2.1 Modified base flow

As the secondary disturbance is calculated with respect to the modified base flow (i.e. Couette flow undergoing TG), it is convenient to designate the latter as

Substituting the velocity field into the Navier–Stokes equations and collecting the terms associated with $O(\unicode[STIX]{x1D700})$ and $O(\unicode[STIX]{x1D700}^{2})$ leads to the governing equations of $\boldsymbol{u}_{1}$ and $\boldsymbol{u}_{2}$ , respectively. The $O(\unicode[STIX]{x1D700})$ equations are solved by assuming that

where $\unicode[STIX]{x1D6FD}$ is the spanwise wavenumber, $\unicode[STIX]{x1D714}_{1}$ is the complex eigenvalue and $\boldsymbol{F}(y)$ is the eigenfunction. As mentioned above, for the streamwise independent case the Orr–Sommerfeld and Squire equations ( $O(\unicode[STIX]{x1D700})$ ) can be solved analytically. The analytical expressions for $\boldsymbol{u}_{1}$ based on four modes are given in appendix A. These analytical expressions have been verified in our previous work by obtaining excellent agreement with DNS results, calculated for very small values of the initial energy, $E_{0}\approx 10^{-6}$ (see figure 3 in Karp & Cohen Reference Karp and Cohen2014).

The $O(\unicode[STIX]{x1D700}^{2})$ equations are non-homogeneous with the right-hand side given by interactions between the four modes constituting $\boldsymbol{u}_{1}$ . The nonlinear interaction $\boldsymbol{u}_{2}$ may be written as a sum of a mean flow modification $\boldsymbol{u}_{m}$ and a harmonic term $\boldsymbol{u}_{h}$ having a spanwise periodicity of $2\unicode[STIX]{x1D6FD}$ :

This set of equations is solved analytically with the use of Duhamel’s principle. The need to calculate the nonlinear modification of the TG arose in our previous work (Karp & Cohen Reference Karp and Cohen2014), where its inclusion was essential for obtaining instability in the case of odd TG. The importance of $\boldsymbol{u}_{2}$ is further discussed in the following section. The full expressions for $\boldsymbol{u}_{m}$ and $\boldsymbol{u}_{h}$ have been omitted from this article and can be found in Karp & Cohen (Reference Karp and Cohen2014).

2.2 Multiple time scales method

It is well known that the TG phenomenon occurs over a long time scale $O(Re)$ (this is also seen from (A 1)). Initially the growth is algebraic, followed by an approach towards maximal growth and finally slow viscous decay. Except the initial stage, one can utilize the slow time scale associated with the TG, and perform a multiple time scales analysis in order to calculate the evolution of an infinitesimal 3-D secondary disturbance on top of the TG. In addition, the cross-stream velocity components are $O(Re)$ smaller than the streamwise component (as is evident from (A 3)). Accordingly, the velocity field and pressure are written as

where $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D707}t$ is the slow time scale ( $\unicode[STIX]{x1D707}=O(Re^{-1})\ll 1$ ), $\boldsymbol{U}(\unicode[STIX]{x1D70F},y,z)$ and $P(\unicode[STIX]{x1D70F},y,z)$ are the modified base-flow velocity and pressure which depend only on the slow time scale, $\boldsymbol{u}_{d}(t,\unicode[STIX]{x1D70F},x,y,z)$ and $p_{d}(t,\unicode[STIX]{x1D70F},x,y,z)$ are the secondary disturbance velocity and pressure which depend on both time scales and $\boldsymbol{u}_{dd}(t,\unicode[STIX]{x1D70F},x,y,z)$ and $p_{dd}(t,\unicode[STIX]{x1D70F},x,y,z)$ are used to correct the long-time evolution of the secondary disturbance. It should be noted that our assumption that $(U,V,W)\sim O(1)$ is justified only when $\unicode[STIX]{x1D700}\sim O(1)$ in (2.3). Therefore, the validity of the scaling $(U,\unicode[STIX]{x1D707}V,\unicode[STIX]{x1D707}W)$ in (2.6) has been confirmed by DNS of the modified base flow for $\unicode[STIX]{x1D700}\sim O(1)$ .

