Shock Waves (1999) 9: 95 105
Shock wave deformation in shock-vortex interactions
A. Chatterjee"
Faculty of Aerospace Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
Received July 28, 1997 / Accepted November 17, 1997
Abstract. An initially planar shock wave can undergo significant distortion to its shape along with changes
in its strength during the period of its interaction with a compressible vortex. This phenomenon is studied
by numerically simulating the shock wave-vortex interaction with a high resolution shock-capturing scheme.
Incident shock waves of various Mach numbers are made to interact with a compressible vortex and the
dependence of the shock wave distortion on the strength of the incident shock wave is studied in detail.
It is known that the type of complex shock structure formed in the later stages of a compressible vortex-
shock wave interaction is dependent on the Mach number of the incident shock wave. A simple physical
model based on the principle of shock wave reflection is proposed to explain this complex shock structure
formation and its dependence on the relative strengths of the interacting vortex and shock wave.
Key words: Shock-vortex interaction, Shock deformation, Shock reflection, Euler equations, ENO schemes
1 Introduction It is known (Ellzey et al. 1995; Guichard et al. 1995)
that an initially planar shock wave involved in a strong
interaction with a compressible vortex undergoes consid-
Shock wave-vortex interactions have been a fertile field
erable deformation and goes through successive phases of
of investigation for several decades. The previous exper-
symmetric deformation, nonsymmetric deformation lead-
imental approach to these investigations has given way
ing to the formation of a complex shock structure which
to a more numerical approach in both two and three di-
includes secondary shocks, and a gradual return to the pla-
mensions during this decade. These numerical investiga-
nar configuration as the vortex is left behind. Numerical
tions have been greatly facilitated in the recent past by
simulations by Ellzey et al. (1995) have also shown that
the evolution of high resolution shock-capturing schemes.
a Mach structure develops in the later stages of the inter-
Shock-vortex interactions are of fundamental importance
action involving a strong shock wave with a compressible
in such diverse fields as noise production in high-speed
vortex while a relatively weaker shock wave interacting
aircraft, in turbulence amplification by shock waves and
with a compressible vortex produces a configuration de-
in the interaction between mixing zones and shock waves
void of a Mach stem where the secondary shock structures
among other applications.
join up with the primary shock at a single point, but the
Shock-vortex interactions can be broadly classified into
reasons behind this transition have not been fully estab-
weak and strong interactions. Weak interactions involve
lished.
slight deformation of the shock wave and the acoustic
This investigation concentrates on studying in detail
wave generation can be predicted by linear theories and
the deformation of an initially planar shock wave as it in-
simulated using simple numerical techniques. Strong inter-
teracts strongly with a compressible vortex with a view
actions involve significant deformation of the shock wave
to developing a physical model to explain the complex
due to the vortex and may include the production of sec-
shock structure formation in the later stages of this inter-
ondary shocks. Linear theories are no longer applicable
action. We consider two-dimensional cases in which inci-
and high resolution shock-capturing schemes are required
dent shock waves of various Mach numbers are made to
for accurate numerical simulation of this strongly nonlin-
interact with a compressible vortex and the dependence
ear phenomenon. Both the strong and weak shock-vortex
of the shock wave distortion on the strength of the inci-
interactions have been studied numerically in the frame
dent shock wave is studied. The complex shock structure
work of the Euler equations (Meadows et al. 1991; Ellzey
that develops during the later stages of the shock-vortex
et al. 1995) and by a direct numerical simulation of the
interaction is examined and the dependence of the type of
Navier-Stokes equations (Guichard et al. 1995).
complex shock structure formed on the strength of the in-
"
cident shock wave sought to be explained. A simple model
Presently, CFD Group, Aeronautical Development Agency,
PB 1718, Vimanpura Post, Bangalore 560 017, India; based on the shock wave reflection principle is proposed
e-mail: [e-mail] to explain the complex shock structure formation, its de-
96 A. Chatterjee: Shock wave deformation in shock-vortex interactions
pendence on the incident shock wave Mach number and
on the relative strengths of the vortex and the incident
shock wave in general.
