12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

FLOW PAST AN INCLINED SPHEROID IN HOMOGENEOUS AND
STRATIFIED ENVIRONMENTS

Sheel Nidhan
Mechanical and Aerospace Engineering

Jose L. Ortiz-Tarin
Mechanical and Aerospace Engineering

University of California San Diego
CA 92093
snidhan@ucsd.edu

University of California San Diego
CA 92093
jlortiztarin@gmail.com

Sutanu Sarkar
Mechanical and Aerospace Engineering
University of California San Diego
CA 92093
sarkar@ucsd.edu

ABSTRACT

ies have also been performed at higher Re with tripped BL by
Fu et al. (1994) and Chesnakas & Simpson (1994). Jiménez
et al. (2010) and Ashok et al. (2015a) studied the high Re wakeof a DARPA SUBOFF at 𝛼 = 0◦ and in pitch configurations,
respectively. While Jiménez et al. (2010) focused on the selfsimilarity and scalings in the near to intermediate wake, Ashok
et al. (2015a) characterized the wake asymmetries induced dueto the pitch configuration.
Numerical simulations have also been used to study slender body flows. Constantinescu et al. (2002) and Wikström
et al. (2004) conducted RANS and LES studies of flow pasta 6:1 prolate spheroid at Re = 4.2 × 106 and 𝛼 = 10◦ , 20◦ .
Tezuka & Suzuki (2006) carried out a three-dimensional stability analysis study for the flow past a 4:1 spheroid for varying
𝛼. For 𝛼 ≠ 0◦ , as Re was increased, they found transition froma symmetric to an asymmetric flow configuration in the longitudinal center-plane. The flow asymmetry at nonzero angle
of incidence was later studied in more detail using DNS by
Jiang et al. (2015). In the last few years, the wake of a DARPASUBOFF has also been studied through LES by Posa & Balaras(2016); Kumar & Mahesh (2018) at Re ∼ 𝑂 (105 ). Ortiz-Tarin
et al. (2021) was the first study of an unstratified slender bodyflow that probed far wake statistics, extending to 𝑥/𝐷 = 80.
The LES study by Ortiz-Tarin et al. (2019) is the firstto take background stratification into account for the study ofslender body flows. Their high-resolution LES study was conducted for a 4:1 spheroid at zero angle of attack, Re = 104 and
Fr = ∞, 3, 1 and 0.5. They analyzed the laminar BL evolution,
force distribution, and the near- and far-field characteristics ofthe steady lee waves. In the current work, we build upon theirwork studying the effect of varying Fr on the flow past a 6:1
spheroid placed at a moderate angle of incidence (𝛼 = 10◦ ). We
analyze the effect of stratification on: (i) the variation of 𝐶 𝑝 ,𝐶 𝑓 , (ii) the forces on the body, and (iii) the flow separation. We
also present a brief analysis of forces on the body at 𝛼 = 0◦ forcomparison with 𝛼 = 10◦ cases. To the best of our knowledge,
this is the first study exploring the flow characteristics of an
inclined slender body flow in stratified environments. In future
work, we plan to extend this study to the analysis of vorticity
and wake dynamics in these flows.

Large eddy simulations (LES) are performed to study theflow past a 6:1 prolate spheroid placed at an angle of incidence of 𝛼 = 10◦ . The diameter-based Reynolds number(Re = 𝑈∞ 𝐷/𝜈) is set to a value of 5 × 103 and four values ofdiameter-based Froude numbers (Fr = 𝑈∞ /𝑁 𝐷) are analyzed:
Fr = ∞, 6, 1.9, and 1. Visualizations of coefficient of pressure (𝐶 𝑝 ) and friction (𝐶 𝑓 ) contours reveal asymmetry in the
Fr = ∞ and 6 flows while, at Fr = 1 and Fr = 1.9, the flow overbody does not have any visible asymmetry. This finding isfurther corroborated through the analysis of force coefficientson the body. The changes in the pressure coefficients (𝐶 𝑝 ),friction coefficients (𝐶 𝑓 ), and drag coefficients (𝐶𝑑 ) with the
Froude number are described in detail for 𝛼 = 10◦ . We alsopresent the analyses of forces on the body at 𝛼 = 0◦ angle ofincidence for comparison with the 𝛼 = 10◦ cases.