Substituting expression (2.6) into the Navier–Stokes equations and collecting the terms of order $O(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}^{2})$ , $O(\unicode[STIX]{x1D6FF})$ and $O(\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF})$ one obtains the equations governing the evolution of the modified base flow $\boldsymbol{U}(\unicode[STIX]{x1D70F},y,z)$ , the secondary disturbance $\boldsymbol{u}_{d}(t,\unicode[STIX]{x1D70F},x,y,z)$ and its slow time correction $\boldsymbol{u}_{dd}(t,\unicode[STIX]{x1D70F},x,y,z)$ , respectively. Following the multiple time scales method, we assume the following expression for $\boldsymbol{u}_{d}(t,\unicode[STIX]{x1D70F},x,y,z)$ :

where $\unicode[STIX]{x1D6FC}$ is the streamwise wavenumber of the secondary disturbance, $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})$ is the slowly varying secondary disturbance eigenvalue, $\hat{\boldsymbol{u}}_{d_{k}}(\unicode[STIX]{x1D70F},y)$ are the Fourier expansion coefficients of the slowly varying secondary disturbance eigenfunction and $M_{z}$ is the number of Fourier harmonics in the expansion. The amplitude $A(\unicode[STIX]{x1D70F})$ is the slowly varying amplitude of the secondary disturbance. The secondary stability problem ( $O(\unicode[STIX]{x1D6FF})$ ) has already been solved in our previous work (Karp & Cohen Reference Karp and Cohen2014) where the modified base flow has been assumed to be parallel and frozen in time (see equation (2.6) and appendix C therein). The $O(\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF})$ equations for $\boldsymbol{u}_{dd}$ are used to obtain the long time scale correction of the amplitude $A(\unicode[STIX]{x1D70F})$ . The left-hand side is identical to the secondary linear stability problem (Karp & Cohen (Reference Karp and Cohen2014), equation (2.6)), whereas the right-hand side contains forcing terms which depend on $\boldsymbol{U}$ and $\boldsymbol{u}_{d}$ . The $O(\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF})$ equations are:

where the right-hand side expressions are given by: Solutions of (2.8) are possible only if its right-hand side is orthogonal to the conjugate of the adjoint variables which are the solutions of the homogenous adjoint equations. The corresponding solvability condition is:

where $\hat{u} _{a_{k}}^{\ast },\hat{v}_{a_{k}}^{\ast },{\hat{w}}_{a_{k}}^{\ast }$ are the conjugate adjoint variables, the derivation of which is detailed in appendix B. Substituting the expression for the secondary disturbance (2.7) into the solvability condition leads to an equation for the amplitude $A(\unicode[STIX]{x1D70F})$ :

where $M(\unicode[STIX]{x1D70F})$ and $N(\unicode[STIX]{x1D70F})$ are given by:

Consequently, the amplitude is given by

where $A_{0}$ is the initial amplitude at time $\unicode[STIX]{x1D70F}_{0}$ . Substituting (2.13) back into the expression for the secondary disturbance (2.7) and replacing $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D707}t$ we obtain:

where $\unicode[STIX]{x1D6E9}(t)$ is the secondary disturbance phase, corrected for long times, and $\tilde{\boldsymbol{u}}_{d}(t,y,z)$ is the eigenfunction. When calculating the evolution of the secondary disturbance, first the eigenvalues $\unicode[STIX]{x1D714}(t)$ and eigenfunctions $\tilde{\boldsymbol{u}}_{d}(t,y,z)$ are obtained using the secondary stability analysis. The eigenfunctions are normalized by the value of the streamwise eigenfunction at a certain location $(y_{0},z_{0})$ to ensure that they change smoothly with time. It should be noted that as the multiple scales analysis is linear, the results are independent of the location of the normalization $(y_{0},z_{0})$ .

The next stage is the calculation and normalization of the adjoint eigenfunctions, in a manner similar to the regular eigenfunctions. It should be noted that the adjoint eigenfunctions can be normalized at a location different from the one corresponding to the regular eigenfunctions. The adjoint eigenfunctions are used to calculate the phase correction term $N(t)/M(t)$ , providing the full solution of the secondary disturbance from (2.14a ).

