Shock Waves (1999) 9: 419 422
Pressure measurements on cone surface
in 3-D shock reflection processes
H. Zhao1, X.Z. Yin2, H. Grönig1
1
Stosswellenlabor, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany
2
Dept. of Modern Mech., Univ. of Sci. and Tech. of China, Hefei, Anhui, 230026, P.R. China
Received 17 November 1997 / Accepted 5 December 1997
Abstract. In a shock tube the pressure distribution was measured on a cone with an angle of attack when
a shock wave passed the cone. The cone has a semi-apex angle of 35ć%, the angle of attack varied from
0ć% to 25ć% and the shock Mach numbers from 1.05 to 3.0. A series of pressure distributions on the cone
circumference are given.
Key words: Shock wave reflection, Pressure measurements, Three-dimensional flow
1 Introduction circumference of the cone and determined a transition an-
gle of Śt =43ć% ą 0.5ć% for the experiments and Śt =46ć%
for the numerical simulation. Yin et al. (1996) made an
Although the shock reflection is an old topic which can
experiment on the 3-D shock reflection over a cone in a
be traced back to a century ago and almost one thousand
shock tube. They determined the 3-D shape of triple point
papers about it have been published, most of the previous
trajectory from a series of schlieren pictures which were
researches were focused on shocks passing over a wedge
taken by rotating the cone model around the shock tube
or other 2-D objects (Ben-Dor 1991), a few of the studies
axis and gave the triple point trajectory for a shock wave
deal with 3-D shock reflection.
of Ms =1.51 passing over a cone with a semi-apex angle
For axisymmetric reflection Bryson and Gross (1961),
of 30ć% and an angle of attack of 15ć%.
Takayama and Sekiguchi (1977), Yang et al. (1995) and
Milton et al. (1995) studied the shock reflection over ex- Up to now, most of the research methods were based
ternal and internal cone surfaces, respectively, and found on optical visualization. In this paper our purpose is to
some differences of triple point trajectory from the case measure the pressure distribution on the cone with an
of a wedge. In mathematics the axisymmetric flow still angle of attack and to find some features of 3-D shock
represents a 2-D problem. reflection.
As to 3-D shock reflection, Suzuki et al. (1995) investi-
gated the shock reflection on a cone with an angle of attack
which is one of the simplest 3-D reflections, and obtained
2 Experimental setup
schlieren pictures in the plane of symmetry of the flow
field. They found that the phenomenon was self-similar,
Experiments were conducted using a shock tube in the
the triple point trajectory was a straight line and the tran-
Shock Wave Laboratory of RWTH Aachen whose inner
sition angle between regular reflection (RR) and Mach re-
diameter was 96 mm, driver and driven section length were
flection (MR) and triple point trajectory angle in the plane
2 m and 4 m, respectively. The test gas was air. The shock
of symmetry were different from both the 2-D and axisym-
Mach numbers varied from 1.05 to 3.0.
metric cases. Hookham et al. (1995) studied numerically
Figure 1 shows the cone model attached to the end
and experimentally a 3-D shock reflection phenomenon in
wall of the shock tube. The model shown in Fig. 2 is a
which a cone with a semi-apex angle of 35ć%, an angle of
cone with a semi-apex angle ´c of 35ć% which is composed
attack of 25ć% and shock waves of Ms = 1.28 were used.
of three parts: a small forebody tip and two half cones.
Under certain conditions regular reflection on the wind-
The parts were connected using mini-screws and formed a
ward side and Mach reflection on the leeward side of the
hollow model in which pressure transducers were installed.
cone occurred and RR-MR transition took place at a cer-
The cone could be positioned at four angles of attack,
tain azimuthal angle. They got RR-MR transition in the
ą =0ć%, 10ć%, 20ć% and 25ć%.
Fourteen piezoresistive transducers (Kulite XCQ-080-
Correspondence to: H. Zhao, Messer North China Industrial
100A, 0-7 bar) which were 2 mm in diameter and 5 mm
Gas Co., Ltd., Xi an Branch, No. 6, the 2nd Electric Road,
Hi-tech Zone, Xi an 710065, P.R. China long, were mounted flush with the wall in 5 cross sections
420 H. Zhao et al.: Pressure measurements on cone surface in 3-D shock
Fig. 1. Schematic arrangement of the cone model attached to
the end wall of the shock tube
Fig. 3. Pressure distribution as function of the azimuthal angle
Ś for the angle of attack ą =0ć%
of the cone, respectively. Before each test the 14 piezoresis-
tive sensors were calibrated in situ by changing the pres-
sure in the shock tube. In the tests it is found that the
response of the piezoresistive sensors depends on the initial
pressure in the driven section of the shock tube. In order
to obtain shorter rise times an initial pressure as high as
possible should be used. In our experiments most of the
piezoresistive transducers had a rise time of 30 µs while
the shortest one was less than 10 µs. For comparison a
piezoelectric transducer (Kistler 703B) was mounted into
the windward side on the plane of symmertry (position
15).