INTRODUCTION
Despite their widespread presence in applications, slenderbody flows have received significantly less attention comparedto the flows past bluff bodies. There are some particularities
that make the study of slender body flows especially challenging. In a numerical simulation, resolving the boundary layer(BL) of a long body is computationally more expensive thanresolving the BL of a blunt body. Additionally, in experimental
studies, slender bodies can take up a significant portion of themeasurement section and, since they normally generate thinwakes, they are potentially harder to probe and measure.
The first experimental study of flow past a slender bodydates back to Chevray (1968) who investigated the wake ofa 6:1 prolate spheroid at Re = 4.5 × 105 . The wake measurements spanned a streamwise distance of 𝑥/𝐷 = 18. Han & Patel(1979) conducted an experimental study of flow past a 4.3:1
spheroid at different angles of incidence (𝛼) at Re ≈ 2 × 104 .
They primarily focused on the flow separation pattern, identifying two regimes as 𝛼 was changed: (i) closed or bubble
separation and (ii) open or free-vortex type separation. Wanget al. (1990) extended the study of Han & Patel (1979) toprolate spheroids with aspect ratios of 2:1, 3:1, and 4:1 for awide range of 𝛼 in the range [0◦ , 90◦ ]. Experimental stud-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
NUMERICAL METHODOLOGY
𝛼 = 0◦ , Fr
Fr = ∞Fr = 6
Fr = 1.9
Fr = 1

The numerical solver used for these simulations have been
extensively validated in the past for body-inclusive simulationsof stratified wakes (Ortiz-Tarin et al., 2019; Chongsiripinyo& Sarkar, 2019; Ortiz-Tarin et al., 2021; Nidhan et al., 2019,2020, 2022). For a detailed description of the solver, boundary
conditions and immersed boundary method (IBM) approach,
we refer the interested readers to Chongsiripinyo & Sarkar(2019).
In the present work, Re = 5 × 103 and four different Fr =∞, 6, 1.9, and 1 are simulated for two angles of incidence 𝛼 =0◦ and 10◦ . For the stratified cases at 𝛼 = 10◦ , radial andstreamwise domains span 0 ≤ 𝑟/𝐷 ≤ 53 and −30 ≤ 𝑥/𝐷 ≤ 50,
respectively. A large radial extent together with a sponge layeron the boundaries weaken the IGWs before they hit the end of
the computational domain and hence control the amplitude ofspurious reflected waves. Numbers of grid points in different
directions are as follows: 𝑁𝑟 = 1000 in the radial direction,𝑁 𝜃 = 128 in the azimuthal direction, and 𝑁 𝑥 = 3584 in thestreamwise direction. For the unstratified wake at 𝛼 = 10◦ ,radial and streamwise domains span 0 ≤ 𝑟/𝐷 ≤ 21 and −11 ≤𝑥/𝐷 ≤ 47 while 𝑁𝑟 = 718, 𝑁 𝜃 = 256, and 𝑁 𝑥 = 2560. At 𝛼 = 0◦ ,we simulate the wakes till a downstream distance of 𝑥/𝐷 = 30
and up to 𝑟/𝐷 = 17 and 53 for unstratified and stratified cases,
respectively. The grid point distribution for 𝛼 = 0◦ cases are
as follows: 𝑁𝑟 = 1000 and 910 for stratified and unstratified
cases, respectively, 𝑁 𝜃 = 128, and 𝑁 𝑥 = 3584.

𝑓

𝐶𝑑
0.24
0.26
0.32
0.52

𝐶𝑑
0.22
0.23
0.25
0.29

𝐶𝑑𝑝
0.02
0.03
0.07
0.23

Δ𝐶𝑑 = 𝐶𝑑 − 𝐶𝑑 (∞)
0
0.02
0.08
0.28

Table 1: Drag coefficients (𝐶𝑑 ) and corresponding pres𝑓sure (𝐶𝑑𝑝 ) and friction contributions (𝐶𝑑 ) for 𝛼 = 0◦ at
different Fr.

