2.1. Large-eddy simulation database

The study is based on an LES data set of a jet flow at Reynolds number $UD/\nu =50\,000$ and Mach number of $U/a=0.4$, where $U$ and $a$ denote the centreline velocity and speed of sound at the nozzle exit, $D$ the nozzle diameter and $\nu$ the viscosity. The computational domain is based on an experimental set-up and includes a converging nozzle and the jet (Maia et al. Reference Maia, Jordan, Cavalieri, Martini, Sasaki and Silvestre2021; Nekkanti et al. Reference Nekkanti, Maia, Jordan, Heidt, Colonius and Schmidt2022). The boundary layer inside the nozzle is tripped 2.5 diameters upstream of the outlet. For the following analyses, the data is considered in a cylindrical region extending from the nozzle outlet up to 30 diameters downstream and 6 diameters in the radial direction. A total of 10 000 snapshots with a non-dimensional time step of $\Delta t = 8.336 \times 10^{-2}$ are considered. The simulation was performed using the solver CharLES. For numerical details we refer the interested reader to Brès et al. (Reference Brès, Ham, Nichols and Lele2017, Reference Brès, Jordan, Jaunet, Le Rallic, Cavalieri, Towne, Lele, Colonius and Schmidt2018). In the following, all quantities are non-dimensionalized with $U$ and $D$. Consequently, frequencies $f$ are given as Strouhal numbers ${\rm St}=fD/U$. Figure 1 shows the time- and azimuthal-averaged mean velocity field in the domain of interest. Streamlines and the contour of the mean axial velocity component are shown.

Figure 1. The time- and azimuthal-averaged jet flow velocity mean-field in the domain of interest. Mean velocity streamlines are shown as black lines with arrows, the contour in the background shows the axial velocity component $\bar {u}_x/U$.

The Reynolds decomposition is applied to separate the velocity fields $\boldsymbol {u} = [u_x,u_r,u_\theta ]$ into a temporal mean (denoted with bar) and a fluctuating (denoted with prime) component such that $\boldsymbol {u} = \bar {\boldsymbol {u}} + \boldsymbol {u}'$. The fluctuating Reynolds-stress tensor is given as $\boldsymbol {R}'=\boldsymbol {u}'\otimes \boldsymbol {u}' - \overline {\boldsymbol {u}'\otimes \boldsymbol {u}'},$ with the outer product denoted with the $\otimes$-symbol.

2.2. Spectral proper orthogonal decomposition

In order to extract coherent structures from the broadband turbulent flow we apply SPOD (Picard & Delville Reference Picard and Delville2000; Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018); SPOD is similar to space-only POD (Lumley Reference Lumley1970; Berkooz, Holmes & Lumley Reference Berkooz, Holmes and Lumley1993), but extends the methodology to frequency space. For the analysis of the jet flow snapshots the fluctuating velocity component as well as the fluctuating Reynolds stresses are first decomposed into a Fourier series in the periodic azimuthal direction

(2.1a)$$\begin{gather} \hat{\boldsymbol{u}}_m'(x,r,t) = \mathcal{F}(\boldsymbol{u}'(x,r,\theta,t)), \end{gather}$$

(2.1b)$$\begin{gather}\hat{\boldsymbol{R}}_m'(x,r,t) = \mathcal{F}(\boldsymbol{R}'(x,r,\theta,t)), \end{gather}$$

where $\mathcal {F}$ denotes the discrete Fourier transform operator and $m$ the real valued azimuthal wavenumber. Due to the periodicity, $m$ is an integer. This yields two-dimensional Fourier modes for each time step, $\hat {\boldsymbol {u}}_m'$ and $\hat {\boldsymbol {R}}_m'$, respectively. Large-scale dominant coherent structures can only be observed for broadband jet flows at low wave numbers. Since modelling large-scale coherent structures is the objective of this study, we limit our analysis to the $m=0$ (axis symmetric) and $m=1$ mode.

The 10 000 snapshots of the velocity Fourier modes $\hat {\boldsymbol {u}}_m'$ are divided into $N_b$ blocks of 256 snapshots each, where the blocks overlap with $50\,\%$. Each block is transformed into Fourier space. For a given azimuthal wavenumber and temporal frequency the matrix

(2.2)\begin{equation} \hat{\boldsymbol{U}}_{m,\omega} = [ \hat{\boldsymbol{u}}_{m,\omega}^{(1)}, \hat{\boldsymbol{u}}_{m,\omega}^{(2)}, \hat{\boldsymbol{u}}_{m,\omega}^{(3)},\ldots,\hat{\boldsymbol{u}}_{m,\omega}^{(N_b)} ] \end{equation}

is constructed where $\hat {\boldsymbol {u}}_{m,\omega }^{(n)}$ denotes the Fourier velocity mode at azimuthal order $m$, temporal frequency $\omega$ and block $n$. From $\hat {\boldsymbol {U}}_{m,\omega }$ the sample cross-spectral density matrix can be computed, whose eigenvectors are the SPOD modes. However, we consider here the entire domain extending 6 diameters in the radial direction and 30 diameters in the axial direction, discretized with 138 and 656 points, respectively. The total number of degrees of freedom is therefore much larger than the number of blocks $N_b$, which makes the computation of the SPOD modes via the method of snapshots much less expensive. This involves first solving the eigenvalue problem

(2.3)\begin{equation} \hat{\boldsymbol{U}}_{m,\omega}^H \boldsymbol{W} \hat{\boldsymbol{U}}_{m,\omega} \boldsymbol{\varPsi} = \boldsymbol{\varPsi} \boldsymbol{\varLambda},\end{equation}

where the superscript $H$ denotes the complex conjugate transpose and $\boldsymbol {W}$ is a weight matrix that takes the volume of each cell into account. The $j$th SPOD velocity mode $\hat {\boldsymbol {\upsilon }}^{(j)}_{m,\omega }$ follows from expanding the Fourier modes with the $j$th eigenvector

(2.4)\begin{equation} \hat{\boldsymbol{\upsilon}}^{(j)}_{m,\omega} = \hat{\boldsymbol{U}}_{m,\omega} \boldsymbol{\psi}^{(j)}, \end{equation}

where $\boldsymbol {\psi }^{(j)}$ denotes the $j$th column (eigenvector) of $\boldsymbol {\varPsi }$. In total, the decomposition provides $N_b$ SPOD modes for each azimuthal order and frequency, which can be ranked according to modal energy content which is represented by the magnitude of the corresponding eigenvalue $\lambda ^{(j)}$. Applied to velocity data, the eigenvalue is a measure of the kinetic energy of the mode. Low-rank dynamics, also known as high-gain separation, are present when one mode has a particularly high energy in a given frequency range compared with the others. The dynamics of the flow is then dominated by one coherent structure in the corresponding frequency range. For more information we refer to the guide to SPOD by Schmidt & Colonius (Reference Schmidt and Colonius2020).

