1. Introduction
Turbulence is stochastic. Despite varying instantaneous motions, the flow is ultimately ergodic, having only one mean flow state. Though this holds for many flows, others break ergodicity. For example, Huisman et al. (Reference Huisman, van der Veen, Sun and Lohse2014) found a hysteretic effect in their experiments of highly turbulent Taylor–Couette flow which led to multiple stable states: despite identical driving parameters in the system, the turbulent state was path-dependent. Existence of multiple flow states has also been observed in Rayleigh–Bénard convection at low Reynolds numbers (Xi & Xia Reference Xi and Xia2008; van der Poel et al. Reference Vreman, Geurts and Kuerten2011; Weiss & Ahlers Reference Weiss and Ahlers2013) and Taylor–Couette flow (Xia et al. Reference Xia, Shi, Cai, Wan and Chen2018; Yang & Xia Reference Yang and Xia2021). In these flows, the multiple stable states feature different mean flow and turbulence characteristics. Alternatively, a flow may break ergodicity through mean flow asymmetry, where a single asymmetric state forms despite the absence of any asymmetry in the boundary conditions. For flow over symmetric bodies, the mean turbulent flow is expected to reflect the constraining symmetry of the geometry. However, studies have demonstrated instances where the mean symmetry of turbulent flow over symmetric geometry is broken. For example, Zhang et al. (Reference Zhang, Yang, Zhu, Wan and Chen2023) found mean lateral asymmetry in their direct numerical simulations (DNS) of turbulent flow over surfaces composed of geometrically symmetric cube arrays.
Mean asymmetry has also been demonstrated in crossflow over slender bodies of revolution. Such flows give rise to complex three-dimensional phenomena of great interest to aerodynamics and hydrodynamics communities as they occur frequently in engineering applications. Bridges (Reference Bridges2006) reviewed many experimental studies and some early computational analyses that sought to identify the source of the mean asymmetry and develop techniques for flow control. Variations in angle of attack
$\alpha$
result in flows that can be divided into three regimes. In the low angle of attack regime, the mean flow is largely symmetric about the meridian of the body, and the mean side force is zero. At intermediate angles of attack, the mean wake is asymmetric, but in the highest angle of attack regime, the mean flow returns to symmetry via unsteady vortex shedding. These angle of attack effects were demonstrated by Lamont (Reference Lamont1982) for an ogive cylinder at Reynolds numbers in the range
$2.0\times 10^5\leqslant \textit{Re}_{\!D}\leqslant 4.0\times 10^6$
.
Flow behaviour associated with these three angle of attack regimes has also been observed for blunt axisymmetric bodies. The prolate spheroid has long served as a useful approximation of blunt slender bodies in general (Constantinescu et al. Reference Constantinescu, Pasinato, Wang, Forsythe and Squires2002; Xiao et al. Reference Xiao, Zhang, Huang, Chen and Fu2007; Ortiz-Tarin et al. Reference Ortiz-Tarin, Chongsiripinyo and Sarkar2019, Reference Ortiz-Tarin, Nidhan and Sarkar2021, Reference Ortiz-Tarin, Nidhan and Sarkar2023); in particular, the 6 : 1 prolate spheroid, where 6 : 1 indicates the ratio of the major axis length to that of the minor axis, has served as a useful representation of slender bodies of revolution in many previous studies. Various aspects of the flow over a 6 : 1 prolate spheroid at low
$\alpha$
have been investigated. Several experimental studies (Fu et al. Reference Fu, Shekarriz, Katz and Huang1994; Chesnakas & Simpson Reference Chesnakas and Simpson1997; Wetzel, Simpson & Chesnakas Reference Wetzel, Simpson and Chesnakas1998; Wetzel & Simpson Reference Wetzel and Simpson1998; Goody, Simpson & Chesnakas Reference Goody, Simpson and Chesnakas2000) were conducted to investigate a range of angles of attack from
$\alpha =0^\circ$
to
$\alpha =30^\circ$
. These studies explored the unsteady surface quantities and near wake of the 6 : 1 prolate spheroid in this range of
$\alpha$
. What data were provided across the meridional plane, the plane coincident to the major axis of the spheroid and the freestream flow vector, showed the mean flow near the body to be symmetric. Thus flow data were typically presented for only one half of the prolate spheroid, and the mean flow was assumed to be everywhere symmetric about the meridional plane. More recently, mean symmetry throughout the flow has been further demonstrated through large-eddy simulations (LES) at
$\alpha =10^\circ$
and
$\alpha =20^\circ$
by Plasseraud, Kumar & Mahesh (Reference Plasseraud, Kumar and Mahesh2023).
Symmetry is also preserved at high attack angles. Crossflow over a 6 : 1 prolate spheroid at angle of attack
$\alpha =90^\circ$
was investigated via DNS by El Khoury, Andersson & Pettersen (Reference El Khoury, Andersson and Pettersen2010, Reference El Khoury, Andersson and Pettersen2012). Here, the wake behind the prolate spheroid bridged the behaviour of an infinite cylinder in crossflow (Zdravkovich Reference Zdravkovich1997) and a sphere (Johnson & Patel Reference Johnson and Patel1999). The unsteady wake exhibited large-scale low-frequency asymmetry in the instantaneous flow field, with alternating von Kármán vortex shedding. However, the mean flow field was effectively symmetric about the meridional and equatorial planes, where the equatorial plane is normal to the major axis of the prolate spheroid.
Intermediate angles of attack between those investigated in these studies result in wakes that exhibit surprising asymmetric mean flow behaviour. Jiang et al. (Reference Jiang, Gallardo, Andersson and Zhang2015, Reference Jiang, Andersson, Gallardo and Okulov2016) reported persisting mean asymmetry in their DNS of flow over a 6 : 1 prolate spheroid at
$\alpha =45^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
. Rather than alternate von Kármán vortex shedding as observed for a 6 : 1 prolate spheroid at
$\alpha =90^\circ$
, the wake skewed to one side only, such that the side force did not change sign. These studies were primarily phenomenological, focusing on features of the wake in a stable state of asymmetry. Indeed, they presented a detailed account of the near and intermediate wake behaviour, and used their observations to validate the self-similarity law for the primary vortex tube in the far wake. Through analysis of the unsteady wake, they also identified the very low dominant frequency in the flow, which they attributed to three-dimensional effects, though no culpable phenomenon was identified in the wake. The results in Jiang et al. (Reference Jiang, Gallardo, Andersson and Zhang2015) led them to comment on the debate described by Bridges (Reference Bridges2006) on attribution of mean flow asymmetry over slender bodies of revolution to a convective or global instability. As their simulations included no disturbance to trip the mean flow asymmetry, they attributed the asymmetry to global instability, i.e. the meridional asymmetry was an inherent property of the flow. Few data were given regarding the onset of the mean flow asymmetry, however, as the emphasis was given to flow structures well after the development of asymmetry.
In this study, we investigate the asymmetry of the vortex wake behind a 6 : 1 prolate spheroid at intermediate angles of attack using high-fidelity numerical simulations. Several aspects of this work represent novel contributions. First, rather than focusing solely on fully developed asymmetric states, we examine the onset of asymmetry. Second, we establish direct connections between global aerodynamic quantities and underlying flow structures. Third, we investigate the physical mechanisms responsible for the generation and destruction of wake asymmetry through the Reynolds-averaged Navier–Stokes equations. Finally, we demonstrate the effect of wake turbulence on the onset of asymmetry by observing the trend in critical angle of attack at which asymmetry emerges at
$\textit{Re}_{\!D}=3.0\times 10^3$
and
$\textit{Re}_{\!D}=6.0\times 10^3$
.
