tion from
printed
In other
arrays
user may
to make
wish
small
to the given
accordingly.
between
of points
mass
second
derivative,
blade surface
changes
The
ZMRSP
before
(NLOSS
nonlinear
The
in the BETI
serve
need
and
is printed
as input for TSONIC
revision.
curvature)
alongside
and BETO
The
informa-
of the THSP-
these arrays.
tangency
to smooth
as they are printed,
in them.
Any
The
BESP
The
angles,
which
the resultant first and
calculated by MERIDL
BESP
array
input if REDFAC
> I) can be used
is sometimes
the blade
13th output is analogous
ISTATL,
and ITSON
unless
The
WOWCR,
is calculated
for ZMSFL
bumpy
The
is chosen
of the mass
and may
derivatives
to re-
to correspond
if BESP
is 1 percent
is necessary.
rather
BESP,
the user wishes
< 1.0 is used for TSONIC.
for TSONIC
through
RMSP,
array
value could be used
and will indicate if smoothing
as TSONIC
ISLINE,
data - ZMRSP,
it can be used as input to TSONIC.
loss distribution
(13) The
(NRSP)
value of ZMSFL
are printed
only needed
be used
flow ZMSFL.
any two blades.
smoothed
IMESH,
of the input may
of stream-channel
- can in general
the number
array
some
cases
for these spline fits, in order
final five arrays
and PLOSS
duce
can in some
derivatives.
The
5O
for each
as end conditions
second
cases,
spline fits (derivative,
against-ZMSP
serve
by MERIDL
flow
also have
to be
of BESP
against
WOWCR
array
The
printed
than a single value for PLOSS
is
PLOSS
if a
is used in MERIDL.
to outputs
8, 9, 10, 11, and 12 for the variables
but is given after the transonic
velocity-gradient
5]6
loin
20[21 25]26
15116
30131
35
36
40141
45146
60
5o151
61
70
71
80
TITLE
DATA12 - AXIAL COMPRESSOR ROTOR - HUB - STREAMLINE NO. I - U = 0.00
GAM
AR
1.4
287.053
REDFAC
VELTOL
1.0
MBI
RHOIP
I.
22534
[
OMEGA
-826.55
ZMSFL
O.025194
0.001
MBO
10
LRVB
TIP 15
288.
MM
50
LOSS
0
MBBI
60
LWCR
I'
20/
_._
0
RV1-HO
-9.88533
NSL 0
171
LIPS
0
RVTHI
NBL]NSPLi]NSLP2NRSP
20
FSMI
FSMO
SSM1
SSM2
-27.6253
CHORD
STGR
O. 113429
O.0.54930
RII
ROI
BETII
O. 001346
BETO1
0. 0009226
29.516
-23. 859
ZMSP1 ARRAY
0. 00068
O. 00629
O. 01247
0. 01863
O. 02479
O. 03093
O. 03707
0. 04320
0. 04933
.0. 05545
0. 06156
0. 06767
0. 07378
0. 07989
0. 08599
0. 09209
O. 09819
0. 10429
0. 11039
0. 11288
O. 00762
O. 02745
0. 04682
0. 06365
0. 07815
0.09046
0. 10067
0. 10889
0.11517
0.11954
0.12199
0.12249
0.12103
0.11747
0.11181
0. 104(X)
0. 09397
0.08159
0.06672
O.06009
THSPI ARRAY
RI2
BETI2
0.001346
RO2
17.593
BETO2
0.0009226
-9.602
ZMSP2 ARRAY
0.00175
O.00629
0.01247
0. 01863
O. 02479
0. 03093
0. 03707
0. 04320
0. 04933
0. 05545
0. 06156
0. 06767
0. 07378
0. 07989
0. 08599
0. 09299
0. 09819
0. 10429
0. 11039
O. 11235
-9. 00835
0.00077
0.01221
O.02263
0. 03198
0.04027
0. 04751
0. 05369
0.05879
0.06283
0. 06581
0. 06775
0. 06866
0. 06847
0. 06720
0. 06487
0.06145
O.05695
0. 05137
0. 04937
0. 04320
0. 06156
0.15968
O.16130
O.002163
O. 002110
0.00432
O. 00634
THSP2 ARRAY
ZMRSP ARRAY
-0. 04741
I
-0. 03285
-9. 01860
-9. 00455
]
0. 00629
0. 02479
0.07989
[
0. 09819
0.11648
0. 13173
]
0. 14544
0. 15916
0.14107
]
O.14594
0.14979
0.15283
0.15478
O.15753
0. 16245
[
0.16322
0. 16372
0. 16399
0. 16417
0. 16431
O. 002268
O. 002233
0.002205
O.002208
O.002228
0. 002199
0.002049
0. 001989
0. 001926
0. 001902
0. 001892
0. 001885
O.00065
0.00249
0.01447
0.01447
RMSP ARRAY
BESP ARRAY
PLOSS ARRAY
0.00
0.00
0. 00851
0.01119
IBSURF
IMESH5 I ISLINE5
i
0.00
0.
01447
]
0.00
0.
01447
IPLOT
I
Figure 13. - TSONIC input form.
(Data shown are for the numerical
example at the hub. )
51
soluUon.
Two
"supersonic"
of these
solutions
and
the
smaller
is obtained.
tion).
The
If REDFAC
cause
can be obtained
the
final
or 'tsubsonic."
solution
= 1.0,
by the
listed
The
here
of the
input
is the
no velocity-gradient
iteration
velocity-gradient
variable
smaller
solution
stream-function
method:
solution
ISUPER
(i. e.,
will
the larger
which
the subsonic
be obtained
is the
controls
or
correct
or
solu-
printed
solution
be-
to the
problem.
Plotted
are
Since
the printed
made
to enable
data.
and
The
would
of input
the
coding
have
data
input
iteration
data
are
ample
are
These
are
14(f)
supplied
trailing
the
user
plotted
of the
microfilm
The
description
the input
(V)0i,
the
data
are
the
and
optional
or
input
radius,
variable
Then
for
solutions,
plots
along
each
of his
input
smooth.
by the numerical
plots
downstream
flow
(rV0) °
whichever
ex-
is as follows:
Po' ortotalpressureloss,
function
by the
velocities
generated
upstream
sections
trailing-edge
transonic
check
of these
routines
mesh.
surface
carefully
plots
and
orthogonal
and blade
of these
plot
is controlled
and for
output
of flow properties;
leading-
generated
should
fits
14(i}
and
(h} indicate
hub
user.
All blade
shows
the
hub and
and
sections
was
of total-pressure-loss
conditions.
or(V0)o)
used
as input.
distribution
against
shroud
shroud
blade
given
as input
channel
sections
are
geometry
plotted
from
input
plotted.
and
the blade
leading
edges.
(5) Figure
140)
shows
the
generated
(6) Figure
14(k)
shows
the
streamline
mass-flow
52
14(g}
by the
(4) Figure
and
shows
stream
are
and
The principal
distributions
by IPLOT)
spline
14.
or
the
input
plots
of chord.
(3) Figures
data
either
with
The
to (e) present
t F'I' Pi" k
against
(2) Figure
fraction
in figure
14(a)
quantities
plotted
to shroud.
of some
data
microfilm
in-house
system.
and blade
as midchannel
the program's
presented
output
(indicated
Lewis
downstream
shroud,
large,
of both his
NASA
on another
and
begin
as well
examples
(1) Figures
data
solution
hub
that
Selected
at which
output
uses
operation
the hub,
can be quite
the quality
plots
upstream
and
run
check
these
for
the
plotted,
from
to ensure
to quickly
frequency
of the
streamlines
streamline
are
plotted
a MERIDL
generates
sections;
The
The
user
that
plotted
geometry.
each
the
from
to be recoded
blade
IPLOT.
output
Output
solution.
orthogonal
pattern
mesh.
for
the final
iteration
of the
reduced-
.50
.28
.26
.24
,,et
.22
.2O
.18
.16
.14
25O
260
270
280
2go
300
310
32O
330
340
350
INPUT ARRAY - TIP
(a) Inlet absolute
Figu[e 14. - Microfilm
total temperatu[e.
plots of input and output..