Figure 2 shows the SPOD spectrum of the jet flow velocity data for $m=0$ (figure 2a) and $m=1$ (figure 2b). A high gain separation can be observed for $m=0$ at intermediate frequencies and for $m=1$ at low frequencies. For a more detailed investigation on the energy content of the leading SPOD mode, we show the energy share of each mode in figure 3. It can be observed, that around ${St}=0.5$ the flow is dominated by a $m=0$ mode and between ${St}=0.2\unicode{x2013}0.4$ by a $m=1$ mode, both containing approximately 20 % of the total kinetic energy at the corresponding wavenumber.

Figure 2. The SPOD spectrum of the broadband jet flow for $m=0$ (a) and $m=1$ (b). The eigenvalue of the first (leading) mode is shown in black, eigenvalues of subleading modes are shown in graduating shades of grey.

Figure 3. Energy share of the SPOD modes for $m=0$ (a) and $m=1$ (b). The energy share of the leading mode is shown in black, for subleading modes in graduating shades of grey.

2.3. Identification of the coherent component of Reynolds stresses based on eSPOD

In regions of high-gain separation, we hypothesize that the coherent structure has sufficient amplitude to extract information on the higher-order statistics, and in particular the coherent Reynolds stresses. Specifically, we are interested in the part of the Reynolds-stress fluctuation that interacts with the dominant coherent structure. We refer to this as the coherent component of the Reynolds-stress tensor. For a broadband flow, it is difficult to extract this quantity from the data as it is of comparatively low amplitude compared with the coherent structure. We thus identify the coherent Reynolds-stress component by using the same eigenvectors as in (2.4) to also expand Fourier-blocks of the Reynolds-stress fluctuations. This method is also known as eSPOD (Boree Reference Boree2003).

For this purpose, the 10 000 snapshots of Reynolds-stress Fourier modes $\hat {\boldsymbol {R}}_m'(x,r,t)$ analogue to the velocity modes are first divided into $N_b$ blocks, transformed into frequency space and stacked into a matrix for each frequency and azimuthal order

(2.5)\begin{equation} \hat{\boldsymbol{O}}_{m,\omega} = [ \hat{\boldsymbol{R}}_{m,\omega}^{(1)}, \hat{\boldsymbol{R}}_{m,\omega}^{(2)}, \hat{\boldsymbol{R}}_{m,\omega}^{(3)},\ldots,\hat{\boldsymbol{R}}_{m,\omega}^{(N_b)} ], \end{equation}

where $\hat {\boldsymbol {R}}_{m,\omega }^{(n)}$ denotes the blockwise Fourier mode of the fluctuating Reynolds-stress component. In the second step, the Fourier modes are expanded with the same eigenvectors $\boldsymbol {\psi }^{(j)}$ as in (2.4),

(2.6)\begin{equation} \hat{\boldsymbol{\varGamma}}^{(j)}_{m,\omega} =\hat{\boldsymbol{O}}_{m,\omega} \boldsymbol{\psi}^{(j)}. \end{equation}

Since the eigenvectors were calculated on the basis of velocity data, the approach guarantees extraction of only the part of the Reynolds-stress tensor that is correlated with the corresponding coherent structure (mode $j,m,\omega$). The following analysis will focus on frequencies between ${St}=0.1$ and $1$. However, following our reasoning above, caution should be taken when considering results that are based on eSPOD outside of the low-rank region, see figure 3.

In the context of this study, we want to investigate the dominant coherent structures and are therefore only interested in the most energetic mode ($j=1$). For the sake of clarity, we drop the superscripts and subscripts and only refer to the velocity SPOD mode as $\hat {\boldsymbol {u}}_{{S}}$ and the corresponding coherent Reynolds-stress component as $\hat {\boldsymbol {R}}_{{S}}$ in the following. The frequency and wavenumber are always given in context.

As an example, figure 4 shows the real part of the axial velocity SPOD mode (figure 4a) and the corresponding real part of the eSPOD axial-radial component of the coherent Reynolds-stress tensor (figure 4b) for $m=0$ and ${St}=0.56$. The axial velocity mode corresponds to the well known Kelvin–Helmholtz mode, which dominates the dynamics of the jet in this frequency range (Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020). The axial-radial component of the coherent Reynolds-stress tensor computed based on eSPOD shows a higher noise level, however, a clear mode shape can be observed, especially in the region where the velocity mode has a high amplitude.

Figure 4. The SPOD and eSPOD mode for $m=0$ and ${St}=0.56$. (a) Real part of the SPOD axial velocity mode (2.4). (b) Real part of the eSPOD axial-radial component of the coherent Reynolds-stress tensor (2.6). Both modes are normalized with their respective maximum absolute value.

4.1. Applying the energy budget to SPOD and resolvent modes

In this study, we make use of the energy budget to gain a deeper insight into the coherent structures. For this purpose, the individual energy terms are analysed in the following sections for both SPOD and resolvent modes. Assuming that the leading SPOD mode approximates the Fourier mode at the respective frequency, which is justified in regions of high-gain separation, the energy terms can be determined as shown in (4.3). This means that (4.3) is evaluated based on the leading SPOD mode $\hat {{\boldsymbol {u}}}_{{S}}$ and the deviatoric component of the coherent Reynolds-stress tensor $\hat {\boldsymbol {D}}$ is computed from the eSPOD mode $\hat {\boldsymbol {R}}_{{S}}$.

When considering the leading resolvent modes, the energy budget is evaluated using the same equations, with the exception of the energy transfer term $\hat {{N}}$, which is modelled in the resolvent analysis based on the Boussinesq model with eddy viscosity. Consequently, the corresponding modelled term $\hat {{N}}_\varTheta$ is considered, (4.5). As the resolvent modes are a solution to (3.5), the energy terms can be evaluated based on the leading resolvent velocity mode $\hat {{\boldsymbol {u}}}_{R}$. Strictly speaking, the energy budget (4.2) of the leading resolvent mode contains an additional term related to the optimal forcing vector $\hat{\boldsymbol{f}}_{R}$, see (4.5). The term reads

(4.6)\begin{equation} \hat{{N}}_{f} = {\rm Re} \left( \hat{{\boldsymbol{u}}}_{R}^* \boldsymbol{\cdot}\hat{\boldsymbol{f}}_{R} \right) . \end{equation}

Since the leading resolvent mode can also be interpreted as the maximum response at minimum forcing, it is intuitive to assume that the energy term associated with the forcing term is small. This is to be expected in particular for the jet flow, which system operator has a high degree of non-normality due to the strong shear and mean flow advection, resulting in a high sensitivity to forcing (Chomaz Reference Chomaz2005; Symon et al. Reference Symon, Rosenberg, Dawson and McKeon2018). On the other hand, resolvent analysis without eddy viscosity is based on the premise that the coherent component of the Reynolds stresses is contained in the forcing vector, see (4.5). As will be shown in the remainder of this study, the resolvent forcing vector does not compensate for non-modelled dissipation caused by the coherent component of the Reynolds stresses. In fact, its contribution to the global energy budget equation (4.4) is negligible for the considered dominant coherent structures.

