[定常射流 马赫环]Steady-State Jets

SIDEBAR 3

Steady-State

Jets

s upersonic gas jets are created in the

laboratory by allowing a highly com-

pressed gas to escape through a nozzle

into the atmosphere or some other ambient

gas. In contrast to astrophysical jets, labora-

tory jets usually have high density ratios

mismatches at the source. Investigations of

laboratory jets have, until recently, focused

on their steady-state behavior rather than the

initial time-dependent behavior of interest in

astrophysical contexts. Nevertheless, our un-

derstanding of the steady-state structures ob-

served in laboratory jets provides a point of

departure for interpreting our numerical

simulations of time-dependent jet flow.

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An Idealized Supersonic Jet

Figure,4 shows the characteristic structure

taken on by a slightly underexpanded super-

sonic jet, that is, one for which the pressure

of the gas at the nozzle orifice. P.. is slightly

greater than the ambient gas pressure, P

a

. (A

jet is referred to as underexpanded if

pressure matched if K = 1.)

This complicated axisymmetric structure

has several remarkable features. First, the jet

boundary oscillates as the jet gas periodically

overexpands and reconverges in its attempt

to match the ambient pressure. The gas con-

tinually overshoots the equilibrium position

because the effects of the boundary arc com-

municated to the interior of the jet by sound

waves, which, by definition, travel more

slowly than the bulk supersonic flow. The

characteristic paths of the sound waves con-

verge to form the second remarkable feature

of the jet, the network of crisscrossed shock

waves, or shock diamonds (red lines). These

standing shocks alternate with rarefaction

fans (blue lines). The gas in the jet interior

expands and cools (shades of blue) as it flows

through the rarefaction fans and is com-

pressed and heats (shades of red) as it passes

through the shock diamonds. The figure

clearly illustrates that the jet interior is

always out of step with the jet boundary. For

example, the positions of greatest gas com-

pression (dark red) do not coincide with the

Spring/Summer 1985 LOS ALAMOS SCIENCE

Supersonic Jets

SIDEBAR 3

Temperature

Jet Boundary Streamline

Fig. A. Idealized steady-state structure of a slightly under-and the blue lines indicate the beginnings of rarefaction expanded (P n slightly greater than P a ) supersonic jet. The fans. The gas temperature varies according to the key. Black red lines represent incident and reflected shocks (see Fig. B),streamlines follow the oscillating flow path of the jet gas.

Fig. B. Regular reflection in a slightly underexpanded jet.(blue) form rarefaction zones. Converging characteristics The pattern of crisscrossed shocks in Fig. A can be under-(red) from the boundary form the incident cortical shocks,stood in terms of characteristics. Diverging characteristics

which reflect off the jet axis to form the reflected shocks.

positions of minimum jet diameter. The black streamlines in the figure indicate the flow paths of the gas. The gas bends out toward the boundary as it passes through rare faction fans and bends back toward the axis as it passes through shock fronts.

Regular Reflections

Figure B shows how the shock structure in Fig. A can be understood in terms of called incident shock reaches the jet axis it characteristics. As the gas leaves the nozzle,undergoes a regular reflection; that is, it it expands and a rarefaction fan (diverging forms a diverging shock. At the point where blue characteristics) emanates from the this reflected shock reaches the jet boundary,nozzle orifice. The gas overexpands, and the it knocks the boundary outward, creating a pressure of the ambient gas at the boundary,new rarefaction fan, and the process begins acting like the piston in Sidebar 1, pushes the all over again. The geometric pattern of the jet gas back toward the axis, creating the red (x, y) characteristics that form the shocks characteristics. These characteristics form a and rarefaction fans in this steady-state two.converging conical shock. When this so-

dimensional flow is similar to that of the

LOS ALAMOS SCIENCE Spring/Summer 1985

47

Mach Disk

(1) Shock Triple Point

reflects at the perimeter of a Mach disk—a strong shock normal to the flow direction (so named because Ernst Mach was the first to record its existence). The angle between the incident shock and the jet axis determines the type of reflection: small angles of in-cidence yield the regular reflections shown in Figs. A and B, and large angles of incidence yield Mach reflections. When gas passes through a shock, its velocity component nor-

slows down the beam much more than passage through the supersonic jet. A Mach reflection creates a shock triple point incident and reflected shocks (from point 1 through points 2where three shocks meet. Figures C1 and C2 show how and 3 to point 4'. As a result, a slip discontinuity is formed passage through the Mach disk {from point 1 to point 4){dashed line).

