Strong regular reflection of steady shock waves
Dmitry Khotyanovsky1 , Alexei Kudryavtsev1 , Gabi Ben-Dor2 , and Mikhail
Ivanov1
1
2
Institute of Theoretical and Applied Mechanics,
Novosibirsk 630090, Russia
Pearlstone Center for Aeronautical Research, Ben Gurion University
Beer-Sheva 84105, Israel
Abstract. The existence of an asymmetric steady regular reflection with a strong reflected shock wave has been documented numerically with a high-order shock capturing
Euler code. Two different regular reflection configurations have been obtained with the
same flow deflection behind the reflection point. One of these regular reflections has
two weak reflected shock waves, and the other has a weak and a strong reflected shock
waves. The pressures behind the reflection point differ significantly for these regular
reflection configurations of shock waves.
1
Introduction
For steady shock wave reflection, two wave configurations are possible: regular
reflection (RR) and Mach reflection (MR). From the theoretical point of view,
two RR configurations are possible: an RR with a weak reflected shock wave
and an RR with a strong reflected shock wave. In practice, however, only an
RR configuration with a weak reflected shock wave is usually observed. There
are some arguments that a shock wave configuration with strong reflected shock
waves is unstable (e.g. [1]). Nevertheless, a convincing proof of this instability
has never been given. It should be noted that a strong shock wave cannot be
obtained without imposing special downstream pressure boundary conditions
[2]. In a recent study [3] it has been demonstrated numerically that a steady RR
with strong reflected shock waves in the symmetric shock wave reflection may be
obtained as a result of an interaction of a steady weak RR configuration with an
artificial shock wave moving upstream. When the pressure at the downstream
boundary of the computational domain was then set at an appropriate value,
a steady RR with strong reflected shocks occurred. Some special geometry of
the problem was used in [3], provided that strong shocks impinge exactly on the
channel shoulders. Any shift of the strong reflected shocks from their position
would most likely result in the destruction of the configuration (e.g. choked flow).
Most of the studies on steady shock wave reflections consider a symmetrical
case where two incident shock waves of equal and opposite angles, α1 = α2 ,
interact with each other at the plane of symmetry. Asymmetric shock wave reflection, i.e., the interaction of two shock waves generated by two wedges having
different angles of attack, have some specific features, which makes it an interesting subject for a detailed investigation. Regular reflection (see Fig.1) consists
2
Khotyanovsky et al.
of two incident shock waves (IS), whose angles of incidence α1 and α2 are determined by the angles of attack θ1 and θ2 of the wedges, two reflected shock waves
(RS), and a slipstream (SS) emanating from the reflection point. The MR configuration is more complicated: in addition to the incident and reflected shocks,
there is a Mach stem (MS) with a subsonic flow behind it and two slipstreams
emanating from the triple points T. In contrast to RR, the MR configuration is
geometry-dependent (see [4]). The length scale information can reach the Mach
stem via the path along the leading characteristic of the expansion fan, and then
through subsonic flow behind the Mach stem. Hence, the Mach stem height is
determined by the normalized distance between the wedges 2g/w, where w is
the length of the wedge. In some cases, however, the flow is subsonic behind
the reflection point in the RR case too. This case will be discussed in more detail below. All the shock wave configurations typical for the asymmetric case
were considered in detail in [5] using pressure-deflection diagrams. An interesting feature of the asymmetric shock wave reflection found theoretically in [5] is
