Research Journal
J
of Appllied Sciences, Engineering annd Technologyy 9(6): 428-4333, 2015
ISSN: 2040-7459; e-ISSN
N: 2040-7467
© Maxwelll Scientific Orrganization, 2015
Submitted:: October 12, 2014
2
Accepted: November 10,, 2014
Publishedd: February 25, 2015
Sphericaal Shock-waave-2D Surfface Interacction
1
Paveel Viktorovich
h Bulat, 2Mikhhail Vladimirrovich Silnikoov and 2Mikhaail Viktorovicch Chernysheev
1
Univeersity ITMO, Kronverkskyy pr., 49, Saintt-Petersburg, 197101, Russia
2
Saint-P
Petersburg Staate Politechniical University, 29 Politekhhnicheskaya Str.,
S
Saint-Peterssburg 195251, Russia
Abstract: The purpose of
o research is thhe study of the transformationn of the shock--wave configurration, caused by the
o a spherical shock
s
wave froom a flat surfaace. The blast of
o HE charge heightened
h
oveer earth surfacee leads
reflection of
to formatioon of shock-waave triple confi
figuration. In sppite of static prressure equalitty of gas stream
ms after the diffferent
wave sequuences, the vellocities, densitties and other flow parameteers are not eqqual. In view of
o the fact thaat flow
velocities are
a sufficiently
y different, winnd loads on obbjects subjectedd to blast wavee action differ also. So blast shock
wave hazaard degree (in particular,
p
for human organissm at body traanslation) depeends on both obbject and HE charge
c
blast heighht. The mathem
matical model to calculate and
a analyze thhe propagatingg shock-wave triple
t
configurrations
occurring at
a the heighten
ned blast is provvided in this sttudy. The moddel is useful forr calculation annd comparison of the
velocities and dynamic pressures of the streams behind
b
the diffferent sequencces of shock waves in the triple
ws us to estimaate the basic paarameters charracterizing the tertiary blast wave
w
hazards.
configuratiion, i.e., it allow
Keywordss: Dynamic preessure, heightenned blast, triple configurationn
INTRO
ODUCTION
Objectt of study is the process of reeflection from the
t
surface of the Earth sph
herical shock wave
w
that occuurs
when the air blast. Heree we considerr the task of the
t
space shocck-2D surfacee interaction. The task of the
t
shock-wavve protection in-situ
i
occurs, particularly, for
f
an aircraft flying at the supersonic speeed. At the sam
me
time, the shock
s
front leaaves a trace on the surfaace
(Fig. 1).
ot flat, the trace from the shoock
If the substrate is no
front (Fig.. 1 in yellow
w), apparently, will experiennce
modificatioons (metamorp
phoses). At thhat, there can be
special poiints on the sho
ock trace. The contact
c
geometry
studies sim
milar modificaations (Arnold,, 1996). Its ow
wn
parameter space geometrry lays at the bottom of eveery
section off the sciencee. The Minkoowski geometry
describes the
t space of the
t Special Reelativity Theory.
The Riemaannian geomettry is the basiss for the Geneeral
Theory off Relativity. Th
he symplectic geometry is the
t
phase spacce of classical mechanics. It always has evven
dimension.
The geometries
g
difffer in how thee space metric is
introducedd. The metric is
i expression for
f the length of
the arbitrarry curve elem
ment the vectorr length shall not
n
depend on choice of the coordinate sysstem. The surfaace
concept geeneralization in
n the modern geometry is the
t
term ‘diveersity’. The diiversity is the arbitrary set of
points introduced as the union of the finite number of
areas of thee Euclidean sp
pace with the loocal coordinattes
Fig. 1: Supersonic
S
airplaane blast action on the place
preset in
i every area. The
T smooth (L
Lagrangian) maapping
is usuaally understood as a projecction of the surface
s
(diversity) of gas-dynnamic parameteers on the planne. The
