Weak and strong oblique shock waves

Journal of Physics: Conference Series
OPEN ACCESS
The Weak and Strong Oblique Shock Waves
Appeared on the Carbon Dioxide Two-phase Flow
in the Ejector Refrigeration Cycle
To cite this article: Y Kawamura and M Nakagawa 2013 J. Phys.: Conf. Ser. 433 012015
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The Weak and Strong Oblique Shock Waves Appeared on the
Carbon Dioxide Two-phase Flow in the Ejector Refrigeration
Cycle
Y Kawamura1
, M Nakagawa1
1
Department of Mechanical Engineering, Toyohashi University of Technology, 1-1
Hibarigaoka, Tenpaku-cho, Toyohashi-city, Aichi, 441-8580, Japan
E-mail: kawamura@nak.me.tut.ac.jp
Abstract. After the Great East Japan Earthquake, the saving energy became the one of the
most important issues. Especially, saving the electrical power for air-conditioning at summer
time is a problem of great urgency. We have been developing the ejector refrigeration system
which can improve coefficient of performance of refrigeration systems by converting
exhausted expansion energy into useful pressure energy. The performance of the ejector is
greatly affected by the pressure recovery at the mixing section in the ejector. It is elucidated by
our recent research on the carbon dioxide refrigeration cycle that the pressure recovery at the
ejector is performed by oblique shock waves appeared in the mixing section. The purpose of
this research is to clarify the characteristics of the two-phase flow oblique shock waves in the
supersonic carbon dioxide two-phase flow. It is shown by the theoretical analyses that the two
types of oblique shock waves occur on the supersonic two-phase flow. One is the week shock
wave behind of which the flow is still a subsonic state. The other is the strong oblique shock
wave which has a large pressure recovery. And the experimental research also carried out by
using the carbon dioxide two-phase flow channel. The theoretical results are compared with the
experiment.
1. Introduction
The speed of sound in high-speed two-phase flow, which has been used extensively in areas such as
total flow systems for geothermal power plants and in refrigeration cycles, is extremely low compared
to that of single-phase flow [1]; moreover, a characteristic of the two-phase flow is its easy
acceleration into a supersonic state. In general, both shock and expansion waves occur with rapid
changes in the flow path of the supersonic state, and the effect of these waves on the flow field cannot
be neglected. In addition, the shock and expansion waves that occur in the two-phase flow differ from
those in a single-phase gas flow, and the pressure field tends to disperse. Therefore, we can expect a
combination of more complex phenomena than those in a single-phase gas flow. We have studied the
ejector-refrigeration cycle [2], which can improve the refrigeration cycle by converting the expansion
energy that has so far been wastefully discarded into useful compression energy. The performance of
the ejector [3] is considerably affected by pressure recovery in the mixing section inside the ejector. In
past studies, because the speed of sound for the two-phase flow is low compared to that for the single-
1
To whom any correspondence should be addressed.
The Irago Conference 2012 IOP Publishing
Journal of Physics: Conference Series 433 (2013) 012015 doi:10.1088/1742-6596/433/1/012015
Published under licence by IOP Publishing Ltd 1

