A Numerical Study of Shock Reflection Phenomena in Shock/Turbulence InteractionMOHAMMAD ALI JINNAH Department of Mechanical and Chemical Engineering Islamic University of Technology (IUT) Board Bazar, Gazipur-1704 BANGLADESH Abstract: - Shock reflection phenomena have been studied numerically in shock/turbulence interaction where different types of shock reflector are used for the partial reflection of the shock wave. The three-dimensional Reynolds-averaged Navier-stokes equations with k-ε turbulence model are solved and the results have been compared with the Navier-Stokes Simulation (NS) results. The comparisons indicate that the present turbulence model is working very well in shock reflection phenomena for the reflection from different shockreflectors. Different strengths of reflected shock wave after reflection from the shock reflectors of 49.0 % opening area, 26.5 % opening area and from the plane end wall interact with the same turbulence field. The outcomes of shock/turbulence interaction are highly influenced by the strength of the reflected shock wave and the longitudinal velocity across the shock wave. The longitudinal velocity behind the reflected shock wave increases due to the partial reflection from the shock reflectors and the higher longitudinal velocities are obtained in the downstream of the reflected shock wave after reflection from the shock reflector of higheropenings. In the case of partial reflection, the static temperature deviations are observed in the downstream region and the deviations are higher for the interaction of stronger reflected shock wave. The turbulent length scales are measured in the upstream and downstream of the reflected shock wave and it is observed that the amplification of the turbulent length scales decrease after the shock/turbulence interaction. The rate ofdissipation of turbulent kinetic energy decreases after the interaction of different strengths of shock wave with turbulent field. Key-Words: - Shock wave, Turbulent flow, Navier-Stokes equations, Turbulence model 1 Introduction The study of shock reflection phenomena is the important part in shock/turbulence interaction. Many researchers were used the plane end wall of the shock tube for the reflected shock wave and the reflected shock wave later was used to interact with the shock-induced turbulent flow. In the present study, special types of shock reflector are used forthe different strengths of reflected shock wave to interact in the turbulent field. The incident shockafter passing through turbulence-generating grids becomes distorted and converges again to become a plane shock wave. Due to shock wave distortion, a homogenous turbulent field is appeared after a certain distance in the wake of the turbulence grids. The transmitted shock wave, which is weaker than the incident shock wave, reflects from the shockreflector and start moving through the turbulent field in the upstream of the reflected shock wave. The strength of the reflected shock wave depends on the open area ratio of the shock reflector. Due to partial reflection from the shock reflector, both deflection and distortion are appeared in the reflected shockwave at the initial stage of the reflection and aftertraveling a short distance; the non-plane reflected shock wave converges to form again the plane normal shock wave and interacts with the grid- generated turbulent field. The reflected shock wave strength is measured numerically for the shockreflector of 26.5% openings and 49.5% openings and for the plane end wall. The fully reflected shockwave is observed from the plane end wall and in the case of the full reflection; the longitudinal velocity in the downstream of the reflected shock wave is nearly zero. So using the shock reflector is the advantage of the increasing the longitudinal velocity in the downstream of the reflected shock wave. Due to increasing the longitudinal velocity behind the reflected shock wave, it is possible to avoid the non-flow turbulent field after the shock wave interaction. Different longitudinal velocities are obtained for the reflection from the shock reflectorProceedings of the 4th WSEAS International Conference on Fluid Mechanics, Gold Coast, Queensland, Australia, January 17-19, 2007 21
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8/3/2019 Mohammad Ali Jinnah- A Numerical Study of Shock Reflection Phenomena in Shock/Turbulence Interaction
Abstract: - Shock reflection phenomena have been studied numerically in shock/turbulence interaction where
different types of shock reflector are used for the partial reflection of the shock wave. The three-dimensional
Reynolds-averaged Navier-stokes equations with k-ε turbulence model are solved and the results have been
compared with the Navier-Stokes Simulation (NS) results. The comparisons indicate that the present
turbulence model is working very well in shock reflection phenomena for the reflection from different shock
reflectors. Different strengths of reflected shock wave after reflection from the shock reflectors of 49.0 %opening area, 26.5 % opening area and from the plane end wall interact with the same turbulence field. The
outcomes of shock/turbulence interaction are highly influenced by the strength of the reflected shock wave and
the longitudinal velocity across the shock wave. The longitudinal velocity behind the reflected shock wave
increases due to the partial reflection from the shock reflectors and the higher longitudinal velocities are
obtained in the downstream of the reflected shock wave after reflection from the shock reflector of higher
openings. In the case of partial reflection, the static temperature deviations are observed in the downstream
region and the deviations are higher for the interaction of stronger reflected shock wave. The turbulent length
scales are measured in the upstream and downstream of the reflected shock wave and it is observed that the
amplification of the turbulent length scales decrease after the shock/turbulence interaction. The rate of
dissipation of turbulent kinetic energy decreases after the interaction of different strengths of shock wave with
turbulent field.
