## Journal of the Brazilian Society of Mechanical Sciences

*Print version* ISSN 0100-7386

### J. Braz. Soc. Mech. Sci. vol.22 n.1 Rio de Janeiro 2000

#### http://dx.doi.org/10.1590/S0100-73862000000100006

**Effect of Wave Frequency on the Nonlinear Interaction Between Görtler Vortices and Three-Dimensional Tollmien-Schlichting Waves**

**Márcio T. Mendonça **Centro Técnico AeroespacialSubdivisão de Propulsão - CTA/IAE/ASA-P, 12228-470, São José dos Campos, SP, Brazil

**The Pennsylvania State University Dept of Mechanical Engineering 16802 University Park, PA USA**

Laura L. Pauley

Philip J. Morris

Laura L. Pauley

Philip J. Morris

The nonlinear interaction between Görtler vortices (GV) and three-dimensional Tollmien-Schlichting (TS) waves nonlinear interaction is studied with a spatial, nonparallel model based on the Parabolized Stability Equations (PSE). In this investigation the effect of TS wave frequency on the nonlinear interaction is studied. As verified in previous investigations using the same numerical model, the relative amplitudes and growth rates are the dominant parameters in GV/TS wave interaction. In this sense, the wave frequency influence is important in defining the streamwise distance traveled by the disturbances in the unstable region of the stability diagram and in defining the amplification rates that they go through.

Keywords: Görtler vortices, Tollmien-Schlichting waves, boundary layer stability, instability, transition.

Introduction

Due to centrifugal effects, a laminar boundary layer over a concave surface may develop counter rotating longitudinal vortices called Görtler vortices (GV). These vortices develop inflectional velocity profiles that are sensitive to other types of instabilities leading to transition to turbulence. The transition may be undesirable since it increases skin friction and heat transfer rates and, if inevitable, it must be predicted with accuracy to allow, for example, for the correct design of cooling systems. Among the aerospace engineering applications where laminar flow over concave surfaces is important one can highlight the flow over the pressure side of turbine blades and the flow inside supersonic converging diverging nozzles. Besides the presence of the vortices, other types of instabilities may also be present and the nonlinear interaction between the Görtler vortices and these other instabilities may hasten the transition to turbulence. More specifically, when the curvature of the wall is small the flow may become unstable also to Tollmien-Schlichting (TS) waves which interact nonlinearly with the vortices.

Tani and Aihara (1969) presented experimental results for the interaction between GV and TS waves generated by a vibrating ribbon. They concluded that the main effect of the vortices on the TS waves is through the spanwise change in boundary layer thickness.

Nayfeh (1981) used the method of multiple scales to study the effect of GV on the development of TS waves. He found that the vortices strongly destabilize TS waves having spanwise wavelength twice the wavelength of the vortices. His results were not confirmed by Malik (1986) who used a temporal, parallel model and found an inconsistent length scale in Nayfeh's formulation. Malik (1986) found that TS waves with spanwise wavelength half the wavelength of the vortices are destabilized by the nonlinear interaction.

Srivastava and Dallmann (1987) used the method of multiple scales to study the same problem, but also allowed for TS wave amplitudes of the same order of magnitude as the vortices. Their results showed good agreement with Nayfeh's results despite the fact that Nayfeh's formulation was incorrect. This result raises doubts about their other findings.

To correct the problem in his previous paper, Nayfeh reworked his formulation and presented new results in Nayfeh and Al-Maaitah (1988). They solved the stability equations using both Floquet theory and the method of multiple scales. This time, their results agreed with Malik (1988) in the sense that resonance occurs when the spanwise wavelength of the oblique wave is half the wavelength of the vortices. They also presented some parametric studies on the effect of Reynolds number and frequency.

Malik and Hussaini (1990) extended Malik's (1986) temporal, parallel formulation to allow TS wave amplitudes of the same order of magnitude as the vortices. They studied the interaction between GV and two-dimensional TS waves and concluded that the growth rate of the TS wave is larger than the growth of the unperturbed wave. They confirmed Nayfeh and Al-Maaitah's (1988) results that interactions take place at a relatively large amplitude of the vortices. Although the model could be used for amplitudes of the waves of the same order of magnitude as the amplitude of the vortices, they only presented results for small amplitude waves.