phase flow, an oblique shock wave was generated at the nozzle exit of the ejector interior that affected
the flow field of the supersonic two-phase flow inside the mixing section. Furthermore, it was found
that most of the pressure recovery in the ejector depends on the kinetic energy of the two-phase flow
in the mixing section. Theoretical studies have shown that in supersonic flow, there are two types of
oblique shock waves with two-phase flow. One exists at the mach angle and is a weak oblique shock
wave, which remains supersonic behind the shock wave. The other is a strong oblique shock wave,
which is subsonic behind the shock wave, whose pressure is considerably recovered similar to that in
case of a normal shock wave. The conditions under which the strong and weak oblique shock waves
are generated were not discussed in these studies, thereby clarifying that the characteristics of these
oblique shock waves is important for elucidating the pressure-recovery phenomenon of the ejector.
Accordingly, in this study, both strong and weak oblique shock waves were generated on an analytical
semi-infinite plane to determine the conditions under which they arise and their characteristics, with
the aim of comparing them to experimental results.
2. Basic equations for compressible two-phase flow
In our analysis, transport phenomena such as momentum, heat, and mass transports were considered
between the gas and liquid phase. Two-phase flow is assumed to consist of a gas phase in the
continuous phase and droplets in the dispersal phase by using the following assumptions:
• Phase changes occur in the flow field according to the heat-transfer limit.;
• The effect of friction other than interphase interactions is neglected.
• All droplets are spheres with diameter d , and any breakup or coalescence due to collisions
among droplets is not considered.
• Both gas and liquid phases are assumed to refer to real gases.
Now, we provide the respective equations for the conservation of mass for gas and liquid phases:
( ) ( ) ,mv
xt
gjg
j
g &=
∂
∂
+
∂
∂
αραρ (1)
( ){ } ( ){ } .mv
xt
ljl
j
l &−=−
∂
∂
+−
∂
∂
ραρα 11 (2)
Here, α expresses the void fraction represented by the volume fraction of the gas phase in the two-
phase flow, ρ is density, m& is the rate of evaporation, and v is flow velocity with the respective
suffixes g and l denoting gas and liquid phases, and i represents each of the directions in two
dimensions.
Similarly, the momentum-conservation equations for each of the gas and liquid phases are as
follows:
( ) ( ) ( ) ,
x
pvv
vmvv
x
v
t iV
ligi
lkigigjg
j
gig
∂
∂
−
−
−−=
∂
∂
+
∂
∂
α
τ
ρααραρ 1& (3)
( ){ } ( ){ } ( ) ( ) .
x
pvv
vmvv
x
v
t iV
ligi
lkililjl
j
lil
∂
∂
−−
−
−+−=−
∂
∂
+−
∂
∂
α
τ
ραραρα 1111 & (4)
The velocity of evaporation is ( )0;0; <≥= mv,mvv giliki && , and the momentum relaxation time is
( )gV d µρτ 182
= .
The total-energy equation is:
.
v
hv)(
v
hv
x
v
e)(
v
e
t
lj
lljl
gj
ggjg
j
lj
ll
gj
gg 0
2
1
22
1
2
2222
=
















+−+








+
∂
∂
+
















+−+








+
∂
∂
ρααρρααρ (5)
2

Here, e represents internal energy, and h represents enthalpy. This equation does not contain a term
for transport between phases.
From the heat-transfer equation for each phase, we can write the following equations considering
entropy:
,
TT
C
tD
sD
T
Tg
gs
pg
g
gg
g
τα
α −−
=
1
(6)
.
TT
C
tD
sD
T
Tl
ls
pl
l
ll
l
τ
−
= (7)
Here, pgC and plC represent the specific heat of gas and liquid phases, respectively, and then, the
thermal-relaxation times are ( )TgpggTg hdC 6ρτ = and ( )TlpllTl hdC 6ρτ = . Please refer to the reference
section for details.
We used the NIST REFPROP v.8 software for calculating the thermophysical properties.
3. Relational expression for oblique shock waves in extreme states
Though momentum, heat, and mass transports are considered in this study, because phase changes
occur according to the heat-transfer limit, only momentum and heat transports are important. Extreme
states exist depending on the velocity at which these transport phenomena progress, and for extremely
high transport velocity, they are referred to as equilibrium states. On the other hand, for low or
negligible transport velocity they are referred to as frozen states. In this section, we illustrate the
method to determine the relational expressions for the shock waves in each of these extreme states.
3.1. Jump equation for the normal shock wave in momentum and thermal-equilibrium states
Interphase velocity is equal to temperature in the momentum and thermal-equilibrium states. If we
denote the forward state for the normal shock wave with suffix 1 and the rear state with suffix 2, the
conservation equations for mass, momentum, and energy for a uniform steady flow are given by
integrating Eqs. (1) + (2), (3) + (4), and (5):
• mass-conservation equation
( ){ } ( ){ } ,ww nlgnlg 2222211111 11 ραραραρα −+=−+ (8)
• momentum-conservation equation
( ){ } ( ){ } ,pwpw nlgnlg 2
2
222221
2
11111 11 +−+=+−+ ραραραρα (9)
• energy-conservation equation
.h
w
w)(h
w
wh
w
w)(h
w
w l
n
nlg
n
ngl
n
nlg
n
ng 