Key-Words: - Shock wave, Turbulent flow, Navier-Stokes equations, Turbulence model
1 IntroductionThe study of shock reflection phenomena is the
important part in shock/turbulence interaction. Many
researchers were used the plane end wall of the
shock tube for the reflected shock wave and the
reflected shock wave later was used to interact with
the shock-induced turbulent flow. In the present
study, special types of shock reflector are used for the different strengths of reflected shock wave to
interact in the turbulent field. The incident shock
after passing through turbulence-generating grids
becomes distorted and converges again to become a
plane shock wave. Due to shock wave distortion, a
homogenous turbulent field is appeared after a
certain distance in the wake of the turbulence grids.
The transmitted shock wave, which is weaker than
the incident shock wave, reflects from the shock
reflector and start moving through the turbulent field
in the upstream of the reflected shock wave. The
strength of the reflected shock wave depends on theopen area ratio of the shock reflector. Due to partial
reflection from the shock reflector, both deflection
and distortion are appeared in the reflected shock
wave at the initial stage of the reflection and after
traveling a short distance; the non-plane reflected
shock wave converges to form again the plane
normal shock wave and interacts with the grid-
generated turbulent field. The reflected shock wave
strength is measured numerically for the shock
reflector of 26.5% openings and 49.5% openings
and for the plane end wall. The fully reflected shock
wave is observed from the plane end wall and in the
case of the full reflection; the longitudinal velocity
in the downstream of the reflected shock wave is
nearly zero. So using the shock reflector is the
advantage of the increasing the longitudinal velocity
in the downstream of the reflected shock wave. Due
to increasing the longitudinal velocity behind the
reflected shock wave, it is possible to avoid the
non-flow turbulent field after the shock wave
interaction. Different longitudinal velocities areobtained for the reflection from the shock reflector
Proceedings of the 4th WSEAS International Conference on Fluid Mechanics, Gold Coast, Queensland, Australia, January 17-19, 2007 21
8/3/2019 Mohammad Ali Jinnah- A Numerical Study of Shock Reflection Phenomena in Shock/Turbulence Interaction
2.1 Governing equations The three-dimensional unsteady, compressible,
Reynolds-averaged Navier-stokes equations with k-ε turbulence model are solved by shock capturing
method. Without external forces and heat sources,the conservative form of non-dimensionalized
governing equation in three-dimensional Cartesian
coordinate system is,
)()()()(
QS z Hv H
yGvG
x Fv F
t Q
=∂−∂
+∂−∂
+∂−∂
+∂∂
where, Q = [ ρ , ρu, ρv, ρw, e, ρk, ρε ],
F = [ ρu, ρu2 , ρuv, ρuw, u(e+p) , ρuk, ρuε ],
G = [ ρv, ρuv, ρv2 , ρvw, v(e+p) , ρvk, ρvε ],
H = [ ρw, ρuw, ρvw, ρw2 , w(e+p) , ρwk, ρwε ] and
Fv = [0, τ xx , τ xy , τ xz , uτ xx+vτ xy+wτ xz –q x , k x , ε x ],Gv = [0, τ xy , τ yy , τ yz , uτ xy+vτ yy+wτ yz –q y , k y , ε y ], Hv = [0, τ xz , τ yz , τ zz , uτ xz +vτ yz +wτ zz –q z , k z , ε z ]
Here Q is the vector of conservative variables which
contain mass, momentum and energy. All variables
are calculated in per unit volume. Three momentum
terms in three-dimensional Cartesian coordinates
system are ρu, ρv and ρw per unit volume. Total
energy, e, turbulent kinetic energy, ρk and turbulent
dissipative energy, ρε are the energy terms per unit
volume. F , G and H are the three inviscid flux
vectors in X -, Y-, and Z -axis respectively. Similarly F v, Gv and H v are the three viscous flux vectors in
X -, Y-, and Z -axis respectively. Each flux vectors
contain mass flux, momentum flux and energy flux.