thinner BL over the body (not shown here for brevity) compared
to Fr = 6 and ∞. In the horizontal plane, for Fr = 1 (compared
to Fr = ∞), BL thickens between 𝑥/𝐷 ≈ −2 to 0 and gets thinner
beyond 𝑥/𝐷 ≈ 1, explaining the trend of 𝐶 𝑓 in the horizontal
plane for Fr = 1.
2 𝐴) for different Fr at
Table 1 presents 𝐶𝑑 = 𝐹𝑑 /(0.5𝜌𝑜 𝑈∞
𝛼 = 0◦ . Here, 𝐴 = 𝜋𝐷 2 /4. There is a monotonic increase in𝑓𝑝
𝐶𝑑 , 𝐶𝑑 , and 𝐶𝑑 with increasing stratification levels. Friction
contributes more to the drag than pressure (except at Fr = 1),
as expected for a slender body flow. However, compared toflow past a 4:1 spheroid (Ortiz-Tarin et al., 2019), we find thatthe effect of stratification on the overall drag, quantified by
Δ𝐶𝑑 = 𝐶𝑑 − 𝐶𝑑 (∞), is weaker in the 6:1 spheroid. Between
Fr = ∞ and Fr = 1.9, 𝐶𝑑 changes by ≈ 33% in the present casewhile there was a 100% increase in the 4:1 spheroid. This
smaller increase in 𝐶𝑑 for 6:1 spheroid is primarily due to a𝑝smaller increase in 𝐶𝑑 at Fr 𝑐 = 1.9 (compared to Fr = ∞),unlike in the 4:1 prolate spheroid. It is only when Fr = 1 that𝑝
we see a sharp jump in 𝐶𝑑 leading to a ≈ 100% increase in 𝐶𝑑 ,consistent with the differences we observe in 𝐶 𝑝 curves (figure1a) between Fr = 1 and Fr ≥ 1.9 cases.
It is a well-established observation that the amplitude ofsteady lee-waves decreases with increasing value of Fr in the
Fr ≥ 1 regime for a variety of wake generators, including 6:1
prolate spheroid (Bonneton et al., 1993; Meunier et al., 2018).
Specifically, Meunier et al. (2018) found that the lee-wave
amplitude for a 6:1 spheroid (based on 𝜕𝑤/𝜕𝑧) decayed as
Fr−2 . On the other hand, it should be noted that the definitionof critical Fr 𝑐 given by Ortiz-Tarin et al. (2019) is based purely
on kinematic considerations, i.e., by equating half-wavelengthof the lee-wave to the length of the body. As a result, Fr 𝑐 =𝐿/𝜋𝐷 increases linearly with the aspect ratio. Thus, for the
6:1 spheroid, we see a weaker effect of stratification on thedrag at its critical value of Fr 𝑐 = 6/𝜋 = 1.9 compared to the
4:1 spheroid for which Fr 𝑐 = 4/𝜋 = 1.27 is lower. That the lee
wave field in the Fr = 1.9 case is weaker than in the Fr = 1 caseis also confirmed also by quantification of the pressure field,
not shown here for brevity.