Malik and Godil (1990) presented another paper using the same formulation used by Malik (1986). They showed that the nonlinear interaction between GV and two-dimensional TS waves leads to the development of oblique waves with a spanwise wavelength equal to that of the vortices. Again, they limited their study to small amplitude TS waves. Their results indicate that the upper branch TS waves are excited while the lower branch waves are relatively insensitive to the vortices.

All these investigations have used local models or temporal, parallel models. Local models are not suitable to study the development of GV which are governed by parabolic equations that, rigorously, can not be simplified to ordinary differential equations, except at large wavenumbers. In this way, local models have been used to study the development of TS waves in boundary layer with embedded streamwise vortices. Temporal models are not the most appropriate to describe the physics of spatially developing vortices. Besides, nonparallel effects are important for both low spanwise wavenumber vortices and three-dimensional TS waves. Only results for TS waves with small amplitudes have been presented in previous works.

Bertolotti (1991) used Parabolized Stability Equations (PSE) to investigate receptivity and nonlinear development of Görtler vortices. Among other results he presented some results for the interaction between Görtler vortices and two-dimensional Tollmien-Schlichting waves. The study considered vortices with initial amplitudes smaller than the TS waves initial amplitudes. The results showed that the nonlinear interaction leads to the development of small amplitude oblique waves with the same frequency and streamwise wavenumber as the two-dimensional wave, which in turn leads to K-type breakdown. Breakdown was defined as the onset of rapid spectrum filling. Different relative initial amplitudes of vortices and TS waves may lead to different breakdown processes.

Mendonça, Morris and Pauley (1997) used a spatial, nonparallel model to verify the conclusions obtained in previous investigations that used local or temporal, parallel models. Their model is based on the (PSE) (Bertolotti, 1991). They showed that the conclusions obtained in previous investigations are valid, but the assumption of parallel mean flow does influence the results. They also presented results for TS wave amplitudes of the same order of magnitude as the vortices which result in significant nonlinear interaction. In this case the breakdown to turbulence may be hastened. Their results show the importance of growth rates and initial amplitudes as controlling parameters in GV/TS wave interaction.

In a second paper Mendonça, Morris and Pauley (1998a) used the same spatial model based on the PSE to investigate the effect of Görtler number and spanwise wavenumber on the nonlinear interaction between GV and two-dimensional TS waves. They showed that it is not possible to isolate the effects of initial amplitude, growth rate, Görtler number and wavenumber. This controlling parameters are interrelated and the nonlinear interaction is strongly dependent on the relative amplitude of the vortices and TS waves. Two types of interactions have been identified; if the TS wave amplitude is of the same order of magnitude as the vortices, the growth of the mean flow distortion (MFD) and of the vortices higher harmonics are greatly enhanced. If the vortices are stronger than the TS wave, the vortices damp the development of the TS wave. These two different types of nonlinear interaction have been called "Type I" and "Type II" interactions.

The effect of wave frequency on the interaction between GV and two-dimensional waves has been studied by Mendonça, Morris and Pauley (1999). They concluded that the longer the path of the TS wave under the unstable region of the TS wave stability diagram, the stronger the disturbance and the higher the nonlinear interaction with the vortices. As observed in previous studies, when the TS wave amplitudes are of the same order of magnitude as the vortices, very strong nonlinear interaction takes place, resulting in earlier breakdown to turbulence or strong destabilization of the vortices.

This paper expands the results presented by Mendonça, Morris and Pauley (1999). It uses the same spatial, nonparallel model to study the influence of frequency on the nonlinear interaction between GV and three-dimensional TS waves. Again, the model allows TS wave amplitudes of the same order of magnitude as the vortices so that the influence of the TS waves on the development of the vortices can be accounted for.

Formulation

The coordinate system used in the present work is the same coordinate system presented by Floryan (1980). It is based on the streamlines *(*y* ^{*})* and potential lines

*(*f

*of the inviscid flow over a constant radius of curvature wall. This coordinate system has the advantage of producing a decay of the curvature away from the wall; at the wall it is surface oriented, but away from the wall it approaches a Cartesian system. In the normal direction a transformation is applied in order to cluster grid points close to the wall and to map the domain such that as y*

^{*})^{*}goes to infinity y goes to 1.

The Navier-Stokes equations for an incompressible flow of a Newtonian fluid are simplified by assuming that the dependent variables are decomposed into a mean component and a fluctuating component as follows:

where * [u ^{*},v^{*},w^{*}]^{T}* is the velocity vector and

*p*is the pressure. The superscript * indicates dimensional variables.

