1. Introduction
1.1. Slender body wakes in unstratified fluid at zero angle of attack
The study of wakes behind slender bodies is a crucial area of fluid mechanics research with significant implications for various engineering applications. From the design of streamlined vehicles and structures to the understanding of biological locomotion, the behaviour of fluid flow in the wake region of elongated objects continues to present both challenges and opportunities. To the best of our knowledge, Chevray (Reference Chevray1968) conducted the first experimental investigation of a slender body wake, employing a 6:1 prolate spheroid with a tripped boundary layer at an angle of attack
$\alpha = 0^{\circ }$
as the wake generator. Further experimental and computational studies by Jimenez et al. (Reference Jimenez, Hultmark and Smits2010), Posa & Balaras (Reference Posa and Balaras2016) and Kumar & Mahesh (Reference Kumar and Mahesh2018) have explored the on-body and near-wake dynamics of slender bodies at
$\alpha = 0^{\circ }$
and
${Re}_L \sim O(10^6)$
, using the DARPA-SUBOFF with a tripped boundary layer (BL) as the wake generator. Due to limitations in the experimental set-up and the prohibitively expensive cost of performing wall-resolved simulations for slender bodies, these studies cover downstream distances of up to
$x/D \approx 20$
. In terms of the near wake, coherent vortex shedding was not detected. The absence of near-wake vortex shedding at a high
${Re}_L$
with a tripped BL configuration was also found by Ortiz-Tarin et al. (Reference Ortiz-Tarin, Nidhan and Sarkar2021) for a 6:1 prolate spheroid. This is in stark contrast to moderate- to low-
${Re}_L$
configuration for slender bodies, 4:1 spheroid by Ortiz-Tarin et al. (Reference Ortiz-Tarin, Chongsiripinyo and Sarkar2019) at
${Re}_L = 4 \times 10^4$
and 6:1 spheroid by Ohh & Spedding (Reference Ohh and Spedding2024), where clear vortex shedding from the slender wake generator is observed. It is worth noting that the high-
$Re$
spheroid wake of Ortiz-Tarin et al. (Reference Ortiz-Tarin, Nidhan and Sarkar2021) at
${Re}_L = 6 \times 10^5$
exhibited a region where the flow evolved self-similarly but with a wake deficit law of
$U_d \sim x^{-6/5}$
(different from the classical
$U_d \sim x^{-2/3}$
) due to non-equilibrium dissipation scaling (Nedić et al. Reference Nedić, Vassilicos and Ganapathisubramani2013; Dairay et al. Reference Dairay, Obligado and Vassilicos2015) that has been attributed to unsteadiness in the energy cascade.
1.2. Unstratified slender body wakes and streamwise vortices at non-zero pitch angle
Another widely studied configuration involves unstratified flow past a spheroid, tilted at a non-zero pitch angle (
$\alpha$
) relative to the incoming flow. While the separation region behind a slender body at
$\alpha = 0^{\circ }$
is relatively small compared with that of a bluff body, the separation patterns can change significantly as
$\alpha$
increases. At even moderate angles of attack (
$\alpha \in [10^{\circ }, 20^{\circ }]$
), the separation line shifts upstream from the tail of the body and forms streamwise vortices on both sides as the flow detaches from the surface (Han & Patel Reference Han and Patel1979; Fu et al. Reference Fu, Shekarriz, Katz and Huang1994; Patel & Kim Reference Patel and Kim1994). In addition to the primary streamwise vortices, secondary vortices with weaker circulation can also develop in these configurations (Fu et al. Reference Fu, Shekarriz, Katz and Huang1994).
The vortices shed off a slender body may not be axisymmetric even in the mean although the body is axisymmetric. The asymmetric nature of a slender-body wake at high angles of attack (
$\alpha \gt 40^{\circ }$
) has been extensively documented for slender bodies with a sharp nose (Nelson & Pelletier Reference Nelson and Pelletier2003; Bridges Reference Bridges2006; Nelson et al. Reference Nelson, Corke and Matsuno2006). Direct numerical simulation (DNS) at moderate
${Re}_L \sim O(10^4)$
and high
$\alpha = 45^{\circ }$
(Jiang et al. Reference Jiang, Gallardo, Andersson and Zhang2015; Strandenes et al. Reference Strandenes, Jiang, Pettersen and Andersson2019) for a 6:1 prolate spheroid reports that the flow separation, resulting wake and streamwise vortex pair are asymmetrical in the lateral direction, despite the configuration being symmetrical. This asymmetry induces a considerable lateral force on the spheroid. In the near wake, these studies do not find vortex shedding (Jiang et al. Reference Jiang, Gallardo, Andersson and Zhang2015). Instead, they report an asymmetric vortex pair where the vortex tube structure with larger circulation magnitude persists throughout the computational domain. At further higher
${Re}_L \sim O(10^6)$
, DNS into the far wake becomes infeasible. Consequently, previous studies on flow past slender bodies at non-zero angle of attack have employed either Reynolds-averaged Navier–Stokes (RANS) simulations (Constantinescu et al. Reference Constantinescu, Pasinato, Wang, Forsythe and Squires2002) or large eddy simulations (LES) (Wikström et al. Reference Wikström, Svennberg, Alin and Fureby2004; Plasseraud et al. Reference Plasseraud, Kumar and Mahesh2023). These computational studies do not report any asymmetry in the flow separation or the resulting wake at
${Re}_L \sim O(10^6)$
and
$\alpha$
within the range of
$10^\circ$
–
$20^\circ$
.
The high-
$Re$
experimental study by Ashok et al. (Reference Ashok, Van Buren and Smits2015a
) does report asymmetry in the flow. The authors conducted detailed velocity measurements in the wake of a DARPA-SUBOFF model (an axisymmetric body) over a wide range of
${Re}_L \in [2.4 \times 10^6, 30 \times 10^6]$
at low to moderate angles of attack, up to
$\alpha = 12^\circ$
. They observed that even at these low to moderate
$\alpha$
values, the streamwise vortex pair and the wake structure were asymmetrical with the circulation of one vortex exceeding the other by 10 %–20 %. To assess the robustness of this asymmetry, they repeated their experiments under various conditions, such as changing the physical model and wind tunnel, and consistently found that the asymmetry remained a distinctive feature of the wake for such configurations. Some experimental studies (Grandemange et al. Reference Grandemange, Gohlke and Cadot2013; Rigas et al. Reference Rigas, Oxlade, Morgans and Morrison2014) have reported the presence of an asymmetric mode in bluff-body wakes, even at zero pitch and yaw. In these wakes, the time scale of the asymmetric mode is almost
$O(10^2) {-} O(10^3)$
times larger than that of the vortex shedding mode.
1.3. Stratification effects on slender-body wakes at zero pitch angle
Despite their widespread use in naval and aerospace applications, the effect of stratification on slender-body wakes (Ortiz-Tarin et al. Reference Ortiz-Tarin, Chongsiripinyo and Sarkar2019, Reference Ortiz-Tarin, Nidhan and Sarkar2023; Ohh & Spedding Reference Ohh and Spedding2024) is not as well studied as the effect of stratification on bluff bodies (Lin et al. Reference Lin, Lindberg, Boyer and Fernando1992; Spedding et al. Reference Spedding, Browand and Fincham1996; Spedding Reference Spedding1997; Dommermuth et al. Reference Dommermuth, Rottman, Innis and Novikov2002; Meunier & Spedding Reference Meunier and Spedding2006; Brucker & Sarkar Reference Brucker and Sarkar2010; de Stadler & Sarkar Reference de Stadler and Sarkar2012; Pal et al. Reference Pal, Sarkar, Posa and Balaras2017; Zhou & Diamessis Reference Zhou and Diamessis2019; Chongsiripinyo & Sarkar Reference Chongsiripinyo and Sarkar2020; Rowe et al. Reference Rowe, Diamessis and Zhou2020; Madison et al. Reference Madison, Xiang and Spedding2022; Li et al. Reference Li, Yang and Kunz2024). For a uniformly stratified medium, the effect of ambient stratification on the wake is quantified by the body-based Froude number,
${\textit {Fr}} = U_\infty /ND$
, where
$U_\infty$
,
$N$
and
$D$
are the free stream velocity, buoyancy frequency and body length scale (diameter in the case of a sphere or disk and minor-axis diameter for a spheroid), respectively. The value of body
$\textit {Fr}$
depends not only on the application through the submersible parameters (relative speed
$U$
and size
$D$
), but also strongly on the environment since
$N$
in s
$^{-1}$
can vary from
$10^{-3}$
in the deep ocean to
$10^{-2}$
in the upper ocean to values as large as
$10^{-1}$
in a sharp pycnocline.
${\textit {Fr}} = 1$
to 500 brackets most applications with lower values corresponding to low-speed operation of large vehicles or to ocean observation floats that have low relative velocity with respect to the background currents. Irrespective of the body
$\textit {Fr}$
, the wake is inevitably affected by buoyancy since the progressive decrease (increase) of wake velocity (size) reduce the local Froude number to
$O(1)$
and smaller. For bluff-body stratified wakes, a three-stage decay is well established: (a) 3-D regime with essentially unstratified wake behaviour until
$Nt \sim O(1)$
; (b) non-equilibrium (NEQ) regime from
$Nt \approx 2{-}50$
where the defect velocity
$U_d \sim x^{-0.25 \pm 0.04}$
and the wake continues to adjust to the buoyancy effects; and (c) quasi-two-dimensional (Q2-D) regime beyond
$Nt \approx 50$
, where
$U_d \sim x^{-0.75}$
and wake organises progressively into horizontally meandering pancake vortices.
To the best of our knowledge, Ortiz-Tarin et al. (Reference Ortiz-Tarin, Chongsiripinyo and Sarkar2019) was the first numerical study of the stratified flow past a slender body (4:1 spheroid) and it was conducted at a moderate Reynolds number
${Re}_L = 40\,000$
. A major finding of their work is that the slender body exhibits stronger buoyancy effects at the same body-based Froude number (
$\textit {Fr}$
) compared with a bluff body. For example, at a critical
${\textit {Fr}}_c = \mathrm {AR}/\pi$
, the flow separation past the spheroid is suppressed due to modulation by steady lee waves, leading to wake relaminarisation. Furthermore, even at
${\textit {Fr}} = \infty$
, representing the unstratified configuration, the wake and flow separation remain azimuthally symmetrical unlike its non-zero
$\alpha$
counterparts (Ashok et al. Reference Ashok, Van Buren and Smits2015a
; Jiang et al. Reference Jiang, Gallardo, Andersson and Zhang2015, Reference Jiang, Andersson, Gallardo and Okulov2016). They find that the topology of the separation line is strongly affected by the stratification level. Building upon this study, Ortiz-Tarin et al. (Reference Ortiz-Tarin, Nidhan and Sarkar2023) studied the far wake of a prolate 6:1 spheroid at three different stratification levels
${\textit {Fr}}=\infty ,10, 2$
,
$\alpha =0^{\circ }$
and a higher
${Re}_L = 6 \times 10^{5}$
. For a prolate 6:1 spheroid at
${\textit {Fr}} = 2 \approx 6/\pi = {\textit {Fr}}_c$
, a strong effect of buoyancy on the mean wake is observed, causing the wake to transition to a Q2-D regime earlier than expected (at
$Nt \approx 15$
instead of
$Nt \approx 40-50$
for the sphere).