(x, t) characteristics for one-dimensional

time-dependent flow described in Sidebar 2.

Mach Reflections

A striking change in flow structure occurs when the pressure mismatch at the orifice is for an underexpanded jet, the incident shock,rather than converging to a point on the axis,

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mal to the shock is greatly reduced but its parallel component remains unchanged.Thus shocks with large angles of incidence relative to the flow axis are much more effec-tive at slowing down the flow than shocks with small angles of incidence. The critical angle for transition from regular reflections to Mach reflections is approximately the angle that yields a sonic relative velocity for the gas downstream of the shock.

Spring/Summer 1985 LOS ALAMOS SCIENCE

Supersonic Jets

SIDE BAR 3

Fig. D, Realistic steady-state structure of an overexpanded overexpanded jet, shocks rather than rarefactions emanate from the nozzle orifice. Shear instabilities create a mixing Mach reflections and regular reflections. Since this is an

layer that grows until it reaches the beam axis.

A prominent feature of Mach reflections is the emergence of a slip discontinuity (dashed line) from the shock triple point where the incident shock, reflected shock, and Mach disk meet, The flow velocity, density, and temperature are discontinuous across this contact surface. This slip discontinuity arises because. the thermodynamic pathway through the incident and reflected shocks does not equal the pathway through the Mach disk.

In Fig. Cl we display two adjacent streamlines, one on each side of the shock triple point, and in Fig. C2 we display the corresponding thermodynamic pathways.Adjacent fluid elements at some initial point on either streamline have the same state variables and the same total (kinetic plus internal) specific energy, that is, energy per unit mass. This quantity is given by 1

where y is the ratio of the specific heat of the

LOS ALAMOS SCIENCE Spring/Summer 1985

fluid at constant pressure to the specific heat at constant volume.

By Bernoulli’s principle the total specific energy remains constant along a streamline,Therefore, when the two adjacent fluid ele-ments arrive at points 4 and 4' in Fig. Cl they must still have the same total specific energy. They must also have the same pres-sure because they are still adjacent. However,we see from Fig. C2 that their densities and entropies are different. The element at point 4 has been shock-heated along a Hugoniot to a higher entropy and lower density than the element that passed through the incident and reflected shocks along points 2 and 3 to point 4'. The lower density of the fluid element at point 4 implies that its internal energy (which is proportional to P/p) is greater than that of the fluid element at point 4'.Bernoulli’s principle then implies that the fluid element at point 4 must have a cor-respondingly lower kinetic energy, and hence flow velocity, than the adjacent element at point 4’. A slip discontinuity results from this difference in flow velocities,

Laboratory Supersonic Jets

The jet structures shown in Figs. A, B, and Care idealizations in that real supersonic jets do not have sharp, stable boundaries but turbulent boundaries where jet and ambient gases mix. Figure D shows a more realistic steady-state structure for an overexpanded laboratory jet. Near the orifice, where the pressure mismatch is large, Mach reflections occur, but farther downstream the reflections are regular. The mixing layer, which grows as a result of Kelvin-Helmholtz (shear) in-stabilities, progressively eats its way into the supersonic core of the jet. When the mixing layer reaches the axis of the jet. the flow is subsonic and fully turbulent. It is then susceptible to twisting and bending motions,like a smokestack plume in a crosswind,Strictly speaking the wave structures with-in the supersonic core are not steady since they are buffeted by the turbulent boundary layer. However, their average positions,those in Fig, D, are well defined. s

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