2.3 Secondary disturbance growth rate components

A useful method for identifying the sources of the growth rate is by performing a production–dissipation analysis of the secondary disturbance kinetic energy. This is achieved by taking the volumetric integral of the disturbance kinetic energy equation. We follow here the analysis presented in Cossu & Brandt (Reference Cossu and Brandt2004) and write the following equation:

where $\unicode[STIX]{x1D714}_{i}$ is the growth rate, $E_{d}$ is the kinetic energy of the secondary disturbance, $T_{y}$ and $T_{z}$ are the energy production terms associated with the wall-normal and spanwise shear, respectively, and $D$ is the viscous dissipation given by the square norm of the perturbation vorticity $(\unicode[STIX]{x1D709}_{d},\unicode[STIX]{x1D702}_{d},\unicode[STIX]{x1D701}_{d})^{\text{T}}$ . The integral expressions are

where

Equation (2.15) provides insight into the mechanism of instability by comparing between the relative strength of each of the growth rate components $T_{y},T_{z}$ and $D$ .

2.4 Direct numerical simulation

The theoretical predictions are compared with results obtained by the ‘Channelflow’ DNS code (Gibson Reference Gibson2012) initiated by the analytical expressions. The simulation is pseudospectral, utilizing Fourier modes in the $x$ and $z$ directions and Chebyshev modes in $y$ . The results are obtained using $N_{x}=64$ , $N_{y}=65$ and $N_{z}=64$ . The effect of increasing the number of modes to $N_{x}=128$ , $N_{y}=101$ and $N_{z}=128$ was found to be negligible. The domain contains two walls at $y=\pm 1$ , a single wavelength in the streamwise direction, $L_{x}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}$ , and a single wavelength in the spanwise direction, $L_{z}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FD}$ . Since the $3/2$ rule is applied to remove aliasing, the number of corresponding Fourier modes is $N_{x,z}^{\prime }=2/3N_{x,z}$ . The time step is chosen to obtain an initial Courant–Friedrichs–Lewy (CFL) number of ${\sim}0.1$ . Reducing the time step had no significant effect on the results.

3.1 Inflection points

We propose the maximal strength (shear) inflection points as an alternative to maximal energy growth. The analysis begins by finding the conditions in the parameter space ( $Re,E_{0},\unicode[STIX]{x1D6FD},t$ ) yielding the strongest inflection points. Neglecting nonlinear interactions in the TG, the streak amplitude and the shear vary linearly with the Reynolds number and the square root of the initial energy (see (A 3) and (A 4)). Therefore, for linear TG it is sufficient to investigate only combinations of $(\unicode[STIX]{x1D6FD},t)$ for a given $Re$ and $E_{0}$ . We begin the analysis by considering only linear TG ( $u_{1}$ , based on 4 modes) and neglecting the nonlinear TG interactions ( $u_{2}$ ). The streamwise component of the modified base flow assuming linear TG for the arbitrary chosen values $Re=1000$ , $E_{0}=10^{-4}$ , $\unicode[STIX]{x1D6FD}=1$ and $t=30$ is presented in figure 1 for the even (1 a) and odd (1 b) TG. The wall-normal inflection points are marked by the black curves while the spanwise inflection points are marked by magenta. The strongest wall-normal inflection points, having the maximum wall-normal shear, are indicated by the black circles and the strongest spanwise inflection points, having the maximum spanwise shear, by the magenta circles. For the even TG the wall-normal inflection points are near $y=\pm 0.5$ in between the vortices at $z\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0,1$ , respectively, whereas the spanwise inflection points are at the centre of the channel close to the cores of the vortices ( $y=0,z\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.5,1.5$ ). For the odd TG the strongest wall-normal inflection point is at the centre of the channel $(y=0)$ in between the vortices $(z\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=1)$ , whereas the spanwise inflection points are located on the sides of the channel close to the cores of the vortices ( $y=\pm 0.5,z\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.5,1.5$ ). The locations of the inflection points along with their strength (shear) are summarized in the top two rows of table 1. It can be seen that both types of TG generate wall-normal inflections with similar strength and that the even TG produces stronger spanwise inflections. The streamwise component of the modified base flow, taking into account the nonlinear interactions between the 4 TG modes, is presented in figure 1(c) and 1(d) for the even and odd TG, respectively. The locations of the inflection points along with their strength (shear) are summarized in the bottom two rows of table 1. It can be seen that the nonlinear modification has some effect on the streaks and changes the locations of the inflection points. Nevertheless, this effect can be considered small since qualitatively the streamwise component is similar to the one obtained using linear TG and the shift in the locations of the inflection points is relatively small. The nonlinear modification also enhances the strength of the inflection points and the most noticeable effect is on the wall-normal shear for the even TG. There is no effect on the spanwise shear for the even TG since the nonlinear modification of the streamwise velocity is antisymmetric with respect to $y=0$ and consequently the strongest spanwise inflection remains unchanged. The effect on the spanwise shear for the odd TG is small. To summarize, linear TG can be used as an adequate approximation for the calculation of the strongest inflection points for various combinations of  $(\unicode[STIX]{x1D6FD},t)$ .