Apart from a recent experimental study (Ohh & Spedding Reference Ohh and Spedding2024) of a spheroid at pitch in stratified flow, there is no other work reported on a stratified wake in this configuration. Ohh & Spedding (Reference Ohh and Spedding2024) use particle image velocimetry (PIV) to investigate
${\textit {Fr}} \sim O(10)$
wakes at non-zero inclinations of
$\theta = 10^{\circ }$
and
$20^{\circ }$
. Notably, the pitched wake exhibited significant vertical displacement that was followed by an oscillatory return to its hydrostatic equilibrium position with a frequency of
$2\pi /N$
. Similar to the canonical straight-on flow, the spheroid at pitch exhibited a reduced decay rate of
$U_d$
and a reduced vertical wake extent (
$L_V$
) compared with its unstratified counterpart. The authors also reported lateral asymmetry in the streamwise shed vortices.
1.4. Open questions and objectives of this study
The previous subsections highlight a clear knowledge gap on the flow past slender bodies at non-zero angles of attack (
$\alpha$
) in situations where the environment is stratified. The focus of the present work is on a 6:1 prolate spheroid at pitch angle
$\alpha$
, where there is no comparable work apart from the PIV-based experimental study of a towed spheroid (Ohh & Spedding Reference Ohh and Spedding2024). The present work, which is based on high-resolution (to be demonstrated shortly) LES, complements the previous experimental study. Apart from
${\textit {Fr}} = 6$
that is close to their lowest
${\textit {Fr}} = 8$
(converting from their
$R$
-based definition to our
$D$
-based
$\textit {Fr}$
), our work considers more strongly stratified
${\textit {Fr}} = O(1)$
wakes too. The near-body flow is also examined and the detailed database is interrogated to understand and quantitatively characterise buoyancy effects on the vortex wake that ensues when a slender body is in pitch.
Several questions are addressed as part of this work. (a) Is the notable vertical oscillation of the wake that was reported by Ohh & Spedding (Reference Ohh and Spedding2024) found in the present work? (b) How do the streamwise vortices, characteristic of inclined spheroids, and flow separation topology respond to buoyancy? (c) How do the mean fields for a spheroid at pitch evolve as a function of
$\textit {Fr}$
, including the hitherto unreported
${\textit {Fr}} = O(1)$
regime? (d) Does the lateral asymmetry in the streamwise vortex pair, previously observed at either small
$\alpha$
with high
${Re}_L$
(Ashok et al. Reference Ashok, Van Buren and Smits2015a
) or large
$\alpha$
with moderate
${Re}_L$
(Jiang et al. Reference Jiang, Gallardo, Andersson and Zhang2015) under unstratified conditions, also occur at moderate
$\alpha$
with moderate
${Re}_L$
of the present unstratified case? (f) How do the characteristics of said asymmetry change with varying
$\textit {Fr}$
– from unstratified to weak-stratification conditions of
${\textit {Fr}} = 6$
to strongly stratified conditions of
${\textit {Fr}} = 1.9$
and
$1$
?
To this end, the paper is organised as follows. Section 2 details the numerical methods used and includes a study of grid quality. Section 3 presents an overall effect of buoyancy through flow visualisations at various
$\textit {Fr}$
. Section 4 describes the evolution of the pressure coefficient (
$C_p$
) and friction coefficient (
$C_f$
) on the body, as well as flow separation. Sections 5 and 6 discuss the evolution of the mean wake (velocity and length scales) and the mean streamwise vorticity field, respectively. The study is concluded in § 7.
2. Methodology
2.1. Governing equations and numerical scheme
Figure 1. Schematic of the flow configuration that is simulated in a cylindrical domain (not to scale).
$L_x^{-}$
,
$L_x^{+}$
and
$L_r$
refer to the upstream, downstream and radial domain distance, respectively.
$Re$
and
$\textit {Fr}$
denote the minor-axis based Reynolds number and Froude number, respectively, and
$\alpha$
is the pitch angle.
${Re}_L$
is the major-axis based Reynolds number. The stratified cases correspond to a background with uniform thermal stratification. The centre of the spheroid is at the origin of the coordinate system.
Figure 1 is a schematic of the simulation set-up wherein a 6:1 prolate spheroid, with major and minor axis given by
$L=6D$
and
$D$
, respectively, is placed at a pitch angle
$\alpha = 10^{\circ }$
in a cylindrical computational domain. The background is thermally stratified with density variation inversely proportional to temperature variation. The non-dimensional filtered Navier–Stokes equations under the Boussinesq approximation, in conjunction with the continuity and density transport equation, are solved in a cylindrical coordinate system to simulate the flow past a prolate spheroid:
-
(i) continuity,
(2.1)
\begin{equation} \frac {\partial u_{i}}{\partial x_{i}} = 0; \end{equation}
-
(ii) momentum,
(2.2)
\begin{equation} \frac {\partial u_{i}}{\partial t} + \frac {\partial (u_{i}u_{j})}{\partial x_{j}} = -\frac {\partial p}{\partial x_{i}} + \frac {1}{{Re}}\frac {\partial }{\partial x_{j}}\left [\left (1 + \frac {\nu _{s}}{\nu }\right )\frac {\partial u_{i}}{\partial x_{j}}\right ] - \frac {1}{{\textit {Fr}}^{2}}{\rho _d} \delta _{i3}; \end{equation}
-
(iii) density,
(2.3)
\begin{equation} \frac {\partial \rho }{\partial t} + \frac {\partial (\rho u_{j})}{\partial x_{j}} = \frac {1}{{Re} {Pr}}\frac {\partial }{\partial x_{j}}\left [\left (1 + \frac {\kappa _{s}}{\kappa }\right )\frac {\partial \rho }{\partial x_{j}}\right ]. \end{equation}
Equations (2.1)–(2.3) are solved to obtain non-dimensional velocity
$u_{i}$
, density
$\rho$
and pressure
$p$
as the flow evolves. Following the Boussinesq approximation,
$\rho$
is decomposed as follows:
\begin{equation} \rho (x_i, t) = \rho _0 + \rho _{b}(x_i) + \rho _d (x_i,t), \end{equation}
where
$\rho _0$
corresponds to the base density,
$\rho _{b}(x_i)$
is the background density variation and
$\rho _d (x_i,t)$
is the density fluctuation. For linear stratification,
$\partial \rho _{b}(x_i)/\partial z = C$
, where
$C$
is a constant. Under the Boussinesq approximation, it is assumed that
$\rho _{b}/\rho _0 \lt \lt 1$
and
$\rho _d/\rho _0 \lt \lt 1$
, which lead to the simplified continuity and momentum equations (2.1) and (2.2), respectively.
The non-dimensionalisation employs the following parameters: (i) free stream velocity
$U_{\infty }$
for
$u_{i}$
; (ii) minor axis length
$D$
for
$x_{i}$
; (iii) dynamic pressure
$\rho _{0}U_{\infty }^{2}$
for
$p$
; (iv)
$D/U_{\infty }$
for time
$t$
; (v)
$-DC$
for density; (vi) kinematic viscosity
$\nu$
for the subgrid kinematic viscosity
$\nu _s$
; and (vii) molecular diffusivity
$\kappa$
for the subgrid molecular diffusivity
$\kappa _s$
. This non-dimensionalisation procedure results in the following non-dimensional parameters: (i) Reynolds number
${Re} = U_{\infty }D/\nu$
; (ii) Froude number
${\textit {Fr}} = U_{\infty }/(ND)$
, where
$N$
represents the buoyancy frequency defined as
$N = \sqrt {-gC/\rho _0}$
; and (iii) Prandtl number
${Pr} = \nu /\kappa$
. In this study,
${Re}_L$
is fixed and
$\textit {Fr}$
-based stratification (see table 1) is moderate (
${\textit {Fr}} = 6$
), critical (
${\textit {Fr}} = AR/\pi = 6/\pi \approx 1.9$
) insofar as lee wave/wake interaction, and quite strong (
${\textit {Fr}} =1$
). The Reynolds number
${Re}_L = U_\infty L / \nu$
, which is based on the major axis and widely used in the literature on slender bodies, takes the value of 30 000. The pitch angle
$\alpha$
is held constant at
$10^{\circ }$
. The Prandtl number is fixed at
${Pr} =1$
for simplicity. For comparison,
${Pr}$
is approximately 0.7 for temperature in air, 5–7 for temperature in water and 700 for salt in water.
Table 1. Simulation parameters.
$N_r, N_\theta , N_x$
correspond to the number of grid points in radial, azimuthal and streamwise directions, respectively. Here,
${Re}_L$
and
$\textit {Fr}$
are the major-axis Reynolds number and minor-axis-based Froude number, respectively.
The 6:1 prolate spheroid is represented in the computational domain using the immersed boundary method (IBM) as described by Balaras (Reference Balaras2004) and Yang & Balaras (Reference Yang and Balaras2006). The governing equations, as given in (2.1)–(2.3), are solved in cylindrical coordinates for the pressure (
$p$
), density (
$\rho$
) and the three velocity components (
$u_{r}$
,
$u_{\theta }$
,
$u_{x}$
). Here,
$r$
,
$\theta$
and
$x$
correspond to the radial, azimuthal and axial directions, respectively. These directions, along with the Cartesian coordinate system, are illustrated in figure 1. Spatial derivatives are calculated using second-order finite difference schemes, while temporal stepping is done using a fractional step method that combines the Crank–Nicolson method with a low-storage Runge–Kutta scheme (RKW3). In the predictor step, the pressure Poisson equation is solved using a direct solver (Rossi & Toivanen Reference Rossi and Toivanen1999). The kinematic subgrid viscosity (
$\nu _{s}$
) and density diffusivity (
$\kappa _{s}$
) are determined using the dynamic Smagorinsky model (Germano et al. Reference Germano, Piomelli, Moin and Cabot1991). For a more detailed description of the stratified-flow solver employed in this study, interested readers are referred to Pal et al. (Reference Pal, Sarkar, Posa and Balaras2017) and Chongsiripinyo & Sarkar (Reference Chongsiripinyo and Sarkar2020).