The numerical simulations are two-dimensional and
carried out by solving the Euler equations. The signifi-
cantly larger time scales for viscous dissipation as com-
pared to other important time scales involved in shock-
vortex interactions makes it possible to neglect viscous
effects in numerical computations (Ellzey et al. 1995). A
high resolution shock-capturing scheme in the form of the
ENO (essentially non-oscillatory) scheme (Harten and Os-
her 1987; Harten et al. 1987) is used to solve the Euler
equations in the conservative form.
2 Numerical technique
The two-dimensional Euler equations of gas dynamics in
conservative form are solved by the ENO (Harten and
Osher 1987; Harten et al. 1987) scheme. This scheme be-
longs to the class of high resolution numerical schemes
developed to deal with flowfields containing shock waves
Fig. 1. The physical and computational domain
and are able to maintain high order accuracy in smooth
regions of the flow as well as provide for nonoscillatory
shocks. The ENO scheme is able to achieve this dual ca-
Ellzey et al. (1995). The vortex model consists of two re-
pacity by employing an adaptive stenciling procedure. The
gions of vortical flow; an inner core region and a surround-
adaptive stenciling attempts to make use of the smoothest
ing region where the velocity gradually goes to zero. In
possible information in the computation of the numerical
the numerical experiments carried out, two different ini-
fluxes at the cell interfaces. The points in the stencil that
tial distributions of tangential velocity in the vortex core
contribute to the computation of numerical fluxes at cell
region were considered. The first distribution is that of a
interfaces for the next time step are chosen in a nonlinear
constant tangential velocity and is represented by
manner and depend on the instantaneous solution. The
ENO scheme used here is the ENO-Roe form (Shu and
U¸(r) =Uc, r < Rc, (1)
Osher 1988; 1989) which is an efficient implementation of
the original ENO methodology and has the ENO construc-
the second distribution is that of a linear tangential ve-
tion procedure based on the numerical fluxes rather than
locity profile and is represented by
on cell averages of the state variables.
U¸(r) =Ucr/Rc, r < Rc, (2)
An explicit form of the mentioned scheme is used to
solve this time-dependent flow and advancement in time
the outer distribution was always given by
is performed by a Runge-Kutta type integration (Shu and
Osher 1988; 1989). The method has a third order accu-
B
U¸(r) =Ar + , Rc d" r d" Ro, (3)
racy in space and a second order accuracy in time. All nu-
r
merical investigations involved a moving shock wave that
was planar in the initial configuration. The initial con- where
ditions behind the shock wave are prescribed according
U¸ = tangential velocity,
to the Rankine-Hugoniot relations while ambient condi- Uc = constant core velocity,
tions are given ahead of it. Characteristic boundary condi- r = distance from vortex center,
tions based on Riemann invariants are prescribed at open
Rc = vortex core radius,
boundaries. Equally-spaced cartesian grids were used, and
Ro = outer radius.
unless mentioned otherwise consisted of 300 × 300 grid
The coefficients A and B in Eq. (3) are chosen so that U¸
points. Numerical simulation of complex flowfields includ- equals Uc when r equals Rc and U¸ equals 0 when r equals
ing compressible vortex-shock wave interactions using the
Ro.
above mentioned numerical technique have been shown to
The initial configuration is shown schematically in
be in good agreement with experimental results (Chatter- Fig. 1. The computational domain measures 4.0 in the
jee et al. 1997).
x-(horizontal) and y-(vertical) directions. The vortex has
The shock wave proceeds into ambient air and inter- a core radius Rc= 0.5 and an outer radius Ro= 1.0, the
acts with a compressible vortex superimposed on the am- vortex-center was placed at the center of the computa-
bient state. The vortex generated is similar to those in tional domain and the vortical velocity was counterclock-
the numerical investigations of Meadows et al. (1991) and wise. The origin is taken to be at the vortex center, and in
A. Chatterjee: Shock wave deformation in shock-vortex interactions 97
Fig. 2a d. Symmetric deformation (Incident Mach number 1.5). Pressure contours with 30 equally spaced levels. a With
initially stepwise distribution in vortex core. b Pressure variation along the x-direction at y = Ä…Rc. c With initially linear
distribution in vortex core. d Pressure variation along the x-direction at y = Ä…Rc
all cases the straight planar shock wave was at x = 1.66 interaction involving a compressible vortex and incident
at the start of the simulations and it proceeds from right shock waves of comparable strengths in the investigations
to left. The size of the vortex core with respect to the of Ellzey et al. (1995).
