5.1. Local LSA

For the LSA, the time- and space-dependent flow $\boldsymbol {u}(\boldsymbol {x},t)$ is decomposed into a time-averaged part $\bar {\boldsymbol {u}}(\boldsymbol {x})$, and a fluctuating part $\boldsymbol {u}'(\boldsymbol {x},t)$:

(5.1)\begin{equation} \boldsymbol{u}(\boldsymbol{x},t)=\bar{\boldsymbol{u}}(\boldsymbol{x})+\boldsymbol{u}'(\boldsymbol{x},t). \end{equation}

For the stability analysis we aim to describe the fluctuating part as normal modes. Therefore, (5.1) is substituted into the full nonlinear equations of motion and the continuity equation for an incompressible isothermal flow, which yields the following set of differential equations in non-dimensional form:

(5.2)\begin{gather} \frac{\partial{\boldsymbol{u}'}}{\partial t} + (\boldsymbol{u}' \boldsymbol{\cdot}\boldsymbol{\nabla}) \bar{\boldsymbol{u}} +(\bar{\boldsymbol{u}} \boldsymbol{\cdot}\boldsymbol{\nabla}) \boldsymbol{u}' + \boldsymbol{\nabla} {p}'-\nu \nabla^2 \boldsymbol{u}' ={-}(\boldsymbol{u}' \boldsymbol{\cdot}\boldsymbol{\nabla}) \boldsymbol{u}' + \overline{(\boldsymbol{u}'\boldsymbol{\cdot}\boldsymbol{\nabla}) \boldsymbol{u}'}, \end{gather}

(5.3)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}' = 0, \end{gather}

with $\nu =\nu ^\ast / u_{\mathrm cl} r_{1/2}$ as the non-dimensional kinematic viscosity. The first nonlinear term on the right-hand side of (5.2) is neglected due to linearization ($(\boldsymbol {u}'\boldsymbol {\cdot }\boldsymbol {\nabla })\boldsymbol {u}'=0$) and the second nonlinear term $\overline {(\boldsymbol {u}'\boldsymbol {\cdot }\boldsymbol {\nabla })\boldsymbol {u}'}$ vanishes for frequencies $\omega \neq 0$. Furthermore, we employ an eddy viscosity which modifies the linear operator. Based on dimensional considerations, one may assume $\nu _t^\ast \sim u_{\mathrm cl} r_{1/2}$, which implies $\nu _t$ to be constant in $x$ and dependent on $r$. For the sake of simplicity, we employ an eddy viscosity that is also constant in $r$. A radially dependent eddy viscosity was also tested which only had marginal impact on the results of the LSA (not shown here). The effective $\nu _t$ was obtained by a least-square fit of the Boussinesq equation to the measured mean profile and Reynolds stresses according to

(5.4)\begin{equation} \nu_t={-}\frac{\overline{u'v'}}{\textrm{d}\bar{u}/\textrm{d} r}, \end{equation}

yielding $\nu _t=0.0259$, which corresponds to an effective Reynolds number of ${Re_{eff}}=\nu _t^{-1}=38.6$. The two orders of magnitudes lower molecular viscosity $\nu$ was subsequently neglected for simplicity. The eddy viscosity is incorporated in (5.2) and (5.3), yielding

(5.5)\begin{gather} \frac{\partial{\boldsymbol{u}'}}{\partial t} + (\boldsymbol{u}' \boldsymbol{\cdot}\boldsymbol{\nabla}) \bar{\boldsymbol{u}} +(\bar{\boldsymbol{u}} \boldsymbol{\cdot}\boldsymbol{\nabla}) \boldsymbol{u}' + \boldsymbol{\nabla} {p}'-\nu_t \nabla^2 \boldsymbol{u}'=0 \end{gather}

(5.6)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}' = 0. \end{gather}

For the solution of (5.5) and (5.6) we employ a quasi-parallel spatial stability approach. The fluctuating quantities are expressed as spatially growing or decaying normal modes which are periodic in the axial and azimuthal directions, i.e.

