The Combined Effects of Channel Spacing and Core Size

Chapter 4 Assessment of the Computational Results 4.1 Qualitative Description of the Flow

5.3 A Model for the Motion of the Vortex Pair

5.3.3 The Combined Effects of Channel Spacing and Core Size

and normalized velocities are the same in either case. The inner velocities for the piecewise constant density flow are simply constant multiples of the inner velocities found for the uniform density case.

The above analysis assumes the flow to be incompressible. For the higher Mach number cases discussed in Chapter 3, although the flow behind the shock is certainly subsonic, the assumption of incompressibility is no longer guaranteed. Moore and Pullin (1987) demonstrated numerically and analytically that the effect of compressibility is to reduce the velocity of a finite core size vortex pair in an unbounded domain. They found that the normalized velocity depends on two parameters: 1) the ratio of boundary velocity to vortex pair velocity, v b o u n d a r y / v l and 2) the ratio of vortex pair velocity to the sound speed far behind the shock,

P/z2.

The worst (limiting) case is the so-called 'evacuated vortex,' corresponding to zero pressure inside the vortex. In that case, the normalized velocity is only a function of

T/z2.

This result can be used as an (upper bound) estimate of compressibility effects in the flows of the present investigation. The velocity is available from Tables 5.2 to 5.4 and the speed of sound far behind the shock is given by one-dimensional normal shock theory as

Consider the highest compressibility case of the present investigation, M=2 and

FLIPH =

0.138.

In that case,

=

0.070 and T2

=

1.30, so that

v/z2 =

0.070/1.30

=

0.054. An upper bound for the compressibility effects is given by an 'evacuated vortex' with

r/& ⁼

0.06, which Moore and Pullin found t o have a normalized velocity 1.6% below the classical 1/(4?r) value. Therefore, the assumption of incompressibility in the present study is quantitatively justified.

This completes the perturbation solution for the effect of vortex size. It is appropriate to compare the analytical results against Pierrehumbert's numerical result as a check on the accuracy of the solution. As a matter of convenience, the present results are plotted according to the convention used by Pierrehumbert, i-e., the extreme points of the boundary, normal to the direction of motion, are taken to be f 1. Figure 5.29(a) shows the family of boundary shapes from the perturbation analysis for E

=

0.048,0.100,0.159,0.225,0.390,0.500,0.639, and 0.844. These agree quite well with Pierrehumbert's shapes (Figure 5.22). Figure 5.29(b) is a comparison of normalized velocity,

cjjm/F,

Figure 5.29(c) is a comparison of Pierrehumbert's so-called 'intervortex gap ratio,' and Figure 5.29(d) is a comparison of aspect ratio. In all cases, for values of x/pw less than about 1.1, the results agree quite well. This agreement both verifies the perturbation analysis and provides an independent check of Pierrehumbert's numerical analysis.

cally, this introduces a second perturbation parameter,

J,/X,

making the problem a two parameter perturbation analysis. This complicates the already intricate mathematics in the one-parameter so- lution. To carry out a two parameter perturbation analysis to more than just lowest order involves truly intractable mathematics, and one would be well advised to consider parametric studies using numerical solutions, instead of perturbation theory, in those cases. For this reason, the present anal- ysis will deal with only the lowest order solution to the two-parameter perturbation, which should be expressible as a superposition of the effects of channel spacing and core size taken separately.

Considering the effect of channel spacing alone, the normalized velocity may be represented as

where

represents the unperturbed velocity (in the limit of infinite channel spacing), and

represents the effects of finite channel spacing ( c - f . Subsection 1).

Considering the effect of core size alone, the normalized velocity may be represented as

where

represents the unperturbed velocity (in the limit of infinitesimal core size), and

represents the effect of finite core size (from the perturbation solution above).

One expects to obtain the lowest order solution in both effects by simply substituting either of the above solutions for the 'unperturbed velocity' term of the other solution, i. e.,

The result is

This represents a lowest-order, but completely analytical, model for the effects of both channel spacing and core size. One can test this model using data from the computations of Chapter 3. Besides the circulation, spacing, and velocity, it is also necessary to know size and location parameters. These are computed as follows. The effective vortex core radius,

7

is based on the mass fraction-weighted area,

From these one can calculate the parameters necessary for use in the model, the ratio of core size to vortex spacing, ?Elgoo, and the ratio of vortex spacing to channel spacing,

gw /x.

The first set of data consists of fixed

p L / ~ H =

0.138 and 1.05

5

M

<

2.0. As the Mach number is increased, the ratio of effective core size to vertical spacing decreases, and the qualitative dependence of

vjj,/F

vs. M should look like Figure 5.30(a) (the vortices move faster as they become more compact). Also, as the Mach number is increased, the ratio of vortex spacing to channel spacing increases, and the qualitative dependence of

p ~ / r

^{us. M} should look like Figure 5.30(b) (the vortices slow down as they get closer to the walls). Considering both effects together, one expects the qualitative trend to look like Figure 5.30(c). The initial rise in velocity due to the vortex size effects should be offset by a decrease due to channel spacing effects. The net effect should be approximately constant or perhaps have a slight concave downward curvature, depending on the exact values of vortex size, vortex spacing, and channel spacing.

Table 5.2 lists the computed and predicted normalized velocities as a function of the incident shock strength, so that the dependencies of core size and channel spacing are implicit in the single parameter M.

Computed Predicted

- - -

M R

/

U P ,

ujiwP

Table 5.2

-

Computed and predicted

vij, ^{f l}

for canonical flows w i t h

pL/pH =

0.138

These data are also plotted in Figure 5.31. The computed data are indicated by a, and the predicted values are indicated by x

.

The second set of data is for M=l.l and

PL/pH =

0.138 and 0.354 (two higher density ratio cases are not included because those cases were not carried out to late enough times to give a steady state vortex pair). Table 5.3 lists the computed and predicted normalized velocities as a function of the density ratio.

Computed Predicted

- - -

pL/pH

R

fim/K u?imlr u P w / r

Table 5.3

-

Computed and predicted

v ~ / r

for canonical flows with M = l . l

These data are also plotted in Figure 5.32.

The last set of data is for M=2 and

&/pH

= 0.138, 0.354, and 0.569 (again, omitting one higher density ratio case which lacks a steady-state vortex pair). Table 5.4 lists the computed and predicted normalized velocities as a function of the density ratio.

Table 5.4

-

^{Computed and predicted}

g&/F

for canonical flows with M=2.0

These data are also plotted in Figure 5.33. The agreement is quite good throughout the range of Mach numbers and density ratios, verifying the ability of even this lowest order model to accurately capture the physics of the flow.