4.1 Base spectrum

To give a general overview of the spectrum, the base flow plane conceived with 500 instantaneous snapshots is considered as a reference baseline case. The branches of eigenvalues corresponding to the type I and II modes are shown in figure 9. The temporal global stability problem is solved for the $\unicode[STIX]{x1D6FC}$ -range $[0.5,18]/\unicode[STIX]{x1D706}_{r}$ with a spacing of $0.1/\unicode[STIX]{x1D706}_{r}$ . In the figure, the branches are shown for the $\unicode[STIX]{x1D6FC}$ -range over which the branches are unstable.

Figure 9. Type I (circles) and II (squares) temporal frequency (a) and growth rate (b) versus the wavenumber using the $N_{z}\times N_{y}=55\times 55$ (dashed line) and $90\times 90$ (solid) grid nodes.

Figure 10. $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|$ for type I (a) and II (b) (levels span $[1/6,5/6]$ with $\unicode[STIX]{x1D6E5}=1/6$ ).  $90\times 90$ (filled contours) and $55\times 55$ (dashed) grid resolution. Near-wall region (dash-dotted line). $\overline{U}_{w}/\overline{U}_{e}$ levels $0.1,0.2,\ldots ,0.9$ (dotted).

Table 1. Parameters of the (most temporally unstable and POD wavelength) modes in the base spectrum for the strong vortex.

Several grid resolutions are used to compute the spectra with the focused grid for the type I mode, going up to $N_{z}\times N_{y}=90\times 90$ nodes. The mode branches are found to be converged already for $N_{z}\times N_{y}=55\times 55$ nodes, with eigenvalue errors of $O(10^{-5})$ in absolute sense for the type I instability. Despite the lower grid density in the region of dominance of the type II mode, the spectral discretization captures this mode properly as well. The most unstable type II eigenvalue experiences a $O(10^{-4})$ absolute error, which is deemed sufficiently small for the purposes of this analysis.

The type I mode attains the maximum temporal growth rate and is therefore locally most unstable. The type II mode is found to be locally more unstable for $\unicode[STIX]{x1D714}_{r}>7.77\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}=5.0~\text{kHz}$ . The spectral information associated with the locally most unstable type I and II modes is given in table 1. It is important to note that these indications do not directly imply these modes are the largest perturbation at this station. To investigate that, the local results, in terms of the spatial amplification rate, have to be integrated in space, i.e.  $N$ -factors should be considered.

The $|\tilde{u} |$ -eigenfunctions corresponding to the most unstable type I and II modes are shown in figure 10. The spatial distributions of the modes are superimposed over the isocontours of the streamwise velocity of the base flow. Previous investigations from both numerical and experimental perspectives, see Malik et al. (Reference Malik, Li and Chang1994), White & Saric (Reference White and Saric2005), Bonfigli & Kloker (Reference Bonfigli and Kloker2007), indicate the type I mode is positioned on the outer upwashing side of the primary vortex, close to the in-plane saddle point, while the type II mode rides on top of the vortex. As evident, these characteristics are well captured by the stability analysis. Two spatial mode distributions are to be distinguished in figures 10(a) and 10(b) and correspond to different grid resolutions. The difference is small, further confirming that the $55\times 55$ grid yields converged eigensolutions.

Using the Reynolds–Orr equation (3.3), the most unstable type I and II eigenmodes can be decomposed into the most dominant contributions shown in figure 11, ordered from absolute largest to smallest top to bottom. The remainder is composed out of terms that are individually smaller than the dominant terms in absolute value. The eigenvalues themselves are indicated with dashed lines. Two bars are given for each term, again corresponding to different grid resolutions. The differences in the contributions are consistent with the errors in the eigenvalues.

Figure 11. (a,c) $\unicode[STIX]{x1D714}_{r}$ - and (b,d) $\unicode[STIX]{x1D714}_{i}$ -budgets (bars) of type I (a,b) and II (c,d), $\unicode[STIX]{x1D714}$ eigenvalue (dashed line). Reynolds stress, viscous dissipation and advection terms are coloured red, blue and black, respectively, see equation (3.6) for symbol definitions. Upper bars: $90\times 90$ , lower bars: $55\times 55$ grid resolution.

The $\unicode[STIX]{x1D714}_{i}$ -budget for both mode types is most dominantly dictated by the $\overline{U}_{w}$ -shear and viscous dissipation. As per the definitions proposed by Malik et al. (Reference Malik, Li and Chang1996), the type I and II (or, $z$ - and $y$ -) modes are produced by the $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}z_{w}$ and $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}y$ shear components, respectively. This is found in the current case as well, as shown in figure 11(b,d). The other shear components are usually unimportant and can have a net destructive nature, as is the case for the type I mode here. Figure 11(d) illustrates this is not the case for the type II mode; in that case the $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}z_{w}$ shear also has a significant net productive role. Modes for which both production terms have comparable contributions are referred to as $y/z$ -modes, see Li et al. (Reference Li, Choudhari, Duan and Chang2014).

Having pinpointed the Reynolds stress terms as most prominent in the $\unicode[STIX]{x1D714}_{i}$ -budget, further insight into their spatial topology is sought. Figure 12 shows the integrands of $R^{y}$ and $R^{z_{w}}$ for both modes in the plane. After integration over the plane, these functions yield the contributions shown in figure 11(b,d). These terms have their origin in the $x_{w}$ -momentum equation (3.2a ) and hence directly produce the $\tilde{u} _{w}$ (energy) component. The integrands therefore indicate which part of the $\tilde{u} _{w}$ eigenfunction they produce. For the type I mode, the integrands clearly reflect the integral values. Interestingly, the downward protrusion located about $(z_{w},y)/\unicode[STIX]{x1D706}_{r}=(0.45;0.22)$ is produced by the $R^{y}$ -term; as illustrated with the additional contours in figure 12(a). The integrands for the type II mode are surprising, because the integrand of $R^{z_{w}}$ attains the largest value, while the $z_{w}y$ -integral value is smaller. The shape of the positive $R^{y}$ - and $R^{z_{w}}$ -integrand contours is comparable to that by Malik et al. (Reference Malik, Li, Choudhari and Chang1999) (cf. figure 10).

As mentioned, the participation of the advection terms $A^{\overline{V}}$ and $A^{\overline{W}_{w}}$ is not restricted to the real dispersion dynamics; as pointed out in figure 11(b,d), they are the next terms in line enhancing or reducing the growth rates of the modes, confirming the inclusion of the $\overline{V}$ - and $\overline{W}_{w}$ -components in the analysis is essential. With respect to $\unicode[STIX]{x1D714}_{i}$ , $A^{\overline{V}}$ and $A^{\overline{W}_{w}}$ respectively exert 2.4% and $-14.1\%$ contributions for the type I and 7.0% and 2.8% contributions for the type II mode. Figure 11(b,d) shows some of these values are larger than the Reynolds stress terms associated with these velocity components.

Figure 12. Imaginary part of the (a,c) $R^{y}$ - and (b,d) $R^{z_{w}}$ -integrands for type I (a,b) and II (c,d) (9 levels span (b) $[6.71,63.0]$ , (c) $[-3.00,10.4]$ and (d) $[-3.07,14.1]$ , negative contours are dashed). Levels in (a) span $[-27.5,-3.41]$ , with $\unicode[STIX]{x1D6E5}=4.83$ , and $[0.350,1.05]$ , $\unicode[STIX]{x1D6E5}=0.350$ . All values are given in $\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}^{3}$ -units. Eigenfunction contour $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|=1/6$ (dash-dotted line). $\overline{U}_{w}/\overline{U}_{e}$ levels $0.1,0.2,\ldots ,0.9$ (dotted).

The origin of the growth induced by the in-plane advection is traced by visualizing the integrands of the related terms in the Reynolds–Orr equation. The previous numbers illustrate $A^{\overline{V}}$ and $A^{\overline{W}_{w}}$ individually yield a predominant decrease and increase in the type II and type I growth rates, respectively. Figure 13(a,b,d,e) shows the integrands associated with $A^{\overline{V}}$ and $A^{\overline{W}_{w}}$ for both modes. The sign of the sum of these terms is illustrated in figure 13(c,f), indicating the (de)stabilizing regions. The latter panels clearly reflect the criterion based on the term (3.7); whenever the in-plane flow is directed away from elevated perturbation levels, the contribution is destabilizing (black arrows). Conversely, whenever the in-plane velocity is aligned with the perturbation level gradient, the contribution is stabilizing. Contours of the sum of the amplitudes $|\tilde{u} _{w}|+|\tilde{v}|+|\tilde{w}_{w}|$ are shown, because (3.7) features the gradient of the velocity amplitudes, not the (square root of the) perturbation energy. In the case of the type I mode, the in-plane flow has the tendency to focus the perturbation energy along a spanwise line and therefore has the major effect of increasing the spanwise extent of the eigenfunction. On the other hand, the eigenfunction’s maximum is located close to the in-plane flow saddle. This location is thus affected to a minor extent only. For the type II mode, the main effect is advection in the $z_{w}$ -direction. Therefore it is expected that the location of the eigenfunction’s maximum is sensitive to small changes in the productive Reynolds stress. Note that the integrand magnitudes of the advection terms for the type II mode are comparable with those of the Reynolds stresses, while they are an order of magnitude smaller for the type I mode.

Figure 13. (a,b,d,e) Imaginary part of (a,d) $A^{\overline{V}}$ - and (b,e) $A^{\overline{W}_{w}}$ -integrands for type I (a,b) and II (d,e) (9 levels ranging (a) $[-6.84,5.78]$ , (b) $[-8.74,6.68]$ , (d) $[-6.43,5.93]$ and (e) $[-8.99,4.32]$ in $\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}^{3}$ -units, negative contours are dashed). Eigenfunction contour $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|=1/6$ (dash-dotted line). (c,f) ( $\overline{V},\overline{W}_{w}$ )-vectors showing where $\text{Im}\{A^{\overline{V}}+A^{\overline{W}_{w}}\}<0$ (white arrows) and ${>}0$ (black arrows) for type I (c) and II (f). Amplitude sum $|\tilde{u} _{w}|+|\tilde{v}|+|\tilde{w}_{w}|$ (6 filled contours from 0 to maximum). $\overline{U}_{w}/\overline{U}_{e}$ levels $0.1,0.2,\ldots ,0.9$ (dotted).