vortex as a whole has been exaggerated when compared
to that measured experimentally for a two-dimensional
3 Numerical results  initial deformation
freely-moving compressible vortex by Kao et al. (1996),
and is done to get a better resolution of the flowfield in
The deformation experienced by the shock wave in the
this critical core region.
early stages of a strong interaction and the appearance
Three different sets of simulations were carried out by of secondary shock structures in the later stages are ini-
varying the strength of the incident shock wave and the tially discussed for an incident shock wave of Mach num-
corresponding incident Mach numbers were 1.5, 1.375 and ber 1.5 and the results obtained are then compared with
1.25. Unless otherwise stated in all the cases considered, that involving relatively weaker incident shock waves of
the maximum core velocity Uc is always set to be the Mach numbers 1.375 and 1.25. It is noted that this de-
velocity of the fluid behind an incident planar shock wave formation process has also been the subject of previous
of Mach number 1.5 and corresponds to approximately numerical investigation by Ellzey et al. (1995), Guichard
237.2 m/s. This value for the maximum core velocity has et al. (1995) and is repeated here in greater detail as it
also been the basis for a compressible vortex in the strong also serves as a basis for developing a physical model to
98 A. Chatterjee: Shock wave deformation in shock-vortex interactions
across the shock wave at y = Ä…Rc for the second distribu-
tion with the shock wave centered at x =0.0 are shown in
Figs. 2c and 2d respectively. The profile of the shock wave
is similar to that in Fig. 2a (but for obvious differences in
the core region) while the pressure jumps at y = Ä…Rc are
almost identical to the previous case.
The initially specified discontinuity in the tangential
velocity across the vortex center in the first distribution
given by Eq. (1) gets smeared with time across the vor-
tex center and results in a peak tangential velocity at an
approximate distance of Rc from the vortex center. The
location and the magnitude of the peak tangential velocity
are almost identical in both the distributions and since the
maximum and minimum pressure ratios across the shock
as well as the maximum shock distortion depend on this
Fig. 3. Nonsymmetric deformation (Incident Mach number
peak tangential velocity, both distributions give almost
1.5)
identical results. The first distribution of a discontinuous
tangential velocity (Eq. (1)) across the vortex center con-
tinues to get smeared with time and tends towards a lin-
explain the complex shock structure formation in the later
ear distribution for the tangential velocity in the vortex
stages of a strong interaction between a shock wave and a
core given by Eq. (2) and the shock profile obtained in
compressible vortex.
both cases becomes more similar. Subsequent numerical
When the shock wave moves into the vortical region,
results are presented using an initial distribution for the
the part of the shock wave in the interaction region where
tangential velocity in the vortex core given by Eq. (1) as
the shock velocity is opposed by the vortex rotation prop-
this initial distribution was found to be more compatible
agates slower than the part where the shock velocity is
with experimental observations in a freely-moving two-
supported by the vortex rotation. In this particular case,
dimensional compressible vortex (Kao et al. 1996), espe-
the bottom half of the shock wave in the interaction region
cially for the exaggerated size of the vortex core compared
meets an opposing flow and propagates slower than the top
to the vortex as a whole considered here. But it is to be
half which encounters a flow in the same direction as the
emphasised that all numerical experiments were also done
propagating shock wave. There is also a strengthening of
using an initial linear distribution of tangential velocity
the part of the shock wave in the interaction region where
in the vortex core given by Eq. (2), and for all practical
the advancing shock wave faces an opposing flow due to
purposes almost identical results were obtained.