^{*}The equations are nondimensionalized using d* _{0}^{*}* and

*U*

^{*}_{¥}

*as the length and velocity scaling parameters, where d*

^{ }*n*

_{0}^{* }=(^{* }f

_{0}^{* }U^{*}_{¥}

*is the boundary layer thickness parameter,*

^{ })^{1/2}*U*

^{*}_{¥}

*is the free stream velocity, f*

^{ }*is a reference length taken as the streamwise location where initial conditions are applied, and n*

_{0}^{*}^{*}is the kinematic viscosity.

Floryan (1980) derived the equations for the zeroth order and first order approximations for the mean flow and for the perturbation quantities. He concluded that for the zeroth order approximation the mean flow equations reduce to the Prandtl boundary layer equations for the flow over a flat plate. The only remaining curvature term for the perturbation equations zeroth order approximation is the term in the momentum equation in the normal direction given by:

Go is the Görtler number, *k ^{*}* is the curvature of the wall, and

*Re*is the Reynolds number.

The resulting momentum and continuity equations may be written in vector form as:

where F*’=[u’,v’,w’,p’] ^{T}*, and the expressions for the coefficient matrices can be found in Mendonça(1997).

The boundary conditions are given by:

The boundary condition for pressure at the wall is given by the momentum equation in the normal direction applied at y *=0* _{y =0}.

Parabolized Stability Equations

The governing equations for the perturbation variables may be simplified, leading to the Parabolized Stability Equations (PSE) developed by Herbert and Bertolotti (Bertolotti, 1991). The resulting set of equations describes the spatial evolution of disturbances, and allows nonparallel, nonlinear effects to be accounted for without the heavy computational demands of a direct numerical simulation. The simplifications leading to the PSE are presented below.

The set of equations represented by Eq. (3) are elliptic and the perturbations propagate in the flow field as wave structures. The governing equations can be simplified if the wavelike nature of the perturbations are represented by their frequency w, a and b, and growth rate g. The perturbation F’ is assumed to be composed of a slowly varying shape function F_{n,m} and an exponential oscillatory wave term c_{n,m}. It is represented mathematically as a Fourier expansion truncated to a finite number of modes:

where F* _{n,m}(*f

*,*y

*)=[u*is the complex shape function vector, and

_{n,m },v_{n,m },w_{n,m }, p_{n,m}]^{T}This procedure is similar to a normal mode analysis but, in this case, the shape function F_{n,m} is a function of both f and y.

The streamwise growth rate g_{n,m}, the streamwise wavenumber a, and the spanwise wavenumber b are nondimensionalized using the boundary layer thickness parameter d* _{0}^{*}*. The frequency w is nondimensionalized using the free stream velocity

*U*

^{*}_{¥}

*and the boundary layer thickness parameter d*

^{ }

_{0}^{*}.

For linear problems only the fundamental mode is significant. With the growth of the amplitude of the fundamental, higher harmonics become significant as well as the mean flow distortion (MFD*) n=0, m=0.* As the nonlinearities become stronger, higher harmonics may be considered by increasing the number of modes N, M in the truncated Fourier expansion. The form of *a _{n,m}* (Eq. 8) reflects thefact that the phase speed of higher harmonics should be the same as the phase speed of the fundamental to avoid dispersion of the wave structure.

The perturbation variable F’, as defined in Eq. (6), is substituted in Eq. (3). The equation is then simplified by assuming that the shape function, wavelength, and growth rate vary slowly in the streamwise direction. In this way, second order derivatives and products of first order derivatives may be neglected in the streamwise direction.

After substitution of these terms into Eq. (3) and with a harmonic balance of the frequency, a set of coupled nonlinear equations is obtained. For each mode (*n,m*) the equation is given in vector form by:

where the coefficient matrices can be found in Mendonça (1997).

The resulting equations are parabolic in f and the solution can be marched downstream given initial conditions at a starting position f_{0}. This is true as long as the instabilities are convective instabilities such that they propagate in the direction of the mean flow and do not affect the flow field upstream.