A large stabilizing pocket is visible in figure 13(b), for which $\overline{W}_{w}>0$ and $\tilde{\boldsymbol{q}}^{\ast }\boldsymbol{\cdot }\unicode[STIX]{x2202}\tilde{\boldsymbol{q}}/\unicode[STIX]{x2202}z_{w}>0$ , above the in-plane flow saddle. That region largely contributes to the net $\overline{W}_{w}$ -advection towards the shear layer’s core causing the negative integral value for the type I mode. The magnitude of $A^{\overline{V}}$ in the energy budget for the type II mode is approximately 3 times smaller than $A^{\overline{W}_{w}}$ for type I. This is reflected by more evenly matched levels in figure 13(d). The positive contribution in figure 11(d) indicates a net $\overline{V}$ -advection away from the core of the shear layer. The largest contours in figure 13(d) indicate this is largely associated with an imbalance of vertical in- and outward advection on the left-hand side of the vortex core. The contours in figure 13(a,e) display a symmetric shape with respect to the absolute eigenfunction contour itself relative to the other cases, explaining negligible integral values. All contours in figure 13 are contained within the region of dominance of the Reynolds stress terms and hence do not generate additional eigenfunction features. Nonetheless, it is noteworthy that the contours in figure 13(a,b,d,e) reach the outer limits of the eigenfunction, especially the top right of the type I mode at which the Reynolds stress terms are an order of magnitude smaller.

For both modes, $A^{\overline{U}_{w}}$ yields the largest contribution to $\unicode[STIX]{x1D714}_{r}$ . The action of the in-plane velocity components is to retard the secondary vortices’ advection in the $x_{w}$ -direction. In total, $A^{\overline{U}_{w}}$ is cancelled to 3.1 % and 7.1 % by other terms for the type I and II modes, respectively. The large retardation in the case of the type II mode, considering that the other terms cancel out, is solely caused by $\overline{W}_{w}$ . This is interpreted to be the consequence of the fact that the eigenmodes travel in the opposite direction of $\overline{W}_{w}$ . No singular such term can be pointed out for the type I mode. An example of a Reynolds stress term participating in the $\unicode[STIX]{x1D714}_{r}$ -budget is $R^{y}$ , that slightly reduces the type I mode frequency.

4.2 Effect of ensemble size

The measured mean flow is subject to an uncertainty of $0.1\overline{U}_{e}/\sqrt{500}=4.5\times 10^{-3}\overline{U}_{e}$ , based on the maximum r.m.s. amplitude of $0.1\overline{U}_{e}$ obtained from the 500 instantaneous PIV snapshots (see § 4.6). The reported uncertainty stems from both systematic errors (such as tomo-PIV correlation errors), as well as from physical fluctuations of the instantaneous flow. For instance, Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) (cf. § 6.3.1) show the r.m.s. field is dominated by a low-frequency spanwise shake of the whole primary crossflow vortex, obtained as the most energetic POD mode. While these are acceptable uncertainty levels for flow diagnostics, their effect on the stability analysis should be carefully identified. In this section an effort is provided towards quantifying the effect of the ensemble average on the eigensolutions.

The uncertainty is quantified by deploying a Monte Carlo approach to the stability analysis. More specifically, stability analysis is performed on mean flows produced by varying the ensemble size, $N_{fr}$ , ranging from 300 to 475, with steps of 25. One hundred different random combinations are made per $N_{fr}$ from the total pool of 500 snapshots, resulting in 800 cases in total plus the single case possible for $N_{fr}=500$ ; used as the baseline case. Stability simulations were performed on the $45.6\,\%$ chord plane using the $55\times 55$ grid. For both modes, the respective most unstable wavenumber in table 1 was used as input.

The results are shown in terms of the mean and 2 standard deviations ( $\pm 2\unicode[STIX]{x1D70E}$ ) of the growth rate in relation to the ensemble size in figure 14. While the solution undergoes large fluctuations for small ensemble sizes, which is expected, a clear convergence trend is established for both the mean value and fluctuations. The difference between the estimated mean growth rate for $N_{fr}=450$ and the single case growth rate for $N_{fr}=500$ is given in table 2 as $\unicode[STIX]{x0394}\unicode[STIX]{x1D707}$ , illustrating convergence of the mean to errors at most one order of magnitude larger than the grid truncation errors. The percentages in the table are the relative errors with respect to the value for the $N_{fr}=500$ case.

Figure 14. Growth rate uncertainty versus ensemble size $N_{fr}$ for both instability modes. Mean (dots) and 2 $\unicode[STIX]{x1D70E}$ (bars) based on 100 random combinations for fixed $N_{fr}$ . The dots at $N_{fr}=500$ and horizontal lines are the calculated eigenvalue itself. Linear fit to the 2 $\unicode[STIX]{x1D70E}$ -bars (dashed lines).

Table 2. Uncertainty and errors in type I and II mode parameters with ensemble size (§ 4.2) and domain extrapolation parameters (handled in § 4.4). $^{\dagger }z_{w}$ is based on $N_{fr}=475$ , avoiding extrapolation.

The growth rate fluctuations within the 100 random cases are relatively large as shown by the standard deviation bars. Nevertheless, they also show an evident linear converging trend, illustrating that the mean value is approached in the limit of large $N_{fr}$ . The linear trend is extrapolated to obtain a measure at $N_{fr}=500$ , also reported in table 2. The demonstrated trend allows discarding the fluctuations, under the condition that a minimum threshold $N_{fr}$ can be defined beyond which the integrity of the solution structure can be demonstrated.

Figure 15. (a,c) $\unicode[STIX]{x1D714}_{r}$ - and (b,d) $\unicode[STIX]{x1D714}_{i}$ -budgets of type I (a,b) and II (c,d), mean value (bars) and $2\unicode[STIX]{x1D70E}$ (slanted lines) per $N_{fr}$ , ranging from 300 (bottom bar) to 500 (top bar) every 25 samples. Vertical lines indicate the single eigensolution for $N_{fr}=500$ as a reference. Small terms (mean and $2\unicode[STIX]{x1D70E}$ ) are magnified with a factor 10, the slanted lines then indicate $20\unicode[STIX]{x1D70E}$ .

The terms in the Reynolds–Orr equation are shown as a function of the ensemble size in figure 15. Fluctuations appear, as expected, but the mean term values per $N_{fr}$ are well defined and indicate a consistent balance for all $N_{fr}$ . Note that some contribution values, including the standard deviations with respect to the mean value, are enlarged by a factor 10 for clarity. For the type II mode the mean value of the Reynolds stress terms $R^{y}$ and $R^{z_{w}}$ changes considerably, albeit only for $N_{fr}\leqslant 375$ and the same relative size is retained with respect to the other terms.

Prior to physical underpinning of the energy term fluctuations, it is necessary to confirm whether these are influenced by the numerical treatment and discretization of the problem. To this goal, preliminary simulations were performed by altering several parameters. These included increasing the grid resolution from $N_{z}\times N_{y}=55\times 55$ to $90\times 90$ and changing the mean flow differentiation method from a sixth- to fourth-order finite difference scheme. Both changes yielded negligible differences in the stability results, corroborating to a physical mechanism as source of the energy term fluctuations.

Based on the previous, the fluctuations are concluded to be caused by the physical response of the instability modes to base flow changes. Figure 15(b,d) indicates that growth rate fluctuations are mainly induced by the fluctuations in the Reynolds stresses $R^{y}$ and $R^{z_{w}}$ and viscous dissipation $D$ . Correlation analysis is used to quantify the link, using all 800 simulation results (all random combinations for all $N_{fr}$ ). The correlation coefficient between the combination of the Reynolds stress and dissipation terms ( $R^{y}+R^{z_{w}}-D$ ) on one hand and the growth rate $\unicode[STIX]{x1D714}_{i}$ on the other are found to be 0.988 and 0.995 for the type I and II modes, respectively. The dissipation evidently adapts itself to the Reynolds stress terms, which is supported by correlation coefficients larger than 0.958 when omitting the dissipation terms (correlating $R^{y}+R^{z_{w}}$ to $\unicode[STIX]{x1D714}_{i}$ ).

For the type I mode, the fluctuations in the Reynolds–Orr terms are relatively small and never change the energy balance structurally. Additionally, the advection terms for this mode, although small in the mean, display weak fluctuations. The fluctuations are small enough that the relative size of the terms in the energy balance is fixed qualitatively; they do not break its structure. This is not the case for the type II mode. Especially the $R^{z_{w}}$ -term experiences fluctuations large enough to drive the term to negative values ( $R^{z_{w}}<0$ ) on the one hand and larger values than the dominating Reynolds stress term ( $R^{z_{w}}>R^{y}$ ) on the other for different random ensembles for fixed $N_{fr}$ . Nevertheless, all fluctuations show a linear convergence trend with $N_{fr}$ , similar to the eigenvalue in figure 14. $A^{\overline{V}}$ displays relatively small fluctuations for type II.

The shear values at the $(z_{w},y)$ -location of the eigenfunction maximum were extracted (see § 4.9) and inspected based on the correlation between different energy terms. For both modes, $R^{z_{w}}$ is highly correlated to the $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}z_{w}$ values (type I: $-$ 0.954, type II: $-$ 0.978), as expected. The $R^{y}$ -term is also most correlated with the $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}z_{w}$ values for mode II, yielding the coefficient 0.763 with respect to $-$ 0.617 for the $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}y$ value. This indicates the dominant role of the $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}z_{w}$ component in the fluctuations of the type II mode.

Figure 16. $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|$ for type I (a) and II (b) (levels span $[1/6,5/6]$ with $\unicode[STIX]{x1D6E5}=1/6$ ) for $N_{fr}=500$ (filled contours) and $450$ (dashed lines). Centre of gravity of $\tilde{\boldsymbol{q}}^{\ast }\boldsymbol{\cdot }\tilde{\boldsymbol{q}}$ for every case (symbols), the case $N_{fr}=500$ (black circle), in (b): $y$ is determined along the $z_{w}$ centre of gravity location, $R^{z_{w}}>R^{y}$ (◃) and $R^{z_{w}}<0$ (▹). $\overline{U}_{w}/\overline{U}_{e}$ levels $0.1,0.2,\ldots ,0.9$ (dotted); in (a), $\overline{U}_{w}/\overline{U}_{e}=0.744$ (white dashed) and $\text{Im}\{A^{\overline{V}}+A^{\overline{W}_{w}}\}=0$ (grey solid). Insets: zooms on rectangles.

