## FLUID DYNAMICS APPLICATION

Published in Interface. Vol. 2, No. 3, August 1994

**Vortices are a distinctive feature of Jupiter's fluid upper surface layers.**

*This issue of Interface features fluid dynamics
computations carried out at the University of South Alabama
by Dr. Robert Spall who serves on the faculty of the Mechanical
Engineering Department. His research interests are in
the area of computational fluid dynamics and heat transfer,
particularly in the modleing of strongly swirling flows.
This article is reprinted from the AIAA Journal, Vol.32.
No. 4, pp. 878-881.*

* Suggestions for future articles are welcome and should
be directed to asnsav01@asnmail.asc.edu. We would
also appreciate feedback regarding the overall content
and coverage of Interface*.

**Effect of Imposed Pressure Gradients on the Viscous
Stability of Longitudinal Vortices**

Robert E. Spall

*University of South Alabama*

**I. Introduction **

VISCOUS stability calculations of the "*q*-vortex"
[derived from the far-wake solution for a trailing line
vortex (Batchelor (1)] have been performed by Lessen and
Paillet(2), Stewartson(3), and Khorrami(4). Several other
researchers have investigated the inviscid stability of
these profiles (Lessen et al.,(5)Duck and Foster,(6) Leibovich
and Stewartson, (7) Stewartson and Capell,(8) Duck(9)).
Resul ts show that the stability of the vortex is strongly
dependent on the value of *q* (related to the ratio
of the maximum swirl velocity to the maximum axial velocity).
For instance, the vortex is completely stabilized to inviscid
disturbances for val ues of *q* > 1.5, with the
most unstable wave obtained in the limit | *n* |
approaches infinity (where *n* is the azimuthal wave
number).

In the present work, the effects of pressure gradients on the viscous stability of longitudinal vortices are investigated. Mean flow profiles computed as numerical solutions to the quasicylindrical equations of motion are employed. This procedure allows one to impose pressure gradients through the boundary conditions at the radial edge of the dom ain. To this author's knowledge, the effect of imposed pressure gradients on the stability of longitudinal vortices has not yet been studied.

The stability calculations are performed using a normal mode analysis. Although, strictly speaking, a nonpar allel stability theory is appropriate [multiple scales (Saric and Nayfeh(10)) or parabolized stability equation approach (Bertolotti(11))], the quasiparallel flow assumption is made. The quasiparallel assumption is justified on the grounds that the spati al development of the vortex is relatively slow; that is, at high Reynolds numbers and under the influence of weak pressure gradients, significant changes occur over many wavelengths. Provisions are made to include some nonparallel effects by retaining r adial velocity and streamwise derivative terms in the mean flow. The stability formulation is based on second-order-accurate finite difference approximations on a staggered grid, with the accuracy of the computer eigenvalues being improved through Richar dson extrapolation.

**II. Problem Formulation Mean Flow**

The linear stability of solutions to the quasicylindrical
equations of motion is considered. The quasicylindrical
equations are derived from the laminar, incompressible,
axisymmetric equations of motion as a result of boundary-layer-type
assumptions (Hall(12)) and are given as

where *r* and *z* are cylindrical-polar coordinates;
*u*, *v*, and *w* are the radial, swirl,
and axial ve locity components; *p* is the pressure,
and *Re* is the Reynolds number. Lengths have been
made nondimensional with respect to a length scale *l*
[defined following Eq. (11)], velocities with respect
to a velocity scale W infinity, and pre ssure with respect
to r*W*^{2} infinity .

The preceding equations are solved subject to the following
boundary conditions:

In addition, initial conditions are specified at z =
0 as

which are derived from Eqs. (10) and (11) later.