The pressure gradient in the streamwise momentum equation also makes the system of equations nonparabolic. For incompressible flow Malik and Li (1993) suggest that sufficiently large steps in the streamwise direction will avoid the elliptic behavior of the problem. They also show that dropping the pressure gradient term altogether does not change the results for the level of approximation given by the PSE. In the present model the pressure gradient term is not included.

The boundary conditions for Eq. (9) are derived from Eqs. (4) and (5). At the wall, homogeneous Dirichlet no-slip conditions are used. In the far field, Neumann boundary conditions are used for the velocity components and a homogeneous Dirichlet condition is used for pressure.

For the parabolic formulation, it is necessary to specify initial conditions at a starting position f_{0} downstream of the stagnation point at the leading edge of the curved plate. For TS waves the initial conditions are obtained from the solution of the eigenvalue problem posed by the Orr-Sommerfeld equation. For the GV the initial conditions are also given by a local normal mode analysis.

Although the use of local analysis for Görtler vortices can only be formally justified for high spanwise wavenumbers, the resulting velocity distribution is hydrodynamically possible and represent physically plausible initial profiles for streamwise vortices. Day, Herbert and Saric (1990) have shown that there is good agreement between marching solutions and normal mode eigenfunctions, and that different initial conditions collapse into the same eigenfunctions as they progress downstream.

Lee and Liu (1992) studied the development of linear and nonlinear Görtler vortices and presented comparisons with experimental results. The initial conditions for the spatial parabolic marching were obtained from a local normal mode analysis. Their results compare very well with experimental data. They state that the initial local analysis can be considered at least as a curve fit for the unknown spanwise and normal velocity components to the given streamwise velocity distribution and fixed spanwise wavenumber and Görtler number.

Since the nonlinear development of Görtler vortices depends on the upstream history of the flow, the results presented in this study are restricted by the choice of initial velocity distribution. The exact specification of initial conditions for Görtler vortices would depend on the solution of the receptivity problem.

Normalization Condition

The splitting of F*’(*f* ,*y*, z, t)* into two functions, F* _{n,m}(*f

*,*y

*)*and c

*f*

_{n,m}(*,*y

*, z, t)*, is ambiguous, since both are functions of the streamwise coordinate f. It is necessary to define how much variation will be represented by the shape function F

*f*

_{n,m}(*,*y

*)*, and how much will be represented by the exponential function c

*f*

_{n,m}(*,*y

*, z, t)*. This definition has to guarantee that rapid changes in the streamwise direction are avoided so that the hypothesis of slowly changing variables is not violated. To do this, it is necessary to transfer fast variations of F

*f*

_{n,m}(*,*y

*)*in the streamwise direction to the streamwise complex wavenumber

*a*f

_{n,m}(*)=*g

*f*

_{n,m}(*)+in*a

*(*f

*)*. If this variation is represented by b

_{n,m}, for each step in the streamwise direction it is necessary to iterate on

*a*f

_{n,m}(*)*until b

_{n,m}is smaller than a given threshold. At each iteration k,

*a*f

_{n,m}(*)*is updated according to:

The variation b_{n,m} of the shape function can be monitored in different ways. In the present implementation the following is used:

where ' is the complex conjugate of The integral of | || |_{ }^{2} was used to assure that the variation is independent of the magnitude of .

Numerical Method

The system of parabolic nonlinear coupled equations given by Eq(9) is solved numerically using finite differences. The partial differential equation is discretized implicitly using a second order backward differencing in the streamwise direction, and fourth order central differencing in the normal direction. The resulting coupled algebraic equations form a block pentadiagonal system which is solved by LU decomposition.

To start the computation a first order backward differencing is used. The first order approximation is also used in a few subsequent steps downstream in order to damp transients more efficiently. For the points neighboring the boundaries, second order central differencing in the normal direction is applied.

The nonlinear terms are evaluated iteratively at each step in the streamwise direction. The iterative process is used both to enforce the normalization condition and to enforce the convergence of the nonlinear terms. A Gauss-Siedel iteration with successive over-relaxation is implemented. The nonlinear products are evaluated in the time domain. To do this, the dependent variables in the frequency domain are converted to the time domain by an inverse Fast Fourier Transform subroutine. The nonlinear products are evaluated and the results are transformed back to the frequency domain.

The complex wavenumber is updated at each iteration according to Eq.(10), and the variation in the shape function is monitored through Eq.(11). The iteration is considered converged when the normalization condition is no larger than a given small threshold. In the present implementation this threshold is 10^{-8}.