The type I eigenfunctions corresponding to $N_{fr}=500$ and 450 (single random sample from the pool of 100 cases) are compared in figure 16(a), confirming the eigenfunction is converged. Furthermore, the centre of gravity of the perturbation velocity contours in the plane is given for all $N_{fr}$ . All points are clustered densely about the indicated $\overline{U}_{w}=0.744\overline{U}_{e}$ contour, slightly lower than the predicted phase speed. Compared to figure 12(b), the points are located close to the maximum of the $R^{z_{w}}$ -integrand. Furthermore, when comparing to figure 13(c), all points turn out to lie in the narrow band where the advection terms are destabilizing. This is indicated by the boundary between the white and black arrows in figure 16(a). The oscillations in this mode appear to be constrained such that the maximum of the eigenfunction remains confined to this narrow band.

The type II eigenfunctions are shown in figure 16(b), corresponding to the $N_{fr}=500$ and 450 cases. Similar to type I, these eigenfunctions display negligible differences. However, the centre of gravity shows a larger spread. The larger spread reconciles well with the topology of the in-plane advection terms in figure 13(f). Indeed, there is no focus point towards which the maximum of the eigenfunction gravitates, in contrast to the case for the type I mode. Nonetheless, the eigenfunction always displays the characteristic shape shown in figure 10; i.e. overarching the entire crossflow vortex. In many cases within the random Monte Carlo pool, however, the eigenfunction distinctively leans to the left or right. Selecting two such eigenfunctions with their maximum located at the left- and rightmost position, the $R^{z_{w}}$ -term was found to have a very high and low (negative) value, respectively. Testing the correlation between the $z_{w}$ -location of the centre of gravity with the $R^{z_{w}}$ -term yields the coefficient $-0.994$ , indicating a direct link between their respective fluctuations. The interpretation follows directly from figure 12(d). Whenever the eigenfunction leans to the left, the destabilizing region of the $R^{z_{w}}$ -integrand increases and vice versa. The shift can be sufficiently large that $R^{z_{w}}$ becomes negative (right shift) or exceeds the $R^{y}$ -term (left shift). In the Monte Carlo framework, out of the 800 solutions these extreme right and left shifts occur 64 and 27 times, respectively. These special occurrences are indicated with the triangles in figure 16. The last occurrences of the right and left shift are observed for $N_{fr}=425$ (once) and 450 (trice), respectively.

The $z_{w}$ -position of the maximum of the eigenfunction gives a direct handle on the convergence of the mode, which is more conclusive than figure 15(d) can show. The fluctuation amplitudes based on 2 standard deviations and difference in the mean values for $N_{fr}=475$ and 450 are reported in table 2. For $N_{fr}>400$ the amplitude becomes smaller than the local grid spacing, $1.2\times 10^{-2}\,\unicode[STIX]{x1D706}_{r}$ and $2.9\times 10^{-2}\,\unicode[STIX]{x1D706}_{r}$ for mode I and II, respectively. Again displaying approximate linear convergence, at $N_{fr}=475$ the amplitudes attain the values $5.48\times 10^{-3}\,\unicode[STIX]{x1D706}_{r}$ and $1.68\times 10^{-2}\,\unicode[STIX]{x1D706}_{r}$ for mode I and II, respectively.

4.3 Divergence of the in-plane flow

As mentioned in § 2.2, the stability approach requires the in-plane velocity field to be solenoidal. However, this cannot be expected from experimentally measured data. It is known from previous work, see Bonfigli & Kloker (Reference Bonfigli and Kloker2007), that the stability results depend on how this issue is approached. However, the order of magnitude of the growth rates is usually preserved. Properly adjusting the fields is out of the current scope. Nevertheless, for completeness, the in-plane divergence is characterized and the effect on the growth rate of the type I mode is estimated in this section.

The maximum divergence levels are attained at the locations in the near-wall region as shown in figure 17(a), where the in-plane $\overline{U}_{w}$ -shear is maximal. The type I eigenfunction displays an overlap for $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|<1/3$ . Outside the near-wall region, the overall magnitude drops significantly. Figure 17(b) illustrates the overall statistical distribution of the divergence in the PIV domain. The standard deviation is 1 % of the maximum in-plane shear in the $\overline{U}_{w}$ -field.

Figure 17. (a) In-plane flow divergence (red contours, levels from $\pm$ 2 to $\pm 8\,\%$ with steps of 2 % of the maximum in-plane $\overline{U}_{w}$ -shear:  $\unicode[STIX]{x1D709}_{m}=16\,\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}$ , negative levels are dashed). In-plane $\overline{U}_{w}$ -shear magnitude of the strong vortex for $N_{fr}=500$ (8 filled contours from 0 to 7 in $\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}$ -units).  $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|$ of most unstable type I mode from figure 10 (a) (dash-dotted contours). Near-wall region ( $y/\unicode[STIX]{x1D706}_{r}\leqslant 0.061$ ) and upper limit PIV domain ( $y/\unicode[STIX]{x1D706}_{r}=0.433$ ) (dotted lines). (b) Occurrence histogram of divergence values, bar width: 0.4 %.

Basic tests were performed to assess the effect of the terms related to the in-plane divergence for the type I mode, with $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{r}=6.2$ . Artificial manipulations of the $\unicode[STIX]{x2202}\overline{V}/\unicode[STIX]{x2202}y$ and $\unicode[STIX]{x2202}\overline{W}_{w}/\unicode[STIX]{x2202}z_{w}$ fields were performed, independently of the other fields, to gauge the change in the eigenvalues. By setting $\unicode[STIX]{x2202}\overline{V}/\unicode[STIX]{x2202}y=\unicode[STIX]{x2202}\overline{W}_{w}/\unicode[STIX]{x2202}z_{w}=0$ and replacing the $\unicode[STIX]{x2202}\overline{V}/\unicode[STIX]{x2202}y$ field by $-\unicode[STIX]{x2202}\overline{W}_{w}/\unicode[STIX]{x2202}z_{w}$ , the growth rate changed by $-0.0142$ and 0.0018 units of $\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}$ , respectively. The former change lies within the error bound indicated by the $2\unicode[STIX]{x1D70E}$ uncertainty specified in table 2 for $N_{fr}=500$ . The latter destabilizing change is qualitatively consistent with the comparison of the $w_{v}$ - and $v_{v}$ – $w_{v}$ -fixed approaches of Bonfigli & Kloker (Reference Bonfigli and Kloker2007, cf. figure 15(a)).

Figure 18. Overlap region definition and variations. In-plane $\overline{U}_{w}$ -shear magnitude of the strong vortex for $N_{fr}=500$ (8 filled contours from 0 to 7 in $\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}$ -units).  $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|=0.5$ for most unstable type I and II modes (white contours). Near-wall region ( $y/\unicode[STIX]{x1D706}_{r}\leqslant 0.061$ ) and upper limit PIV domain ( $y/\unicode[STIX]{x1D706}_{r}=0.433$ ) (dotted lines).

Figure 19. Type I (a) and II (b) eigenvalues ( $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{r}=8.2$ ) for different domain extrapolation parameters (legend indicates $\unicode[STIX]{x1D6FF}_{p}$ ; $\unicode[STIX]{x1D6FF}_{o}$ in units of $\unicode[STIX]{x1D6FF}_{mp}$ ):  $10^{-6}$ (solid line), $10^{-5}$ (dashed) and $10^{-4}$ (dash-dotted) neighbourhoods of the eigenvalues with $\unicode[STIX]{x1D6FF}_{p}=\unicode[STIX]{x1D6FF}_{o}=\unicode[STIX]{x1D6FF}_{mp}$ .

4.4 Effect of wall-normal extrapolation

Another question related to the use of the measured mean flow is what impact the free-stream PIV data extrapolation method has on the results. Specifically, the effect of the overlap region’s parameters is to be quantified. To this end, tests were performed applying significant variations in its position, through $\unicode[STIX]{x1D6FF}_{p}$ , and size, with $\unicode[STIX]{x1D6FF}_{o}$ , see figure 2 for their definition. The largest value for $\unicode[STIX]{x1D6FF}_{p}$ is the height of the PIV domain, $\unicode[STIX]{x1D6FF}_{mp}=0.433\unicode[STIX]{x1D706}_{r}$ . By setting $\unicode[STIX]{x1D6FF}_{p}<\unicode[STIX]{x1D6FF}_{mp}$ , the upper part of the PIV data are artificially altered, which is to be avoided. By increasing $\unicode[STIX]{x1D6FF}_{o}$ , the shear caused by the discontinuity is reduced. Increasing both parameters $\unicode[STIX]{x1D6FF}_{p}$ and $\unicode[STIX]{x1D6FF}_{o}$ should therefore yield converging eigenvalues.

To test this, a $(\unicode[STIX]{x1D6FF}_{p},\unicode[STIX]{x1D6FF}_{o})$ -test matrix was set up, setting $\unicode[STIX]{x1D6FF}_{p}/\unicode[STIX]{x1D6FF}_{mp}=0.9$ , 0.95 and 1 and $\unicode[STIX]{x1D6FF}_{o}/\unicode[STIX]{x1D6FF}_{mp}=0.2$ , 0.6 and 1, see figure 18. For $\unicode[STIX]{x1D6FF}_{p}=0.90\unicode[STIX]{x1D6FF}_{mp}$ the type II mode is covered significantly and $\unicode[STIX]{x1D6FF}_{o}=0.2\unicode[STIX]{x1D6FF}_{mp}$ is comparable to the vorticity thickness of the shear layer, which is expected to influence the results significantly.

