Axisymmetric supersonic flow patterns withMach disk in convergent conical ducts

and in over-expanded jets

Gounko Yu.P.ITAM SB RAS, Novosibirsk, E-mail: gounko@itam.nsc.ru

Workshop on Non-equilibrium Flow

Phenomena in Honor of Mikhail Ivanov's 70th Birthday

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.

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

streamlines and local flow Mach numbers

initial bell-shapedshock

reflectedshock

sonic point

relative diameter of subsonic core at M = 1.6

funnel angle, degree

The Mach disc cannot be determined !

Just only possible cross size of a central zone of subsonic flow was estimated

Computed flow pattern at M = 1.6

subsonic flow core

hypotheticalMach disc

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 plotp(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.

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 triple-shock 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

On explanation of possible irregular interactions of

shock waves

1 – initial incident shock polar,δ = 3 … δ = 12 – secondary (reflected) shock waves with different angles of initial flow deflections

Possible types of

Mach triple-shock configuration-15 -10 -5 0 5 10 15

1

1.5

2

2.5

3

7

1

2 3 4 5

66

p/pМ = 1.6

The inverse (or inverted) and direct Mach triple-shock configurations are separated by the stationary Mach configuration when the second shock polar intersects the strong branch of the initial shock polar 1at 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 SSTwas 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 2106. 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.

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

М = 1.6, δс = 5Irregular triple-shock configuration of the type which

occurs in conditions of the von Neumann paradox

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 atM = 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 = 10single Mach triple-shock configuration

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

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 !

-20 -10 0 10 20 301

2

3

4

1

2

c

p /ps

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 triple-shock solution.

As concerning with the both presented computed flow examples, it should be noted formation of a finite-thickness shear layer with the cross-varying 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 bell-shaped 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 sizeof the Mach stem at the steady irregular reflection ofwedge-generated shock waves in the two-dimensionalsupersonic flows or the position and size of the Mach discin the axisymmetric supersonic jets.

This problem was of long-standing interest.

As for the two-dimensional flows, the hypothesis based on theone-dimensional consideration of a virtual stream formingdownstream of the Mach stem was common (Hornung 1986, Ben-Dor

1992).

This stream is primarily subsonic, theflow in it accelerates to forming further avirtual throat where the flow velocity issonic, further the stream becomessupersonic. It is assumed that anexpansion wave emanating from thewedge rear edge affects on the saidvirtual stream so that the position of theMach stem depends on reaching thecritical sonic condition in its throat.

This effect was investigatedanalytically by Li, Schotz, and Ben-Dor(1995).

This problem was of long-standing interest.

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 over-

expanded 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 one-

dimensionality 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.

The flow in the virtual jet accelerates just behind the Mach disk, a virtual “sonic” throat forms downstream of which the jet becomes supersonic.

М = 2, δC = 5

Inviscid flow pattern in the funnel-shaped duct

M = 1.6, δс = 5DМ = 0.1730.012

andхМ 0.396

Viscid flow patternin the funnel-shaped duct

М = 1.6, δс = 5Irregular triple-shock configuration of the same type which occurs in conditions of the von Neumann

paradox

DМ 0.1950.005 and хМ 0.38

There is the boundary layer displacement effect !

Turbulent over-expanded jet at Мj = 1.6, pj/pa = 0.7

М= 1.6

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

RM = 0.13…0.145

XM = 0.313

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.

CONCLUSIONS

Thank you for attention!