ρu is the mass flux and ρu2 , ρuv, ρuw are the
momentum flux and u(e+p) , ρuk, ρuε are the energy
flux in the X -axis. Similarly ρv is the mass flux and
ρuv, ρv2 , ρvw are the momentum flux and v(e+p),
ρvk, ρvε are the energy flux in the Y -axis. ρw is the
mass flux and ρwu, ρvw, ρw2are the momentum flux
and w(e+p) , ρwk, ρwε are the energy flux in the Z -axis. Also ρ is the fluid density and u, v and w are
velocity components in each direction of Cartesiancoordinates. While e is the total energy per unit
volume, pressure p can be expressed by the
following state equation for ideal gas:
p = ( γ –1)[e – 21 ρ(u
2+ v
2+ w
2 )]
where γ is the ratio of specific heats.
From the relationship between stress and strain and
assumption of stokes, non-dimensional stress
components are as follows
τ xx= Reµ
32 (2.
xu∂∂ –
yv∂∂ –
z w∂∂ ),
τ yy= Reµ
32 (2. yv∂∂
– z w∂∂ – xu∂∂ ),
τ zz= Reµ
32 (2.
z w∂∂ –
xu∂∂ –
yv
∂∂ ), τ xy= Re
µ (
xv∂∂ +
yu∂∂ ),
τ yz= Reµ
( yw∂∂ +
z v∂∂ ), τ xz= Re
µ (
z u∂∂ +
xw∂∂ )
k x= Re1
x
k
k
t l ∂
∂+ )(σ
µ µ , ε x= Re
1
x
t l
∂
∂+ ε
ε σ
µ µ )(
k y= Re1
y
k
k
t l
∂
∂+ )(σ
µ µ , ε y= Re
1
y
t l
∂
∂+ ε
ε σ
µ µ )(
k z = Re1
z
k
k
t l
∂
∂+ )(σ
µ µ , ε z = Re
1
z
t l
∂
∂+ ε
ε σ
µ µ )(
The element of heat flux vectors are expressed by
Fourier law of heat conduction as
q x= Re
ck xT ∂∂ , q y=
Reck
yT ∂∂ , q z=
Reck
z T ∂∂
where T is the temperature and k c is the thermal
conductivity. The expression of the thermal
conductivity is k c /k o= ck (T/T o )1.5
where k o is the
thermal conductivity at the ambient temperature (T o ) and the value of the coefficient, ck depends on the
temperature and the ambient gas. The expression of
laminar viscosity is µl /µo= cv (T/T o )1.5
where µo is the
laminar viscosity at the ambient temperature and the
coefficient, cv depends on the temperature and the
ambient gas. The total viscosity µ=µl +µt where µt is
the turbulent eddy viscosity and the expression of
turbulent eddy viscosity, µt= c µ. ρ ε
2k . The Reynolds
number of the flow is defined by Re=( ρcucl c /µo ) where ρ
c,u
c , l
cand µ
oare respectively a
characteristics density, a characteristics velocity, a
characteristics length and the viscosity of the fluid.
The source term S(Q) of the k-ε turbulence model
is written by,
S(Q)= [0, 0, 0, 0, 0, P k – ρε –Dk ,(cε1.P k – cε2 . ρε ) k ε ]
where the production term P k is given in
Cartesian coordinates as,
P k = { 2 µt xu∂∂ –
32 [ ρk+ µt( x
u∂∂ +
yv∂∂ +
z w∂∂ )]}
xu∂∂ +
{2 µt yv
∂∂ –
32 [ ρk+ µt( x
u∂∂ +
yv∂∂ +
z w∂∂ )]}
yv
∂∂ +{2 µt z
w∂∂ –
32 [ ρk+ µt( x
u∂∂ +
yv∂∂ +
z w∂∂ )]}
z w∂∂ + µt( y
u∂∂ +
xv∂∂ )2+
µt( z u∂∂ +
xw∂∂ )
2+ µt( y
w∂∂ +
z v∂∂ )
2and
the destruction term Dk is given as,
Dk = T .
