Let us now use the results obtained in sections §23 and §24 and apply them to the case of jets that are curved due to any mechanism, for example the interaction of the jet with a cloud as was discussed in Chapter III.
The greatest danger occurs when a jet is bent and forms internal shock waves. This is because, after a shock, the normal velocity component of the flow to the surface of the shock becomes subsonic and the jet flares outward. Nevertheless, as we have seen in section §24, the shock that forms when gas flows around a curved profile (such as a bent jet due to external pressure gradients) does not start from the boundary of the jet. It actually forms at an intermediate point to the flow. In other words, it is possible that, if a jet does not bend too much the intersection of the characteristic lines actually occurs outside the jet and the flow can curve without the production of internal shocks.
As we have seen in section §24 the Mach angle of the flow, relativistic and non-relativistic, does not remain constant in the bend -see for example eq.(24.7) and eq.(24.10). The Mach number monotonically decreases as the bend proceeds.
Eq.(24.7) and eq.(24.10) imply that:
where
As was mentioned above, if the jet is sufficiently narrow, it appears
that it can safely avoid the formation of an internal shock. However,
differentiation of eq.(25.1) with respect to the angle the
velocity vector makes with the axis, that is the deflection
angle
, implies that:
with
The Mach number is given by eq.(11.22) and
eq.(11.20) respectively. As the Mach number
, then the derivative
. This means that the rate of change of the
Mach angle with respect to the deflection angle grows without limit
as the Mach number decreases and reaches unity. On a bend, the Mach
number decreases and care is needed, otherwise characteristics will
intersect at the end of the curvature. There is only one special shape
for which this effect is bypassed and this occurs when the increase
of
matches exactly with the increase of
(Courant & Friedrichs, 1976), but of course, this is quite a unique case. It appears
however, that whatever the thickness of the jet it cannot be bent more
than the point at which
exceeds the rate of change of
with respect to the bending
angle
. In other words,
.
From this last inequality and eq.(25.3) a value of the Mach
number can be obtained (Icke, 1991):
If the Mach number in the jet decreases in such a way that the
value is reached, then a terminal shock is produced
and the jet structure is likely to be disrupted. It is important to note
that this terminal shock is weak since
and so,
it might not be too disruptive. Nevertheless, this monotonic decrease
of the Mach number makes the jet to flare outwards, even if the terminal
shock is weak.
Let us now calculate an upper limit for the maximum deflection angle for which jets do not produce terminal shocks. In order to do so, we rewrite eq.(25.1) in the following way:
To eliminate the constant from all our relations,
we can compare the angle
evaluated at the minimum possible
value of the Mach angle
with
evaluated at its maximum value
. In other words,
the angle
defined as:
is an upper limit to the deflection angle. Jets which bend more
than this limiting value
develop a terminal
shock and the jet will flare outward.
This upper limit however, does not mean that the jet is immune from developing an internal shock if it is bent by a smaller angle. Indeed, let us suppose that the jet bends and that the curvature it follows is a segment of a circle as it is shown in fig.(IV.3). According to the figure, the equation of the characteristic OA that emanates from the point O, where the curvature starts is:
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Once the flow has curved
degrees, the
characteristic at this point is given by:
where R is the radius of curvature of the circular
trajectory. The intersection of this characteristic and that given by
eq.(25.8) occurs when the coordinate has a value:
Substitution of eq.(25.3) gives (Icke, 1991):
Using eq.(25.6) and eq.(25.9) it is possible to make a
plot in which two zones separate the cases for jets which develop shocks
at the onset of the curvature, and the ones that do not. Indeed, we can
plot the ratio of the width of the jet to radius of curvature
as a function of the difference
between the deflection angle
and the maximum deflection
angle
, as is shown
in fig.(IV.4).
Jets for which the ratio lies below the curve do not
develop any shocks at all. For example, consider a jet with a given Mach
number for which its ratio
is given. As the width of the
jet increases (or the radius of curvature of the profile decreases), it
comes a point in which a shock at the onset of the curvature is produced.
In the same way, jets with a fixed ratio
for a given Mach
number which are initially stable -so that they lie below the curve-
can develop a shock at the beginning of the curvature by increasing
the bending angle of the curve.
![]() ![]() |
The relativistic Mach angle is smaller for a given value of the velocity of the flow than its classical counterpart as it was proved in Section §11 -see for example fig.(II.2). This fact is extremely important when analysing the possibility of the intersection of different characteristics in a jet that bends. For a relativistic flow, the characteristics, which make an angle equal to the Mach angle to the streamlines, are always beamed in the direction of the flow. Thus, when a jet starts to bend the possibility of intersection between some characteristic line in the curved jet and the ones before the flow has curved, become more probable than their classical counterpart.
This difference results in a severe overestimation of the maximum
bending angle
. For example, Icke (1991)
used the classical analysis in the discussion of the generation of
internal shocks due to bending of jets. Using the classical equations
described above, but with a polytropic index
,
then
.
This is much greater than the value of
obtained with a full relativistic treatment
which is impossible.
The analysis made by Icke (1991) is important for jets in which the microscopic motion of the flow inside the jet is relativistic, but the bulk motion of the flow is non-relativistic.
Sergio Mendoza Fri Apr 20, 2001