## Abstract

This paper presents studies of a three-dimensional secondary instability of a spatially developing von Kármán vortex street. It develops owing to the nonlinear interaction between a two-dimensional mean far-wake flow and its most unstable disturbances. This forms a nonlinear primary wake flow. Sections of this flow are selected to perform a temporal secondary stability study under the assumption of parallel flow. The eigenvalue characteristics of the secondary instability are compared with the results from the use of a linear primary flow comprising unmodified mean wake flow coexisting with a linear primary fundamental disturbance with an empirical amplitude as a parameter, resulting in a simpler Floquet analysis. The maximum amplification rates occur at about the same spanwise wavenumber for both the nonlinear and linear primary flows, in qualitative agreement. But the amplification rate versus the spanwise wavenumber spectrum are both qualitatively and quantitatively different, the nonlinear primary flow results in a lower magnitude of the amplification rates. Some interpretations of controlled experiments are made, and it is concluded that the two- and three-dimensional disturbances so obtained appeared to be from the primary instability, where the amplification mechanisms come from the unmodified mean flow. A general discussion of the nonlinear interaction between the primary two-dimensional flow and the three-dimensional secondary instability is given, which may well form the basis for further nonlinear studies.

## 1. Introduction

Large-scale coherent structures have been observed in many types of shear flows, particularly free flows such as wakes. Three-dimensional spanwise structures are believed to result from a secondary instability of a prevailing two-dimensional flow; the latter includes a two-dimensional primary instability that dominates over the primary three-dimensional one. The primary flow is susceptible to a three-dimensional secondary instability mechanism, which is the topic of the present paper. Its study can help us better understand the development of three-dimensional disturbances from a two-dimensional nonlinearly unstable primary flow as a prelude to transition to turbulence.

Studies of the two-dimensional primary linear and nonlinear regions in wakes have been performed by many investigators, both theoretically (Mattingly & Criminale 1972; Ko *et al.* 1970) and experimentally (Sato & Kuriki 1961; Sato & Saito 1975). Recently, by performing direct numerical simulations on the transition mechanism in a spatially developing two-dimensional wake flow, Maekawa *et al.* (1992) studied the effects of different types of forcing. Three-dimensional primary disturbances are always weaker than two-dimensional ones in the incipient transition region, supported by measurements (Sato & Kuriki 1961) and by an extension of Squire’s theorem (Squire 1933), which is valid for temporal disturbances, to spatially developing instabilities. These two-dimensional studies provide a valuable basis for the present nonlinear primary flow.

The observations of three-dimensional coherent structures in wake flows were made by Cimbala (1984) and Cimbala *et al.* (1988). They used smoke-wire flow visualization and hot-wire anemometry to study the near and far wakes of two-dimensional bluff bodies. They suggested that the three-dimensional structure is a result of a secondary instability arising from the interaction of spanwise oblique disturbances with the two-dimensional periodic vorticity structures. Fleming (1987) applied Floquet theory in the linear analysis of a secondary instability of the far wake, subjected to a linear primary flow in the sense already discussed. He found that a three-dimensional subharmonic instability can occur as the primary disturbance amplitude reaches an assigned amplitude of about 15 per cent (at which amplitude, the assumption of a linear primary flow would be violated). Other similar applications of Floquet analysis to the wake of a cylinder were reported by Noack & Eckelmann (1994) and Barkley & Henderson (1996). The study of the instability of primary flows that are modulated in time and space (in the streamwise direction) began with the pioneering work of Kelly (1967) and Herbert (1988). Herbert also reviews earlier work on the secondary instability of wall-bounded flows.

Corke *et al.* (1992) experimentally introduced into a far wake controlled three-dimensional subharmonic wave pairs of equal but opposite sign, simultaneous with a two-dimensional wave. They found that there exists, in their words, a resonance between a fundamental two-dimensional mode and subharmonic three-dimensional modes. The experimental threshold amplitudes were approximately 1.3 and 0.3 per cent for the J- and A- conditions for three-dimensional subharmonic development, much smaller than those in Fleming’s (1987) analysis, which were typically 10–15%. A possible explanation of the discrepancy on part of the linear primary flow case is the neglect of the important mechanism of distortion of the mean flow and of the primary perturbation neglected in Fleming’s (1987) analysis, but which is of central importance in nonlinear hydrodynamic instabilities (Stuart 1960).

