Qualitative Description

Chapter 7 Variations on the Canonical Problem

7.2 Elliptical Jets

7.2.1 Qualitative Description

considering that the worst case corresponds to a perturbation wavelength of the same order as the radius!

In summary, Richtmyer-Meshkov instability affects the qualitative details of the tail develop- ment, but not the stability of the vortex core. The instability traps high mass fraction fluid within the tail, resulting in a more efficient partitioning of the initial jet into discrete sub-structures. This phenomenon becomes more pronounced as wavelength or amplitude is increased. Wavelength is the primary factor which affects tail development. The amplitude affects the size of the instability, and particularly its length, but not its character.

again spun off as an upstream tail, which stretches vertically inward and trails behind the faster vortex core. Because of the longer streamwise dimension of the original structure, the tails are longer than in the circular case.

Density contour plots of this case are shown in Figure 7.14. Again, the basic interaction is similar to the circular jet case. In this case, because of the reduced initial horizontal dimension, less swirling is resolvable within the developing vortex, and the tail development is reduced.

Next consider a strong shock, with the same aspect ratios as above.

In the circular jet case, increasing the Mach number from 1.1 to 2.0 resulted in suppression of the tails, because the increased shock strength greatly compressed the jet in the streamwise direction. The same phenomenon occurs with the elliptical jet: compare the contour plots of this M=2.0, AR=0.5 case (Figure 7.15) with those of the M=l.l, AR=0.5 case (Figure 7.13). Another consequence of the compression is that swirling within the lobes, clearly visible in the M=l.l case, can no longer be resolved; there are simply not enough cells in the streamwise direction.

Density contour plots of this case are shown in Figure 7.16. There is now almost no tail, because of the combination of high Mach number and small initial horizontal dimension. Likewise, no swirling within the lobes can be resolved.

7.2.2 Quantitative Description

This section quantitatively describes the elliptical jet interactions. In particular, the analytical models for circulation, timescaling

,

velocity, and spacing of the steady-state vortex pairs are extended to the elliptical jet case.

Table 7.1 summarizes the computed data for the elliptical jet runs. For completeness, the circular jet data are included, treating the circular jets as ellipses of unit aspect ratio. In the usual manner, the velocities of the vortex pairs have been error-corrected according to the prescription of Chapter 4, Section 2, Subsection 2.

- -

Table 7.1

-

Steady-state

I', uvw/i?,

and for the elliptical jet flows

First, consider the circulation. The derivation of the model proceeds in an analogous manner to the circular jet scaling law of Chapter 5, Section 1. The essential difference is that an additional parameter, the aspect ratio of the ellipse, now appears.

Consider a cross section through an elliptical jet, as shown in Figure 7.17. The horizontal and vertical semi-axes are a and b, respectively, and the interface is idealized as a sharp step discontinuity in density. The light jet gas has density p ~ , and the ambient gas has density p ~ . A shock wave of strength M approaches from left to right. The density and pressure ratios across the shock wave are

~ 2 1 ~ 1 and ~ 2 1 ~ 1

The vorticity production term of the vorticity equation is

The vorticity is perpendicular to the plane of the jet, g

=

w i , . Integrating the vorticity equation for w gives

To evaluate the integrand, one works in elliptical coordinates, where and 7 denote coordinates normal and tangent to the ellipse, res{ectively. The ellipse is given by

so that the gradient is

and the unit vector normal to the surface is

The (<, q ) coordinates are related to (x,y) under the transformation z = c cosht cosq, y = c sinht sinq.

This defines a family of ellipses for which

with the particular ellipse of interest being given by

t = to.

Equivalently,

a

=

c c o ~ h ( ~ , b

=

c sinhEo, and c2

=

a2

-

b2, so that

coshto

=

a and sinhto

=

⁶

d m -

The density is given by

and the density gradient is given by

where

h = hC = h,, = c J s i n h 2 t

+

^sin211

Therefore,

D = A p

' ( 6 - t o )

_it

cJsinh2<

+

^sin2q .

The pressure gradient is simply

Q = -Ap ⁶⁽²

-

K t ) i,, so that

a(. -

K t )

' ( 5 - 50)

^sinq

= A p Ap ez .

c J s i n h 2 ~

+

sin2q J(b/a)2cos2q

+

^sin2q

The integral thus becomes

AP AP b e

- t o )

^sinq

w=- p2c dsinh2<

+

sin2q J(b/a)2cosZq

+

^sin2q

lm

^{o '(x}

^-

^{~ , t ) dt}

AP AP

'(t - t o )

--

^sinq

-

p2Kc Jsinh2<

+

sin2q J(b/a)2cos2q

+

sin27 ^'

Having this expression for the vorticity, one integrates over the area of the jet to obtain the circu- lation:

. = / / U ~ A

J(t - t o )

sing

=-

A' p 2 K c A'

J /

J s i n h 2 ~

+

sin2q J(b/a)2cos2q

+

^{sin2q dA}

The transformation of dA from Cartesian to elliptical coordinates is as follows:

dA

=

dxdy

=

JJldtdq, where

d x d y d x d y

I J I = -- - --

t

7 dl7

at

Therefore,

and

Evaluating the delta-function in the usual manner and inserting the appropriate limits for 0

5

q

5

^T

in the upper half plane gives

r ⁼

^{c/o* sinq \/sinh2fo}

+

^sin2q

p2

v,

d(b/a)2cos2q

+

sin2q dt b2/(a2

-

^b2)

+

^sin2g

-

- *

_{p2 V,}

1.

^sing J ( b / ~ ) ~ ( l -

^J

sin2 q )

+

^{sin2q d t}

APAP c sinq d t

In this expression, it is necessary to choose suitable values of p, one for the pressure term, and one for the density term. Since the pressure gradient is due to the shock, choose ^p2 for the pressure term, and since the density gradient is across the interface, choose the average density at the interface, (pL

+

pH)/2, for the density term. The circulation then becomes

Finally, nondimensionalize all lengths by b, all densities by pl, all pressures by pl, all velocities by cl, and the circulation by bcl. The circulation then becomes

This formula is exactly that for the circular jet case divided by the aspect ratio of the ellipse!

Summarized below are the computed and predicted circulation values for the elliptical jet runs.

The computed values are simply copied from Table 7.1 above.

Computed Predicted

M

AR

f f

Table 7.2

-

Computed and predicted

T

for the elliptical jet flows

These data are also plotted in Figure 7.18 (as a matter of convenience, these are actually plots of the ratio of predicted to computed circulation, which should be unity in the case of a perfect scaling law). The model underpredicts the M = l . l circulation at low aspect ratios, and overpredicts the M=2.0 circulation at high aspect ratios. The overall results are within f 30%, considerably worse than the circular jet case.

Besides the circulation, one is interested in the time scaling; Exactly as before, one defines a characteristic timescale as

where H is a characteristic height, W is a characteristic width, and

r

is the circulation. For H, one takes b, the vertical dimension, and for L, one takes (pl/p2)a, the horizontal dimension just after the shock has passed (the factor pl/pz accounts for the streamwise compression due to the shock).

Therefore,

Substituting and nondimensionalizing gives

which is exactly the same result as in the circular jet case. This means that aspect ratio does not affect the time scaling. Thus, at fixed Mach number, the AR = 0.5, 1, and 2 flows should all be at the same stage of development at the same time. This is indeed shown by the computations: see Figure 7.19 for a comparison at M=l.l and Figure 7.20 for a comparison at M=2.0.

Now consider the model for the motion of the vortex pair derived in Chapter 5, Section 3. This model needs no modification for the case of initially elliptical jets, because it is independent of the initial configuration.

Summarized below are the computed and predicted normalized velocity values for the elliptical jet runs. The computed values are simply copied from Table 7.1 above. The predicted values follow from a straightforward determination of the size and spacing parameters,

fi/gm,

and &,/f;.

Computed Computed Computed Predicted

- -

M AR z/jjo & m / x

uvm/F

lJjjm/T

Table 7.3

-

Computed and predicted

u&/F

for the elliptical jet flows

These data are also plotted in Figures 7.21 and 7.22. As in the circular jet case, the model faithfully reproduces the qualitative trends. The quantitative agreement, while reasonable, is not as good as before.

Next consider the model for impulse of the vortex pair. Exactly as in the circular jet case,

Substituting for the vorticity distribution, carrying out the integral, picking appropriate reference values, and nondimensionalizing gives

One correlates the steady-state product of spacing and circulation with the initial impulse deposited by the passing shock,

- I

-

I,

c i,(i= o+),

where again,

7,' r

0.5 gm

r.

Table 7.4 shows a comparison of the quantities

Tx1

and

Tz(f

= 0+).

M AR

TX1 -

I,(?

=

O+)

Table 7.4

-

^Computed

^TSt

and predicted

Tx(,(i =

0+) for the elliptical jet flows These data are also plotted in Figures 7.23 and 7.24. The qualitative agreement is good through- out. The quantitative agreement is very good at moderate and high aspect ratios, but not for low aspect ratios.

Finally, the circulation and impulse models may be combined to give an expression for the half-spacing, jj,. The result is exactly the same as in the circular jet case,

i.e., there is no dependence on aspect ratio. The computed data in Table 7.1, which show very little variation in jja for varying aspect ratio at fixed Mach number, verifies this prediction. However, just as before, the model fails to predict the dependence of jja on the Mach number.

7.3 Reflected Shock Interactions with a Vortex Pair