At the inlet, a uniform inlet velocity boundary condition is prescribed for the velocity components:
$[u_{x}, u_{r}, u_{\theta }] = [U_{\infty }, 0, 0] = \mathbf {U}$
. The outlet boundary condition for velocities corresponds to an Orlanski-type convective boundary condition (Orlanski Reference Orlanski1976). The pressure
$p$
is set to zero at the inlet boundary. On the remaining boundaries (outlet and radial), a Neumann boundary condition is used for pressure. For stratified cases, a Robin boundary condition is applied at the radial boundary, satisfying
$\partial {\rho }/\partial {z} = C$
. This implies that the density far away from the wake generator is assumed to be unperturbed. On the body surface, a no-slip condition is prescribed for velocity, and the normal gradients of pressure and density are set to zero. To prevent the reflection of internal gravity waves from the boundaries, sponge layers are employed for velocity and density at the inlet, outlet and radial boundaries. These layers are defined as
$\phi (x_i)(\mathbf {u}-\mathbf {U})$
for velocity components and
$\phi (x_i)(\rho -\rho _{b})$
for density, where
$\rho _{b}$
corresponds to the background density state. A quadratic damping function of the following form,
$\phi (x_i) = C^* ( (x_i - x_{s,i})/(L_{x_{i}} - x_{s,i}))^{2}$
, is used for the sponge. Here,
$x_{s,i}$
is the start of the sponge layer and
$L_{x_{i}}$
is the extent of the numerical domain, respectively, in the
$i\mathrm{th}$
coordinate axis direction. Additionally,
$C^*=10$
for
${\textit {Fr}} = 1.9$
and
$1$
wakes and
$C^*=5$
for the
${\textit {Fr}}=6$
wake. For the unstratified wake, sponge layers are not necessary.
Flow statistics are obtained by time averaging with an averaging window of approximately two flow-through times
$\approx 100D/U_\infty$
. This time window is sufficient for convergence of mean and second-order statistics of interest here. The time average (equivalent to ensemble or Reynolds average for this flow) is denoted by angle brackets, e.g. the average of
$u_x$
is
$\left \lt u_x \right \gt$
. The simulations have approximately 458 million grid points for the stratified cases and 470 million grid points for the unstratified case (details given in § 2.2). These simulations were conducted on 128 or 256 CPUs, consuming approximately
$500\,000$
compute hours for
$150D/U_\infty$
simulation time units.
2.2. Domain size and grid quality
Table 1 presents the various grid parameters corresponding to the four simulations at
${\textit {Fr}} = \infty , 6, 1.9$
and
$1$
. The radial and upstream extent of the simulation domain,
$L_r$
and
$L_x^{-}$
, respectively, is increased for the stratified simulations to ensure that the steady lee waves weaken before they hit the radial and upstream boundary. Additionally, the azimuthal grid points in the unstratified
${\textit {Fr}} = \infty$
simulation is increased to
$N_\theta = 256$
from
$N_\theta = 128$
in the stratified cases. The increased vertical displacement of the
${\textit {Fr}} = \infty$
wake (figure 13
b) to higher-
$r$
regions requires a smaller azimuthal grid spacing (
$2 \pi r/N_\theta$
) and thus larger
$N_\theta$
to ensure adequate azimuthal resolution away from the centreline.
Figure 2. Variation of
$C_p$
in the
$y/D=0$
plane on the leeside and windside of the spheroid. Potential solution for
$C_p$
in dashed line, obtained from Piquet & Queutey (Reference Piquet and Queutey1992), is also shown.
For stratified wakes, the minimum
$\Delta r/D \approx 7 \times 10^{-4}$
occurs at
$r/D \approx 0$
and remains approximately constant until
$r/D \approx 0.8$
. Beyond this point, the radial grid is stretched until the edge of the radial domain. In the streamwise direction, the minimum
$\Delta x/D = 4 \times 10^{-4}$
occurs at the nose of the spheroid. The streamwise grid is gradually stretched, due to the thickening of the boundary layer, reaching
$\Delta x/D \approx 7 \times 10^{-3}$
at the tail. The stratified simulations were conducted prior to the unstratified simulations and it was found that the minimum
$\Delta r/D$
could be relaxed. Therefore, in the unstratified case,
$\Delta r/D \approx 2 \times 10^{-3}$
. Aside from this adjustment, the grid distribution over the body remains the same as in the stratified case. At
$x/D = 0$
, at the centre of the spheroid, there are approximately
$40$
–
$50$
grid points across the
$\delta _{99}$
thickness of the BL on the leeside of the spheroid for stratified cases. For the unstratified case, this number is
$\sim 22$
. At the tail of the spheroid, there are 90 points and 47 points across the
$\delta _{99}$
thickness of BL for
${\textit {Fr}} = 6$
and
$\infty$
, respectively.
Figure 2 shows a comparison between the
${\textit {Fr}} =\infty$
simulation and the potential flow solution for the coefficient of pressure,
$C_p = (P-P_\infty )/0.5\rho U_\infty ^2$
. The match between the potential solution and the current simulation is excellent until
$x/D \approx 1$
, indicating sufficient grid resolution to capture the boundary-layer evolution. Beyond
$x/D \approx 1$
, pressure recovery does not happen in our simulation due to viscous effects and subsequent boundary-layer separation. Here,
$C_p$
for the
${\textit {Fr}} = 6$
case is very similar to the unstratified case and is not presented for brevity.
To assess grid quality in the wake, the spatial variation of the ratio of grid spacing to the Kolmogorov length, denoted as
$\eta = (\nu ^3/\varepsilon )^{1/4}$
, is computed in all three directions and shown in figure 3. We present results for
${\textit {Fr}} = \infty$
and
$6$
, specifically, for two reasons: (
$a$
) distinct grid resolutions for the unstratified and stratified cases (refer to table 1); and (
$b$
) the relatively strongly stratified simulations at
${\textit {Fr}} = 1$
and
$1.9$
are less turbulent than
${\textit {Fr}} = 6$
at
${Re}_L = 30\, 000$
, and hence have better resolution of the wake turbulence. Figure 3 shows that the ratios of streamwise (
$\Delta x$
), radial (
$\Delta r$
) and azimuthal (
$r \Delta \theta$
) grid spacing to
$\eta$
stay below
$6$
,
$1$
and
$5$
, respectively, for both
${\textit {Fr}} = \infty$
and
$6$
, establishing that the current LES study has been conducted at high resolution. The Ozmidov scale,
$L_O = \sqrt {\varepsilon /N^3}$
, is also well resolved with
$\Delta r/L_o \lt 0.1$
in the turbulent wake.
Figure 3. Ratio of grid size to Kolmogorov length
$\eta$
in all three directions for (a,c,e)
${\textit {Fr}} = \infty$
and (b,d,f)
${\textit {Fr}} = 6$
. For panel (
$a,b$
), the smallest
$\eta$
at the corresponding
$x/D$
is used for calculation. For panel (
$c{-}f$
), dissipation along the line
$\theta = 3\pi /2$
is used at respective
$x/D$
locations, since wake descends as it evolves.
3. Visualisations
Figure 4. Instantaneous axial velocity (
$u_x$
) isovolumes: (
$a$
)
${\textit {Fr}} = \infty$
; (
$b$
)
${\textit {Fr}} = 6$
; (
$c$
)
${\textit {Fr}} = 1.9$
; and (
$d$
)
${\textit {Fr}} = 1$
, with maximum value of
$u_x/U_{\infty } = 0.75$
and viewed from the front-left of the body.
Isovolumes for instantaneous axial velocity (
$u_x$
) for all
$\textit {Fr}$
are shown in figure 4. These visualisations provide insights into buoyancy effects on the wake structure before moving on to quantification through mean velocity and vorticity fields in later sections. It is evident that, as
$\textit {Fr}$
decreases, the topology of flow separation is altered and there is a systematic decrease in wake turbulence. It is also worth noting that, using finer azimuthal grid size, the streak-like ripples in figure 4 have been confirmed to be mere artefacts of visualisation in Paraview and should not be misconstrued as a physical phenomenon.
For the
$Fr = \infty$
wake, two lobes of velocity defect are seen – one for each side of the body (figure 4
a). These lobes break up quickly (at
$x/D \approx 5$
) into multiscale structures, implying transition from the initially laminar separation. Similar multiscale structures are observed in the
$Fr = 6$
wake (figure 4
$b$
), but the breakup is observed further downstream (
$x/D \approx 10$
) relative to
$Fr = \infty$
. The wakes at higher stratification (
${\textit {Fr}} = 1.9$
and
${\textit {Fr}} = 1$
) are significantly different. Although, two velocity defect lobes can also be seen in the
$Fr = 1.9$
wake (figure 4
$c$
), they do not transition to smaller-scale structures until much further downstream (
$x/D \approx 20$
). At
$Fr = 1$
, transition to turbulence is further inhibited by buoyancy and the wake stays laminar throughout with a layered structure. Furthermore, two vertically separated and distinct regions of flow separation are observed at
${\textit {Fr}}=1$
. These distinct regions have an imprint on wake evolution as will be quantified later.
Figure 5. Instantaneous axial velocity (
$u_x$
) isovolumes along with streamlines: (
$a$
)
${\textit {Fr}} = \infty$
; (
$b$
)
${\textit {Fr}} = 6$
; (
$c$
)
${\textit {Fr}} = 1.9$
; and (
$d$
)
${\textit {Fr}} = 1$
. The isovolumes correspond to
$u_x/U_{\infty } \leqslant 0.75$
and are viewed from the rear-left of the body.
Figure 5 presents another view of
$u_x$
-isovolumes, this time with streamlines added. With this view, the two lobes of the
$Fr = \infty$
wake are seen to break down into spiral structures (figure 5
$a$
). Additionally, it is apparent that the unstratified wake descends vertically. The streamlines indicate cross-flow from the windward side towards the leeward side at all stratification levels. A similar spiral structure is observed in the
${\textit {Fr}} = 6$
wake (figure 5
$b$
). However, less vertical drift is observed as will be quantified later. Once again, the
${\textit {Fr}} = 1.9$
and
${\textit {Fr}} = 1$
wakes exhibit notable differences from the wakes at higher
$\textit {Fr}$
. While two structures emerge from either side of the body at
${\textit {Fr}} = 1.9$
(figure 5
$c$
), akin to previous cases, large-scale undulations are observed in the isovolume and the streamlines due to the lee-waves-induced wake oscillation that has been previously reported for cases without angle of attack. Furthermore, the cross-flow from windward to leeward sides that was observed in the streamlines of the previous cases is now mildly suppressed at
${\textit {Fr}} = 1.9$
and even more so at
${\textit {Fr}} = 1$
(figure 5
$d$
). Moreover, the lee-waves-induced oscillations of the wake are stronger at the lower values of
$\textit {Fr}$
.
