Analysis of coherence in turbulent stratified wakes using spectral proper orthogonal decomposition

Turbulent wakes are ubiquitous both in nature and man-made devices. From flow past moving vehicles (Grandemange et al., 2015) to flow past topographic features (Puthan et al., 2021) in oceans, they play an important role in transporting momentum and energy across large distances from the wake generator. In the ocean and the atmosphere, the background density often has a stable density stratification. Buoyancy in a stable background enables the emergence of several distinctive features, e.g., suppression of vertical turbulent motions (Spedding, 2002b), multistage wake decay (Lin & Pao, 1979; Spedding, 1997), appearance of coherent structures in the late wake (Lin & Pao, 1979; Lin et al., 1992a), and formation of steady (Hunt & Snyder, 1980) and unsteady (Gilreath & Brandt, 1985; Bonneton et al., 1993) internal gravity waves, to name a few. A majority of wake studies utilize axisymmetric body shapes (sphere, disk, spheroid, etc.) since such canonical shapes make it convenient to understand the phenomenology of turbulent stratified wakes.

The existence of coherent structures has been established to be an universal feature of both unstratified and stratified turbulent wakes. The Kárman vortex street associated with vortex shedding from the body at a specific frequency is a well known feature of unstratified bluff body wakes which arises from the global instability of the $m=1$ azimuthal mode as was demonstrated for a sphere by Natarajan & Acrivos (1993) and Tomboulides & Orszag (2000). The Strouhal number (St) associated with vortex shedding varies with the shape of the body. Vortex shedding has been investigated in stratified wakes too. Lin et al. (1992b) conducted a detailed experimental investigation of stratified flow past a sphere of diameter $D$ towed with speed $U$ in a fluid with buoyancy frequency $N$ for $5\leq\Rey\ (UD/\nu)\leq 10^{4}$ and $0.005\leq\mbox{{Fr}}\ (U/ND)\leq 20$. At $\mbox{{Fr}}\gtrsim 2$, they found that St in the near wake of the sphere, at $x/D\approx 3$, attained a constant value of $\mbox{{St}}\approx 0.18$, same as in the unstratified wake. For $\mbox{{Fr}}\lesssim 2$, the vortex shedding was two-dimensional and St increased with decreasing Fr in the near wake, similar to the trend in the flow past a circular cylinder. Chomaz et al. (1993) identified four regimes, differentiated by the value of Fr, in the near wake of a sphere. These regimes showed structural differences in the shed vortices and their interactions with the lee wave field.

Another distinctive feature of the stratified wakes is the generation of IGWs which are of two types: (i) body generated steady lee waves and (ii) wake generated unsteady IGWs. In their pioneering work on the wake of a self-propelled slender body, Gilreath & Brandt (1985) noted a coupling between the unsteady IGWs in the outer wake and the wake core turbulence, which suggests that the generation of the unsteady IGWs is inherently nonlinear in nature. Bonneton et al. (1993) and Bonneton et al. (1996) examined IGWs in the flow past a sphere. Lee waves were found to dominate when $\mbox{{Fr}}\lesssim 0.75$ and, for $\mbox{{Fr}}\gtrsim 2.25$, the downstream wake was dominated by the unsteady IGWs. Analysis of the density and velocity spectra in the outer wake showed a distinct peak at the vortex shedding frequency of sphere, $\mbox{{St}}\approx 0.18$. Brandt & Rottier (2015) found wake turbulence to be a dominant source term for IGWs at $\mbox{{Fr}}\gtrsim 1$ in their experimental work on sphere wakes. However, they did not expand on the spectral characteristics of these wake-generated IGWs. Recently, Meunier et al. (2018) also conducted a theoretical and experimental study of waves generated by various wake generators, focusing primarily on the scalings of wavelengths and amplitudes across various Fr and wake generators. Various aspects of IGWs have also been studied through numerical simulations (Abdilghanie & Diamessis, 2013; Zhou & Diamessis, 2016; Ortiz-Tarin et al., 2019; Rowe et al., 2020).