Figure 1.

Streamwise modified base flow for $Re=1000$ , $E_{0}=10^{-4}$ , $\unicode[STIX]{x1D6FD}=1$ and $t=30$ . The wall-normal (black) and spanwise (magenta) inflection points contours are marked on the figures with the strongest inflections encircled. (a) Even, linear TG; (b) odd, linear TG; (c) even, nonlinear TG modification; (d) odd, nonlinear TG modification.

Table 1.

The shear and location of the strongest wall-normal and spanwise inflection points for even and odd TG, $Re=1000$ , $E_{0}=10^{-4}$ , $\unicode[STIX]{x1D6FD}=1$ and $t=30$ . The top two rows are for linear TG whereas the bottom two rows include the nonlinear TG modification.

Figure 2.

Maximal shear at the inflection points versus time and $\unicode[STIX]{x1D6FD}$ for linear TG, $Re=1000$ and $E_{0}=10^{-4}$ . (a) Even TG, maximal wall-normal inflection $|\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y|$ ; (b) even TG, maximal spanwise inflection $|\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}z|$ ; (c) odd TG, maximal wall-normal inflection $|\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y|$ ; (d) odd TG, maximal spanwise inflection $|\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}z|$ . The strongest inflections versus time are given by the dashed lines.

The calculation of the strongest inflection points has been repeated for various wavenumbers and times ( $\unicode[STIX]{x1D6FD},t$ ). The shear at the strongest wall-normal and spanwise inflection points is presented in figure 2 for the even (2 a) and odd (2 b) TG, $Re=1000$ and $E_{0}=10^{-4}$ . It can be seen that the optimal wavenumber depends on time. For the even TG the maximal wall-normal inflection is obtained for $\unicode[STIX]{x1D6FD}=1.7$ , similarly to the optimal energy gain (Butler & Farrell Reference Butler and Farrell1992, $G_{max}(\unicode[STIX]{x1D6FD})$ ). This is due to the fact that most of the energy is contained in the streamwise velocity, suggesting that maximum deviation of the streamwise velocity from the Couette profile is likely to produce the strongest wall-normal inflection as well as the largest energy growth. The optimal spanwise inflection is obtained at $\unicode[STIX]{x1D6FD}=3$ . This wavenumber corresponds approximately to an optimum of $\unicode[STIX]{x1D6FD}\sqrt{G_{max}(\unicode[STIX]{x1D6FD})}$ since for linear TG $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}z=-\unicode[STIX]{x1D6FD}u_{1}\sin (\unicode[STIX]{x1D6FD}z)$ and $u_{1}\sim \sqrt{G}$ . The optimal values for the odd TG are $\unicode[STIX]{x1D6FD}=2.8$ and $\unicode[STIX]{x1D6FD}=5.2$ for the wall-normal and spanwise inflections, respectively. It should be noted that in contrast to the maximal energy growth, which is much larger for the even TG, the strength of the inflection points generated by the even and odd TG is comparable and there exists a region (e.g. $t\leqslant 30$ ) where the odd TG generates inflection points stronger than the ones generated by the even TG. Nevertheless, the spanwise inflections generated by the even TG are always stronger than the ones generated by the odd TG.

Next, the above calculation of the strongest inflection points has been repeated including the nonlinear TG modification. The strongest inflection points, together with the optimal wavenumbers and times when they are obtained are summarized in table 2 in which the top two rows correspond to linear TG, whereas the bottom two rows to TG including nonlinear modification. It can be seen that the nonlinear TG modification affects the wall-normal inflections, changing the optimal parameters and enhancing the maximal wall-normal shear. Nevertheless, its influence on the spanwise shear is negligible. It should be noted that the even TG spans over longer times compared to the odd TG (note the different time scales in figure 2). This is best understood by considering very long times when only one least stable mode is left in the TG. The ratio between the decay rates (see (A 1)) of the least stable even and odd modes is $\unicode[STIX]{x1D714}_{sq_{e}}/\unicode[STIX]{x1D714}_{sq_{o}}=(0.25+(\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0})^{2})/(1+(\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0})^{2})$ , which varies between 0.25 for small $\unicode[STIX]{x1D6FD}$ to $1$ as $\unicode[STIX]{x1D6FD}\rightarrow \infty$ . Consequently, the inflection points generated by the even TG remain in the modified base flow for longer times, making the even TG a better candidate for possible instability.