The remainder of this paper is organised as follows. In § 2, we provide a detailed description of the computational set-up, while § 3 includes the Navier–Stokes-based derivation for the mean flow asymmetry transport equation. Results are presented in § 4, where we demonstrate that the mean flow asymmetry is robust to various control volume discretisation methods and the numerics of the simulations. We then use the asymmetry transport equation in § 5 to gain insight into the nature of the mechanism of mean flow asymmetry itself. In § 6, we investigate the effects of variations in angle of attack and Reynolds number on flow asymmetry. Concluding remarks are given in § 7.
2. Methodology
2.1. Coordinate system
Coordinate systems are specified as indicated in figure 1, with the origin coincident to the spheroid centroid, and the
$x$
,
$y$
and
$z$
axes corresponding to the longitudinal, vertical and lateral directions, respectively. The freestream velocity
$U_{\infty }$
is oriented parallel to the
$x$
axis at zero angle of attack (
$\alpha =0^\circ$
) and in the
$[0.7071\ 0.7071\ 0]$
direction at
$\alpha =45^\circ$
such that there is no lateral component to the freestream flow. Flow coordinates for drag and lift are aligned with the freestream such that the drag force is parallel to
$U_{\infty }$
, and lift is normal to
$U_\infty$
(
$[0\ 1\ 0]$
at
$\alpha =0^\circ$
, and
$[-0.7071\ 0.7071\ 0]$
at
$\alpha =45^\circ$
). The side force is parallel to the
$z$
axis. Force coefficients
$C_L$
,
$C_D$
and
$C_S$
are oriented parallel to the lift, drag and side force vectors. The rolling
$C_X$
, yawing
$C_Y$
, and pitching
$C_Z$
moments are taken about the positive
$x$
,
$y$
and
$z$
axes, respectively. The meridian of the spheroid is coincident to the
$z=0$
plane, and the azimuthal coordinate
$\phi$
is specified as indicated in figure 1, with
$\phi =0^\circ$
located at the intersection of the
$-y$
face of the spheroid and the meridional plane, and proceeding around the
$-x$
axis. When
$\alpha \gt 0^\circ$
,
$\phi =0^\circ$
corresponds to the pressure side of the spheroid, and
$\phi =180^\circ$
to the lee side. Axial station
$x/L$
is defined relative to the bow of the spheroid, such that
$x/L=0$
corresponds to the tip of the bow, and
$x/L=1$
to the stern. Vertical
$y/L$
and lateral
$z/L$
stations are measured with respect to the spheroid centroid.
Figure 1. Coordinate system and computational grid schematic of the simulations.
2.2. Computational grids
The surface of the prolate spheroid is comprised entirely of body-conforming quadrilateral cells. Four radial panels form the majority of the body, with two H-grid end caps at the bow and stern. Cells are extruded from the surface of the prolate spheroid such that the height of the first grid point in the boundary layer is
$\varDelta _{c,wall}\approx 4.0\times 10^{-4}\,\text{L}$
. This initial wall spacing results in
$\varDelta ^+=0.81$
on the pressure side at
$x/L=0.5$
and
$\phi =0^\circ$
for the
$\alpha =45^\circ$
case. Normalising with respect to this
$\varDelta ^+$
at the mid-body, the maximum surface cell dimension on the spheroid is
$5\varDelta ^+$
on the radial panels, while the minimum reaches
$0.62\varDelta ^+$
on the bow and stern H-grids. The wall-normal growth rate of the cells in the boundary layer is
$1.05$
. A wake refinement region is placed downstream of the prolate spheroid to resolve wake features for minimum length
$1\,\text{L}$
. The boundaries of the computational domain are everywhere a minimum distance
$100\,\text{L}$
from the body (figure 1). In total, the computational grid includes 97 million cells. Figure 2 gives a representation of every fourth grid point on the meridional plane and spheroid surface.
Figure 2. An illustration of the computational grid. We show every fourth grid point for clarity. The black lines show the grid in the fluid regime, and the red lines show the grid on the surface of the prolate spheroid.
Two variants of the above described grid are constructed. In one, grid symmetry across the meridional plane is not enforced. Maximum deviation across the meridional plane in position of grid points on the H-grid panels at the bow and stern of the prolate spheroid is
$0.11\varDelta ^+$
. For the second grid, meridional symmetry is enforced by splitting the grid at
$z\approx 0$
and manually projecting all grid points on the cut plane to the meridional plane. Then the grid is reflected about the meridional plane, and both halves are joined. The purpose of this grid variation is to ensure that the results reported here are robust to small perturbations in the transverse direction.
2.3. Solver configuration
The full compressible Navier–Stokes equations are here solved over the discretised control volumes with CharLES (Khalighi et al. Reference Khalighi, Nichols, Ham, Lele and Moin2010), an unstructured finite-volume flow solver. For spatial terms, CharLES uses a second-order non-dissipative discretisation (Mahesh, Constantinescu & Moin Reference Mahesh, Constantinescu and Moin2004) and an explicit third-order Runge–Kutta discretisation (Williamson Reference Williamson1980) for temporal terms. In the present work, the
$\textit{Re}_{\!D}=3.0\times 10^3$
flow is oriented so as to result in angles of attack from
$\alpha =0^\circ$
to
$\alpha =45^\circ$
, dependent on the case. Fluid properties are specified such that the freestream Mach number was
$\textit{Ma}_{\infty }=0.2$
to mitigate compressibility (Wilcox Reference Wilcox2000) effects while avoiding numerical stiffness resulting from low
$\textit{Ma}_{\infty }$
. As for the boundaries of the computational domain (see figure 1), the surface of the prolate spheroid is treated as a no-slip wall, while the outer boundaries are given the characteristic boundary condition (Thompson Reference Thompson1990), with inflow versus outflow determined from the orientation of characteristic velocities at the boundary with respect to the local boundary normal. In the LES cases at
$\textit{Re}_{\!D}=6.0\times 10^3$
, the same grids are used, and Vreman’s dynamic mixed subgrid-scale model (Vreman, Geurts & Kuerten Reference van der Poel, Stevens and Lohse1994) is employed for subgrid-scale modelling.