Finally, we would like to point out that the data-driven SPOD/eSPOD modes only approximate a solution of the energy budget. How good this approximation is depends on how well the SPOD mode approximates the Fourier mode and how well the coherent component of the Reynolds-stress tensor can be determined with the eSPOD method, which is addressed in the next section. In § 7.2 it is validated a posteriori that the dominant SPOD/eSPOD modes satisfy the energy equation. The resolvent forcing-response mode pairs, on the other hand, result from the discretized, linearized equations and thus fulfil the energy budget by definition, regardless of the use of eddy viscosity (Symon et al. Reference Symon, Illingworth and Marusic2021).

4.2. Validation of the coherent component of the Reynolds-stress tensor

We determine the coherent part of the Reynolds-stress tensor using an eSPOD, which extracts the part of the $\hat {\boldsymbol {R}}_m'$ tensor that correlates with the first velocity mode $\hat {{\boldsymbol {u}}}_{S}$, see (2.6) in § 2.3. As mentioned in the discussion above, this assumes that the modes are dominant, i.e. have sufficient energy such that the second-order statistics can be identified. In order to assess how well the coherent part of the Reynolds-stress tensor can be identified in this way, the global energy budget, (4.4), of the SPOD modes is considered. Figure 5 shows the ratio of spatially integrated coherent Reynolds-stress term (nonlinear energy transfer term) and coherent production term $|\hat {N}_\varOmega/\hat {P}_\varOmega|$ computed based on the SPOD and eSPOD results for $m=0$, $m=1$ and frequencies between ${St}=0.1$ and ${St}=1$. According to the global energy budget, (4.4), the value should be approximately $1$. As expected, the energy balance is well respected in regions of high-gain separation. Since the production term is based on first- (SPOD) and zero-order (mean-field) statistics, it is reasonable to assume that the deviation results from the $\hat {N}_\varOmega$ term, or more precisely from the identification of the coherent part of the Reynolds-stress tensor required to determine this term. The fact that the energy balance is met by approximately 80 % in high-gain separation regions (${St}\approx 0.5$ for $m=0$ and ${St}<0.6$ for $m=1$) gives us confidence in our approach. In regions without high-gain separation, especially at low frequencies and $m=0$, the deviation is much higher, indicating that based on eSPOD only a part of the coherent component of the Reynolds-stress tensor has been identified. At low frequencies and $m=0$ the dynamics are governed by the strongly non-modal Orr mechanism (Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020). Corresponding coherent structures are of high-rank, which is likely the reason why the coherent component of the Reynolds-stress tensor cannot be identified well in this frequency range.

Figure 5. Ratio of coherent Reynolds-stress energy term and mean-field production energy term for $m=0$, $m=1$ and frequencies between ${St}=0.1$ and $1$. The marker transparency indicates the energy share of the respective SPOD mode, see figure 3.

We conclude this section by discussing the implications of the energetic considerations for resolvent analysis. The dissipative Boussinesq approach with eddy viscosity enables the energy entering the modes via the mean-field to be dissipated. In resolvent analysis without eddy viscosity, the energy entering the coherent structure from the mean-field must be dissipated either via molecular dissipation or via the resolvent forcing energy term, which then represents the coherent Reynolds-stress term. The difference in considering eddy viscosity is that dissipation is explicitly taken into account. Since both forcing and response modes are part of the resolvent modelling, it can be assumed that better results can be achieved if more information is included in the model and the dissipative part is modelled explicitly. This will be validated a posteriori in § 7.

5.1. Predictive mean-field-consistent eddy viscosity

If a resolvent analysis is to be performed with high-fidelity data, such as PIV measurements or LES data, the question arises as to how the eddy viscosity should be determined. A common method is to calibrate the field using available Reynolds stresses and the mean strain-rate tensor, or to assume a constant eddy viscosity and adjust the magnitude until the resolvent mode is close to the validation data (usually SPOD modes) in the frequency range of interest. Here, we propose an alternative approach that does not require validation data and is therefore referred to as predictive. Furthermore, the approach does not require Reynolds stresses, which are often not available or only available with poor quality, as higher-order statistics require long time-series to converge.

For the predictive approach, the effective viscosity $\varTheta$ for modelling the coherent component of the Reynolds stresses is set to the eddy viscosity consistent with the mean-field, i.e. $\varTheta =\varTheta _{{mean}}$ in (3.4). To find the corresponding field, we take the time-averaged LES snapshots ($\bar {{\boldsymbol {u}}}$ and $\bar {q}$) and assimilate the eddy viscosity under the constraint that the mean-fields together with the eddy viscosity correspond to a solution of the RANS equations. In other words, we formulate the optimization problem: find $\varTheta _{{mean}}$ such that $\varTheta _{{mean}}$, $\bar {{\boldsymbol {u}}}$ and $\bar {q}$ together approximate a solution of the RANS equations. There are various techniques for solving this optimization problem, such as adjoint-based optimization (Foures et al. Reference Foures, Dovetta, Sipp and Schmid2014; Symon et al. Reference Symon, Dovetta, McKeon, Sipp and Schmid2017; Mons et al. Reference Mons, Vervynck and Marquet2024). Here we use the results from a previous study in which the formulated problem was solved for the same data set with a physics-informed neural network (PINN) (von Saldern et al. Reference von Saldern, Reumschüssel, Kaiser, Schmidt, Jordan and Oberleithner2023). As the focus of the study is not on this model, we refer the interested reader to the referenced study and the Appendix (A) in which the PINN method is briefly outlined. Further information on mean-field assimilation in fluid mechanics with PINNs can be found in von Saldern et al. (Reference von Saldern, Reumschüssel, Kaiser, Sieber and Oberleithner2022), Eivazi et al. (Reference Eivazi, Tahani, Schlatter and Vinuesa2022) and Patel et al. (Reference Patel, Mons, Marquet and Rigas2024).

Figure 6 shows the normalized assimilated eddy viscosity field. The red lines mark the locations at which the normalized mean axial velocity $\bar {u}_x/U$ is $0.95$ and $0.5$. The eddy viscosity field is typical for a jet flow, with low values in the potential core region and increased values along the shear layer. For validation of the assimilated field, we consider the Reynolds-stress term in the energy equation of the mean-field, which is to be modelled with the Boussinesq approach. For a turbulent jet, the term can be approximated as

(5.1a)\begin{gather} \bar{N}(x,r) ={-}\overline{u_x'u_r'}\frac{\partial\bar{u}_x}{\partial r}\end{gather}

(5.1b)\begin{gather} \approx \varTheta_{{mean}} \left(\frac{\partial\bar{u}_x}{\partial r} \right)^2, \end{gather}

where in (5.1b) the Boussinesq model replaces the Reynolds-stress term. Figure 6(b) shows the axial profile of the radially integrated term calculated from the LES data, (equation (5.1a)) (solid) and based on the assimilated eddy viscosity, (equation (5.1b)) (dashed). Apart from small deviations, both curves show good agreement, which means that the assimilated eddy viscosity can sufficiently approximate the energy transfer from the mean-field to the turbulence. Downstream in the flow ($x/D>15$) the eddy viscosity slightly underestimates the transfer but still follows the general trend well. The deviation shortly after the nozzle is due to a jump in the assimilated eddy viscosity. Shortly after the nozzle outlet, the gradients of the velocity field are extremely large, which leads to problems when assimilating the eddy viscosity using the PINN method. However, as will be shown below, the coherent fluctuations are still very small so close to the nozzle exit and therefore the assimilation error visible in the eddy viscosity field does not cause any further problems in the resolvent analysis and can be ignored. The velocity mean-fields, however, are very important in this region as they affect the coherent production determined by the resolvent analysis and are therefore taken directly from time averaging the LES snapshots.