In the last two decades, the rise in computing power has enabled a number of numerical studies which have improved our understanding of stratified wakes. A large body of numerical literature employs the temporal model wherein the wake generator is not included (Gourlay et al., 2001; Dommermuth et al., 2002; Brucker & Sarkar, 2010; Diamessis et al., 2011; de Stadler & Sarkar, 2012; Abdilghanie & Diamessis, 2013; Redford et al., 2015; Zhou & Diamessis, 2019; Rowe et al., 2020). Instead, these simulations are initialized with synthetic mean and turbulence profiles mimicking those of a wake. Body-inclusive simulations which resolve the flow at the wake generator and at a high enough $\Rey$ that sustain turbulence are relatively recent (Orr et al., 2015; Pal et al., 2016, 2017; Ortiz-Tarin et al., 2019; Chongsiripinyo & Sarkar, 2020).

The database from the body-inclusive simulation of Chongsiripinyo & Sarkar (2020), hereafter referred to as CS2020, will be interrogated in this paper to analyze spatio-temporal coherence. CS2020 perform large eddy simulation (LES) of flow past a disk at $\Rey=5\times 10^{4}$ and at various values of Fr. The authors find that the wake transitions through three different regimes of stratified turbulence (provided buoyancy Reynolds number $>O(1)$), each with distinctive turbulence properties: weakly stratified turbulence (WST) which commences when the turbulent Froude number $Fr_{h}$ decreases to $O(1)$, intermediately stratified turbulence (IST) when $Fr_{h}$ decreases to $O(0.1)$, and strongly stratified turbulence (SST) when $Fr_{h}$ reduces to to $O(0.01)$. Here $\mbox{{Fr}}_{h}=u_{h}^{\prime}/Nl_{v}$, where $u^{\prime}_{h}$, $N$, and $l_{v}$ are r.m.s. horizontal velocity fluctuations, buoyancy frequency, and a characteristic turbulent vertical lengthscale, respectively. In the WST regime, the turbulence is not yet appreciably affected by buoyancy effects. Anisotropy in turbulent velocity components, which is a key manifestation of stratification, has not kicked in yet (see figure 8 of CS2020). As the flow evolves downstream, turbulence anisotropy keeps increasing and the turbulence transitions to the IST regime at $\mbox{{Fr}}_{h}\sim O(0.1)$. The SST regime, which commences at $\mbox{{Fr}}_{h}\approx 0.03$, is characterized by a strong anisotropy in turbulence. An indication of arrival of this regime is the scaling of the vertical lengthscale with $u^{\prime}_{h}/N$ as derived by Billant & Chomaz (2001) (also see figure 12 in CS2020). In the SST regime, mean defect velocity and $u^{\prime}_{h}$ decay at the same rate of $x^{-0.18}$ while vertical turbulent velocity ($u^{\prime}_{z}$) decays at a faster rate of $x^{-1}$. Regime classification based on turbulence instead of mean velocity was introduced in the context of stratified homogeneous turbulence, e.g., Brethouwer et al. (2007), and was recently extended to stratified turbulent wakes by Zhou & Diamessis (2019) and CS2020.

With the huge amount of numerical and experimental data becoming available, data-driven modal decomposition techniques have also seen an unprecedented rise in their use to understand the dynamics and role of coherent structures in turbulent flows. These techniques have also been used to construct reduced-order models of these flows. One popular technique is proper orthogonal decomposition (POD), proposed by Lumley (1967, 1970) in the context of turbulent flows, which provides a set of modes ordered hierarchically in terms of energy content. Another popular technique is dynamic mode decomposition (DMD), described by Schmid (2010), which decomposes the flow into a set of spatial modes, each oscillating at a specific frequency.

However, applications of modal decomposition to stratified flows are few in number. Diamessis et al. (2010) performed snapshot POD (Sirovich (1987)) on the vorticity field from a temporal simulation at $\Rey=5\times 10^{3}$ and $\mbox{{Fr}}=2$, noting a link between wake core structures and the angle of emission of IGWs in the outer wake. The layered wake core structure, which is a distinctive feature of stratified turbulent wakes, was found in the POD modes with lower modal index (corresponding to higher energy). As the modal index increased, the wake core was found to be dominated by small-scale incoherent turbulence. Xiang et al. (2017) performed spatial and temporal DMD on the experimental data of the stratified wake of a grid showing that DMD modes successfully captured lee waves and Kelvin-Helmholtz (KH) instability in the near wake ($Nt<10$). Nidhan et al. (2019) performed three-dimensional (3D) and planar two-dimensional (2D) DMD on the sphere wake at $\Rey=500$ and $10^{4}$, respectively. At $\Rey=500$ and $\mbox{{Fr}}=0.125$, they found that the 2D vortex shedding in the center-horizontal plane and ‘surfboard’ structures in the center-vertical plane corresponded to the same DMD mode oscillating at the vortex shedding frequency of $\mbox{{St}}\approx 0.19$. At the higher $\Rey=10^{4}$, DMD modes associated with vortex shedding showed IGWs in the outer wake.