the existence of a solution for the asymmetric regular reflection that consists of
one weak and one strong reflected shock waves. Such a feature of the asymmetric shock wave reflection is theoretically possible in a very narrow range of the
(θ1 , θ2 ) angles close to the upper limit of the existence of the RR. In contrast to
the symmetric case, the asymmetric shock wave reflection contains a mechanism
for providing the necessary pressure boundary conditions behind the reflection
point to maintain a strong reflected shock wave. Consider a case where the flow
Mach number is M = 4.96 and two wedges have angles of attack θ1 = −35◦ and
θ2 = 14.58◦ . The corresponding pressure-deflection diagrams are given in Fig.2
(left part). Here I is the incident shock polar; R1 and R2 are the reflected shocks
polars corresponding to θ1 and θ2 . The R2 polar intersects the R1 polar at two
points, each corresponding to an RR solution. The intersection with lower pressure is an RR with two weak reflected shock waves (RRww). The intersection
with higher pressure occurs at the strong portion of the R1 polar (above the
maximum deflection point D) and is, consequently, an RR with one weak and
one strong reflected shock waves (RRws). As mentioned above, a solution with
θ1
α
1
EF
θ1 α 1 EF
IS
M>1
2g
IS
RS
RS
IS α 2 θ2
EF
T
M>1
MS
SS
T
IS
α2
RS
SS
2g
RS
SS
EF
θ2
Fig. 1. Schematics of asymmetric RR (left) and MR (right)
Strong regular reflection of steady shock waves
3
lower pressure usually occurs. Hence, the expected RR configuration in this case
is the RRww. Note, there is also an MR solution in this case corresponding to
polar intersections marked M R1 and M R2 .
Consider in detail the pressure-deflection diagram in the vicinity of the tangency point of the reflected polars tangency point. Figure 2 (right part) shows
the shock polars of the upper and lower reflected shock waves for M = 4.96,
θ1 = −35◦ and two values of θ2 . In this figure, S and D are the sonic point and
the maximum deflection point, respectively, and E is the point of tangency of
two polars, analogous to the detachment criterion in the reflection of symmetrical shock waves. If the R2 polar intersects the R1 polar above the point D,
R 2 polar
50
RRws
60
R 1 polar
50
48
RRws
D
RRww
MR 2
MR 1
30
20
46
p/p0
p/p0
40
.E
I polar
θ2=15.98°
.D
44
42
θ1 = -35°
.S
10
θ2 = 14.58°
0
-50
-40
-30
-20
-10
0
θ
10
20
30
40
50
40
-19
θ1=-35°
-18.5
RRww
-18
θ
θ2=14.58°
-17.5
-17
Fig. 2. Shock polars intersections for M = 4.96, θ1 = 35◦ and different θ2
then even an RR configuration corresponding to lower pressure is the RRws and
includes one weak R2 and one strong R1 reflected shock waves.
It should be noted that for any R1 - R2 intersection point, which lies between
the D and E points, there exist a counterpart intersection point below D (see
Fig. 2). The corresponding RR configuration is an RRww, which deflects the flow
at the same angle as the above-mentioned RRws configuration. For example, for
M = 4.96 and θ1 = −35◦ the flow deflection behind the reflection point is the
same, θ = −18.22◦ , for both RRws at θ2 = 15.98◦ and RRww at θ2 = 14.58◦ .
The possibility of the existence of such an RRws configuration is a unique
feature of the asymmetric shock wave reflection. As mentioned in [5], the complementary weak reflected shock wave corresponding to the R2 polar provides
high pressure behind it to support the strong reflected shock wave corresponding
to the polar R1 . (Note that the pressures at the slipstream must be identical).
For a given θ1 , an RR with one strong reflected shock wave can exist for θ2 in
the range θ2D < θ2 < θ2E . The angle θ2D corresponds to the R2 polar, which
intersects the R1 polar at the point D. The angle θ2E corresponds to the R2
polar, which is tangent to the R1 polar (point E). The regions of existence of
the RRws are given in Fig.3 for several flow Mach numbers M.
4
Khotyanovsky et al.
M = 15
0.6
M = 10
θ2E - θ2D, deg.
0.5
M = 4.96
0.4
0.3
M=4
0.2
0.1
0
-45
-40
-35
θ1, deg.
