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Numerical solution of the converse problem for transonic flow in a curved channel
NUMERICAL SOLUTION OF THE CONVERSE PROBLEM FOR TRANSONIC FLOW IN A CURVED CHANNEL* L. A. DORFMAN Leningrad 7 January
(Received
THE converse
problem
on transonic
1972)
flow in a curved
channel
is solved
by using
spline smoothing of the streamlines, and of the flow parameters along them This device regularizes the process obtained in the course of the computations. of solving the Cauchy problem in the subsonic zone. We pose the problem shape
of one streamline
form of the streamline bourhood of the initial
of constructing
the flow in a curved
and the flow parameters
Notice
above cannot
that,
when obtaining
channel,
given the
Given the analytic
Having constructed this solution, we problem. and the flow parameters on it, and continue the its analytic properties.
In the general case, when the streamline is incorrectly posed in the subsonic (elliptic) outlined
it.
and the analytic data along it, there exists in a neighstreamline, by Kovalevskii’s theorem [lI, a unique
solution of the relevant Cauchy can find the adjacent streamline solution further, while retaining
solution
along
be used.
even if the initial a numerical
is not analytic, the Cauchy problem region, so that the method of
data are analytic
solution
in the subsonic
by a finite-difference
region,
method the approxi-
mation errors and the rounding-off errors may lead to the analytic property being destroyed; this shows itself in the solution surging and deviating away from the true solution. We therefore have to provide a procedure whereby the discrete solution obtained is reduced to an analytic form, i.e. a procedure for regularization on each streamline.
As this procedure
of the streamline
and of the parameters
*Zh.
mat.
vychisl.
Mat.
Fiz.,
we propose
13, 3. 799-802.
347
along it.
1973.
to use spline
smoothing
both
L. A. Dorfman
348
Notice
that,
for computing basis
in the case
of straight
flows involving
of a given distribution
suitable
finite-difference
tion lie in using derivatives.
through
of the parameters
a method was devised
in [2]
the velocity
on the
of sound,
along the channel
method was described.
a non-uniform
This
channels,
a transition
The elements
axis,
mesh and in the method for evaluating
method cannot
be used in the case
and a
of the regularizathe
of substantially
curved
channels. 1.
To construct
equations entropic
the curved
in the natural irrotational
system
streamlines,
it is best to write the flow
of coordinates.
flows of an ideal
gas.
We shall
consider
They are described
the homo-
by the equations
[31: absence
of vorticity
ax/an =
(1)
d,
continuity d$/dn
(2)
= A,
where the dimensionless sionless
velocity
-
(3)
density
E = p/p,, is expressed
in terms of the dimen-
h = u/a* as
-
k--l
l/(k-I) h2
k+l
Here, n is the normal to the streamline streamline, the stream
To avoid the unwieldy more convenient passing through the normals In the
(Fig.
l),
x
is the curvature
k is the isentropic index, a, is the critical function, and p,, is the stagnation density. of the normals
of the
of sound,
to the streamline,
I/I is
it is
to use, instead of the normals, a system of fixed straight lines, the computational points on the initial streamline and close to
(Fig. (s,
construction
velocity
I)
1). coordinates,
all
(4)
-=~xl,sintJ+-cosl3, a1
(5)
d$ / dl = &I.sin 0,
Eq. (1) and (2) become ah
3s
Converse
problem
for transonic
flow in a curved
channel
349
FIG. 1. where 8 is the angle
between
the streamline
and the coordinate
lines
(we shall
call them cross-sections). To solve
our problem,
we write Eqs.
(4) and (5) in the difference
form.
On
the i-th cross-section (6)
E(V) ZzT
1 --
[
i/(k--l)
k-l
(h(V))Z
I
k+l
I
where (AI/J)~ is found from the given (AZ)j at the start of the cross+ection, and &/al is found from the computed value of the right-hand side of (4). At the start
of the iterations
we take (dx/dZ$),
The regularization of the streamline, of the streamline values
process
amounts
(thsin @\~l
in essence
the values
spline
the curvature
of spline
approximation
of h and dX/ds are smoothed.
the right-hand
approximation
= (~Asin@j-
to evaluating
and also sin 8 and cos 8, by means
obtained;
are used to evaluate
We use cubic
= (&/al),,
sides
of Eqs.
These
(4) and (5).