mappinng can have its features andd critical points. The
set of critical
c
points is called the mapping causstic. In
the spaace of gas-dyynamic variablles the shockk front
meets the
t caustic (Arrnold and Giveental, 1985). So,
S the
shock-w
wave is a set of
o features of the
t smooth maapping
of projeecting diversityy of gas-dynam
mic variables.
Thhe contact geoometry studies projective maapping
of diveersity of gas-dyynamic param
meters onto the space
with thhe dimension one less. Suchh mapping is called
Legenddre mapping. The
T contact geoometry is the tw
win of
the syymplectic geoometry, but with the uneven
u
dimenssion of space. The shock front trace on
o the
substraate (Fig. 1) is called the Legendre
L
cobordism.
When travelling the shock front trrace (cobordism) on
the unneven surfacee can be suubject to inntricate
modificcations, incluuding with formation
f
of selfintersecction points, cllosed loops andd «bows» (Fig.. 2).
Thhe soviet matthematicians of the V. Arnold
A
scientiffic school deevoted their researches to this
Correspond
ding Author: Paavel Viktorovichh Bulat, University ITMO, Kronnverksky pr., 49, Saint-Petersburrg, 197101, Russsia
428
Res. J. App. Sci. Eng.. Technol., 9(6)
6): 428-433, 20015
mples of nontriviaal modifications of the shock-waave
Fig. 2: Sam
tracee in-situ
Fig. 3: Irreggular Mach refl
flection (on the left) and reguular
refleection of the sho
ock-wave from the
t sloping walll in
the experiments
e
of Mach
M
(1878)
(a)
(b))
Exxperimental research of interaction of the
progresssive shock-w
wave with thee immobile wedge
w
executeed later by Smith
S
(1945) and White (1951)
(
discoveered an opporttunity of existeence of a few kinds
of irreggular reflectioon which can appear at diffferent
values of the wave velocity
v
and itss contact anglee with
the surfface (Fig. 4).
Thhe analogous effects of forrmation of moovable
triple configurationns at the shock-wave
s
s
sloped
reflectiion are well-knnown (Bazhennova and Gvozzdeva,
1977; Arutyunyan and Karcheevsky, 1973)) and
stationaary configurattions at the Mach reflectiion of
compreession shockks in the supersonic jets
(Omel’chenko et al., 2003; Hadjaddj et al., 2004). The
model of triple coonfigurations of shock-wavves is
studiedd in detail by V.
V Uskov scienntific school (U
Uskov
et al., 1995;
1
Uskov and
a Mostovykhh, 2008). It is shown
s
that thhe triple confiigurations cann be extreme under
some gas-dynamic
g
paarameters (Taoo et al., 2005; Uskov
U
and Chhernyshov, 20006), includingg under the rate
r
of
velocityy head follow
wing them, whhich is actuallly for
study of
o the ways of protection agaainst the shockk-wave
effect (Gelfand
(
and Silnikov,
S
2006).
MATERIA
ALS AND ME
ETHODS
C
(b) dou
uble Mach reflecction
Fig. 4: (a) Complex,
problem. In
I Arnold’s prroduction workk (Arnold, 19776)
the approaach to the sho
ock-waves andd shock fronts is
formulatedd according to the featuures of smoooth
mapping. The
T classificattion of the feattures and criticcal
(special) points
p
of diffferentiated maapping is givven
(Arnold et al., 1982).
Todayy, the theory of the featuures of smoooth
mapping modifications by force of the famoous
y, Arnold, Tom
m, Poincaré has
h
mathematicians Whitney
been deveeloped very go
ood. For practtically importaant
cases of coodimension (diimension of thhe tangent spacce)
2 and 3 the
t complete classification of the mappiing
features annd their modifications is made
m
up (Arnoold,
1978).
ve shock-wavees and generatted
First, the progressiv
shock-wavve structures were discoveered by (Macch,
1878). In his
h work he desscribed two kinnds of the shocckwave mappping from the sloping
s
surfacee (Fig. 3):