-30
-25
Fig. 3. Regions where RRws can exist for several flow Mach numbers and γ = 1.4
The goal of this study is to verify numerically the possibility of the existence
of such an asymmetric regular reflection configuration with one strong reflected
shock. The range of θ2 where the RRws exists is very narrow (e.g., at M = 4.96
and θ1 = 35◦ , the strong reflected shock wave is possible for 15.595◦ < θ2 <
15.983◦ ). Moreover, this region is very close to the upper bound of the existence
of the RR. Hence, the accuracy and robustness of the numerical method are very
important for obtaining such a shock wave configuration.
2
Numerical method
Two-dimensional unsteady Euler equations for a perfect gas with γ = 1.4 were
solved numerically with a high-order shock capturing finite volume algorithm.
The fourth-order MUSCL reconstruction formula [6] is utilized to reconstruct
the cell face values of the primitive variables (i.e., density, pressure, and velocity components) from cell averaged ones. Numerical fluxes are calculated with
the HLLE (Harten—Lax—van Leer—Einfeldt) approximate Riemann solver [7],
which is very robust for modeling high-speed flows. Time integration was accomplished by the third-order explicit TVD Runge-Kutta scheme [8].
The flow around two wedges immersed in a uniform supersonic stream was
simulated. The wedges (Fig. 4), which were shaped as right-angle triangles, had
an angle of 15◦ at their leading edge. The distance between the trailing edges of
the two wedges, 2g, was 0.6w (where w is the length of the wedge hypotenuse).
The origin was located at the half-distance between the traling edges of the
wedges. All lengths were non-dimensionalized to the wedge length w. The computational domain was divided into four zones (see Fig. 4), and the total number
of quadrilateral cells in four zones was approximately 660,000. A uniform flow
with a Mach number M = 4.96 was specified on the left (inflow) boundary. The
right (outflow) boundary was located far enough downstream of the wedges so
that the flow was supersonic on it. The flow variables were extrapolated from
Strong regular reflection of steady shock waves
5
within the domain on the outflow as well as on the bottom and top boundaries.
The use of extrapolation on the bottom and top boundaries is similar to imposing
some kind of non-reflective conditions and results only in a very weak artificial
reflection from these boundaries. Inviscid solid wall conditions were imposed on
the wedge surface. The simulations started with a uniform M = 4.96 flow filling
the entire computational domain. The time integration continued until a steady
state solution was achieved.
3
Results and Discussion
A shock wave configuration with one weak (lower one) and one strong (upper
one) reflected shocks, i.e. RRws, was obtained in our simulations for M = 4.96,
θ1 = 35◦ and θ2 = 15.98◦ . Figure 5 (left part) shows an enlarged view of the
flowfield in the vicinity of the reflection point. For comparison, an RRww shock
wave configuration, across which the flow deflection was identical to the abovementioned RRws case, was obtained for M = 4.96, θ1 = 35◦ , and θ2 = 14.58◦ ,
(shown in right part of Fig. 5). Thus, two different shock wave configurations
have been obtained numerically, an RRws and an RRww, behind which the flow
deflections are identical but the pressures are different (Fig.6): pw /p∞ = 40.7
for the RRww case, and ps /p∞ = 46.0 for the RRws case. While the flow behind
the reflected shocks is supersonic in the RRww case, it is subsonic behind the
strong shock wave in the RRws case. This is illustrated in Fig. 5, where the
subsonic region behind the upper reflected shock wave in the RRws case is
evident. Farther downstream, the flow is accelerated to a supersonic velocity
due to the influence of the expansion fan emanating from the trailing edge of
the upper wedge. It should be noted that the above-mentioned pressure value
ps /p∞ = 46.0 in the RRws case is somewhat lower than the analytically predicted
value for this case (i.e., 46.8). This discrepancy is probably associated with the
upstream influence of the expansion fan through the subsonic zone, which cannot
be taken into account in an analysis based on the shock polar technique. Note a
slight curvature of the strong reflected shock wave in Fig. 5. The existence of this
Zone 3
θ1
w
15
O1
Y
M = 4.96
g
Zone 2
X
Zone 4
g
w
θ2
O
15
2
Zone 1
Fig. 4. Schematic of the computational domain
6
Khotyanovsky et al.