Figure 6. Streamlines of the secondary flow on a domain cross-section at
$X/D = 1$
with normal as the major axis of the body. The bottom half of each panel corresponds to the windward side of the spheroid at
$\alpha = 10^\circ$
. Contours of the vertical velocity (
$u_z$
) are shown: (
$a$
)
$Fr = \infty$
; (
$b$
)
$Fr = 1.9$
; and (
$c$
)
$Fr = 1$
. Here,
$X$
is the coordinate along the major axis of the body, where
$X = 0$
corresponds to the centre of the body. The contours are in the
$Y{-}Z$
plane normal to the major axis of the body. Top-right inset shows the
$(X,Y,Z)$
coordinate system.
The influence of stabilising buoyancy on the cross-flow and the resulting wake structure is made clearer by visualising streamlines of the secondary flow on a cross-section perpendicular to the inclined major axis of the body in figure 6. Contours of vertical velocity (
$u_z$
) are also depicted. Note that the cross-sections shown in figure 6 are in the plane normal to the major-axis of the body. In the case of
${\textit {Fr}} = \infty$
, the cross-flow is conspicuous, with the flow curving as it moves from the windward side to the leeward side (figure 6
$a$
). Atop the body, there is an attached circulation zone with two oppositely rotating streamwise vortices on either side. The imprint of stratification begins to manifest at
${\textit {Fr}} = 1.9$
as follows: (a) streamlines that show flattening of the attached recirculation zone (figure 6
$b$
); and (b)
$u_z$
contours that show a higher upward velocity at the sides, whereas the downward velocity diminishes at the top. These features can be attributed to buoyancy whereby lighter fluid that is brought downward on the windward side by the inclined spheroid tends to rise to its neutral position and then cannot curve downward across isopycnals. These effects become even more pronounced at
${\textit {Fr}} = 1$
, where the secondary-flow pattern (figure 6
$c$
) changes: the secondary streamlines rise almost vertically from the windward side before traversing almost horizontally to the leeward side along isopycnals. Consequently, some of the
${\textit {Fr}} = 1$
flow separates prematurely from the top, as will be shown quantitatively through the skin friction. This region of separated flow – a secondary wake – remains distinct from the primary wake formed after the flow separates further downstream, as observed in figures 4(
$d$
) and 5(
$d$
).
Figure 7. Isocontours for
$\lambda _2$
criterion (top-view) are shown: (
$a$
)
$Fr = \infty$
; (
$b$
)
$Fr = 6$
; (
$c$
)
$Fr = 1.9$
; and (
$d$
)
$Fr = 1$
. Contour levels for each case are mentioned on top of each panel.
To visualise vortical flow structures, isocontours of the
$\lambda _2$
criterion are shown in figure 7. For all four cases, the isocontour threshold at
$x/D\gtrapprox 15$
is smaller than at
$x/D\lessapprox 15$
to ensure similar visibility between the near and intermediate wake. For
${\textit {Fr}}=\infty$
and
$6$
(figure 7
a,b), there is a distinct signature of a helical vortical structure developing from each lateral side of the body. These structures stay asymmetrical in the spanwise direction, a phenomenon consistent with the asymmetry previously observed in the velocity field (figures 4 and 5) and will be discussed further in the following sections. Coherent vortical structures become difficult to distinguish from small-scale turbulence when
$x/D\gtrapprox 15$
. At the higher stratification of
${\textit {Fr}}=1.9$
and 1 (figure 7
c,d), layering of the instantaneous
$\lambda _2$
isocontours is observed in the near wake with no distinguishable helical structures. These layers subsequently break down into small-scale structures as the wake evolves.
4. Body forces
At
$\alpha = 0$
, the buoyancy force affects the acceleration/deceleration of the flow through its tangential component and, through its normal component, influences the confinement of the boundary layer. The steady lee wave also imposes its pressure gradient and velocity on the boundary layer. In addition, at
${\textit {Fr}} \leqslant 1$
, the fluid tends to move sideways instead of over and below the body, owing to the potential energy barrier imposed by the stable stratification. These effects of buoyancy on the force coefficients, the boundary-layer structure and its separation at
$\alpha = 0^{\circ }$
, which were described in detail by Ortiz-Tarin et al. (Reference Ortiz-Tarin, Chongsiripinyo and Sarkar2019) for a straight-on 4:1 spheroid, are also operative at a non-zero pitch angle. Importantly, for a pitched spheroid, there is an additional consequence of buoyancy whereby it influences the cross-flow and secondary circulation, as will be shown in this section.
Body forces are diagnosed by (a) the streamwise variation of pressure and shear-stress on the lee and windside as contour plots and (b) the streamwise variation of the force coefficients (
$C_p$
and
$C_f$
) along the body surface in the central plane (
$y=0$
) as line plots. Here,
$C_p = (P-P_{\infty })/0.5\rho U^2_{\infty }$
and
$C_f=Re^{0.5}|\tau _x|/0.5\rho U^2_{\infty }$
. It is found that the previously noted buoyancy effects on the flow are strongly reflected by differences in the streamwise variation of body forces among the different stratification levels. A summary of these differences follows.
4.1. Pressure coefficient
Figure 8. Pressure contours on the (a,c,e,g) leeside and (b,d,f,h) windside of the spheroid. Dashed line in panels (
$a$
) and (
$c$
) corresponds to
$y/D=0$
. Variation of
$C_p (x)$
on the (i) leeside and (j) windside of the body surface in the
$y/D=0$
plane. The potential flow solution for
$C_p (x)$
, obtained from Piquet & Queutey (Reference Piquet and Queutey1992), is also shown as a dashed line in panels (i) and (j).
Figure 8(a,c,e,g) shows the pressure contours on the leeside and figure 8(b,d,f,h) shows those on the windside of the body. The streamwise variation of
$C_p$
in the centreplane (
$y=0$
) on both lee and windside is shown in figure 8(i,j) and is discussed first. For comparison,
$C_p$
obtained from the potential-flow solution (Piquet & Queutey Reference Piquet and Queutey1992) is also presented in figure 8(i,j). For both sides, LES solutions agree well with the potential solution till
$x/D\approx 1$
, beyond which they deviate from the potential-flow solution owing to viscous effects and subsequent flow separation. On both sides,
$C_p$
rapidly drops from its stagnation value near the nose of the body. The value of this initial drop of
$C_p$
decreases systematically from its largest value at
${\textit {Fr}} = \infty$
as
$\textit {Fr}$
decreases (figure 8
$i$
).
On the leeside, the initial drop in pressure near the nose is followed by a slow and continuous recovery that is similar among the
${\textit {Fr}} \geqslant 1.9$
cases (figure 8
$i$
). In contrast, at
${\textit {Fr}}=1$
, the pressure recovery on the lee side does not occur until
$x/D \approx 1.5$
so that the front part of the body (
$x/D \lt 0$
) has higher pressure than the other cases, which is a contributor to the increased drag in this case. The delayed pressure recovery can be observed both from the pressure contours (figure 8
$g$
) and the
$C_p$
evolution at the centreline (figure 8
$i$
). Mean pressure contours in the vicinity of the spheroid (not shown here for brevity), reveal a strong low-pressure region on the entirety of the leeside. This low-pressure region is imposed by the strong steady lee wave field at
${\textit {Fr}}=1$
. It can also be seen that the leeside pressure contours for
${\textit {Fr}}=\infty$
and
$6$
(figure 8
a,c) show lateral asymmetry about the centreline (
$y=0$
), which is consistent with the results of the previous section. The windside of the body exhibits more pronounced differences between unstratified (
${\textit {Fr}}=\infty$
) and strongly stratified (
${\textit {Fr}}=1.9,1$
) cases as compared with the leeside. Here,
$C_p$
of
${\textit {Fr}}=1.9$
and
$1$
falls significantly below those of
${\textit {Fr}}=\infty$
and
$6$
after
$x/D\approx 2$
, and recovers sharply towards the end of the body. Pressure contours in figure 8(b,d,f,h) confirm this trend of higher and lower pressure at the nose and the tail of the body, respectively, for the
${\textit {Fr}}=1$
and
$1.9$
cases compared with the
${\textit {Fr}}=6$
and
$\infty$
cases.
4.2. Friction coefficient
Figure 9. Contours of
$Re^{0.5}|\tau _x|$
on the (a,c,e,g) leeside and (b,d,f,h) windside of the spheroid for all
$\textit {Fr}$
at
$\alpha =10^\circ$
. Dashed lines in panels (
$a$
) and (
$c$
) correspond to
$y/D=0$
. Variation of
$C_f$
on the (i) leeside and (j) windside of the body at the
$y/D=0$
plane.
Figure 9(a,c,e,g) displays the scaled shear stress contours,
$Re^{0.5}|\tau _x|$
, on the leeside (left) and figure 9(b,d,f,h) those on the windside of the body across all
$\textit {Fr}$
. These contours reveal that flow separation primarily occurs on the leeside, identifiable by regions in green where
$|\tau _x| \rightarrow 0$
. For the unstratified (
${\textit {Fr}}=\infty$
) and weakly stratified (
${\textit {Fr}}=6$
) cases, the wake separates mainly from the lateral portions of the leeside (figure 9
a,c), forming a streamwise vortex pair to be discussed in § 6.
A central region (in red) near
$y=0$
remains attached until nearly the tail, with a noticeable lateral asymmetry. At
${\textit {Fr}}=1.9$
, separation occurs both from the sides and the central region around
$y=0$
(figure 9
$e$
). However, at
${\textit {Fr}}=1$
, separation is primarily central, with no clear lateral separation (figure 9
$g$
). Recall that a distinct streamwise vortex pair was present at
${\textit {Fr}}=1.9$
(figure 7
$c$
) but, at
${\textit {Fr}}=1$
, the vortex pair is less evident.
The variation of
$C_f$
on the body surface at the
$y=0$
plane is shown in figure 9(i,j). For all cases,
$C_f$
varies quite similarly on both sides till
$x/D \approx 0$
. For
$x\geqslant 0$
,
${\textit {Fr}}=\infty$
and
$6$
show higher
$C_f$
on the leeside (figure 9
$i$
) than their strongly stratified counterparts of
${\textit {Fr}}=1.9$
and
$1$
. On the windside,
${\textit {Fr}}\leqslant 1.9$
cases show elevated levels of
$C_f$
when compared with
${\textit {Fr}}\geqslant 6$
cases (figure 9
$j$
). This region of elevated surface shear in
${\textit {Fr}}=1.9$
and
${\textit {Fr}}=1$
coincides with the region of steep pressure drop observed at
$x/D\gt 1$
(figure 8
$j$
). The elevated surface shear is related to an accelerating external flow that thins the boundary layer.
4.3. Overall forces
Figure 10. Force coefficients decomposed between pressure and shear contribution. Coefficient of (a) drag (
$C_d$
) and (b) lateral force coefficient (
$C_y$
).