The eigenvalue problem was solved, fixing $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{r}=8.2$ and using the $55\times 55$ grid resolution on the $N_{fr}=500$ mean flow. Figure 19 shows the resulting eigenvalues. Both modes converge as $\unicode[STIX]{x1D6FF}_{p}$ and $\unicode[STIX]{x1D6FF}_{o}$ are increased. As expected, the type II mode is affected more than type I, but the absolute eigenvalue errors are smaller than the discretization error. This is attributed to the small eigenfunction magnitudes in the overlap region. It is concluded that, when taking $\unicode[STIX]{x1D6FF}_{p}=\unicode[STIX]{x1D6FF}_{o}=0.433\unicode[STIX]{x1D706}_{r}$ , the base flow extrapolation influences the results negligibly. Table 2 reports the errors for $(\unicode[STIX]{x1D6FF}_{p},\unicode[STIX]{x1D6FF}_{o})/\unicode[STIX]{x1D6FF}_{mp}=(0.95,1.0)$ and $(1.00,0.6)$ . These results justify using the Blasius profile for the extrapolation.

Figure 20. (a) Group (dashed lines) and phase (solid lines) speeds for both modes: $55\times 55$ (lines) and $90\times 90$ (symbols) grid resolution. (b) Gaster transformed temporal amplification rates, $-\unicode[STIX]{x1D714}_{i}/c_{g}$ , (solid, solving (3.9),  $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ ) and spatial growth rates, $\unicode[STIX]{x1D6FC}_{i}$ , (symbols, solving (3.10),  $\unicode[STIX]{x1D714}\in \mathbb{R}$ ).

4.5 Applicability of the Gaster transformation

The secondary vortices are known to be convective instabilities (Bonfigli & Kloker Reference Bonfigli and Kloker2007). They grow in space subject to an imposed frequency, which corresponds to the case where $\unicode[STIX]{x1D6FC}\in \mathbb{C}$ is unknown and $\unicode[STIX]{x1D714}\in \mathbb{R}$ is given; i.e. the spatial problem. Up to now, only the solutions of the temporal problem have been handled. Equation (3.10) illustrates that the spatial stability problem is twice as expensive, because $\unicode[STIX]{x1D6FC}$ appears quadratically in the equations. Solving this problem can be circumvented by applying the Gaster transformation, see Gaster (Reference Gaster1962), based on the fact that spatial and temporal growth are inter-related for convective perturbations. To this end, the simple formula:

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}_{i}|_{\unicode[STIX]{x1D714}_{i}=0}=c_{g}^{-1}\unicode[STIX]{x1D714}_{i}|_{\unicode[STIX]{x1D6FC}_{i}=0}+O(\unicode[STIX]{x1D714}_{i}|_{\unicode[STIX]{x1D6FC}_{i}=0}^{2}), & \displaystyle\end{eqnarray}$$

can be used, where $c_{g}$ is the group speed shown in figure 20(a) for both mode types and compared to the phase speeds. The Gaster transformation is valid for small $\unicode[STIX]{x1D714}_{i}$ -values only, see Gaster (Reference Gaster1962). The inviscid instabilities considered here have relatively large $\unicode[STIX]{x1D714}_{i}$ , which renders its application questionable. Nevertheless, it is well established in the literature that the transformation yields near-exact results in this case, see Malik et al. (Reference Malik, Li, Choudhari and Chang1999) and Koch et al. (Reference Koch, Bertolotti, Stolte and Hein2000). Here, this check is reproduced to rule out different sensitivities of the spatial and temporal stability problems to measurement noise in the modified base flow. Furthermore, the difference is regarded from the point of view of the eigenfunctions and the Reynolds–Orr decompositions of the eigenvalues.

The comparison between the spatial amplification rates, obtained by solving (3.10), and the Gaster transformed temporal growth rates, obtained by solving (3.9) and applying equation (4.1), are shown in figure 20(b), using the $55\times 55$ grid on the $N_{fr}=500$ mean flow at $45.6\,\%$ chord. The eigenvalue error at the most unstable frequencies is $O(10^{-4})$ ; which is in line with the grid resolution accuracy. Thus, next to the agreement with the literature, the spatial and temporal problems do not display a relative sensitivity to the used measured flow field.

Figure 21. Temporal (solid lines, $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ ) and spatial (dashed lines, $\unicode[STIX]{x1D714}\in \mathbb{R}$ ) eigenfunctions ( $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|$ levels span $[1/6,5/6]$ with $\unicode[STIX]{x1D6E5}=1/6$ ) for type I (a, $\unicode[STIX]{x1D714}_{r}\unicode[STIX]{x1D706}_{r}/\overline{U}_{e}=4.5967$ ) and II (b, $\unicode[STIX]{x1D714}_{r}\unicode[STIX]{x1D706}_{r}/\overline{U}_{e}=7.354$ ).  $\overline{U}_{w}/\overline{U}_{e}$ levels $0.1,0.2,\ldots ,0.9$ (dotted).

Figure 22. (a,c) $\unicode[STIX]{x1D714}_{r}$ - and (b,d) $\unicode[STIX]{x1D714}_{i}$ -budgets (bars) of type I (a,b) and II (c,d), complex conjugate eigenvalue (dashed line). Temporal (top bar, $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ ) and spatial (bottom bar, $\unicode[STIX]{x1D714}\in \mathbb{R}$ ) problem.

Although the eigenvalues are virtually identical, this does not warrant similarity of the eigenfunctions or $\unicode[STIX]{x1D714}$ -budgets. Both features are compared in figures 21 and 22 for the most temporally unstable type I and II modes. It is to be noted that the most amplified modes (maximal $\unicode[STIX]{x1D6FC}_{i}$ ) have a slightly lower frequency than the most unstable modes (maximal $\unicode[STIX]{x1D714}_{i}$ ). The spatial and temporal eigenfunctions match closely, the only difference is the slightly larger extent of the temporal eigenfunction. Additionally, the phase distribution, accounting for the direction reversal, is found to be identical.

The $\unicode[STIX]{x1D714}$ -budgets are the same qualitatively, but individual terms show noticeable changes. A new contribution is that of $\unicode[STIX]{x1D6FC}\overline{U}_{w}$ to $\unicode[STIX]{x1D714}_{i}$ . The contribution due to the $2\unicode[STIX]{x1D6FC}_{i}\tilde{u} ^{\ast }\tilde{p}$ -term is negligible; the double integral over $\tilde{u} ^{\ast }\tilde{p}$ evaluates to (numerical) zero. For this particular case, the changes in the individual contributions for the type I mode cancel, to yield $-\unicode[STIX]{x1D714}_{i}\approx \unicode[STIX]{x1D6FC}_{i}\iint \overline{U}_{w}\,\tilde{\boldsymbol{q}}^{\ast }\boldsymbol{\cdot }\tilde{\boldsymbol{q}}\,\text{d}y\,\text{d}z_{w}/||\tilde{\boldsymbol{q}}||^{2}$ . This is not the case in general, shown by the type II case. The variation in the individual contributions adds up to the difference between the $-\unicode[STIX]{x1D714}_{i}$ , indicated by the dashed line, and $\unicode[STIX]{x1D6FC}\overline{U}_{w}$ -terms. For the type I mode, this difference turns out to be small. The dominant advection terms in the $\unicode[STIX]{x1D714}_{i}$ -budget change negligibly, which is expected given the similar eigenfunctions.

Figure 23. Velocity fluctuation fields associated with the eigenmode (solid contours, $|\tilde{u} _{w}|$ , levels: $6.67\,\%$ and $20\,\%$ ), the tomo-PIV total r.m.s. (dashed, $x_{w}$ -component at $x_{w}=15.76~\text{mm}$ , level: $33.3\,\%$ ) and the temporal r.m.s. of the POD mode couple (filled, $\sqrt{\unicode[STIX]{x1D6F7}_{9}^{2}+\unicode[STIX]{x1D6F7}_{10}^{2}}$ at $x_{w}=15.76~\text{mm}$ in Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016), level: $20\,\%$ ) and the hot-wire bandpass filtered fluctuation field (red dashed, effective velocity in the $(X,y)$ -plane at $x_{w}=0~\text{mm}$ , band 2 in Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016), level: $20\,\%$ ). All percentages are relative to the in-plane maximum. Near-wall region for the tomo-PIV and $\overline{U}_{w}/\overline{U}_{e}$ levels $0.5,0.6,\ldots ,0.9$ at $x_{w}=15.76~\text{mm}$ (dotted lines).

4.6 Comparison with experiments

A detailed account is given on the spatial structure of the type I mode by Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016), through means of spectral and POD analysis. In particular, the POD analysis presented by Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) undeniably confirms the presence of the type I mode. This allows a detailed comparison with the retrieved eigenmode in terms of flow structure and spatial growth. Despite the limited temporal resolution of the tomo-PIV technique, the power of the POD method is to extract prominent wavelengths from the experimental data. Based on this, the eigenmode with the same spatial wavelength as the POD mode representing the type I secondary instability reported by Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) is considered. The wavelength is $\unicode[STIX]{x1D706}=4.6~\text{mm}$ , which corresponds to $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{r}=2\unicode[STIX]{x03C0}\times 9\cos 40^{\circ }/4.6\approx 9.4$ . Note that this corresponds to a larger wavenumber than the locally most unstable mode reported in table 1. This is expected; the mode with the largest amplitude at a given location is usually situated closer to the neutral curve at a larger wavenumber.

First, a quantitative comparison with the experimentally measured in-plane amplitude distributions is discussed. Figure 23 shows the absolute amplitude of the $x_{w}$ -velocity component of the type I eigenmode versus both the hot-wire and tomo-PIV measurement results presented by Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016). On the one hand, the eigenmode is compared against the bandpass filtered r.m.s. field associated with the type I mode frequency obtained from hot-wire measurements, see their figure 20 (top centre). The considered band corresponds to slightly higher frequencies, $5$ – $6~\text{kHz}$ , when compared to the frequency associated with the POD mode pair, $4.6~\text{kHz}$ , see table 1. On the other hand, it is compared with the magnitude of the total r.m.s. and the POD pair associated with the type I mode obtained with the tomo-PIV measurements, see their figure 28.

The hot-wire was oriented in the $Z$ -direction, measuring the effective velocity in the $(X,y)$ -plane. The $y$ -velocity component is small, as indicated by the base flow and the eigenfunction, and $x_{w}$ and $X$ deviate only by $5^{\circ }$ , so the measured velocity is representative of the $x_{w}$ -velocity component. Furthermore, the hot-wire was traversed in the $z$ -direction, the data corresponding to the 45 % chord station are here projected onto the $z_{w}$ -coordinate.