Figure 6. (a) Physics-informed neural network assimilated mean-field-consistent eddy viscosity field ($\varTheta _{{mean}}$). Red dotted lines indicate the locations at which $\bar {u}_x/U$ is $0.95$ and $0.5$. (b) Radially integrated Reynolds-stress term ((5.1a) and (5.1b)) in mean-field energy equation.

Finally, we note that an eddy viscosity was also determined via calibration of the Boussinesq model to the turbulence statistics using the LES Reynolds stresses and the mean strain-rate tensor, as is common in many linearized analyses (Viola et al. Reference Viola, Iungo, Camarri, Porté-Agel and Gallaire2014; Rukes et al. Reference Rukes, Paschereit Oliver and Oberleithner2016; Kuhn et al. Reference Kuhn, Soria and Oberleithner2021). Since the resulting field is similar to the field assimilated by PINN, apart from noise resulting from the calibration method, and there are hardly any differences in the resolvent modelling (not shown), this approach is not considered further in the context of this study. Instead, we also consider the naïve approach $\varTheta =0$ as a baseline configuration. As is clear from the findings in the previous section, this approach implies that any coherent energy is either dissipated by molecular viscosity, by the resolvent forcing or goes back to the mean-field. Taking into account the energy cascade and the high turbulence intensity of the flow, this should be directly ruled out. However, since the approach is frequently used, it is also pursued and discussed in further detail in this study.

5.2. Data-driven eddy viscosity modelling

The Boussinesq model relates the coherent component of Reynolds stresses to the coherent strain-rate tensor via the eddy viscosity. Since the coherent strain-rate tensor can be approximated based on the SPOD modes and the coherent Reynolds-stress tensor via the eSPOD modes, two data-driven models for $\varTheta$ are presented in the following.

5.2.1. Spatially resolved eddy viscosity field

The first approach is inspired by the eddy viscosity calibration method commonly applied to extract an eddy viscosity from turbulence statistics. However, instead of applying it to the mean statistics, we consider the corresponding coherent components and invert the Boussinesq-like model, (3.4) in frequency space. More specifically we solve the optimization problem

(5.2)\begin{equation} \hat{\varTheta}_{{local}} (x,r) = \text{argmin} || \hat{\boldsymbol{D}}_{S} + \varTheta \hat{\boldsymbol{S}}_{S} ||_F \end{equation}

for individual frequencies $\omega$ and mode numbers $m$, where $||{\cdot }||_F$ denotes the Frobenius norm. As introduced above, the index ${S}$ indicates that the deviatoric part of the coherent Reynolds-stress tensor as well as the coherent strain-rate tensor are computed based on the eSPOD and SPOD modes. For this least-squares problem the closed-form solution is given by

(5.3)\begin{equation} \hat{\varTheta}_{{local}}(x,r) = \frac{\langle -\hat{\boldsymbol{D}}_{S} , \hat{\boldsymbol{S}}_{S} \rangle_F}{\langle \hat{\boldsymbol{S}}_{S} , \hat{\boldsymbol{S}}_{S} \rangle_F}, \end{equation}

where $\langle {\cdot } \rangle _F$ denotes the Frobenius scalar product, $\langle \boldsymbol {A}, \boldsymbol {B} \rangle _F = \sum _{l,b} A_{lb}^* B_{lb}$. Equation (5.3) is solved on a pixel-by-pixel basis resulting in a spatially resolved eddy viscosity field at each frequency and azimuthal wavenumber. It should be noted at this point that the fields determined in this way are complex-valued. However, as discussed in § 4, the nonlinear energy transfer term is primarily determined by the real part of the term $-\varTheta || \boldsymbol {\nabla }\hat {{\boldsymbol {u}}} ||_F$ when approximated with the Boussinesq model. Consequently, it is also the real part of the complex eddy viscosity fields that is decisive.

Figure 7 shows the real part of the $\hat {\varTheta }_{local}$ fields for $m=0$ (figure 7a) and $m=1$ (figure 7b) and frequencies within the respective high-gain separation region. Black dashed lines indicate locations at which $\bar {u}_x/U$ is 0.95 and 0.5 and black dotted lines mark locations at which the magnitude of the respective SPOD velocity mode $|\hat {\boldsymbol {u}}_{S}|$ reaches $20\,\%$ and $80\,\%$ of its maximum value. To highlight the relevant regions, the transparency outside the $20\,\%$ magnitude line is set to $50\,\%$. The shown fields are smoothed with a moving-average filter in order to eliminate high-frequency noise. Nevertheless, a high level of noise can be observed in regions where the magnitude of the SPOD mode is low. The reason for this is that the signal-to-noise ratio is low in regions in which the velocity mode has a low magnitude. In addition, (5.3) is based on gradients of statistical quantities, which poses a further challenge. However, as the a posteriori validation in § 7.2 will show, the noise has only a minor influence on the results of the resolvent analysis. For this reason, no further noise reduction measures were taken.

Figure 7. Real part of spatially varying frequency dependent eddy viscosity field $\hat {\varTheta }_{{local}} (x,r)$ retrieved by inverting the Boussinesq-like model in frequency space, (5.3). Fields are shown for the respective frequency range of high-gain separation for $m=0$ (a) and $m=1$ (b). The black dashed lines show the positions where $\bar {u}_x/U$ is 0.95 and 0.5. Black dotted lines show the positions at which the magnitude of the respective SPOD velocity mode $|\hat {{\boldsymbol {u}}}_{S}|$ reaches $20\,\%$ and $80\,\%$ of its maximum value. Outside the $20\,\%$ line the transparency is set to $50\,\%$ to highlight the relevant region.

Figure 7 shows clear eddy viscosity structures in the regions in which the respective SPOD velocity modes are located (dotted lines). These structures differ between $m=0$ and $m=1$. For $m=0$, the $\hat {\varTheta }_{local}$ fields show a small negative region close to the centre of the mode and a larger, elongated positive structure located at the outer upper- and downstream end of the mode. With increasing frequency, both the mode shapes and the eddy viscosity fields hardly show any changes. For $m=1$, a positive structure with large radial and axial expansion can be recognized, which expands over the entire range of the mode. With increasing frequency the mode as well as the structure of increased eddy viscosity move upstream. The intensity of the eddy viscosity structure slightly decreases with increasing frequency.