The aforementioned changes in
$C_p$
and
$C_f$
lead to a buoyancy effect on overall drag. Also, in cases with lateral asymmetry, there is an additional sideways force. Since the flow separation and the boundary layer developing over the body remain laminar at
${Re}_L = 30\, 000$
, the fluctuation root mean square (r.m.s.) of force coefficients is negligible. Hence, only mean force coefficients are discussed in this section.
Figure 10 shows the drag
$C_d = F_x/(1/2\rho U_\infty ^{2}A)$
and spanwise force
$C_y = F_y/(1/2\rho U_\infty ^{2}A)$
coefficients as well as their partition into pressure and frictional coefficients. Here,
$A = \pi D^2/4$
. Additionally,
$C_d$
exhibits a systematic and moderate increase as stratification increases until
${\textit {Fr}} = 1.9$
. At
${\textit {Fr}} = 1$
, there is a relatively larger increase in
$C_d$
. The increased drag at
${\textit {Fr}} = 1$
is primarily associated with a rise in the pressure contribution that is a consequence of the extended zone of high pressure on the windside that is seen at
${\textit {Fr}} =1$
(figure 8
$i$
) and not in the other cases.
Figure 10(
$b$
) shows the dependence of the lateral force coefficient (
$C_y$
) on stratification. The weakly stratified (
${\textit {Fr}} = 6$
) and unstratified (
${\textit {Fr}} = \infty$
) flow exhibit a sideways force consistent with the previously noted lateral asymmetry of
$C_p$
and
$C_f$
distributions. It is worth noting that the
$C_y$
of
${\textit {Fr}} = \infty$
and
${\textit {Fr}} = 6$
are similar in magnitude but flipped in sign. We hypothesise that the
${\textit {Fr}} = \infty$
and
${\textit {Fr}} = 6$
flows are bi-stable with two different reflectional-symmetry-breaking states, each state being equally probable. The lower
${\textit {Fr}} = 1.9$
and 1 cases do not show asymmetry (associated with one streamwise vortex being larger than its partner) since buoyancy changes the cross-flow secondary circulation strongly (figure 6) to inhibit the rollup into streamwise vortices. Corresponding to the lateral symmetry,
$C_y \approx 0$
for these cases.
5. Mean velocity and geometry of the wake
The variation of mean defect velocity,
$ U_\infty - \left \lt u_x (x,y,z)\right \gt$
, hereafter denoted by
$U_d$
, as a function of
$\textit {Fr}$
is of interest. Buoyancy effects that alter the flow at the body and its separation were discussed in §§ 3 and 4, and are manifested as persistent differences in the near wake. These differences in the near wake and those in the far wake are elaborated below. The streamwise evolution will be presented as a function of
$x/D$
, where
$x$
is measured from the centre of the body. Buoyancy time,
$Nt = N x/U_\infty = (x/D) /{\textit {Fr}}$
, which is often used to gauge the strength of buoyancy effects, will also be used to discuss results as appropriate.
5.1. Evolution of mean velocity on streamwise-constant planes
Figure 11. Mean defect velocity (
$U_d$
) contours and isopycnals at
$x/D = 3, 10, 20$
and
$30$
(row-wise) for
${\textit {Fr}} = \infty , 6, 1.9$
and
$1$
(column-wise). The extent of the radial domain is shown adjacent to each contour plot. For the stratified cases, body-generated steady lee waves in the domain are not visible due to the choice of contour ranges to focus specifically on the wake defect.
Figure 11 presents mean defect velocity
$U_d$
contours at various streamwise locations (
$x/D = 3, 10, 20\ \text {and}\ 30$
). In stratified scenarios, mean isopycnals are superimposed on the
$U_d$
contours. In the
${\textit {Fr}} = \infty$
unstratified wake, two distinct lobes are evident as the flow leaves the body at
$x/D \approx 3$
(figure 11
$a$
). These lobes are also observable in the instantaneous visualisations of
${\textit {Fr}} = \infty$
(figure 5
$a$
). The asymmetry of flow at the body and its separation persist downstream leading to the asymmetry in
$U_d (y,z)$
profiles between the left and right lobes. Asymmetric flow separation also occurs for
${\textit {Fr}} = 6$
, but with the direction of asymmetry reversed. No preferred direction for lateral asymmetry has been imposed by us; however, small asymmetries in the evolving flow have resulted in the observed polarity of the asymmetry. The asymmetry observed in the
${\textit {Fr}} = \infty$
wake persists until at least
$x/D = 10$
(figure 11
$e$
). However, by
$x/D = 20$
, the
$U_d$
contours for the
${\textit {Fr}} = \infty$
case no longer exhibit the distinct lobes observed in the near wake. Another noteworthy observation is the continuous drift of the
$U_d$
contour in both the vertical (
$z$
) and spanwise (
$y$
) directions due to the body’s angle of inclination and the wake’s asymmetry, respectively. These vertical and horizontal drifts will be further quantified in subsequent sections of the manuscript.
Despite its weak stratification, the evolution of the
${\textit {Fr}} = 6$
wake is affected early on. At
$x/D = 10$
or
$Nt = 1.67$
(figure 11
g), its vertical downward displacement is significantly less than in the unstratified case. The lower boundary of the wake is seen to lie along one of the isopycnals displaced by the flow. The restoring force of buoyancy comes into play so that, with the downstream weakening of the wake deficit and the associated recovery of the boundary isopycnal to its neutral position, the mean wake tends to return towards the centreline (
$z = 0$
).
In the strongly stratified cases of
${\textit {Fr}} = 1.9$
and 1, the mean flow leaving the body is no longer laterally asymmetric, as seen in figure 11(c,d). It was shown previously that the sideways force coefficient (
$C_y$
) approaches zero at
${\textit {Fr}} = 1.9$
and
${\textit {Fr}} = 1$
(figure 10
$b$
). Similar to the
${\textit {Fr}} = \infty$
and
${\textit {Fr}} = 6$
cases, most of the wake at
$x/D = 3$
is below the
$z = 0$
plane for both
${\textit {Fr}} = 1.9$
and 1. Similar to the
${\textit {Fr}} = 6$
wake, the
${\textit {Fr}} = 1.9$
wake and its corresponding isopycnals are constrained by the restoring force of buoyancy, which is stronger and is manifested as lee-waves-induced oscillations as the wake progresses downstream. By
$x/D = 30$
(figure 11
$o$
), an elliptical
$U_d$
topology appears for the
${\textit {Fr}} = 1.9$
wake.
The downstream evolution of the
${\textit {Fr}} = 1$
wake differs significantly from the other three cases. Immediately after the body, the wake assumes a double-lobed structure with isopycnals extending both upward and downward. These two lobes evolve independently of each other. By
$x/D = 30$
, the
${\textit {Fr}} = 1$
wake has a significantly greater vertical extent than the
${\textit {Fr}} = 1.9$
wake due to the initial emergence and persistence of the double-lobed structure, which eventually transitions into a double-layered wake, as shown in figure 11(
$p$
). The upper lobe – a secondary wake – originates from the region of separated flow from the top, recall the discussion of figure 9(g).
5.2. Evolution of peak defect velocity and mean kinetic energy
Figure 12. (a) Decay of the mean defect velocity peak,
$U_d^{\mathrm {\textit {peak}}}$
and (b) area-integrated mean kinetic energy,
$\{E^{M}_K\}$
, decay for different
$\textit {Fr}$
.
Figure 12(
$a$
) shows the decay of peak mean defect
$U_d^{\mathrm {\textit {peak}}}$
as a function of
$x/D$
for all four cases. The magnitude of
$U_d^{\mathrm {\textit {peak}}}$
for strongly stratified wakes (
${\textit {Fr}} = 1.9$
and
$1$
) is significantly higher than in the
${\textit {Fr}} = \infty$
and
$6$
wakes, verifying that the longstanding hypothesis that stratified wakes ‘live’ longer (Spedding Reference Spedding1997). The
$U_d^{\mathrm {\textit {peak}}}$
values for
${\textit {Fr}} = 1.9$
and
${\textit {Fr}} = 1$
are
$3{-}4$
times higher than for
${\textit {Fr}} = \infty$
and
$6$
by
$x/D=40$
. We refrain from attempting to fit decay rates for the
$U_d^{\mathrm {\textit {peak}}}$
evolution since the streamwise domain available for curve fitting, approximately
$30D$
from
$x/D = 10$
onward, is relatively small for a high-confidence fit. However, for the
${\textit {Fr}} = 6$
wake, a careful reader can observe that the decay rate of
$U_d^{\mathrm {\textit {peak}}}$
slows down compared with the
${\textit {Fr}} = \infty$
wake around
$x/D \approx 20$
(
$Nt \approx 3.33$
). Ohh & Spedding (Reference Ohh and Spedding2024) also reported a similar slowdown in the decay of
$U_d$
for
${\textit {Fr}} \geqslant 8$
wakes compared with the
${\textit {Fr}} = \infty$
baseline around
$Nt \approx 2$
(see their figure 17
$b$
).
In figure 12(
$b$
), the decay of area-integrated mean kinetic energy,
$\{E_K^{M}\}$
=
$\int (U_d^{2} + \langle u_y \rangle ^{2} + \langle u_z \rangle ^{2})/2 \ \mathrm {dA}$
, is compared among the four cases. Circles of radii
$r/D=2$
and
$r/D = 3.5$
are used as cross-sections for the stratified cases and the unstratified case, respectively, ensuring that the wakes remain inside the domain of integration for all
$x/D$
. The area-integrated energy in both
${\textit {Fr}}=1.9$
and
$1$
wakes is of comparable magnitude throughout the domain. Lee-waves-induced oscillations with wavelength approximately equal to
$2\pi {\textit {Fr}}$
are clearly visible in both
${\textit {Fr}} = 1.9$
and
${\textit {Fr}} = 1$
wakes. These lee-wave-induced oscillations were identified by Bonnier & Eiff (Reference Bonnier and Eiff2002) as an ‘accelerated’ collapse in the
$U_d$
evolution profile. DNS simulations by Pal et al. (Reference Pal, Sarkar, Posa and Balaras2017) showed that this ‘accelerated’ collapse region in the
$U_d$
profiles is actually an imprint of the surrounding steady lee waves, having (a) a time scale of
$Nt = 2\pi$
and (b) a wavelength of
$\lambda /D = 2\pi {\textit {Fr}}$
. This effect is particularly pronounced at
${\textit {Fr}} \sim O(1)$
, where the lee waves are the strongest. Ortiz-Tarin et al. (Reference Ortiz-Tarin, Chongsiripinyo and Sarkar2019, Reference Ortiz-Tarin, Nidhan and Sarkar2023) observed similar lee-wave-induced modulations in the wake of 4:1 and 6:1 spheroids at zero-degree pitch angles. More recently, Ohh & Spedding (Reference Ohh and Spedding2024) reported the presence of these lee-wave-induced oscillations in the wake trajectory of non-zero pitch angle spheroids at
${\textit {Fr}} \geqslant 8$
(see figures 13 and 14 in their manuscript), a phenomenon also present in figure 13.