an increase in the relative Mach number (relative to the
oncoming flow) of this part of the shock wave and a similar
This nonuniform pressure jump across the shock wave
weakening of the part of the shock wave in the interaction
in the initial stages also means a distortion of the ini-
region where the vortical velocity is in the same direction
tial vortex and is accompanied by a more nonsymmetric
as that of the shock wave due to a decrease in its relative
deformation of the shock wave. During this phase of non-
Mach number. The initial stage of the shock-vortex inter- symmetric deformation there is a further increase in the
action involves an almost symmetric deformation of the
deviation of both the forward and backward part of the
initially planar shock wave. The pressure contours corre- deformed shock wave from its initial straight planar con-
sponding to this almost symmetric deformation is shown
figuration. The pressure contours corresponding to a non-
in Fig. 2a with the shock wave centered at x = 0.0. At
symmetric deformation with the shock wave centered at
this stage the shock wave is locally almost normal to the -0.3 are shown in Fig. 3. This nonsymmetric de-
x =
x-axis at distances y = Ä…Rc (also see Fig. 1) from the vor- formation ends with the formation of secondary shock
tex center. Figure 2b shows the pressure variation along
structures which are related to the appearance of addi-
the x-direction at y = Ä…Rc. Local changes in the relative
tional pressure gradients in a direction parallel to the
Mach number of the shock wave in its interaction with
shock wave. In this case of a relatively strong shock wave
the vortex cause a local increase in the shock strength at
interacting with a compressible vortex a Mach structure
y = -Rc and a decrease in the local shock strength at
is formed. The Mach stem has two secondary shocks at-
y =+Rc, as can be seen in Fig. 2b. This goes along with
tached at either end. The pressure contours correspond-
a decrease in the local shock velocity at y = -Rc and an
ing to this Mach structure with the shock wave centered
increase in the local shock velocity at y =+Rc contribut- at x = -1.07 are shown in Fig. 4a. This Mach structure
ing to the deformation of the shock wave in Fig. 2a.
is shown in terms of density contours in Fig. 4b where a
The results shown above pertain to an initial distri- slipstream at either end of the Mach stem can be clearly
bution of an initially constant tangential velocity in the seen. This Mach structure was also captured on a finer
vortex core region given by Eq. (1). Numerical experi- discretization of 600 × 600 grid points and the pressure
ments were also done using the second distribution that and density contours for the Mach structure are shown
of a linearly varying tangential velocity in the vortex core in Figs. 4c and 4d, respectively. The shock wave gradually
as in Eq. (2). The pressure contours and the pressure ratio returns to its initial planar configuration following the for-
A. Chatterjee: Shock wave deformation in shock-vortex interactions 99
Fig. 4a d. Mach Structure (Incident Mach number 1.5). a Pressure contours. b Density contours. 60 equally spaced contour
levels. c Pressure contours (600 × 600 grid points). d Density contours (600 × 600 grid points)
mation of the complex shock structure as the vortex is left and the short plateau that follows correspond to the ap-
behind. pearance of secondary shocks and the formation of the
complex shock structure. Thereafter, the decrease in this
Similar numerical simulations were also carried out, in
deviation indicates a gradual return of the shock wave to
which weaker incident shocks of Mach numbers 1.375 and
its original straight planar configuration. The maximum
1.25 interact with the same vortex as in the previous sim-
deformation is seen to occur shortly after the vortex cen-
ulation. As the Mach number of the incident shock wave is
ter is crossed at around x = -0.3 and persists till about
decreased, it gets deformed to a greater degree by the same
x = -0.8.
