Download Axisymmetric supersonic flow patterns with Mach disk in convergent

Workshop on Non-equilibrium Flow
Phenomena in Honor of Mikhail Ivanov's
70th Birthday
Axisymmetric supersonic flow patterns with
Mach disk in convergent conical ducts
and in over-expanded jets
Gounko Yu.P.
ITAM SB RAS, Novosibirsk, E-mail: [email protected]
The attribute of the considered flows is formation
of an initial bell-shaped shock wave the intensity of
which increases with it passing downstream and sloping
towards the duct axis so that eventually a terminal
central shock wave close to the normal one – the Mach
disk forms near the axis.
There is no the regular reflection of an oblique
shock wave from the flow axis in the axisymmetric
steady supersonic flows!
Let us firstly to discuss results of a numerical
simulation of axisymmetric steady supersonic flows in
convergent conical ducts.
Computed flow pattern at M = 1.6
An early theoretical investigation
of axisymmetric compression
flows in convergent conical duct
with the Mach disc which was
fulfilled with the method of
characteristics by Ferri, 1946.
grid
hypothetical
Mach disc
sonic point
relative diameter of subsonic core at M = 1.6
streamlines and local
flow Mach numbers
subsonic
flow core
reflected
shock
initial bell-shaped
shock
funnel angle, degree
The Mach disc cannot be determined !
Just only possible cross size of a central zone of subsonic flow was estimated
Analogous computations of the axisymmetric
compression flows in convergent conical ducts with the
Mach disc in Mach number range М = 1.6-12 were
fulfilled quite later by Gutov and Zatoloka at ITAM in
1975.
Now there are works on numerical computations
of such flows but features of forming these flows
adequately are not studied as before.
The structure of the considered flows is
determined by interaction of the initial bell-shaped
shock wave and the Mach disk. This interaction was
analyzed using the known three-shock theory based on
Rankine-Hugonio jump relations with plotting the shock
polars.
The theory developed by von Neumann derives
from solution of problem on an irregular reflection – the
so-called Mach reflection of an oblique shock wave from
the plane in the steady two-dimensional inviscid flow.
Therewith postulated formation of a reflected shock
wave and a slipline issued from the point of triple
intersection of the shocks. All the shocks and slipline
are assumed to be of negligible thickness and curvature.
The shock polar is a plotp(δ) of dependence of relative
pressure p = ps /p on angle δ of flow deflection by the shock. Here
ps is static pressure of flow immediately downstream of the shock,
p is static pressure of the free stream flow.
As for the considered axisymmetric flows the angle δ is
assumed positive if the flow downstream of the shock deflects to the
axis.
p/p
М = 1.6
2
3
3
4 5
6
On explanation of possible
irregular interactions of
shock waves
2.5
7
2
   
  
   
1.5
1
  
  
1
-15
-10
-5
1 – initial incident shock polar,
δ = 3 … δ = 12 – secondary
(reflected) shock waves with different
angles of initial flow deflections
0
5
10
15

Possible types of
Mach triple-shock configuration
2 – shock polar intersection corresponding to inverse Mach triple-shock
configuration (it is not occur in steady axisymmetric compression flows),
3 – shock polar intersection corresponding to direct (single) Mach tripleshock configuration,
4 – shock polar intersection corresponding to Mach triple-shock
configuration with a normal reflected shock,
5 – shock polar intersection corresponding to triple-shock configuration of
von Neumann type,
6 – reflected shock polar in conditions of von Neumann paradox,
7 – sonic point
The inverse (or inverted) and direct Mach tripleshock configurations are separated by the stationary
Mach configuration when the second shock polar
intersects the strong branch of the initial shock polar 1
at δ = 0.
The inverse Mach triple-shock configuration does
not realize in steady two-dimensional flows.
As for the considered steady axisymmetric
supersonic compression flows, this solution disagrees
with the known patterns of these flows since in this
case the Mach disc should be convex toward the
oncoming flow, the flow behind the disc should be
diverging and should be decelerating.