We recall that a negative value of the eddy viscosity means that energy is transferred into the coherent structure via the coherent component of the Reynolds stresses, see § 4. This cannot be taken into account in the gradient-type Boussinesq model in resolvent analysis as it leads to numerical instability. Therefore, the eddy viscosities are clipped to positive values when used in the resolvent model. However, the data-driven determined eddy viscosity fields in figure 7 show predominantly positive structures – regions in which energy is dissipated (transferred to other scales) by the coherent component of Reynolds stresses. For modelling dissipation, the Boussinesq approach is well suited.

The differences in the various $\hat {\varTheta }_{local}$ fields observed in figure 7 indicate that the eddy viscosity is wavenumber and for $m=1$ also frequency dependent. In comparison with the mean-field-consistent eddy viscosity $\varTheta _{mean}$, see figure 6(a), it can be observed that, apart from very different shapes, the fields are all of the same order of magnitude with a maximum value of the order of $10^{-2}$.

5.2.2. Spatially constant global eddy viscosity

In the second approach, we follow a more robust methodology that takes into account the fact that the SPOD and eSPOD modes are statistical quantities that, as seen above, contain noise when calculating a spatially resolved eddy viscosity. The second data-driven approach is therefore based on an energetic consideration and the identification of an eddy viscosity that is constant in space. The use of spatially constant eddy viscosities is common in resolvent analysis and can lead to results comparable with those obtained using spatially resolved fields (Oberleithner et al. Reference Oberleithner, Paschereit and Wygnanski2014; Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018; Pickering et al. Reference Pickering, Rigas, Schmidt, Sipp and Colonius2021; Kuhn et al. Reference Kuhn, Müller, Knechtel, Soria and Oberleithner2022). Here we follow this approach to understand more about the role of eddy viscosity in modelling coherent structures.

Inspired by consideration of the energy budget of coherent structures, the objective is to find a global eddy viscosity that best represents the nonlinear energy transfer term, (4.3e) and (4.5), at frequency $\omega$ and mode number $m$ which can be formulated as

(5.4)\begin{equation} \hat{\varTheta}_{{global}} = \text{argmin} || \hat{{N}} - \hat{{N}}_\varTheta ||_{L_2} = \text{argmin} || \boldsymbol{\nabla}\hat{\boldsymbol{u}}_{S}^* \boldsymbol{\cdot} \hat{\boldsymbol{D}}_{S} + \varTheta \boldsymbol{\nabla} \hat{\boldsymbol{u}}_{S}^* \boldsymbol{\cdot} \hat{\boldsymbol{S}}_{S} ||_{L_2}. \end{equation}

In other words, the eddy viscosity is determined which, multiplied by the coherent strain-rate tensor, best approximates the energy transfer to other structures (dissipation caused by the coherent component of Reynolds stresses). To ensure that only the dissipative part is approximated by the eddy viscosity, the real part of $\hat {{N}} = {\rm Re} (\boldsymbol {\nabla } \hat {\boldsymbol {u}}_{S}^* \boldsymbol {{\cdot }}\hat {\boldsymbol {D}}_{S})$ is clipped to be purely negative before solving the problem.

Note that the summands in (5.4) are scalar fields such that an eddy viscosity field could be computed by a simple division of the two fields at every location (pixel). However, we are interested in determining a global quantity that minimizes the least-squares problem which follows from

(5.5)\begin{equation} \hat{\varTheta}_{{global}}= \frac{\langle -\boldsymbol{\nabla} \hat{\boldsymbol{u}}_{S}^* \boldsymbol{\cdot} \hat{\boldsymbol{D}}_{S} , \boldsymbol{\nabla} \hat{\boldsymbol{u}}_{S}^* \boldsymbol{\cdot} \hat{\boldsymbol{S}}_{S} \rangle_{L_2}}{\langle \boldsymbol{\nabla} \hat{\boldsymbol{u}}_{S}^* \boldsymbol{\cdot} \hat{\boldsymbol{S}}_{S}, \boldsymbol{\nabla} \hat{\boldsymbol{u}}_{S}^* \boldsymbol{\cdot} \hat{\boldsymbol{S}}_{S} \rangle_{L_2}}, \end{equation}

where $\langle {\cdot } \rangle _{L_2}$ denotes the inner product, $\langle A,B\rangle _{L_2} = \sum _{l,b} A(x_l,r_b)^* B(x_l,r_b)$.

Figure 8 shows the real and imaginary part of the identified eddy viscosities for $m=0$ and $m=1$ in the frequency range between ${St}=0.1$ and ${St}=1$. The transparency of the markers indicates the energy content of the corresponding SPOD mode, see figure 3. For modes with high energy content, we have great confidence in the results, as the determination of the coherent part of the Reynolds-stress tensor works particularly well for these, see figure 5. The imaginary parts show a very similar behaviour for both azimuthal wavenumbers, they are close to $0$ over the entire frequency range. This observation is consistent with the fact that the energy transfer/dissipation is primarily determined by the real part of the eddy viscosity, as already discussed above. Considering the real parts, clear differences can be observed: for $m=1$, the values are significantly higher than for $m=0$ especially for frequencies below ${St}=0.6$. A result that was perhaps to be expected considering that the data-driven eddy viscosity fields in figure 7 show significantly larger areas with increased values for $m=1$ compared with $m=0$. For wavenumber $m=1$, a decrease in eddy viscosity can be observed for increasing frequency, whereby the values for $m=0$ are largely constant.

Figure 8. Real (a) and imaginary (b) part of data-driven global eddy viscosity. The values are determined by calibrating the modelled nonlinear energy transfer term (dissipation), see (5.4). The marker transparency indicates the energy share of the respective SPOD mode, see figure 3.

Consistent with the spatially resolved data-driven eddy viscosities $\hat {\varTheta }_{{local}}$ in figure 7, the real parts of the global data-driven eddy viscosities $\hat {\varTheta }_{{global}}$ in figure 8 are all positive, which means that the nonlinear energy transfer term has a dissipative effect on the energy balance. In the following section, the ability of the different eddy viscosities to approximate the nonlinear energy transfer term is investigated in more detail.

6.1. Wavenumber, frequency and spatial dependence of eddy viscosity

The results from the a priori analysis in the previous section will be briefly discussed in this section with respect to the wavenumber and frequency dependence of the data-driven eddy viscosities. In figure 8 it was observed that the data-driven global eddy viscosity $\hat {\varTheta }_{{global}}$ changes with wavenumber and for $m=1$ also with frequency. However, the a priori analysis, figure 9, demonstrated that the energy transport term for two considered modes with different azimuthal order and frequency can be approximated well not only for the data-driven, mode-specific eddy viscosities (figure 9a ii,iii,b ii,iii), but a good approximation is also achieved when the spatially varying eddy viscosity $\varTheta _{{mean}}$ is used for both modes (figure 9a iv,b iv). The reason for this is the structure of the Boussinesq model, in which the eddy viscosity appears as a factor in the product with the coherent strain-rate tensor. This results in the eddy viscosity only being effective in the regions where the coherent strain-rate tensor has a significant amplitude. In simple terms, the multiplication limits the region in which the eddy viscosity is effective to the region of the mode. Consequently, if modes are located in different regions, a spatially varying eddy viscosity has a mode-specific effect. For the modes considered in figure 9, the $m=1$ mode is located further downstream compared with the $m=0$ mode. Due to its spatial distribution, the mean-field-consistent eddy viscosity $\varTheta _{{mean}}$ therefore affects the corresponding energy transfer terms differently.