+−+







+=







+−+







+ 2
2
2
2222
2
2
2221
2
1
1111
2
1
111
2
1
22
1
2
ραραραρα (10)
Because the physical quantities for the forward shock wave are known, the unknowns terms in Eqs. (8),
(9), and (10) are 2α , 2gρ , 2lρ , 2nw , 2p , 2gh , and 2lh . However, 2gh , 2lh , 2gρ , and 2lρ can be calculated from
the pressure corresponding to the saturation state of thermal equilibrium in the two-phase flow.
Therefore, the three unknown terms 2α , 2nw , and 2p can be obtained using Eqs. (8), (9), and (10).
3.2. Jump equation for the normal shock wave in momentum equilibrium and thermal frozen states
For the thermal frozen state in which only a small amount of heat is transferred between gas and liquid
phases, because the thermodynamic variable is a function of pressure and entropy, when adiabatic
conditions are applied to Eqs. (8) and (9), we get Eq. (11) in addition to Eqs. (8), (9), and (10).
.ss,ss llgg 2121 == (11)
3

From the tabulated thermodynamic properties, 2gρ , 2lρ , 2gh , and 2lh can be estimated using the pressure,
2p , and entropy, 2s , for mutual phases. Therefore, the five unknown terms 2α , 2nw , 2p , 2gs , and 2ls can
be found by solving Eqs. (8), (9), (10), and (11).
3.3. Jump equation for the normal shock wave in the momentum frozen and thermal equilibrium states
If the mutual forces acting on the phases are reduced to approximately zero, the mutual phases flow
independently. Therefore, the momentum equation is required for both gas and the liquid phases.
( ) ( ) ,wwww lnlgnglnlgng 222222111111 11 ραραραρα −+=−+ (12)
• total momentum-conservation equation
( ) ( ) ,pwwpww lnlgnglnlgng 2
2
222
2
2221
2
111
2
111 11 +−+=+−+ ραραραρα (13)
• droplet momentum-conservation equation
,dT
dT
dp
ww
l
lnln 0
1
2
1
2
1 2
1
2
1
2
2 =+− ∫ ρ
(14)
.h
w
w)(h
w
wh
w
w)(h
w
w l
ln
lnlg
gn
gngl
ln
lnlg
gn
gng 







+−+








+=







+−+








+ 2
2
2
2222
2
2
2221
2
1
1111
2
1
111
2
1
22
1
2
ραραραρα (15)
Similar to the momentum and thermal-equilibrium states, 2gρ , 2lρ , 2gh , and 2lh can be calculated from
the pressure corresponding to the saturation state of themal equilibrium in the two-phase flow.
Therefore, the four unknown terms 2α , 2gnw , 2lnw , and 2p can then be found from Eqs. (12), (13), (14),
and (15).
3.4. Jump equation for the normal shock wave in the momentum and thermal frozen states
By assuming that all transport phenomena can be neglected, mass, momentum, and energy for each
phase are conserved.
,ww gnggng 222111 ραρα = (16)
( ) ( ) ,ww lnllnl 222111 11 ραρα −=− (17)
• momentum-conservation equation
,pwpw gnggng 2
2
2221
2
111 +=+ ραρα (18)
( ) ( ) ,pwpw lnllnl 2
2
2221
2
111 11 +−=+− ραρα (19)
,h
w
wh
w
w g
gn
gngg
gn
gng








+=








+ 2
2
2
2221
2
1
111
22
ραρα (20)
( ) ( ) .h
w
wh
w
w l
ln
lnll
ln
lnl 







+−=







+− 2
2
2
2221
2
1
111
2
1
2
1 ραρα (21)
The terms 2gh and 2lh can be estimated using the mutual phase pressure 2p and the densities 2gρ and
2lρ . Therefore, the six unknown terms 2α , 2gnw , 2lnw , 2gρ , 2lρ , and 2p can be estimated using Eqs. (16)–
(21).
4