The direct Mach triple-shock configuration is quite
pertinent for the considered axisymmetric flows, the
Mach disc in this case should be convex away the
oncoming flow, the flow behind the disc should be
converging and should be accelerating.
One could note a limiting case of a degenerate
polar of the secondary shock with δ = δМ=1
corresponding to sonic point 7 of initial shock polar 1.
The Mach disc in this case, if it existences, should be
highly convex away the oncoming flow, the pressure
behind it should change in radial direction from
p = 2.15 at δ = δМ=1 = 14.24 up to p = 2.82 for normal
shock along the flow axis. Theoretically, the secondary
shock in this case could be initiated from the Mach line
normal to the sonic velocity behind the initial chock.
This case corresponds per se to assumptions under
which the axisymmetric supersonic flow in convergent
conical duct was considered by Ferri (1949).
The numerical simulation and investigation of the
considered flows in the presented work was fulfilled with the
Navier-Stocks density based, time-explicit codes provided by the
program package FLUENT. The turbulent flow model k-ω SST
was used.
The problem on determination of the axisymmetric supersonic steady flow in
the duct was solved with step by step iterations. A computational domain began from a
certain section ahead of the inlet on the left and external boundaries of which initial flow
parameters corresponding to the free stream were preset. The same parameters were
imposed as initial ones inside of the computational domain. In the duct exit cross section,
the boundary outlet conditions were set by specifying the static pressure and total
temperature which corresponded to the free stream undisturbed parameters. If the flow
at the outlet is locally supersonic then the given pressure is not used and is determined by
the extrapolation from the interior flow region as the remaining parameters of the flow.
Performance of computations of the limit high resolution was not pursued. The grid with
uniform spacing was used, a cross step was from D0/1000 to D0/2000 where D0 is the duct
entrance diameter, overall number of the grid sell was up to 2106. In the beginning of the
steadying process the first-order approximation of dissipation terms was used, the
solution subsequently was refined with the second-order approximation.
М = 1.6, δс = 5
Irregular triple-shock configuration of the type which
occurs in conditions of the von Neumann paradox
1 – polar of incident bell-shaped shock, 2 – reflected shock polars along
shown streamline, diamonds – flow parameters behind of Mach disc and
reflected shock by numerical computation, 3 – polar of reflected normal shock,
4 – sonic point of incident bell-shaped shock polar
The Mach disc size is quite larger than assumed by Ferry !
The obtained irregular triple-shock configuration of the von Neumann
paradox type in the axisymmetric flows with the Mach disc and features of
variation of flow parameters behind it are analogous to those of the work
Ivanov et al. (2010) for the steady reflection of wedge-generated shock
wave with forming a Mach stem in two-dimensional supersonic flow at
M = 1.7,  = 5/3, wedge angle δw = 12 :
Ivanov M. S., Bondar Ye.A., Khotyanovsky D.V., Kudryavtsev A.N.,
Shoev G.V. Viscosity effects on weak irregular reflection of shock waves in
steady flow // Progress in aerospace sciences, 2010. Vol. 46. P. 89-105.
The problem in that work was solved numerically with the Navier-Stocks
solver in comparison with the direct simulation Monte Carlo method. The
irregular reflections of the oblique shock wave was considered in conditions
of the von Neumann paradox when the polars of incident and reflected
shocks do not intersect and there is no triple-shock solution. There were
confirmed the assumptions of Sternberg (1959) on a buffer zone between
the areas where the Rankine-Hugoniot jump relations are true.
Sternberg J. Triple-shock-wave interaction. Phys. Fluids, 2 (1959),
p. 179-206
М = 2, δс = 10
single Mach triple-shock configuration
ps /p
4
2
3
2
-20
Transition of flow parameters behind
of the Mach disc to those behind of the
reflected shock by the numerical
computation takes place with a
deflection from the polar of incident
bell-shaped shock !
