The Effect of Variations in Channel Spacing

5.5 Additional Tests of the Vortex Pair Models

5.5.2 The Effect of Variations in Channel Spacing

This section examines the effect of channel spacing on the motion of the vortex pair. The Mach number is M=l.l, and the density ratio is

%/pH

= 0.138. Three cases are computed: Channel spacing

=

16, 8, and 4 (the spacing

=

8 data are those of Chapter 3, Section 1). The explicit variation of channel spacing provides additional checks of the circulation, impulse, and velocity models. The circulation and impulse models assume an unbounded domain, and it is interesting to see if and how and

Ts'

are affected by variations in channel spacing. The velocity model predicts an explicit dependence of the normalized velocity on the ratio of vortex spacing to channel spacing. Previously, this effect was tested only through variations in vortex spacing as either the shock strength or the density ratio was changed. The present computations provide an opportunity to test the model through variations in channel spacing.

5.5.2.1 Qualitative Description

The primary contribution to the velocity of the upper-half-plane vortex is the induced motion from the vortex in the lower-half-plane (and vice versa). This contribution is in the sense of a downstream velocity. However, there are also contributions from the infinite array of image vortices that constitute the mathematical equivalent to the perfectly reflecting channel walls. The net con- tribution from these vortices is a much smaller upstream velocity. Therefore, changes in channel spacing can have a slight influence on the behavior of the system, but no major changes are expected.

Contour plots for the three cases are shown in Figures 5.42, 3.1, and 5.43. Qualitatively, they look very similar. The details of the structures are essentially the same in all three cases. However, there are minor differences in the tails. As the channel spacing is decreased and the strength of the vortex pair decreases, these is less induced motion of the tails. This means there is less inward and upstream stretching of the tails, relative to the cores, and consequently one expects less straining of the fluid within the tails relative to the ambient fluid. This may have implications for mixing enhancement. The case of narrow channel spacing shows more high mass fraction fluid in a more well-defined structure than does the case of wide channel spacing.

5.5.2.2 Q u a n t i t a t i v e Description

Table 5.10 shows steady-state computational data and corresponding predictions from the an- alytical models.

Computed Predicted Computed Predicted

- - - - - - -

~ h . Spac.

I' u

^YOO

R u u 7,)

I,(? = o+)

Tabla 5.10

-

C o m p u t e d and predicted

v ~ / r

^and

^TS

^and

7,(5 =

0+) for variations in channel spacing

The retarding influence of the image vortices on the motion of the actual vortex pair is a strong function of the wall spacing. As the channel walls are brought closer together, one expects a monotonic decrease in the velocity of the vortex pair. In Section 3, the velocity model was only tested against computed data spanning a relatively small range of vortex spacing/channel spacing (less than 30% variation). The present four-fold variation in channel spacing provides a more rigorous and explicit test of the model. From the table above, note the reasonable agreement between the computed and predicted values of normalized velocity (see also Figure 5.44). The model again slightly overpredicts the measured values, but the difference is slight.

Now, since the motion of the vortex is related to its strength, the circulation may change as well. Note, however, that Kelvin's Theorem,

DT/DT =

0, must still be valid, because this result was derived independently of any boundary conditions. The decrease in channel spacing affects the interaction between the shock and the circular jet, and hence the value of the circulation at

f ₌

0+, but then this circulation remains essentially invariant thereafter. The table shows only a negligible change in circulation in reducing the channel spacing from 16 to 8, but the value drops about 11% if the spacing is again halved. Also note that there are essentially no variations in the size or spacing of the vortices, the maximum difference in these quantities being less than 4%. These differences are too small to be meaningful. The size of the vortex pair,

z,

is primarily a function of the initial shock strength, which does not change. Similarly, the spacing, jj,, is primarily a function of the light/heavy gas density ratio, which, too, does not change.

Finally, recall that a measure of the impulse is

7,'

^E 0.5 ^jjoO

F,

which is to be compared with the predicted initial impulse,

-i,(i = O+). 7,'

is essentially unchanged as the spacing is halved from

16 to 8, but shows a large drop as the spacing is further halved to 4. Since jj- is essentially constant, the decrease in impulse appears to be manifested primarily through a decrease in circulation. The findings may be summarized as follows. The computed circulation and impulse values are essentially the same for the two wide channel spacings but drop about 13% and 17% respectively for the narrow channel spacing. The circulation and impulse models do not capture this dependence on channel spacing, but they are still satisfactory as these are relatively small differences relative to a factor of four variation in channel spacing. The computed normalized velocities decrease monotonically with decreasing channel spacing, and this decrease is quite well represented by the velocity model.