3.5. Relationship between normal and oblique shock waves
The equation for the oblique shock wave can be calculated from its relationship with the normal shock
wave. As shown in figure 1, because the velocity in the tangential direction does not change for the
oblique shock wave, Eq. (22) is obtained:
.
w
w
.e.i
w
w)(
n
n
n
n








−==
− −
φφθ
φ
θφ
tantan
tan
tan
1
21
1
2 (22)
Here, θ is the turning angle of the inclined wall, φ is the shock-wave angle for the oblique shock
wave, and nw is the normal velocity. The shock-wave angle φ can be calculated from Eq. (22) using
the turning angle of the inclined wall, θ , and the ratio of normal velocities, 12 nn ww . The ratio of the
normal velocities is similar to that for the normal shock wave. Therefore, if the turning angle of the
inclined wall is given, the oblique shock-wave angle is determined using the relational expression for
the normal shock wave.
Oblique shock wave
1tw 1nw
1w
2w
2nw 2tw
φ
θ
θφ −
Inclined wall
φ
Inlet
Outlet
Wall
Wall
Oblique shock wave
(Slip for gas, outlet for liquid)
(vg = vl)
L
Inlet Inlet
4. Relationship between strong and weak oblique shock waves
4.1. Weak oblique shock wave
4.1.1. Analysis method for the weak oblique shock wave. Weak oblique shock waves are generated by
mach waves, and in supersonic flow, entropy production is small. In addition, with only slight
disturbance to the flow field of weak oblique shock waves, the back of the shock wave is supersonic at
the inlet.
In past studies [4][5], weak oblique shock waves were simulated using numerical analysis. In
numerical analysis, Constrained Interpolation Profile Scheme method is applied to the basic equations
of the compressible two-phase flow, which considers momentum, heat, and mass transports. Non-
steady calculations are performed, and a steady-state solution is obtained at which a constant value is
reached. The calculation domain in figure 2 has the same shape as that of the inclined wall so that
oblique shock waves are generated over a semi-infinite plane. The inclined wall has an angle of θ . For
the domain, we make 70 divisions in the direction horizontal to the inclined wall and 100 divisions in
the direction perpendicular to the wall. In the calculation domain, we define a typical length of L for
Figure 2. Calculation domain for the
weak oblique shock wave.
Figure 1. Velocity vector diagram for the oblique
shock wave.
5