53
,54
•28
.24
N
""
.22
.20
.18
.16
.14
e.N|10|
10 06
o.lleeoe
I
ii
e.lo_eoe
06
!
Io N
(b) IntetabsoJutetotalpressure
Figure14.- Continued.
54
e. 11elllle
11 le
I1(I
,2B
.26
\
\
•2,4
\
.2.2
.20
.1B
.IB
-40
-2O
(c) Inlet absolute
tangential
-15
-!0
-5
0
V+_IN
velocity.
Figure 14. - Continued.
55
,3O
,28
.26
.2.4
ql
.22
.2O
.18
.t6
.14
l.t
IIi
t_fltt
N
O.o_ldlOO
|, oal4el
u IO Ol
s _|
IId%PT ARRAY - PROP
(d) Outlet absolute
total pressure.
Figure 14. + Continued.
56
04
.5O
.28
/
.26
.24
/
¢::b
.22
B"
.2O
.18
.16
.14
-170
-!(;5
-160
-155
-150
-145
INPUT _RAV
-140
-
-155
-I]0
-1 5
-_20
V'rI.tOUT
(e) Outletabsolutetangentialvelocity.
Figure14.- Continued.
57
1.0
/
.9
.8
.7
.6
/
.5
/
r
U'3
c_')
. J
L_
Jl
/
/
.4
,3
,,,
.2
/
• OI
.0
,I
•2
.3
._,
.5
.6
.:'ERCRD
(f) Pressure loss distribution.
Figure 14.- Continued.
58
.7
.8
.9
!.0
,lq
.16
,14
.q[
I..--
0,"-
.12
.4[
.I0
!
I,.J
-4[
Z
N
•O8
0
0
r
..4
.04
p-l.J
0
-. 02
• O0
.02
.04
.06
BLADE
.08
SECTION
.10
.12
tqERIDIONAL
.14
.16
• 18
.20
COORDINATE
(g)inputbladesectionsfrom
ZBL, RBL,THBL,andTNBL- blade
section1.
Figure
]4.-
Continued.
59
,111
I
,1G
.14
i'm
mr
.12
.10
!
W
I--
• 08
Z
o
o
. OG
-.,I
• 04
• 02
.00
-. 02
.00
• 02
.Oa
.06
BLAi_
.08
_CTION
(h) Input blade sections
.12
NERIDIOMAL
from ZBL,
fisu[e
6O
.10
RBL, THBL,
.14
.IG
COORDINAT[
and TNBL - blade section 11.
14. - Continued.
.18
.20
Z
o
w
Z
DIRECTION
(i) Hub, shroud, and blade boundaries
Figu[e
in meridional
plane.
14.- Continued.
61
I i I ! ! _ i l_H-H-H--i
o
N
u
w
Z
(j) 0rthogonal
D IRECT I01_
mesh in meridional
Figure 14.-
62
Continued.
plane.
Z
o
N
U
w
Z
(k) Streamline
DIRECTION
plot in meridional
plane.
Figure]4. - Continued.
63
340
520
300
2_0
220
_-
2O0
160
.025
.050
.0_
.tO0
.125
I'IL"R
ID I0_I_
C_
.150
64
:200
.225
INJI,
T[
(I) Meridional and blade surface relative velocities-streamline
Figure ]4
.I_
Continued
1 Normalized stream function, 0
.25O
340
32O
3OO
290
2,40
u
0
22O
200
_GO
140
.000
i
,025
.050
.075
.I00
,125
HL"RIDION_.
.150
,175
.200
._5
.250
COORDINATE
(m) Meridionaland blade surface relative velocities- streamline ll.
Normalized stream function, 1.0.
Figure 14.- Concluded.
65
(7) Figures
for
the hub
along
14(/)
and
each
shroud
of the
and
and
iteration
also
blade
show
meridional
sections
for
the
from
velocity
(indicated
after
(m)
streamlines
Streamline
each
and
the
plots
similar
by IPLOT)
error
messages
cause
of the
are
iteration.
surface
A similar
velocities
plot
is made
to figures
14(k),
(l),
and
(m) are
stream-function
made
after
solution
solution.
messages
listed
blade
of the finite-difference,
velocity-gradient
of error
same
and
hub to shroud.
Error
A number
velocities
have
here,
Messages
been
followed
incorporated
into
by suggestions
for
the program.
finding
and
These
correcting
the
error:
M_eMHT_NHUB#NTIPwNIN*N_UT*N_LPLrNPP_*N_STATeNSL*NL_SSeLSF_wLTPLrLAMVTwLR_TwLBL_D*_
IS
The
TCC
input
LARGE
TOO
OR
dictionary
LETFAN
SMALL
gives
the maximumandminimum
values
for
all these
input
varia-
bles.
WHEN
UPSTREAM
THE_E
AND
AND
MUST
A
BE
A
DOWNSTREAM
CHANGE
CORRESPONDING
differ
must
in the
RADIN
(and
RADOUT)
shroud
lines
at the
upstream
I
This
This
I
message
INPUT
from
CANNOT
=
message
PROGRAM
occur
ARE
is for
THE
(and
GIVEN
AS
AND
NTIN
RHIN
RADIN
AND
RHOUT
arrays,
which
RADOUT
from
A
FUNCTION
AND
ALSO
OF
FADIUS,
BETWEEN
FHOUT
AND
RTOUT
ARRAYS
RTOUT)
are
and downstream
used
input
so that
to identify
there
will
points
along
be a variation
the
hub--
stations.
VALUE
only
LARGER
IN
is usually
caused
flow
solution.
Try
NEWRHO
erroneous
IDEAL
only,
DUE
by having
a smaller
with
THAN
information
This
66
ARE
2
VALUES
STOPPED
DATA
BETWEEN
INTERPOLATED
=
should
PROP
IN
RTIN
FIND
J
VALUE
CHANGE
RHIN
LININT
INPUT
IN
TO
TOTAL
the
EXCESSIVE
supersonic
value
for
geometry
PRESSURE,
program
STREAM
flow
REDFAC.
will
FUNCTION
in some
input.
RESULTING
continue
IN
NEGATrVE
LOSS
normally.
GRADIENT
region
of the
reduced-mass-
THE
This
UPSTREAM
message
The
velocity
upstream
THERE
LOCATIONS
OF
OUTPUT
ON
this
line
if the
choking
GRADIENT
found
for
sages
will
This
cannot
reason,
COULD
(TIP,
PRIP,
NOT
HAD
THAN
ITERATION
FOE
_AY
BE
corrected.
TRANSONIC
OUTPUT.
MAY
cautious
there
VERTICAL
IN
BE
IN
in using
may
EEBO_.
station
be little
station
line
problem
is not near
ORTHOGONAL
MESH
LINE
I
:
I
ERROR
vertical
be printed.
Some
the problem.
whirl,
blade
The
mesh
line
cannot
of the following
program
will
be
error
proceed
mesto the
ITERATIONS.
transonic
flow
PROP,
BE
solution
for
in 100 iterations.
or impossible
upstream
LOSOUT,
RESTARTED
TO
OR
TEMPERATURE
OF
for
input
or VTHOUT
ADJUSTMENT
mesh
causes
or downstream
NEGATIVE
AFTER
vertical
Possible
LAMOUT,
AVOID
VELOCITY,
some
line
this
quantities
arrays).
OR
VELOCITT
STAGNATION
TEMPERATURE,
WHIBL.
TO
when
in 1000
impossible
or wall
100
CONTIN
PROCEDDRE
is printed
are
or erroneous
temperature.
LINES
or if the
any particular
will
IN
the
TO
(LOOP
TVELCY
input
for
SOLUTION
TANGENTIAL
OR
I_ERATIONS
by subroutine
high
A
VTHIN,
DENSITY,
message
flow
gives
CHOKED.