3. Mean flow asymmetry and its conservation equation
Mean asymmetry in the flow may be quantified by taking the difference between the mean velocity
$U^+_i$
at a given position in the
$z\gt 0$
region and its counterpart
$U^-_i$
at the same location reflected across the meridional plane in the
$z\lt 0$
region:
\begin{equation} \varPhi _i(x,y,\pm z)=U^+_i(x,y,+z)-U^-_i(x,y,-z). \end{equation}
Here,
$\varPhi _i$
measures the difference in the
$i$
th mean velocity component at the conjugate points
$(x, y, \pm z)$
, where
$U^+_i$
and
$U^-_i$
are the mean velocities at the two locations symmetric about the meridional plane. The transport equations that govern the behaviour of
$U^+_i$
and
$U^-_i$
read
\begin{align} \frac {\partial U_i^+}{\partial t} + \frac {\partial U_{\!j}^+U_i^+}{\partial x_{\!j}} = -\frac {1}{\rho } \frac {\partial P^+}{\partial x_i} + \nu \frac {\partial ^2 U_i^+}{\partial x_{\!j}\, \partial x_{\!j}} - \frac {\partial \overline {u_i^+ u_{\!j}^+}}{\partial x_{\!j}}, \\[-12pt]\nonumber \end{align}
\begin{align} \frac {\partial U_i^-}{\partial t} + \frac {\partial U_{\!j}^-U_i^-}{\partial x_{\!j}} = -\frac {1}{\rho } \frac {\partial P^-}{\partial x_i} + \nu \frac {\partial ^2 U_i^-}{\partial x_{\!j}\, \partial x_{\!j}} - \frac {\partial \overline {u_i^- u_{\!j}^-}}{\partial x_{\!j}}, \end{align}
where
$P$
is mean pressure. These are Reynolds-averaged equations, where capital letters denote mean quantities, and lowercase letters denote fluctuating components. Taking the difference between the two equations, we get
\begin{align} &\underbrace {\frac {\partial \left (U^+_i-U^-_i\right )}{\partial t}}_{\textit {1}}+\underbrace {\frac {\partial \left (U^+_{\!j}U^+_i-U^-_{\!j}U^-_i\right )}{\partial x_{\!j}}}_{\textit {2}} \nonumber \\ &\quad =\underbrace {-\frac {1}{\rho }\frac {\partial \left (P^+-P^-\right )}{\partial x_i}}_{\textit {3}} \,\,\underbrace {+\nu \frac {\partial ^2 \left (U^+_i-U^-_i\right )}{\partial x_{\!j}\,\partial x_{\!j}}}_{\textit {4}} \,\,\underbrace {-\frac {\partial \left (\overline {u_{\!j}^+ u_i^+} - \overline {u_{\!j}^- u_i^-}\right ) }{\partial x_{\!j}}}_{\textit {5}}. \end{align}
We can expand terms 2 and 5 of (3.3a ) into
\begin{align} \underbrace {\frac {\partial \left (U^+_{\!j}U^+_i-U^-_{\!j}U^-_i\right )}{\partial x_{\!j}}}_{\textit {2}} =\underbrace {\frac {\partial \left [\left (U^+_i-U^-_i\right )\left (U^+_{\!j}+U^-_{\!j}\right )\right ]}{\partial x_{\!j}}}_{\it 2a} - \underbrace {\frac {\partial \left (U^+_iU^-_{\!j}-U^+_{\!j}U^-_i\right )}{\partial x_{\!j}}}_{\it 2b}, \end{align}
\begin{align} \underbrace {\frac {\partial \left (\overline {u_{\!j}^+ u_i^+} - \overline {u_{\!j}^- u_i^-}\right ) }{\partial x_{\!j}}}_{\it 5}=\underbrace {\frac {\partial \left [\overline {\left (u_i^+-u_i^-\right )\left (u_{\!j}^++u_{\!j}^-\right )}\right ] }{\partial x_{\!j}}}_{\it 5a}-\underbrace {\frac {\partial \left (\overline {u_i^+u_{\!j}^-}- \overline {u_{\!j}^+u_i^-}\right ) }{\partial x_{\!j}}}_{\it 5b}, \end{align}
then substitute (3.4a ,b ) into (3.3a ). Bringing term 2b to the right-hand side and combining with term 5b leads directly to
\begin{align} &\qquad \frac {\partial \varPhi _i}{\partial t}+\underbrace {\frac {\partial V_{\!j}\varPhi _i}{\partial x_{\!j}}}_{\textit {convection}} ={} \underbrace {-\frac {1}{\rho }\frac {\partial \left (P^+-P^-\right )}{\partial x_i}}_{\textit {production}} \nonumber \\ \underbrace {{}+\nu \frac {\partial ^2 \varPhi _i}{\partial x_{\!j}\,\partial x_{\!j}}}_{\textit {viscous destruction}} \,\,&\underbrace {{}-\frac {\partial \overline {v_{\!j}\varphi _i}}{\partial x_{\!j}}}_{\textit {turbulent destruction}} \,\,\underbrace {{}-\frac {\partial }{\partial x_{\!j}}\left [U_{\!j}^+U_i^- -U_{\!j}^-U_i^+ +\overline {u_{\!j}^+ u_i^-} - \overline {u_{\!j}^- u_i^+}\right ]}_{\textit {cross-correlation}}, \end{align}
where
$\varPhi _i=U_i^+-U_i^-$
is the difference mean velocity,
$V_{\!j}=U_{\!j}^++U_{\!j}^-$
is a new mean convective velocity,
$\varphi _i=u_i^+ - u_i^-$
is the fluctuation of the velocity difference across the meridian, and similarly
$v_{\!j}=u_{\!j}^+ + u_{\!j}^-$
.
We propose
$\varPhi _x$
as a measure of mean flow asymmetry, with the
$x$
-component of (3.5) describing the spatial variation of the asymmetry. The terms on the right-hand side are responsible for the generation and destruction of the asymmetry, as indicated in (3.5). It should be noted that some of the terms of (3.5) are non-local as they include the products of velocity components from two locations and derivatives with respect to
$z$
, where
$z=\Delta z/2$
effectively represents the distance between corresponding points. As
$U_i^+\equiv U_i^-$
in the far field, the last term, i.e. the cross-correlation term, integrates to 0 in the region
$z\gt 0$
per Green’s theorem, and is a redistribution term. We will provide more detailed physical interpretation of the terms in § 4.
As the present approach involves comparison of flow quantities at distinct points in the flow, we must distinguish between (3.5) and similar formulations such as those presented in Hill (Reference Hill2002), which also examine the velocity relationship between two points in a flow field. First, the present formulation focuses on a first-order statistic of the mean flow field, while Hill (Reference Hill2002) addresses a second-order statistic of the instantaneous flow. Second, (3.5) interrogates two points that are symmetric about the meridional plane, thus involves only three degrees of freedom, i.e.
$x$
,
$y$
and
$\Delta z=z^+-z^-$
. In contrast, the two-point formulation of Hill (Reference Hill2002) compares arbitrary pairs of points in the flow, and thus involves six degrees of freedom
$x_1$
,
$y_1$
,
$z_1$
,
$x_2$
,
$y_2$
and
$z_2$
in its most general scenario. Given these two fundamental differences – in both statistical order and spatial dimensionality – the conclusions drawn in Hill (Reference Hill2002) are not directly applicable to the present analysis.
4. Development and characteristics of the asymmetric flow
4.1. Robustness to small transverse perturbations
We first verify that the results are robust to small transverse perturbations. At
$\alpha =45^\circ$
, simulations are performed with computational grids that are symmetric and asymmetric about the meridional plane to isolate the effects of numerical asymmetry in the flow. Both simulations are allowed to run sufficient time
$t$
for the flow to enter a stable and persistent asymmetric state and accrue mean statistics. Figure 3 gives the force and moment histories for each simulation normalised as
\begin{align} C_{D, L, S}=\frac {F_{D, L, S}}{\frac {1}{2}\rho U_{\infty }^2 S}, \quad C_{X,Y,Z}=\frac {M_{X,Y,Z}}{\frac {1}{2}\rho U_{\infty }^2 SD}, \end{align}
where
$\rho$
is the fluid density,
$S=\pi D^2/4$
is the projected cross-sectional area based on the minor diameter of the prolate spheroid
$D$
, and
$F$
and
$M$
are the given forces and moments integrated over the surface of the prolate spheroid. The agreement between the present results and those reported by Jiang et al. (Reference Jiang, Gallardo, Andersson and Zhang2015) validates the grids.
