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 rW2 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 r0, W, or P may be arbitrarily specified. An iterative solution procedure is then implemented such that r0 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 ortex1 and are given in dimensionless form as

where

lambda and a are constants, and W0 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, 4N + 3 equations for 4N + 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 r0 = 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

(1) Batchelor, G.K., "Axial Flow in Trailing Line Vortices," Journal of Fluid Mechanics, Vol 20, 1964, pp. 645-658.
(2) Lessen, M., and Paillet, F., "The Stability of a Trailing Line Vortex. Part 2. Viscous Theory," Journal of Fluid Mechanics, Vol. 65, 1974, pp. 769-779.
(3) Stewartson, K., "The Stability of Swirling Flows at Large Reynolds Numbers When Subjected to Disturbances at Large Azimuthal Wavenumber," Phys ics of Fluids, Vol. 25, Nov. 1982, pp. 1953-1957.
(4) Khorrami, M., "On the Viscous Modes of Instability of a Trailing Line Vortex," Journal of Fluid Mechanics, Vol. 225, 1991, pp. 197-212.
(5) Lessen, M., Singh, P.J., and Paillet, F. , "The Stability of a Trailing Line Vortex. Part 1. Inviscid Theory," Journal of Fluid Mechanics, Vol. 63, 1974, pp. 753-763.
(6) Duck, P.W., and Foster, M.R., "The Inviscid Stability of a Trailing Line Vortex," Journal of Applied Mathema tics and Physics, Vol. 31, 1980, pp. 523-530.
(7) Leibovich, S., and Stewartson, K., "A Sufficient Condition for the Instability of Columnar Vortices," Journal of Fluid Mechanics, Vol. 126, 1983, pp. 335-356.
(8) Stewartson, K., and C apell, K., "On the Stability of Ring Modes in a Trailing Line Vortex: the Upper Neutral Points," Journal of Fluid Mechanics, Vol. 156, 1985, pp.369-386.
(9) Duck, P., "The Inviscid Stability of Swirling Flows: Large Wavenumber Disturbances," Journal of Applied Mathematics and Physics, Vol. 37, May 1986, pp. 340-360.
(10) Saric, W.S., and Nayfey, A.H., "Nonparallel Stability of Boundary-Layer Flows," Physics of Fluids, Vol. 8, Aug. 1975, pp. 945-950.
(11) Bertolotti, F .P., "Compressible Boundary Layer Stability Analyzed with the PSE Equations," AIAA 22nd Fluid Dynamics, Plasma Dynamics and Lasers Conference, AIAA Paper 91-1637, Honolulu, HI, June 1991.
(12) Hall, M.G., "Vortex Breakdown," Annual Review of Fluid Mechanics, Vol. 4, 1972, pp. 195-218.
(13) Hall, M.G., "A Numerical Method for Solving the Equations for a Vortex Core," Royal Aircraft Establishment, Tech. Rept. No. 65106, Farnborough, England, UK, May 1965.
(14) Batchelor, G.K., and Gill , A.E., "Analysis of the Stability of Axisymmetric Jets," Journal of Fluid Mechanics, Vol. 14, 1962, pp. 529-551.
(15) Malik, M.R., and Orszag, S.A., "Efficient Computation of the Stability of Three-Dimensional Compressible Boundary Layers," A IAA 14th Fluid and Plasma Dynamics Conference, AIAA Paper 81-1277, Palo Alto, CA, June 1981.