(5.7)\begin{gather} {\boldsymbol{u}'}(\boldsymbol{x},t)=\hat{\boldsymbol{u}}(r)\exp({\textrm{i}(\alpha x+ m \theta -\omega t)}) + c.c., \end{gather}

(5.8)\begin{gather}{p'}(\boldsymbol{x},t)=\hat{p}(r)\exp({\textrm{i}(\alpha x+ m \theta -\omega t)}) + c.c., \end{gather}

where $\alpha$ is the complex streamwise wavenumber, $\omega$ the real frequency, $m$ the real azimuthal wavenumber, $\hat {\boldsymbol {u}}$ the complex radial amplitude function and $c.c.$ represents the corresponding complex conjugate.

With the normal mode formulation (5.7) and (5.8), (5.5) and (5.6) can be rearranged into an eigenvalue problem

(5.9)\begin{equation} \boldsymbol{D}\hat{\boldsymbol{q}}(r)=\alpha \boldsymbol{E} \hat{\boldsymbol{q}}(r), \end{equation}

with complex eigenvalues $\alpha =\alpha _r+\textrm {i}\alpha _i$ and complex eigenfunctions $\hat {\boldsymbol {q}}=[\boldsymbol {\hat u},\hat p]^{\mathrm {T}}$. Within the analysis, homogeneous Dirichlet boundary conditions are set at $r\rightarrow \infty$. On the jet axis, the boundary conditions depend on the azimuthal wavenumber $m$. All boundary conditions are set in accordance with the method used in Khorrami, Malik & Ash (Reference Khorrami, Malik and Ash1989), which also involves a matrix companion method to reduce the quadratic eigenvalue problem to a linear problem in $\alpha$. The resulting eigenvalue problem (5.9) is solved numerically using a spectral method (Khorrami et al. Reference Khorrami, Malik and Ash1989). Detailed explanation about the numerical approach can be found in Oberleithner et al. (Reference Oberleithner, Sieber, Nayeri and Paschereit2011).

To obtain an overall assessment of the linear stability of the mean flow, figure 9(a) depicts the spatial growth rate $-\alpha _i$ obtained by the LSA for azimuthal wavenumbers ranging from $m=0$ to $m=5$. The growth rates correspond to those of the leading (least stable) mode at each azimuthal wavenumber and are shown as a function of frequency $\omega$. The results reveal that only mode $m=1$ is linearly unstable with the neutral point at $\omega _n=0.33$. A spatio-temporal analysis was further conducted to ensure that the $m=1$ mode is not absolutely unstable (not shown here). All other azimuthal wavenumbers are stable over the entire frequency range. For $m\geq 2$, the growth rates of the leading mode continuously decrease with increasing $m$. This trend is in line with the SPOD eigenvalue spectrum that shows less pronounced low-rank behaviour for higher azimuthal wavenumbers.

Figure 9.

(a) Growth rate $-\alpha_i$ of least stable eigenmode vs frequency $\omega$ for azimuthal wavenumbers $m=0$ to $m=5$. (b) LSA eigenvalue distribution for $m=1$: growth rate $-\alpha _i$ vs streamwise wavenumber $\alpha _i$ at $\omega =0.2$ (unstable) and (c) at $\omega=1$ (stable). Colour coding corresponds to the alignment metric $\mathcal {M}$ between the first SPOD mode and eigenfunctions from LSA.

To exemplify the LSA results, the eigenvalue spectrum of azimuthal wavenumber $m=1$ is shown in figure 9(b) at a frequency of $\omega =0.2$ (unstable regime) and in figure 9(c) at $\omega =1$ (stable regime). Both spectra show several discrete eigenvalues marked by colour coded circles and a continuous spectrum. For $\omega =0.2$ two continuous branches merge along $\alpha _r \approx 0$ and two continuous branches merge at higher axial wavenumbers that originate at $-\alpha _i=2$ and $\alpha _r=-2$. The latter two branches strongly depend on the frequency and move outside the shown domain for $\omega =1$ (Salwen & Grosch Reference Salwen and Grosch1981; Rodríguez et al. Reference Rodríguez, Cavalieri, Colonius and Jordan2015). The eigenvalues within these branches depend on the numerical discretization and contain core and shear-layer modes, as shown by Rodríguez et al. (Reference Rodríguez, Cavalieri, Colonius and Jordan2015). In contrast, the discrete eigenmodes are numerically converged and reveal, for $\omega =0.2$, an unstable mode that is separated from the remaining sub-leading modes. At $\omega =1$, the leading mode is also separated from the sub-leading modes but features a negative growth rate, corresponding to a spatially decaying mode. The overall structure of the eigenspectra as shown here is the same for most frequencies and azimuthal wavenumbers considered in this work, with a clear separation of continuous and discrete modes, even at very high frequencies where the discrete modes are strongly damped. Only at near-zero frequencies is it found that the discrete modes interfere with the third and fourth continuous branches.