-10
1
1
0
10
20
 c 30
1 – polar of incident bell-shaped shock,
2 – reflected shock polar along shown
streamline,
diamonds – flow parameters behind of
Mach disc and reflected shock by
numerical computation
The obtained single Mach triple-shock configurations in the axisymmetric
flows with the Mach disc and features of variation of flow parameters behind
it are analogous to those of the work Khotyanovsky et al. (2009) for the
steady reflections of wedge-generated shock waves with forming a Mach
stem in two-dimensional supersonic flows (wedge angle δw = 25, M = 4):
Khotyanovsky D.V., Bondar Y.A., Kudryavtsev A.N., Shoev G.V., Ivanov
M.S. Viscous effects in steady reflection of strong shock waves // AIAA
Journal, 2009. Vol. 47, № 5. P. 1263-1269.
The problem in that work was solved numerically with the Navier-Stocks
solver in comparison with the direct simulation Monte Carlo method.
Transition of flow parameters behind of the Mach stem to those behind of the
reflected shock by the numerical computation of the viscous flow also takes
place with a deflection from the incident bell-shaped shock polar and does
not describes by the Rankine-Hugoniot jump relations. However,
computations with enlarging Reynolds number in that work evidence on the
convergence of the Navier-Stocks solution to the theoretical Mach tripleshock solution.
As concerning with the both presented computed flow examples, it
should be noted formation of a finite-thickness shear layer with the crossvarying entropy which develops downstream of a local zones of the irregular
triple-shock interaction. A thickness of this layer accounts for a several
thicknesses of the interacting shocks.
This layer in the case of the single triple-shock Mach configuration
develops instead of the slipline of theoretically zero thickness.
This layer in the case of the irregular triple-shock configuration of the
von Neumann paradox type develops as a zone of transition from the state
behind the Mach disc described by the Rankine-Hugoniot jump relations to
the state behind the reflected shock also described by these relations.
Such resolution of these interactions in the performed computations
is explained according to Sternberg work (1959).
The shocks in the viscid flows have a finite thickness so that a zone of the
triple-shock interaction has a cross size of the same order. Just downstream of this
interaction, transition of flow parameters in the cross direction from the state behind
the Mach disk to the state behind the reflected shock wave occurs not in a jump
described by the Rankine-Hugoniot relations but continuously.
It should be noted the following essential diversity of forming the
Mach triple-shock configurations in the considered axisymmetric supersonic
compression flows as compared to the two-dimensional flows.
An incident wedge-generated shock in the two-dimensional flow is
straight linear and its parameters do not change downstream so that a type
of its reflection is determined uniquely and does not depend on a position of
the Mach stem.
Parameters of an incident longitudinally-curved shock in the
axisymmetric flow change downstream, that is why a continuous series of
particular irregular triple-shock solutions are possible depending on a
position of the Mach disc. The numerical solution of the problem with step
by step iterations gives a particular irregular triple-shock configuration, its
quantities, including the Mach disc position.
The supersonic axisymmetric flows in
converging funnel duct are analogous to
those in the initial section of over-expanded
jets exhausting into still air or into an
external cocurrent air stream
Turbulent jet exhausting into still air at Мj = 1.6, pj/pa = 0.7
The irregular triple-shock configuration occurs of the type which occurs
in conditions of the von Neumann paradox
1 – polar of incident bell-shaped shock, 2 – reflected shock polars, 3 –
polar of reflected normal shock, 4 – sonic point of incident bellshaped shock polar, diamonds – flow parameters behind of Mach disc
and reflected shock by numerical computation
Turbulent jet exhausting into still air at Мj = 2, pj/pa = 0.6
Single triple-shock Mach configuration occurs
1 – polar of incident bell-shaped
shock, 2 – reflected shock polars,
diamonds – flow parameters behind
of Mach disc and reflected shock by
numerical computation
Let us consider one more problem on position and size
of the Mach stem at the steady irregular reflection of
wedge-generated shock waves in the two-dimensional
supersonic flows or the position and size of the Mach disc
in the axisymmetric supersonic jets.
This problem was of long-standing interest.
This problem was of long-standing interest.