BODY FORCES AT 𝛼 = 0◦
Figure 1(a) and (b) show the variation of the pressure co2 and skin-friction coefficientefficient 𝐶 𝑝 = (𝑃 − 𝑃∞ )/0.5𝜌𝑈∞
2 , respectively, for different values of
𝐶 𝑓 = Re0.5 |𝜏𝑥 |/0.5𝜌𝑈∞
Fr at 𝛼 = 0◦ . In the reported 𝐶 𝑓 , the raw skin friction coefficient is multiplied by Re0.5 to obtain an 𝑂 (1) value. The
𝜃 = 0◦ and 90◦ labels correspond to the horizontal 𝑥 − 𝑦 andvertical 𝑥 − 𝑧 plane, respectively. An averaging time windowof 30𝐷/𝑈∞ is used to obtain these results.
For Fr ≥ 1.9 cases (figure 1(a)), the drop of pressure andits recovery primarily occur towards the beginning and the endof the body, respectively. For −2 / 𝑥/𝐷 / 1, 𝐶 𝑝 remains
approximately constant for Fr ≥ 1.9. On the other hand, 𝐶 𝑝 in
Fr = 1 case shows a monotonic decay till 𝑥/𝐷 ≈ 2 and a slightrecovery for 𝑥/𝐷 ' 2, indicating strong effect of buoyancy
on the flow over the body. For Fr ≥ 1.9, 𝐶 𝑝 variations are
qualitatively very similar. The differences between Fr = ∞ andthe critical Fr 𝑐 curves are less pronounced in the 6:1 spheroid(Fr 𝑐 = 1.9) than what was observed in the 4:1 spheroid (Fr 𝑐 ≈1) as reported by Ortiz-Tarin et al. (2019). Anisotropy between
horizontal and vertical plane 𝐶 𝑝 curves appears at Fr = 1.9
and increases slightly at Fr = 1. Pressure visualizations (notshown here) confirm that the low pressure region in the tail ofthe spheroid (in the vertical plane) at Fr = 1 is imposed by thesteady lee-wave field, indicating a strong influence of buoyancy
on the flow at this value of Fr.
Figure 1(b) shows the variation of 𝐶 𝑓 for different Fr at
𝛼 = 0◦ . Similar to the behavior of 𝐶 𝑝 , the variation of 𝐶 𝑓 for
Fr = 6 and ∞ are very similar. The flow separates at 𝑥/𝐷 = 2.34
for the unstratified flow (marked by 𝐶 𝑓 → 0). This value isin excellent agreement with the result of Patel & Kim (1994).
Fr = 1.9 shows elevated and suppressed levels of 𝐶 𝑓 in verticaland horizontal planes, respectively, compared to Fr = 6 and ∞.
At Fr = 1, 𝐶 𝑓 in the vertical plane further increases compared
to Fr = 1.9. Moreover, in the horizontal plane as well, Fr = 1
shows higher 𝐶 𝑓 than Fr = ∞ and 6 for 𝑥/𝐷 ' 0.5. Increased
𝐶 𝑓 in the vertical plane for Fr = 1 and 1.9 is a consequence of

BODY FORCES AT 𝛼 = 10◦
Introducing even a moderate angle of incidence, namely
𝛼 = 10◦ , significantly changes the characteristics of flow on thebody, reflected by significantly different trends of 𝐶 𝑝 , 𝐶 𝑓 andforce coefficients in 𝛼 = 10◦ flows compared to their 𝛼 = 0◦
counterparts. In what follows, we discuss the trends of abovementioned quantities when 𝛼 = 10◦ and Fr = ∞, 6, 1.9, and1. For 𝛼 = 10◦ , a time averaging window of approximately
50𝐷/𝑈∞ is used.

Coefficient of Pressure 𝐶 𝑝
Figure 2(a-h) show the pressure contours on the leesideand windside of the body. We also present the variation of 𝐶 𝑝
along 𝑦 = 0 line on the leeside and windside in figure 2(i,j),2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1: Variation of (a) pressure coefficient 𝐶 𝑝 and (b) skin-friction coefficient 𝐶 𝑓 for different Fr at zero angle of attack(𝛼 = 0◦ ). The 𝜃 = 0◦ and 90◦ curves correspond to variations on the surface in the horizontal (𝜃 = 0◦ ) plane and the vertical(𝜃 = 90◦ ) planes on the spheroid. Plotted value of 𝐶 𝑓 is Re0.5 times the friction coefficient.

respectively. 𝐶 𝑝 obtained from potential flow solution is alsopresented for comparison in figure 2(i,j). On both sides, theagreement between the potential solutions and LES simulations
is excellent till 𝑥/𝐷 ≈ 1. Beyond 𝑥/𝐷 ≈ 1 LES simulations
deviate from potential solutions, presumably due to the threedimensionality of BL evolution and effects of flow separation.
Pressure drop on both sides primarily happens near to the noseof the body and in the ascending order of Fr, i.e., the pressure
drop in the nose for Fr = 1 > 1.9 > 6 > ∞ (figure 2(i)). It isalso worth noting that the Fr = 6 and ∞ cases show asymmetry
in the leeside pressure contours about the 𝑦 = 0 plane as shownin figure 2(a,c).
On the leeside, for Fr ≥ 1.9 cases, pressure recovers continuously, albeit slowly, after the initial drop in the nose (figure2i). On the contrary, pressure in the Fr = 1 case, which shows
the lowest initial drop (from the stagnation value at the nose)of all cases in the fore of the body (𝑥/𝐷 < 0) drops below thatof the remaining cases for 𝑥/𝐷 > 0. This is evident both fromcontours (figure 2g) and 𝐶 𝑝 plots (figure 2i). Mean pressure
contours in the vicinity of the spheroid (not shown here forbrevity), reveal a strong low-pressure region on the entiretyof the leeside. This low-pressure region is imposed by thesteady lee wave field at Fr = 1. The differences between the
unstratified case and strongly stratified cases (Fr = 1.9 and 1)
are even more pronounced on the windside compared to theleeside of the spheroid. Beyond 𝑥/𝐷 ≈ 2, 𝐶 𝑝 of Fr = 1.9 and
1 fall significantly below those of Fr = ∞ and 6 (figure 2j),followed by a sharp recovery towards the end. The contours infigure 2(b,d,f,h) confirm this trend of higher and lower pressure
at the head and tail, respectively, for the Fr = 1 and 1.9 cases
compared to Fr ≥ 6.