In order to assess the congruence of the LSA eigenspectrum with the SPOD modes, the discrete modes are colour coded with an alignment metric $\mathcal {M}$. It is defined as (Cavalieri et al. Reference Cavalieri, Rodríguez, Jordan, Colonius and Gervais2013)

(5.10)\begin{equation} \mathcal{M}=\frac{|\left\langle \hat{\boldsymbol{u}}_{m,\omega}, \boldsymbol{a}^1_{m,\omega}\right\rangle|}{\left\|\hat{\boldsymbol{u}}_{m,\omega}\right\| \left\| \boldsymbol{a}^1_{m,\omega}\right\|}, \end{equation}

where the mode shapes are weighted by the radial coordinate $r$ and $\langle \cdot , \cdot \rangle$ denotes the inner product between the weighted mode shapes from LSA and the first SPOD mode and $\|\cdot \|$ denotes the corresponding Euclidean norm. A value of $\mathcal {M}=1$ indicates perfect alignment between the LSA eigenfunctions and the first SPOD mode while $\mathcal {M}=0$ corresponds to completely uncorrelated mode shapes. As shown in figure 9(b,c) for both frequencies, the leading mode agrees very well with the first SPOD mode. In the unstable regime ($\omega =0.2$), the second least stable eigenvalue is also in fairly good agreement with the first SPOD mode. In contrast, the leading mode in the stable regime ($\omega =1$) is the only mode that shows excellent alignment with the empirical mode. The dominance of the leading mode in relation to the sub-leading modes can be interpreted as a low-rank behaviour of the LSA modes. This behaviour applies to a wide frequency and azimuthal wavenumber range, as will be demonstrated in the following.

Figure 10 shows the alignment between the first SPOD mode and the five leading LSA modes for various frequencies and azimuthal mode numbers. The congruence between the first SPOD and leading LSA mode of azimuthal wavenumber $m=0$, shown in the top row of figure 10(a), is excellent over a wide range of frequencies. However, the agreement drops at low and very high frequencies. At low frequencies, the third least stable mode (third row) is in better agreement with the empirical mode shapes. For $m=0$ the second and fourth least stable modes are only weakly correlated with the SPOD modes with values of $\mathcal {M}<0.2$. The poor alignment of these eigenmodes with the first SPOD mode is reasonable since the second and fourth least stable LSA modes represent shear-layer modes which are uncorrelated with the first $m=0$ SPOD mode that represents a core mode. The blank areas in the second, fourth and fifth lines correspond to regions where the eigenvalues could not be distinguished from one another properly. For azimuthal wavenumber $m=1$, the leading LSA mode is in excellent agreement with the empirical modes up to the highest resolved frequencies. At the lower end of the resolved frequency range, the leading LSA mode does not represent the first SPOD mode and neither is any sub-leading mode in agreement with the empirical modes. Thus, the leading LSA mode shows dominant low-rank behaviour over almost the entire frequency range except for very low frequencies. The blank areas in the alignment contours of the fourth and fifth least stable modes in figure 10(b) are again due to difficulties in tracking the eigenvalues due to their interference with the continuous spectrum. For azimuthal wavenumber $m=2$, the leading LSA mode is in very good agreement with the first SPOD mode over the entire frequency range and additionally the second leading mode is in good agreement at low frequencies. Higher azimuthal wavenumbers $m>2$ generally feature very similar trends as observed for $m=2$. In summary, at mid-range to high frequencies, the empirical mode shapes of all azimuthal wavenumbers are represented very well solely by the leading LSA mode which is clearly prominent in terms of alignment and separation from the sub-dominant modes. This also holds for high azimuthal wavenumbers and frequencies where the growth rate is negative and very low.