141: the curve f(x),
given at the points
xi, i = 0, 1,. . . , N, which occur at unequal intervals, is replaced by the curve g (x), which is continuous along with its first and second derivatives, so that we satisfy the conditions
L. A. Dorfmm
350
at
N
s
g(G)-_i
[g”(x) I2 ds = min,
<
Z(
x0
6yi
i-0
where 6y, and S determine equal
the degree
’
of smoothing
s,
1
(we chose
S = N + 1 and
6yi (= 8~)).
The relevant ALGOL program [4] also gives the required information for evaluating g’ and g”, i.e. we can find x, sin 0, cos 8, and the length of the arc s. When using this program we have to employ an (x, y) coordinate system, in which the streamline y = f(x) is expressible by a single-valued function. Notice also that the spline
curve g(x) has straight
ends of the interval. zero curvature
Hence,
pieces,
in order to obtain
at the ends of the streamline,
g”(x,) = g” (x,)
reasonably
= 0, at the
accurately
we have to extrapolate
beyond the interval (x,, 3~~) (this can be done by continuing a circle through the last three points at the ends of the interval) and plotting curve in the extended 2.
The problem
approximated
the nony = f(x) drawn the spline
interval. is solved
as follows.
by a curve smoothed
and cos 8 are evaluated.
The initial streamline by cubic spline approximation,
In addition,
we smooth the initial
values
(j = 0) is and x, sin 8, of h and find
&/as. To evaluate the next streamline, spaced (AZ), away at the initial crosssection (i = 0), we find the flow rate (AI,+), from Eq. (7) and X1(l) from Eq. (6) to a first approximation. We then find from Eqs. (6) and (7) the coordinates of the next point, streamline.
and the value This
of X at this point,
streamline
etc. travelling
is found more accurately,
along the entire
by evaluating
x,
and cos 8, and the smoothed X and dh/& by means of spline smoothing, performing the computations to the second approximation, starting with
sin 8, and
improvement of the value of (A$)0 at the input, etc. until the positions of the streamline become coincident. On replacing the streamline obtained by the smoothed
streamline,
we pass
on to computing
the next streamline.
Our method was checked against exact solutions. In particular, we considered the example of the complicated Ringleb flow [3]. In spite of the fact that a large part of the flow takes place in the subsonic region (in Fig. 2 we show half the channel separated from the flow) and the coarse, good agreement was obtained between the last line streamline b. The velocity distribution on this streamline It is vital that the solution deteriorates considerably even
mesh is fairly a and the exact is in exact agreement if the approximation
Converse problem
fortrunsonic
351
flowin a curved channel
FIG. 2.
-i.
u
u
1.n
x
FIG. 3.
of the streamline and velocity are replaced by interpolation. During interpolation, the velocity distribution (curve c) is strongly oscillatory, and becomes more so as we move away from the supersonic zone. In Fig. 3 we show computational velocity
distribution
results
on a plane transonic
nozzle
A = I + 0.9 th (1.75x) on the axis and 25 stream-tubes
with with
352
L. A. Dorfman
sy = 1.5 x lo-+, &I = 4 x 10-a. Notice
in conclusion
more complicated
case
that the present of isentropic
but also
for axisymmetric
twisting
of the flow can be taken
magnetohydrodynamic first proposed The author performing
ring-shaped
flows.
curved
channels
(in the latter
It is also possible
solution
to the supersonic
F. P. Belitskii
directly
flow, not just for plane
into account).
The computer
in [s], as applied thanks
method can be extended
rotational
to the
channels
case to compute
of the converse
problem was
part of a nozzle.
for programming
the computer
and
the computations. Tran&ted
by D. E. Brown
REFERENCES 1.
PETROVSKII, I. G. Lectures uravneniyakh
s chastnymi
OR Partial Differential Equations (Lektsii proizvodnymi), Fizmatgiz, Moscow, 1961.
Calculation of the flow in a Lavalle zhidkosti i gaza, 5, 1042, 1967.
2.
PIRUMOV, U. G. SSSR, Mekhan.
3.
~LNE-THOMSON,
4.
REINSCH, C. I-I. Smoothing by spline functions,
5.
ROSLYAKOV, G. S. and TELENIN, G. F. Survey of papers on the computation of stationary axisymmetric gas flows, performed at the computational centre of
L. Theoretical
hy~rod~ami~s,
nozzle
ob
Izu. Akad.
Nauk
MacmiIIan, 1960. Namer. Math., 10, 172463,
MOSCOWState University. in: Numerical Methods in Gas Dynamics metody v gazovoi dinamike), MGU, Moscow, 5-19. 1963.