We conclude that an eddy viscosity field can exhibit a certain frequency/mode number dependence via spatial distribution if the corresponding coherent structures are located in different regions. How well a frequency and mode number dependence can be represented depends on how far the modes are spatially separated, how large the modes are and how much the eddy viscosity field changes in space.

7.1. Mean-field-consistent eddy viscosity and molecular viscosity

We begin with a qualitative comparison of the axial velocity mode shape between SPOD and resolvent mode based on mean-field-consistent eddy viscosity and pure molecular viscosity. Figure 10 shows the real part of the two dominant axial velocity modes for $m=0$ and $St=0.56$ (figure 10a) and $m=1$ and $St=0.28$ (figure 10b). Figure 10(a i,b i) show the SPOD modes. Figures 10(a ii,b ii) and 10(a iii,b iii) show the resolvent modes based solely on molecular dissipation and the mean-field-consistent eddy viscosity, respectively.

Figure 10. Real part of axial velocity mode for $m=0$, ${St}=0.56$ (a) and $m=1$, ${St}=0.28$ (b). Panels (a i) and (b i) show the SPOD modes. Panels (a ii,b ii) and (a iii,b iii) show the resolvent modes for the purely molecular $\nu$ and mean-field-consistent $\varTheta _{{mean}}$ eddy viscosity, respectively. The black dashed lines indicate the location at which $\bar {u}_x/U = 0.95$ and $0.5$.

It can be observed that neglecting the modelling of the coherent part of the Reynolds-stress tensor and thus considering only molecular dissipation gives less satisfactory results than modelling the term with a mean-field-consistent eddy viscosity. If only molecular viscosity is considered, the resolvent modes agree fairly well in the upstream region, but show strong qualitative deviations for $x/D >5$ and both modes considered. With mean-field-consistent eddy viscosity, a high degree of agreement between the resolvent mode and the SPOD mode can be observed throughout the domain, both for the $m=0$ and for the $m=1$ mode. Slight differences are only observed in the radial expansion and in the range $x/D > 6$ where the SPOD mode is more pronounced, which could indicate that the mean-field-consistent eddy viscosity leads to too much dissipation.

Figure 11 shows how the radially integrated energy terms (7.1) evolve with $x/D$, in figure 11(a) for the $m=0$ mode and in figure 11(b) for the m=1 mode. The curves are smoothed with a moving average filter to eliminate unphysical high-frequency noise. The solid lines show the energy terms for the SPOD mode (7.1), the dotted lines for the resolvent modes with only molecular dissipation ($\varTheta =\nu$ in (7.1d)) and the dashed lines for the resolvent modes with mean-field-consistent eddy viscosity ($\varTheta =\varTheta _{{mean}}$ in (7.1d)). Since the scaling of the SPOD and resolvent modes is arbitrary, all terms belonging to one mode are normalized with the respective maximum absolute value of all considered energy terms.

Figure 11. Radially integrated production (green, $\hat {{P}}_x$), dissipation (red, $\hat {{N}}_x$) and convection (blue, $\hat {{C}}_x$) energy terms (7.1) for $m=0$, ${St}=0.56$ (a) and $m=1$, ${St}=0.28$ (b). The solid lines show the terms for the SPOD modes. The dashed and dotted lines show the terms for the modes of the resolvent analysis (RA) with mean-field-consistent eddy viscosity and molecular viscosity, respectively. Terms belonging to one mode are normalized with the respective maximum absolute value of all considered energy terms.

The energy terms of the $m=0$, ${St}=0.56$ SPOD mode (figure 11a; solid) show the following behaviour. Shortly after the nozzle outlet, the mode begins to receive energy from the mean-field (production) and grows with the direction of the flow. At $x/D \approx 3$, the point of maximum energy input is reached, after which it decreases again. Interestingly, the production term also becomes slightly negative at $x/D\approx 5$ before it becomes zero, which means that a small share of energy from the mode is returned to the mean-field. The turbulent dissipation term grows with a slight offset. It reaches less high values (negative), but is active over a larger axial range. The convection term transfers energy in the axial direction and is negative in regions from which energy is transported away and positive in regions to which energy is transported. As discussed above, the convection term is of conservative from – it can be observed that the corresponding axially integrated term is equal to zero. The $m=1$, ${St}=0.28$ SPOD mode (figure 11b; solid) shows a similar behaviour. The mode is located slightly further downstream and the dissipation term has a smaller axial offset with respect to the production term in comparison with the $m=0$ mode, which also leads to a less pronounced convection term. Moreover, the production term does not become negative, thus all energy, except for a very small portion related to molecular dissipation, is dissipated through the coherent component of Reynolds stresses.

A comparison of the distribution of the energy terms between the SPOD and resolvent modes clearly shows that the consideration of molecular dissipation alone leads to very poor modelling results. As was to be expected, SPOD and the corresponding resolvent modes show a completely different energy budget. For the $m=0$ resolvent mode the dissipation term is negligibly small, and all the energy that enters the mode from the mean-field also returns to the mean-field. In contrast, for $m=1$, the energy is mainly dissipated by molecular dissipation in the downstream region. Thereby, the small value of the molecular viscosity is compensated by a correspondingly large coherent strain-rate tensor, which leads to the clear differences in mode shape, see figure 10(b ii). We note that figure 10(b ii) does not directly show the coherent strain-rate tensor, but high spatial gradients are visible, especially in the range $x/D=10\unicode{x2013}15$ and $r/D\approx 0.5$.

The fact that the resolvent mode shape for $m=0$ and molecular dissipation agrees qualitatively well with the SPOD mode in the upstream region, as observed in figure 10(a ii), can be explained well by the distribution of the energy terms. The $m=0$ SPOD mode shows a small region in which energy is returned from the mode to the mean-field. If in the resolvent analysis the dissipation through the coherent part of the Reynolds-stress tensor is underestimated and only molecular dissipation is considered, the energy feedback to the mean-field is inflated and used as energy sink instead. Although this leads to qualitatively similar mode shapes, the physics of the mode is misrepresented.