In the present work, we use spectral proper orthogonal decomposition (SPOD), originally proposed by Lumley (1967, 1970) and recently revisited by Towne et al. (2018), to identify and analyze the coherent structures in the turbulent stratified wake of a disk at $\Rey=5\times 10^{4}$. In its original form, POD is prohibitively expensive to apply on today’s large numerical databases with high space-time resolution. The form put forward by Towne et al. (2018) leverages the temporal symmetry of statistically stationary flows to improve computational tractability. SPOD decomposes statistically stationary flows into energy-ranked modes with monochromatic frequency content, thus separating both the temporal and spatial scales in the flow, unlike the popular snapshot variant given by Sirovich (1987). SPOD has been used extensively in recent times for analysis of coherent structures and reduced-order modeling in a variety of unstratified flow configurations: (i) turbulent jets (Semeraro et al., 2016; Schmidt et al., 2017, 2018; Nogueira et al., 2019; Nekkanti & Schmidt, 2020b), (ii) turbulent wakes (Nidhan et al., 2020), (iii) channel (Muralidhar et al., 2019) and pipe (Abreu et al., 2020) flows, (iv) flow reconstruction (Nekkanti & Schmidt, 2020a) and low-order modeling (Chu & Schmidt, 2021), (v) wakes of actuator disks in turbulent environments (Ghate et al., 2018, 2020), etc.

The formation of coherent pancake vortices in the Q2D late wake does not necessarily require vortex shedding from the body as was demonstrated by Gourlay et al. (2001) whose temporally evolving model at $\mbox{{Fr}}=10$ did not include the vortex shedding mode but still exhibited Q2D-regime pancake vortices. Our interest is also in coherent structures but in a region of the far wake which is at large $x/D$ but still not in the Q2D regime. We ask how does buoyancy affect the space-time coherence as the flow progresses from the near wake to the far wake? What are the salient differences between the unstratified ($\mbox{{Fr}}=\infty$) and stratified wakes in the context of coherent structures? We will address these questions by analyzing the LES dataset of CS2020, specifically the wakes at $\mbox{{Fr}}=2$ and 10. We adopt SPOD for the data analysis since it is well suited to extract modes which have spatial and temporal coherence and thus track the evolution of specific modes, e.g. the vortex shedding (VS) mode, as the wake evolves downstream. The SPOD analysis also allows us to address a second set of questions: (i) are coherent modes linked to unsteady IGWs and (ii) how is the energy in dominant coherent structures distributed across the wake cross-section during downstream evolution? SPOD modes can also be useful for constructing reduced-order models prompting the third question: what is the efficacy of different SPOD modal truncations in regard to the reconstruction of various second-order turbulence statistics in turbulent stratified wakes?

The rest of the paper is organized as follows. §2 and §3 give a brief overview of the numerical methodology and SPOD technique. Visualizations of $\mbox{{Fr}}=2$ and $10$ wakes are presented in §4. The characteristics of SPOD eigenvalues and eigenspectrum are discussed in §5. The VS mode and its link to the unsteady IGWs are discussed in detail in §6. Sections §7 and §8 discuss the spatial structure of SPOD eigenmodes and trends in the reconstruction of second-order statistics by sets of truncated SPOD modes, respectively. Finally, the discussion and conclusions are presented in §9.

We use the numerical database of the wake of a circular disk at $\Rey$ = $5\times 10^{4}$ from CS2020. In particular, we analyze the datasets of stratified wakes at $\mbox{{Fr}}=2$ and $10$ from their numerical database. CS2020 use high-resolution large eddy simulation (LES) to numerically solve the filtered Navier-Stokes equations system along with density diffusion equation under the Boussinesq approximation.