vortex, with decreasing incident shock strength. It is ob-
served that there is a greater deviation for the shock wave Figure 6a shows the pressure contours for the complex
from its straight planar configuration during all stages of shock structure formed in the later stages of the inter-
its interaction with the compressible vortex. The defor- action involving an incident shock wave of Mach number
mation on either side of the initially straight planar shock 1.375 with the shock wave centered at x = -1.04 while
wave was tracked for all the three cases till the time when the same involving an incident shock wave of Mach num-
the shock wave starts relaxing to its initial planar configu- ber 1.25 with the shock wave centered at x = -1.02 can
ration following the appearance of secondary shock struc- be seen in Fig. 6b. A still finite, albeit shorter Mach stem
tures. Figures 5a and 5b, respectively, show the variation than that involving the relatively stronger incident shock
of maximum deviation for the shock wave to the right and wave can be seen in Fig. 6a. The complex shock structure
left of its initially straight planar shape during its interac- involving the weakest shock with an incident Mach num-
tion with the vortex. The initial increase in the deforma- ber 1.25 in Fig. 6b shows a nonexistent Mach stem with
tion of the shock wave pertains to the symmetric defor- the secondary shocks attached to the primary shock wave
mation and the early stages of nonsymmetric deformation at a single point. Thus, decreasing the incident shock wave
experienced by the shock wave. The maximum deviation Mach number while keeping the vortex unchanged results
100 A. Chatterjee: Shock wave deformation in shock-vortex interactions
Fig. 6a,b. Complex shock structure. a Incident Mach number
1.375. b Incident Mach number 1.25
act with a weaker vortex as compared to that in the pre-
vious simulations. The vortex in this case has the same
dimensions as the previous vortex, but is weaker in the
sense that the constant core velocity Uc equals the velocity
of the fluid behind an incident planar shock wave of Mach
Fig. 5a,b. The history of maximum deformation of shock number 1.25 (corresponding to approximately 128.1 m/s)
wave. a To the right of initially straight planar configuration. as opposed to Uc equaling the velocity behind a planar
b To the left of initially straight planar configuration
shock wave of Mach number 1.5 in previous simulations.
Figure 7 shows the pressure contours corresponding to the
complex shock structure with the shock wave centered at
in a transition to a complex shock structure completely
x = -0.91 which in this case is a Mach structure. Thus,
devoid of a Mach stem from an earlier well-defined Mach
an incident shock of Mach number 1.25 which formed a
structure.
complex shock structure entirely devoid of a Mach stem
while interacting with an earlier stronger vortex forms a
well defined Mach structure when it interacts with a rel-
4 Complex shock structure  atively weaker vortex. This shows that a shock wave is
required to be strong relative to the interacting vortex to
numerical results
form a Mach structure.
The process leading to the formation of the complex shock This suggests an analogy, also noted briefly by Ellzey
structure in the later stages of the shock-vortex interac- et al. (1995), with the well known case of a planar prop-
tion was examined in detail. To show the dependence of agating incident shock wave interacting with a rigid sur-
the complex shock structure formed on the strength of the face where depending upon the incident shock wave Mach
vortex, numerical simulations were carried out in which an number and the angle of incidence (between the rigid sur-
incident shock wave of Mach number 1.25 is made to inter- face and the incident shock wave) two types of shock struc-
A. Chatterjee: Shock wave deformation in shock-vortex interactions 101
Fig. 7. Mach Structure (Incident Mach number 1.25 and
weaker vortex)
tures or reflections are possible (Ben-Dor 1992). For small
angles of incidence, the reflected shock wave meets the in-
cident shock wave at the reflecting point, and is known
as a regular reflection. Increasing the angle of incidence
while keeping the incident shock wave Mach number con-
stant results in a transition to a configuration where the
intersection of the reflected and incident shock wave lies
above the surface and the intersection is joined to the re-
flecting surface by a Mach stem, which results in a Mach
reflection.
In this case of a planar shock wave interacting with
a compressible vortex, the resulting complex shock struc-
tures with and without the Mach stem can be likened to
Fig. 8a,b. Velocity vectors for the complex shock structure.
the shock structures in Mach and regular reflection respec-
Every 4th point in each dimension considered. a Incident Mach
tively with a change in the vortex strength analogous to a
number 1.5. b Incident Mach number 1.25
change in the angle of incidence of the planar propagating
incident shock wave with the rigid surface. In the present
experiment, keeping the incident shock wave Mach num-
ture formation in the later stages of the interaction of a
ber constant while reducing the vortex strength results in
shock wave with a compressible vortex.
the transition to a Mach structure from an earlier complex
The situation in terms of a velocity vector plot just
shock structure devoid of a Mach stem.