Figure 10.

Alignment between first SPOD mode and five least stable LSA modes. Alignment of leading mode is shown in the top row and subsequent modes below.

For a visual comparison with the experiments, the LSA mode shapes for azimuthal wavenumber $m=1$ are shown in figure 11(a) along with the first SPOD mode (b). The displayed mode shapes correspond to the tangential velocity component and are shown at five discrete frequencies. The black dashed circle marks the location of the half-width radius $r=1$. As observed in the alignment metric in figure 10(b), the match between the LSA and SPOD modes is very good for a large frequency range, except for the lowest displayed frequency of $\omega =0.1$. In figure 11(c), the normalized magnitudes of axial $u$, radial $v$ and tangential $w$ velocity components are shown at a frequency of $\omega =1$. The velocity is normalized such that the energy norm ($\int _0^\infty (|u|^2+|v|^2+|w|^2)r\, \textrm {d} r$) for SPOD and LSA modes equals 1. In figure 12 the same plot is shown for an azimuthal wavenumber of $m=3$. As quantified in figure 10(c) for $m=2$ (which is also applicable for $m>2$), very good agreement between the LSA and SPOD mode shapes are observed over the entire resolved frequency range.

Figure 11.

Mode shapes of tangential velocity component from LSA (a) and SPOD (b) at several frequencies for $m=1$. Black dashed circle marks characteristic length scale $r_{1/2}$. (c) displays normalized magnitudes of axial, radial and tangential velocity components at $\omega =1$. Solid lines correspond to LSA results and dashed lines represent SPOD modes.

Figure 12.

Mode shapes of tangential velocity component from LSA (a) and SPOD (b) at several frequencies for $m=3$. Black dashed circle marks characteristic length scale $r_{1/2}$. (c) displays normalized magnitudes of axial, radial and tangential velocity components at $\omega =1$. Solid lines correspond to LSA results and dashed lines represent SPOD modes.

5.2. Local resolvent analysis

LSA predicts the mean flow to be linearly stable for all modes except for the $m=1$ mode at low frequencies. However, SPOD reveals low-rank dynamics for structures within a wide frequency and wavenumber range that correlates very well with discrete but stable LSA modes. The occurrence of coherent structures in the flow that are in line with strongly damped stable LSA eigenmodes suggests that the empirical modes are driven by nonlinear forcing which cannot be accounted for by the LSA. This calls for a resolvent analysis (RA) that is capable of determining the optimal response modes in linearly stable flows.

RA is carried out in a local framework following the approach of McKeon & Sharma (Reference McKeon and Sharma2010) with the addition of an eddy viscosity model. In accordance with the LSA, the linear operator is modified by introducing a constant eddy viscosity $\nu _t=0.0259$ and the remaining nonlinearities are cast into a forcing term $\boldsymbol {f}$ that is interpreted as intrinsic turbulent forcing. The resulting system of equations reads

(5.11)\begin{gather} \frac{\partial{\boldsymbol{u}'}}{\partial t} + (\boldsymbol{u}' \boldsymbol{\cdot}\boldsymbol{\nabla}) \bar{\boldsymbol{u}} +(\bar{\boldsymbol{u}} \boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{u}' + \boldsymbol{\nabla} {p}' -\nu_t \nabla^2\boldsymbol{u}'=\boldsymbol{f}'. \end{gather}

(5.12)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}' = 0. \end{gather}

As for the LSA, a normal mode ansatz is introduced. This ansatz applies to both response and forcing modes

(5.13)\begin{gather} {\boldsymbol{u}'}(\boldsymbol{x},t)=\hat{\boldsymbol{u}}(r)\exp({\textrm{i}(\alpha_r x+ m \theta -\omega t)}) \end{gather}

(5.14)\begin{gather}{p'}(\boldsymbol{x},t)=\hat{p}(r)\exp({\textrm{i}(\alpha_r x+ m \theta -\omega t)}) \end{gather}

(5.15)\begin{gather}{\boldsymbol{f}'}(\boldsymbol{x},t)=\hat{\boldsymbol{f}}(r)\exp({\textrm{i}(\alpha_r x+ m \theta -\omega t)}). \end{gather}

Here, $\alpha _r$ is a real axial wavenumber, $\omega$ a real frequency and $m$ the real azimuthal wavenumber.