2γ
ρ k ε
The mass average turbulent kinetic energy and
homogeneous component of turbulent kinetic energy
dissipation rate are defined by as,
k=21 .ct
2.(u
2+v
2+w
2 ) and ε=cm.k
2.
100Re
Proceedings of the 4th WSEAS International Conference on Fluid Mechanics, Gold Coast, Queensland, Australia, January 17-19, 2007 23
8/3/2019 Mohammad Ali Jinnah- A Numerical Study of Shock Reflection Phenomena in Shock/Turbulence Interaction
treated as the grid-data planes and the grids inside
the turbulent region cut by the grid-data planes are
the grids on the grid-data plane. The value of the
turbulent parameter on the center line of the
turbulent region is the average value of all the grid
values of that parameter on the grid-data plane and
in the present computations, the grids adjacent to the
boundary are not taken into account due to viscous
effect. The pressure, velocity and temperature etc,
are determined across the reflected shock wave
when the position of the reflected shock wave is in
the turbulent region and the characteristics profiles
of these parameters across the reflected shock wave
are plotted to observe the reflection phenomena. The
longitudinal distance ( X/m) of any point on the
centerline of the turbulent region are determined
from the turbulence-generating girds where m = 5.0
mm, the maximum dimensional length of a grid inthe grid system. The distance, X = 0.0 mm at the
position of turbulence-generating grids. The value,
X/m = 6.9 is the starting point of the centerline and
the value, X/m = 18.2 is the ending point of the
centerline of the turbulent region.
After the shock wave is diffracted through the
turbulence grids, the fluid flow causes the formation
of unsteady, compressible vortices and the vortices
separate from the grids, then merges, dissipates and
forms a compressible turbulent field at some
distance downstream of the grids. It is observed that
near the turbulence grids, the unsteady vorticityfluctuations in the lateral direction are high which
cause more flow fluctuations in the lateral direction.
The vortex fluctuations as well as vortex interaction
change to fully developed turbulence field in the
wake of the turbulence grids. Fig.3 shows the
vorticity contour in the lateral planes at different
positions along the longitudinal direction from the
turbulence grids. It is observed in Fig.3 that the
interaction of vortices is strong enough near the
turbulence grids and the vorticity diminishes in the
far distance, which indicates the change of unsteady
vorticity fluctuations to homogeneous turbulence.The location where X/m>3.78, is taken as the
uniform and small vorticity fluctuations region.
The average longitudinal velocity, U av /cvariations are determined across the reflected shock
wave when the position of the reflected shock wave
is in the turbulent region. The average longitudinal
velocity, U av = nU n
ii∑
=1
, U i is the instantaneous
velocity for any grid on the grid-data plane and n is
the number of grid on the grid-data plane avoiding
grids near the boundary. c is the local soundvelocity. It is observed that longitudinal velocities
behind the reflected shock wave are observed for the
shock wave reflection from different shock
reflectors of 49.0 %, 26.5 % opening area and from
the plane end wall. Fig.4 shows the longitudinal
velocity profiles across the reflected shock wave and
a good agreement for the longitudinal velocity
profiles is observed between the present simulation
results and the NS results. It is observed that the
longitudinal velocity behind the reflected shock
wave increases and the longitudinal velocity
difference across the reflected shock wave is higher
for the plane end wall reflection and this difference
decreases for the shock reflector of higher openings.