Of related importance is the work of Stuart (1962*a*,*b*) on three-dimensional nonlinear interaction effects on the stability of parallel flows. A brief discussion is warranted to render its relation to the present work. Denote (*m*,*n*) as the combination of interacting two- and three-dimensional instability modes, where *m* is an integer and multiple of the basic streamwise wavenumber *α*, and *n* is a multiple of the spanwise wavenumber *β*. The nonlinear interaction problems described by Stuart (1962*a*,*b*) after Fourier representation include groups (i) through to (viii) in that paper. The (0,0) problem is the mean motion, and its accompanying Reynolds stresses (1,0) denotes the basic nonlinear two-dimensional mode, (1,1) denotes the basic nonlinear three-dimensional mode with the same streamwise wavenumber as that of the two-dimensional mode, higher harmonics are represented by the (2,0), (2,2) and (2,1) problems, representing nonlinear effects among the modes; the (0,2),(0,1) problems are the three-dimensional harmonics that are not modulated by the streamwise wavenumber. In the present work, the nonlinear two-dimensional primary flow is numerically computed and is similar in spirit to Stuart’s (1962*a*,*b*) problems (i) for (0,0), (ii) for (1,0) and (iv) for (2,0) and for still higher harmonics. The numerical results for the nonlinear primary flow in the present work is Fourier analysed, and it is found that it is adequately represented by the fundamental two-dimensional component and the mean flow; in this case, both are modified by their mutual strong nonlinear interactions. This was the point argued and anticipated by Stuart (1958) in nonlinear hydrodynamic stability problems, which appear to have generalities beyond weakly nonlinear theory in hydrodynamic stability of parallel flows. The present linear secondary instability problem is equivalent to Stuart’s (1962*a*,*b*) group (iii) for (1,1), but which is extended to include the subharmonic (1/2,1). In Stuart (1962*a*,*b*), it is argued that the streamwise phase velocities of (1,0) and (1,1) are not necessarily equal; in the second-order theory of Benney (1961), they were set equal to each other for simplicity. In the simultaneously imposed two- and three-dimensional disturbances in the wake flow, Corke *et al.* (1992) measured their respective phase velocities and found that they rapidly approach each other and remained so downstream. In the present analysis, the phase velocities of the fundamental primary disturbance and that of the three-dimensional disturbance are thus taken to be equal.

## 2. Formulation of the nonlinear three-dimensional secondary instability of two-dimensional unstable primary flow

In the general formulation, nonlinear interactions between the secondary instability and the primary flow will be taken into account. The primary flow consists of a two-dimensional mean flow with a nonlinear two-dimensional primary instability. The schematic of the mean flow, characterized by the Goldstein (1930) wake, and the mean-flow nomenclature are given in appendix A of the electronic supplementary material.

We begin with the dimensionless continuity and the Navier–Stokes equations for an incompressible fluid, where the velocities ** u**, coordinates, and time

*t*are made dimensionless by the free-stream velocity

*U*

_{0}, and the wake half width

*b*

_{0}at the streamwise location

*x*

_{0},

*Re*=

*U*

_{0}

*b*

_{0}/

*ν*is the Reynolds number and

*ν*is the kinematic viscosity. The pressure is made dimensionless by the free-stream dynamical pressure. The subscript notation follows that of Herbert (1988), and the following representation is made:2.1where

*q*is a total flow quantity

**,**

*u**p*. The secondary instability is denoted by

*q*

_{3}, which is three-dimensional and spanwise periodic. The two-dimensional primary flow is denoted by the sum

*q*

_{2}=

*q*

_{0}+

*q*

_{1}, where

*q*

_{0}(

*x*,

*y*) is the steady, mean flow and

*q*

_{1}(

*x*,

*y*,

*t*) is the primary instability. These will be given further discussion after the general formulation. Upon substituting equation (2.1) into the continuity and Navier–Stokes equations and spanwise

*z*-averaging, denoted by an overbar, the two-dimensional primary flow equations are obtained,2.2and2.3In shorthand notation, the second vector velocity in the Reynolds stresses is indicative of the vector representation of the primary flow and thus becomes a scalar in the component form of equation (2.3); while the divergence operates on the vector quantity formed by the first of the velocities. Because of the presence of

*u*_{1}, the primary flow is modulated in both (

*x*,

*t*).

Upon subtracting equations (2.2) and (2.3) from the respective full continuity and Navier–Stokes equations, the nonlinear form of the secondary instability equations is obtained. In this form, *u*_{2} is the advection velocity,2.4and2.5The shorthand notation is again applied to the excess Reynolds stresses in equation (2.5). For finite amplitude *u*_{3}, the two-dimensional nonlinear system (2.2), (2.3) and the three-dimensional nonlinear system (2.4), (2.5) are interacting and are coupled through the Reynolds stresses and the advective effects (*u*_{2}⋅∇)*u*_{3} and −(*u*_{3}⋅∇)*u*_{2}. They describe the nonlinear interaction between the spanwise-averaged two-dimensional mean field *u*_{2},*p*_{2} and the three-dimensional spanwise-fluctuation field *u*_{3},*p*_{3}. The source of the hydrodynamic instability of *u*_{3} lies in its advection of momentum sources coming from *u*_{2}.