By introducing these ansatz functions, (5.11) and (5.12) can be reformulated as

(5.16)\begin{equation} \hat{\boldsymbol{q}}=\boldsymbol{C}(-\textrm{i} \omega \boldsymbol{I} - \boldsymbol{L})^{{-}1} \boldsymbol{B}\hat{\boldsymbol{f}} = \mathcal{H}\hat{\boldsymbol{f}}, \end{equation}

where $\mathcal {H}$ refers to the resolvent operator. The resolvent analysis constitutes an optimization problem whereby the optimization parameter is the gain defined as

(5.17)\begin{equation} \sigma^2=\frac{\left\| \tilde{\boldsymbol{\varPsi}} \right\|^2}{\left\| \tilde{\boldsymbol{\varPhi}} \right\|^2}. \end{equation}

Here, $\tilde {\boldsymbol {\varPhi }}$ and $\tilde {\boldsymbol {\varPsi }}$ correspond to forcing and response modes, respectively. The norm in (5.17) is defined as the $L_2$ norm. The optimal response modes $\widehat {\boldsymbol {\varPsi }_i}$ and forcing modes $\widehat {\boldsymbol {\varPhi }_i}$ form an orthonormal basis and are obtained by a singular value decomposition of the resolvent operator $\mathcal {H}$. A system is called to be of low rank, if the gain assigned to the optimal mode is significantly higher than the remaining ones.

Within the present study, the linear operators in (5.16) are constructed via an finite element method (FEM)-based formulation following the procedure of Kaiser, Lesshafft & Oberleithner (Reference Kaiser, Lesshafft and Oberleithner2019). Analogous to the LSA approach, homogeneous Dirichlet boundary conditions are set at $r\rightarrow \infty$ and the boundary conditions on the jet axis depend on the azimuthal wavenumber.

In figure 13, the optimal and sub-optimal resolvent gains are shown as a function of the axial wavenumber $\alpha _r$ at a selected frequency of $\omega =1$. For the three shown azimuthal wavenumbers, a peak in the resolvent gain curves is observed, which flattens with increasing azimuthal wavenumber, and the gap between the optimal and first sub-optimal mode decreases. These trends are qualitatively in line with the growth rates $-\alpha _i$ from LSA, which continuously decrease with increasing $m$ for $m>1$.

Figure 13.

Gain $\sigma$ vs axial wavenumber $\alpha _r$ at $\omega =1$ for $m=1$, $m=2$ and $m=3$. Gain of the optimal mode is plotted in black and the subsequent 9 sub-optimal modes are plotted in grey scale in sequential order.

In order to compare the results of the RA with the SPOD modes, an axial wavenumber $\alpha _r$ has to be selected for each frequency $\omega$. Within the present study, the axial wavenumber associated with the largest gain is selected, which corresponds approximately to the point of maximum alignment between the SPOD and RA mode. Figure 14 shows the resulting resolvent gain curves $\sigma$ for the optimal and subsequent sub-optimal response modes as a function of frequency. The trends observed in figure 13 are even more pronounced in this representation. For $m=1$, a very large gap between the gain of the optimal and first sub-optimal mode can be observed almost over the entire frequency range. Only at very high frequencies does the low-rank behaviour diminish. Furthermore, $m=1$ is the only azimuthal wavenumber for which a distinct peak in the gain spectrum is observed. The gain of the optimal mode peaks at $\omega =0.33$. This corresponds to the neutral point determined from LSA. At this point, the resolvent gain is infinite by definition. In the unstable regime at frequencies below $\omega =0.33$, the RA is not properly defined and neither the gain nor respective mode shapes are considered fully reliable. As the present analysis targets a broad range of frequencies and azimuthal wavenumbers, the inconsistency in this small frequency range appears acceptable. For $m>1$ the gain of the optimal mode continuously decreases with increasing frequency and azimuthal wavenumber. Furthermore, the gain separation between the optimal and sub-optimal modes decreases, showing less pronounced low-rank behaviour.