Table 2.

The shear at the strongest wall-normal and spanwise inflection points and the optimal wavenumbers $\unicode[STIX]{x1D6FD}$ and times $t$ when it is attained for even and odd TG, $Re=1000$ and $E_{0}=10^{-4}$ . The top two rows are for linear TG whereas the bottom two rows include the nonlinear TG modification.

3.2 Secondary stability analysis

To verify that the strongest secondary instability is indeed associated with the strongest inflection points, a 3-D stability analysis of the two-dimensional (2-D) modified base flow is performed. A Floquet decomposition is utilized in the spanwise direction, with $M_{z}=5$ Fourier modes and $N_{y}=100$ Chebyshev modes over the 2-D domain $z\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=[-1,1];y=[-1,1]$ . There are two types of modes – varicose and sinuous. Usually the varicose modes are associated with wall-normal inflections whereas the sinuous modes are associated with spanwise inflections. However, there may exist other combinations, such as varicose modes associated with spanwise inflections or sinuous modes associated with wall-normal inflections, depending on the relative strength of the spanwise and wall-normal inflection points. The analysis is based only on the streamwise component as the cross-stream components are an order of magnitude smaller than the streamwise component (as is evident from (A 3)), with the time being ‘frozen’ and used as a parameter. The nonlinear TG modification is included in the modified base flow since in our previous work (Karp & Cohen Reference Karp and Cohen2014) it was found to be influential for initial energies of order $O(10^{-4})$ and $Re=1000$ . An additional parameter required for the stability analysis is the streamwise wavenumber of the secondary disturbance,  $\unicode[STIX]{x1D6FC}$ .

The 3-D stability analysis for even and odd TG, $Re=1000$ and $E_{0}=10^{-4}$ has been performed in the parameter space $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},t)$ . The combinations yielding the highest growth rate are given in table 3 for the varicose and sinuous modes, together with the values of $T_{y}/(2E_{d}),T_{z}/(2E_{d})$ and $D/(2E_{d})$ . It can be seen that the optimal spanwise wavenumbers are different compared to the ones maximizing the energy gain ( $\unicode[STIX]{x1D6FD}=1.67$ , 2.8 for the even/odd TG, respectively). The most dangerous secondary disturbance is sinuous and the even TG is more unstable than the odd TG. For both sinuous disturbances the wavenumbers are very similar and $\unicode[STIX]{x1D6FC}\approx 0.5\unicode[STIX]{x1D6FD}$ . Examining the growth rate components $(T_{y},T_{z},D)$ it is evident that the sinuous modes, as well as the varicose mode of the even TG, are associated with spanwise inflection points while the varicose mode of the odd TG is associated with wall-normal inflections.

Table 3.

The optimal parameters yielding the maximal growth rate for $Re=1000$ and $E_{0}=10^{-4}$ for even and odd TG including nonlinear modifications based on a 3-D analysis of a 2-D base flow.

The streamwise magnitude of the above four secondary disturbances is presented in figure 3. The even-varicose mode (figure 3 a) is maximal near the locations of the strongest spanwise inflections. The combination of a varicose mode and spanwise inflections is somewhat surprising. Therefore this disturbance is compared with a sinuous disturbance calculated for the same parameters ( $\unicode[STIX]{x1D6FC}=2.5,\unicode[STIX]{x1D6FD}=2.1$ and $t=81$ ). It is found that for these parameters, which maximize the varicose instability, the sinuous disturbance is more unstable than the varicose disturbance. This implies that the varicose disturbance should not be treated as a separate route to transition, as it is completely masked by the sinuous disturbance. Therefore, only three instability mechanisms remain relevant, even sinuous, odd varicose and odd sinuous. The sinuous disturbances (figure 3 b,d) are maximal near the locations of the strongest spanwise inflections. The mode associated with the even TG spans through the entire channel (figure 3 b) whereas the one associated with the odd TG is concentrated in the upper half of the channel (figure 3 d) (the latter mode has a complex conjugate mode concentrated in the lower half of the channel). As the spanwise shear is stronger for the even TG, the growth rate is also higher for this case. The physics behind both these modes is very similar as their instability is triggered via the spanwise inflection points between a single pair of vortices. This type of instability has been investigated by Waleffe (Reference Waleffe1995) and Reddy et al. (Reference Reddy, Schmid, Baggett and Henningson1998). The odd-varicose mode (figure 3 c) is maximal near the location of the strongest wall-normal inflection. This type of disturbance has been investigated in our previous work (Karp & Cohen Reference Karp and Cohen2014).