prior to the appearance of the secondary shock structures,
Figures 8a and 8b respectively show the velocity vec-
in the nonsymmetric deformation phase for the shock wave,
tor plots corresponding to complex shock structures with
is shown in Fig. 9. The shock wave as seen at this stage
and without a Mach stem. Figure 8a represents the sit-
consists of three distinct parts; the upper curved region,
uation when a shock wave of incident Mach number 1.5
the almost straight planar region (albeit small) in the
interacts with the relatively strong vortex and the pressure
middle and the lower curved region. The secondary shock
and density contours for that situation were presented in
structures are seen to form as the upper and lower curved
Figs. 4a d. Figure 8b represents the situation when a rel-
part of the deformed shock wave reflects from the vorti-
atively weaker shock wave with an incident Mach number
cal flow stream ahead of the locally almost straight planar
1.25 interacts with the same vortex and the correspond-
part of the shock wave in the middle, as seen in velocity
ing pressure contours for the complex shock structure were
contours in Fig. 9. This almost straight planar region in
presented in Fig. 6b. The velocity field induced behind the
the middle is the region of the shock wave locally facing
relatively stronger incident shock wave alters the original
the most opposition to its propagation due to the vorti-
vortical velocity field considerably even causing a flow re-
cal flowfield. Thus, this region of the shock wave is locally
versal in the strong core region as can be seen in Fig. 8a.
the region of maximum strengthening and minimum shock
wave velocity, and lies at a distance of approximately Rc
(the vortex core radius) below the vortex center.
5 Complex shock structure  physical model
Figure 10a shows the tangential velocity profile and
A simple model based on the shock wave reflection phe- Fig. 10b the density and pressure profile (normalized with
nomenon is proposed to explain the complex shock struc- respect to the ambient values) that develops along the
102 A. Chatterjee: Shock wave deformation in shock-vortex interactions
Fig. 9. Velocity vectors prior to appearance of secondary
shocks. Close up view with every 4th point in each dimension
considered
radius of the vortex undistorted by a shock wave at the
time of complex shock structure formation. The vortex is
the relatively stronger one considered in the previous two-
dimensional numerical simulations. The tangential veloc-
ity peaks at a distance Rc from the vortex center, as seen
in Fig. 10a. As mentioned previously, the prescribed vor-
tex model has Rc = 0.5 which is half the vortex outer
radius Ro.
Assuming a vortex undistorted by the shock wave dur-
ing the interaction, the maximum strengthening and min-
imum shock velocity for the propagating shock wave can
be expected at a distance Rc directly below the vortex
center (x = 0, y = -Rc) during the entire interaction
process. This obviously assumes a counterclockwise rota-
Fig. 10a,b. Undisturbed vortex at complex shock structure
tion for the vortex with the initially straight planar shock
formation. a Tangential velocity distribution. b Pressure and
wave proceeding from right to left and the shock wave
density variation
being locally straight and planar at this location. At this
shock wave location, the conditions ahead of the shock
wave would be that of the peak tangential Mach number
of the vortex exactly opposing the shock wave velocity. wave velocity Us. The conditions ahead of the shock wave
It was shown previously in the two-dimensional numerical were changed to that corresponding to the peak vortex
simulation that the shock wave locally almost retains its tangential velocity opposing the shock wave velocity along
straight planar shape at this location. with representative values of density and pressure from
Fig. 10b. The ambient state (1) ahead of the shock wave
To get an estimate of this maximum increase in shock
strength (and maximum decrease in the shock wave veloc- was altered to state (1 ) and the altered flowfield (state
ity) under such conditions, numerical experiments were
(1 )) had a velocity u1 (the peak vortex tangential ve-
carried out in which a straight planar shock wave was
locity opposing the shock wave velocity), pressure p1 and
made to propagate into a uniform flowfield having a flow
density Á1. This consequently alters the state behind the
Mach number corresponding to that of the peak tangen-
shock wave from state (2) to state (2 ). This altered state
tial Mach number of the vortex opposing the shock wave
behind the shock wave was obtained by numerically solv-
propagation, and this involved the solution of the one-
ing the time-dependent one-dimensional Euler equations
dimensional Euler equations.