Figure 14.

Gain $\sigma$ vs $\omega$ for $m=1$, $m=2$ and $m=3$ for optimal (black) and following 9 sub-optimal modes (grey scale) in sequential order. Axial wavenumber $\alpha _r$ corresponds at each frequency $\omega$ to the wavenumber associated with the largest gain.

To assess the congruence between RA and the empirical modes, the alignment between the first five SPOD modes and corresponding five RA modes is shown in figure 15 for three azimuthal wavenumbers. The alignment between the first SPOD and optimal RA mode for $m=0,1,2$ shows very similar trends as for the least-stable LSA mode shown in figure 10. Again, the agreement is excellent over the entire frequency range, except at very low frequencies for $m=0$ and $m=1$ and at very high frequencies for $m=0$. Interestingly, the very good agreement in the high frequency regime is also present where the RA gain curves cannot be considered low rank. The alignment between the sub-optimal RA and SPOD modes is relatively weak throughout, except for $m=0$, where a slight increase in correlation towards lower frequencies is observable. The same applies to the sub-optimal RA modes for $m=1$ around $\omega =0.4$. However, note that the sub-leading SPOD modes are substantially less energetic than the leading one, and the correlations are noisy and inconclusive.

Figure 15.

Alignment between first five SPOD modes and five optimal and sub-optimal RA modes. Alignment of the optimal mode is shown in the top row and subsequent modes below.

Overall, SPOD, LSA and RA show consistent characteristics. All three methods indicate low-rank behaviour of the flow dynamics over a wide frequency and azimuthal wavenumber range. From the empirical prospective, the azimuthal wavenumber $m=1$ is the most energetic and the SPOD mode shows the most pronounced low-rank behaviour. The LSA reveals a corresponding single eigenmode that is the least stable compared to all other azimuthal wavenumbers. Likewise, the RA reveals the largest gain and most pronounced low-rank behaviour. For higher frequencies and azimuthal wavenumbers, a reduction in SPOD mode energy and SPOD and RA low-rank behaviour is observed, which goes along with the more stable LSA modes.

5.3. Accuracy of linear models in the context of turbulent scales

Since both modelling approaches, LSA and RA, show good agreement with the empirical modes, we seek to relate them to the scales of the turbulent spectrum. Furthermore, the congruence and discrepancies between the two models is assessed to potentially identify mechanisms governing the far field dynamics.

To generally classify the turbulent scales resolved in this study, the empirical and analytical data are presented as a function of the axial wavenumber. For the empirical data, a wavenumber spectrum is determined from a spatial Fourier transform of the SPIV data taken in streamwise sections of the flow, following the method proposed by Wänström (Reference Wänström2009). On the jet axis, the wavenumber spectrum reduces to

(5.18)\begin{equation} \boldsymbol{\hat{u}}(\kappa) = \int_{\xi}{\boldsymbol{u}'(\xi)\,\textrm{e}^{-\textrm{i} \kappa \xi}\,\textrm{d}\xi}, \end{equation}

where $\xi =\ln (x-x_0)$ is the logarithmically scaled axial coordinate as proposed by Ewing et al. (Reference Ewing, Frohnapfel, George, Pedersen and Westerweel2007) and $\kappa$ the corresponding non-dimensional axial wavenumber. The latter is related to $\alpha _r$, with $\alpha _r=b \kappa$ and $b$ being the jet spreading rate. The fluctuating velocity $\boldsymbol {u}'$ in (5.18) is normalized by $u_{cl}$. Based on the wavenumber spectra, the spectral density estimates are determined for the axial normal stresses $\varPsi _{uu}$ according to

(5.19)\begin{equation} \varPsi_{uu}(\alpha_r) = \frac{1}{2{\rm \pi}}\overline{{\hat{u}^{{{\dagger}}}(\alpha_r) \hat{u}(\alpha_r)}}, \end{equation}

and respectively for the radial normal stresses $\varPsi _{vv}$.

