Progress in the use of the Hodograph method of aerodynamic design is discussed. It was found that there are some restricted conditions in the application of Hodograph design to transonic turbine and compressor cascades. The Hodograph method is suitable not only to the transonic turbine cascade but also to the transonic compressor cascade. The three dimensional Hodograph method will be developed after obtaining the basic equation for the three dimensional Hodograph method. As an example of the Hodograph method, the use of the method to design a transonic turbine and compressor cascade is discussed

Chen, Zuoyi

Guo, Jingrong

English

NASA Technical Reports Server

Third In_mational Conference on Inverse Design Concepts and Optim_a_on in Engineenng Sciences
_CIDES-HI). Editor. G.S. Dulik_vich. Washington D.C.. October 23-25. 1991.
[- The Research Progress on Hodograph Method
of Aerodynamic Design at Tsinghua University
Chen Zuoyi Gut Jingrong
(Tsinghua University )
'? "i '" 0 1
1. Introduction |"
The Hodograph Method isa classicalmethod influiddynamics. Because itcan transform
the nonlinear equation to the linear.So itisalways used to resolvethe fluiddynamic equation
in the early time. For example, the well known Karmen-Jein formula is obtained from
Hodograph equation. Then the Hodograph method iswidely applicated in the research of
transonic flow. Due to the Mixed-Type character of transonic flow equation, the Hodograph
method may be the only accurate method to solve the transonic flow equation. In the 1970's,
the Hodograph method startto be used in the inverseproblem of fluiddynamics. (i.e.design
problem, such as the work by Hobsen and karadimas (Ref.l)In the 1980's.,we have done sys-
tzmatic reserch work on Hodograph method in Tsinghua University, and have taken much
progress as the follows.
1. Research progress on the analytical solution of Hodograph Mixed-Type equation
The various analytical solutions of Hodograph Mixed-Type equation are presented in
Ref.3. If we use these solutions to solve the inverse problem, especially design the cascade pro-
file, there are limits on various boundary condition. We can't always find right solution under
any boundary condition. So that the application scope may be restricted. The differences
among various analytical solution of Hodograph Mixed-Type equations and their applica-
tion conditions have been presented. It shows the differences are obvious (see Fig. 1).
The application scope of the four nozzle-solution of Hodograph is shown in Table. 1.
All the nozzle-solutions of Hodograph were straight line corresponding with the mean
stream line before. If we use them in the inverse problem of cascade, the best one is prefered to
be curve mean streamline. The nozzle-solution of Hodograph corresponding with the curve
mean streamline is presented. We get the analytical solution of Mixed-Type Hodograph
equation that is corresponding with the parabolic coordinates, as shown in Fig.2.
The relationship between the parabolic and rectangular coordinates is:
| 2
-n =) (1)
y = Cn (2)
Thus the vorticity of V may be represented by:
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Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences
(ICIDES-III). Editor: G.S. Dulikmvich. Washington D.C.. October 23-25. 1991.
F
1 (a(v.'J_ _ n_ _ n 2+ ) _(v_4 + )) (3)
It satisfies:
age = _ _2 2 (4)
a--_ + rl p V
2
_¢ ,/_ + _ (5)
arl - p V ¢
If the equivalent velocity is introduced:
I) = + 17 V (6)
I?_ = _= + r/= V¢ (7)
the similar Mixed-Type Hodograph equation can be obtained:
2 2
a ¢ )__ff^----i-+ K(a = 0 (8)
ao-
2. Design of the transonic turbine and compressor cascade using Hodograph method
we have presented the new method to design the transonic turbine and compressor cas-