POD of the tomo-PIV measurement data gives two phases per advecting mode (shifted by $\unicode[STIX]{x03C0}/2$ ), representing all velocity components in the entire measurement volume. Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) reported this pair as $\unicode[STIX]{x1D6F7}_{9}$ and $\unicode[STIX]{x1D6F7}_{10}$ ; the ninth and tenth POD modes. The Euclidean sum of these modes, weighted with the variance of their respective time coefficients, yields an amplitude distribution with the least phase modulation. The total r.m.s. distribution corresponding to the tomo-PIV measurements is also considered for reference. The attention is focussed on the $x_{w}$ -velocity component, the symbol $\unicode[STIX]{x1D6F7}$ will therefore be used to indicate the spatial structure of that component only. The tomo-PIV data are extracted at the location where the POD mode attains its maximum amplitude, at $x_{w}=15.76~\text{mm}$ . The maximum value of the total r.m.s. $x_{w}$ -velocity component is equal to $0.10\,\overline{U}_{e}$ ; the number used in the uncertainty arguments treated before.

The shape of the $|\tilde{u} _{w}|$ amplitudes shows a qualitative agreement and can also be compared to the results of White & Saric (Reference White and Saric2005) and Serpieri & Kotsonis (Reference Serpieri and Kotsonis2018). The bandpass filtered r.m.s. field is found to have an overall similar spatial structure, but it displays a larger longitudinal extent at the leeward side of the primary vortex (towards $z_{w}/\unicode[STIX]{x1D706}_{r}=1$ ). The POD mode $\unicode[STIX]{x1D6F7}_{9}$ has a significantly lower magnitude than $\unicode[STIX]{x1D6F7}_{10}$ for $z_{w}>0.5\unicode[STIX]{x1D706}_{r}$ , consistent with the lower amplitude of the corresponding total r.m.s. distribution. This corroborates the segmented shape of the Euclidean sum of the POD modes. The total r.m.s. distribution shows perturbations are supported in a broader spanwise range under the primary vortex when considering all frequency content.

Effectively, figure 23 demonstrates the merits of the stability analysis technique as a tool for experimental data reduction. The method is able to isolate the pertinent monochromatic eigenmodes based on the mean measurement data. Especially in the case of advanced flow diagnostic techniques such as tomo-PIV, it is very challenging to distinguish between the physical r.m.s. field of different modes as well as measurement noise. The (most unstable) eigenmodes give an indication of the most dominant frequencies and the expected spatial topology. The proposed methodology extends the information on the perturbation field and, furthermore, it enables enhancing the measurability of desired features by focusing the experimental set-up accordingly.

The shape of the eigenfunctions is found to be wavelength independent. However, the relative magnitudes of the $|\tilde{v}|$ and $|\tilde{w}_{w}|$ components change significantly for different wavelengths. The relative magnitudes of the velocity components of the eigenmode with the wavelength extracted from POD are in close agreement with the total r.m.s. values. The ratios of the in-plane maxima of $|\tilde{v}|$ and $|\tilde{w}_{w}|$ for the eigenmode are: 23 % and 50 %, relative to the maximum of $|\tilde{u} _{w}|$ . The same quantities for the total r.m.s. are: 21 % and 44 %, respectively. These maxima for the eigenmode are located at $(z_{w},y)/\unicode[STIX]{x1D706}_{r}=(0.38;0.23)$ for $|\tilde{u} _{w}|$ and $(0.39;0.25)$ for $|\tilde{v}|$ , the total r.m.s. has both maxima at $(0.36,0.21)$ . The wavelength from POD is used to facilitate this specific comparison. When determined for the most unstable wavenumber reported in table 1, for example, the relative maxima of $|\tilde{v}|$ and $|\tilde{w}_{w}|$ are 14 % and 33 %, respectively, which are lower than the values corresponding to the POD wavelength. This suggests that the mode’s wavelength can be estimated by identifying the eigenmode that has approximately the same amplitude ratios as observed in the total r.m.s. data; POD is not required for that.

Figure 24. $x_{w}$ -velocity isosurfaces of (a) the type I eigenmode at $\unicode[STIX]{x1D714}/2\unicode[STIX]{x03C0}=4.6~\text{kHz}$ ( $\unicode[STIX]{x1D6FC}_{r}=9.4/\unicode[STIX]{x1D706}_{r}$ ), having maximal amplitude 1 at $x=-17.6~\text{mm}$ , plotting the $\pm 84\,\%$ levels, and (b) the tomo-PIV POD mode ( $\unicode[STIX]{x1D6F7}_{10}$ in Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016)), plotting the $\pm 0.086\overline{U}_{e}$ levels, the near-wall region is cut. $\overline{U}_{w}/\overline{U}_{e}$ levels $0,0.05,\ldots ,1$ (contours).

A three-dimensional representation of the eigenmode and POD mode associated with the type I instability is shown in figure 24(a) and (b), respectively, illustrating their spatial structure. The most unstable eigenmode is extrapolated in space, incorporating the exponential growth in space calculated using the Gaster transformation: $-\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D706}_{r}/c_{g}=-0.09702/0.7805=-0.1243$ . These structures are compared to the 10th POD mode, $\unicode[STIX]{x1D6F7}_{10}$ , of Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016). Upstream of $x_{w}=-4~\text{mm}$ the isosurfaces are absent in the POD mode. This is a consequence of the limited dynamic range of this particular tomo-PIV experiment and of the very low perturbation amplitude.

Overall, a qualitative match of the topology is established between the modes, the largest difference being the structures’ length. Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) documented the orientation of the secondary instability structures in terms of their azimuthal angle and inclination: $-18.2^{\circ }$ and $21^{\circ }$ , respectively, with respect to the stationary vortices. The eigenmode displays a comparable azimuthal angle, $-17.8^{\circ }$ , but a smaller inclination: $12^{\circ }$ . The latter angle agrees with the value reported by Janke & Balakumar (Reference Janke and Balakumar2000) and Wassermann & Kloker (Reference Wassermann and Kloker2002); who also report an inclination angle of $12^{\circ }$ . A similar difference in the inclination is observed in the application to the instabilities in the wake of a micro-ramp Groot et al. (Reference Groot, Ye, van Oudheusden, Zhang and Pinna2016); the structures as observed in the tomo-PIV experiment also display a larger inclination in that case.

Figure 25. Eigenmode $\text{Re}\{\tilde{u} _{w}\}/\max |\tilde{u} _{w}|$ (solid contours) (levels: $\pm 6.67\,\%$ and $\pm 20\,\%$ ) and POD mode $x_{w}$ -velocity component ( $\unicode[STIX]{x1D6F7}_{10}$ in Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016)) at $x_{w}=15.76~\text{mm}$ (dashed) (levels: $\pm 20\,\%$ of in-plane maximum), level signs are indicated. Near-wall region and $\overline{U}_{w}/\overline{U}_{e}$ levels $0.5,0.6,\ldots ,0.9$ at $x_{w}=15.76~\text{mm}$ (dotted lines).

The instantaneous flow structures are compared to a higher degree of detail in the $z_{w}y$ -plane in figure 25. The POD mode is extracted at $x_{w}=15.76~\text{mm}$ , maximizing its absolute amplitude. Both modes show the same arrangement of positive and negative perturbation velocity pockets, even in locations where the velocity maxima are small, despite a slight misalignment. The orientation of the contours is the same and can be compared to phase-locked hot-wire measurement observations presented by Kawakami et al. (Reference Kawakami, Kohama and Okutsu1998) and Serpieri & Kotsonis (Reference Serpieri and Kotsonis2018) and the computations of Janke & Balakumar (Reference Janke and Balakumar2000). Two contour levels are shown for the eigenmode, of which the largest corresponds to the POD mode contour level. The lowest level shows that the structure corresponding to the POD mode is broader than the eigenmode in the direction perpendicular to the shear layer, as was already apparent in figure 24. But both modes have the same qualitative shape; both show a contour in the centre that has a large downwards protrusion. This illustrates the stability analysis effectively describes the perturbation flow topology. The broader structures observed in the measurement could be explained by the limited capability of the tomo-PIV experiment in capturing complicated flow structures in the presence of strong shear. From the perspective of the (de)stabilizing action of the in-plane advection terms in (3.7), the broader structure would be more stabilized as this corresponds to a larger white region in figure 13(c).

Having identified the correspondence between the structures of the eigenmode and POD mode, the exponential growth can be analysed. It should be highly stressed here that there is no reason to expect that the eigenmode and POD mode should display the same growth rate. The POD mode is a data-driven, energy-maximizing coherent structure having a broad spectral content, that is, to a degree, corrupted with systematic and random measurement noise. As stated by Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016), other POD modes showed similar structures to that in figure 24 and this corroborates with the broad frequency band in the hot-wire spectrum for type I mode fluctuations. An eigenmode, on the other hand, is purely monochromatic and represents a rigorous solution of the governing equations, making it an entirely different entity. Growth rates are moreover notoriously hard to match, as pointed out with the executed sensitivity study and by verification studies in the computational literature, see Bonfigli & Kloker (Reference Bonfigli and Kloker2007). For this reason, the scope of this comparison serves more as a qualitative comparison for the methodology, rather than a strict validation.

Figure 26. $N$ -factors based on the: POD mode couple r.m.s. (dotted line), linear fit to POD r.m.s. (solid line), bandpass filtered hot-wire anemometry velocity fluctuation fields reproduced from figure 21(b) of Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016, band 2) (red), eigenmode: most unstable (dashed, $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{r}=6.2$ ) and POD wavelength (dash-dotted, $9.4$ ). Uncertainty in linear fit slope and growth rate based on $2\unicode[STIX]{x1D70E}$ in table 2 (shaded).