Figure 3.

Magnitude of the streamwise component of the secondary disturbance corresponding to the parameters in table 3, obtained by stability analysis of a 2-D base flow, including the nonlinear TG modification for $Re=1000$ and $E_{0}=10^{-4}$ . (a) Varicose, even TG; (b) sinuous, even TG; (c) varicose, odd TG; (d) sinuous, odd TG.

Comparing the most unstable secondary disturbances to the inflection point strength in table 2 several interesting observations can be made: (i) although the wall-normal inflection points are stronger than the spanwise inflection points, the latter trigger instability more efficiently, since the sinuous modes are more unstable than the varicose modes. A possible reason for the dominance of the spanwise inflections is the absence of walls in the spanwise direction; (ii) the even TG generates a stronger wall-normal inflection point compared to the odd TG. Nevertheless, only the odd TG has an instability associated with a wall-normal inflection. This is attributed to the location of the wall-normal inflection points associated with the even TG, which varies with time and located closer to the wall; and (iii) the optimal spanwise wavenumbers which maximize the secondary instability are not exactly the same as the ones maximizing the strength of the inflection points. The differences between them result from the two-dimensionality of the modified base flow which is not taken into account in the inflection point strength study. The effect of the two-dimensionality of the modified base flow can be evaluated by looking at the dependence of the growth rate components ( $T_{y},T_{z},D$ ) on $\unicode[STIX]{x1D6FD}$ . These are shown in figure 4 for the odd-sinuous mode ( $Re=1000,E_{0}=10^{-4},\unicode[STIX]{x1D6FC}=1.5$ and $t=36$ ). It can be seen that indeed the $T_{z}$ component, associated with the spanwise shear, is stronger at $\unicode[STIX]{x1D6FD}=5$ (corresponding to the strongest spanwise inflections). However, the effect of $T_{y}$ , associated with the wall-normal shear, and the dissipation $D$ results in a shift of the optimal spanwise wavenumber to $\unicode[STIX]{x1D6FD}=3.5$ . Thus, the difference in the spanwise wavenumber is due to 2-D effects and not due to a different mechanism. For the other secondary modes similar effects are observed (not shown).

To understand the reasons for some of these observations stability analysis of one-dimensional (1-D) velocity profiles containing the strongest inflection points (wall-normal or spanwise, in accordance with the inflection point direction) can be performed. This analysis is carried out in appendix C, leading to the following conclusions: (i) the highest growth rate corresponds to the spanwise 1-D profiles of the even and odd TG, suggesting that the spanwise inflections are more dominant than the wall-normal inflections; and (ii) for the wall-normal 1-D profiles, the odd TG is much more unstable than the even TG, suggesting that the wall-normal inflections of the even TG are not capable of introducing a significant instability. It can be seen that analysis of the 1-D velocity profiles predicts the tendencies observed in the full analysis of the 2-D flow. Nevertheless, the 1-D stability analysis is clearly insufficient to find the optimal parameters for the instability and the growth rate.

Figure 4.

Growth rate components versus $\unicode[STIX]{x1D6FD}$ for the odd TG with nonlinear modification, sinuous disturbance, $Re=1000,E_{0}=10^{-4},\unicode[STIX]{x1D6FC}=1.5$ and $t=36$ .