In general, the specification of the outer boundary conditions and the radius of the outer boundary may be arbitrarily specified. An exception occurs if the outer boundary represents an inviscid stream surface. In that case, the relationship

is implied and only one of *r*_{0}, *W*,
or *P* may be arbitrarily specified. An iterative
solution procedure is then implemented such that *r*0
takes on a value appropriate for the stream surface defined
by Eq. (9). This is the procedure followed in this paper
for the pressure gradient cases. That is, the pressure
is specified along the outer boundary, which is defined
as an inviscid stream surface. The problem is parabolic
in the streamwise direction and thus the equations may
be solved by marching in *z*. The quasicylindrical
equations were first solved by Hall(13) using finite difference
methods for several sets of initial and boundary condit
ions.

The objective of the present study is to investigate
the linear stability of a longitudinal vortex under the
effect of a weak pressure gradient. Toward this end, initial
conditions are derived from the similarity solution to
the trailing line v ortex^{1} and are given in
dimensionless form as

where

*lambda* and *a* are constants, and *W*0
is the centerline axial velocity. The length scale was
chosen as the vortex core radius (the radius at which
the swirl velocity is a maximum), and thus

The velocity scale was chosen as the far-field axial
velocity *W* infinity.

The preceding equations are solved using second-order-ac curate finite-difference approximations, marching in the streamwise direction. Grid resolutions used in the solution procedure were sufficient to ensure that the stability characteristics of the mean flow profiles are grid independent.

**Stability E quations**

Briefly, the stability equations are derived
by first linearizing the equations of motion and then
expressing the linearized perturbation variables in the
form

Here, alpha is the real axial wave numbe r, *n*
is the real azimuthal wave number, and w is complex (thus,
we are considering temporal stability). These equations,
consisting of three second-order momentum equations and
the continuity equation, have previously been given in
Lessen and Paill et(2) and may be expressed in the form

where is a four element vector defined
by

,

where *r* is the radial coordinate. The preceding
equations are solved subject to th e following boundary
conditions(14):

The solution technique employed is based on a second-order-accurate
finite difference approximation on a staggered mesh. The
staggered mesh eliminates the need for pressure boundary
con ditions. In addition, Richardson extrapolation is
employed to increase the accuracy of the computed eigenvalues.
The scheme is analagous to that employed quite successfully
in the compressible boundary-layer stability code COSAL.(15)
The discretized sys tem represents, along with six boundary
conditions, 4*N* + 3 equations for 4*N* + 3
unknowns (where *N* is the number of grid points).
The system constitutes a generalized eigenvalue problem
of the form

which i s solved for the eigenvalues omega using the IMSL QL routine CXLZ.

**III. Results**

The linear stability of solutions to the quasicylindrical
equations of motion under the influence of weak pressure
gradients is investigated. To provide a baseline for comparison,
results for a zero pressure gradient case are examined
first. The Reynolds number at the inflow plane *Rei*
was set to 1000 and the parameters
and were set to 0.5 and 1.0, respec
tively. Note that the Reynolds number is based on the
local vortex core radius, and thus changes in the streamwise
direction.

In the solution of the quasicylindrical equations for
the mean flow, 400 points were used in the radial direction,
with a st retching function employed to cluster points
near the vortex core. The far-field boundary was set to
*r*0 = 40 (i.e., 40 core radii as defined at the
inflow plane), and the streamwise step size was Delta
*z* = 1. The radius was chosen large en ough so that
the zero perturbation radial boundary conditions required
for the solution of the stability equations would be satisfied.

Figure 1. Mean flow solutions computed as solutions to the quasicylindrical equations of m otion; Rei = 1000, = 0.5, = 1.0.

Figure 1a shows the swirl velocity distribution as a
function of radius for axial locations up to *z*
= 600. This figure reveals that the vortex core has doubled
in radius over an axial distance of *z* = 600, and
that at this location, the maximum swirl velocity has
decreased to approximately 50% of that at inflow. The
corresponding axial velocity profiles are shown in Fig.
1b. Note that the jetlike profile st ill exists at *z*
= 600. Clearly, decay due to laminar diffusion is a slow
process (in relation to other physical mechanisms such
as vortex breakdown or turbulent diffusion) and thus provides
much of the incentive for studying the stability of these
vortices. This also provides some justification for the
quasiparallel stability formulation.