the inlet-section height and a typical velocity of u for the gas-flow rate in the inlet section. The two-
phase flow enters into the domain from the left side and flows out from the top and right sides. On the
bottom side of the wall surface, the component of the gas velocity in the direction perpendicular to the
wall is 0, and the horizontal component is symmetrical with the wall. It has been assumed that, after
collision with the wall, the liquid flows as a liquid film with no reflection. Because the volume of such
a liquid film is small, the volume of the liquid-film flow and the boundary layer were neglected in the
analysis. Therefore, as for the outlet, the boundary conditions of the free outflow for the liquid are
given. The inflow conditions were that constant pressure and velocity were maintained, and for an
equilibrium state, the velocities of both gas and liquid phases at the inlet were assumed equal.
This analysis was performed by simulating the four extreme states of the transport phenomena for
the two-phase flow, total-equilibrium state, thermal frozen state, momentum frozen state, and total
frozen state. The momentum relaxation time, Vτ , and the thermal relaxation times, Tgτ and Tlτ , are
important in simulating these extreme states. The relaxation times are functions of the droplet size.
Therefore, the analysis of weak oblique shock waves proceeds by assuming each non-dimensional
relaxation time as shown in table 1.
Table 1. Supposition of the non-dimensional relaxation time.
Extream state
Momentum relaxation time Thermal relaxation time
Droplet size d '
Vτ
'
Tgτ , '
Tlτ
a
m:eq , t:eq µm10. -4
10421 ×. -4
10421 ×.
b
m:eq , t:fr µm10. -4
10421 ×. 4
10421 ×.
c
m:fr , t:eq µm1000 4
10421 ×. -4
10421 ×.
d
m:fr , t:fr µm1000 4
10421 ×. 4
10421 ×.
a
Momentum and thermal equilibrium state.
b
Momentum equilibrium and thermal frozen state.
c
Momentum frozen and thermal equilibrium state.
d
Momentum and thermal frozen state.
4.1.2. Characteristics of weak oblique shock waves. Figure 3 shows the results of the numerical
analysis for weak oblique shock waves and the differences in the oblique shock waves by assuming an
inflow equilibrium Mach number 1721 .Me = for the three extreme states: (a) the total equilibrium state,
(b) the thermal frozen state, and (c) the momentum frozen state. In figure 3, according to the
differences between these extreme states, differences appear in the generated shock-wave angle of the
oblique shock wave. To compare the results of the numerical analysis with the theoretical values for
the oblique shock waves in the extreme states, the relationship between the shock-wave angle φ and
the inflow equilibrium Mach number 1eM is shown in figure 4, and the relationship between the
pressure ratio 2112 ppp = for the front and back of the shock wave and the inflow Mach number, 1eM ,
is shown in figure 5. Figures 4 and 5 are the results for a constant turning angle of the inclined wall,
°=10θ , and for the calculations in the previous section, which uses the relational expression for the
oblique shock waves in the extreme states. The curves in the diagram are shown as a red solid line for
the momentum and thermal equilibrium state, blue dashed line for the momentum equilibrium and
thermal frozen state, green dashed–dotted line for the momentum frozen and thermal equilibrium state,
and purple dashed-two dotted line for the momentum and thermal frozen state. The domain for the
weak oblique shock wave is the region ranging from the point of inflection in figure 4 with a smaller
shock-wave angle φ , but in figure 5, it is the region ranging from the point of inflection for which the
pressure ratio, 21p , tends to increase by a smaller amount with increasing inflow equilibrium Mach
number 1eM . In addition, the domain for the strong oblique shock wave, which will be discussed later,
is the region in figure 4 ranging from the point of inflection with the larger shock-wave angle φ , but in
6

figure 5 is the region ranging from the point of inflection for which the pressure ratio 21p tends to
increase by a larger amount with increasing inflow equilibrium Mach number, 1eM .
The angles of the oblique shock waves obtained from the numerical analysis are shown in figure 4
as circles. From figure 4, because the numerical-analysis results match approximately with the
theoretical curves for the oblique shock waves in the extreme states, the weak oblique shock waves
may be predicted using this type of analysis.
35.25[o] 41.20[o] 56.66[o]
Momentum and Thermal
Equilibrium
Momentum equilibrium
and
Thermal frozen
Momentum frozen
and
Thermal equilibrium
1.0 1.5 2.0 2.5 3.0
20
30
40
50
60
70
80
90
1.0 1.5 2.0 2.5 3.0
o
Extreame states : , , ,
m:fr , t:frm:fr , t:eqm:eq , t:fr
Obliqueshockwaveangleφ[]
Front equilibrium Mach number Me1
m:eq , t:eq
Numerical analyses (weak range) : , , ,
1.0 1.5 2.0 2.5 3.0
1.0
1.5
2.0
2.5
3.0
Extreamstates : , , ,
m:fr , t:eq
Strengthofobliqueshockwavep21
Front equilibriumMach number Me1
m:eq , t:eq
m:eq , t:fr
m:fr , t:fr
4.2. Strong oblique shock waves
4.2.1. Analysis method for strong oblique shock waves. Another solution for oblique shock waves are
strong oblique shock waves, as is the case for normal shock waves, with distinctive large changes in
the physical quantities in the front and back of the shock wave and within its interior. Phenomena such
as viscous stress and heat transfer occur in a non-equilibrium state. In addition, an increase in the
Figure 5. Strength of oblique shock wave
versus front Mach number for extreme states.
Figure 4. Shock wave angles versus front
Mach number for extreame states.
(a) (b) (c)
Figure 3. Weak oblique shock waves for 1721 .Me = .
7