The r.m.s. field of the coupled POD modes is integrated in both wall-normal and spanwise directions, towards producing a relative amplitude. The $N$ -factor is defined as the natural logarithm of the resulting quantity:

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle N(x_{w})=\ln \left(\iint |\unicode[STIX]{x1D6F7}(x_{w},y,z_{w})|\;\text{d}y\,\text{d}z_{w}\right), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6F7}$ corresponds to the $x_{w}$ -velocity component of the POD mode couple. The integral can be evaluated in the $x_{w}$ -range $[-4,17]$ in millimetres, where the dynamical range of the experiment was sufficient to resolve the mode couple. Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) reported $N$ -curves in their figure 21(b), based on the bandpass filtered r.m.s. data corresponding to the type I frequency range, measured using hot-wire anemometry. That figure illustrates that the former streamwise range does not include the upstream neutral point. For that reason, the currently extracted $N$ -curve is shifted to the value ( $N=2.44$ ) extracted from their results at the location currently investigated. The resulting $N$ -curve is shown in figure 26. The oscillation in the dotted curve, with a wavelength comparable to the individual POD mode of $4.6~\text{mm}$ , reflects the underlying phase undulation of the spatial velocity maxima. A clear growth trend is obtained nonetheless. A linear fit is used to obtain a quantitative means of comparison for the eigenmode growth. The fitted slope corresponds to $-\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D706}_{r}=0.230\pm 0.008$ . This value is significantly larger compared to the growth rate of the eigenmode with the same wavelength, for which $-\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D706}_{r}=0.1243$ . This is reflected in the mismatch of the slopes of the $N$ -curves in figure 26. The grey area about the $N$ -curve for the eigenmode indicates the uncertainty ( $\pm 2\unicode[STIX]{x1D70E}=\pm 0.0106$ units) given in table 2. Including the worst error estimate, the growth rate does not match that extracted from the POD mode. The $N$ -curve corresponding to the most unstable eigenmode is also included, for which $-\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D706}_{r}=0.2041$ . This value only slightly underestimates the value corresponding to the POD mode.

The $N$ -curve corresponding to the bandpass filtered fluctuation field associated with the type I mode (band 2) shown in figure 21(b) of Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016), corresponding to hot-wire measurements of the same vortices, is repeated here. This curve reflects the growth rate $-\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D706}_{r}=0.1270$ , including the projection onto the $x_{w}$ -direction (uncorrected value: $0.1272$ ), which does match the considered eigenmode’s growth to within the uncertainty. The latter match should, however, be interpreted with caution. The experimental curve corresponds to the r.m.s. amplitude averaged over 3 neighbouring vortices, amongst which are both the currently investigated ones, and the curve has a sample spacing of 0.025 % chord, corresponding to $29~\text{mm}$ in the $x_{w}$ -coordinate. In summary, qualitative agreement between the stability analysis and experimental measurements further demonstrates the applicability of the proposed methodology towards enhancing and extending the experimental measurability.

4.7 Effect of primary vortex strength

As mentioned in § 2.3, two neighbouring vortices are measured, where the left-hand side vortex is slightly weaker (27.3 % $\overline{U}_{e}$ ) than the right-hand side vortex (28.7 % $\overline{U}_{e}$ ) considered up to now. Next to the reduced strength, the perturbations have been experimentally identified to be much weaker by using the POD technique in the vicinity of the weak vortex, suggesting that the lower primary amplitude results in a reduced growth of the secondary instability modes.

The mild difference in amplitude between the primary crossflow vortices provides an ideal case in demonstrating the ability of the global stability approach to identify pertinent stability features based on the measured mean flow alone. The analysis performed so far on the baseline stronger vortex is here repeated for the weaker vortex using the domain $-1\leqslant z_{w}/\unicode[STIX]{x1D706}_{r}\leqslant 0$ indicated in figure 3 and a mean field constructed with 500 instantaneous snapshots. For ease of comparison purposes, the domain is translated in the $z_{w}$ -direction, so the $z_{w}$ -coordinate again spans $[0,\unicode[STIX]{x1D706}_{r}]$ . In figure 27, the type I and II mode branches are shown and compared with those corresponding to the stronger vortex. The branches are given for two grid resolutions. The most unstable type I and II modes are again found to be subject to $O(10^{-5})$ and $O(10^{-4})$ eigenvalue errors, respectively.

Figure 27. Type I (circles) and II (squares) temporal stability branches for the weak vortex using $N_{z}\times N_{y}=55\times 55$ (dashed lines) and $90\times 90$ (solid) grid nodes. Strong vortex branches (solid without open symbols). Arrows link type I and II modes with $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{r}=6.2$ and 8.6 (filled symbols), respectively, on the different vortices. All most unstable eigenvalue symbols are filled. Eigenvalues corresponding to domain containing both strong and weak primary vortices using $N_{z}\times N_{y}=140\times 70$ (red symbols). Branches corresponding to the interpolated base flows with $\unicode[STIX]{x1D712}=0.2,0.4,0.6$ and $0.8$ (dotted lines, bottom to top).

Despite the mild differences in the base flow, the stability characteristics of the weaker vortex are drastically changed towards a more stable state. The type I mode is stable for all wavenumbers and type II is marginally unstable, indicating the weak vortex amplitude of 27.3 % $\overline{U}_{e}$ based on (2.1) is close to the neutral secondary instability limit for both currently considered modes. This further corroborates the low perturbation amplitude observed in the experimental flow field. The most unstable modes’ characteristics for both vortices are compared in table 3.

Table 3. Parameters of the most temporally unstable modes in the base spectrum corresponding to the weaker (left) and stronger (right) vortex in figure 3.

Due to the apparent extreme sensitivity of stability on the vortex strength, an order-of-magnitude check is performed by comparing the order of magnitude of the growth rates to the work of Koch et al. (Reference Koch, Bertolotti, Stolte and Hein2000) and Bonfigli & Kloker (Reference Bonfigli and Kloker2007). The current Reynolds number, $Re=\overline{U}_{e}\unicode[STIX]{x1D706}_{r}/\unicode[STIX]{x1D708}=1.32\times 10^{4}$ , while the latter authors’ simulations correspond to $1.34\times 10^{4}$ and $0.87\times 10^{4}$ , respectively. Assuming a comparable integral effect of the pressure gradient, the current results are comparable with those of Koch et al. (Reference Koch, Bertolotti, Stolte and Hein2000), while relatively larger shear levels and therefore growth rates are expected in the case of Bonfigli & Kloker (Reference Bonfigli and Kloker2007). By converting the maximal growth rates in these reference into $\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}$ units, one respectively retrieves the values 0.49 and 1.5, which, compared to the currently found maximal value of 0.16, are significantly larger. (In their nomenclature, for Koch et al. (Reference Koch, Bertolotti, Stolte and Hein2000, cf. figure 18): $(\unicode[STIX]{x1D70E}_{r}L_{ref}^{\ast }/Q_{ref}^{\ast })\times (\unicode[STIX]{x1D706}_{z_{c}}^{\ast }/L_{ref}^{\ast })\times$ $(Q_{ref}^{\ast }/Q_{e}^{\ast })=0.029\times 12/0.7092\times 1=0.49$ and for Bonfigli & Kloker (Reference Bonfigli and Kloker2007, cf. figure 13): $(\text{Im}(\unicode[STIX]{x1D714})L_{ref}^{\ast }/u_{\infty }^{\ast })\times (\unicode[STIX]{x1D706}_{0,z}^{\ast }/L_{ref}^{\ast })\times (u_{\infty }^{\ast }/u_{b,e}^{\ast })=10\times 12/100\times 14/11=1.5.$ ) Similarly, the maximal growth rate of the primary instability reported by Koch et al. (Reference Koch, Bertolotti, Stolte and Hein2000) is 0.12. This is taken as an indication that the currently considered strong vortex lingers close to neutral conditions. This is a reasonable explanation for the apparent large decrease of the growth rate of the strong, as opposed to the weak vortex. The near-neutral conditions are also reasonable in the perspective of the small difference in the vortex amplitudes.

The estimate of the primary amplitude leading to neutral secondary modes of Fischer et al. (Reference Fischer, Hein and Dallmann1993) of 11 % $\overline{U}_{e}$ is rather low compared to the value found here. The order of magnitude is comparable with the results of Wassermann & Kloker (Reference Wassermann and Kloker2002), reporting 30 % $\overline{U}_{e}$ based on the maximum deceleration imposed by the mean flow distortion. Instances of the type I eigenmode being more stable than type II for all wavenumbers are uncommon, e.g. see Koch et al. (Reference Koch, Bertolotti, Stolte and Hein2000). As elaborated on in § 2.3, the in-plane velocity components show a small increase relative to the primary perturbation amplitude based on $\overline{U}_{s}$ . Based on figure 20 of Bonfigli & Kloker (Reference Bonfigli and Kloker2007), this effect should render the type II mode more stable than type I.

For the stronger vortex, the most unstable wavenumbers for the type I and II modes are $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{r}=6.2$ and 8.6, respectively. The arrows in figure 27 link the modes corresponding to these wavenumbers for both vortices. Bonfigli & Kloker (Reference Bonfigli and Kloker2007) show (cf. figure 36b) the frequencies for fixed wavenumbers are proportional to the primary vortex strength, i.e. a decrease of approximately 1.4 % is expected. Instead, the frequency at a fixed wavenumber increases 2.6 % for type I and 1.2 % for type II. By inspection of the terms in the $\unicode[STIX]{x1D714}_{r}$ -budget, the increase of the frequency for a fixed wavenumber cannot be associated with an individual term; it is the integral effect of small changes in all terms. For the type I mode, $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{r}=6.2$ is again most unstable. For the type II mode, the most unstable wavenumber is smaller for the weaker vortex. This behaviour for the type I mode agrees with the results of Koch et al. (Reference Koch, Bertolotti, Stolte and Hein2000), who report a type I branch (cf. figure 18) that has an invariant most unstable frequency at different streamwise locations, although that type I branch is not the most unstable type I harmonic over the considered streamwise range.

To further assess the reliability of the decrease in the growth rate from the strong to the weak vortex, intermediate temporal stability branches are computed based on the flow obtained by artificially interpolating the two vortices considered here. The strength parameter $\unicode[STIX]{x1D712}$ is introduced, defining the interpolated solution as follows:

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{Q}_{\unicode[STIX]{x1D712}}(z_{w},y)=\unicode[STIX]{x1D712}\overline{Q}_{[0,1]}(z_{w},y)+(1-\unicode[STIX]{x1D712})\overline{Q}_{[-1,0]}(z_{w}-\unicode[STIX]{x1D706}_{r},y), & \displaystyle\end{eqnarray}$$

where $\overline{Q}$ denotes any mean flow variable, $\overline{Q}_{\unicode[STIX]{x1D712}}$ is the interpolated flow variable and $\overline{Q}_{[0,1]}$ and $\overline{Q}_{[-1,0]}$ denote the strong and weak vortices, respectively. A similar approach is deployed by Piot (Reference Piot2008) to investigate the effect of a bump on a boundary layer flow. The values $\unicode[STIX]{x1D712}=$ 0.2, 0.4, 0.6 and 0.8 are considered and the attention is restricted to the type I mode. The results are shown as the dotted lines in figure 28. Evidently, transitioning from the strong to the weak vortex corresponds to a consistent and monotonic decrease of the branch in the $\unicode[STIX]{x1D714}$ -plane, further confirming that the decreased growth rate is not a random artefact of the measured flow representation.