As for the two-dimensional flows, the hypothesis based on the
one-dimensional consideration of a virtual stream forming
downstream of the Mach stem was common (Hornung 1986, Ben-Dor
1992).
This stream is primarily subsonic, the
flow in it accelerates to forming further a
virtual throat where the flow velocity is
sonic, further the stream becomes
supersonic. It is assumed that an
expansion wave emanating from the
wedge rear edge affects on the said
virtual stream so that the position of the
Mach stem depends on reaching the
critical sonic condition in its throat.
This
effect
was
investigated
analytically by Li, Schotz, and Ben-Dor
(1995).
An analogous hypothesis was used for determination of the Mach
disc in axisymmetric non-isobaric jets (Dash and Thorpe, 1981) and
recently for determination of the Mach stem in two-dimensional overexpanded jets (Omel'chenko, Uskov, and Chernyshov, 2003).
Note, it was stated experimentally and convincingly
that the Mach stem height at the irregular reflection of
wedge-generated shock waves in the two-dimensional
flow does not depends on the influence of downstream
flow conditions (Chpoun and Leclerc 1999).
Independence of the position and size of the Mach stem or the
Mach disc from downstream flow conditions one can ground by the
integral equations of conservation lows (flow rate, energy, momentum)
for the steady flow. These equations have to be true for whatever
selected closed volume of the flow, particularly for a volume not
including a section of the virtual stream with the “sonic throat” or a
section in which the effect of an expansion wave emanating from the
rear edge of the duct wall manifests itself.
From this view point, the consideration based on the onedimensionality of the virtual stream and taking into account the
expansion wave emanating from the wedge rear edge is no more than
approach allowing a simple approximate particular solution.
In numerical simulations of flows with its determination in step
by step iteration process the said integral equations should kept with
the solution completed. However, as for the problem of steadying the
supersonic flow with Mach disk, sizes of the computational domain
should be preset so that it includes sub domain in which Mach disk
forms.
Let us to present computed examples confirmed the said
statements.
An example of the flow in a conical funnel duct when there is no
action of the expansion wave emerging from the duct rear edge on
the virtual jet developing downstream of the Mach disk.
М = 2, δC = 5
The flow in the virtual jet accelerates just behind the Mach disk, a
virtual “sonic” throat forms downstream of which the jet becomes
supersonic.
Viscid flow pattern
in the funnel-shaped duct
М = 1.6, δс = 5
Irregular triple-shock configuration
of the same type which occurs in
conditions of the von Neumann
paradox
DМ  0.1950.005 and
хМ  0.38
There is the boundary layer
displacement effect !
Inviscid flow pattern in
the funnel-shaped duct
M = 1.6, δс = 5
DМ = 0.1730.012
andхМ  0.396
М= 1.6
The position and size of the Mach stem
or the Mach disc is independent from
downstream flow conditions.
Particularly it does not depend on
action of the expansion wave emerging
from the jet boundary bending.
RM = 0.134…0.142
XM = 0.313
Turbulent over-expanded jet at Мj = 1.6, pj/pa = 0.7
М= 1.6
RM = 0.13…0.145
XM = 0.313
CONCLUSIONS
Thus, two types of irregular triple-shock interactions of the initial
longitudinally-curved shock and the Mach disk in the steady
axisymmetric supersonic compression flows are presented. One type
is analogous to that forming in conditions of the von Neumann
paradox; another corresponds to the analytical solution for the single
Mach triple-shock reflection.
These interactions are bounded by the stationary Mach
configuration on one side and by the hypothetical limiting Mach
configuration of Ferry with a degenerated polar on another side. Note,
in the case of the stationary Mach configuration, the Mach stem has to
be straight linear and the Mach disc has to be plane.
Ad hoc problem statement is required for the numerical realization
of the said special Mach configurations in the axisymmetric flows.
As for the direct Mach triple-shock interactions one can
contemplate that, besides the single Mach triple-shock configuration,
other types such as transitional and double Mach reflections can form
with increasing free stream Mach numbers M > 2…3.
Thank you for attention!