## 3. Generalities of the linear secondary instability formulation. Distinctions between choices of the primary flow

In the event that |*u*_{3}|≪|*u*_{2}|, the secondary instability system follows from equations (2.4) and (2.5) and reduces to the familiar linearized form in Herbert (1988). In this case, the primary flow (*u*_{2},*p*_{2}) is independent of the secondary instability problem and is a self-standing input.

The primary flow in the case of the linear secondary instability is obtained from equations (2.2) and (2.3) by omitting the Reynolds stresses of the linear secondary instability. As such, it is described independently by the two-dimensional Navier–Stokes equations prior to assumptions and simplifications.

There are two avenues in describing the primary flow. One is termed the *linear primary flow*, where *q*_{0} is a given local parallel shear flow and *q*_{1} is the most unstable linear eigensolution. These are non-interacting. The sum in this case, *q*_{2}=*q*_{0}+*q*_{1}, involves an assigned constant amplitude *A* for the normalized *q*_{1} (Herbert 1988). The secondary instability properties *q*_{3} are then assessed as a function of increasing values of the empirical parameter *A*. This results in a simple Floquet theory applied to the far-wake flow performed by Fleming (1987) and is briefly discussed in appendix A in the electronic supplementary material.

The other avenue is termed the *nonlinear primary flow*. It comes from the realization that for finite amplitude (e.g. *A*=0.10), *q*_{1} and its interactions with *q*_{0} become the essential nonlinear mechanism (Stuart 1960) in determining the nonlinear primary flow *q*_{2}, even as the secondary instability *q*_{3} is incipiently linear. The features of the nonlinear primary flow are addressed in §4, which forms the basis of a linear secondary instability study in §5.

## 4. Summary of the nonlinear spatial two-dimensional primary wake flow

The scenarios associated with the two-dimensional nonlinear primary wake are well described by the experiments of Sato & Kuriki (1961) and theoretically in terms of a developing wake flow by Ko *et al.* (1970). Maekawa *et al.* (1992) performed direct numerical simulations of the unstable two-dimensional far wake, based on the algorithm developed by Buell (1991), related to the methodology developed at the Transition and Turbulence Group, Universität Stuttgart (e.g. Rist & Fasel 1991; Kloker & Bestek 1992). The nonlinear primary flow is here, by necessity, recomputed from Maekawa *et al.* (1992) for the desired parameter values for secondary instability studies. For details, we refer to the originators of the method. We shall only state the results of the nonlinear primary flow, which are obtained for the two representative cases.

Case I: *Re*=360.60, *λ*=0.33, *ω*=0.684, *α*_{fund}=0.791, where the subscript ‘fund’ indicates the fundamental streamwise wavenumber corresponding to the fundamental frequency *ω* of the inlet conditions. This subscript is omitted later. Case I corresponds to the experimental conditions of Cimbala *et al.* (1988), Corke *et al.* (1992) and the analysis of Fleming (1987).

Case II: *Re*=818.18, *λ*=0.55, *ω*=0.65, *α*_{fund}=0.857, corresponds to the experimental conditions of Sato & Kuriki (1961) and the nonlinear analysis of the primary flow of Ko *et al.* (1970).

In the above, the frequency and streamwise wavenumber are related via linear theory (the Orr–Sommerfeld equation) for the most unstable fundamental mode of the local mean Gaussian velocity profile. The inlet fluctuation amplitude is taken to be 2 per cent; 768 uniformly distributed grid points are taken in the *x*-direction within 0≤*x*≤200 and 128 spectral modes in the *y*-direction. The fluctuation input involves only the fundamental mode. The implementation of the numerical scheme requires about 0.6 s of the CPU per time step on the Cray YMP8/8128 for a 768×128 grid.

### (a) The von Kármán vortex street

The simulated results are shown in terms of a ‘snapshot’ of the spanwise vorticity contours in the *x*,*y*-plane in figure 1*a*,*b* for cases I and II, respectively. The vorticity contours oscillate from the inlet plane and develop into a von Kármán vortex street downstream. There are few changes in the vorticity structure further downstream, which indicates that the nonlinear primary velocity field develops periodically in the downstream direction. Quantitatively, time traces of the velocity field were taken at several locations in the wake for case I, the results (though not shown here) indicate that the wake is time periodic and that the response frequency corresponds very closely to the frequency of the fundamental mode. In the vicinity of the inlet plane, the Fourier decompositions show that the fluctuations are primarily responding to the fundamental input mode; further downstream, the harmonic modes are also present, but the fundamental mode is always the dominant mode. However, this fundamental mode is modified from its behaviour at the inlet though nonlinear interactions with the (modified) mean flow.