These equations are as follows:

continuity,

momentum,

and density diffusion,

where $u_{i}$ corresponding to $i=1,2,$ and $3$ refer to velocity in the streamwise ($x_{1}$ or $x$), lateral ($x_{2}$ or $y$), and vertical ($x_{3}$ or $z$) directions, respectively. Gravity acts in the vertical direction (2). The density field is decomposed into a background profile, $\rho_{b}(z)=\rho_{o}+(\mathop{}\!\mathrm{d}\rho_{b}/\mathop{}\!\mathrm{d}z)z$ (where $K$ is a constant), and density deviation ($\rho^{\prime}$). Thus $\rho(x,y,z,t)=\rho_{b}(z)+\rho^{\prime}(x,y,z,t)$. In (2), $\nu_{s}$ and $\nu$ refer to the subgrid kinematic viscosity obtained from LES and kinematic viscosity of the fluid, respectively. Likewise, $\kappa_{s}$ and $\kappa$ in equation (3) refer to the subgrid density diffusivity and density diffusivity of the fluid, respectively.

(1) - (3) are non-dimensionalized using the following parameters: (i) free stream velocity ($U_{\infty}$) for velocity field, (ii) diameter of disk ($D$) for spatial locations $x_{i}$, (iii) dynamic pressure ($\rho_{o}U_{\infty}^{2}$) for pressure field, (iv) advection timescale ($D/U_{\infty}$) for time $t$, and (iv) $-(\mathop{}\!\mathrm{d}\rho_{b}/\mathop{}\!\mathrm{d}z)D$ for density deviation. There are three non-dimensional parameters of interest: (1) body-based Reynolds number ($\Rey$) defined as $U_{\infty}D/\nu$, (2) body-based Froude number (Fr) defined as $U_{\infty}/ND$ where $N$ is the buoyancy frequency, $N^{2}=-g/\rho_{o}(\mathop{}\!\mathrm{d}\rho_{b}/\mathop{}\!\mathrm{d}z)$, and (3) Prandtl number ($\Pran$) defined as $\nu/\kappa$ which is set as 1 in CS2020 simulations. $\kappa_{s}$ is also set equal to $\nu_{s}$ for the LES simulations.

A cylindrical coordinate system is adopted and the disk is represented using the immersed boundary method (IBM) of Balaras (2004); Yang & Balaras (2006). Spatial derivatives are computed using second-order central finite differences and temporal marching is performed using a fractional step method which combines a low-storage Runge-Kutta scheme (RKW3) with the second-order Crank-Nicolson scheme. The kinematic subgrid viscosity ($\nu_{s}$) and density diffusivity ($\kappa_{s}$) are obtained using the dynamic eddy viscosity model of Germano et al. (1991). At the inlet and outlet, Dirichlet inflow and Orlanski-type convective (Orlanski (1976)) boundary conditions are specified, respectively. The Neumann boundary condition is used at the radial boundary for the density and velocity fields. To prevent the spurious propagation of internal waves upon reflection from the boundaries, sponge regions with Rayleigh-damping are employed at radial, inlet, and outlet boundaries.

The radial and streamwise domains span $0\leq r/D\leq 80$ and $-30\leq x/D\leq 125$, respectively. A large radial extent facilitates weakening of the IGWs before they hit the boundary and thereby also controls the amplitude of spurious reflected waves. The distribution of grid points are as follows: $N_{r}=531$ in the radial direction, $N_{\theta}=256$ in the azimuthal direction, and $N_{x}=4608$ in the streamwise direction, resulting in approximately 530 million elements. The grid resolution is excellent by LES standards in all three directions. Readers may refer to Chongsiripinyo & Sarkar (2020) for more details on the grid resolution and numerical scheme.

In this work, we employ spectral POD (SPOD) to study the dynamics of coherent structures in stratified wakes, rather than the more commonly employed snapshot POD (Sirovich, 1987). SPOD enables the identification of dominant structures evolving coherently in both space and time by exploiting temporal correlation among flow snapshots. This approach is particularly well-suited for flow configurations like turbulent wakes which are known to be dominated by mechanisms operating at specific frequencies, e.g., vortex shedding, pumping of recirculation bubble, shear layer breakdown, to name a few (Berger et al., 1990). On the contrary, snapshot POD assumes each snapshot of the flow to be an independent realization. As a result, the temporal coherence of POD modes is not guaranteed. Furthermore, it can be also shown that the coefficients dictating the temporal evolution of snapshot POD modes are broadband, i.e., containing contributions from a range of frequencies (Towne et al., 2018). SPOD requires a larger amount of time-resolved data compared to snapshot POD. Hence, snapshot POD has dominated the literature compared to SPOD.