cade using Hodograph. This method is that applieated the Chaplygin equation numerical so-
lution to solve the subsonic area and applieated the analytical solution of Mixed-Type
Hodograph equation to solve the supersonic area.
The differential equation for numerical solution of Chaplygin equation is:
( Q ,,.k T,.k0,,._= B,.,,+P"* • + --7 + 2q:-,._f,,,+,
S i,k _ t_+l.k [i,_ ,
, , /[2pjk/S_. k +2Q,.k/l_. k]2 0,, + 2 21t. _ ,,.,_,S_, -'_ 1_,k ' '
(9)
L
The profile coordinates equations are:
p Ma
X - X o = - h ,o ® cosO P + Q --
• (30
p Ma
,o),l,pdMa * -_ cosOdMa (10)
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Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences
('ICIDES-III). Editor: G.S. Dulikravich. Washineton D.C.. October 23-25. 1991,
F
I
sinOdMa (! 1)
The profile coordinates equations which are obtained by the analytical solution of
Mixed-Type Hodograph equation are:
xo[; )]...... + K°(a) 0 (12)
Ma °_ ,.
aft
y--yo = If: I" sinO (ad/aa a0}_ ) 1.... + K° (a) 0
. Ma or/ ,.
aft
(13)
The location of shock wave is the importantance in design. The method of profile design
by the given location of shock wave is also presented. The comparision between the design
shock wave locatgion and the experiment is shown in Fig.3.
3. The basic equation of three dimensional Hodograph method
On the basis of the general theory in three dimensional flow and the correspondence be-
tween the physical surface and Hodograph, the basic equations of three dimensional
Hodograph method have been obtained.
Using the concept of equivalent physical surface and equivalent Hodograph, we obtain
the different equation of streamfunction with Hodograph coordinates and integral equation
for returning from the Hodograph to the physical surface.
The correspondence between S1 flow surface and equivalent physical surface is shown in
Fig.4.
The streamfunction equation of Hodograph corresponding to the S 1 flow surface is
here _ is equivalent velocity. 0 is the equivalent flow angle.
The relationship between the equivalent velocity and true velocity is
L
toe = A + Bi
B=_. (ro9 +r oJ-roo ) (15)
1¢_3 o r n t
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602
Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences
CICIDES-IID. Editor: G.S. DuIikravich, Washintzton D.C., October 23-25. 1991.
F
A = _ (_o + flo0)
_2
q
Similarly, we can obtain the streamfunction equation of Hodograph corresponding to the $2
flow surface, that is:
1--2-7 + + + + As-_ 0
ao a?o a_oaO aO
(16)
The correspondence between $2 flow surface and equivalent physical surface is shown in
Fig.5.
The relationship between the equivalent velocity and actual velocity is:
n
A.K 6 = o - -z-' (o e + or) (17)
n o
n
BK 5 = o, - _-0 (o0 + or) (18)
The integral equations for returning from the Hodograph to the SI surface are:
z-z, = _.t_co L _ -Z+_-_<_i
ao
+ -_9-9& dO (19)
" . (.reo L aO a_ + bp_o (a__O) a_) (a____) + V 6_) dO (20)
a(.O aO
The integral equation for returning from the Hodograph to the $2 surface is:
z-z = -- - a0 d_ (22)
aO OO
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603
Third International Conference on Inverse Design Concepts and Opumization in Engineering Sciences
(ICIDES-III). Editor: G.S. Dulikravich. Washinmon D.C.. October 23-25. 199 I.
V 1
_o
9(.o
--I
+ r'°K4 a---_J (23)
4. The aerodynamic design of Hodograph is revolutionary surface
On the basis of the three dimensional flow Hodograph method, the aerodynamic design
of Hodograph in revolutionary surface is presented.
The Hodograph equation corresponding to the revolutionary surface can be obtained
from Hodograph equation corresponding to the SI flow surface.
M
a
r • •a d/ M. ab M° a-_LaMa.2 + 1 b aM -P aM
° °
--g,o,Mj,o ."oo+p_ +,M"
+ +_-a_ +- =0P aMa" ]aO 2
o o
ag a_ a-p 1 1 a2E ]a¢,
a0 +°0aM; ff-p' + ....baM; .O -_