Code Validation

A comparison between the experimental results from Kachanov and Levchenko (1984) for subharmonic breakdown and PSE results has been conducted to validate the numerical procedure. The subharmonic breakdown is characterized by the nonlinear interaction between a finite amplitude two-dimensional TS wave and small amplitude three-dimensional waves with half the frequency of the 2D TS wave. The starting conditions are:Re=400, frequency w_{2,0}=0.00439, spanwise wavenumber b_{1,1}=0.1333, frequency w_{1,1}=0.0248 and initial amplitudes e_{2,0}=0.000439%, , and e_{1,1}=0.000039%.

Figure 1 presents a comparison between the PSE results and the experimental results from Kachanov and Levchenko (1984) for the amplitude of different harmonics. It shows that the PSE is able to reproduce the development of all harmonics with good accuracy. Joslin, Street and Chang (1993), showed that the small differences between the experimental results from Kachanov and Levchenko (1993) and the PSE computational results observed for higher harmonics can be attributed to small differences between the experimental conditions reported by Kachanov and Levchenko and the actual experimental conditions. Those differences were due to a small streamwise pressure gradient and a larger frequency.

Good comparisons were also obtained with numerical results from Bertolotti (1991) for K-type breakdown and with numerical and experimental results from Malik and Li (1993) and Swearingen and Blackwelder (1987) respectively for nonlinear GV development.

Results

In this section the effect of TS wave frequency on the nonlinear interaction between GV and three-dimensional TS waves is studied. The following computational parameters are used in the calculations: the number of grid points used in the normal direction is 250 with 200 grid points clustered inside the boundary layer region, the step size *dx* is 10, the number of Fourier modes N in the streamwise direction is 6 and in the spanwise direction M is 5 (given the symmetry conditions, a total of 143 modes are considered, but only 42 modes are stored). For a typical case, 180 steps in the streamwise direction takes 133.1 minutes of CPU time, with 8.5 seconds per iteration on an IBM RS6000 workstation Model 560.

The following test cases consider vortices specified by Go/Re^{3/2}=6.25x10^{-4}, which corresponds to a constant wall curvature for every test case, b=b/Re^{1000}=0.1 with an initial amplitude e_{GV}=0.5%, interacting with TS waves of different frequencies. The TS waves initial amplitude is e_{TS}. Four different frequencies are considered: F=w/Re× 10^{6}=40,50,75 and 100. The starting streamwise positions are defined by the lower branch of the neutral curve and are given, with respect to the frequencies above, by Re=660, 600, 490 and 425. The vortices and the TS waves are followed to a streamwise position past the upper branch of the neutral curve. Both fundamental resonance and subharmonic resonance are investigated.

Before investigating the effect of wave frequency on GV/TS waves interaction it is helpful to look at the general conclusions about GV/3D-TS wave interaction. A study on the nonlinear interaction between GV and three-dimensional TS waves was presented by Mendonça, Morris and Pauley (1998b). For fundamental resonance the nonlinear interaction results in the following: the nonlinear interaction has little effect on the development of the fundamental modes; a Fourier spectrum broadening is observed, resulting in the development of Fourier modes that would not grow without interaction (e.g. modes (1,0), (1,2), (2,0), (2,2), etc.); the development of mode (0,2) is governed by the development of the TS wave; the development of the MFD (mode (0,0)) is governed by the development of either the vortices or the TS waves, depending on which one results in the stronger MFD. For subharmonic resonance, with the spanwise wavenumber of the vortices two times the spanwise wavenumber of the TS waves, the nonlinear interaction results in a strong effect on the development of the GV. Again, the development of the MFD is governed by either the vortices or the TS waves, depending on which one results in the stronger MFD. The TS dominance on the development of mode (0,2) for fundamental resonance and the strong effect on the development of the GV for subharmonic resonance are due to the fact that the evolution of three-dimensional TS waves results in a strong growth of longitudinal vortices with spanwise wavenumber two times the wavenumber of the fundamental TS wave.