(Chisl.
1967.
(Received
THE converse
problem
on transonic
1972)
flow in a curved
channel
is solved
by using
spline smoothing of the streamlines, and of the flow parameters along them This device regularizes the process obtained in the course of the computations. of solving the Cauchy problem in the subsonic zone. We pose the problem shape
of one streamline
form of the streamline bourhood of the initial
of constructing
the flow in a curved
and the flow parameters
Notice
above cannot
that,
when obtaining
channel,
given the
Given the analytic
Having constructed this solution, we problem. and the flow parameters on it, and continue the its analytic properties.
In the general case, when the streamline is incorrectly posed in the subsonic (elliptic) outlined
it.
and the analytic data along it, there exists in a neighstreamline, by Kovalevskii’s theorem [lI, a unique
solution of the relevant Cauchy can find the adjacent streamline solution further, while retaining
solution
along
be used.
even if the initial a numerical
is not analytic, the Cauchy problem region, so that the method of
data are analytic
solution
in the subsonic
by a finite-difference
region,
method the approxi-
mation errors and the rounding-off errors may lead to the analytic property being destroyed; this shows itself in the solution surging and deviating away from the true solution. We therefore have to provide a procedure whereby the discrete solution obtained is reduced to an analytic form, i.e. a procedure for regularization on each streamline.
As this procedure
of the streamline
and of the parameters
*Zh.
mat.
vychisl.
Mat.
Fiz.,
we propose
13, 3. 799-802.
347
along it.
1973.
to use spline
smoothing
both
L. A. Dorfman
348
Notice
that,
for computing basis
in the case
of straight
flows involving
of a given distribution
suitable
finite-difference
tion lie in using derivatives.
through
of the parameters
a method was devised
in [2]
the velocity
on the
of sound,
along the channel
method was described.
a non-uniform
This
channels,
a transition
The elements
axis,
mesh and in the method for evaluating
method cannot
be used in the case
and a
of the regularizathe
of substantially
curved
channels. 1.
To construct
equations entropic
the curved
in the natural irrotational
system
streamlines,
it is best to write the flow
of coordinates.
flows of an ideal
gas.
We shall
consider
They are described
the homo-
by the equations
[31: absence
of vorticity
ax/an =
(1)
d,
continuity d$/dn
(2)
= A,
where the dimensionless sionless
velocity
-
(3)
density
E = p/p,, is expressed
in terms of the dimen-
h = u/a* as
-
k--l
l/(k-I) h2
k+l
Here, n is the normal to the streamline streamline, the stream
To avoid the unwieldy more convenient passing through the normals In the
(Fig.
l),
x
is the curvature
k is the isentropic index, a, is the critical function, and p,, is the stagnation density. of the normals
of the
of sound,
to the streamline,
I/I is
it is
to use, instead of the normals, a system of fixed straight lines, the computational points on the initial streamline and close to
(Fig. (s,
construction
velocity
I)
1). coordinates,
all
(4)
-=~xl,sintJ+-cosl3, a1
(5)
d$ / dl = &I.sin 0,
Eq. (1) and (2) become ah
3s
Converse
problem
for transonic
flow in a curved
channel
349
FIG. 1. where 8 is the angle
between
the streamline
and the coordinate
lines
(we shall
call them cross-sections). To solve
our problem,
we write Eqs.
(4) and (5) in the difference
form.
On
the i-th cross-section (6)
E(V) ZzT
1 --
[
i/(k--l)
k-l
(h(V))Z
I
k+l
I
where (AI/J)~ is found from the given (AZ)j at the start of the cross+ection, and &/al is found from the computed value of the right-hand side of (4). At the start
of the iterations
we take (dx/dZ$),
The regularization of the streamline, of the streamline values
process
amounts
(thsin @\~l
in essence
the values
spline
the curvature
of spline
approximation
of h and dX/ds are smoothed.
the right-hand
approximation
= (~Asin@j-
to evaluating
and also sin 8 and cos 8, by means
obtained;
are used to evaluate
We use cubic
= (&/al),,
sides
of Eqs.
These
(4) and (5).