The decay of the unstratified wake and the
${\textit {Fr}} = 6$
wake
$\{E_K^{M}\}$
remain similar until
$x/D \approx 4$
, after which the decay rate of the
${\textit {Fr}} = 6$
wake increases compared with the unstratified wake until
$x/D \approx 25$
. At
$x/D \approx 25$
, the decay rate of
$\{E_K^{M}\}$
reduces, implying that ultimately, the
${\textit {Fr}} = 6$
wake will be more ‘long-lived’ than the
${\textit {Fr}} = \infty$
wake, in agreement with previous experimental and numerical studies on stratified wakes.
Figure 13. Evolution of wake centre in the (
$a$
) horizontal (
$\langle y^c \rangle$
) and (
$b$
) vertical (
$\langle z^c \rangle$
) directions. Evolution of wake (
$c$
) horizontal (
$L_y$
) and (
$d$
) vertical (
$L_z$
) length scales. In panel (
$a$
),
$-\langle y^c \rangle$
is plotted for
${\textit {Fr}}=6$
to compare the magnitudes of lateral displacement of wake of
${\textit {Fr}} = \infty$
and
$6$
wakes.
5.3. Evolution of wake geometry
Figure 11 makes it clear that the wake of a prolate spheroid at a pitch angle exhibits a pronounced buoyancy effect both in terms of its centre and its length scales. Figure 13 presents the evolution of the wake centre and length scales in the spanwise and vertical direction as a function of
$x/D$
.
These quantities are calculated as follows:
\begin{equation} \langle y^{c} \rangle = \frac {\int yU_d^{2}\mathrm {dA}}{\int U_d^{2}\mathrm {dA}}, \qquad \langle z^{c} \rangle = \frac {\int zU_d^{2}\mathrm {dA}}{\int U_d^{2}\mathrm {dA}}, \end{equation}
\begin{equation} L_y^{2} = \frac {\int (y-\langle y^{c} \rangle )^{2}U_d^{2}\mathrm {dA}}{\int U_d^{2}\mathrm {dA}}, \qquad L_z^{2} = \frac {\int (z-\langle z^{c} \rangle )^{2}U_d^{2}\mathrm {dA}}{\int U_d^{2}\mathrm {dA}}, \end{equation}
where
$\langle y^{c} \rangle$
,
$\langle z^{c} \rangle$
are the temporally averaged wake-centre coordinates and
$L_y$
,
$L_z$
are the mean wake length scales in the cross-sectional planes normal to the streamwise coordinate
$x$
.
Figure 13(a,b) presents the streamwise evolution of the wake centre in both the spanwise (
$y$
) and vertical (
$z$
) directions as a function of
$x/D$
. The four wakes under consideration exhibit distinct evolutions. The unstratified wake moves away from the centreline (
$r/D = 0$
axis) as the flow evolves, with more pronounced movement in the vertical (
$z$
) direction compared with the spanwise (
$y$
) direction. Its drift in the
$z$
direction is associated with the induced downward velocity at the centre of the counter-rotating streamwise vortex pair in the separated flow, as will be discussed in § 6. The drift in the
$y$
direction is due to the initial lateral asymmetry of the wake vortices with one vortex being larger than the other.
Despite its relatively weak stratification, the descent of the
${\textit {Fr}} = 6$
wake is strongly modified by buoyancy as seen in the trajectory of
$\langle z^{c} \rangle$
. The descent rate is already smaller than its unstratified counterpart at
$x/D = 4$
, a mere body diameter behind the trailing edge of the spheroid. At
$x/D \approx 10$
, the wake descent changes to an ascent until the wake reaches its neutral configuration. The locations of maximum descent and ascent of the wake are separated by
$\pi {\textit {Fr}} \approx 18.8$
, demonstrating that even at a relatively weaker stratification of
${\textit {Fr}} = 6$
, the wake is strongly influenced by the surrounding lee waves. The lee-wave-induced modulation of the wake trajectory in the vertical direction aligns with the findings of Ohh & Spedding (Reference Ohh and Spedding2024), who reported a similar modulation with a wavelength of
$\lambda /D = 2\pi {\textit {Fr}}$
for their
${Re}_L = 30\, 000$
,
${\textit {Fr}}=8$
and
$\alpha =20^{\circ }$
wake (see their figure 8
$d$
). To compare the lateral drift magnitude with that of the unstratified wake, we plot
$-\langle y^c \rangle$
in figure 13(
$a$
) for the
${\textit {Fr}} = 6$
wake. While the lateral drift is comparable in magnitude near the body, the two cases diverge significantly beyond
$x/D \approx 5$
, highlighting notable qualitative and quantitative differences in the wake trajectory evolution between the
${\textit {Fr}}=\infty$
and
$6$
wakes.
The values of
$\langle y^{c} \rangle$
of the
${\textit {Fr}} = 1.9$
and
${\textit {Fr}} = 1$
wakes remain zero throughout their evolution (figure 13
$a$
). In the vertical direction, the
${\textit {Fr}} = 1.9$
wake shows a lee-waves-induced oscillation in the
$\langle z^{c} \rangle$
trend (figure 13
$b$
), also evident in contours of
$U_d$
of the
${\textit {Fr}} = 1.9$
wake (figure 11
c,g,k,o). Interestingly,
$\langle z^{c} \rangle$
of the
${\textit {Fr}} = 1$
wake also stays close to zero, which implies that the
$U_d^{2}$
weighted mean of the vertical coordinate of the two distinct lobes of the double wake in figure 11(d,h,l,p) approximately balance each other out. Both
${\textit {Fr}} = 1$
and
$1.9$
wakes show a clear lee-waves-induced oscillation in the
$\langle z^{c} \rangle$
evolution with wavelength equal to
$2\pi {\textit {Fr}}$
.
5.4. Evolution of wake length scales
Figures 13(c) and 13(d) show the evolution of horizontal (
$L_y$
) and vertical (
$L_z$
) length scales, respectively. Here,
$L_y$
and
$L_z$
of the
${\textit {Fr}} = \infty$
and
$6$
wakes evolve similarly in rate and magnitude until
$x/D \approx 15$
(
$Nt \approx 2.5$
). Beyond
$x/D \approx 25$
(
$Nt \approx 4$
), the growth of
$L_z$
in the
${\textit {Fr}} = 6$
wake slows down owing to the increasing influence of stratification. The
$Nt$
location of this reduction in the growth rate of
$L_z$
is consistent with the findings of Ohh & Spedding (Reference Ohh and Spedding2024), who reported that the
$L_z$
of their stratified wakes deviated from that of the unstratified wake at
$Nt \sim 2{-}4$
for
$\alpha = 20^\circ$
and
${\textit {Fr}} \geqslant 8$
(see their figure 17). Furthermore, to preserve momentum deficit, i.e.
$U_dL_yL_z \sim \text {constant}$
,
$L_y$
increases for the
${\textit {Fr}} =6$
wake as compared with the
${\textit {Fr}} = \infty$
wake (see figure 13
$c$
). The streamwise location of the reduction in the growth rate of
$L_z$
for the
${\textit {Fr}} = 6$
wake aligns with the decrease in the decay rates of
$U_d$
and
$E_K^M$
(figure 12). This contraction of
$L_z$
(compared with the
${\textit {Fr}}=\infty$
wake) in conjunction with the decreased decay of
$U_d$
has been widely reported in stratified wake literature of spheroids (Ortiz-Tarin et al. Reference Ortiz-Tarin, Chongsiripinyo and Sarkar2019, Reference Ortiz-Tarin, Nidhan and Sarkar2023) as well as other bluff bodies (Chongsiripinyo & Sarkar Reference Chongsiripinyo and Sarkar2020; Pal et al. Reference Pal, Sarkar, Posa and Balaras2017) for a variety of
$Re$
and
$\textit {Fr}$
values. The contours of wake defect velocity (figure 11) revealed the strong coupling of the wake topology to the isopycnal geometry, which in turn is significantly modified by the steady lee wave field (Bonnier & Eiff Reference Bonnier and Eiff2002; Pal et al. Reference Pal, Sarkar, Posa and Balaras2017) at
${\textit {Fr}} = 1.9$
and 1. The behaviour of
$L_y$
and
$L_z$
reflects this coupling. Additionally, the buoyancy effect on flow separation in the cases with higher relative stratification is also important. For example,
$L_z$
at
${\textit {Fr}} = 1$
is significantly larger than in the other cases until almost the end of the domain owing to the structure of the
${\textit {Fr}} = 1$
wake (figure 11
d,h,l,p) with an upper and a lower lobe. Owing to the peculiar geometry of the
${\textit {Fr}} = 1$
wake, calculation of
$L_z$
using (5.2) underestimates its vertical extent. Visually, the
${\textit {Fr}} =1$
wake is discernible within
$z = \pm 0.7$
in figure 11(d,h,l,p).
6. Evolution of mean streamwise vorticity
Figure 14. Instantaneous streamwise vorticity (
$\omega _x$
) on the spheroid at
$x/D = 0, 1, 2$
and
$2.75$
for all
$\textit {Fr}$
at
$\alpha = 10^{\circ }$
. Views from the (a,c,e,g) left and (b,d,f,h) right.
Shifting focus from the mean wake evolution, we now delve into the dynamics of streamwise vortices that develop at the body and into the wake. Figure 14 presents the distribution of instantaneous streamwise vorticity
$\omega _x$
at positions
$x/D = 0, 1, 2$
and
$2.75$
on the body. Views at these four locations from both the left and right perspectives are provided to provide a complete picture. The flow remains laminar over the body for all
$\textit {Fr}$
values at the present
${Re}_L = 30\, 000$
until it separates.
At both
${\textit {Fr}} = \infty$
and
${\textit {Fr}} = 6$
, the counter-rotating vortex pair is evident as early as the body centre (
$x/D = 0$
), see figure 14
$(a{-}d)$
. The development is similar on both sides until
$x/D = 1$
, maintaining symmetry about the vertical centreplane of the spheroid. However, at
$x/D = 2$
, lateral asymmetry emerges as one side undergoes a stronger vortex rollup than the other. As was surmised from the
$U_d$
contours, the polarity of the stronger vortex is reversed at
${\textit {Fr}} = 6$
relative to
${\textit {Fr}} = \infty$
. Specifically, the positive vortex filament (figure 14
$b$
) is larger at
${\textit {Fr}} = \infty$
in contrast to
${\textit {Fr}} = 6$
(figure 14
$c$
) where the negative vortex filament is larger. At
$x/D = 2.75$
, the vortex asymmetry becomes even more pronounced in both cases, consistent with the asymmetry in
$U_d$
(figure 11) and force coefficients (figure 10). Previous studies on slender bodies have documented the presence of asymmetry in a counter-rotating vortex pair under two conditions: (
$a$
) at high pitch angle
$\alpha$
and moderate Reynolds numbers
$Re$
(Jiang et al. Reference Jiang, Gallardo and Andersson2014, Reference Jiang, Gallardo, Andersson and Zhang2015, Reference Jiang, Andersson, Gallardo and Okulov2016); or (
$b$
) at moderate
$\alpha$
and high
$Re$
(Ashok et al. Reference Ashok, Van Buren and Smits2015a
). The present study reports lateral vortex asymmetry in yet another unstratified wake, namely moderate
$\alpha$
and moderate
$Re$
. Furthermore, the study reveals that this asymmetry persists for mild stratifications too, until
$\textit {Fr}$
decrease to 6.