The results for the effect of wave frequency on GV/3D-TS waves interaction for fundamental resonance are presented in Figures 2 through 5. The general characteristics of fundamental resonance described above are observed for all wave frequencies considered. The figures show the development of the Fourier modes (0,0), (0,1), (0,2) and (1,1) due to the nonlinear interaction (symbols) and due to the development of the GV (solid lines) and TS waves without interaction (dashed lines). The variation of the TS wave frequency changes the strength of the TS waves and the relative amplitude of the vortices and TS waves. Increasing the frequency from F=40 to F=50 the amplitude of the TS wave with respect to the amplitude of the GV increases. As a consequence the dominance of the TS wave over the development of the MFD increases. From F=50 to F=75, again, the amplitude of the TS wave with respect to the amplitude of the vortices increases, and the dominance of the TS wave over the development of the MFD increases. On the other hand, increasing the frequency to F=100 results in a decrease in the amplitude of the TS wave. But since the development of the MFD distortion of this disturbance is stronger than the development of the MFD due to the vortices, the TS wave dominates the development of the MFD due to the interaction. In all four cases the nonlinear mechanism that controls the growth of mode (0,2) is governed by the development of the TS wave.

The results for subharmonic resonance are presented in Figures 6 through 9. These figures show the development of modes (1,1), (0,2) and (0,4). Again the general conclusions described above for the nonlinear subharmonic interaction between GV and TS waves are observed. The TS wave is responsible for the development of longitudinal vortices with the same spanwise wavelength of the GV. The stronger these longitudinal vortices are the stronger is the nonlinear interaction. Increasing the frequency from F=40 to F=50 the amplitude of the TS wave with respect to the amplitude of the GV increases and the strength of mode (0,2) due to the development of the TS waves also increases. As a result the nonlinear effect on the development of the GV (mode (0,2)) and on its first harmonic (mode (0,4)) is stronger. For F=75 the final TS wave amplitude is even closer to the amplitude of the GV and the strength of mode (0,2) due to the development of the TS wave is also stronger. As a consequence, the nonlinear effect on the development of the GV and mode (0,4) is stronger. Increasing the frequency to F=100 results in a reduction in the final amplitude of the fundamental TS wave and a reduction on the development of mode (0,2) due to the TS waves. That results in a reduction in the nonlinear effect on the development of the GV.

Discussion

The results for the interaction between GV and two-dimensional TS waves can help interpret the present results. For two-dimensional TS waves the results from Mendonça, Morris and Pauley (1998a) indicate that the most important controlling parameter in GV/TS wave interaction is the relative amplitude of the vortices and TS waves. The relative amplitude is determined by the initial amplitudes and growth rate of the disturbances. In the instability diagram presented in Figure 18 a given TS wave follows a line of constant frequency F as it travels downstream. For two-dimensional TS waves (b=0) the higher the frequency F the weaker the TS wave. The TS wave travels a shorter streamwise distance in the unstable region in the stability diagram and is subject to lower growth rates. The weaker the TS waves, the stronger the dominance of the vortices in the nonlinear interaction. Lower frequencies result in stronger two-dimensional waves which may grow to amplitudes of the same order of magnitude as the vortices, resulting in a TS wave dominance over the nonlinear interaction.

For three-dimensional TS waves (b¹0), the relative amplitude of the disturbances is also one of the dominant parameters. But for three-dimensional waves the instability diagram shows that the unstable region may define a closed region as seen in Figure 18. In this case, there is a range of frequencies for which the TS waves are unstable. frequencies outside this range, lower or higher, will result in no disturbance growth. There is a given frequency that results in the strongest growth of the TS waves. Lower or higher frequencies will result in weaker TS waves, and weaker TS wave effect on the development of the vortices. For the range of frequencies studied in this investigation the frequency F=75 results in the strongest TS wave and in the strongest TS wave effect on the nonlinear development of the vortices. Higher and lower frequencies result in weaker TS waves.

Conclusions

As observed in previous investigations the relative amplitude of the vortices and TS waves is one of the most important parameters in GV/TS wave nonlinear interaction. In this way, the most important effect of TS wave frequency is the definition of the total amplification rate of the disturbance. Three-dimensional TS waves are characterized by stability diagrams that may define a closed unstable region. In these cases there is a certain frequency that results in the strongest growth. Increasing or decreasing this frequency will result in weaker TS waves and weaker TS wave effect on the nonlinear interactions. This conclusion is valid both for fundamental resonance and subharmonic resonance.

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Article presented at the 1st Brazilian School on Transition and Turbulence, Rio de Janeiro, September 21-25, 1998.

Technical Editor: Atila P. Silva Freire.