### (b) The nonlinear primary mean-flow velocity distribution

The development of the mean streamwise velocity is shown in figure 2 for case II. Although difficult to distinguish in the plot, the dimensionless mean velocity exceeds unity in the neighbourhood of *y*=±3.30 at the downstream station *x*=82. The overshoot feature was observed in Sato & Kuriki (1961) in the nonlinear region. The simulated nonlinear primary flow has all the symptoms of experimental observations in terms of centreline mean velocity and wake-thickness development in the two-dimensional region. These features are theoretically interpreted in Ko *et al.* (1970) and are thus not repeated here.

### (c) The nonlinear primary flow r.m.s. velocity fluctuations

Distributions of the r.m.s. primary velocity fluctuation are given in figure 3*a*–*d* for case II. The two peaks found in the *u*_{r.m.s.} distribution move away from each other as the flow develops downstream. The development of maximum is shown in figure 3*e*, it grows exponentially at small values of *x* until it ‘saturates’ as it attains the maximum value, then decays in an undulating manner. The undulating behaviour is attributable to exchange of energy between the modes and the mean motion. The further gradual decay is attributable to viscous dissipation overcoming the production mechanism. In case I (which is not shown), occurs at about *x*=47 and at *x*=35 for case II (figure 3*e*).

### (d) Choosing the local streamwise section for secondary stability analysis

The secondary instability analysis is performed for chosen sections of the nonlinear primary flow under parallel-flow assumption at a location where quasi-equilibrium occurs. For case II, the location is at *x*=150, for case I, the section chosen is at *x*=100.

The Fourier-analysed distributions of the nonlinear primary flow, obtained by numerically fitting the simulation results (e.g. the velocity components of the von Kármán vortex street, figure 1), are represented by4.1where and asterisk denotes the complex conjugate. The steady part of the mean flow comprising of *u*_{0} and *u*_{1,0} are the streamwise velocity components in the parallel-flow approximation, where *u*_{1,0} is the nonlinear modification of the mean flow. In turn, the primary perturbations *u*_{1n} (*n*>0) is modified from its inlet behaviour through interaction with the mean flow. These are the important generic mechanisms of nonlinear hydrodynamic instabilities (Stuart 1960).

The velocity amplitudes from Fourier analysis of the computed nonlinear primary flow are shown in figure 4 for case II. Higher order velocity corrections *u*_{1,n} for *n*>1 are small compared with the mean flow and the *n*=1 fundamental mode. Terms of *n*>1 are thus negligible in the performance of secondary instability analysis. The nonlinear primary flow representation reduces to the modified mean flow and modified fundamental. This again is an anticipation from earlier work on nonlinear hydrodynamic stability (Stuart 1958, 1960).

## 5. The linear secondary instability of the two-dimensional nonlinear primary wake flow

The linear secondary stability partial differential equations are obtained from equations (2.4) and (2.5) for *u*_{3},*p*_{3} in component form, with boundary conditions *u*_{3} → 0 as .

The nonlinear primary flow plays the role of advection and secondary instability momentum sources and is modulated in (*x*,*t*). It is not presumptuous to impose a single phase velocity fixed from the inlet conditions and which remains the same under the nonlinear interactions determining *u*_{2}. This amounts to the observation that nonlinear interactions modify the wave amplitude and shape of the mean flow, but that the wavy characteristics remain robustly the same as in observations and as suggested in earlier hydrodynamic stability studies (e.g. Stuart 1960). In this case, a phase velocity *c*_{r} is associated with *u*_{1}. Herbert (1988) suggests the Galilean transformation *ξ*=*x*−*c*_{r}*t* to render the primary flow independent of time, so that *u*_{2}(*ξ*,*y*)=*u*_{2}(*ξ*+*λ*,*y*), where *λ* is the streamwise wavelength of the wavy solution *u*_{2}. In this case, the secondary instability is represented in the general form (Herbert 1988)5.1where the periodicity of the amplitude function depends on the type of disturbance: for fundamental modes and for subharmonic modes. Using a truncated Fourier series, these two modes can be written for a temporal instability as5.2where are the truncated eigenfunctions to be determined, in principle, by a tuncated system of ordinary differential equations, and where *σ*=*σ*_{r}+i*σ*_{i},*γ*=*γ*_{r}+i*γ*_{i}. The spanwise wavenumber is *β*=2*π*/*λ*_{z}, and *α*=2*π*/*λ* is the streamwise wavenumber already defined. As in the Orr–Sommerfeld problem, only two of the four parameters in *σ* and *γ* are solvable as part of the eigenvalue problem, the other two are assigned. The preference here is to pick *γ*_{r} and *γ*_{i} in the following context: *γ*_{r} is the spatial amplification rate and is set equal to zero in favour of the temporal amplification rate *σ*_{r}. The detuning parameter *γ*_{i} is picked to be zero for fundamental modes and 1/2 for subharmonic modes. There remains *σ*_{i}, which is a frequency shift with respect to that of the primary instability and will be shown to be very nearly zero for the fundamental component as part of the solution in the cases considered. We study the temporally growing modes since it is less demanding and it is easy to access the eigenvalue spectra. Two uncoupled sets of equations for the fundamental components (*γ*_{i}=0) and subharmonic components (*γ*_{i}=1/2) are obtained from this procedure in the following.