From optical measurements (Suzuki et al. 1995) it is
known that the shock reflection phenomenon considered
is self-similar and symmetric about the vertical plane. In
this paper Ś =0ć% corresponds to the plane of symmetry
of the windward side of the cone and the angle Åš increases
clockwise facing the tip of the cone.
3 Experimental results
Figure 3 shows the pressure distribution on the cone sur-
face as function of the azimuthal angle Åš for the angle of
attack ą =0ć%, in which p1 and T1 indicate the pressure
and temperature ahead of the moving shock respectively,
and the measured values of p were taken as the first high-
est pressure behind the incident shock wave. It can be seen
that the pressure ratio remains constant in circumferen-
tial direction, because the flow is axisymmetric. The solid
lines denote results calculated by axisymmetric equations
of shock dynamics for Ms = 1.05, 1.9 and 2.5, respectively.
The results show that the measurements are slightly lower
than the theoretical predictions.
Figure 4 shows the pressure distribution on the cone
surface as function of the azimuthal angle Ś for ą =25ć%
and Ms = 1.25 (p1 = 400 mbar), 1.73 (p1 = 260 mbar),
2.50 (p1 = 60 mbar) and 3.26 (p1 = 20 mbar), respectively.
Due to the non-uniform pressure distribution behind the
Fig. 2. Schematic geometry of the cone model
incident shock wave, the first peak pressure behind the
H. Zhao et al.: Pressure measurements on cone surface in 3-D shock 421
Fig. 4. Pressure distribution as function of the azimuthal angle
Ś for ą =25ć%
Fig. 6. Pressure distribution as function of the azimuthal angle
Ś for ą =20ć%
Fig. 5. Pressure distribution on the cone surface in different
sections
Fig. 7. Pressure distribution as function of the azimuthal angle
Ś for ą =10ć%
shock was taken. In order to get more data, in the ex-
periments we rotated the cone model around its axis to
achieve different locations relative to the position Ś =0ć%.
For the 3-D shock reflection, 3-D two-shock theory can
It is evident in Fig. 4 that the pressure is symmetric about
be only used to predict the pressure behind the reflected
the vertical plane of the cone (Ś = 0ć%), the pressure in
shock in RR part. The predicted pressures is higher than
smaller azimuthal angles is higher than that in the larger
the measured values, for example, p/p1 = 26.8 is given
azimuthal angles, and a pressure drop exists.
in the theory for Åšt =0ć%, Ä… =25ć%, Ms =2.5 and ´c =
35ć% which is much higher than the experimental results
Figure 5 shows in detail results of Ms =2.5 and Ä… =
(see Fig. 5). A possible reason could be that the dynamic
25ć% in Fig. 4. The measured results for different sections
response of the pressure sensors is not very high. Figure 8
were superimposed. We can see in Fig. 5 that a pressure
shows the typical pressure curves recorded by the piezore-
drop exists between | Ś | H"42ć% and 70ć%, and the pressure
sistive and the piezoelectric transducers, respectively, in
stays constant for | Ś |e" 90ć%. For smaller azimuthal angles
which we can see that the rise time of the piezoresistive
the angle between the incident shock wave and the cone
sensors is about 10-30 µs, while that of the piezoelectric
surface is small and regular reflection may occur, while
sensor is about 5 µs.
for larger azimuthal angles Mach reflection may occur.
A pressure drop exists during the transition from RR to In the curves the first pressure jump represents the
MR for strong shock reflection. The 3-D two-shock theory pressure on the cone surface behind the incident shock,
predicts the transition azimuthal angle of Śt = 42.62ć% the second step represents re-reflection of the reflected
for Ms = 2.5, ´c = 35ć% and Ä… = 25ć%, which basically shock from the shock tube wall, and the third step is the
agrees with the experimental results. Figures 6 and 7 show pressure behind the reflected shock from the end wall of
measured results for ą =20ć% and ą =10ć%, respectively. the shock tube. Therefore, we are only interested in the
422 H. Zhao et al.: Pressure measurements on cone surface in 3-D shock
ranging from 1.05 to 3.0 in air. The cone has a semi-apex
angle of 35ć%. The pressure was measured on the cone cir-
cumference. The error of the pressure measurement is es-
"p
timated as µp = d" 1%. For an angle of attack of
m
p
ą =0ć% the results show that the measurements are lower
than the theoretical predictions. The 3-D two-shock the-
ory predicts the transition azimuthal angle of Śt =42.62ć%
for Ms = 2.5, ´c = 35ć% and Ä… = 25ć%, which basically
agrees with the experimental results. The experimental
results show that it is difficult to precisely measure the
pressure distributions in 3-D shock reflections and pres-
sure transducers with high response should be used.
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