THE
ORTHOGONAL
However,
OBTAINED
LINE
determine
be unusual
LESS
OF
TOT&L
BE
MESH
message
when
PROCEDURE
STAGWATION
THAT
by subroutine
LAMIN,
MAGNITUDE
RZSTART
FIND
could
ITERATION
ARE
OF
be extremely
mass
be
whirl
line.
be obtained
message
full
WHICH
CHOKED
upstream
stagnation
should
BEGINNING
THESE
if the
relative
LINES
THE
should
to the
solution
this
is printed
error
AT
solution.
CANNOT
FOR
to help
mesh
message
the user
transonic
be printed
CONTIN
NEAR
SOLUTION
OUTPUT
some
vertical
LOCATED
is close
TVELCY
orthogonals.
SUBSEQUENT
approximate
next
flow
LARGE
as input
MESH
ABOVE
velocity
or
given
GIVEN
LINES
TOO
in a negative
ORTHOGONAL
ARE
transonic
mass
If the
VERTICAL
IS
NEWREO
velocity
is printed,
the
VELOCITY
AMY
I
LINES
STATION
of the choked
A
This
ARE
after
VELOCITY
as to result
or tangential
message
output
TANGENTIAL
by subroutine
THESE
ANY
OR
so large
whirl
BEWARE.
When
WHIRL
is printed
a tangential
one
INPUT
(LOOP
TO
STATEMENT
the
STATEMENT
transonic
iterations
upstream
90)
HAS
ABORTED
AFTER
1000
OR
MORE
150).
velocity-gradient
for
or
some
vertical
downstream
solution
mesh
flow
line.
condition,
cannot
be found
Possible
and
causes
complex
geometry.
67
THE
MAXIMUM
MASSFLOW
MASSFLOW
WHICH
MINIMUM
THE
MAXIMUM
VALUE
OF
W
THE
MINIMUM
VALUE
OF
W AT
This
information
be found
for
apparent
some
for
ITERATION
INCREASED
NHUB
WAS
DECREASED
the
apparent
CHOKING
less
than
printed
flow
mass
been
MASSFLOW
end
row
CORRECTION
checks
the
subsonic
sonic.
I£ the
mation
only.
lO1.1_00
WAS
99.7999g
the
line.
transonic,
This
RESTARTED
velocity-gradient
information
380
LAMBDA)
whenever
has
solution
debug
value
cannot
if there
is no
TIMES
WERE
the
This
34
ADJUSTED
transonic
TIMES
solution
information
has
FLOW
FOR
VERTICAL
ORTHOGONAL
if the choking
mass
flow
The
following
INPUT
MASSFLOW
MSFL.
OF
=
was
cannot
debug
be found
value
if there
after
re-
is no
MESH
LINE
I
=
I
flow
calculated
by subroutine
message
will
TVELCY
be printed
is
after
all
THE
0.14_00E-02
printed
the
LEADING
solution
calculating
vertical
choking
EDGE
when
the flow
procedure;
if this
solution
flow,
of the
estimate
J
FOR
=
< 1,
is subsonic
is found,
this
which
CHANEL
of the
choking
the blockage
by an iterative
fails,
line,
message
is the
will
choking
program
mass
be
mass
(ref.
13)
flow.
2
REDFAC
the density
mesh
mass
Use
accurate
on whether
supersonic
any
by MERIDL.
a more
-
for
minimum
estimated
to obtain
depends
WAS
OBTAINED
0.96000
to give
the blade
procedure
OBTAINED
BE
RHOIP,
MASSFLOW
requires
BE
COULD
0.15000E-02
the full-mass-flow
ing edge
COULD
SOLUTION
checked:
message
SUPERSONIC
For
=
CHOKING
is recommended
SOLUTION
A
0.950000E-03
IS
MASSFLOW
at the
for
A
WHICH
MASS
have
previous
WHICH
FOR
TIMES
=
input
MINIMUM
If the
FLOW
the
INPUT
FOR
TIMES
is printed
CHOEING
0.120C0COE-02
0.I01000_E-02
difficulty.
lines
vertical
WAS
WAS
HUB
procedure.
EXCEEDS
message
OBTAINED
OBTAINED
HUB
WAS
56
for
This
BE
BE
THE
mesh
is printed
MASS
COULD
COULD
THE
(TIPBD¥,
iteration
C_OEING
SOLUTION
SOLUTION
whenever
112
reason
MSFL
A
difficulty.
VALUES
information
starting
AT
PROCEDURE
WAS
A
WHICH
vertical
WHUB
BOUNDARY
This
FOR
is printed
reason
THE
68
FOR
THE
or supersonic,
it is usually
the
preceding
correction
procedure.
because
message
Since
at the
the
the
program
the
flow
is printed
lea&
iterative
first
is superfor
infor-
SUPERSONIC
This
is the
NO
same
DENSITY
If after
that
CORRECTION
as the
with
near-sonic
The
program
may
be poor.
This
is the
INRSCT
HAS
as the
FAILED
TOLERANCE
and
INRSCT
tolerance
tween
factory
sive,
the
last
there
SPLINE
the
--
ONE
OF
ADJACENT
X
2.
SOME
%
POINTS
3.
SONE
X
POINTS
THREE
and
in the
or blade
the leading
geometry.
edge
edge.
of input
edge
by an iterative
message
slightly
station
is printed.
larger
than
loss
of accuracy.
geometry
input.
lines,
mesh
method.
If the
the
If the
lines,
If the
distance
tolerance,
distance
be-
a satisis exces-
CAUSES
DUPLICATES
ARE
OUT
OF
ARE
UNDEFINED.
OF
EACH
OTHER.
SEQUENCE.
3
ARRAY
12.000
sage,
some
POSSIBLE
=
Y
the
ARE
0.00000
fit subroutine
associated
0.678900E-05
coordinates
is only
with
error
10.00¢
spline
problems
near
trailing
=
or trailing
points
0.00000
points
fails,
to be the free-stream
wall
output
also
2
POINTS
20 iterations,
POINTS
POINTS
1.0000
spline
leading
some
I.
ARRAY
The
blade
after
is probably
OF
and
ITERATIONS
INTERSECTION
will be obtained,
NUMBER
20
is taken
or erroneous
although
=
is tried
of numerical
but at the
intersection
two intersection
ERROR
because
J
message
two
the
be met
solution
the blade
complex
FOR
edge.
procedure
within
EDGE
trailing
2
message,
IN
LAST
finds
with
cannot
=
supersonic
and
TRAILING
CONVERGE
at the
occur
this
2
0.ICI000E-05
BETWEEN
streamlines
the
J
density
previous
TO
=
DISTANCE
Subroutine
-
=
but
probably
after
CORRECTION
J
FOR
high blockage,
continue
same
the
would
flows,
DENSITY
EDGE
fails
and
problem
FOR
message
procedure
will
EDGE
LEADING
is aborted
This
NO
-
subsonic
procedure
TRAILING
previous
CORRECTION
the
density.
-
15.000
for
the program
a spline
curve
must
be distinct
(either
SPLINE,
SPLINT,
will
terminate.
Since
and
SPLISL,
spline
given
in sequence.
or SPINSL)
curves
are
used
will
print
If not,
the
this
mes-
so extensively
in
69
the program, it is difficult to state the possible cause. However, the printout of the
spline points should assist in pinpointing the causeof the error.
SPLINE
ERROR
NUMBER
X
--
NUMBER
OF
SPLINE
OF
POINTS
=
-3
ARRAY
¥
must
(SPLINE,
gram
will
be given
SPLINT,
terminate°
to determine
An example
and
midchannel
SPLISL,
The
one
or SPINSL)
printed
0.5;
the
the
stream
aspect
ple
spline
printed
direction,
This
rotor
was
done
have
(velocity
the
was
REDFAC
finite-difference
approximate
exact
the
7O
case
run
The
reduction
be very
factor
close
example
number
twice,
with
to test
the
in comparison
solutions
of mesh
with
will
be obtained
is obtained.)