In figure 16(a) the derived spectral density estimates are shown. From the shape of the wavenumber spectrum, the turbulent scales can be classified in the outer productive range with strong anisotropic turbulence ($\alpha _r<4$) and the inner scales that behave nearly isotropic, characterized by the ${\alpha _r}^{-5/3}$ slope, which is characteristic of the inertial subrange. The approximate isotropy of the turbulent scales is also indicated by the equality of the axial and radial normal stresses. The increase of the energy at very high frequencies/wavenumbers is due to the noise floor of SPIV. Overall, the spectrum shows that the scales resolved by SPIV reach far into the inertial subrange.

Figure 16.

(a) spectral density estimates from spatial Fourier analysis on jet axis. Markers in (a) represent an estimate of the highest resolved axial, radial and tangential wavenumbers associated with the first SPOD mode for $m=5$. (b) dispersion relation for $m=0$ to $m=5$ for LSA (solid) and RA (dashed).

In order to display the LSA and RA modes as a function of the axial wavenumber, the dispersion relation is required, which is shown in figure 16(b). The solid lines correspond to the LSA results and are directly derived from the least stable LSA mode. The dashed lines represent the RA results, which correspond to the axial wavenumber associated with maximum gain (refer to figure 13). The LSA and RA dispersion relations agree generally very well.

To assess the LSA- and RA-based modelling with respect to the turbulent scales, the alignment metrics (5.10) between the leading LSA and optimal RA modes and the first SPOD mode are displayed in figures 17(a) and 17(b), respectively, as a function of the axial wavenumber for azimuthal wavenumbers $m=0$ to $m=5$. Both types of analyses show strikingly similar trends. Particularly for axial wavenumbers higher than $\alpha _r=4$, which are associated with the inertial subrange in the averaged turbulent spectrum, a surprisingly good correlation between both models and the first SPOD modes is observed. A slight deterioration of the alignment at the upper end of the wavenumber range can be attributed to aliasing effects/the PIV noise floor that affects the turbulent spectrum. The discrepancies between models and empirical modes at higher wavenumbers for $m=0$ seem to be of a physical nature.

Figure 17.

Alignment of mode $m=0$ (top) to $m=5$ (bottom) of first SPOD mode and leading mode from LSA (a) and for first SPOD mode and optimal response mode from RA (b). Alignment between leading LSA mode and optimal RA mode is shown in (c).

As the SPOD modes are anisotropic, their axial wavenumber is not necessarily the same as the radial and azimuthal wavenumbers. In fact, as can be inferred from figure 12, the SPOD and analytic modes feature lower radial and azimuthal wavenumbers in comparison to the axial wavenumbers. To account for this, the dominant radial and tangential wavenumbers of the first SPOD mode are estimated from the shape of the $m=5$ mode at a frequency of $\omega =10$, which corresponds to the mode with the smallest scale resolved. The wavelength in radial direction is estimated by visually inspecting the shape of the radial and tangential velocity components in dependence on the radial coordinate. The wavelength in the azimuthal direction is evaluated at the radial location where the radial and tangential velocity magnitudes reach their peak values, which approximately coincide. The resulting wavenumbers are $\alpha _{rad}=2{\rm \pi} /\lambda _{{rad}}\approx 5.24$ and $\alpha _{tang}=2{\rm \pi} /\lambda _{tang}\approx 5$. The axial wavenumber is determined from LSA as $\alpha _{{ax}}=\alpha _r=10.1$. All three values are indicated by yellow markers in figure 16(a). It shows that, although the radial and tangential wavenumbers are lower than the axial wavenumber, all three scales reach into the inertial subrange.