changes in the physical quantities means that the flow changes from supersonic to subsonic. The
calculation domain created to predict the strong oblique shock waves on a semi-infinite plane is shown
in figure 6. For the calculation domain, we defined the number of divisions in the direction horizontal
to the inclined wall as 120 and the divisions in the direction perpendicular to the inclined wall as 100.
Moreover, in this domain, we defined a typical length, L, for the height of the inlet section and a
typical velocity, u , for the gas-flow rate in the inlet section. The boundary and inlet conditions are
defined similarly for the weak oblique shock waves.
The flow at the back of the shock wave, in case of strong oblique shock waves, is subsonic. In
general, within the supersonic flow, because the flow exceeds maximum propagation velocity, the
effects on the downstream cannot be propagated forward. However, within the subsonic flow, the
effects on the downstream can be propagated forward. Therefore, not only the inflow conditions were
given but also the initial distribution as well as the physical quantities derived from the relational
expression for the oblique shock wave in extreme states are given for the back of the shock wave
shown in the gray region in figure 6. The non-dimensional relaxation times used in the analysis of the
strong oblique shock wave in the extreme states were assumed to be the same as those shown in table
1.
Inlet
Outlet
Wall Wall
Oblique shock wave
(Slip for gas, outlet for liquid)
Outlet
(vg = vl)
L
4.2.2. Establishing the initial distribution for the calculation domain. The physical quantities that are
important in the initial distribution for the calculation domain are the pressure ratio, 21p , for the front
and back of the shock wave and the shock-wave angle, φ . Both values are computed from the
previously mentioned shock-wave relational expression. However, if the analysis is performed using
these two values, an extremely unstable solution is obtained for given arbitrary values. The analysis
results are given below. First, figure 7 shows the results for the case in which the pressure ratio is large
for initial conditions, rather than just the required pressure ratio, to sustain an oblique shock wave at a
given shock-wave angle. In figure 7, we can observe that the fundamental oblique shock wave moves
in front of the inclined wall and is generated with an angle approximately equal to the normal shock
wave. On the other hand, figure 8, which is the opposite case of figure 7, shows the results for a small
pressure ratio. In figure 8, from the turning starting point of the inclined wall, a weak oblique shock
wave is generated in the backward direction with a small shock-wave angle. Then, at some arbitrary
position, a shock wave that is close to a normal shock wave is generated from the inclined wall, and
the oblique shock waves that are weak in the forward direction overlap to generate a comparatively
strong oblique shock wave. This configuration conforms to the λ -shaped shock wave in gas dynamics.
Figure 6. Calculation domain for strong
oblique shock wave.
8

If the time-step progresses further, the results in figure 7 proceed forward, and those in figure 8
proceed backward. Because the domain for the strong oblique shock wave is an extremely unstable
region, these waves are sensitive to very slight differences in the initial values and tend to migrate
toward a stable solution.
4.2.3. Characteristics of strong oblique shock waves. The numerical-analysis results for strong
oblique shock waves are shown in figure 9. In figure 9, extreme states are shown for the total
equilibrium state with ( )511 .Me = in (a), the thermal frozen state with ( )711 .Me = in (b), and the total
frozen state with ( )621 .Me = in (c). From figure 9, it can be observed that, even if the inflow Mach
number for each of the extreme states is different, there is no considerable difference in the shock-
wave angle for the generated oblique shock wave. By plotting the shock-wave angle obtained by this
analysis as triangles on the theoretical curves, as shown in figure 4, we get figure 10. From figure 10,
given that the numerical-analysis results are an approximate match with the theoretical curves for the
oblique shock waves in extreme states, this demonstrates the possibility of predicting strong oblique
shock waves by using this type of analysis.
82.05[o] 68.84[o] 75.46[o]
Equilibrium
Momentum equilibrium
and
Thermal frozen
Equilibrium
Figure 7. Case of large pressure ratio. Figure 8. Case of small pressure ratio.
(a) (b) (c)
Figure 9. Strong oblique shock waves from numerical analysis.
9