(REDFAC
= 0. 99999)
method
is 0. 99999
if the methods
both
compare
fit sub-
the proerror.
the
with
of the
is analyzed
the
computer
hub-tip
tip relative
sonic,
use
on the
program
radius
Mach
there
ratio,
number
at
are
no locally
in table
points
I.
used
The
was
for
this
861:41
(REDFAC)
factor
and
input
in the
of 0. 99999
also
the
0.70.
accuracy
Since
difference)
in each
of them.
(If REDFAC
= 1.0,
a comparison
to be made
between
to obtain
(used
solutions
well.
transonic
for
the
axial
and
method.
and
exam-
(finite
solution
(<1.0),
as to the
Flow
the finite--difference
permits
(used
stream-function
a clue
and
to illustrate
is near
factors
subsonic
solution
the
of reduction
the
spline
message,
the inlet
and
is given
both
give
the
surface.
reduction
effect
this
designed
number
stream
direction.
< 1.0,
1.0;
Mach
radial
velocity-gradient
finite-difference,
should
table.
this
If not,
can be obtained.
is 1. 275,
tip solidity,
midchannel
print
is used
rotor
ratio
tip relative
method
first
the
curve.
may
rotor
that
pressure
from
in order
gradient)
The
1.5,
21 in the
velocity-gradient
cases
design
output
and
TWO
EXAMPLE
of an axial-fiow
on the
1 in that
of results
will
point
compressor
type
Although
regions
is item
This
The
ratio,
0.9.
supersonic
the
surface
11.
inlet,
The
of an axial-flow
to show
of reference
THAN
a spline
NUMERICAL
program
LESS
I0.000
two points
routine
IS
GIVEN
ARRAY
0.00009
At least
POINTS
solutions)
subsonic
will be obtained,
of the
both
transonic
and
solutions).
but the
only
the
the more
Since
answers
-----......
26C
Finite difference (REDFAC• O.99999)
Velocity gradient (REDFAC= O.99999)
Velocity gradient (REDFAC= 0.70)
22C
_ edee'
_
Blade i
18(]
l
I
"_gllade
__edge
trailintleadine
(a)Hub section.
28O
E
200
160
I
'ng
edle
t
I
Bladetrailingledge
(b) Mean section.
280--
2111--
200 --
__
Blade leading edge
IBlade-t_ n]'_-e_e
16o
-.08
l
II
-.04
0
I
t
I
.04
.08
z-Coordinate,
meters
I
I
.12
.]6
]
•20
(c)Tipsection.
Figure15.- Midchannelvelocities
foraxial-flow
compressor
example.
The
Since
second
the
and
be
Figure
types
sented
for
all
parts
difference
most
15
of
of
(the
values
this
figure,
of
The
of
the
results
now
the
than
midchannel
solution
permits
is
E),
accurate
shows
iteration
accurate.
(appendix
less
three
= 0.70)
solution
speed
will
both
(REDFAC
finite-difference
rotational
solution
On
case
when
the
REDFAC
presented
for
surface
the
line
= 0.
the
(0.0,
70
0.5,
percent
of
both
1.0)
velocities
a much
is
This
better
mass
flow
used.
reduction
factors
gradient).
and
the
effect.
velocity-gradient
factor
for
solution.
show
reduction-factor
full-mass-flow,
velocity
represents
here
only
velocities
99999
of
reduction
and
function
solid
at
a higher
difference
stream
comparison
obtained
results
flow
finite
a
Results
at
hub,
from
solution
are
mean,
the
final
is
comparison
and
and
pretip.
finite-
mathematically
of
velocity-
71
440--
MERIDL (REDFAC• O.99999)
TSONIC
.....
400
35O
P__.
eL,
E
320 t
|
-- \\\
280
\
___s_
S_
\\
\
240 _
200
I
160
.04
i
.08
J
.12 0
.04
m-Coordinate, meters
la) Hub section.
.08
.12
(b) rip section.
Figure 16. - Blade surface velocities for axial-flow compressor example.
gradient
to finite-difference
gram
in reference
much
better
Figure
16 shows
program
TSONIC
velocities
Research
National
than that
two different
presented
previously
velocity-gradient
a comparison
of blade
surface
(appendix
G) and
calculated
are
accurate
for
more
from
the design
Center,
Aeronautics
Cleveland,
505-04.
72
results
The
for
solutions
are
the
MERIDL
likewise
pro-
now
in
agreement.
MERIDL
Lewis
4.
Ohio,
and
July
Space
28,
Administration,
1976,
velocities
the
estimated
TSONIC
blade
shape.
program
from
the
(ref.
3).
The
APPEND_
GOVERNING
The
obtained
used
cylindrical
coordinate
on an orthogonal
to denote
For
32u
8s 2
along
solution,
stream-function
METHOD
mesh,
distance
subsonic
system
differential
EQUATIONS
is shown
as illustrated
the
the
A
equation
4.
in figure
streamwise
stream
in figure
The
the
variables
solution
s
orthogonals,
respectively.
is used.
In appendix
B we obtain
from
the
assumptions
given
is
and
and normal
function
derived
3.
However,
in the
t
are
the
section
OF ANALYSIS:
+02u
+i
8u{slncp
Ot 2
0s
8_BB
+l __
B 8s
_
p 8s
8t
+i _B +l _
8t\
r
B
8t
+
p 8t
+4
+_
+wW s L r O(rVo)
w2
ot
+F
t
=0
(A 1)
where
(A2)
2 \Cpp"
8t
2
= co r cos
gt-
_o
Equations
face
B.
except
The
(A1) to (A4) are
Note
for
that
all the
derived
partials
3p/0 0 in equation
derivatives
of the stream
from
Op_
p"
Ot
8u _
0S
(A3)
(A4)
p 80
the hub-shroud
which
function
/
RT"
in equations
(A4),
8t
80 1 _p
8t
appendix
T"
(A1)
momentum
to (A4) are
is at constant
satisfy
the
rBpWt
z
and
equation
on the
stream
in
sur-
r.
equations
(A5)
W
73
0u_
rBpWs
0t
For
line
the
final
curvatures
first
from
The
transonic
and
flow
solution
angles
a reduced-flow,
the
needed
equation
\
gion,
in the
given
as follows:
Blade-
region
a,
b,
upstream
for
c,
and
region,
d,
and
method
the velocity-gradient
is used.
equation
The
are
stream-
obtained
solution.
is
/
dW = _aW+b
coefficients,
velocity-gradient
subsonic-stream-function
velocity-gradient
The
(A6)
w
are
\
+_+dcosHIdt+_+Wf
W
]
given
in the
(AT)
W
by different
downstream
expressions
region.
in the blade
These
coefficients
reare
coefficients:
a = c°s2_ cos(_ - _) _ sin2_ cos _p+ sin
r
_ sin fl cos/3
r
v__=
0t
c
b=cos/3
dmm
dW
sin(a-
go)-
2w sin/3
c=O
d=O
Upstream-region
74
coefficients:
cos
_o +r
('dWo
cosfl_-_m
+2w
sin
a/ 3 0
(AS)
a=C°S(_-
_)
r
c
b=O
(A 9)
cos(_
C
- _) + _ + wr2
r
dW
d =---_msin(_
dm
Downs tream-
region
- ¢)
co efficients:
a = cos(a
- _)
r
e
b=0
(rV°+wr)
C
_
m
cos(c_I!rVot_:°zr21IrVoIolc
2
(A10)
(p) +
w2
r
dW
d
D
m sin(c_- _)
dm
Finally,
in all
three
regions,
we have
e = CpT}-
w dh-
CpdT"
RT r,
+-dp"
p"
(A11)
f_ dT"
2T "
Rdp"
2Cpp"
J
75
Equations
initial-value
t-line
{A7) to (All)
problem,
running
the
hub,
from
a solution
are
where
hub
the initial
to tip.