and corresponds to a flow velocity u2, pressure p2 and
Let Us be the shock wave velocity corresponding to
density Á2. Let Ms be the modified Mach number for the
the incident shock wave Mach number Ms. Initially the
shock wave under these conditions; this has also been re-
propagating planar shock wave had state (1) in front of
ferred earlier as the relative Mach number (relative to the
the shock and state (2) behind. State (1) corresponds to
ambient conditions of pressure p1, density Á1 and veloc- oncoming flow). The modified shock wave velocity Us can
ity u1 = 0. State (2) corresponds to conditions prescribed in turn be easily calculated by applying the well-known
according to the Rankine-Hugoniot relations for a shock Galilean transformation to this propagating shock wave.
A. Chatterjee: Shock wave deformation in shock-vortex interactions 103
Fig. 11. Model for shock wave reflection
Under the Galilean transformation the shock wave is Table 1. Parameters for the reflection process under proposed
considered stationary and the flowfield ahead of the sta- model; Ms - incident shock Mach number, Us - incident shock
velocity, Ms - effective incident shock Mach number, Us - mod-
tionary shock wave (state (1 )) has a velocity Us-u1 along
ified shock velocity, ¸ - angle of incidence
with pressure p1 and density Á1. In state (2 ) behind the
stationary shock wave the flow velocity is Us - u2 along
Ms Us(m/s) Ms Us(m/s) ¸(deg.) Config.
with the pressure p2 and density Á2. The shock velocity
1.500 512.34 1.768 397.79 49.18 MR
Us and Mach number Ms for the propagating shock wave
1.375 469.65 1.626 351.93 44.90 MR
under the second configuration can be easily obtained
1.25(1) 426.95 1.486 301.07 38.57 RR
from normal shock relations involving the stationary shock
1.25(2) 426.95 1.387 353.33 57.99 MR
wave.
The model proposed assumes the vortex to be undis-
torted by the shock wave in the interaction process lead-
ing to the formation of a complex shock structure. In the at this location. Ms and Us are, respectively, the (relative)
model proposed, shown schematically in Fig. 11, the shock Mach number and shock wave velocity calculated accord-
wave reflection takes place with the locally almost straight ing to the above described methodology at point C. It is
planar part of the deformed shock wave represented here assumed that the shock wave has a uniform Mach num-
by point B located at x = -Rc and y = -Rc, and this ber and velocity corresponding to Ms and Us, respectively,
location is partly based on observations from the two- along the horizontal line BD (the line parallel to the x-axis
dimensional numerical simulations for the shock-vortex in- at y = -Rc) passing through C.
teraction process discussed in the previous sections. The
The shock wave is completely planar at x = Ro and
two curved (here inclined) part of the shock wave reflects
the time t elapsed between this planar configuration and
from the reflecting surface AB in Fig. 11 which repre- that at reflection shown in Fig. 11 is
sents the vortical flow-stream in front of the locally al-
most straight planar part of the deformed shock wave. t =(Ro + Rc)/Us =3Rc/Us, (4)
In keeping with the prescribed model for the vortex and
and represents the time required for the shock wave to
assumption of an undistorted vortex, the Mach number
reach point B along the horizontal line AD, with the as-
and velocity of the incident shock wave are unchanged
sumed uniform velocity Us, starting from an initial po-
along the horizontal lines EF and GH which are lines par-
sition D. The distance traveled by the shock wave along
allel to the x-axis at y = -Ro and y = 0 respectively.