Three-dimensional visualizations of the $Q$ criterion and planar views of the vorticity and velocity fields in this section provide a first look at the vortical and unsteady IGW structure of the simulated wakes. The structure of the steady (in a frame attached to the disk) lee wave field is not discussed in this paper.

To emphasize the large-scale coherent structures, the instantaneous velocity fields have been filtered using a SciPy Gaussian low-pass filter (gaussian_filter) in all three directions with standard deviation $\sigma=5$ before calculating the $Q$ criterion and vorticity fields.

Figure 1 shows that, in both wakes, circular vortex rings appear immediately downstream of the disk. At $\mbox{{Fr}}=2$, the buoyancy induced anisotropy between horizontal and vertical directions commences in the near wake. The wake contracts in the vertical at $x/D\approx 5$ (visible in the side view given in figure 1(b)) owing to the oscillatory modulation by the lee wave. The top view (figure 1(a)) shows a distinct large-scale waviness in the intermediate wake, shown by the dashed black line. Its approximate wavelength is $\lambda/D\approx 1/\mbox{{St}}_{VS}$, where $\mbox{{St}}_{VS}$ is the vortex shedding (VS) frequency. Likewise, large-scale VS structures separated approximately by $\lambda/D\approx 1/\mbox{{St}}_{VS}$ can also be identified in the $\mbox{{Fr}}=10$ wake (figure 1(c)). The value of $\mbox{{St}}_{VS}$ and the spatial behavior of the VS mode will be made precise formally using SPOD in the subsequent sections.

Figure 2 shows the instantaneous vertical vorticity ($\omega_{z}D/U_{\infty}$) on the central horizontal plane ($z=0$) for the $\mbox{{Fr}}=2$ (top) and $\mbox{{Fr}}=10$ (bottom) wakes. Similar to figure 1, $\omega_{z}$ is calculated using filtered velocity fields to emphasize large-scale features. In both wakes, the complex spatial distribution of vorticity in the immediate downstream of the disk gives way to a well-defined coherent distribution of opposite signed vortices in the intermediate to late wakes. For the $\mbox{{Fr}}=2$ wake, spatial coherence is visible as early as $x/D\approx 20$. Beyond $x/D\approx 20$, the regions of opposite-signed $\omega_{z}$ remain separated till the end of the domain. On closer inspection, a streamwise undulation of length $\lambda/D\approx 1/\mbox{{St}}_{VS}$ can be observed in figure 2(a). At this point, it is important to emphasize that the $\mbox{{Fr}}=2$ wake remains actively turbulent throughout the computational domain as demonstrated by CS2020 through spectra and visualizations of the turbulent dissipation rate. From $x/D\approx 40$ onward, the $\mbox{{Fr}}=2$ wake resides in the strongly stratified turbulent (SST) regime. Different regimes of stratified turbulence are discussed briefly in §1. The strong signature of coherence in the $\mbox{{Fr}}=2$ wake is not a consequence of the transition into the weakly turbulent state of the Q2D regime noted in previous works, e.g. by Spedding (1997).

The $\mbox{{Fr}}=10$ wake also shows a distinct wavy motion with non-dimensional wavelength $\approx 1/\mbox{{St}}_{VS}$. However, the separation between the regions with opposite signed vorticity is not as well defined as in the $\mbox{{Fr}}=2$ wake. According to CS2020, the $\mbox{{Fr}}=10$ wake stays in the weakly stratified regime (WST) from $x/D\approx 10$ to $50$ and thereafter stays in the intermediately stratified regime (IST) till the end of the domain.