Figure 3. Histories of force coefficients, i.e. lift
$C_L$
, drag
$C_D$
and side
$C_S$
, and moment coefficients, i.e. rolling
$C_X$
, yawing
$C_Y$
and pitching
$C_Z$
for (a) the symmetric grid and (b) the asymmetric grid at
$\alpha =45^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
, with mean results from Jiang et al. (Reference Jiang, Gallardo, Andersson and Zhang2015) (indicated with dashed lines). The signs of the side force and yawing moment from Jiang et al. (Reference Jiang, Gallardo, Andersson and Zhang2015) are changed from (a) to (b) to facilitate comparison with the present results.
Initially, the side forces
$C_S$
are effectively zero for both grids. Eventually, the side force deviates from zero and reaches a new non-zero value that persists through the remainder of the simulations, indicating development of a stable asymmetric state of the flow. Notably, the side force develops in each simulation with opposite signs:
$+z$
in the symmetric grid case, and
$-z$
in the asymmetric grid case. Despite this and the yawing moment that also changes sign, the magnitudes of all forces and moments are effectively equivalent for both grids, each force within 1 % of the corresponding value from the other grid, and each moment within 2 %, with means calculated over the final
$20{tU_\infty }/{L}$
of each simulation. This verifies that our numerics do not prefer one transverse direction to the other, and that our results are robust to small numerical asymmetries in the transverse direction.
Besides verifying the robustness of the results to slight transverse asymmetry in the numerics, several interesting trends emerge in figure 3. First, with the onset of asymmetry, the side force and the moments
$C_Y$
and
$C_Z$
acting on the body change. However, this response is delayed by approximately
$10tU_\infty /L$
for the in-plane forces (
$C_D$
and
$C_L$
) and the pitching moment (
$C_M$
). Specifically, significant changes in those forces and moments associated with the onset of asymmetry occur at approximately
$tU_\infty /L = 26$
and
$tU_\infty /L = 5$
in figures 3(a) and 3(b), while
$C_D$
,
$C_L$
and
$C_Z$
exhibit significant changes at
$tU_\infty /L = 36$
and
$tU_\infty /L = 15$
. Despite this delay, for a given case, all forces and moments reach their final values in the asymmetric wake state at approximately the same time:
$tU_\infty /L \approx 43$
for the symmetric grid, and
$tU_\infty /L \approx 23$
for the asymmetric grid. Second, as mentioned in § 1, there is some debate in the literature as to whether the mean flow asymmetry rises due to a convective or global instability of the flow; see Bridges (Reference Bridges2006). If convective instability is responsible for the flow, then a space-fixed, time-invariant, symmetry-breaking perturbation is required for sustained mean flow asymmetry to occur. Upon removal of the perturbation, the flow will relax to a state of mean flow symmetry. Conversely, the results of Jiang et al. (Reference Jiang, Gallardo, Andersson and Zhang2015) demonstrated sustained asymmetry in flow over a 6 : 1 prolate spheroid in the absence of any such perturbation, thus they concluded that the asymmetry occurs due to a global instability of the flow. In the present work, we enforce grid symmetry and demonstrate a lack of directional preference within the numerics, supporting and strengthening the conclusion of global instability. Third, despite differing sources of initial perturbation, the exponential side force growth rate is effectively identical in both cases, albeit with opposite signs. This reinforces that the observed flow asymmetry is governed by flow physics rather than minor numerical asymmetries (or imperfections in the experimental set-up).
4.2. Coherent motions
The discussion so far is limited to force and moments statistics. In this subsection, we identify the flow structures resulting from the asymmetry. In both the symmetric and asymmetric flow states, flow over the forward section of the prolate spheroid is laminar and nearly steady. Figure 4 gives sample instantaneous Q-criterion isosurfaces (
$ {QL^2}/{U_\infty ^2}=250 $
) coloured by normalised velocity magnitude (
$ U/U_\infty $
) representative of the transitional wake before and after the onset of asymmetry at
$\alpha =45^\circ$
. Laminar counter-rotating body vortices form from vortex sheets at the bow of the spheroid, breaking down to turbulence near the stern on the lee side. In the symmetric flow field, the coherent structure of both counter-rotating primary vortices breaks down shortly after detachment from the aft region of the spheroid. In the asymmetric flow field, however, the wake structures skew in the
$z$
direction. Here, we use ‘outboard’ and ‘outward’ to refer to the side and direction opposite the direction of lateral wake skew, i.e. the
$+z$
side in figure 4, while ‘inward’ and ‘inboard’ refer to the side and direction corresponding to the direction of lateral wake skew. When the wake symmetry breaks, the outboard vortex shifts inwards, and the inboard vortex is lifted away from the spheroid surface and pushed to the inboard side. As the wake skews, the outboard vortex is accelerated due to the suction on the outboard lee side. This acceleration results in the abrupt breakdown of the vortex tube at the stern. Conversely, the inboard vortex lifts away from the body and is preserved downstream, entraining the smaller remnant structures of the outboard vortex. The downstream preservation of the inboard vortex is a consequence of the low Reynolds number and does not occur at higher Reynolds numbers, as will be shown in a subsequent section.
Figure 4. Instantaneous Q-criterion isosurfaces (
${QL^2}/{U_\infty ^2}=250$
) coloured by normalised velocity magnitude from the symmetric grid case at
$\alpha =45^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
before and after the onset of mean flow asymmetry.
Mean pressure and skin friction coefficient contours are given in figure 5. Here, the spheroid surface is unwrapped into a two-dimensional plane such that the azimuthal coordinate
$\phi$
is zero at
$z/L=0$
on the pressure side. Note that in the symmetric wake case (figure 5
a), the lee side (
$90^\circ \leqslant \phi \leqslant 270^\circ$
) pressure recovery is gradual, corresponding to early and gradual decomposition of the vortical structures over the spheroid. In the asymmetric flow field, however, we see a distinct difference in the pressure distributions on the inboard and outboard sides (figure 5
b) leading to a large side force in the
$+z$
direction (see figure 3
a). This difference in pressure results primarily from the skewed wake, which leads to a suction region on the outboard side, accelerating the outboard vortex and contributing to its rapid breakdown with the abrupt pressure recovery at the stern (figure 4). Contrastingly, the inward wake skew results in an adverse pressure gradient on the inboard side of the spheroid surface, leading to early detachment of the inboard vortex, and contributing to its downstream preservation.
Figure 5. Mean surface streamlines over contours of mean pressure coefficient (a) before and (b) after the onset of persisting mean asymmetry, and mean skin friction coefficient (c) before and (d) after asymmetry on the surface of the prolate spheroid for
$\alpha =45^\circ$
and and
$\textit{Re}_{\!D}=3.0\times 10^3$
. Mean flow streamlines are seeded arbitrarily to aptly represent flow features. Primary separation lines are solid red, and secondary separation lines are dashed red.
Also featured in figure 5 are the primary and secondary separation lines associated with the symmetric and asymmetric wakes. Here, the separation lines are defined as the curves to which the local mean skin friction vectors asymptotically converge in the downstream direction (Wetzel et al. Reference Wetzel, Simpson and Chesnakas1998). Asymptotic convergence of surface streamlines may also indicate reattachment of the flow, but in such cases, the surface streamlines converge in the upstream direction and diverge in the streamwise direction. The curves in figure 5, highlighted in red, converge in the streamwise direction and are therefore associated with flow separation. These lines indicate the azimuthal location at which the primary and secondary flows separate from the spheroid to create a recirculation region inside which the body vortices form. A region of high skin friction is visible between the secondary separation lines where the counter-rotating body vortices recirculate flow back to the lee-side surface of the spheroid. The axial station
$x/L\approx 0.69$
, where the inboard secondary separation line curls in the
$-\phi$
azimuthal direction, is the same axial station of the maximum sectional side force on the prolate spheroid.