141: the curve f(x),
given at the points
xi, i = 0, 1,. . . , N, which occur at unequal intervals, is replaced by the curve g (x), which is continuous along with its first and second derivatives, so that we satisfy the conditions
L. A. Dorfmm
350
at
N
s
g(G)-_i
[g”(x) I2 ds = min,
<
Z(
x0
6yi
i-0
where 6y, and S determine equal
the degree
’
of smoothing
s,
1
(we chose
S = N + 1 and
6yi (= 8~)).
The relevant ALGOL program [4] also gives the required information for evaluating g’ and g”, i.e. we can find x, sin 0, cos 8, and the length of the arc s. When using this program we have to employ an (x, y) coordinate system, in which the streamline y = f(x) is expressible by a single-valued function. Notice also that the spline
curve g(x) has straight
ends of the interval. zero curvature
Hence,
pieces,
in order to obtain
at the ends of the streamline,
g”(x,) = g” (x,)
reasonably
= 0, at the
accurately
we have to extrapolate
beyond the interval (x,, 3~~) (this can be done by continuing a circle through the last three points at the ends of the interval) and plotting curve in the extended 2.
The problem
approximated
the nony = f(x) drawn the spline
interval. is solved
as follows.
by a curve smoothed
and cos 8 are evaluated.
The initial streamline by cubic spline approximation,
In addition,
we smooth the initial
values
(j = 0) is and x, sin 8, of h and find
&/as. To evaluate the next streamline, spaced (AZ), away at the initial crosssection (i = 0), we find the flow rate (AI,+), from Eq. (7) and X1(l) from Eq. (6) to a first approximation. We then find from Eqs. (6) and (7) the coordinates of the next point, streamline.
and the value This
of X at this point,
streamline
etc. travelling
is found more accurately,
along the entire
by evaluating
x,
and cos 8, and the smoothed X and dh/& by means of spline smoothing, performing the computations to the second approximation, starting with
sin 8, and
improvement of the value of (A$)0 at the input, etc. until the positions of the streamline become coincident. On replacing the streamline obtained by the smoothed
streamline,
we pass
on to computing
the next streamline.
Our method was checked against exact solutions. In particular, we considered the example of the complicated Ringleb flow [3]. In spite of the fact that a large part of the flow takes place in the subsonic region (in Fig. 2 we show half the channel separated from the flow) and the coarse, good agreement was obtained between the last line streamline b. The velocity distribution on this streamline It is vital that the solution deteriorates considerably even
mesh is fairly a and the exact is in exact agreement if the approximation
Converse problem
fortrunsonic
351
flowin a curved channel
FIG. 2.
-i.
u
u
1.n
x
FIG. 3.
of the streamline and velocity are replaced by interpolation. During interpolation, the velocity distribution (curve c) is strongly oscillatory, and becomes more so as we move away from the supersonic zone. In Fig. 3 we show computational velocity
distribution
results
on a plane transonic
nozzle
A = I + 0.9 th (1.75x) on the axis and 25 stream-tubes
with with
352
L. A. Dorfman
sy = 1.5 x lo-+, &I = 4 x 10-a. Notice
in conclusion
more complicated
case
that the present of isentropic
but also
for axisymmetric
twisting
of the flow can be taken
magnetohydrodynamic first proposed The author performing
ring-shaped
flows.
curved
channels
(in the latter
It is also possible
solution
to the supersonic
F. P. Belitskii
directly
flow, not just for plane
into account).
The computer
in [s], as applied thanks
method can be extended
rotational
to the
channels
case to compute
of the converse
problem was
part of a nozzle.
for programming
the computer
and
the computations. Tran&ted
by D. E. Brown
REFERENCES 1.
PETROVSKII, I. G. Lectures uravneniyakh
s chastnymi
OR Partial Differential Equations (Lektsii proizvodnymi), Fizmatgiz, Moscow, 1961.
Calculation of the flow in a Lavalle zhidkosti i gaza, 5, 1042, 1967.
2.
PIRUMOV, U. G. SSSR, Mekhan.
3.
~LNE-THOMSON,
4.
REINSCH, C. I-I. Smoothing by spline functions,
5.
ROSLYAKOV, G. S. and TELENIN, G. F. Survey of papers on the computation of stationary axisymmetric gas flows, performed at the computational centre of
L. Theoretical
hy~rod~ami~s,
nozzle
ob
Izu. Akad.
Nauk
MacmiIIan, 1960. Namer. Math., 10, 172463,
MOSCOWState University. in: Numerical Methods in Gas Dynamics metody v gazovoi dinamike), MGU, Moscow, 5-19. 1963.
(Chisl.
1967.