The nonlinear primary instability, subjected to a similar Galilean transformation *ξ*=*x*−*c*_{r}*t*, appears in the parallel-flow form,5.3Substituting equations (5.2) and (5.3) into the linear partial differential equations for the secondary instability, summing all terms with like exponentials and setting them equal to zero, the linear ordinary differential equations for the secondary instability, subjected to a nonlinear primary flow, are obtained,5.4
5.5
5.6and
5.7where and the already defined *α*_{m}=(*m*+*γ*_{i})*α*. Terms indicated bydenote convolution. Equations (5.4)–(5.7) comprise two sets of uncoupled equations, one for the fundamental (*γ*_{i}=0) and the other for the subharmonic (*γ*_{i}=1/2). The relevant boundary conditions become5.8Equations (5.4)–(5.7) with boundary conditions (5.8) lead to an eigenvalue system and are solved by using the standard spectral method with mapping function. The physical interpretation of the instability mechanisms are discussed in §6.

### (a) The eigenvalue results

The secondary instability eigenvalue results of the nonlinear primary flow are shown in figure 5 for case II with the streamwise section taken at *x*=150. The distribution of *fundamental* eigenvalues *σ* versus *β* is shown in figure 5*a* and that of the *subharmonic* mode (*α*_{sub}=0.43) in figure 5*b*. The most unstable fundamental mode is *three dimensional*, with the growth rate of *σ*_{r}=0.014 at a spanwise wavenumber of about *β*=0.464. The two-dimensional fundamental mode, with *σ*_{r}=0, at *β*=0, is a stable mode. The frequency-shift parameter *σ*_{i} for all of the unstable fundamental modes is nearly zero and hardly visible in the plot. Thus, there is no frequency shift with respect to the primary disturbance and the three-dimensional waves are phase locked with the two-dimensional field. The most unstable growth rate of the secondary instability subharmonic mode is *σ*_{r}=0.0223 at *β*=0.81 (*θ*_{sub}=65^{°}, equation (5.9)) in figure 5*b*, which is larger than that of the most unstable fundamental mode. In addition, the three-dimensional subharmonic mode is found to be more unstable than the two-dimensional mode at *β*=0 for which *σ*_{r}=0.008. The detuning frequency or frequency shift is found to be weakly dependent on the spanwise wavenumber.

The secondary instability eigenvalue results for case I are shown in figure 6. The three-dimensional subharmonic modes (*α*_{sub}=0.40) are more unstable than the two-dimensional modes. The most unstable mode has the growth rate of *σ*_{r}=0.0078 and the frequency shift of 0.023 for a spanwise wavenumber *β*=0.64 (*θ*_{sub}=62^{°}).

### (b) Comparisons of secondary instability eigenvalue results from linear primary flow

The secondary instability analyses for a linear primary flow (Fleming 1987), for consistent comparison purposes, have been recomputed (appendix A and figures S1–S3 in the electronic supplementary material) using a similar numerical framework as that for the nonlinear primary flow. The empirically assigned amplitude, *A*, is the maximum streamwise r.m.s. disturbance of the primary flow. For the nonlinear primary flow, such an empirical amplitude was neither assigned nor was it necessary. A consistent and effective amplitude, *Ae*, would be the maximum streamwise r.m.s. disturbance of the nonlinear primary flow, obtained from figure 3*e* (case II at *x*=150), *Ae*=0.05. Similarly, case I at *x*=100 gives *Ae*=0.03. For the linear primary flow, the amplitudes selected for comparison are *A*=0.10 and 0.15 for both cases. The instabilities produced by the lower amplitudes *A* are totally insignificant.

The first comparison is the propagation angle of the three-dimensional subharmonic mode, which is more unstable than the fundamental mode. The spanwise wavenumber corresponding to the maximum amplification rate, , is denoted by . The propagation angle is calculated from5.9measured away from the *x*-axis. It is perpendicular to the wavefront; the wavefront can be interpreted as a wedge-like front advancing into the free-stream direction at half angles ± (90^{°}−*θ*_{sub}) from the *x*-axis.