satisfying
_0
When
equation
to-tip
mesh
The
for
face
the
(A7) has
line
solution
midchannel
velocities.
change
velocities
surface
of whirl.
in the
been
These
ttip pWrB
cos(a
the
entire
the
solutions
that
hub for
any
values
of
varying
the
solution
will
given
W
distribution
satisfy
for
velocity-gradient
interest
loading
of velocity
G the following
(A12),
every
hub-
is obtained.
or
the blade
at
(A12)
equation
Of greater
variation
In appendix
is,
as an
_ dt = w
finite-difference
since
at the
for
to satisfying
velocity
(A7) is solved
is specified
- (p)cos
the blades.
a linear
W
Equation
be found;
subject
can be estimated
can be calculated.
velocities
is derived:
of
C.
several
will
between
By assuming
value
continuity
by either
surface
in appendix
By finding
solved,
region,
obtained
derived
are
the blade
depends
between
equation
method
on the
blade
for
is
surrate
of
surfaces,
calculating
blade
d(rV 0)
Wl
= Wmid
-} cos -2
dm
(A13)
Wt r = Wmid
76
+ B cos//
2
d(rV0)
dm
B
APPEND_
DERIVATION
Wu derives
the
on a meridional
OF STREAM-
following
stream
radial-
and
FUNCTION
EQUATION
axial-momentum
equations
(96),
(eq.
ref.
1)
surface:
Wo
r
3(rVo)
Or
+ Wzk
[aWr
_z
OWz/_
_r]
_3I +TSS+
8r
Or
F
r
(B1)
_OWr
- Wr k az
The
partial
derivatives
surface.
since
the
Wu used
momentum
Our
flow solution
hub-tip
direction.
s-direction
t-direction.
function
This
equation.
radial
and
cos
¢p and
rule
for
a partial
3 illustrates
the
the
axial
derivative
the
Wo
r
momentum
a(rVo)
at
and
will
is,
first
We want
be used
and
the
is obtained
also
the
by adding
the first
of (B1) multiplied
by
and
shows
momentum
to derive
of (B1) is
on an orthogonal
direction
distances
here
equation.
methods
throughflow
t
stream
equation
momentum
by multiplying
equation
sin
equation
in
stream-
t--components
equation
q_.
in the
the angle
desired
the
t
of
of (B1) by
Also,
the
chain
is
a
--=
3t
Therefore,
s
equation
that
second
in the
midchannel
but it is not necessary
The
is the axial
direction.
equation
momentum;
this,
surface.
s
3s +F z
3z
on the
finite-difference
are
t--momentum
the
second
+T
of change
stream
by using
t-momentum
axial
subtracting
the
3I
az
to indicate
on the
distances
and
The
the
and
obtained
mesh
rate
symbol
only
equation
Figure
the
to the
derivative
is
WO 3(rVo)_
r
az
all refer
equation
orthogonal
_p between
the
the
radial
The
the
a bold
we consider
mesh.
here
OZz._
Or /
equation
+ (W z cos
3
cos _--Or
in the
_o + W
t-direction
sin
r
3
sin q_-Oz
_o)
can
l,w
be written
r
_
_r /
aI
as
a'_ + T_-_
+ Ft
(B2)
77
where Ft is the t-component of the vector F, which is given by equation (97) of reference 1 as
nt ap
Ft -
(B3)
n0r p a 0
Here
n t and
Note
that
stream
angle
n o are
ap/a0
is in the
surface
to the
ponents
as
t- and
normal
0-components
blade-to-blade
theother
meridional
of the
the
direction
partials
plane.
The
of the unit
are.
at constant
of this
angle
surface
must
satisfy
no_
to the
r
In the t-direction,
tangent
to the stream
normal
the
is
and
stream
surface.
and
not on the
z
stream
r(a0/at).
surface
Hence,
1
nt
has
the
an
com-
(B4)
r(a0_
\at/
We can
substitute
this
into
equation
(B3) to get
Ft_a0
at
It is desired
t-components
and
to express
their
the
t-momentum
derivatives.
leaves
First,
we will
r
az
(B2) to be expressed
in s-
W r=w
W z=w
78
(B2) entirely
in s- and
use
cp + W r sin
_p
(B6)
only
aw
in equation
(B5)
equation
W s = W z cos
This
1 ap
p aO
and
aw
z
ar
t--components.
tcos
s cos
e+W
_p-
s sin
W tsin
For
this
)
we use
(B7)
.nd the chain rule (noting that Os/Or= sin
O_.. = sin
Or
q_, etc.)
cp O__ + cos
_s
¢pO__
Ot
(B8)
_.a = cos ¢_---0z
3y using
equations
(B7) and
(B8) we can
0Wr
0z
the
;ions
desired
t-momentum
(B6) and
(B9) into
It is desired
nents
momentum
the equation
tion
the
u will
stream
equation
from
function
That
u
and
function.
two
8Ws
0t
+ W
t
stream-function
(W s
to a second-order
be used.
s
after
combining
_
s _s
and
simplifying,
+ W t 0(p
0t
could
now be given
equation
expressing
(B9)
by substituting
equa-
(B2).
the
of a stream
0t
express,
0Wt
0s
in
equation
to obtain
as partials
0WzOr
equation
sin ¢0_.
Os
This
and
reduce
the
(the
stream
W t) to one
partial
is,
will
differential
u is zero
can be obtained
at the
number
equation.
hub and
by integrating
velocity
function).
But
shroud.
a vertical
compo-
of unknowns
A normalized
1 at the
along
the
in the
it changes
stream
The
mesh
func-
value
of
line.
,t
u=lw
By differentiating
Similarly,
the
line
given
to any
this
stream
mesh
]0
expression,
function
point
u
_u
r__
Ot
w
/0
and
then
tl
w
differentiate
this
expression
w
s
can be obtained
s 1, t I
u = 1--
We can
pwsrBdt
pW s rB
by integrating
integrating
1
dt - -W
with
respect
along
/s
along
a vertical
a horizontal
PWtrB
mesh
mesh
line:
ds
Sl
to
s
to obtain
79
ORIGINAL
PP,CE _,'_
OF POCR
QUALITY
au _
as
Hence,
the
derivatives
related
to the
of the normalized
velocity
components
rB 0 Wt
w
stream
function
in the s-
and
t-directions
are
by
au
as
_
rBp
w
Wt
(B10)
au_
at
The
flow
quantity
w.
B
is the
However,
the overall
channel.
stream-surface
the
variation
blade-to-blade
B
of
in local
z
and
By solving
0l
are
the
equation
and
r,
for
sheet
and
r) = 0tr(Z,
Wt
corresponding
so that
0-coordinates
(B10)
S
stream
is used,
B(z,
where
0tr
spectively.
W
thickness
spacing
is a function
rBp
w
w
and
thickness
is the
we can
use
r) - 0 l(z,
r)
of the
Ws
trailing
and
to the
mass
overall
mass
is not known,
flow
through
so that
one blade
(BII)
and
leading
differentiating,
blade
re-
surfaces,
we obtain
V
aWt
_
w
as
102u
rBp Las 2
+± a_BB
+1
_U
B as
_S
p 0s
(B12)
aWs_
w
ot
Equations
Wt
8O
in favor
(B10)
and
of the
(B12)
stream
[_a2u
at
rBp la_2
are
now
function
substituted
u:
B at
into
equation
P at
(B9) to eliminate
W
S
and
0W r
0W z _
Oz
ar
w
[_ 02u
rBPL
02 u + 0u (sin_
Os2
Ot2
-_s\
_
+1
8B +_i Op_
B 8s
r
p Os
0t
(B13)
Now substitute
u
equations
....