On the other hand, if we consider the resolvent modes with mean-field-consistent eddy viscosity (figure 11, dashed), a completely different picture emerges. The modelled energy terms for both $m=0$ and $m=1$ qualitatively follow those of the SPOD modes. Although an axial offset of approximately two diameters can be observed, the physics of the dominant structures seems to be well represented by the resolvent mode when using the mean-field-consistent eddy viscosity to represent the coherent component of Reynolds stresses. The axial offset is caused by the mean-field-consistent eddy viscosity $\varTheta _{{mean}}$, which, as shown in figure 9(a iv,b iv), overestimates the dissipation effect of the energy transfer term.

To summarize, the following conclusions can be drawn from these results. The energetic dissipation through the coherent part of the Reynolds-stress tensor must be modelled in order to model the dominant coherent structures with resolvent analysis – a simple consideration of the molecular dissipation is not sufficient, as also shown in previous studies (Pickering et al. Reference Pickering, Rigas, Schmidt, Sipp and Colonius2021; Kuhn et al. Reference Kuhn, Müller, Knechtel, Soria and Oberleithner2022). If the mean-field-consistent eddy viscosity is used, the dominant structures can be modelled qualitatively very well. This result is particularly remarkable as the same eddy viscosity field was used for both structures with different frequencies and wavenumbers.

7.2. Data-driven eddy viscosities

Figures 12(a) and 12(b) compare the real part of the same axial velocity SPOD modes with the corresponding resolvent modes for the two data-driven approaches. Figure 12(a ii,b ii) show the resolvent result for $\hat {\varTheta }_{{local}}$, the spatially resolved and frequency and wavenumber dependent eddy viscosity field (see figure 7) and the figure 12(a iii,b iii) for the global, space-independent, frequency and wavenumber dependent eddy viscosity $\hat {\varTheta }_{{global}}$ (see figure 8). We recall that the eddy viscosities used were determined from the SPOD data at the respective frequency and wavenumber, meaning that they differ for the shown $m=0$ and $m=1$ modes. The qualitative comparison of the axial velocity mode shapes in figure 12 shows an almost perfect match for both wavenumbers and both eddy viscosities. The two resolvent results show hardly any differences. For $m=0$ and $\hat {\varTheta }_{{local}}$ (figure 12a ii), a small amount of noise can be detected near the axis at $x/D = 10$ and just above the $\bar {u}_x/U =0.95$ line, which results from the noisy $\hat {\varTheta }_{{local}}$ field used in the analysis (see figure 7). Apart from small local deviations, however, the strong noise of the eddy viscosity field observed in figure 7 does not appear to have a major influence on the resolvent modes. For a more in-depth analysis, the corresponding distributions of the dominant energy terms (7.1) are compared in figure 13, as in the previous section for the predictive eddy viscosity models. Especially for $m=0$ (figure 13a) but also for $m=1$ (figure 13b) a very high qualitative and quantitative agreement of the axial profiles of the radially integrated energy terms can be observed for both resolvent results.

Figure 12. Real part of axial velocity modes for $m=0$, ${St}=0.56$ (a) and $m=1$, ${St}=0.28$ (b). Panels (a i) and (b i) show the SPOD modes. Panels (a ii,b ii) and (a iii,b iii) show the resolvent modes for the two data-driven eddy viscosities $\hat {\varTheta }_{{local}}$ and $\hat {\varTheta }_{{global}}$, respectively. The black dashed lines indicate the location at which $\bar {u}_x/U = 0.95$ and $0.5$.

Figure 13. Axially resolved production (green, $\hat {P}_x$), dissipation (red, $\hat {N}_x$) and convection (blue, $\hat {C}_x$) energy terms, (7.1) for $m=0$, ${St}=0.56$ (a) and $m=1$, ${St}=0.28$ (b). The solid lines show the terms for the SPOD modes. The dashed and dotted lines show the terms for the modes of the resolvent analysis (RA) with data-driven spatially varying and global constant eddy viscosity, respectively. Terms belonging to one mode are normalized with the respective maximum absolute value of all considered energy terms.

For $m=0$ (figure 13a) and the spatially resolved $\hat {\varTheta }_{\text {local}}$ field (dashed), the modelled energy terms closely follow those of the SPOD terms (solid). If the modelling of the coherent component of the Reynolds-stress tensor is based on the global eddy viscosity $\hat {\varTheta }_{{global}}$ (dotted), slight differences can be observed in the axial profiles. With $\hat {\varTheta }_{{global}}$, the turbulent dissipation seems to be slightly underestimated, which leads to the modelled mode dissipating slightly too much energy via the production term (negative production). This behaviour is similar to that observed above for purely molecular dissipation, but to a much lesser extent, since the energy dissipation due to the coherent part of the Reynolds-stress tensor is largely accounted for. Apart from these small differences, the overall distribution of energy terms is also very well represented by the resolvent mode when using $\hat {\varTheta }_{{global}}$.

For $m=1$ (figure 13b) and $\hat {\varTheta }_{{local}}$ (dashed), the axial distribution of the radially integrated energy terms between resolvent and SPOD (solid) mode agrees fairly well, although a slight axial offset can still be observed. This offset is slightly higher when using $\hat {\varTheta }_{{global}}$, but an improvement can still be observed compared with the result based on the predictive mean-field-consistent eddy viscosity, compare with figure 11. These results are in perfect agreement with the results of the a priori analysis in § 6: the better the ability of the eddy viscosity to represent the energy dissipation (figure 9), the better the overall energy budget of the SPOD mode is modelled by the resolvent mode (figures 11 and 13).

Since the eddy viscosities used for these results were determined from the data itself, it was perhaps to be expected that the modelling would also work well. We would therefore like to elaborate a little more on the relevance of the results at this point. The results presented in the a priori and a posteriori validation clearly show that if the turbulent dissipation is well approximated by a suitable choice of eddy viscosity, the corresponding dominant coherent structure can be modelled almost perfectly with the resolvent analysis. This leads to the following three main conclusions. First, the coherent component of the Reynolds-stress tensor can be well represented by the eSPOD modes for dominant large-scale coherent structures. Second, the high agreement between the energy budget of the data-driven SPOD with the physics-based resolvent modes validates that the dominant SPOD modes approximate a solution of the energy equation. Third, the results show that the coherent part of the Reynolds-stress tensor indeed acts as a dissipative term on the energy budget, confirming the commonly used Boussinesq approach. Consequently, the role of eddy viscosity in jet flow resolvent analysis is to ensure that all the energy entering the mode through the mean-field is appropriately dissipated.

Moreover, the results show that the correct choice of eddy viscosity for the resolvent analysis depends on the mode considered. However, a specific spatial distribution of the eddy viscosity is not essential, as a comparison of the resolvent results for the spatially resolved $\hat {\varTheta }_{{local}}$ and the global $\hat {\varTheta }_{{global}}$ eddy viscosity show. The reason for this is the high spatial alignment between the true turbulent dissipation term and the corresponding modelled term based on the coherent strain-rate tensor, as discussed in § 6.1. On the other hand, this high alignment also results in the fact that a certain frequency or mode-number dependence of the eddy viscosity can be represented via a spatially distributed eddy viscosity, as here with $\varTheta _{{mean}}$.