Vortex asymmetry is absent at the stronger relative stratifications of
${\textit {Fr}} = 1.9$
and
$1$
. Additionally, the vortex-pair geometry changes. For example, the two filaments at
$x/D = 2.75$
in the
${\textit {Fr}} = 1.9$
wake appear thinner and less curved than at
${\textit {Fr}} = \infty$
or
$6$
. At
${\textit {Fr}} = 1$
, the vorticity field becomes even more complex. There is a primary vortex pair whose filaments roll up even less than that at
${\textit {Fr}} = 1.9$
. Furthermore, the region above the vortex pair also contains
$\omega _x$
, unlike the other three cases (compare the
$x/D = 2.75$
slice between the bottom row and other three rows). This distinct upper region of
$\omega _x$
corresponds to the upper lobe of the double-lobed mean wake structure in figure 11(
$d$
).
6.1. Streamwise vortex pair in the unstratified wake
Figure 15. Mean streamwise vorticity
$\langle \omega _x \rangle$
contours at
$x/D = 3, 10, 20$
and
$30$
for
${\textit {Fr}} = \infty$
.
$r/D$
denotes the radial extent of the contour at the respective
$x/D$
locations. The black dot and cross represent the centre of the negative and the positive vortex filament, respectively.
The downstream evolution of the counter-rotating vortex pair is of interest. Figure 15 presents
$\langle \omega _x \rangle$
contours at four locations in the wake,
$x/D = 3, 10, 20$
and
$30$
. The black circle and cross show the centroid of the positive and negative vortex filaments, which are computed following Jemison et al. (Reference Jemison, Delaney and Kannepalli2020):
\begin{equation} \langle y^{c(+,-)}_\omega \rangle = \frac {\int y\langle \omega _x \rangle ^{(+,-)}\mathrm {dA}}{\int \langle \omega _x \rangle ^{(+,-)}\mathrm {dA}}, \qquad \langle z^{c(+,-)}_\omega \rangle = \frac {\int z\langle \omega _x \rangle ^{(+,-)}\mathrm {dA}}{\int \langle \omega _x \rangle ^{(+,-)}\mathrm {dA}}, \end{equation}
where
$+$
and
$-$
denote the positive and the negative vortex filament, respectively.
The asymmetric vortex pair on the body propagates downstream as two distinct patches of
$\langle \omega _x \rangle$
. The centroids exhibit downward descent as well as lateral shifts. It is worth noting that, despite the spatial asymmetry in the organisation of the negative and positive filaments, the magnitudes of their time-averaged circulation remain equal throughout the computational domain, i.e.
$\langle \Gamma ^{+} \rangle = -\langle \Gamma ^{-} \rangle$
, owing to the conservation of angular momentum.
Figure 16. Location of the streamwise vortex pair in the
$y{-}z$
plane as a function of
$x/D$
.
$\langle z ^{c}_\omega \rangle$
and
$\langle y^{c}_\omega \rangle$
denote the evolution in the vertical and spanwise direction, respectively. Results from the point and Lamb dipole modelling are shown for the geometrical centre of the centroids of the negative and positive vortex.
Figure 16 provides a comparison of the actual trajectoriy of the geometric centre of the two vortices with the predictions of point vortex and Lamb dipole models. In an unstratified medium, according to the point vortex approximation, the vortex pair travels in its plane with the following vortex-induced velocity (Hill Reference Hill1975):
\begin{equation} \boldsymbol {V} (x) = \frac {\langle \Gamma \rangle (x)}{2\pi d} \boldsymbol {n}. \end{equation}
Here,
$\langle \Gamma \rangle (x)$
represents the time-averaged circulation in constant-
$x$
planes,
$\boldsymbol {n}$
is the unit normal to the vector
$\boldsymbol {d}$
between the positive and negative vortex centroids, and
$d$
is the magnitude of
$\boldsymbol {d}$
. Assume that the vortex pair moves with constant speed
$U_\infty$
in the
$x$
direction so that the vortex travel time is
$t = x/U_\infty$
and integrate
$\boldsymbol {V} (t= x/U_\infty )$
with respect to
$t$
. The result is the vortex centroid displacement in
$y$
- and
$z$
-directions under the point vortex modelling assumption. The Lamb dipole modelling follows the same approach, e.g. Flór et al. (Reference Flór, Van and Gertijan1994):
\begin{equation} \boldsymbol {V} = \frac {\langle \Gamma \rangle }{(6.83^2/2\pi) d} \boldsymbol {n} \, . \end{equation}
Both the point vortex and Lamb dipole models overestimate the vertical displacement magnitude (figure 16
$a$
) until
$x/D \approx 15{-}20$
followed by an overestimate. These models satisfactorily capture the spanwise drift of the geometric centre of the vortex pair (figure 16
$b$
). While the simplified models capture the order of magnitude of the vertical and horizontal vortex displacement in the current unstratified simulation until
$x/D = 40$
, the accuracy of these simplified models is marginal in capturing the streamwise trend, indicating that the spatial organisation of the vortices shed from the body is more complex than that of a point-vortex pair.
Figure 17. Variation of
${\textit {Fr}}_{\omega }$
, the hypothetical Froude number of the counter-rotating streamwise vortex pair in the
${\textit {Fr}} = \infty$
wake if subjected to
${\textit {Fr}} = 6$
stratification, as a function of streamwise distance
$x$
for both positive and negative vortex filaments.
6.2. Buoyancy effects on the streamwise vorticity
Figure 18. Mean streamwise vorticity
$\langle \omega _x \rangle$
at
$x/D = 3, 10, 20, 30$
for
${\textit {Fr}} = 6, 1.9$
and
$1$
.
$r/D$
denotes the radial extent of the contour at the respective
$x/D$
locations.
The
${\textit {Fr}} =6$
wake showed a significant reduction in the wake descent (
$\langle z^c \rangle$
in figure 13(b) as early as
$x/D = 4$
, only 1 body diameter behind the trailing edge of the spheroid, or at
$Nt = 2/3$
in buoyancy units behind the body centre. This relatively early buoyancy effect, despite a relatively weak initial stratification of
${\textit {Fr}} = 6$
, can be understood by consideration of the hypothetical Froude number (
${\textit {Fr}}_{\omega }(x)$
) if the vortex pair in the unstratified wake was subjected to the buoyancy frequency (
$N$
) of the
${\textit {Fr}} = 6$
case. Here,
${\textit {Fr}}_{\omega }(x) = W_c(x)/N{d}(x)$
, where
$W_c(x) = \mathrm {d}\langle z^c_{\omega } \rangle /\mathrm {d}t = U_\infty \mathrm {d}\langle z^c_{\omega } \rangle /\mathrm {d}x$
is the measured descent velocity, and
$d(x)$
is the distance between the centroids of the positive and negative vortex. Remarkably,
${\textit {Fr}}_{\omega }$
is already
$O(1)$
by
$x/D \approx 5$
(figure 17), indicating that the streamwise vortices of the
${\textit {Fr}} = \infty$
cases would be strongly impacted by buoyancy, right from the beginning, if exposed to the
${\textit {Fr}} =6$
stratification. In what follows, we will delve deeper into the effect of buoyancy on streamwise wake vortices.
Figure 18 displays the contours of
$\langle \omega _x \rangle$
for the stratified wakes. At
$x/D = 3$
, the presence of counter-rotating streamwise vorticity is evident across all three stratified cases. Buoyancy significantly affects the subsequent streamwise variation of
$\langle \omega _x \rangle$
. With decreasing
$\textit {Fr}$
, not only does the magnitude of
$\langle \omega _x \rangle$
increase, but also its spatial organisation.
Notably, in the
${\textit {Fr}}=6$
wake (figure 18
a,d,g,j), the integrity of the streamwise counter-rotating vortex filaments disintegrates rapidly, unlike the
${\textit {Fr}} = \infty$
wake. This buoyancy effect will be attributed to the baroclinic torque shortly. Given the absence of distinct positive and negative vortex filaments in the
${\textit {Fr}} = 6$
wake, a simplified point vortex or Lamb vortex model prediction is not attempted. In the case of stronger relative stratifications of
${\textit {Fr}} = 1.9$
and
$1$
, the
$\langle \omega _x \rangle$
fields exhibit increased complexity at
$x/D = 10$
and beyond. These wakes display a larger spatial extent of
$\langle \omega _x \rangle$
owing to steady lee waves outside the wake core. Also, the interaction between the wake vorticity and the lee waves vorticity is intricate, making it challenging to discern the fate of the shed vortices based solely on the contours, in contrast to the
${\textit {Fr}} = \infty$
wake.
Figure 19. Mean streamwise baroclinic torque
$-1/{\textit {Fr}}^{2} {\partial \langle \rho \rangle }/{\partial y}$
at
$x/D = 3, 10, 20, 30$
for
${\textit {Fr}} = 6, 1.9$
and
$1$
.
$r/D$
denotes the radial extent of the contour at the respective
$x/D$
locations. Note that as
$x/D$
increases, colourbar limits are reduced to accentuate the spatial structure of the mean streamwise baroclinic torque.
The baroclinic torque, which measures the direct influence of buoyancy on vorticity, is useful to understand the observed dependence of
$\left \lt \omega _x \right \gt$
on
$\textit {Fr}$
. On taking the curl of the Reynolds average of (2.2), it follows that there is a baroclinc torque
\begin{align} \boldsymbol {\omega }_{BT} &= - \frac {1}{Fr^2} \boldsymbol {\nabla} \times \langle \rho _d \rangle \boldsymbol {e}_3 \nonumber \\ (\omega _{BT})_ i &= - \frac {1}{Fr^2} \epsilon _{ijk} \frac {\partial \langle \rho _d \rangle }{\partial x_j} \delta _{k3} = - \frac {1}{Fr^2} \epsilon _{ij3} \frac {\partial \langle \rho _d \rangle }{\partial x_j}. \end{align}
The mean streamwise contribution of the baroclinic torque is
$\omega _{BTx} = - (1/{\textit {Fr}} )^2 \partial \langle \rho \rangle /\partial y$
and follows from (6.4) with
$i = 1$
. It involves only the lateral (spanwise) mean buoyancy gradient, apart from the
$\textit {Fr}$
-dependent normalisation factor.