0u
Ot 2
Os
0s 2
(B6) and
+1
0B
(B13)
+1
B 0s
into
equation
(B2) and
Op_ 0___ _ 8u
p 0s
8t
+i__ __
0B
_
B
+ rBp_
O(rV0)
WWs/ t.
entropy
change
dS
can be calculated
dS=
at
the
particle
entropy
of a particle
at stagnation
at actual
conditions
(either
takes
The
less
static
equation
calculation
than
temperature
T
flow
velocity
absolute
(B15),
since
___aI +T__Os_
at
at
using
I = CpT!-I
a(CpTI
static
is calculated
wh,
- wXat
p 0t
0s
at
+F
= 0
0s
tl
as the
entropy
at
(B14)
p
is the
or
dT"
dS = Cp _-];
This
Op +0__
R d]2
P T
Since
0t
to obtain
from
dT
C
+i
OI +T__
IW0
The
rearrange
relative),
same
we can
also
of that
use
R dp"
P"
(BI 5)
values.
from
T = T"
- W2/2C
P
.
Using
this
with
yields
Cp T")
RT"
p"
_
at
+ K__
2
Rap"
_Cpp"
at
81
But
T!- T"-2w_'1
__wrp2
2C
P
and
Or
--=COS
(p
0t
so that
_ aI +Ta__2_s =w 2 rcos
at
0t
Now,
substitute
equation
(B16)
(p _ RT'_____'
p"
at
2
_p"+W2_cR_____
pp"
into
equation
(B14)
(B16)
8p"
0t
to obtain
\
__02u+_____82u
_u_
8s 2
at2
3s
+___10B
B as
p 0s
Ot
- _t
B
p 0t
as]
a(rVo)
+ rBpI!0
WWs[
0t
r
at
=0
+}W
2 +_
(BIT)
+F 1
where
(B18)
_ =I_cR
8p"
2
pp" 8t
= w 2 r cos ¢
and
culate
82
F t is given by equation
In the program,
sin _
the partials
of
(B5).
and cos
_p in equation
_0 are
(B17)
1
T"
RT"
0p"
p"
Ot
stored,
by
OT"/
Ot
so that
(BI9)
it is more
convenient
to cal-
O__ _
1
as
cos
a (sin
_o
_o) _
1
as
sin
O (cos, (p)
¢
0s
(B 20)
O__ =
1
at
Either
sin
the middle
q_ is near
(B17),
and within
difference
are
input
whirl
the
right
Ft
and
a(rV0)/at
the blade.
assumed
Within
using
is
of the
From
stream
dmwnstream
must
Outside
the
the
input,
function
u.
J
on whether
cos
_p or
can be calculated
blade,
at
the
whirl
are
is specified
the
by a finite-.
the first
partials
upstream
at
outside
is calculated
For
au_dX
du
differently
iteration,
calculated
(or
the
par-
from
the
earl be estimated)
of the blade,
rBpWs
du
(B21)
w
of the blade,
is caused
calculate
atrv0)/at
Hence,
d (rV0)o
du
by the hub-shroud
F t from
from
equation
the blade
rBpW s
(B22)
at
Ft
at
be calculated
the
o (rV o)
The
_p)
depending
iteration.
_Ok_dR
Ot
We can
used,
the blade,
O(rVo)
Similarly,
term
O(cos
sin (p
the previous
to be zero.
distribution.
as a function
1
at
or
approximation
rials
¢
_p) _
zero.
In equation
blade
term
cos
a(sin
w
pressure
(B5).
loading
The
gradient
blade-to-blade
by assuming
constant
induced
pressure
entropy
by the blade
gradient
blade
lean.
ap/aO
to blade,
so that
dp = p Cp dT
Using
the fact
that
T = T"
- W2/2Cp,
we get
-ow aw
O0
since
T"
Now,
is constant
substitute
from
this
blade
in equation
(B23)
(B24)
DO
to blade.
(B5) to get
83
Ft
The
blade-to-blade
pendix
sure
G, by using
gradient.
velocity
gradient
equation
(G2).
= _ ___00
W 0W
at
00
is calculated
Outside
the blade,
84
the blade.
d(rV0)/dm,
there
as explained
is no blade-to-blade
in appres-
Hence,
Ft = 0
outside
from
(B2 5)
(B26)
APPENDIX
DERIVATION
The
tional
general
from
of the
Newton's
equation
relative
general
the
on the
The
approximate
velocity-gradient
In this
velocity
midchannel
equation
for
the
velocity-gradient
equation.
meridional
EQUATION
is an expression
velocity.
force
to determine
velocity
The
equation
three-dimensional
is used
supersonic
OF VELOCITY-GRADIENT
velocity-gradient
derivative
C
value
equation
program
the
distribution
stream
of the
direc-
is derived
velocity-gradient
when
there
is locally
surface.
can be written
(C1)
where
W cos
a'
r
cos 2
_
sin 2
W
/3 + sin
_ cos
_
dW
m - 2w sin/3
dm
r-
r
C
dW
b -
W cos2/jr
sin a
+ cos
a
cos/3
m
dm
C
= W sin
C
o_ sin/3
cos/3
+ r cos/3
(dw0
(C2)
+ 2w sin _
7
Equation
tional
(C1) is
term
from
the
tion
along
these
input
specification
for
are
primarily
to neglect
viscous
the blade
Using
this
as equations
to allow
streamlines
Outside
region.
sion
same
T(dS/dq),
forces
sistent
the
row,
identity
for
(B13)
variation
of relative
adiabatic
and
in entropy.
flow
along
is possible
the
c = 0.
This
and
relation
the
of reference
The
total-pressure-loss
in Me streamwise
forces
(B14)
direction
hub-shroud
follows
from
2 but with
entropy
variation
distribution.
only with
viscous
so that
mesh
the fact
W 0 = V 0 - cvr,
the
is known
Entropy
forces.
it is reasonable
addi-
variaHowever,
and
con-
lines.
that
we can
d(rV0)/dm
derive
= 0
the
in this
expres-
85
dW0
_
W 0 + 2wr
dm
When
this
The
mesh
is substituted
general
lines
into
the
expression
velocity-gradient
in the
t-direction,
tion of the orthogonal
for
equations
so that
mesh.)
sin
o_
r
c, we find
(C1) and
q = t.
that
(C2) will
(See appendix
c = 0.
be applied
B and fig.
along
3 for
vertical
a descrip-
Then
dr
- cos
(¢
dq
(c3)
dz_
--
sin
q_
dq
Note
that,
by using
equation
(C3),
sin(a
- (p) = sin c_ dr + cos
dq
adz
dq
"_
(C4)
cos(_ - _) = cos _-
dr
dz
sina--
dq
Using
dW
dt
equations
_
(C3)
W COS2fl cos(c_
rc
and
(C4) in equations
- 9o) + cos/3
dm
dW
In sin(_
(CI)
and
- (¢) -
dq
(C2),
J
we can
r
/ w sin2/3
get
+ 2w sin
fit
cos W
+cd0+l(
at°
(C5)
We can
conditions.
86
express
We use
T
dS
in terms
of the
relative
velocity
W
and
relative
stagnation
W2
W
_
W Tt
2C
By using
equations
0315)
and
TdS=
Now we can
write
(C6),
we get
dT"
RT"
Cp
(C6)
mm
P
dp" +W2_
P"
R--_-_
Cp \2p"
the velocity-gradient
equation
dW = (aW + b)dt
dT "_
--
C
-
(C7)
P2T"/
(C5) as
+e
+ Wf
W
(C8)
where
2
a = cos
/3 cos(a,
. 2
sm /3 cos
- _)
rc
b=cosl?-_mm
sin(o_-_o)-2a_
Equations
(C8) and
(C9) can
the
blade
row.
angle
of
r,
in favor
_,
and
W.
This
cos fi dO
dt
sin_
f = dT"
T"
2
the
c_ sin/3
r
cos
e = Cp dT_ - co dk-
side
_ + sin
be used
can
Cp dT"
easily
+ _
/dw0 +2co
sin
(C9)
dp"
_ _
_C P"
2 P
directly
/3 is not known
e)+reos/3/--_m
within
directly
be done
the blade
so that
row.
we desire
However.
out-
to eliminate
since
k
W0 -
o0r
r
and
87
Wo
sin _ =
W
Then
cos
3
is obtained
from
cos2fi
Also,
c = 0
substitutions,
in equation
equation
(C2) outside
= 1 - sin2_
the blade
as was
shown
previously.