A comparison of the most unstable eigenfunctions of the different vortices is shown in figure 28. Both eigenfunctions display a broader support about the vortex. This behaviour is qualitatively comparable to the findings of Koch et al. (Reference Koch, Bertolotti, Stolte and Hein2000). Furthermore, the maxima of the functions have a higher position relative to the distorted base flow contours, which is reflected by slightly higher phase speeds, see table 3. In turn, this is directly linked to the slight increase of the frequencies at constant wavenumber discussed before. The difference in the orientation of the highest level contour of the type II mode is important to note. For the stronger vortex, this is located to the left of the primary vortex core and tilted to the left, whereas for the weaker vortex it is located and tilted to the right.

Figure 28. $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|$ for type I (a) and II (b) (levels span $[1/6,5/6]$ with $\unicode[STIX]{x1D6E5}=1/6$ ) for the weak (filled contours) and strong (dashed) vortex. $\overline{U}_{w}/\overline{U}_{e}$ levels $0.1,0.3,\ldots ,0.9$ for the weak (dash-dotted) and strong (dotted) vortex.

Figure 29. $\unicode[STIX]{x1D714}_{i}$ -budgets (bars) of type I (a) and II (b), $\unicode[STIX]{x1D714}$ eigenvalues (dashed lines). Bar triplets: strong vortex, nominal $Re$ (top, ‘1’); weak vortex, nominal $Re$ (middle, ‘2’); strong vortex, lower Reynolds number (bottom, ‘3’): $10^{-0.54}Re$ (a) and $10^{-0.55}Re$ (b).

Figure 30. Imaginary part of the (a,e) $R^{y}$ -, (b,f) $R^{z_{w}}$ -, (c,g) $A^{\overline{V}}$ - and (d,h) $A^{\overline{W}_{w}}$ -integrands for type I (a–d) and II (e–h) on the weaker vortex (9 levels span (b) $[-0.300,47.2]$ , (c) $[-0.873,7.62]$ , (d) $[-4.69,5.59]$ , (e) $[-6.74,4.42]$ , (f) $[-8.76,6.21]$ , (g) $[-3.20,4.00]$ , (h) $[-5.01,3.56]$ , negative contours are dashed). Levels in (a) span $[-27.8,-3.63]$ with $\unicode[STIX]{x1D6E5}=4.84$ , and $[0.300,0.900]$ , $\unicode[STIX]{x1D6E5}=0.300$ . All values are given in $\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}^{3}$ -units. Eigenfunction contour $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|=1/6$ (dash-dotted lines). $\overline{U}_{w}/\overline{U}_{e}$ levels $0.1,0.2,\ldots ,0.9$ (dotted) for the weak vortex.

The previous analysis clarifies that while the spatial topology of the type I and type II modes is rather insensitive to mild changes in the base flow, their respective growth rates are strongly affected. To identify the physical mechanism that renders the eigenmodes more stable for the weaker vortex, the energy balances corresponding to the most unstable modes is displayed in figure 29 (top and middle bars per term). For the type I mode, the vortex strength difference leads to a decrease of both $R^{y}$ and $R^{z_{w}}$ in the $\unicode[STIX]{x1D714}_{i}$ -budget. Note that the size of $A^{\overline{W}_{w}}$ persists. The topology of the production terms related to the Reynolds stress and advection terms is represented in figure 30. The highest level contours are nearly identical to those observed in figures 12 and 13, which explains the similarity of the $\unicode[STIX]{x1D714}_{i}$ -budgets. The shape displayed by the lower contour levels is quite different, however, and explains the differences in the eigenfunction shape. As the $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}z_{w}$ shear component is smaller for the weaker vortex, other productive contributions come into play in the region located above the primary vortex. Figure 30(a,c,d) shows productive contributions by the integrands of $R^{y}$ , $A^{\overline{V}}$ and $A^{\overline{W}_{w}}$ , respectively. The downwards protrusion of the eigenfunction about the point $(z_{w},y)/\unicode[STIX]{x1D706}_{r}=(0.45;0.22)$ in figure 10(a), associated with the marginally positive $R^{y}$ contribution in figure 12(a), is absent in figure 28(a). In figure 30(a), the equivalent $R^{y}$ -integrand contours have a smaller magnitude and extend less in the direction orthogonal to the shear layer. The local maximum of the $R^{y}$ -integrand in the neighbourhood of the protrusion has dropped from 1.41 to $0.92\,\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}^{3}$ for the strong and weak vortices, respectively.

For the type II mode, unexpectedly, the main Reynolds stress production term, $R^{y}$ , exerts a virtually identical contribution in the $\unicode[STIX]{x1D714}_{i}$ -budget shown in figure 29. In fact, the production term $R^{z_{w}}$ is largely responsible for the stabilization relative to the stronger vortex. This illustrates that, although the type II instability is mainly generated by the Reynolds stress associated with the $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}y$ shear component, in this case the other component is the main translator of the vortex strength. The apparent link between the spanwise location of the eigenfunction’s maximum and the Reynolds stress $R^{z_{w}}$ , first encountered in § 4.2, reappears here; as the maximum of the eigenfunction moves in the positive $z_{w}$ -direction, this production term decreases. The relation to the topology of the production term can be deduced by comparing figure 30(f) with 12(d). The negative contours have approximately the same magnitude, but the positive productive contours change quite considerably, the maximum reducing from 16.2 to $6.9\,\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}^{3}$ for the strong and weak vortices, respectively. Lastly, although it has a small overall magnitude, $A^{\overline{V}}$ reduces significantly in the $\unicode[STIX]{x1D714}_{i}$ -budget; it is comparable to the decrease in $D$ . Comparing figures 30(g) and 13(d), the levels corresponding to the weaker vortex are smaller and are more balanced in the $z_{w}$ -direction than those corresponding to the stronger vortex.

4.8 Effect of periodic boundary conditions

As mentioned in § 3, the measured flow fields of the single strong and weak vortices have not been periodized. The coefficients are left discontinuous across the boundary, such that no artificial shear layer is introduced. To assess the impact of this approach on the solutions, the problem was set up for the domain containing both vortices, as shown in figure 3, herein denoted as the double-vortex domain. The problem was evaluated at the most unstable wavenumbers presented in table 3. Given the domain is twice as large, the resolution had to be increased accordingly. The currently available resources maximally allowed $N_{z}\times N_{y}=140\times 70$ , which, for this domain, represents a resolution in between the cases $55\times 55$ and $90\times 90$ on the single-vortex domains.

Figure 31. $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|$ for type I (a) and II (b) (levels span [ $1/6~5/6$ ] with $\unicode[STIX]{x1D6E5}=1/6$ ) of respective modes for both vortices, computed on the single (filled contours) and double (dashed) domain. To emphasize: the eigenfunctions on the different vortices correspond to different eigenvalues; each has a support limited to one vortex. $\overline{U}_{w}/\overline{U}_{e}$ levels $0.1,0.2,\ldots ,0.9$ (dotted). Domain separation for strong (right) and weak (left) vortex (vertical dotted line, $z_{w}/\unicode[STIX]{x1D706}_{r}=0$ ).

The resulting eigenvalues match up to $O(10^{-4})$ absolute errors with those presented in table 3, and effectively collapse in figure 27. The corresponding eigenfunctions are shown in figure 31. Note that each eigenfunction on the different vortices corresponds to a different eigenvalue in the spectrum of the double-vortex domain problem. They are compared to the eigenfunctions retrieved with the single-vortex domains, adjusted to appropriately illustrate their support on the double-vortex domain, using the periodic boundary conditions. All eigenfunctions match perfectly, even the smaller amplitude contours of the type II mode on the weak (left) vortex. Moreover, despite the fact this eigenfunction significantly protrudes the $z_{w}/\unicode[STIX]{x1D706}_{r}=0$ boundary, it does not experience distortion due to the minor discontinuity in the coefficients for the single-vortex domain. This illustrates that, in this case, using discontinuous coefficients across the boundary is justified.

These results are very similar to those presented by Bonfigli & Kloker (Reference Bonfigli and Kloker2007, cf. § 6.2) and Choudhari et al. (Reference Choudhari, Li and Paredes2016). The current results corroborate the notion that the type I and II eigenmodes on different vortices do not participate in the same resonance; they correspond to different modes in the collective spectrum. In this sense, they are proper discrete modes. In addition, in this case, the single-vortex domain problems are representative of the dynamics in the double-vortex domain. So, it is deduced that, given the primary vortices are reasonably separated in space, it is not necessary to consider the more expensive double-vortex domain problem. Neighbouring primary vortices do not contribute crucial information, in that case.

4.9 Reynolds number dependence

Following the analysis of Bonfigli & Kloker (Reference Bonfigli and Kloker2007), the type I and II modes are Kelvin–Helmholtz instabilities and hence display an independency of the Reynolds number when large enough; this is as opposed to viscous Tollmien–Schlichting instabilities that are stable in the Rayleigh limit, see Schlichting et al. (Reference Schlichting, Krause, Oertel, Gersten and Mayes2003) and Drazin & Reid (Reference Drazin and Reid2004). The stability problem for the strong vortex is solved, varying the Reynolds number artificially and fixing all other parameters. The type I and II modes are each evaluated at the most unstable wavenumber ( $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{r}=6.2$ and 8.6, respectively). With increasing Reynolds number, the eigenfunction shapes become very slender. Therefore a $90\times 90$ grid is used to ensure accurately capturing all structural details.