7.3. Energetic consideration of the resolvent forcing vector

Up to this point, the contribution of the resolvent forcing vector to the energy budget was neglected. This section briefly examines the corresponding energy term for the various resolvent models with different eddy viscosities. For a compact comparison, we again consider the radially integrated term, and label the term as the forcing term, reading

(7.2)\begin{equation} \hat{{N}}_{f,x} (x) = \int \hat{{N}}_f (x,r)r\,\mathrm{d}r. \end{equation}

Figure 14 shows the forcing term for $m=0$, ${St}=0.56$ (figure 14a) and $m=1$, ${St}=0.28$ (figure 14b). The term is shown for the resolvent results based on pure molecular viscosity $\nu$ (solid), mean-field-consistent eddy viscosity $\varTheta _{mean}$ (dashed) and the two data-driven local $\hat {\varTheta }_{local}$ (dashed–dotted) and global $\hat {\varTheta }_{global}$ (dotted) eddy viscosities. Each curve is normalized with the same value as the corresponding energy terms in figures 11 and 13 in order to establish direct comparability.

Figure 14. Radially integrated resolvent forcing energy term for $m=0$, ${St}=0.56$ (a) and $m=1$, ${St}=0.28$ (b) and various eddy viscosities. Same normalization as in figures 11 and 13.

As already anticipated, figure 14 shows that the contribution of the forcing terms to the overall energy balance is very small for all considered cases. For $m=0$ and $m=1$, the terms for different eddy viscosities show a qualitatively similar trend over the axial coordinate. Although the amplitude is generally small, it appears to depend strongly on the underlying eddy viscosity. For $m=0$ the forcing is strongest near the nozzle exit. For $m=1$ the forcing is also highest in the upstream region, but shows a less pronounced decrease with increasing distance from the nozzle. Moreover, the forcing term for molecular viscosity is the lowest for both $m=0$ and $m=1$.

It is striking that the forcing energy terms remain positive and represent a source of energy and not a sink. This is particularly important for the case without eddy viscosity. In resolvent analysis without eddy viscosity, the entire nonlinear energy term is represented by the forcing term, see (4.5). The analysis of the SPOD mode has shown that the true nonlinear energy transfer term mainly has a dissipative effect on the energy budget of the coherent structures under consideration, see figures 9, 11 and 13. However, the resolvent model does not capture any dissipation with the forcing term, but a pure energy source, albeit with low amplitude. This observation clarifies why dissipation due to the nonlinear energy transfer must be explicitly modelled in resolvent analysis. The resolvent optimal forcing vector does not compensate for the non-modelled nonlinear energy transfer.

7.4. Alignments

So far, our analysis has focused on two coherent structures ($m=0$, ${St}=0.56$ and $m=1$, ${St}=0.28$) for which a large gain separation can be observed in the SPOD spectrum. In this section, resolvent and SPOD modes for both azimuthal orders are compared over a larger frequency range, ${St}=0.1$ to ${St}=1$. For this purpose, we consider the alignment between the leading resolvent and SPOD velocity mode, defined as

(7.3) \begin{equation} A = \frac{| \langle \hat{\boldsymbol{u}}_S, \hat{\boldsymbol{u}}_R\rangle_{L_2} |} {\sqrt{ \langle \hat{{\boldsymbol{u}}}_{S}, \hat{{\boldsymbol{u}}}_{S} \rangle_{L_2} \langle \hat{{\boldsymbol{u}}}_{R}, \hat{{\boldsymbol{u}}}_{R} \rangle_{L_2}}}, \end{equation}

where the indices ${R}$ and ${S}$ denote leading resolvent or leading SPOD velocity mode, respectively. Considering the alignment between the resolvent and SPOD modes is the most common method for a qualitative comparison (Cavalieri et al. Reference Cavalieri, Rodríguez, Jordan, Colonius and Gervais2013; Pickering et al. Reference Pickering, Rigas, Schmidt, Sipp and Colonius2021). A perfect match between the modes results in an alignment of $A=1$, an alignment of $A=0$ means that the modes are orthogonal.

Figure 15 shows the alignment of the velocity modes for $m=0$ (figure 15a) and $m=1$ (figure 15b) in the frequency range ${St}=0.1\unicode{x2013}1$ and the different eddy viscosities indicated by the markers. The transparency of the markers indicates the energy content of the respective SPOD mode, see figure 3. It can be observed that the alignment for the axisymmetric $m=0$ mode is high for all considered resolvent results in the frequency range in which the leading SPOD mode has a high energy content. Due to the high qualitative agreement of the velocity modes, the alignment takes fairly high values even for purely molecular dissipation, although, as shown in the previous section, the physics is not correctly represented. It can be concluded from this that the consideration of alignment alone should be treated with caution and may also be misleading. For the mean-field-consistent eddy viscosity, the alignment in this region reaches values between 0.8 and 0.9, for the data-driven fields even values of up to 0.95. Outside the region of high-gain separation the alignment drops, although it can still be observed that the data-driven eddy viscosity fields lead to acceptable results over large parts of the frequency range. The reason that the alignment for the data-driven fields is not even higher is that the validity of the data-driven eddy viscosity based on eSPOD decreases as the gain separation decreases, as explained above. The poor alignment for $m=0$ and $\hat {\varTheta }_{{local}}$ at low frequencies is also due to noise and resulting convergence problems of the resolvent analysis.

Figure 15. Alignment of velocity modes for $m=0$ (a), $m=1$ (b) and different eddy viscosities indicated by the markers. The marker transparency indicates the energy share of the respective SPOD mode, see figure 3.

For $m=1$ (figure 15b), a similar picture can be observed with high alignments in the region of high-gain separation. Here, in contrast to the $m=0$ case, a poor alignment can be observed for the case with pure molecular dissipation. This means that the qualitative measure of alignment in this case correctly reflects the poor physical representation of the mode in the case of pure molecular dissipation. For the mean-field-consistent and the data-driven eddy viscosities, the differences in alignment are smaller than for $m=0$, but the slightly better representation of the mode for the data-driven viscosities can be recognized. Outside the range of high-gain separation, the alignment decreases, but also for $m=1$, fairly good results are still achieved with data-driven eddy viscosities.

From the results shown, we can conclude that the mean-field-consistent eddy viscosity is overall a good choice to predict the dominant coherent structures, i.e. modes that are associated with much energy. A certain mode number and frequency dependence is thereby modelled via the spatial distribution of the eddy viscosity, as described above. Outside the high-gain separation region, the mean-field-consistent eddy viscosity cannot compensate for the wavenumber and frequency dependency observed in the data-driven eddy viscosities. This makes a predictive approach to modelling the coherent component of the Reynolds stresses outside the high-gain separation region difficult overall. To find the corresponding frequency- and mode-dependent eddy viscosities outside the high-gain separation region, for example, a data-driven optimization within the resolvent model is suitable (Pickering et al. Reference Pickering, Rigas, Schmidt, Sipp and Colonius2021).