The use of the Hodograph Method in the design of a transonic compressor cascade is discussed. The design of the flow mode in the transonic compressor cascade must be as follows: the flow in the nozzle part should be uniform and smooth; the location of the sonic line should be reasonable; and the aerodynamic character of the flow canal in the subsonic region should be met. The rate through cascade may be determined by the velocity distribution in the subsonic region (i.e., by the numerical solution of the Chaplygin equation). The supersonic sections A'C' and AD are determined by the analytical solution of the Mixed-Type Hodograph equation

Chen, Zuoyi

Guo, Jingrong

English

NASA Technical Reports Server

593
ThirdIn_mafio_l Conferenceon bwerse Design Conceptsand Op_iza_on inEngineenng Sciences
(ICIDES-TIIh. Editor: G.S. Dulikravich. Washington D.C.. October 23-25. 1991.
r  T92"139731
Design of Transonic Compressor Cascades Using
Hodograph Method
Chen Zuoyi Gut Jingrong
(Dept. of Thermal Engineering, Tsinghua University)
(Beijing, China)
1. Introduction _..
The design of transonic turbine cascade using Hodograph Mathod is presented in Ref. 1,
2, 3, etc. But up to now, there are no published papers about the design of transonic
compressor cascade using Hodegraph Mathod. It is given in this article.
The design of flow mode in the transonic compressor cascade must be as follows: (1)
the flow in nozzle part should be uniform and smooth. (2) the location of sonic
line should be resonable, and (3) aerodynamic character of the flow canal in
subsonic region should be met. The rate through cascade may be determined by velocity dis-
tribution in subsonic region. (i.e. by the numerical solution of Chaplygin equation). The su-
personic sections A/C / and AD are determined by the analystical solution of Mixed-Type
Hodograph equation. If the shock wave exist that we should consider the flow turn by
the shock wave. (The Ushock wave-Mean Stream Line Turn Method" has been used).
2. The compressor canal design using the analytical solution of Mixed-Type Hodograph
equation
In general, the analytical solution of Mixed-Type Hodograph equation (i.e. the
nozzle solution) is used in the design of transonic turbine cascade. Can it be used in the design
of compressor? Our research shows that it is sure.
For example, if we use the generalization Tricomi approximation, the approximate
compressible function is:
• i
/
/
K ,(u) = b ' (a) / (1 - cba) s (1)
For this approximation, comparing the approximate compressible function Kt(a) to
the true compressible function K(a), the difference is very small in supersonic region but not
in subsonic region. This is suitable for the design of transonic turbine cascade because the ana-
lytical solution only apply in the supersonic region. But for the design of compressor cascade,
the analytical solution must be used in subsonic region. In order to.decrease the difference,
we should used the other boundary condition to determine the coefficients b and c, this is:
K. (_)1u," .o.3 = K(_)I M," .o.3 (2)
dK, (a) I dK(a)•°- ..0 (3)
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Third International Conference on Inverse Design Concepts and Opumization in Engineering Sciences
CICIDES-IID. Editor: G.S. Dulikravich. Washineton D.C.. October 23-25.1991.
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and we obtain the c, b as follows:
If y = 1.4 c =-0.28236
b = 1.15709
The comparision between K(a) and Ka(a) is shown in Fig.2.
3. To determine the leading edge shock wave and coming flow parameters of transonic
compressor cascade
On the basis of the sonicline location which has been given in design and according to the
application of analytical solution in compressor canal, we may obtain the profile C'A' and
DA, along with the velocity distribution in C'A ' and DA. (i.e. obtain the geometry of leading
edge and the Ma^, MaA,(see Fig.3)). If the computation model in Fig.3 is used, we can consid-
er that the flow turn proceeded from coming flow Mare0 at the leading edge of suction and
pressure surface respectively. If the turn angles are 6 and 6_,
thus di+ _i/ =
here • is geometric angle of leading edge.
Two shock waves are produced when the flow turn suddenly. The shock wave angles are
ill, f12, fl'_ and fl'2 respectively. Thus, from the relationship of oblique shock wave, we
obtain:
tg(fll - fl2)= 2ctgfll Malsin fll
tg(fl'l fl'2)2ctgfl'l[ Ma2 . 2_' IMam(y+ 1)+-- = lstn //1--1] / cos2fl' 2] (5)
fl'l -fl'2=O-(_1 -f12) (6)
Ma2 + 2
y-- 1 Ma 2 2- co5 fll
Ma22 = + i (7)
2 2 y-I 2 2
2y Ma i sin fl - 1 Ma i sin [3] + 1
y--I 1 2
Ma21 + 2 2
"2 y---_ Ma l c°s2 fl'l
= +
Ma2 2___yMalsin li l -- I Malsin +12 2 _/ y -- I 2 2fll
(8)
in previous five equations, the known parameters are Ma2, Ma 2' and q_, and unknown
parameters are _t, f12, fl'l, fl'2, Mare, so the solutions are completely dertermined
by the _ve equations.
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Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences
CICIDES-IID, Editor: G.S. Dulikravich. Washington D.C., October 23-25. 1991.
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If the shock wave is strong, the flow turn passing the shock wave should be
considered. The model of design is shown in Fig.4.
We consider that the shock wave strength is determined by the flow turn on
o
the meanstreamline, i.e. the shock wave angles /_1 and P2" are determined by
the flow turn angle 8 ", thus the follwoing relations may be obtained:
M2 = 2ctg_: + 2tg([3: --B2 ) (9)
°,, 2ctg_;sin2_; --,g(_ --_)(3,+co,2B;)
Ma2BI + 2
__ Ma2BI 2 _COS 1
M 2 = 3,--1 +
°_2 23' Ma 2 2 3,- l 2 2_
-- Ma BIsiny-1 Blsin #_ -1 2
+1
(10)
Now, the flow field can be divided into two parts, the region before the
shock wave and after the shock wave. In the region A'BA, applied the Hodograph
Mixed-Type Equation to determine the profile and velocity distribution of the C'A' and DB.
There are four parameters MaB1, MaB2, [3_ , and fl_ in the Eq.9 and Eq.10. If the MaB_
is determined, thus the relation between/_ _ and/_ can also be determined, and the design
of Hodograph Method that consider the sudden change proceeded from the shock wave may
be solved with the alternative method. For example, we assume the/?_ and therefore the lo-
cation of shock wave can be determined, i.e. the location of point B is determined, and the
MaB2 can also be determined. From the Eq.9 and Eq.10, the MaB_ and /?_' may be ob-
tained, and the _" may also be obtained. After this, we can obtain the turn-meanline EF. On
the basis of the EF we can solve the velocity distribution on the BA, and the Ma 2 can be ob-
tained. So that from the Eq.4-- Eg.8 the new B2" can be determined. Put the new//2" in pre-
vious calculation and till it is satisfied.
In the subsonic region, the numerical solution of Chaplygin Eguation can be used. i.e.
+ -_Q = .
oM M
o G il
(11)
4. Design example
Using previous theory and mathod, two compressor cascade have been designed. The one
is called J-3-Type cascade, and the other is called Ma-Type cascade which is designed on
the basis of velocity distribution of the compressor cascade which is provided by Ref.4.
The designed parameters of transonic compressor-3 profil are: Math number at outlet is
62, flow angle at outlet is 45 ° , cascade pitch is 50mm, the Math number and flow angle I..d
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Third Int_madonal Conference on Inverse Design Concepts and Optimization in Engineering Sciences
(ICIDES-IID. Editor:. G.S. Dulikr'avich. Washincton D.C.. October 23-25. 1991,
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at inlet are 50 ° and 1.22 respectively. The designed cascade profile is shown in Fig 5. The
profile coordinates and its velocity distribution are shown in Table 1.
Comparing the velocity distribution of J-3-Type cascade with the calculating of the
Time Marching method, we find the results are in agreement (see Fig.6).
The profile of Ma-Type cascade comparing with the cascade in Ref.4 is also in agree-
ment (see Fig.7).
5. Conclusion
1. The Hodograph Method may be used to the design of transonic compressor cascade.
2. The flow field may be divided into two parts for using Hodograph method design, to solve
the Hodograph Mixed-Teyp Equation in supersonic region and the Chaplygin Equation in
subsonic region.
3. If the strength of shock wave is large, the flow turn by shock wave, the UShock
wave-Meanstreamline turn Method u is suitable for this design.
Reference
1. Chen Zuoyi, Journal of Tsinghua Univ., Vol.21, No.4, 1981.
2. Chen Zuoyi, Journal of Tsinghua Univ., Vol.22, No.4, 1982.
3. G.Karadimas, Institute for Fluid Dynamics lecture Series 59.
4. William. T. Thompkins, Jr, GT and PDL Report, No. 162, Sept. 1981, MIT.
blal.,
f
\
1/
Fig.1 The flow model of
transonic eomprcssor cascade
I 1
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Third International Conference on Inverse Design Concepts and 09timizarion in Engineering Sciences
• " v' " r o .9
o •
F
7
-0°|
,I
-!
Fig.2 Comparision between Ka(_) and K(_) respecting
to various coefficients a and b
-' true K(o)
Ka(o): b -- 0.024 c =-0.78 / b
-s Ka(o): b=1.15609 c=-0.29236
..i:
._,.[_ streamline
Fig.3 The relationbetween theleadingedge
shock wave and coming flow parameters
Fig.4Calculationmodel
incompressor cascade
J
598
Third International Conference on Inverse Design Concepts and Optimizaaon in Engineering Sciences
OCIDES-UD. Editor: G.S. Dulikravich. Washington D.C.. October 23-25. 1991.
F
_'Cma,)
'2GO
]
Fig.5
Fig.6
Transoniccompressor cascadeJ-3-Typc with
Hodograph Method obliqueshock
Velocity distribution comparison between
Hodograph Method and Time Marching
method
Hodograph Method
----- Time Marching Method
.v
go
/ "GO
/ 64
40
' 30
Z0
-,0--t+---it-ritz+_,o _,
ii"
- YO
Fig.7 Comparison between the transonic
compressor cascade (Ma-Type) pro
ccedatfrom inverted calculation with
Hodograph Method and ReL4
-- Profile in Ref.4
.... Profile by Hodograph Method
L J


Design of transonic compressor cascades using hodograph method

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