To conclude this section, instantaneous snapshots of fluctuating spanwise velocity ($u^{\prime}_{y}/U_{\infty}$) are shown in figure 3 at locations in the near, intermediate and far wake at $\mbox{{Fr}}=2$ (top row) and $\mbox{{Fr}}=10$ (bottom row). An ellipse with major and minor axes equal to $2L_{Hk}$ and $2L_{Vk}$, where $L_{Hk}$ and $L_{Vk}$ are the TKE-based wake widths in horizontal and vertical directions, respectively, is also shown. $L_{Hk}$ is defined by $\mathrm{TKE}(x,y=L_{Hk},z=0)=\mathrm{TKE}(x,r=0)/2$ and $L_{Vk}$ by $\mathrm{TKE}(x,y=0,z=L_{Vk})=\mathrm{TKE}(x,r=0)/2$. Here, $r=0$ denotes the disk centerline. It is worth noting that using the sum of TKE and TPE to define the wake widths (not shown here) result in values similar to $L_{Hk}$ and $L_{Vk}$ for both $\mbox{{Fr}}=2$ and $\mbox{{Fr}}=10$ wakes. Following CS2020, we use the TKE-based definitions in the rest of the results and discussions. This ellipse, based on $L_{Hk}$ and $L_{Vk}$, is used to approximately demarcate the wake core from the outer wake. In subsequent sections, this definition of the wake core will prove to be useful for the interpretation of some SPOD results.

At $\mbox{{Fr}}=2$, an appreciable effect of buoyancy is already present in the near wake as shown in figure 3(a) for $x/D=10$, which corresponds to $Nt_{2}=5$ in buoyancy time units. At the same streamwise location, the $\mbox{{Fr}}=10$ wake still has a circular cross-section with an imprint of the $m=1$ azimuthal mode which was found to be energetically important in the unstratified wake (Nidhan et al. (2020)). As both the wakes evolve downstream, buoyancy has a progressively increasing effect on the the wake core as well as the surrounding outer wake. By $x/D=50$, vertically flattened wake cores can be observed in figure 3(b,d) for both the wakes, more so at $\mbox{{Fr}}=2$ than at $\mbox{{Fr}}=10$. It is also worth noting that the wake core of $\mbox{{Fr}}=2$ consists of distinct layers by $x/D=50$. The $\mbox{{Fr}}=2$ wake also shows a significant amount of IGW activity in the outer region, i.e. outside the ellipse in figure 3(b). Farther downstream at $x/D=100$, the $u^{\prime}_{y}$ field of $\mbox{{Fr}}=2$ (figure 3(c)) shows IGWs occupying a significant portion of the outer wake with the wake core being further flattened and comprising an increased number of horizontally oriented layers. The $\mbox{{Fr}}=10$ wake core also starts showing appreciable IGW activity in the ambient by $x/D=100$ ($Nt_{10}=10$), as shown in figure 3(f).

We start the discussion of SPOD modes by evaluating their overall contribution to fluctuation energy and by their eigenspectra. There are significant effects of buoyancy as elaborated below.

A comparison between SPOD eigenspectra of the stratified wakes (figure 6) and the unstratified wake (figure 3 in Nidhan et al. (2020)) reveals that both types of wakes are dominated by vortex shedding which gives rise to a distinct spectral peak in the vicinity of $\mbox{{St}}\approx 0.13$. For the unstratified wake, besides the VS structure, which appears in the azimuthal mode $m=1$, a double helix ($m=2$) mode with a peak at $\mbox{{St}}\rightarrow 0$ is also found to be energetically important (Johansson & George, 2006; Nidhan et al., 2020). In the stratified wake, as elaborated below, we find that the VS mode is persistent, is linked to unsteady internal gravity waves (IGWs), and is thereby responsible for the accumulation of fluctuation energy outside the wake core.

Stratification qualitatively affects the streamwise evolution of the energy in different frequencies. The evolution of the frequency-binned energy is shown for the stratified wakes in figure 8 (a)-(b). For the unstratified case ($\mbox{{Fr}}=\infty$), the azimuthal modes $m=1$ and $m=2$ are shown in figure 8(c) and (d), respectively. For the stratified wakes in figures 8 (a)-(b), the spectral peak in the vicinity of $\mbox{{St}}\approx 0.13$ remains prominent for significant downstream distances, especially for $\mbox{{Fr}}=2$. A somewhat wide band ($0.1\leq\mbox{{St}}\leq 0.2$), centered around $\mbox{{St}}\approx 0.13$ of the VS mode, is excited for the stratified wakes. Furthermore, this band persists into the far wake. Even at $x/D=100$, this band has larger energy density than at other frequencies. Such persistence in the energetic dominance of the VS mode (and neighboring frequencies) is absent in the unstratified $\mbox{{Fr}}=\infty$ case where the energy at the two peaks of: (i) $\mbox{{St}}=0.135$ in the $m=1$ mode (figure 8(a)) and (ii) $\mbox{{St}}=0$ in the $m=2$ mode (figure 8(b)) declines sharply with increasing $x/D$.