It is well known that viscosity has a significant impact on the stability of free shear layers when the parameter

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}{\displaystyle \frac{\unicode[STIX]{x1D6FF}_{v}}{2}}\,Re_{\unicode[STIX]{x1D6FF}_{v}}\equiv \unicode[STIX]{x1D6FC}{\displaystyle \frac{\unicode[STIX]{x1D6FF}_{v}}{2}}\,{\displaystyle \frac{\unicode[STIX]{x0394}\overline{U}_{w}\,\unicode[STIX]{x1D6FF}_{v}}{4\unicode[STIX]{x1D708}}}=\unicode[STIX]{x1D6FC}{\displaystyle \frac{\unicode[STIX]{x1D6FF}_{v}}{2}}\,{\displaystyle \frac{\unicode[STIX]{x0394}\overline{U}_{w}/2}{\overline{U}_{e}}}{\displaystyle \frac{\unicode[STIX]{x1D6FF}_{v}/2}{\unicode[STIX]{x1D706}_{r}}}Re, & \displaystyle\end{eqnarray}$$

is of $o(10^{2})$ (small- $o$ notation) (Tatsumi, Gotoh & Ayukawa Reference Tatsumi, Gotoh and Ayukawa1964; Michalke Reference Michalke1972). Here, $\unicode[STIX]{x1D6FF}_{v}$ and $\unicode[STIX]{x0394}\overline{U}_{w}$ are the vorticity thickness and velocity difference relevant for the particular instability. In the present case, these parameters are determined by quantifying the in-plane shear components $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}z_{w}$ and $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}y$ at the location where the $|\tilde{u} |$ -amplitude is maximal. Along the direction indicated by the shear components, denoted by $\unicode[STIX]{x1D703}$ , the closest minimum or, if no minimum exists, the closest ‘favourable’ inflection point of $\overline{U}_{w}$ is determined under the layer of interest. The difference between the free-stream velocity and $\overline{U}_{w}$ at this point is $\unicode[STIX]{x0394}\overline{U}_{w}$ . The vorticity thickness is defined as:

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}_{v}\equiv {\displaystyle \frac{\unicode[STIX]{x0394}\overline{U}_{w}}{\sqrt{\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{U}_{w}}{\unicode[STIX]{x2202}z_{w}}}\right)^{2}+\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{U}_{w}}{\unicode[STIX]{x2202}y}}\right)^{2}}}}. & \displaystyle\end{eqnarray}$$

The inset in figure 32 illustrates the location at which the shear components are extracted together with the vorticity thickness $\unicode[STIX]{x1D6FF}_{v}$ and orientation $\unicode[STIX]{x1D703}$ . All relevant parameters are reported in table 4, including: the parameter $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}_{v}Re_{\unicode[STIX]{x1D6FF}_{v}}/2$ , the extraction location, the $\overline{U}_{w}$ -shear components and the angle $\unicode[STIX]{x1D703}$ . Using these scales, customized for each mode, insight is gained into the relative ‘efficiency’ of the type I with respect to the II mode.

The growth rates in the $\unicode[STIX]{x1D6FF}_{v}$ -scaling are presented with full black symbols in figure 32. It is evident that the growth rates saturate with increasing Reynolds number. Viscosity has a significant effect in the nominal case. This is as expected, because all values $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}_{v}Re_{\unicode[STIX]{x1D6FF}_{v}}/2<100$ . The parameter values for the type II mode are smaller than those for type I, causing the type II growth rate to saturate more slowly.

Figure 32. Type I ( $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}_{r}=6.2$ , circles) and II ( $8.6$ , squares) growth rate in $\unicode[STIX]{x1D6FF}_{v}$ -scaling versus $Re$ for the strong vortex. Eigenvalues: computed (filled symbols), excluding $A^{\overline{V}}$ and $A^{\overline{W}_{w}}$ (open symbols), weak vortex (grey symbols). Nominal $Re$ (vertical dashed line), interpolation of weak vortex onto strong vortex data (arrows). Inset: $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|=0.5$ for both modes at nominal $Re$ and $\overline{U}_{w}/\overline{U}_{e}$ levels $0.4,0.5,\ldots ,0.9$ for the strong (solid respectively dotted lines) and weak (dashed respectively dash-dotted) vortex. $|\tilde{u} |$ -maximum location (crosses) and $\unicode[STIX]{x1D6FF}_{v}$ in the $\unicode[STIX]{x1D703}$ -direction (bars).

It is striking that the type II mode is the most unstable of the two, in the custom per-mode scaling. The type II mode is the more efficient mechanism. This suggests that the Reynolds stress production terms do not contribute to the growth rate as in the case of a one-dimensional shear layer. For the strong vortex, both shear components act constructively for the type II mode, while the $\unicode[STIX]{x2202}\overline{U}_{w}/\unicode[STIX]{x2202}y$ component acts destructively for the type I mode. However, next to these Reynolds stresses, there are effects associated with the in-plane velocity components; i.e. the equivalents of non-parallel flow effects. The largest contributions in the $\unicode[STIX]{x1D714}_{i}$ -balance, except $D$ , are found to vary mildly with the Reynolds number as shown in figure 29 (compare top and bottom bars per term). Only the Reynolds stress production terms in the type II budget show an increase, but have the same character. The non-parallel contributions also retain the same character; $\overline{V}$ -advection destabilizes the type II mode and the $\overline{W}_{w}$ -term stabilizes type I.

Table 4. Vorticity thickness parameters for the nominal most unstable modes. Shear component values are in units of $\overline{U}_{e}/\unicode[STIX]{x1D706}_{r}$ .

Based on the previous observations, further links are sought between growth, base flow strength and Reynolds number. Figure 32 suggests the eigensolutions corresponding to the weak vortex have common features with those on the strong vortex at a lower Reynolds number. The horizontal arrows in figure 32 indicate the interpolation of the weak vortex growth rates onto the curve with varying Reynolds number of the strong vortex. This interpolation yields nearly matching Reynolds numbers for both mode types, $10^{-0.54}Re=3770$ for type I and $10^{-0.55}Re=3700$ for type II. The mild variation in the energy budgets with the Reynolds number mentioned before implies that the increased viscous dissipation term is directly equivalent to the net reduction of the Reynolds stress terms for the weaker vortex case. Figure 29 visualizes this; the energy budgets are given for these specific Reynolds numbers. For the type II instability, for example, the change in $D$ closely matches the change in the $R^{z_{w}}$ -term.

Despite these equivalences, the eigenfunctions for the two cases are different. As mentioned before, the main dependency of the eigenfunctions on the Reynolds number is their respective width orthogonal to the shear layer. The eigenfunctions corresponding to the strong vortex are shown in figure 33 for various Reynolds numbers. For an increasing Reynolds number, the eigenfunction focuses about the region where the Reynolds stresses are active. The increase in width is caused by viscous diffusion. The distinction is attributed to the fact that the dissipation acts on all velocity components, while both dominant Reynolds stress terms produce or destroy the $\tilde{u} _{w}$ component directly. In the weak vortex case the latter is reduced causing a redistribution of the energy balance at local points in the $z_{w}y$ -plane. The decreased Reynolds number instead has a global impact on all terms, which does not cause a significant redistribution within the plane.

Using the energy decomposition, the solely parallel effects can be separated from the total contributions in the eigenvalue information. This is done by subtracting all contributions involving the $\overline{V}$ - and $\overline{W}_{w}$ -components from the computed eigenvalue, i.e. all associated advection and Reynolds stress terms, including those in the remainder. The results are the empty symbols shown in figure 32. It is revealed that, in close proximity to the nominal Reynolds number for the strong vortex case, the modes are equally matched. In terms of the growth rate, this demonstrates both mode types are the offspring of the same parallel instability mechanism. This is a non-trivial result regarding the productive and destructive character of the Reynolds stress terms. Nevertheless, the cumulative effect is the same in this particular range.

Figure 33. $|\tilde{u} _{w}|/\max |\tilde{u} _{w}|=1/6$ for Reynolds number: $10^{-0.8}Re$ (dashed line), nominal (solid) and $10^{0.8}Re$ (dash-dotted) for type I (a) and II (b). $\overline{U}_{w}/\overline{U}_{e}$ levels $0.1,0.2,\ldots ,0.9$ (dotted).

The previous analysis demonstrates that the eigensolutions incorporating all non-parallel effects, including non-trivial redistribution, generation and destruction effects imposed by the $\overline{V}$ - and $\overline{W}_{w}$ -components, can be recast into a self-similar parallel form that only depends on the details associated with the main shear layers, viz.  $\unicode[STIX]{x1D6FF}_{v}$ and $\unicode[STIX]{x0394}\overline{U}_{w}$ . Capturing those details sets the main physical basis of the perturbation and the non-parallel velocity components are extra effects. Note that the reverse approach, i.e. performing the stability analysis on the $\overline{U}_{w}$ -field only, does not necessarily yield the same result due to the redistribution imposed by the in-plane velocity components.

From the relationship governing the inviscid stability of the piecewise linear shear layer, see Drazin & Reid (Reference Drazin and Reid2004):

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle (2\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}_{v}/\unicode[STIX]{x0394}\overline{U}_{w})^{2}=(1-2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}_{v})^{2}-(\text{e}^{-2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}_{v}})^{2}, & \displaystyle\end{eqnarray}$$

the maximal temporal growth rate $\unicode[STIX]{x1D714}_{i}=0.2012\unicode[STIX]{x0394}\overline{U}_{w}/\unicode[STIX]{x1D6FF}_{v}$ is found, which is significantly larger than the limiting values shown in figure 32. Although it is not in the scope of the current paper, next to the destructive nature of the Reynolds stress terms, other effects like the two-dimensionality imposed by the shear layer’s finite spanwise extent and wall proximity have to be carefully factored before the results can be expected to be comparable to the one-dimensional shear layer characteristics, see Groot et al. (Reference Groot, Ye, van Oudheusden, Zhang and Pinna2016) (cf. figure 6 (left)). Both effects are stabilizing and not accounted for in the simple scaling. Drazin & Reid (Reference Drazin and Reid2004) show this for the wall proximity. Measured orthogonally with respect to the shear layers, their centres are located more than $2\unicode[STIX]{x1D6FF}_{v}$ from the wall, which indicates no significant effect. The modes have a wavelength in the $z_{w}y$ -plane parallel to the shear layer, which has a stabilizing effect through viscous dissipation, but is not accounted for in the one-dimensional case. In the case of the strong vortex, the most unstable modes have approximately equal such wavelengths ( ${\approx}0.3\unicode[STIX]{x1D706}_{r}$ ) for the larger part of the domain and therefore has an equal impact for both modes.