Effect of surface roughness on Mach reflection

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XXI ICTAM, 15-21 August 2004, Warsaw, Poland
EFFECT OF SURFACE ROUGHNESS ON MACH REFLECTION
Susumu Kobayashi*, Takashi Adachi*, Klaus Debatin**, Torsten Schenkel**, Herbert Oertel, Jr.**
* Department of Mechanical Engineering, Saitama Institute of Technology, 369-0293 Saitama, Japan
** Institut für Strömungslehre, Universität Karlsruhe, 76128 Karlsruhe, Germany
Summary The effects of surface roughness and transport properties have been compared experimentally to investigate non-self-
similar Mach reflection phenomena. The surface roughness was given by pasting a piece of sand paper on the model surface. The
results were compared with those for smooth surfaces. The effect of surface roughness turned out to be small compared with
viscosity effect so that the effect of transport properties is proved to be dominant.
INTRODUCTION
It was first reported by Walenta [1] some twenty years ago that, under rarefied-gas conditions, the transition from
regular to Mach reflection takes place during the incident shock propagation over a wedge. This non-self-similar
phenomenon was first considered to be a phenomenon characteristic of extremely low-pressure, low-density
atmosphere, because such dynamic transition phenomena have never been observed in ordinary atmospheric conditions.
However, the authors observed a same kind of phenomenon even in atmospheric pressure for the first time [2], proving
that such non-self-similar phenomenon is not restricted to low-pressure conditions.
The disruption of self-similarity suggests that a length scale has been introduced into the system. There are two
candidate causes: viscosity and surface roughness. According to Henderson et al.’s numerical experiment [3], the
condition on the solid boundary plays a key role in the phenomenon. The authors’ experiment in Karlsruhe found
different wave angles from those obtained in Saitama [4]. In the Karlsruhe experiment, the initial pressure p1 of the
driven section was below atmospheric pressure and ranges from 280 mmHg to 610 mmHg depending on shock Mach
number Mi. In addition, the model surface roughness was estimated from 6 to 12 µm. In contrast, in the Saitama
experiment, p1 was atmospheric pressure and the surface roughness ranged from 1 to 2 µm. When the pressure is low,
the effect of viscosity is enhanced, as seen by the definition of kinematic viscosity. Therefore, the results for both
experiments are subject to mixed effects of viscosity and surface roughness. In this paper, we investigated the effect of
surface roughness on the behavior of Mach reflection.
EXPERIMENT
Experimental Apparatus
We performed the experiments using a conventional shock tube in our institute. The working gas was air, and the
driven section was set at room temperature and atmospheric pressure at each experiment run. The models were ordinary
smooth wedges of 20º and 30º, over which a sheet of sandpaper was firmly pasted to keep it flat. We could select from
two surface roughnesses by changing sandpaper (#60 and #240). This means the surface is not hydraulically smooth
according to Ben-Dor et al [5]. In contrast, the surface roughness in past experiments [4] was 12 µm maximum, and the
wedge surface was hydraulically smooth. The incident shock Mach number Mi was 1.10, 1.20, 1.30 and 1.40. However,
only partial results are presented here due to page restrictions.
Fig. 1 Definition of geometric variables Fig. 2 Visualized reflection configuration
Measurement
The triple-point coordinate (x, y) and the angle
ω
ir made by the incident and reflected shocks at the triple point were
measured (see Fig. 1) directly from photographic negatives enlarged by a factor of about 50 using a profile projector
(V-12, Nikon, Inc.). In measuring the triple-point location, the model corner ahead of the incident shock was taken as a
θ
w
i
r
mT
ω
ir
ω
i
ω
r
Mi
s
x
y
Ο

XXI ICTAM, 15-21 August 2004, Warsaw, Poland
reference point to avoid the influence of optical distortion behind the shock waves. The triple-point location was easily
converted in the coordinate system with the leading edge taken as the origin O, and the incident shock propagation
direction as the x-axis. The y-axis was defined upward normal to the x-axis. The maximum error involved in measuring
the angle
ω
ir is ±2.0º at an early stage of reflection (10 mm < x < 30 mm) where the radius of curvature of the reflected
shock is small. However, the error is reduced within ±1.0º as the incident shock proceeds (x > 40 mm). The error was
very large (around ±5º) for x < 5 mm.