Figure 9 shows the streamwise evolution of energy in the leading 15 SPOD modes and in a frequency band around the VS frequency. The energy in the $\mbox{{Fr}}=2$ wake remains almost constant till $x/D=60$ and starts decaying slowly thereafter. On the other hand, the $\mbox{{Fr}}=10$ wake shows an initial decay in the VS mode energy which closely follows that of the $\mbox{{Fr}}=\infty$ wake till $x/D=20$. Subsequently, buoyancy effects set in for the $\mbox{{Fr}}=10$ wake to slow down the energy decay.

To investigate the reason behind the downstream persistence of the VS spectral peak in stratified wakes, the total energy in the leading 15 SPOD modes is partitioned into two components: (i) energy of the wake core, $E_{core}$ and (ii) energy of the outer wake, $E_{outer}$. The energy in each of the regions is calculated as:

where $\Omega$ denotes the wake core at a given $x/D$, as defined in §4. $\mathcal{H}$ denotes the area of the circular cross-section bounded by $0\leq r/D\leq 10$ at a given $x/D$. Here, $\Phi^{(n)}_{i}$ corresponds to the $n^{th}$ SPOD eigenmode for a given $x/D$ and St.

The energy in the wake core peaks around the VS-mode frequency, $\mbox{{St}}\approx 0.12-0.13$, for both wakes (see figure 10(a,c)). With increasing $x/D$ (or $Nt$), the VS signature in the wake core decays for both wakes. The energetics of the outer wake is remarkably different. $E_{outer}$, which starts off with a small value across all St at $x/D\approx 10$ in the $\mbox{{Fr}}=2$ wake, develops a peak at $\mbox{{St}}\approx 0.15$ at $x/D\approx 20$. Note that this peak is the same as the peak in the SPOD eigenspectrum for the entire wake (figure 6(a)). Farther downstream, there is significant energy content in the outer wake for $x/D\approx 16-80$ ($Nt_{2}\approx 8-40$) with a spectral peak located at $\mbox{{St}}\approx 0.13-0.15$. The spectral peak is broad, i.e. nearby frequencies with $0.1\leq\mbox{{St}}\leq 0.2$ also have comparable energy levels. For the $\mbox{{Fr}}=10$ wake, $E_{outer}$ picks up only beyond $x/D=60$ ($Nt_{10}=6$), and thereafter increases progressively in the vicinity of $\mbox{{St}}\approx 0.13$ till the end of the domain. We also find that the qualitative nature of the variation of energy in the outer wake and wake core found in figure 10 does not change when the number of modes over which energy is summed is decreased from 15 to 3 (not presented here for brevity).

In figure 11, we sum up the SPOD energies across $\mbox{{St}}\in[-0.4,0.4]$ separately for the wake core and the outer wake and compute their percentage contribution to the entire area-integrated fluctuation energy as follows,

where $E^{T}_{k}(x/D)$ and $E^{T}_{\rho}(x/D)$ are area-integrated TKE and TPE in the circular region of $0\leq r/D\leq 10$ at the $x/D$ location under consideration. The streamwise evolution of $\chi_{core}$ and $\chi_{outer}$ are shown in figure 11.

For the $\mbox{{Fr}}=2$ wake (figure 11(a)), $\chi_{outer}$ increases monotonically until $x/D\approx 60$ followed by a slight decrease. At its peak, $\chi_{outer}$ constitutes up to $50\%$ of the total fluctuation energy, becoming even larger than $\chi_{core}$. In the $\mbox{{Fr}}=10$ wake (figure 11(b)), $\chi_{outer}$ remains negligible till $x/D=60$, followed by a monotonic increase. The increase in the value of $\chi_{outer}$ is accompanied by a decrease in the wake-core contribution. The percentage of total energy captured by the leading 15 SPOD modes and $|\mbox{{St}}|\in[0,0.4]$, i.e., $\chi_{wake}+\chi_{outer}$ (shown in green), increases for both wakes from its initial value at $x/D=10$. This reinforces a main finding of this work that stratified wakes display an increased coherence of fluctuation energy as they evolve downstream.