3.3 Effect of initial energy, TG nonlinearity and Reynolds number

In this section the effect of initial energy, TG nonlinearity and $Re$ on the three most dangerous secondary disturbances identified above is presented. The most dangerous secondary disturbances have been obtained above including the nonlinear TG modification in the modified base flow. Here, the effect of neglecting the nonlinear TG modification is examined by excluding $u_{2}$ . As the initial energy is increased, more nonlinear TG effects come into play and many other secondary modes may become unstable. However, our analytical model for the TG becomes less valid for higher initial energies and we are interested in working with the smallest initial energy allowing instability. Thus, we are particularly interested in the effect of lowering the initial energy. To isolate the effect of the initial energy, the growth rate of each of the three eigenmodes is calculated while keeping $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$ constant. The growth rates for these three eigenmodes for various times and initial energies are presented in figure 5, where only positive growth rates are shown. It can be seen that the time when the growth rate is optimal is almost constant. This is in accordance with the effect of $E_{0}$ on the inflection points. According to linear TG the locations of the optimal inflection points do not change and their strength (shear) is proportional to $\sqrt{E_{0}}$ (see (A 3) and (A 4)). The most significant change in the growth rate is for the even TG (figure 5 a), whereas for the odd TG the changes are similar for both modes (figure 5 b,c). The threshold energy for instability varies between $4\times 10^{-5}$ for the even-sinuous case to $7\times 10^{-5}$ for the odd-varicose case. Nevertheless, the optimal parameters ( $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ ) may vary with $E_{0}$ , so lower thresholds for instability may be found. It should be noted that Reddy et al. (Reference Reddy, Schmid, Baggett and Henningson1998) have calculated thresholds for instability based on a numerical TG representation with many modes, albeit only for the even TG.

Figure 5.

Effect of initial energy on the growth rate of the optimal secondary disturbances in table 3, for $Re=1000$ and various times. The growth rate is obtained by stability analysis of a 2-D base flow, including nonlinear TG modifications. Maximal growth rate curves are indicated by the dashed lines and only positive growth rates are shown. (a) Sinuous, even TG, $\unicode[STIX]{x1D6FC}=1.8$ , $\unicode[STIX]{x1D6FD}=3.6$ ; (b) varicose, odd TG, $\unicode[STIX]{x1D6FC}=1.2$ , $\unicode[STIX]{x1D6FD}=1.6$ ; (c) sinuous, odd TG, $\unicode[STIX]{x1D6FC}=1.5$ , $\unicode[STIX]{x1D6FD}=3.5$ .

The effect of TG nonlinearity is shown in figure 6, where the growth rates versus initial energy for linear TG (dashed) and including nonlinear TG modification (solid) are presented for the three eigenmodes, while keeping $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},t)$ constant. It can be seen that for values of initial energy lower than $E_{0}\approx 3\times 10^{-5}$ the TG nonlinearity has a minor effect on the growth rate. For higher values of the initial energy the growth rate including TG nonlinearity is considerably larger than the one associated with the linear TG. The linear TG becomes unstable only for initial energies of $E_{0}\approx 1.5\times 10^{-4}$ , for which the TG nonlinearity is important. It should be noted that even for initial energies of $O(10^{-5})$ the nonlinear TG modification has a minor effect, expressed by the small deviations between the curves.

Figure 6.

Effect of TG nonlinearity on the growth rate of the optimal secondary disturbances in table 3 for $Re=1000$ . The growth rate ( $\unicode[STIX]{x1D714}_{i}$ ) as a function of initial energy ( $E_{0}$ ) is obtained by stability analysis of a 2-D base flow. (a) Sinuous, even TG, $\unicode[STIX]{x1D6FC}=1.8$ , $\unicode[STIX]{x1D6FD}=3.6$ , $t=61$ ; (b) varicose, odd TG, $\unicode[STIX]{x1D6FC}=1.2$ , $\unicode[STIX]{x1D6FD}=1.6$ , $t=43$ ; (c) sinuous, odd TG, $\unicode[STIX]{x1D6FC}=1.5$ , $\unicode[STIX]{x1D6FD}=3.5$ , $t=36$ .