subsonic region implies that the shock reflection configuration (the size of the
subsonic region, variation of the reflected shock angle, variation of the pressure
along the subsonic zone) may depend on the geometry of the problem, i.e. on
the normalized distance between the wedges 2g/w. That is, in contrast to the
RRww, the RRws contains a characteristic length scale. Additional computations
performed with different 2g/w = 0.5 have shown that the extent of the subsonic
region increases significantly compared to the previous case 2g/w = 0.6. The
results of computations at different 2g/w show that the locus of the sonic line
is defined by the expansion fan emanating from the trailing edge of the upper
wedge. The flow becomes supersonic just behind the intersection point of the
leading characteristic of the fan with the strong reflected shock (see Fig. 5).
Certainly, the narrow size of the region of the existence of the RR with a
strong reflected shock wave makes it difficult to capture such a configuration in
experiments. However, the above-mentioned range of the wedge angles 15.595◦ <
θ2 < 15.983◦ at M = 4.96 and θ1 = 35◦ (see also Fig.3) is not something
insensible and might be resolved in a thorough experiment. The only problem
that will most likely arise in the wind tunnel test is that the RRws existence
conditions are very close to the upper limit of the existence of a regular reflection.
It is very likely that, in a wind tunnel experiment, Mach reflection would occur
under these conditions rather than regular reflection due to the free stream
disturbances (see, for example, [4].
The existence of a subsonic region behind the reflection point of the RRws
configuration implies that the flow is aware of the geometry of the problem, i.e.
there is a path, along which the information about the distance between the
wedges can propagate. Although the existence of such an information path is
supposed to cause the transition from RR to MR [4], the present study shows
the possibility of a steady RR with a subsonic flow behind the reflected shock.
0.1
0.05
M = 1.71
0
Y
-0.05
M = 4.96
M = 1.46
M = 3.52
-0.1
0
1.48
1.54
-0.2
X
0.1
7
1.18
3
-0.15
1.2
1.03
1.09
M = 1.0
1.6
-0.1
0.2
Fig. 5. RRws (left) and RRww (right) with identical flow deflection. Mach number
contours. The white lines correspond to the analytically predicted angles of the discontinuities
Strong regular reflection of steady shock waves
0.05
0.05
p = 16.7
p = 16.7
0
0
43.43
p=1
p = 40.7
41.39
44.79
-0.1
42.07
42.75
39.69
38.2
-0.15
44.1
-0.15
Y
-0.05
p=1
45.47
Y
-0.05
-0.1
7
40.38
9
1
-0.2
-0.2
p = 5.13
-0.25
p = 4.57
-0.25
-0.1
0
X
0.1
0.2
-0.1
0
X
0.1
0.2
Fig. 6. RRws (left) and RRww (right). Pressure contours
4
Conclusions
Asymmetric reflection of steady shock waves between two wedges has been simulated numerically with a high-order shock-capturing Euler code. For the first
time, a steady regular reflection configuration with a strong reflected shock wave
has been obtained without imposing any artificial pressure boundary conditions
downstream. It is also shown that there are two different regular reflection configurations, which deflect the flow behind the reflection point at an identical angle.
One of these configurations has two weak reflected shocks, and the other has a
weak and a strong reflected shocks. The pressures behind these configurations
differ significantly. The flow behind the strong reflected shock is subsonic in the
vicinity of the reflection point and becomes supersonic downstream. The size of
the subsonic region is found to depend on the normalized distance between the
wedges.
Acknowledgements This work was supported by the Russian Foundation for
Basic research (RFBR), Grant No. 00-01-00824 and Grant No. 01-01-06253, and
INTAS, Grant No.99-0785. This support is gratefully acknowledged.
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