*Propagation angles*. In case II, the empirical amplitude *A*=0.10 for linear primary flow is sufficiently large to produce the same result for the propagation angle (*θ*_{sub}=65^{°}) as for *A*=0.15. This propagation angle gives a wavefront inclination angle of 35^{°}. The amazing coincidence of the propagation angles for the nonlinear and linear primary flows apparently indicates that the maximum amplification rate, , regardless of its value, lies in the same vicinity of .

For case I, the amplitude *A*=0.10 in the linear primary flow case is sufficient to enable the propagation angle to mimic that of the nonlinear primary flow case (67.5^{°} compared with 65^{°}), but that at the higher *A*=0.15 the occurrence of in the linear primary flow case has shifted to a larger spanwise wavenumber so that the propagation angle lies closer to the *x*-axis, 62^{°}. The strict comparisons are not possible as the linear and nonlinear primary flows are very different in their physical implications.

*Growth rates*. In general, the results from the linear primary flow have much larger growth rates and larger empirical amplitudes, *A*=0.10–0.15, than those pertaining to the nonlinear primary flow (*Ae*=0.03–0.05). The nonlinear mechanisms of mean-flow modification by the finite-amplitude disturbances are accounted for in the nonlinear primary flow, as is the distortion of the fundamental mode. Thus, there is much less necessity to have a large local amplitude to force this issue. As the mean flow has felt the depletion of its energy in the nonlinear primary flow case, there is less energy available to supply to the disturbances, including the secondary instability, which is reflected by its lower amplification rates (–0.020). The lesser local amplitude (*Ae*∼0.03–0.05) is also due to the unavailability of the modified mean flow to supply energy to the primary disturbance.

In the case of the linear primary flow, the mean flow remains unaffected by the primary disturbance, in which case, it has infinite energy to supply to the linear primary disturbance as well as to the secondary instability. This is understood from linear hydrodynamic stability theory in light of the nonlinear theory (Lin 1955, Stuart 1960). Thus, it is no surprise that amplification rates of the secondary instability for a linear primary flow are considerably larger (–0.070).

*The σ_{r} versus β spectrum*. For case II, the nonlinear primary flow yields a three-dimensional spectrum for both the fundamental and the subharmonic secondary instability (figure 5

*a*,

*b*), with the subharmonic dominating, which is the expected case. The linear primary flow results are quite different qualitatively: the fundamental remains two dimensional and dominates over the three-dimensional subharmonic at least for

*A*=0.15, while for

*A*=0.10, the subharmonic two-dimensional and most amplified three-dimensional modes have comparable amplification rates. For case I, the fundamental is clearly two dimensional (figure S1 in the electronic supplementary material), except for the vigorous probably unrealistic

*A*=0.15 case, both two- and three-dimensional modes have comparable amplification rates. For the subharmonic in case I (figure S1 in the electronic supplementary material), increasing

*A*drives the spectrum towards three dimensionality. For the nonlinear primary flow case I, the subharmonic is three dimensional.

### (c) Interpretations of experiments pertaining to secondary instabilities in the far wake

It is not intended to extensively survey experimental wake measurements, but to select those that have a much closer relation to the present analysis than others. Observations of Cimbala (1984) and Cimbala *et al.* (1988), although for a cylinder wake, showed that the two-dimensional vortex street, which is ‘nonlinear’ eventually developed into a three-dimensional structure. They attributed the three dimensionality as a secondary, rather than primary, instability. This essentially opened the possibility of analysing the secondary instability of a basic spatially and temporally modulated primary wake flow, such as that of Fleming (1987), and the controlled experiments of Corke *et al.* (1992). However, the forcing in Corke *et al.* (1992) did not account for the possibility that the secondary instability also supports a fundamental that could be three dimensional. The forcing-wave *angles* were not explicitly defined. One can however, assume that their A-condition 60^{°} corresponds to that defined in equation (5.9) and thus corresponds to the predicted wave angle of 62^{°}. From Corke *et al.*’s fig. 9*b*, the two-dimensional fundamental mode and the three-dimensional subharmonic mode have overlapping linear regions. In the present analysis, the fundamental two-dimensional primary mode is already developed into a nonlinear stage where it interacts with the mean wake flow and where the linear secondary instability is incipient. Thus, there is somewhat of a *disparity* between the experiments and the analysis. The experimental fundamental may well correspond to the incipient linear fundamental mode of the primary two-dimensional wake in the present analysis. In this case, the instability characteristics are given by the Orr–Sommerfeld problem for the sinuous mode, where the maximum dimensionless-spatial amplification rate is about −*α*_{i}=0.06.