4.3. Scale composition and inter-scale interactions
Despite the low Reynolds number, the forces and moments exhibit strong multi-scale features similar to those at higher Reynolds numbers. In the previous subsection, we examined large-scale coherent motions. Here, we examine the inter-scale interaction in the flow, which manifests as amplitude modulation. First, we identify the scale content of the flow through spectral analysis of the temporal variation of forces and moments. Then we identify the coherent motions in the flow associated with the dominant frequency. Finally, we quantify the amplitude modulation of the small-scale motions by the large-scale motions.
Fluctuating force and moment histories sampled every ten time steps at Strouhal number (
$ \textit{fL}/{U_\infty } $
)
$St_s\approx 3.2\times 10^3$
for the
$\alpha =45^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
case are presented in figure 6 along with the corresponding energy spectra. We see a dominant frequency in the integrated side force at
$St_1=0.434$
. This agrees with the results of Jiang et al. (Reference Jiang, Gallardo, Andersson and Zhang2015). In these spectra, however, we see a distinction between the dominant Strouhal numbers of the forces and moments that are zero and non-zero before and after development of mean asymmetry. Those forces and moments that only reach non-zero values after development of mean asymmetry, e.g. coefficients of side force
$C_s$
, rolling moment
$C_X$
and yawing moment
$C_Y$
, are dominated by the
$St_1$
mode in the asymmetric state. Conversely, the forces and moments that pre-date the mean asymmetry are dominated by
$St_2=0.866$
. This dominant Strouhal number,
$St_2\approx 2St_1$
, is present in the coefficients of drag
$C_D$
, lift
$C_L$
and pitching moment
$C_Z$
, because the local minima and maxima of these values correspond to the moments in time where the side force crosses its mean magnitude, which occurs twice per period of
$St_1$
.
Figure 6. (a) Force and moment coefficient histories from the asymmetric grid at
$\alpha =45^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
, with mean values subtracted, and (b) power spectral density of force and moment histories for
$tU_\infty /L\gt 25$
.
Though prior work, e.g. Jiang et al. (Reference Jiang, Gallardo, Andersson and Zhang2015), has also identified that the dominant Strouhal number is caused by very-low-frequency three-dimensional effects, they did not report the physical mechanism within the flow responsible for the oscillation. Analysis of the temporal variation of the wake structures reveals a cyclic meandering of the inboard vortex tube associated with the frequency
$St_1$
. This meandering propagates upstream along the inboard vortex, resulting in a cyclic perturbation on the surface of the prolate spheroid. Figure 7 provides representative Q-criterion isosurfaces at four instances in time, equally spaced by
$\Delta t\,U_\infty /{L}={1}/({4\,St_1})$
. Here, meandering of the inboard vortex tube forms a perturbation in the rolling vortex sheet of the inboard vortex at the stern. A local temporal maximum occurs in the side force at the moment when the perturbation is closest to the stern (
${1}/({4\,St_1})$
). Upstream propagation of the perturbation results in a side force local minimum at
${3}/({4\,St_1})$
, after which the dissipation of the feature leads to a rise in the side force, and the process repeats with frequency
$St_1$
.
Figure 7. Instantaneous Q-criterion isosurfaces (
${QL^2}/{U_\infty ^2}=250$
) with time history of side force coefficient
$C_S$
with its mean value subtracted for the
$\alpha =45^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
case. Isosurfaces are provided at four instances in the simulation at an increment of
$\Delta {t\,U_\infty }/{L}= {1}/({4\,St_1})$
to visualise the upstream propagation of the vortex sheet perturbation.
Figure 8 gives longitudinal distributions of sectional side force (
$C_s$
) and contours of the difference between instantaneous and mean pressure coefficient (
$\Delta C_{\!p}$
) on the surface of the prolate spheroid at the four instances in time featured in figure 7. The sectional side force is computed as
\begin{equation} C_s=\frac {r}{{1}/{2}\rho U_\infty ^2D}\int ^{2\pi }_{0}\left [\left (p\boldsymbol{n}_c+{\boldsymbol{\tau }}\boldsymbol{\cdot }\boldsymbol{n}_c\right )\boldsymbol{\cdot }\boldsymbol{n}_z\right ] \text{d}\phi, \end{equation}
where
$r$
is the local spheroid radius,
$\boldsymbol{n}_c$
is the spheroid surface unit normal,
$p$
and
$\boldsymbol{\tau }$
are the static pressure and shear stress tensor acting at the wall, and
$\boldsymbol{n}_z$
is the unit normal of the
$z$
axis. At the moment of peak side force (
$1/4\,St_1$
), we see in figure 8(a) that the instantaneous
$C_{\!p}$
exceeds the mean
$C_{\!p}$
in the
$\phi \gt 180$
and
$0.3\lt x/L\lt 0.8$
region. This leads to a distribution of instantaneous sectional side force that exceeds that of the mean over this same region (figure 8
e). As the inboard vortex perturbation moves to its most advanced upstream station (
$3/4\,St_1$
), the high
$\Delta C_{\!p}$
zone on the
$\phi \gt 180$
side is reduced, causing the instantaneous sectional side force to fall below the mean (figures 8
c,g). At the transitional instances
$2/4\,St_1$
and
$4/4\,St_1$
, the instantaneous
$C_s$
distributions (figures 8
f,h) approximately line up with the mean, though some hysteresis effect is visible in the
$\Delta C_{\!p}$
contours (figures 8
b,d).
Figure 8. (a–d) Contours of
$\Delta C_{\!p}$
, the difference between the instantaneous and mean pressure coefficients on the surface of the prolate spheroid. (e–h) Longitudinal distributions of instantaneous and mean sectional side force (
$C_s$
), and the difference between the two. The four instances in time from the
$\alpha =45^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
simulation shown here are separated by
$\Delta {t\,U_\infty }/{L}= {1}/({4\,St_1})$
, and are the same as in figure 7.
Finally, we quantify the amplitude modulation of the high-frequency modes by the low-frequency modes identified above. Amplitude modulation is well studied in wall-bounded flows, where the amplitude of small-scale motions in a turbulent boundary layer is modulated by the large-scale motions. The large-scale motions are typically defined as those responsible for the outer peak of the premultiplied velocity power spectrum, whereas the small-scale motions are responsible for the inner peak (Hutchins & Marusic Reference Hutchins and Marusic2007; Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2009). Here, we assess the amplitude modulation in the present flow. Rather than calculating the amplitude modulation locally in the boundary layer, we calculate the amplitude modulation of the global flow through analysis of the temporal variation of integrated forces and moments. In this analysis, a low-pass filter (LPS) with cut-off frequency
$f_c$
is used to decompose the source signal
$F(t)$
into large-scale
$F_L(t)=\textit{LPS}(F(t))$
and small-scale
$F_S(t)=F(t)-\textit{LPS}(F(t))$
components. Then a Hilbert transformation is used to calculate the envelope of the small-scale component
$env(F_S(t))$
as in Mathis et al. (Reference Mathis, Hutchins and Marusic2009). A low-pass filter is then used to extract the large-scale component
$env_L(F_S(t))=\textit{LPS}(env(F_S(t)))$
of
$env(F_S(t))$
, where
$f_c$
is the same cut-off frequency as used in the decomposition of the original signal. Finally, amplitude modulation strength is computed as
\begin{equation} R_{\textit{AM}}=\frac {\overline {F_L(t)\,env_L(F_S(t))}}{\sigma _{F_L(t)}\,\sigma _{env_L(F_L(t))}}, \end{equation}
where
$\overline {F_L(t)\,env_L(F_S(t))}$
is the mean of the product of the large-scale components
$F_L(t)$
and
$env_L(F_S(t))$
, and
$\sigma _{F_L(t)}\,\sigma _{env_L(F_L(t))}$
is the product of the standard deviations of the large-scale components. This is repeated with two cut-off frequencies,
$f_c=St_1$
and
$f_c=St_2$
.