Figure 10 suggests that the unsteady IGWs in the outer wake radiate from the VS mode at intermediate to late $Nt$. Nevertheless, to further establish causation between the unsteady IGW emission and the VS mode, we perform additional SPOD analyses for the $\mbox{{Fr}}=2$ wake. In these analyses, we replace the fluctuating density field $\rho^{\prime}$ with the fluctuating pressure field $p^{\prime}$. These SPOD analyses are performed at $x/D=10,20,\cdots,90,100$. At all locations, the eigenspectra obtained from these modified SPOD analyses show a prominent peak at the VS frequency with a large gap between $\lambda^{(1)}$ and $\lambda^{(2)}$ for $\mbox{{St}}<0.2$, qualitatively akin to the left column of figure 6.

Using $p^{\prime}$ along with $u^{\prime}_{i}$ enables us to reconstruct the pressure transport term in the radial direction, $\langle p^{\prime}u^{\prime}_{r}\rangle$, which accounts for the energy transferred radially from the wake core to the IGW dominated outer wake region through pressure-work (de Stadler & Sarkar, 2012; Rowe et al., 2020). We reconstruct $\langle p^{\prime}u^{\prime}_{r}\rangle$ contours using leading 15 SPOD modes and frequencies in the range of (i) $\mbox{{St}}\in[0.1,0.2]$ and (ii) $\mbox{{St}}\in[0.1,0.3]$ as follows:

Figure 12 shows the actual (top row) and reconstructed (middle and bottom rows) $\langle p^{\prime}u^{\prime}_{r}\rangle$ at three streamwise locations $x/D=20,40,$ and $60$ for the $\mbox{{Fr}}=2$ wake. The actual $\langle p^{\prime}u^{\prime}_{r}\rangle$ shows a strong signature of IGW flux in the outer wake region at all three downstream locations in figure 12. We found that the nonlinear transport term was negligible outside the wake core (not shown here). Hence the primary source of the energy transfer to the outer wake is the pressure-work term due to the IGW radiation. The reconstructed $\langle p^{\prime}u^{\prime}_{r}\rangle$ using $\mbox{{St}}\in[0.1,0.2]$ (middle row) shows qualitative agreement with the spatial distribution of actual $\langle p^{\prime}u^{\prime}_{r}\rangle$, both in the wake core as well as outer wake region, at all downstream locations. As more frequencies are included (bottom row), adjacent to the VS frequency, the accuracy of reconstruction increases. The key spatial characteristics of $\langle p^{\prime}u^{\prime}_{r}\rangle$ remain similar in both reconstructions, showing that frequencies in the vicinity of the VS frequency satisfactorily capture the key dynamics of unsteady IGW generation.

To further elucidate the causal link between the VS mode and the IGW generation, we plot the $x/D-\mbox{{St}}$ variation of the integrated $\langle p^{\prime}u^{\prime}_{r}\rangle$ in the outer wake region (similar to 18), reconstructed using 3 and 15 SPOD modes (figure 13). The evolution of integrated $\langle p^{\prime}u^{\prime}_{r}\rangle$ in the outer wake of the $\mbox{{Fr}}=2$ wake is very similar to that of the energy in the outer wake region (figure 10(b)). It starts off at a small value at $x/D\approx 10$, develops a broad peak centered at $\mbox{{St}}\approx 0.15$ between $20\leq x/D\leq 80$, and gradually starts declining beyond $x/D=80$. Increasing the number of modes from $3$ to $15$ makes the active St region broader while intensifying the reconstructed values.

Figure 10, 12, and 13 firmly establish that the VS mode energy radiates out of the wake core instead of being acted on by nonlinear interactions in the turbulent wake responsible for the usual energy cascade. Therefore, unlike their unstratified counterpart, the stratified wakes exhibit a persistent VS spectral peak when the energy in the full domain of influence (denoted by $\mathcal{H}$) of the wake is taken into account as in the SPOD results of figure 8(a,b).