From fig. 9*b* of Corke *et al.* (1992), the open-circled data in the linear region gives a dimensional amplification rate of about 0.021 mm^{−1}. The intent of the experiments is to force the instability at less than the most amplified mode. Using their local-wake half-thickness of *b* approximately 2.2 mm at the *x* approximately 110 mm station and constant free-stream velocity U_{0}≈2.59 m s^{−1} and forcing frequency of 100 Hz gives a value of the dimensionless amplification rate of about −*α*_{i}=0.045. The corresponding dimensionless frequency is 0.40, which gives a theoretical Orr–Sommerfeld amplification rate of about −*α*_{i}=0.045, which is identical to that obtained from the experiments.

This value is less than the theoretical maximum of about −*α*_{i}=0.06, which corresponds to *ω*=0.70 of the Orr–Sommerfeld results for the sinuous mode. The experimentally forced three-dimensional subharmonic mode, which coexists with the linear two-dimensional fundamental over a lengthy streamwise region, may well be a forced primary three-dimensional mode and not necessarily the secondary instability of the primary fundamental plus mean wake flow as envisioned in Fleming (1987) as well as in the present analysis. Their amplification rates (Corke *et al.*’s fig. 9*b*) are comparable and thus comparisons with the theoretical secondary instability amplification rates, which are consistently much smaller, may not be meaningful.

Two important lessons from Corke *et al.*’s experiment can be drawn. One is that the streamwise wave speeds of the two-dimensional fundamental and that of the three-dimensional subharmonic become equal, even in the linear region and long before their subsequent nonlinear interactions take place. The other is that in the nonlinear interaction region, counter-undulations of the respective wave envelops of the two- and three-dimensional disturbances take place. This is strongly reminiscent of nonlinear interactions, represented by wave envelopes, in other nonlinear hydrodynamic instability problems (e.g. Liu 1988; Nikitopoulos & Liu 2001) where there are interactive give and take energy exchanges as the flow develops downstream. This interesting experimental revelation provides the incentive to study such nonlinear interactions, some gross features of which are described in §6.

In the experiments of Julien *et al.* (2003), a forced wake flow behind a flat plate results in the streamwise development of a periodic Kármán vortex street. They found that the streamwise wavelength of the three-dimensional instability is of the same order as the basic primary flow. No subharmonic modes were addressed. It is well known from earlier experiments on streamwise vortices (e.g. Swearingen & Blackwelder 1987; Saric 1994) that in the plane of the flow (the *x*,*z*-plane), varicose and sinuous instability modes exit, in the spirit of Lord Rayeigh, and that such instabilities are indeed convective. This has been found to be the case in Julien *et al.* (2003), however their mode 1 is actually the varicose mode in the *y*,*z*-plane and their mode 2 the sinuous mode. These are more reminiscent to primary instabilities. In the present analysis, as in other works on secondary instability of streamwise vortices (e.g. Yu & Liu 1991, 1994; Saric 1994; Girgis & Liu 2002), the sinuous mode in the *x*,*z*-plane appears as the incipient (and subsequent) dominant mode. Julien *et al.* measured a spatial amplification rate of about 0.37 cm^{−1}; using their initial wake thickness of 0.4 cm, this gives a dimensionless amplification of about 0.14. To convert this into a dimensionless temporal amplification rate (Gaster 1962), they estimated a group velocity to be about 0.8 that of the free-stream velocity, thus giving *σ*_{r}≈0.1. This is much larger than that estimated by the computational model of Julien *et al.* (2004) and that given in appendix A in the electronic supplementary material as well as the results of the present paper. One can speculate that the measured secondary instability in Julien *et al.* (2003) may well be a primary instability and, although three dimensional, the subharmonic is not involved. The temporal model of Julien *et al.* (2004) used a Bickley wake to start the nonlinear time-dependent parallel mean-flow computation. The primary flow develops into a temporal (but not spatially developing) von Kármán vortex street. The temporal linear secondary instability of this primary nonlinear flow is then computed, which yields a zero phase velocity. The domain is limited to the streamwise wavelength of the primary flow and the subsequent symmetry conditions imposed rule out the possibility of subharmonics, which is of central importance (Fleming 1987; Corke *et al.* 1992 and the present paper). The normal domain is also taken as periodic. It is difficult therefore to reconcile with the experimental situation and when the consequences of subharmonic interaction with a basic nonlinear two-dimensional flow are sought.

## 6. Physical mechanisms of the three-dimensional secondary instability developing from two-dimensional primary unstable flow

The physical mechanisms are best understood by considering the energy exchanges between the primary and secondary instability flows. The kinetic energy of the two-dimensional primary flow is obtained from equation (2.3) using equation (2.2),6.1The kinetic energy per unit mass (in dimensionless form) is written as *u*_{2}⋅*u*_{2}/2. The physical mechanisms of energy exchange on the right-hand side are identified. The energy conversion mechanisms involve the Reynolds stresses of the secondary instability doing work against the rates of strain of the primary flow; the first two terms in the square bracket represent exchange between and , whereas the last two represent exchanges between and . The pressure work against the normal rates of strain through *p*_{2}∂*u*_{2}/∂*x* and *p*_{2}∂*v*_{2}/∂*y* sums to zero for an incompressible fluid in the overall energy balance, but in the internal balance it has the tendency to isotropize and . The rate of viscous dissipation of the primary flow is symbolically represented by *Φ*_{2}.