Figure 2. Growth rate as a function of wave number at various streamwise locations.

Figure 2a shows growth rate curves for t he preceding
mean flow as a function of wave number as the vortex evolves
in space; i.e., for *z* = 1, 200, 400, and 600 (corresponding
to local Reynolds numbers of 1016, 1362, 1717, and 1964,
respectively). A substantial decrease in the maximum gro
wth rates is observed as the vortex evolves. This decrease
is due primarily to a decrease in the local swirl ratio
(the magnitude of which may be ascertained from Fig. 1a).
Based on the results of Khorrami, ^{4} the increase
in local Reynolds nu mber may be expected to have a minor
stabilizing effect. The figure also indicates that the
most unstable wave number remains nearly constant at alpha
is approximately 0.575. It is noted that a search over
a wide range of wave numbers failed to reveal any instability
for axisymmetric (*n* = 0) modes.

We now investigate the effect of favorable and adverse
pressure gradients on the stability characteristics of
the vortex. Mean flows are computed for pressure gradients
specified as partialdiff *p* /partialdiff *z*
= +/-0.0005 (imposed along the outer boundary stream tube).
The other parameters are identical with those of the zero
pressure gradient case. The stability results for the
adverse pressure gradient are shown in Fig. 2b, an d results
for the favorable pressure case are shown in Fig. 2c.
Figure 2b indicates that the adverse gradient has only
a minimal effect on the maximum growth rates of the vortex
(when compared to the zero pressure gradient case). However,
the most unstab le modes rapidly shift toward longer wavelength
disturbances (the alpha corresponding to the highest growth
rate decreases from 0.6 at *z* = 0 to 0.18 at *z*
= 750), with the shorter wavelength disturbances becoming
stable.

Results shown in Fig. 2c indicate that the effects of
the favorable gradient on the maximum growth rates of
the vortex are minimal. However, in this case the maximum
growth rates shift toward shorter wavelength disturbances
as the vortex evolves in the streamwise direct ion. The
longer wavelength disturbances also remain unstable. It
is noted that a search failed to reveal any viscous instability
of the *n* = 0 mode for either the favorable or adverse
pressure gradient cases (for the given values of
and ).

The preceding calculations were made using the quasiparallel flow approximation. Additional calculations were performed that included nonparallel terms in the mean flow profiles. That is, stability results were a lso obtained by including terms containing the radial velocity, and streamwise derivative of the axial velocity, in the coefficient matrix [of Eq. (15)]. Although plots are not presented, the results are worth mentioning. The primary effects are in term s of the growth rates. That is, for the adverse pressure gradient case, at some downstream location, the growth rates begin to increase. Conversely, for the favorable pressure gradient case, the growth rates were decreased (compared with the results in Fig. 2c) as the vortex evolved. The trends concerning the shift in wavelengths of the most unstable modes remained unchanged. A properly formulated nonparallel analysis would be required to verify these findings, which have implications in areas such as vortex breakdown.

**IV. Conclusions**

The linear stability of a quasicylindrical vortex evolving
in the streamwise direction was investigated. The mean
flow profiles were computed as numerical solutions to
the quasicylindrical equations of m otion with imposed
pressure gradients. Results indicate that, under the quasiparallel
approximation, the maximum growth rates of the vortex
are not greatly influenced by pressure gradients. However,
under the influence of an adverse pressure gradient, t
he most unstable disturbances rapidly shifted toward longer
wavelengths as the vortex evolved in the streamwise direction.
In all cases, viscous disturbances were noted only for
the *n* = 1 mode. Further work into the stability
of these swirling fl ows is anticipated by the author.
To properly account for nonparallel effects, a multiple-scale
technique will be attempted.

**Acknowledgments**

The author would like to acknowledge the NASA JOVE program and the Alabama Supercomputer Authority for providing the necessary resources.

**References**

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