The magnitude of
$\omega _{BTx}$
is substantial and increases with increasing stratification (figure 19). At
$x/D = 3$
, it is especially large at the wake boundaries as suggested by the organisation of mean isopycnals in figure 11. Further downstream, the contribution of lee waves to
$\omega _{BTx}$
becomes progressively important. For
${\textit {Fr}} = 6$
, the initial distribution of
$-(1/{\textit {Fr}} )^2 \partial \langle \rho \rangle /\partial y$
(figure 19
$a$
) is laterally asymmetric since transport by the laterally asymmetric flow in the
${\textit {Fr}}=6$
wake (evident through the previous analyses) leads to buoyancy asymmetry. In contrast, the
${\textit {Fr}} = 1.9$
and
$1$
cases exhibit lateral symmetry except for a sign change upon spanwise reflection. This contrast is evident when comparing the left column to the right two columns in figure 19. As these wakes progress downstream, the coherence of the baroclinic torque field in the
${\textit {Fr}} = 6$
wake diminishes significantly when compared with the
${\textit {Fr}} = 1$
and
$1.9$
wakes.
7. Summary and conclusions
The stratified wake of a 6:1 prolate spheroid at angle of attack
$\alpha = 10^{\circ }$
is studied using large eddy simulations (LES). Cases with
${\textit {Fr}} = U_{\infty }/ND = 6, 1.9$
and
$1$
are simulated along with its unstratified (
${\textit {Fr}} = \infty$
) counterpart at a length-based Reynolds number
${Re}_L = U_{\infty }L/\nu = 30\, 000$
. The focus of the present manuscript is on the mean wake and vorticity dynamics along with the on-body flow. Subsequent work that includes higher values of
${Re}_L$
will report on turbulence, unsteady coherent structures and mixing aspects.
Visualisations of instantaneous
$u_x$
-isovolumes, streamlines and
$\lambda _2$
-structures (figures 4–7) reveal overall changes in flow topology as
$ {\textit {Fr}}$
decreases. At
$ {\textit {Fr}} = \infty$
and
$ {\textit {Fr}} = 6$
, flow separation at the body aligns with previous findings in the literature. In both cases, a pair of streamwise vortices emerges from the lateral sides of the body, initially laminar, but transitioning to turbulence near the spheroid. Even at the relatively weaker stratification of
$ {\textit {Fr}} = 6$
, stratification effects are evident. Similar to the observations by Ohh & Spedding (Reference Ohh and Spedding2024), the wake at
$ {\textit {Fr}} = 6$
does not descend as much as the
$ {\textit {Fr}} = \infty$
wake, due to buoyancy forces inhibiting vertical displacement. As stratification strength increases, the wake topology and streamline distribution change significantly. Small-scale variability (turbulence) in the wake is inhibited at
$ {\textit {Fr}} = 1.9$
and
$ {\textit {Fr}} = 1$
. At
$ {\textit {Fr}} = 1.9$
, the
$\lambda _2$
-structures and
$u_x$
-isovolumes remain laminar up to
$x/D = 20$
, with even later transition at
$ {\textit {Fr}} = 1$
.
The analysis of averaged pressure and friction contours on the spheroid provides further insight into the effect of stratification on the mean flow at the spheroid. Specifically, the
$C_f$
contours across different
$ {\textit {Fr}}$
values reveal that stratification alters the flow separation topology (figure 9). At
$ {\textit {Fr}} = \infty$
and
$ {\textit {Fr}} = 6$
, flow separates from both sides of the body. However, at
$ {\textit {Fr}} = 1.9$
, flow separation occurs from both sides as well as the central region (
$y/D = 0$
). By
$ {\textit {Fr}} = 1$
, lateral separation is completely suppressed since the cross-stream windward-to-leeward surface flow is arrested by buoyancy. Furthermore,
${\textit {Fr}} = \infty$
and
$6$
exhibit asymmetric separation, quantified further by
$C_y$
in both cases (figure 10). At stronger stratifications, the overturning motion in the vortex rollup weakens as evidenced by the secondary flow streamlines (figure 6) and the vorticity is no longer laterally asymmetric.
The evolution of the mean wake topology, both quantiatively and qualitatively, is also analysed for all four
$\textit {Fr}$
cases. In unstratified conditions and in the
${\textit {Fr}}=6$
case, the initial wake drifts both in the vertical and horizontal direction owing to vertical momentum imparted by non-zero
$\alpha$
and initial lateral asymmetry present in the flow (figure 11
a,e,i,m). Previous studies on flow past slender bodies at both high (Jiang et al. Reference Jiang, Gallardo and Andersson2014, Reference Jiang, Gallardo, Andersson and Zhang2015, Reference Jiang, Andersson, Gallardo and Okulov2016) and low-to-moderate (Ashok et al. Reference Ashok, Van Buren and Smits2015
Reference Ashok, Van Buren and Smits
a
,Reference Ashok, Van Buren and Smits
b
) angles of attack have found lateral asymmetry in the wake. We hypothesise that the
${\textit {Fr}} = \infty$
(and
${\textit {Fr}} = 6$
) flows at
${Re}_L = 30\, 000$
are bistable, being locked in two distinct reflectional-symmetry-breaking states, with each state being equally probable. The above-mentioned studies reporting flow asymmetry also found that the orientation of asymmetry remained constant over time, similar to our findings for both
${\textit {Fr}} = \infty$
and
${\textit {Fr}} = 6$
. A caveat is that the present averaging time span of approximately
$100 D/U_\infty$
is not long and the polarity could conceivably switch at longer times, although it did not do so in the laboratory experiments of Ashok et al. (Reference Ashok, Van Buren and Smits2015a
,Reference Ashok, Van Buren and Smits
b
).
Interestingly, the effect of stratification on the wake is evident in the
${\textit {Fr}} = 6$
wake from the outset. Specifically, focusing on the vertical motion (a descent in the present configuration) of the wake centre
$\langle z^{c} \rangle$
(figure 13) owing to the self-induced velocity of the vortex pair, it is found that the descent of the
${\textit {Fr}} = 6$
wake is inhibited from approximately
$x/D \approx 3$
. A Froude number (
${\textit {Fr}}_\omega$
) based on the baseline unstratified descent velocity and
$N$
is found to become
$O(1)$
in the near wake, explaining the rapid arrest of the wake descent by buoyant deceleration. Subsequent oscillation in the evolution of
$\langle z^{c} \rangle$
with a wavelength
$\lambda = 2\pi {\textit {Fr}}$
, consistent with observations by Ohh & Spedding (Reference Ohh and Spedding2024), is also observed. As
$\textit {Fr}$
is reduced further, the mean wake topology becomes more influenced by stratification. At
${\textit {Fr}} = 1.9$
, the vertical wake drift decreases further and the wake oscillates with a significantly stronger amplitude than at
${\textit {Fr}} = 6$
. The
${\textit {Fr}} = 1$
case exhibits a unique wake topology, featuring two distinct lobes due to the different separation patterns, which persist until at least
$x/D = 40$
. Higher
${Re}_L$
studies at
${\textit {Fr}} = 1$
in this configuration are necessary to assess whether turbulent mixing at high
${Re}_L$
would prevent formation and persistence of the intriguing double-lobed wake found here.
The streamwise evolution trends of key mean wake quantities like
$U_d$
, mean kinetic energy and wake length scales (
$L_y$
and
$L_z$
) are qualitatively similar to previous studies of stratified wakes (Spedding Reference Spedding1997; Bonnier & Eiff Reference Bonnier and Eiff2002; Pal et al. Reference Pal, Sarkar, Posa and Balaras2017; Ortiz-Tarin et al. Reference Ortiz-Tarin, Chongsiripinyo and Sarkar2019): (a) lee-waves-induced modulation is observed in all stratified wakes, with its intensity increasing with decreasing
$\textit {Fr}$
; (b)
${\textit {Fr}} = 1.9$
and
$1$
wakes retain significantly higher MKE and
$U_d$
than
${\textit {Fr}} = 6$
or the unstratified wake; and (c) the deviation in area-integrated quantities in the
${\textit {Fr}} = 6$
wake from the
${\textit {Fr}} = \infty$
wake occurs at
$Nt \sim 1{-}3$
.
Shifting the focus to the mean vorticity dynamics in the wake, it was found that the asymmetry in the
${\textit {Fr}} = \infty$
and
$6$
wakes originates from the earlier roll-up of the vortex on one side compared with the other (see figure 14
$a{-}d$
). At
${\textit {Fr}}=1.9$
and
$1$
, the roll-up is significantly inhibited by buoyancy (see figure 14
$e{-}h$
), leading to a symmetric configuration of the streamwise vortex pair. The negative and positive vortices maintain a clear signature for
${\textit {Fr}} = \infty$
(figure 15), prompting an attempt to model them using point vortex and Lamb dipole approaches to capture the trends in vertical descent and spanwise displacement of the vortex pair. While these rudimentary models provide an order of magnitude estimate for the vertical descent and spanwise displacement (figure 16), they do not accurately capture the streamwise trend since the topology of the relatively close spheroid vortices is more complex than a pair of pointwise vortices. At
${\textit {Fr}} = 6$
, the streamwise vortex pair loses coherence as early as
$x/D = 10$
(figure 18
$d$
), preventing any attempts at similar modelling for the
${\textit {Fr}} = 6$
wake. The
${\textit {Fr}} = 1.9$
and
$1$
wakes exhibit a significantly more complex spatial distribution due to nonlinear interactions between the vorticity carried by the wake and that carried by the steady lee waves. The influence of steady lee waves is evident in both
$\langle \omega _x \rangle$
and baroclinic torque contours for
${\textit {Fr}} = 1.9$
and
$1$
(figure 18 and 19).
The findings of this work suggest that energetic, weakly stratified wakes at
${\textit {Fr}} \sim O(10)$
need to be investigated at a higher
${Re}_L$
to confirm whether, close to the body, there is strong buoyancy-induced modification of the wake descent and structure at these elevated Reynolds numbers, similar to what is observed in the
${\textit {Fr}} = 6$
wake of this study. Future simulations of
${\textit {Fr}} \sim O(1)$
wakes will help better understand how the transition to turbulence might occur at higher
${Re}_L$
and how wake mixing takes place, an aspect that is strongly suppressed in the
${\textit {Fr}} = 1$
case of this study. Finally, larger values of pitch angle are also deserving of study.
Funding.
We gratefully acknowledge the support of ONR grant N00014-20-1-2253. Computational resources were provided by the Department of Defense High Performance Computing Modernization Program.
Declaration of interests.
The authors report no conflict of interest.
Author contributions.
S.N. and S.J. contributed equally to this work.
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