With
these
(C5) becomes
/
dW=_aW+
c +d
W
\
COS
\
/_|dt
/
+ e +Wf
W
(C10)
where
cos(a - _)
a-
r
c
e
cos(a - _) +
_-
r2 r2_/IX
- r cor
2
C
dW
d
m sin(a
__
- _o)
dm
(C_I)
e = C
dT!p
1
co dk-
dT"
p
f_
cos _ =
88
C
dT"
R dp"
2T"
2 Cpp"
RT"
+-p,,
dp"
Downstream
(rVo)
tions
0
of the
isusedinsteadof
blade,
_-.
outlet
whirl
Equations
should
be used.
(A7) to (All)
are
Hence,
obtained
in equation
directly
from
(Cll),
equa-
(C8) to (Cll).
89
APPENDIX
LOSS
An approximate
pressure.
input
This
station
quantities
loss
may
vary
loss
pressure
For
where
the
loss
case
the
from
the
loss
change
fraction
input
stagnation
stagnation
in whirl
T'
o
along
is known
First,
stagnation
at the
downstream
pressure.
These
the
is given
in all
pressure
is calculated.
ideal
pressure
So,
station
input
stagnation
is calculated.
outlet
relative
by the
or the
If the outlet
downstream
pressure
the
is specified
pressure
at the
of stagnation
is calculated
the
hub to tip.
of stagnation
by reducing
pressure
either
from
CORRECTIONS
is made
in stagnation
of stagnation
tional
correction
by specifying
the fractional
loss
loss
D
cases,
from
hub
is given
outlet
as input,
the fractional
to tip.
as input,
total
the frac-
temperature
T'
o
a streamline,
= T: +
I
(DI)
C
P
Then
the
ratio
of actual
to ideal
stagnation
pressure
0o
(";,)
ideal
Within
at the
loss
nation
the blade,
leading
fraction
the loss
edge
is the
pressures.
is distributed
from
('y- l)
(D2)
V;4
linearly
to the fraction
given
same
it is expressed
whether
is calculated
or as specified
or calculated
by the
at the blade
in terms
input,
trailing
of relative
from
edge.
or absolute
zero
The
stag-
Thatis,
(D3)
vt
Pideal
When
this
as follows:
90
ratio
is known,
the density
!
Pideal
can be calculated
as a function
of the
velocity
W.
, T
P = Pi ,'L'T
p"
V{ I
(D4)
P_.tdeal
where
T = T! - w2
I
Equation
sity.
tion
(D4) applies
Equation
means
of equation
loss
(D4) is derived
(3) of reference
is recalculated
the
with
2.
each
The
correction
by calculating
as equation
value
iteration.
of
+ 2w)_ - (c0r) 2
2C
P
(B6)
The
value
the density
in reference
_,_"/_":_ideal varies
of
p
(D5)
13.
at each
is used
from
the
Equation
point
(D5)
of the
in checking
ideal
den-
is equa-
region
and
continuity
by
(A12).
91
APPENDIX
DEFINING
When
REDUCED-MASS-FLOW
midchannel
stream
surface
not be obtained
directly
by solving
the
an approximate
solution
this
the
to the full-flow
depends
strongly
tions
that
will
flow,
if both
ratio
as the
lines
will
lines
throughout
boundaries.
flow,
mass
by the
most
and
flow,
the flow
inlet
outlet
whirl,
and
factor
is specified
outlet
then,
rotational
in the
same
This
speed
That
and
are
is,
the
curvatures
and
full-flow
With
extending
solution
the
a reduced
reduced
condi-
mass
in the
reduced-flow
as
can-
However,
to establish
solution.
result.
angles
is the boundary
the flow
and
method.
rotational
tA1)).
solution
so it is important
the
will
(eq.
the solution
same
stream-
the full-flow
stream-
field.
consideration
Again,
and
flow,
a reduced-flow
stream-function
flow
the
supersonic
equation
by getting
solution,
whirl
a similar
locally
velocity-gradient
statable
outlet
PROBLEM
stream-function
reduced-flow
approximately
In summary,
92
the
the inlet
have
Another
mass
ratio.
on the
give
has
can be obtained
solution
E
distribution
whirl,
for
the
will
are
rotational
all
by REDFAC,
at the upstream
be similar
reduced-mass-flow
speed
input
and
conditions
reduced
speed
to the
are
all
solution,
in the
as explained
reduced
reduced
the
same
inthe
mass
ratio.
section
and
downstream
mass
flow
by the
same
flow,
This
inlet
reduction
INPUT.
if the
and
APPEND_
INCIDENCE
The
and
solution
region
downstream.
be no difference
the leading
condition
would
region.
Since
mean
flow
cause
of this,
trailing
edge.
blade-shape
The
chord
solidity
and
along
ity of 2 or
distance
zero
nonzero
and
actual
incidence
near
the
is made
the blade,
there
should
of the
deviation
lines
angles,
shape
at
this
in the blade
deviation
angles,
the
trailing
edges.
Be-
and
blade
is,
side
blade
and
leading
to the
that
on either
incidence
the
within
other;
of a particle
surface
more,
limits.
to be the
distance.
When
The
line
over
angle
by the
the
partial
appropriate
shape
near
the
leading
to a streamline.
Therefore,
Therefore,
flow
an iterative
iteration,
imate
solution
From
the
From
continuity,
angle
no blade-shape
is obtained,
requirement
the leading
and
edge,
to
is not known
is used
the flow
of continuous
so that
then
angle
the blade.
in the
in advance
of 1/2
and
_0/_t
at the blade
the
f3 is calculated
first
the
is
is linmust
equation
the
leading
correction.
iteration,
be
is
corrected
s-coordinate
is not
throughout
the
between
correction
with
that
a solid-
momentum
correction
Therefore,
blade
or less,
solidity
This
the blade--shape
After
true
stream-function
is changed,
is made.
tangential
the angle
at
For
tangential
to
differs
the
with
It is assumed
to make
a streamline
solidity
a solidity
_0/0t.
_0/_s.
along
on solid-
a streamline.
angle
shape
and
_0/_s
along
linearly
and
depending
the
varies
within
a0/_s
chord
or trailing
in the program
and for
flow
the blade
correction
and
chord;
chord,
in radius,
of that
distance
only
procedure
blade
purposes
instead
to the
is made
true
is a change
distance
derivatives,
from
of the blade
of the
or trailing
However,
correction
The free--stream
The
prescribed
t3.
1/2
of the
is made
leading
and
calculation
is 1/6
correction
a distance
there
is used
chord.
for
ratio
For
distance
at the blade
to the flow
expressed
edges.
of the blade
made
1/6
is defined
the
decreased
is made
between
an s-coordimte
early
first
upstream,
each
by using
correction
varies
trailing
is 1/2
continuous
close
match
were
always
the blade
correction
distance
the leading
the
If there
almost
an empirical
the blade-to-blade
fl,
are
should
automatically
not follow
subregions:
momentum
edges.
be satisfied
there
regions
tangential
trailing
will
This
these
three
into three
CORRECTIONS
edges.
The
ity.
in the
and
AND DEVIATION
is divided
These
F
is
changed.
edge.
On the
an approx-
the
region.
momentum,
93
w4s 0WmB
' b
where
NBL
is the
number
of blades.
Hence,
since
tanfl
= W0/Wm,
B × NBL
Pbf
2_
Pfs
tan flbf = tan fifs
Equation
(F1) is used
varied
linearly
within
of the partials
of
to calculate
flbf"
the blade for the
The difference
fibf - fib (see fig.
specified
distance.