Figure 9 gives plots of the decomposition of the forces and moments into large- and small-scale components, along with the filtered envelope of the small-scale component, for a sample section of the force and moment histories. Decomposition of
$F(t)$
with
$f_c=St_1$
results in
$F_L(t)$
that does not adequately capture the large-scale motions of
$F(t)$
. Using
$f_c=St_2$
results in
$F_L(t)$
that represents the large-scale motions of
$F(t)$
more appropriately. This is reflected in table 1, where the strength of the amplitude modulation for each force and moment is listed. Here, a positive amplitude modulation strength indicates a positive correlation between the amplitude of the large-scale
$F_L(t)$
and the amplitude of the small-scale
$env_L(F_S(t))$
; i.e. when the amplitude of
$F_L(t)$
is large, the amplitude of
$env_L(F_S(t))$
is also large. A negative strength indicates that while the amplitude of the large-scale
$F_L(t)$
is large, the amplitude of the small-scale
$env_L(F_S(t))$
is small. We see that the strength of the amplitude modulation is robust to changes in
$f_c$
. The modulation is high-strength across all forces and moments for
$f_c=St_2$
. In this case, the modulating signal is the very-low-frequency three-dimensional effects shown in figure 7, while the modulated signal is the temporal variation of forces and moments due to small-scale wake structures and turbulence. The amplitude modulation strength is higher for
$f_c=St_2$
than for
$f_c=St_1$
because both frequencies are dominant for certain forces and moments, e.g. side force, yawing moment and rolling moment for
$St_1$
, and lift, drag and pitching moment for
$St_2$
. Consequently, the large-scale component must contain the contributions of both frequencies to reflect the strength of amplitude modulation present in the flow.
Table 1. Amplitude modulation strength
$R_{\textit{AM}}$
in the temporal variation of integrated forces and moments imparted on the 6 : 1 prolate spheroid at
$\alpha =45^\circ$
.
Figure 9. Sample temporal variation of (a) forces and (b) moments with mean values subtracted at
$\alpha =45^\circ$
, along with corresponding
$F_L(t)$
and
$env_L(F_S(t))$
. Signal decompositions are plotted for two cut-off frequency values,
$f_c=St_1$
and
$f_c=St_2$
.
5. Generation and destruction of flow asymmetry
In this section, we explore a Navier–Stokes-based theory, attempting to clarify the mechanism responsible for the generation and destruction of mean flow asymmetry in the flow. We measure the spatial growth of the terms in (3.5) by integrating over a set of 61 equally spaced axial planes spanning
$-1\leqslant x/L \leqslant 2$
, or
$1L$
upstream and downstream of the bow and stern. Mean quantities are taken over a sample period of the final
$25\,tU_\infty /L$
of the symmetric grid simulation. Fluctuating quantities are calculated from samples of the instantaneous flow field taken every
$0.1\,tU_\infty /L$
for the duration of the sample period.
Figure 10(a) gives the longitudinal distribution of the
$x$
-component of the terms in (3.5). As expressed in (3.1),
$\varPhi _i$
is defined as the mean velocity values in the
$-z$
region subtracted from the mean velocity at conjugate points across the meridional plane in the
$+z$
region. As such,
$\varPhi _i$
and the terms of (3.5) are defined and evaluated only in the
$+z$
region. The terms have units of length over time squared, thus are here normalised by
$L/U_\infty ^2$
. As the time rate of change of mean asymmetry is equal to zero (
$ {\partial \varPhi _i}/{\partial t}=0 $
) over the sample period, the axial distribution of
$\varPhi _x/U_\infty$
is instead included to visualise the spatial distribution of the mean asymmetry. Asymmetry in the mean axial velocity extends upstream of the bow, but reaches a peak over the aft section of the prolate spheroid
$0.5\leqslant x/L \leqslant 1$
before changing sign at the stern where the wake is shed from the body. The budget of the asymmetry features a balance between the pressure production and the mean flow convection. The production term
$\varPhi _{\mathcal{P}_x}=-({1}/{\rho })\,{\partial (P^+-P^- )}/{\partial x}$
reaches peak values over the prolate spheroid, though tends to zero away from the body. Notably, the production distribution behaves in a manner similar to
${\partial \varPhi _x}/{\partial x}$
. Peak production values occur in the same axial locations as peak slopes in the
$\varPhi _x$
curve, and production crosses zero at the same axial stations where the slope of the
$\varPhi _x$
curve changes sign. Beyond the stern in the
$x/L\gt 1$
region, the production returns to zero, and the magnitude of
$\varPhi _x$
effectively remains constant. This reflects that downstream of the body, there is nothing to drive change in the axial pressure gradient.
Figure 10. (a) Longitudinal distributions of terms of the
$x$
-component of (3.5) normalised by
$L/U_\infty ^2$
. (b) Contour lines of mean asymmetry
$\varPhi _x/U_\infty$
superimposed onto filled contours of the magnitude of the
$x$
-component of mean asymmetry production
$|{\varPhi _{\mathcal{P}_x}L}/{U_\infty ^2}|=|({L}/{U_\infty ^2})[(-{1}/{\rho })\,{\partial (P^+-P^-)}/{\partial x}]|$
at the axial station of peak production (
$x/L=0.9$
) for
$\alpha =45^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
.
A contour of the
$x$
-component of mean asymmetry production magnitude is given in figure 10(b) at
$x/L=0.9$
, the axial station of peak production. Contour lines of asymmetry
$\varPhi _x /U_\infty$
are superimposed onto the production contour to illustrate their interdependence. At this axial station, the outboard primary vortex tube, which has shifted inwards and lifted the inboard vortex, has broken down due to the abrupt adverse pressure gradient visible in figure 5. Thus there are strong meridional differences in axial pressure gradient at the locations associated with each primary body vortex, as indicated in figure 10(b).