(i) (ii)
(iii) (iv)
(v) (vi)
(vii) (viii)
Fig.3: Vorticity contour field in the lateral plane produced by the shock diffraction at turbulence grids where thecontour jump is 5.0 x 10
-4and minimum vorticity,
maximum vorticity, lateral plane location ( X/m) are (i) -50 x 10 -4, 50 x 10 -4, 0.22; (ii) -65 x 10 -4, 65 x 10 -4, 0.67;(iii) -73 x 10 -4, 73 x 10 -4, 1.22; (iv) -63 x 10 -4, 63 x 10 -4,
1.78; (v) -56 x 10-4
, 56 x 10-4
, 2.38; (vi) -41 x 10-4
, 41 x10 -4, 2.78; (vii) -32 x 10 -4, 32 x 10 -4, 3.18; (viii) -18 x 10
-4, 18 x 10
-4, 3.78 respectively
Proceedings of the 4th WSEAS International Conference on Fluid Mechanics, Gold Coast, Queensland, Australia, January 17-19, 2007 26
8/3/2019 Mohammad Ali Jinnah- A Numerical Study of Shock Reflection Phenomena in Shock/Turbulence Interaction
Fig.4: The longitudinal velocity profiles across thereflected shock wave for the reflection from the differentshock reflectors of 49.0 %, 26.5 % opening area and the plane end wall
1
3
5
7
9
11
6 8 10 12 14 16 18 20
X/m
P a v / P o
Present result, 49.0% opening area
NS result, 49.0% opening area
Present result, 26.5% opening area
NS result, 26.5% opening area
Present result, 00.0% opening area
NS result, 00.0% opening area
Fig.5: The average pressure profiles across the reflectedshock wave for the reflection from different shock
reflectors of 49.0 %, 26.5 % opening area and the planeend wall
The normalized pressure, P av /P o variations are
determined across the reflected shock wave when
the position of the reflected shock wave is in the
turbulent region where the average pressure,
P av = n P n
ii∑
=1
, P i is the instantaneous pressure for
any grid on the grid-data plane and n is the number
of grid on the grid-data plane avoiding grids near the
boundary. P o is the STD atmospheric pressure. Fig.5
shows the pressure profiles across the reflected
shock wave for the shock wave reflection from
different shock reflectors and the pressure profiles
obey the shock reflection theory. The average
pressure difference between downstream and
upstream of the reflected shock wave depends on the
opening area of the shock reflector and the pressure
profiles across the reflected shock wave have the
Table 1
Type of shock
reflector
Incidentshock
Machnumber
Transmittedshock Mach
number after turbulence
grid
Reflectedshock Machnumber after
reflection fromshock reflector
49.0 %
opening
area
2.00 1.83 1.66
26.5 %
opening
area
2.00 1.83 1.73
Plane
end wall
2.00 1.83 1.90
good agreement with the NS results. The strength of the reflected shock wave is decreased for the
reflection from the shock reflector of higher opening
area and the strength is increased as increasing the
blockage ratio of the reflector. The maximum
strength of the reflected shock wave is obtained for
the reflection from the plane end wall. After
measuring the transmitted shock Mach number, it is
shown that the value of transmitted shock Mach
number is 1.83 for the incident shock Mach number,
M s = 2.00. Table-1 shows the value of transmitted
shock Mach number and the partial reflected shock
Mach number for the incident shock Mach number,M s = 2.00.
The total temperature (T/T o ) variations are
determined across the reflected shock wave for the
different shock reflection techniques, which are
shown in Fig.6. It is observed that the temperature
variations are occurred in the interaction region and
the higher temperature variations at different points
in the interaction region are observed for the strong
reflected shock wave. It is also observed that the
measurements of the temperature variations by the
present computational techniques have the good
agreements with the NS results.The dissipative-length scale is defined by the
expression, k 3/2
/ ε where the turbulent kinetic energy,
k= nk n
ii∑
=1
and k i is the instantaneous turbulent
kinetic energy for any grid on the grid-data plane
and n is the number of grid on the grid-data plane
where the grids adjacent to the boundary are not
taken into account due to viscous effect. Similarly
the dissipation rate, ε= nn
ii∑
=1
ε where εi is the
instantaneous turbulent kinetic energy dissipationrate for any grid on the grid-data plane. The
Proceedings of the 4th WSEAS International Conference on Fluid Mechanics, Gold Coast, Queensland, Australia, January 17-19, 2007 27
8/3/2019 Mohammad Ali Jinnah- A Numerical Study of Shock Reflection Phenomena in Shock/Turbulence Interaction