Lowering the Reynolds number has several effects on the eigenvalues. The TG time scale is shortened (A 1), the streamwise velocity of the modified base flow is lowered (A 3) and the effect of dissipation ( $D$ ) on the growth rate of the secondary disturbance becomes more noticeable (2.16a,b ). To isolate each of these effects as $Re$ is lowered, the growth rate of each of the three eigenmodes is calculated while keeping $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$ constant and changing the initial energy according to the relation $E_{0}=10^{-4}(1000/Re)^{2}$ . Changing $E_{0}$ allows keeping the strength of the inflectional points fairly constant (they are not strictly constant as nonlinear TG modifications are included). It should be noted that the dictated increase in initial energy as $Re$ is lowered results in higher amplitudes of the TG which result in more significant nonlinear TG interactions, not taken into account in our simplified model. Therefore, the growth rates should be interpreted as leading-order results with respect to $Re$ . The growth rates for the three eigenmodes for various times and $Re$ are presented in figure 7, where only positive growth rates are shown. It can be seen that the time when the growth rate is optimal is linearly proportional to $Re$ although the analysis includes nonlinear TG modifications. Thus, the TG time scale is linearly proportional to $Re$ even when the TG includes nonlinear modifications. As $Re$ is lowered, the optimal growth rate reduces due to the higher dissipation. The decrease in the optimal growth rate as $Re$ is lowered is similar for all three modes. This suggests that the threshold energy for instability should change with a power $n$ , higher than $-2$ which is used for $E_{0}$ above (i.e. $E_{0}\sim Re^{n}$ , $n>-2$ ). Nevertheless, it can be seen that the growth rate of each one of the three modes continues smoothly to lower $Re$ with the window of instability, i.e. the time range when each of the modes is unstable, shrinking linearly with $Re$ . For transition to occur we would like to choose the most unstable eigenmode which has the largest window of instability. This mode is associated with the even TG as it generates spanwise inflection points for longer times.

Figure 7.

Effect of $Re$ on the growth rate of the optimal secondary disturbances in table 3, for various times. The growth rate is obtained by stability analysis of a 2-D base flow, including nonlinear TG modifications. Maximal growth rate curves are indicated by the dashed lines and only positive growth rates are shown. The initial energy used in the calculation is given by $E_{0}=10^{-4}(1000/Re)^{2}$ . (a) Sinuous, even TG, $\unicode[STIX]{x1D6FC}=1.8$ , $\unicode[STIX]{x1D6FD}=3.6$ ; (b) varicose, odd TG, $\unicode[STIX]{x1D6FC}=1.2$ , $\unicode[STIX]{x1D6FD}=1.6$ ; (c) sinuous, odd TG, $\unicode[STIX]{x1D6FC}=1.5$ , $\unicode[STIX]{x1D6FD}=3.5$ .

3.4 Stability for temporally slowly evolving flow and comparison with DNS

In the previous sections the stability properties of PCF modified by TG have been investigated and the most dangerous disturbances have been identified. To verify our stability analysis and further investigate the mechanism of instability we have simulated the evolution of the disturbances in DNS. Our model assumes a separation of time scales between the slowly varying TG and the rapidly growing secondary disturbance. The TG is initially algebraic and becomes slowly varying only at some finite time. Accordingly, to test our model, the simulation should not be initiated at $t=0$ but rather at a time when the TG is already slowly varying and a separation of time scales exists. The variation of the TG is estimated from the linear energy gain by calculation of the local TG growth rate:

The local TG growth rate has been compared with the growth rate of the secondary eigenmode for several times and the initial time $t_{0}$ for the simulation has been chosen to ensure that $\unicode[STIX]{x1D714}_{i}\gg |\unicode[STIX]{x1D714}_{TG}|$ . Since nonlinear TG effects are non-negligible they have been included in the initial conditions for the DNS. To summarize, the DNS is initiated with the analytical expressions (including linear TG and its nonlinear modification) at time $t_{0}$ and the corresponding secondary disturbance. Three scenarios have been investigated: even TG with a sinuous disturbance, odd TG with a varicose disturbance and odd TG with a sinuous disturbance. The initial times and growth rates for these three cases are summarized in table 4. The transition scenario for each of the cases is analysed separately in the following sections. The amplitude of the secondary disturbance has been chosen so that the fraction of energy in the secondary disturbance is $1\,\%$ of the initial TG energy (e.g. for $E_{0}=10^{-4}$ the secondary disturbance adds the energy $\unicode[STIX]{x1D6E5}E=10^{-6}$ to the TG at time $t_{0}$ ).

The DNS results are compared for each scenario to the corresponding analytical expressions obtained by using the multiple time scales method. To highlight the advantage of the multiple time scales method over the frozen flow analysis, i.e. assuming the secondary disturbance has constant growth rate and eigenfunction, a direct comparison between the two analytical methods has been performed, as detailed in appendix D. It is shown that the multiple time scales better predicts the DNS results.

Table 4.

Summary of the parameters used in the scenarios tested in the DNS for $Re=1000$ . The initial time $t_{0}$ is chosen to ensure $\unicode[STIX]{x1D714}_{i}\gg |\unicode[STIX]{x1D714}_{TG}|$ .