The magnitudes of the axial integration of the
$x$
-components of viscous destruction
$\nu [{\partial ^2\varPhi _x}/{\partial x^2} + {\partial ^2\varPhi _x}/{\partial y^2} + {\partial ^2\varPhi _x}/{\partial z^2} ]$
and turbulent destruction
$- [{\partial \overline {v_x \varphi _x}}/{\partial x} + {\partial \overline {v_y \varphi _x}}/{\partial y} + {\partial \overline {v_z \varphi _x}}/{\partial z} ]$
are orders of magnitude smaller than that of the production term, suggesting little influence on mean asymmetry. Cross-correlation terms
$-{\partial }/{\partial x} [U^+_xU^-_x+U^+_yU^-_x+U^+_zU^-_x ]$
and
${\partial }/{\partial x} [U^-_xU^+_x+U^-_yU^+_x+U^-_zU^+_x ]$
mirror one another across the zero axis, and the fluctuation cross-correlation terms,
$-({\partial }/{\partial x}) [\overline {u^+_xu^-_x}+\overline {u^+_yu^-_x}+\overline {u^+_zu^-_x} ]$
and
$({\partial }/{\partial x}) [\overline {u^-_xu^+_x}+\overline {u^-_yu^+_x}+\overline {u^-_zu^+_x} ]$
, are small, suggesting negligible non-local interaction. The dominance of the production term suggests that the asymmetry is an inviscid process originating from discrepancies in pressure gradients across the meridional plane, thus the overall balance reduces to
\begin{equation} \underbrace {\frac {\partial V_x\varPhi _x}{\partial x} + \frac {\partial V_y\varPhi _x}{\partial y} + \frac {\partial V_z\varPhi _x}{\partial z}}_{\textit {convection}} \approx \underbrace {-\frac {1}{\rho }\frac {\partial \big (P^+-P^-\big )}{\partial x}}_{\textit {production}}. \end{equation}
To summarise, the asymmetry begins with an initial perturbation – numerical asymmetry of the grid itself, or asymmetry associated with the unsteady turbulent wake – which results in a meridional difference in pressure gradient large enough to exceed the weak strength of the viscous and turbulent destruction; when tripped, the asymmetry grows exponentially as discussed in § 4.1; once in the stable asymmetric state, the difference in pressure gradient across the meridional plane sustains the asymmetry.
Finally, figure 11 gives
$\varPhi _x/U_\infty =\pm 0.175$
mean asymmetry isosurfaces and
$\varPhi _{\mathcal{P}_x}L/U_\infty ^2=\pm 0.875$
mean asymmetry production isosurfaces coloured by
$U/U_\infty$
for the symmetric grid case. Here, the flow field is coarsely reconstructed with the 61 equally-spaced planes at which the terms of (3.5) were evaluated. Isosurfaces representing positive isovalues are banded in red, while those of negative isovalues are banded in blue. As in figure 10, we see the mean asymmetry and mean asymmetry production structures in regions associated with the primary body vortices. Due to the abrupt breakdown of the outboard vortex in seen in figure 4, the asymmetry associated with this vortex is strongest near the stern. The inboard vortex is preserved in the wake and therefore produces strong meridional asymmetry well downstream of the body.
Figure 11. Isosurfaces of mean asymmetry
$\varPhi _x/U_\infty =\pm 0.175$
and of mean asymmetry production
${\varPhi _{\mathcal{P}_x}L}/{U_\infty ^2}=({L}/{U_\infty ^2})[(-{1}/{\rho })\,{\partial (P^+-P^-)}/{\partial x}]=\pm 0.875$
coloured by
$U/U_\infty$
for
$\alpha =45^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
. Red and blue bands indicate the positive and negative isovalues, respectively.
6. Critical angle of attack and Reynolds number effects
6.1. At
$\textit{Re}_{\!D}=3.0\times 10^3$
As has been established in previous works (Kumar & Mahesh Reference Kumar and Mahesh2018; Morse & Mahesh Reference Morse and Mahesh2021, Reference Morse and Mahesh2023), mean flow over axisymmetric bodies is symmetric with respect to the meridional plane at
$\alpha =0^\circ$
. In the present work, we verify the presence of mean flow asymmetry at
$\alpha =45^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
. From the results in previous sections, we have the following expectations. First, our analysis in § 5 shows that the primary mechanism responsible for generating mean flow asymmetry is the pressure gradient, an inviscid process. As such, the conclusions drawn from our mean flow asymmetry equation are expected to remain valid at higher Reynolds numbers. Second, we observe in § 4 that the flow remains symmetric for an extended period before being perturbed by wake turbulence, demonstrating that turbulence plays a critical role in introducing perturbations that initiate the onset of mean flow asymmetry. Since wake turbulence intensifies with Reynolds number, we expect the critical angle of attack for asymmetry onset to decrease at higher Reynolds numbers.
In this subsection, we investigate the impact of wake turbulence on the onset of mean flow asymmetry. To do this, we first identify the critical angle of attack at which mean flow asymmetry occurs at
$\textit{Re}_{\!D}=3.0\times 10^3$
. Then we perform simulations at a higher Reynolds number (
$\textit{Re}_{\!D}=6.0\times 10^3$
) to demonstrate the reduction in critical
$\alpha$
. At
$\textit{Re}_{\!D}=3.0\times 10^3$
, we perform simulations at
$\alpha =0^\circ ,20^\circ ,30^\circ ,40^\circ ,42^\circ$
. In figure 12, we see force coefficient histories for each
$\textit{Re}_{\!D}=3.0\times 10^3$
case. Though side force unsteadiness develops in the
$\alpha \leqslant 40^\circ$
cases, oscillations are centred about the zero axis. Only the
$\alpha =42^\circ$
and
$\alpha =45^\circ$
cases develop persisting non-zero side forces. Thus the critical angle of attack at which mean flow asymmetry develops occurs in the range
$40^\circ \lt \alpha \lt 42^\circ$
for
$\textit{Re}_{\!D}=3.0\times 10^3$
investigated here. Besides the critical angle of attack, it is also interesting to note that the side force does not feature clear periodic oscillations in the
$\alpha =42^\circ$
case as in the
$\alpha =45^\circ$
case (where the dominant oscillation at
$\alpha =45^\circ$
is associated with the wake meandering at frequency
$St_1$
as described in § 4.3). We also remark that at
$\alpha =42^\circ$
, any rise in drag coefficient with the onset of asymmetry is indistinguishable. This is because the side force that develops in this case is an order of magnitude smaller than at
$\alpha =45^\circ$
, and the change in drag is smaller than the standard deviation of the drag force history.
Figure 12. Force histories for (a) drag and (b) side force coefficients acting on the 6 : 1 prolate spheroid with varying angles of attack at
$\textit{Re}_{\!D}=3.0\times 10^3$
.
This is the first time the critical angle of attack is measured for the 6 : 1 prolate spheroid, but others have measured this quantity for different geometries. For example, Lamont (Reference Lamont1982) observed critical angles of attack between
$\alpha \approx 20^\circ$
and
$\alpha \approx 30^\circ$
for an ogive cylinder at Reynolds numbers in the range
$2.0\times 10^5\leqslant \textit{Re}_{\!D}\leqslant 4.0\times 10^6$
. The author observed that while Reynolds number affected the magnitude of the measured mean side force in their experiments, it was not evident that the critical angle of attack changed significantly between
$\textit{Re}_{\!D}=2.0\times 10^5$
, where separation was laminar, and
$4.0\times 10^6$
, where separation was fully turbulent. The results in Lamont (Reference Lamont1982) readily suggest that Reynolds number does not play a significant role determining the angle of attack at which mean flow asymmetry develops; however, the large gaps in
$\alpha$
investigated in the experiments leave open the possibility of small variations in critical
$\alpha$
due to increased wake turbulence.
6.2. At
$\textit{Re}_{\!D}=6.0\times 10^3$
The Reynolds number
$\textit{Re}_{\!D}=3.0\times 10^3$
is within a regime where small increments in Reynolds number result in significant changes in flow behaviour. This is demonstrated by the results of Jiang, Gallardo & Andersson (Reference Jiang, Gallardo and Andersson2014) of a 6 : 1 prolate spheroid at
$\textit{Re}_{\!D}=1000$
and
$\alpha =45^\circ$
. They found the near-field flow to be entirely steady and laminar, with transition to turbulence beginning only downstream in the intermediate wake. In this subsection, we present results at
$\textit{Re}_{\!D}=6.0\times 10^3$
, and we will compare these results to those at
$\textit{Re}_{\!D}=3.0\times 10^3$
.