The comparison of the length scales of the leading SPOD mode with the turbulent spectrum allows for a number of observations. Even for the highest resolved frequencies, the first SPOD mode yields non-negligible production and anisotropic structures (confirmed by figure 7). Meanwhile, an increase in isotropic turbulence is reflected by a weakening of the low-rank behaviour of the SPOD modes. Thus, the small-scale isotropic structures associated with the inertial subrange must primarily be represented by subdominant SPOD modes which are neither captured by sub-optimal RA modes (see figure 15) nor by sub-leading LSA modes (not shown here). Nevertheless, the comparison with turbulent scales shows that both RA and LSA are capable of successfully modelling a remarkably large range of frequencies in the jet far field.

In a study by Sasaki et al. (Reference Sasaki, Cavalieri, Jordan, Schmidt, Colonius and Brès2017), high frequency wavepackets were modelled based on parabolized stability equations (PSE) in the jet near field for $m=0$ to $m=4$. In this study, very good agreement in the alignment was reported between the leading PSE mode and SPOD mode, which was obtained in the full spatial domain. This applied to a very wide frequency range and all investigated azimuthal wavenumbers. However, high values of alignment were only found in the spatially unstable regime and the alignment deteriorated quickly in the decaying regime. These contrasting findings with regard to stable modes at high frequencies may be explained by the following aspects of the fully developed turbulence in the jet far field. First, in the present study, an eddy viscosity model is used that is known to work well in the jet far field. Second, the modelled coherent structures show positive production throughout the entire frequency range, which links them to the mean field. This is not necessarily the case for the jet near field. Potentially, this explains why linear models based on the mean flow capture such a wide range of the far field dynamics.

Since both modelling approaches, LSA and RA, yield very similar values in the alignment with the first SPOD mode over a very wide range of turbulent scales, we seek to determine congruence and discrepancies between the two models.

In figure 17(c) the alignment between the least stable LSA and optimal response modes are shown. Astonishingly, the alignment for low and moderate azimuthal and axial wavenumbers is almost perfect. In this regime, the resolvent gains associated with the optimal mode show low-rank behaviour and the LSA growth rate is not too far from zero. Those conditions typically result in a strong agreement between LSA and RA (Beneddine et al. Reference Beneddine, Sipp, Arnault, Dandois and Lesshafft2016; Symon et al. Reference Symon, Rosenberg, Dawson and McKeon2018) unless a pseudo-resonant mechanism (based on non-normality of the linear operator) is present.

Interestingly, figure 17(c) indicates that, for azimuthal wavenumbers $m\geq 3$, the alignment between each model and the first SPOD mode is better than the alignment between the two models. This results from the radial position of maximum velocity fluctuation magnitude that differs between the models. In relation to the first SPOD mode, the LSA eigenfunctions are centred slightly closer to the jet axis whereas the resolvent modes are located further away from the jet axis.

Despite the small difference between the modelling results, very good agreement between both models and the empirical modes is obtained for higher azimuthal and streamwise wavenumbers. This is also true for the high wavenumber regime, where the LSA predicts strongly decaying modes and where the RA modes do not feature low-rank behaviour. The good match between LSA and RA strongly hints toward a predominantly resonant stability mechanism with negligible pseudo-resonant amplification (Symon et al. Reference Symon, Rosenberg, Dawson and McKeon2018).

Figure 17(a,b) further shows a deviation of both models from the empirical SPOD modes at low axial wavenumbers for $m=0$ and $m=1$. This deviation may be linked to non-parallel flow effects that are relevant at very low axial and azimuthal wavenumbers. This is supported by the work of Mullyadzhanov et al. (Reference Mullyadzhanov, Yavorsky and Oberleithner2019), where a local (or quasi-global) LSA of the Landau jet was performed in spherical coordinates which takes the non-parallel base flow into account. They observed strong deviations from a local quasi-parallel approach for modes $m=0$ and $m=1$ at low frequencies that are associated with high axial wavelengths, while the results converged for high frequencies and corresponding short axial wavelengths. This suggests that the current mismatch at low frequencies observed for $m=1$ and $m=0$ is attributed to non-parallel flow effects rather than non-modal stability mechanisms.

On a more general note, discrepancies between the two modelling approaches can also stem from the eddy viscosity model employed in the analysis. The LSA was found to be very robust towards small changes in the eddy viscosity, whereas the local RA reacted quite sensitively, which leaves room for further improvement.