1.0 1.5 2.0 2.5 3.0
20
30
40
50
60
70
80
90
1.0 1.5 2.0 2.5 3.0
o
Extreame states : , , ,
m:fr , t:frm:fr , t:eqm:eq , t:fr
Front equilibriumMach number Me1
m:eq , t:eq
Numerical analyses (strong range) : , ,
4.3. Comparison with the oblique shock waves obtained by experiments
In past studies [5], the authors have experimentally measured the oblique shock waves generated in a
convergent-divergent (Laval) nozzle using carbon dioxide gas as a refrigerant. The static-pressure
distribution inside the nozzle in the experiments is shown in figure 11. For the measured oblique shock
waves, when a third-order approximation was made for the pressure distribution around the points at
which the pressure started rising and a cross-correlation was performed, the average value was found
to be °67 . This result demonstrates that for this shock wave angle, which was shown theoretically for
carbon dioxide gas, the experimentally generated oblique shock waves exhibit a relatively strong
pressure ratio of the front and back of the shock wave. Accordingly, if we plot the angles for the
strong oblique shock waves obtained by numerical analysis and the shock wave obtained from
experiments on the shock-wave-angle-relationship diagram, we get figure 12. We can observe from
figure 12 that the shock-wave angle for the oblique shock wave obtained from experiments is
approximately equal to the strong oblique shock-wave region.
-10 0 10 20 30
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Centerline (T)
Extended wall (T)
Extended wall (P)
Inclined wall (T)
Inclined wall (P)
(T: Thermocouple)
(P : Pressure transducer)
Theoretical curve
Pressure[MPa]
Distance fromthe deflection point [mm]
Deflection point
Oblique shock wave
66.8 [ ]o
1.0 1.5 2.0 2.5 3.0
20
30
40
50
60
70
80
90
Experiment
Extreamstates : , , ,
m:fr , t:fr
m:fr , t:eqm:eq , t:fr
Front equilibrium Mach number Me1
m:eq , t:eq
o
Numerical analyses (strong range) : , ,
Experimental value :
5. Conclusion
We analytically investigated the weak and strong oblique shock waves that are generated in the
supersonic two-phase flow of carbon-dioxide gas and found the following wave characteristics:
• For each respective extreme state, a stable solution for weak oblique shock waves was found
with outflow boundary conditions.
Figure 12. Comparison between experiment
and analysis.
Figure 11. Pressure distribution at the nozzle
outlet from the experiment.
Figure 10. Oblique shock wave angles for the strong shock wave range.
10

• The angles for the weak oblique shock waves varied considerably with differences in the
transport processes, but the solution matches with the theoretical solution and demonstrates
that this analysis was correct.
• Because the back of the strong oblique shock waves is subsonic, it is necessary to define the
pressure at the back of the wave. For a small change in the pressure at the back of the wave, a
solution was obtained in which either the oblique shock wave becomes a normal shock wave
that progresses forward or becomes a wave associated with a λ -shaped weak oblique shock
wave that retreats backward.
• Strong oblique shock waves are moderately unstable, but steady-state solutions were obtained
with this analysis method for each respective extreme state.
• We revealed that the oblique shock waves obtained in the experiments within the two-phase
flow nozzle for carbon dioxide are strong waves.
6. Reference
[1] L.D. Landau E.M. Lifshitz 1952 Fluid Mechanics Pergamon Press 422
[2] Nakagawa, M., Takeuchi, H. and Nakajima, K. 1998 Performance of Two Phase Ejector in
Refrigeration Cycle Transactions of the Japan Society of Mechanical Engineers Series B
Vol 64 No 625 pp 3060-3067
[3] Nakagawa, M. 2004 Refrigeration Cycle with Two-Phase Ejector Refrigeration Vol 79 No 925
pp 856-861
[4] Harada, A. and Nakagawa, M. 2010 Theoretical Analysis of the Two-Phase Oblique Shock
Waves in an Ejector with Momentum and Temperature Relaxation Transactions of the Japan
Society of Refrigerating and Air Conditioning Engineers Vol 27 No 3 pp 13-21
[5] Kawamura, Y. and Nakagawa, M. 2010 Investigation on Oblique Shock Waves Occurred in the
Supersonic Carbon Dioxide Two-phase Flow 9th International Conference on Heat Transfer,
Fluid Mechanics and Thermodynamics pp 939-945
11

Weak and strong oblique shock waves

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