At
$\textit{Re}_{\!D}=6.0\times 10^3$
, we perform LES at the same angles of attack that bound the critical
$\alpha$
at the lower Reynolds number,
$\alpha =40^\circ$
and
$\alpha =42^\circ$
. Through this means, we may demonstrate any change in the critical
$\alpha$
greater than
$1^\circ$
. Figure 13 gives the force histories from these simulations. The side force
$C_S$
and yawing moment
$C_Y$
coefficients clearly indicate the development of mean flow asymmetry as they diverge from zero and enter a new state of stationarity at large non-zero values. The present result suggests that the critical
$\alpha$
at
$\textit{Re}_{\!D}=6.0\times 10^3$
is lower than
$40^\circ$
– indeed, the critical angle of attack reduces as the Reynolds number increases. As with other simulations in the present work, we see the side forces at both angles of attack develop with opposite sign, and notably, the
$\alpha =42^\circ$
case at
$\textit{Re}_{\!D}=6.0\times 10^3$
developed asymmetry with a sign opposite that of the same angle of attack at the lower Reynolds number (figure 12). This again confirms that we may attribute the asymmetry to flow physics rather than to a directional bias of the numerics.
Figure 13. Histories of force coefficients, i.e. lift
$C_L$
, drag
$C_D$
and side
$C_S$
, and moment coefficients, i.e. rolling
$C_X$
, yawing
$C_Y$
and pitching
$C_Z$
for (a)
$\alpha =40^\circ$
and (b)
$\alpha =42^\circ$
cases at
$\textit{Re}_{\!D}=6.0\times 10^3$
.
Isosurfaces of the Q-criterion are given in figures 14 and 15 for the
$\alpha =40^\circ$
and
$\alpha =42^\circ$
cases at both Reynolds numbers. Several significant changes in flow behaviour between the two Reynolds numbers are immediately obvious. First, the decomposition of the primary body vortices is markedly more abrupt at
$Re=6.0\times 10^3$
. Second, neither primary vortex retains coherent structure beyond the stern at the higher Reynolds number. Finally, in the
$\alpha =42^\circ$
and
$\textit{Re}_{\!D}=6.0\times 10^3$
case, the inboard vortex begins to break down at
$x/L\approx 0.3$
as it is lifted away from the body by the inward-shifting outboard vortex. This breakdown occurs at a more advanced upstream location than observed in the other cases, where breakdown occurs at the mid-body. We associate the marked change in turbulent wake behaviour with the shift in critical angle of attack at
$Re=6.0\times 10^3$
. The wake turbulence introduces stronger perturbations that exceed the viscous damping effects that maintain symmetry at lower angles of attack.
Figure 14. Instantaneous Q-criterion isosurfaces (
${QL^2}/{U_\infty ^2}=250$
) coloured by normalised velocity magnitude from the
$\alpha =40^\circ$
cases at
$\textit{Re}_{\!D}=3.0\times 10^3$
and
$\textit{Re}_{\!D}=6.0\times 10^3$
.
Figure 15. Instantaneous Q-criterion isosurfaces (
${QL^2}/{U_\infty ^2}=250$
) coloured by normalised velocity magnitude from the
$\alpha =42^\circ$
cases at
$\textit{Re}_{\!D}=3.0\times 10^3$
and
$\textit{Re}_{\!D}=6.0\times 10^3$
.
In figure 16, we show surface streamlines superimposed on skin friction coefficient contours for the instantaneous flow fields of the
$\alpha =40^\circ$
and
$\alpha =42^\circ$
cases at both Reynolds numbers. The surface streamlines are arbitrarily spaced to best represent the surface flow. At the lower Reynolds number, the surface flow appears effectively symmetric at both angles of attack, even in the
$\alpha =42^\circ$
case where a small non-zero side force develops (figure 12
b). At
$\textit{Re}_{\!D}=6.0\times 10^3$
, however, we see pronounced meridional asymmetry in the surface flow. The inboard (
$\phi \gt 180^\circ$
side for
$\alpha =40^\circ$
, and
$\phi \lt 180^\circ$
for
$\alpha =42^\circ$
) secondary separation lines terminate at axial stations far more advanced than their counterparts on the outboard side. This advanced axial termination of the inboard secondary separation lines seen in figures 16(b) and 16(d) is associated with the primary inboard vortex lifting away from the lee side of the spheroid, contributing to its abrupt decomposition in the
$\textit{Re}_{\!D}=6.0\times 10^3$
cases. This is distinct from the
$\alpha =45^\circ$
case in figures 5(b) and 5(d), where the secondary separation lines continue on the lee side of the spheroid until they converge at the stern.
Figure 16. Instantaneous
$C_{\!f}$
contours with surface streamlines for (a)
$\alpha =40^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
, (b)
$\alpha =40^\circ$
and
$\textit{Re}_{\!D}=6.0\times 10^3$
, (c)
$\alpha =42^\circ$
and
$\textit{Re}_{\!D}=3.0\times 10^3$
, and (d)
$\alpha =42^\circ$
and
$\textit{Re}_{\!D}=6.0\times 10^3$
cases. Streamlines are seeded arbitrarily to aptly represent flow features. Primary separation lines are solid red, and secondary separation lines are dashed red.
7. Conclusions
This paper presents new data and analysis tools, offering fresh insights into the wake asymmetry behind a body of revolution.
Direct numerical simulations are conducted at
$\textit{Re}_{\!D} = 3.0 \times 10^3$
over a range of angles of attack. The critical angle of attack for the onset of mean flow asymmetry is identified to be between
$40^\circ$
and
$42^\circ$
. In cases where asymmetry eventually develops, the flow initially remains symmetric for an extended period before being perturbed by wake turbulence. Since turbulence becomes more vigorous at higher Reynolds numbers, this observation suggests a lower critical angle of attack at higher Reynolds numbers – an expectation confirmed by our wall-resolved LES at
$\textit{Re}_{\!D} = 6.0 \times 10^3$
.
Particular attention is given to the case at
$\alpha = 45^\circ$
and
$\textit{Re}_{\!D} = 3.0 \times 10^3$
, where the wake asymmetry is well developed and the flow exhibits rich unsteady dynamics. For this case, the temporal signals of integrated forces and moments reveal a dominant Strouhal number
$St_1 = 0.434$
. The physical mechanism responsible for this low-frequency variation in side force is traced to the upstream propagation of perturbations associated with wake meandering. Amplitude modulation analysis of the force and moment signals further indicates positive amplitude modulation in drag, and negative amplitude modulation in the rolling moment.
To investigate the mechanism behind the mean flow asymmetry, we derive a new measure of asymmetry by subtracting the Navier–Stokes-based transport equations for the mean velocity field across the meridional plane. Analysis of the resulting asymmetry transport equation reveals that the viscous and turbulent destruction terms have negligible influence on the sustained asymmetry. The primary production of asymmetry is associated with differences in pressure gradients across the meridional plane, leading to the conclusion that the onset and persistence of the asymmetry are driven by inviscid processes. Once established, the asymmetric mean state is maintained by the pressure-gradient-driven production term, while the destruction terms remain small.
Acknowledgements
X.I.A.Y. acknowledges ONR contract N000142012315.
Declaration of interests
The authors report no conflict of interest.
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