the horizontal lines EF and GH in the same time with the
As mentioned previously, the shock wave experiences the
uniform unchanged incident shock wave velocity Us is
maximum strengthening and possesses the minimum ve-
locity at point C located at x = 0 and y = -Rc with
the shock wave being locally almost straight and planar Ust =3RcUs/Us. (5)
104 A. Chatterjee: Shock wave deformation in shock-vortex interactions
The greater distance traveled by the shock wave along the
horizontal lines EF and GH in time t as compared to that
along the horizontal line AD results in an equal inclination
for the shock wave above and below the reflecting surface
AB. Thus, the incidence angle ¸ made by the two inclined
parts of the shock wave with the reflecting surface AB, at
point B is
¸ = tan-1(1/3(Us/Us - 1)). (6)
The shock wave Mach number defining the reflection phe-
nomenon along with the angle of incidence ¸ is the above
mentioned local Mach number Ms for the shock wave at
the reflection point B and this can be considered as the ef-
fective incident shock wave Mach number for the reflection
process in the proposed model. Table 1 lists the effective
incident shock wave Mach number Ms, the corresponding
Fig. 12. Possible reflection configurations
shock wave velocity Us and the angle of incidence ¸ ob-
tained from this proposed model for the various incident
Mach numbers Ms (and incident shock wave velocity Us)
with the vortex. Under the proposed model, a decrease in
of the initially planar propagating shock wave for which
the incident shock wave Mach number (for an unchanged
two-dimensional numerical simulations were discussed in
vortex strength) also means a proportional decrease in the
the earlier section.
effective shock wave incident Mach number at reflection
Figure 12 shows the domain of theoretically possible
while a greater deformation for the shock wave implies a
reflection configurations when a planar propagating inci-
decreasing angle of incidence for the incident shock wave
dent shock wave interacts with a rigid surface. The calcu-
with the reflecting surface. The general tendency to move
lations are done using the iterative technique described in
from a MR to RR configuration in such a case observed
detail by Ben-Dor and Rayevsky (1994). The upper transi-
in the two-dimensional numerical simulations is the same
tion criterion refers to the largest incident angle (the angle
as that can be expected for a shock wave interacting with
between the incident planar shock wave and the rigid sur-
a rigid surface under similar circumstances of a simulta-
face) for which a RR (regular reflection) is possible for a
neous decrease in the incident shock wave Mach number
given incident shock wave Mach number while the lower
and angle of incidence, as can be seen in Fig. 12.
transition criterion refers to the smallest incident angle
The effect of reducing the vortex strength while keep-
for which a MR (Mach reflection) is possible for a given
ing the incident shock wave Mach number constant is a
incident shock wave Mach number. Consequently, the up-
smaller deformation for the shock wave in the shock-vortex
per and lower criterion are, respectively, the largest and
interaction process. Under the proposed model a smaller
smallest possible values of the incident angle for transition
deformation of the incident shock wave implies an increase
for a given incident shock wave Mach number (Ben-Dor
in the incidence angle with the reflecting surface for the
1992).
incident shock wave. The transition of the complex shock
structure from a RR configuration to that of a MR con-
The effective incident shock wave Mach number Ms
figuration observed in such a case in the two-dimensional
and angle of incidence ¸ define the reflection process in the
numerical simulations can be considered similar to that
proposed model, and the resulting reflection configuration
obtained by substituting these values in the domain of the- in the well-known case of the interaction of a shock wave
with a rigid surface under circumstances of increasing the
oretically possible reflection configurations given in Fig. 12
are also listed in Table 1. The reflection configuration ob- incidence angle for a constant incident shock wave Mach
number. The change in the effective incident shock wave
tained from the proposed model agrees with the type of
complex shock structure formed in the two-dimension nu- Mach number at reflection in the proposed model that also
results from a change in the vortex strength (for a con-
merical simulation of the shock-vortex interaction for all
stant incident shock wave Mach number) has a relatively
the cases considered. In this proposed model the MR/RR
less significant effect on this transition process compared
boundaries obtained for steady (pseudo-steady) flows are
to the change in the incidence angle.
applied to an essentially unsteady flow (the shock-vortex
interaction). Thus, the proposed approach can be consid-
ered as a very first approximation towards explaining the
complex shock structure formation in the interaction be-
6 Conclusion
tween an initially planar shock wave and a compressible
vortex.
This investigation concentrated on studying the deforma-
It was shown in the two-dimensional simulations that tion experienced by an initially planar shock wave as it
decreasing the incident shock wave Mach number while interacts strongly with a compressible vortex. The shock-
keeping the vortex unchanged results in a greater defor- vortex interactions were simulated using a high resolu-
mation for the shock wave in all stages of its interaction tion shock-capturing scheme. By varying the Mach num-
A. Chatterjee: Shock wave deformation in shock-vortex interactions 105
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