The kinetic-energy equation for the three-dimensional secondary instability, *u*_{3}⋅*u*_{3}/2, obtained from equation (2.5), after spanwise averaging is6.2The energy exchange between the primary flow and the secondary instability, which is the first term on the right-hand side of equation (6.2), appears as equal but opposite in sign to that appearing in equation (6.1). The first and third terms in the exchange mechanism are the work done by the normal Reynolds stresses of the secondary instability against the normal rates of strain of the primary flow; the second group is the work done by the Reynolds shear stresses against the shear rates of strain.

In an unmodulated primary shear flow, the only energy-conversion mechanism comes from the Reynolds shear stress of the three-dimensional disturbance doing work against the rate of strain of the two-dimensional flow, , which plays the only role towards instability development. Since expressing the advection of *u*_{3}⋅*u*_{3}/2 by *u*_{2} is the preferred form on the left-hand side of equation (6.2), the nonlinear self-advection effect would then appear as the third term on the right-hand side of equation (6.2); together with , they are referred to broadly as ‘self transport’ of energy by the fluctuations and do not contribute to net energy transfer; similarly with the viscous-diffusion effect. The rate of viscous dissipation of the secondary instability energy is written in short as .

It is clear from equation (6.2) that reference to the ‘resonance’ generation (Corke *et al.* 1992) of the three-dimensional secondary instability is meant that the net (volume-integrated) energy-exchange mechanism is positive and overwhelming the effects of viscous dissipation. In order for this to happen, the net effect of energy exchange has to favour the instability to begin with and thus depends on the relative phase between the Reynolds stress and the appropriate rate of strain that it is working against.

In equations (6.1) and (6.2), it is noticed that only exchanges energy with , whereas receives its sustainable development from the isotropizing mechanism of pressure velocity strain,The origin of *w*_{3} is explanable from the stability analysis of §5, which provides the possibility that three-dimensional disturbances could develop within a band of spanwise wavenumbers surrounding the most amplified mode (at the same frequency and streamwise wavenumber of the primary disturbance) from which upstream or free-stream disturbances are selected to amplify. This is similar to the two-dimensional disturbance problem provided by the Orr–Sommerfeld equation (Lin 1955).

## 7. Concluding remarks

The three-dimensional secondary instability of a two-dimensional nonlinear primary wake flow is studied. The nonlinear primary flow is obtained via a numerically simulated flow system where the mean-flow and disturbance components are interacting nonlinearly. The simulated flow, which appears as a von Kármán vortex street, is then Fourier analysed, and it is found that only the modified mean flow and modified fundamental disturbance suffice for inclusion as the primary flow. The nonlinear primary flow is in contrast to the linear primary flow case in many respects, where the unmodified mean wake flow and the linear fundamental instability comprise the primary flow. The temporal secondary instability result between nonlinear and linear primary flows yield similar spanwise wavenumber locations for the most amplified subharmonic mode and similar wave-propagation angles. The qualitative and quantitative behaviour of the amplification rate versus the spanwise wavenumber spectrum shape are very different. The significant difference being that the nonlinear primary flow supports lower maximum-amplification rates. The principal reason for this is that the nonlinear primary flow is not as robust a disturbance energy supplier as the linear primary flow, which, because of the entire lack of nonlinear interactions, remain ‘robust’ suppliers of energy to the secondary instability. In general, amplification rates of the secondary instability are considerably less than those of the upstream primary fundamental instability of the two-dimensional flow. In experiments where two- and three-dimensional disturbances are forced, if the measured amplifications are comparable in their respective linear region, it is probable that they are both primary instabilities of an unmodified mean flow. It would be helpful to quantitatively measure the development of a secondary instability out of an established nonlinear primary wake flow (e.g. Cimbala *et al.* 1988) for understanding the mechanisms discussed in this paper. The general discussions of the nonlinear interaction between two-dimensional primary flow and the three-dimensional disturbance in §§2 and 6 could be the starting point of leapfrog nonlinear parabolic computations, as has been performed in other secondary instability problems in shear flows (Girgis & Liu 2002, 2006; Liu 2008).

## Acknowledgements

The authors thank Professor J. T. Stuart, FRS, for his interest and helpful discussions. This work was initiated through partial support from the US Office of Naval Research, and was monitored by Dr Edwin P. Rood.

- Received March 2, 2010.
- Accepted July 23, 2010.

- © 2010 The Royal Society