r
reflecttion, which consists
c
of tw
wo
The regular
shock--waves: incideent wave cominng onto the soolid
surface and reflectted wave outtgoing from the
t
impact point.
c
of thrree
The irrregular reflecction, which consists
shock--waves: incideent, reflected and basic onnes
havingg the common point; such a kind
k
of reflectiion
is nam
med as simp
ple Mach refflection and the
t
correspond configurration, unless it contains othher
TC)
normaal discontinuitiies, Triple Connfiguration (T
of shoock-waves.
Schem
matic circuit of the shock
k-wave-2D su
urface
under the blast interaction:
i
L
Let’s
consideer the
simplesst sample of thhe shock-wavee-surface interraction
when the
t surface is plane (Fig. 5).
5 Analogous triple
configuurations of moovable shock-w
waves appear at the
surfacee explosion in the open on toop reservoir annd are
formedd with the shhock-wave envveloping the device
d
edges (Gelfand
(
and Silnikov,
S
2003).
Thhe shock-wave s1 formed at the
t elevated blast in
point O at height H (F
Fig. 5) reachess the ground inn point
A and then
t
spreads over
o
the surfacce with regularr (e.g.,
in poinnt B) reflectionn of the wave s2. Under thee wave
front s1 further movinng off the blasst epicenter (inn point
C) its regular refleection transfoorms into irrregular
(Mach)) reflection: a new
n (major, Mach) wave apppear s3.
At every point of tim
me the shock--waves s1, s2 and
a s3
w
they forrm the
have thhe common (trriple) point T, where
triple configuration.
c
T further moovement of thee blast
The
wave front
fr
s1 results in elevation of the triple poiint (its
trajectoory t in show
wn in Fig. 5), in the Mach wave
upsizinng s3 and graduual junction of the incident wave
w
s1
and refflected wave s2.
Thhe cocurrent flows folllowing the triple
configuuration movingg along the conntact discontinnuity 
are coollinear and they
t
have equual static preessure.
Howevver, their veloccities (u2 and u3, Fig. 5), dyynamic
pressurre are not equual to each othher which resuults in
differennce (occasionaally essential) of wind (dyynamic
pressurre) loads on the objects loocated at the same
distance R up to the blast cennter, but at diffferent
height h.
h
429 Res. J. App. Sci. Eng. Technol., 9(6): 428-433, 2015
Fig. 5: Interaction of the shock-wave generated by the blast some height over the plane surface
O: Blast epicenter; s1: Original shock-wave; s2: Shock-wave reflected from the plane surface; s3: Mach stem by height h
in the triple configuration of shock-waves caused by the origin shock; Wave: Reflected shock-wave interaction; Т: Triple
point; t: Triple point travel; τ: Tangential discontinuity following point Т; W: Velocity of the triple point
Just that very dynamic pressure of moving air mass
entrained by the blast shock-wave is the reason of
destructive effect of the shock-wave, instead of increase
of static pressure in its front, as many think.
The model of triple configuration of progressive
shockwaves is built based on the known correlations for
the similar structure of waves at rest (Compression
shocks) in the supersonic jet (Fig. 5). Such a stationary
configuration consists of three stationary shock-waves
(s1, s2 and s3) in the incoming supersonic jet with Mach
number M>1 and also tangential discontinuity
following their common point T. For generalizing the
results of analysis of the triple configurations of
stationary shock-waves on the similar structures of
progressing shock-waves, the principle of inversion of
motion is used (Uskov, 2000) relatively to the triple
point (Fig. 5). At motion inversion the triple point
becomes immovable and the gas flow leaks on it with
velocity W in the direction opposite to the trajectory of
its movement shown in Fig. 5.
Conditions of dynamic compatibility in the triple
point: The parameters of shocks s1…3 are connected to
each other and to the following flow property with the
conditions of equality of pressure and co directional
streams on the discontinuity sides  :
J1 J 2  J 3
(1)
1   2  3
(2)
where, Ji = pi/pj is relation of pressure of streams
following the shock pi and before it ипереднимpj;
i = 1…3 and βi is an angle of the flow turning on the
shock connected to its intensity (J) and Mach number
(Mj) before it:

1    M 2j    J i

Ji  
 i  arctg 

1    J i  1

1    M 2j  1    J i  1 
(3)
The Mach number Mi of the flow following the
shock is also specified by its intensity:
Mi 
 J i    M 2j  1     J i2  1
J i 1   J i 
(4)
Here, ε = (γ - 1) / (γ + 1) and γ is the gas adiabatic
index (in this study we consider the air shock-waves
and γ = 1, 4).
Dependences (1-4), the shock adiabat and the
equations of the medium condition specify also other
parameters of the streams following the triple
configuration shocks. Particularly, velocities u2 and u3
of the gas streams on the sides of the tangential
discontinuity can be found from the correlations:
u2  a0 M 2 E1 E2 J 3 , u3  a0 M 3 E3 J 3
(5)
where, Ei = (1 + εJi) / (Ji + ε) and a0 is the sonic speed
in the unperturbed medium.
The ravel speed, shock intensity and dynamic
pressure following it: Intensity J1 and J3 of shockwaves s1 and s3 spreading on the medium at rest are
430 Res. J. App. Sci. Eng. Technol., 9(6): 428-433, 2015
specified with velocities v1 and v3 of the normal
propagation of these waves fronts:
J1  1    M D2 1   ,
J 3  1    M D2 3   ,
(6)
M D1  V1 a0 ,
M D 3  V3 a0 .
At the same time, the condition of belonging the
triple point to both waves at every point of time is
written as:
V1 sin  1  V3 sin  3  W
(7)
where, W is velocity of the triple point travel,
 1   2     (Fig. 1a, b), sin    H  h  R ,
tg   y   x  and y (x) is the equation of the point
RESULTS AND DISCUSSION
The system (6-10) specifying the velocity of the
streams following the triple configuration of movable
blast shock-waves closes with the help of the
correlation for the intensity of shock-wave in the triple
point and the height of the Mach stem. These values
can be determined numerically and with the help of the
known auto model solutions. Also, suitable empirical
dependences are known which are obtained as a result
of processing of numerous results of computation and
experiments (Henrych, 1979). One of them is the
1 ∙ of the
dependence of the amplitude ∆
incident wave s1 on the distance R to the air blast
epicenter, which specifies, according to (6), velocity V1
of this wave travel:
1.38 ∗
0.543
∆
when 0.05<R*≤0.3
travelling trajectory T. At the known dependence V1 (R)
and the given equation of the point travelling T of
intensity of waves s1 and s3 (therefore, value J2 = J3/J1)
are specified at every point of time.
The intensity of waves s1 and s2 in the equivalent
triple configuration of compression shocks are specified
with correlations:
J1  1   W sin  1 a0    ,
2
J 3  1   W sin  3 a0   
2
(8)
As one can see from (6, 7), they remain invariant
as compared with the considered case of the triple
configuration of movable waves. The system (1-5)
allows, due to correlation (5), to find both the velocities
of streams on the discontinuity sides  and on their
projection on the coordinates in the equivalent
configuration of shocks at rest:
u2 x  u2 cos   3    ,
u2 y  u2 sin   3    ,
(9)
u3 x  u3 cos   3    ,
,
∗
(11)
0.607 ∗
∆
when 0.3<R*≤1.0
0.032
∗
0.209
∗
,
(12)
0.065 ∗
∆
when 1.0≤R*<10
0.397
∗
0.322
∗
,
(13)
where, R*  R G1 3 , m/kg1/3 is the distance in the
Sadovsky-Hopkinson variables; G, kg is the
trinitrotoluene equivalent of the blast energy; ∆ is the
wave amplitude s1, MPa. By analogy with variable R in
this study we use the reduced values of the blast height
H *  H G1 3 , the triple point h*  h G1 3 , the distance
2
x*  x G1 3  R2   H  h  G1 3 between the projections
of the blast epicenter and triple point of the earth
surface.
Also
dimensionless
quantities
R  R* H *  R H , h  h* H *  h H , x  x* H *  x H
are used.
The second empirical relation which closes the
system (6-10) is the equation h  h  x  of the trajectory
of point T (Balagansky and Merzhievsky, 2004):
u3 y  u3 sin   3    .
At back transfer to the coordinates related to the
flow at rest before the progressing waves, vectorial
addition of velocities of the triple point and streams
following the stationary configuration is carried out:
 
 


U 2  u2  W , U 3  u3  W
0.035
∗
(10)


which allows to find the velocity vectors U2 and U 3 of
the streams following the movable triple configuration,
to compare their absolute values.

1 h
R
 ctg  *  1, 2 ln
x
1, 3 

(14)
where,  *  2 9
In accordance with the correlation (13) at h  0
deviation of the triple point from the reflecting surface
starts at x  3, 443 ( R  3, 585 ). The intensity of the
shock-wave transferring from the regular to Mach
reflection according to (14) is close to two known
criteria of the reflection type change (Fig. 6). The
diagram of changing of the triple point dimensionless
431 Res. J. App. Sci. Eng. Technol., 9(6): 428-433, 2015
evaluation of effect of the dynamic pressure following
the shock-wave, on the objects located on in-situ.
Considering the geometry of particular objects with the
help of methods of the theory of features of the caustic
and wave fronts, it is possible to reveal modifications of
the wave front happening at some little distance from
the object.
4.5
4.0
J1
3.5
3.0
2.5
ACKNOWLEDGMENT
2 3
2.0
1.5
1
1.0
0.5
2.0
1.5
H*
Fig. 6: Dependence of the incident shock-wave intensity,
with transition to its irregular reflection, on the
reduced height of the blast (N*, m/kg1/3)
1: Empirical relation (13); 2: Spectral conditioning
criterion; 3: Criterion of “maximal angle of the flow
turning”
4.5
4.0
3.5
3.0
h
2.5
2.0
1.5
1.0
0.5
3.5
4.0
4.5
x
5.0
5.5
6.0
Fig. 7: Trajectory of the triple point specified with the
empirical relation
height h  h  x  depending on horizontal moving off
the blast source is given in Fig. 7.
The system (5-13) allows calculating and
comparing both parameters of the progressing shockwaves in triple configurations appearing at the elevated
blast and the velocity of following gas streams. The
higher the Mach stem, the more area with high dynamic
pressure.
CONCLUSION
The mathematical model built based on the
accurate and semi empirical correlations describes the
shock-wave structure of the elevated blast and allows
evaluating the wind loads following different shockwaves of this structure. Usage of semiempirical
dependences for the wave intensity and the Mach stem
height essentially simplifies calculating with
preservation of acceptable quality. The results of such
calculation can be used as the first approximation for
This study was prepared as part of the "1000
laboratories" program with the support of SaintPetersburg National Research University of Information
Technologies, Mechanics and Optics (University
ITMO) and with the financial support of the Ministry of
Education and Science of the Russian Federation (the
Agreement No. 14.575.21.0057).
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