We can express
DR
do
cos(oz- ¢) + "--=
sin(aat
f
dm
this
is solved
for
(a0/aS)bf,
a0
point.
The
INDEV
for
tion
00/as
uses
calculations
each
the
for
iteration.
same
is not needed.
method
these
When
- _)
(F2)
cos (_- (p)
f
to calculate
sin(_
at
r
(F2) is used
]
q_)]
J
we obtain
tan _bf
Equation
12) is then
fl in terms
O by
tan_bf=r--=r
When
(F1)
(a0/aS)b
f
from
blade-shape
REDFAC
to correct
fl
the
linearly
corrections
is less
within
the
than
blade
varying
are
1,
done
the final
(subroutine
_
at each
mesh
by subroutine
transonic
LINDV),
calculabut
APPENDIX
BLADE
The
tion
blade
the
case,
tropy
velocities
The
desired
AND BLADE-TO-BLADE
can
be calculated
blade--to--blade
relation,
q = 0 and
S are
VELOCITIES
surface
is obtained.
To obtain
this
SURFACE
velocity
we use
the
dr/d0
= dz/d0
in the
blade--to-blade
constant
G
= 0.
once
gradient
AVERAGE
a meridional
_W/a0
velocity-gradient
Also,
midsurface
depends
on
equations
it is assumed
direction.
With
DENSITIES
that
this,
dW0/dm.
(C1) and
TI,
solu-
X, and
equations
(C2).
the
In
en-
(C1) and
(C2)
become
0W
m=Wsin
D0
Using
the
fact
that
_ sin_
V 0 = W 0 + wr
cosfl
and
+r
(GI)
cosfl
W 0 = W sin_,
we can
rewrite
equation
(G1) as
DW
d(rV0)
- cos fl _
00
dm
Since
are
only
a midchannel
constant
blade.
from
(This
assumption,
may
solution
blade
is obtained,
to blade.
This
be in considerable
we assume
means
error
that
near
W
that
cos _
varies
the leading
and
linearly
or
trailing
d(rV0)/dm
from
blade
edge.)
to
With
this
then
_ Wl
Wmid
Integrating
(G2)
(G2) from
blade
+ Wtr
(G3)
2
to blade
(from
0 = Ol
Wtr
= Wmi d +
B cos
to
0 = 0tr ) and
using
(G3),
we obtain
fl d(rVo)
2
dm
(64)
Wl
Note
that
ref.
16).
equation
(G4) is very
= Wmid
close
-
to that
B cos
2
_ d(rV0)
dm
developed
by Stanitz
(eqs.
(16) and
(17),
95
Equation (G4)is used in subroutine BLDVEL to obtain blade surface velocities from
the midchannel meridional solution.
It is desirable to consider the blade-to-blade variation in density to satisfy continuity for the blade passage. Equation (G4) gives irfformation we need to do this. Since the
midchannel solution is considered to be representative of all meridional-plane stream
surfaces, we can consider the solution to be based on average blade-to-blade conditions.
This means that in equation (B10) average blade-to-blade velocity anddensity shouldbe
used. Equation (B10) becomes
0u_
0s
rB @Wt)
w
av
(G5)
0t
If we use
Simpson's
rule
to calculate
PWs )a v - OlWs_
The
velocity
component
WS
w
v
the average
value,
we obtain
l +4PmidWs_mid
6
can be expressed
+OtrWs_
in terms
o£ the
tr
(G6)
velocity
and
flow
angles
by
W
This
relation
constant
holds
also
on the
in the blade-to-blade
= W cos
s
blade
¢ cos((_
surfaces,
since
(0) direction,
W s,l
- _)
(G7)
the flow
angles
are
assumed
to be
so that
= W l cos
_ eos(e-
_)
(G8)
Ws, tr = Wtr
By using
equations
(G3),
(G4),
and
oW s)
= OavWs
av
96
cos ¢_ cos(or
(GS) in equation
+ Pl - Ptr
_-2
cos/3
- _o)
(G6),
cos(s-
we can express
q)){W l - Wtr)
(pW s
v
as
(G9)
where the subscript mid is omitted from Ws, and
- Pl
+ 4Pmid
+ Ptr
Pay
\_en
a solution
of equation
mated
from the previous
tion (Gg) and use equation
(A1) is obtained,
iteration,
except
(G5) to obtain
all
for
quantities
W s.
in equation
So we solve
(G9) can be esti-
for
W S in equa-
_)u
VV_
W
s
This
is the
first
iteration,
values
of
are used.
equation
Omid,
In a parallel
used
Pay
3t
rBp av
Ol'
manner,
is used,
Wl'
Wtr'
we can
_ cos(o ' - _o)(_)
- Wtr)
(GI0)
12p av
in sub_x)utine
= O'i'deal
Ptr'
(Ol - 0t r) cos
NEWRHO
and
cos
5,
derive
the
second
and
the
to calculate
term
eos(a
equation
W
from
is omitted.
- _0) from
for
s
the
0u/0t.
After
previous
On the
this,
the
iteration
Wt:
_u
Wt _
w w
0s
rBp av
(p/
- Ptr)
cos
3 sin(a
- ¢)(_
- Wtr)
(Gll)
120 av
97
APPENDIX
H
SYMBOLS
a
coefficient
in velocity-gradient
B
tangential
b
coefficient
C
P
specific
C
coefficient
in velocity-gradient
equations
d
coefficient
in velocity-gradient
equation
(A7)
e
coefficient
in velocity-gradient
equation
(A7)
F
vector
space
between
at constant
normal
gradient,
I
rothalpy,
C T!p 1
wk,
i
incidence
angle,
rad
m
meridional
streamline
n
unit
normal
P
pressure,
q
distance
R
gas
r
radius
from
radius
of curvature
c
pressure,
(A7} and
(C1)
J/(kg)(I4)
stream
in velocity-gradient
(A7) and
surface
along
meters2/sec
2
distance,
meters
and
(C1)
proportional
(A7)
stream
surface
curve,
meters
space
J/(kg)(K)
axis
entropy,
J/(kg)
S
distance
along
T
temperature,
t
distance
U
normalized
V
absolute
W
fluid
of rotation,
meters
of meridional
streamline,
meters
(K)
orthogonal
mesh
lines
in throughflow
mesh
lines
in direction
direction,
K
along
orthogonal
stream
fluid
velocity
to tangential
2
an arbitrary
constant,
equation
to midchannel
N/meter
S
98
(C1)
N/kg
coefficient
vector
(A7) and
rad
equations
to midchannel
f
r
blades,
in velocity-gradient
heat
sure
equations
function
velocity,
relative
meters/sec
to blade,
meters/sec
across
flow
meters
pres-
w
mass flow, kg/sec
z
axial coordinate, meters
angle betweenmeridional streamline and axis of rotation, rad, see fig. 4
fl
angle
between
_/
specific-heat
angular
prerotation,
(p
angle
w
rad;
meridional
plane,
rad;
see
fig.
4
between
fig.
in eq.
(A3)
in eq.
{A2)
4
equation,
defined
3
s-distance
function,
rotational
see
defined
meters2/sec
in stream-function
kg/meter
stream
and
equation,
coordinate,
(rV0) i,
coefficient
density,
vector
in stream-function
relative
p
velocity
ratio
coefficient
0
relative
line
and
axis
of rotation,
rad,
see
fig.
3
kg/sec
speed,
rad/sec_
see
fig.
4
Subscripts:
av
average
blade--
to-blade
b
blade
bf
blade
cr
critical
fs
free
h
hub
i
inlet
l
blade
m
component
mid
midchannel
O
outlet
r
component
in radial
S
component
in s- direction
t
tip,
tr
blade
flow
stream
surface
or
facing
direction
in direction
of meridional
rotation
streamline
blade-- to-blade
component
surface
of positive
facing
direction
in t-direction
direction
of negative
rotation
99
z
componentin axial direction
0
component
in tangential
direction
Superscripts:
'
absolute
stagnation
"
relative
stagnation
100
condition
condition
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