(23)
The integral equations for returning from the Hodograph to revolutionary surface are
Z--Z
o "_ Map L_ "_ +b-fiMa a_b a¢,
+
° •
aMa aMa
i
aMa
(24)
O- O° = j _ (r--o_ z t---g-__)M--a"-p" L -_ -_ + b-fiM a " _ a gt aM a "
o °
aMa aMa
Ma'__ ag_ , ]dO+ bp aMa
(25)
In order to determine the supersonic region, the analytical solution
Mixed-Type equation corresponding to revolutionary surface is presented.
From the streamfunction equation:
of Hodograph
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604
Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences
(ICIDES-III). Editor: G.S. Dulikravich. Washineton D,C.. October 23-25. 1991.
F
[_ Mo" a-p KaCt [M2aMa " K 2 ad/ dK ] + • +atr dMa" -P aMa atr "
1
,
+_ . = 0 (26)
P aMa Jab 2
If put the
F(_r)-- 2 +PaMa• • (27)
thus the streamfunction equation can be transformed to the analytical solution equation
which is similarly the plane flow
a_2 a2_
F(_)_-_- + 2 - 0
au 9O"
Design example:
Design parameters: r = 1.29 M== 0.2238 0 = 0
OO
T== 1490k 02= 73 M2= 1.15
R 0= 300mm a_ = 26.56 H 0= 50.05mm
The profile in the revoluitonary surface with the Hodograph Method is shown in Fig.6.
5. Summary
The research progress of Hodograph method on aerodynamic design in Tsinghua Uni-
versity has summrized in this article, i.e. (1) There are some restricted conditions in applica-
tion with Hodograph method to design the transonic turbine and compressor cascades. (2)
The Hodograph method design is suitable not only to the transonic turbine cascade but also
to the transonic compressor cascade (3) The three dimensional Hodograph method will be de-
veloped after obtaining the basic equation in three dimensional of Hodograph method, as the
example the transonic turbine cascade design of Hodograph in revolutionary surface is pres-
ented.
Reference
1. D.E.Hobson. CUED/A-Turbo/TR 40. NOV.1972.
2. Karadimas Von-Karman Institute for Fluid Dynamics Lecture Series 59.
3. C. FERRARI and F.G.TRICOMI. Transonic Aerodynamics Academic press York and
Landon.
4. Chen Zuoyi, papers of ISABE. 1983, Paris.
5. Chen Zuoyi. Journal of Engineering Thermophysies, Vol.4. No.4. 1983.
6. Chen Zuoyi, Journal of Tsinghua Univ., Vol.22, No. 1, 1982.
7. Chen Zuoyi, Journal of Tsinghua Univ., Vol.21, No.4, 1981.
L ]
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Third In_.mational Conference on Inverse Design Concepts and OpLimmation in Engineering Sciences
_ICIDES-IID. Editor: O.S. pulik_vich, Washington D.C.. October 23-25. 1991,
]
4-_c J-_.I-1_I Y(_"")
approximation
Tricomi . "_ Ill+
appr°ximati°n _i '
- :4-._-'i '
Fig.l The Comparison between the simple
Tricomi approximation and generalized
Tricomi approximationunder variouscondition
Table IThe Comparison ofapplicationscopeunder
variousnozzlesolution
0 0
0 o 0 o
0 _r 13.
0
_r
D
)
-014
rn
Note:
normal
solution
approximation
Gencralizcd
Homographic
Tomotika-Tamada
Generalized Tomotika-Tamadc
J
/
cosnt
\
Fig.2The paraboliccoordinates in physicplane
Fig.3Comparison of shock wave Location
Between Design and Experiment Q9 cascade
profile
----_xpcrimenta|shock wave
--design shock wave
l
606
Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences
flCIDES-III). Editor: G.S. Dulikr'avich. Washington D.C.. October 23-25. 1991.
[
j<>G--..
°q-',)
S1 flow surface coordinates
Fig.4 S 1
iI
equiv_ilent complex plane
flow surface and equivalent complex plane'
11, ,
$2 flow surface coordinates
Fig.5 S2
L'
equivalent complex plane
flow surface and equivalent complex plane
1
0c')
0.20
0116
O.JZ
O.OS
\\ )-0.0,
\\) -o.-
'_'--0.12
-- O.16
t-o.=o
Fig.6 The transonic cascade profile
in revolutionary surface using
Hodograph method,
]


The research progress on Hodograph Method of aerodynamic design at Tsinghua University

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