RESULTS
Reflected wave configuration
Figure 2 shows a representative image of shock reflection over the wedge with surface roughness (Mi = 1.40,
θ
w = 20º,
#60, x = 64.19mm). A group of compression wavelets issues cylindrically from the wedge surface (see behind the foot
of the Mach stem). The wavelets are more clearly observed when the surface roughness is larger, and their pattern is
quite similar to the shock reflection over a multi-guttered wedge or a step-like wedge [6]. Although the data are omitted
here, the triple-point trajectory proved to be almost independent of the surface roughness.
Variation of the angle between incident and reflected shocks
Figure 3 illustrates the variation of the angle
ω
ir made by the incident and reflected shock waves at the triple point as
the incident shock proceeds (Mi = 1.40). Angle
ω
ir is large near the wedge tip but decreases and approaches an
asymptotic value with the progress of the incident shock wave. The behavior of
ω
ir shows that the Mach reflection is
not self-similar. On the whole, angle
ω
ir is smallest for smooth wedges. However, the difference is not clear for large x.
Therefore, the effect of surface roughness on the wave angles is not strong.
The effect of viscosity can be taken into account by defining the dimensionless variable, x
ur
1
1
µ
ρ
ζ
=, where
ρ
1 and
µ
1
are the density and viscosity behind the incident shock, and ur is the flow velocity there. With this transformation of
space variable x, the difference of
ω
ir obtained in Saitama and Karlsruhe diminishes as in Fig. 4 (Mi = 1.30). The
difference is still large for Mi = 1.20 and 1.40. Possibly the error in angle measurement was larger than expected.
Fig. 3 The angle
ω
ir between incident and reflected shocks Fig. 4 The angle
ω
ir in the dimensionless coordinate
CONCLUSIONS
The results show that the effect of surface roughness is not distinct with smooth (less than the surface roughness of about
10µm) wedges. This negative result leads to the conclusion that the effect of viscosity is dominant in non-self-similar
phenomena of Mach reflection. Consequently, the difference in the two kinds of experiments [4] should be ascribed solely to
viscosity.
References
[1] Walenta Z.: Formation of Mach-type reflection of shock waves, Arch. Mech. 35: 187-196, 1983.
[2] Kobayashi S., Adachi T., Suzuki T.: On the unsteady transition phenomenon of weak shock waves, Theoret. Appl. Mech.: 49: 271-278, 2000.
[3] Henderson L. F., Crutchfield, W. Y., Virgona R. J.: The effects of thermal conductivity and viscosity of argon on shock waves diffracting over
rigid ramps, J. Fluid Mech. 331: 1-36 , 1997.
[4] Kobayashi S., Debatin K., Oertel Jr. H., Adachi T.: Non-selfsimilar nature of Mach reflection over a flat smooth slope, Proc. 23rd Int. Symp.
Shock Waves: 1262-1267, 2002.
[5] Ben-Dor G., Mazor G., Takayama K., Igra O.: Influence of surface roughness on the transition from regular to Mach reflection in pseudo-steady
flows, J. Fluid Mech. 176: 333-356, 1987.
[6] Suzuki T., Adachi T., Kobayashi S.: Nonstationary shock reflection over nonstraight surfaces, Shock Waves 7: 55-62 , 1997.
0
10
20
30
40
50
60
00.511.5
ζ
ω
ir
[deg]
Karlsruhe
SIT
rough(#60)
M
i
= 1.30
0
10
20
30
40
50
60
70
0 20406080
x [mm]
ω
ir
[deg]
smooth(20deg)
#60(20deg)
#240(20deg)
smooth(30deg)
#60(30deg)
#240(30deg)