the lee of the body (figure 3(a,c)). There is a central region
near 𝑦 = 0 which remains attached until nearly the tail. In the
Fr = 1.9 case, separation occurs from the sides as well as thecentral region around the 𝑦 = 0 plane (figure 3e) while in the
Fr = 1 case, separation primarily happens in the central region(figure 3g) and not at the sides. Thus, it can be inferred that thestratification level strongly influences the flow separation evenat the moderate non-zero angles of attack of this study.
In the vertical-center plane (𝑦 = 0), 𝐶 𝑓 varies similarly for
all Fr on both sides (figure 3(i,j)) till 𝑥/𝐷 ≈ 0. On the leeside(figure 3i) and for 𝑥/𝐷 ≥ 0, the Fr = ∞ and 6 cases show higher
𝐶 𝑓 than the strongly stratified cases of Fr = 1.9 and 1. On thewindside, beyond 𝑥/𝐷 ≈ 1, Fr ≤ 1.9 cases show elevated 𝐶 𝑓 ascompared to Fr ≥ 6 (figure 3j). This region of elevated surface
shear in the Fr = 1.9 and 1 flows coincides with the region ofsteep pressure drop observed at 𝑥/𝐷 > 1 (figure 2j).

Force Coefficients
Figure 4(a,b,c) present the force coefficients (𝐶𝑖 =2 𝐴) at 𝛼 = 10◦ , decomposed between pressure and𝐹𝑖 /0.5𝜌𝑈∞friction contributions. Here 𝐶 𝑥 , 𝐶 𝑦 and 𝐶 𝑧 correspond to drag(𝐶𝑑 ), lateral force, and lift (𝐶𝑙 ) on the body, respectively. We
first discuss 𝐶𝑑 and 𝐶𝑙 and follow by noting the unusual characteristics of 𝐶 𝑦 for 𝛼 = 10◦ . Figure 4(d) also shows 𝐶𝑑 for
𝛼 = 0◦ for reference. In 𝛼 = 0◦ cases, 𝐶𝑙 ≈ 0 and 𝐶 𝑦 ≈ 0 at all
Fr indicating no asymmetry in the flow over body.
𝑓𝑝
For all Fr, 𝐶𝑑 , 𝐶𝑑 and 𝐶𝑑 (figure 4a) at 𝛼 = 10◦ increase
relative to the zero degree angle of incidence ((figure 4d).
Similar to 𝛼 = 0◦ , there is a weak monotonic increase in 𝐶𝑑
till Fr = 1.9 and a large jump thereafter at Fr = 1. This jump𝑝
primarily comes from an approximately 100% increase in 𝐶𝑑
from Fr = 1.9 to Fr = 1. Figure 2(g,h) show that the pressure
contours at Fr = 1 are quite different from those of Fr ≥ 1.9. The
tail and nose on the leeside and windside, respectively, are at alower and higher pressure (compared to Fr ≥ 1.9 cases) which𝑝leads to an enhanced 𝐶𝑑 . For 𝐶𝑙 , primary contributor is thepressure rather than shear, as expected for a moderate angle ofincidence. It is interesting to note that 𝐶𝑙 decreases till Fr = 1.9
and then increases significantly at Fr = 1 case, resulting from𝑝
an increase in 𝐶𝑙 . The reason for this increase is the large
difference between the leeside and windside pressure in Fr = 1
flow as shown in figure 2(g,h).
Figure 4(c) shows that 𝐶 𝑦 ≠ 0 for Fr = ∞ and 6 at 𝛼 =10◦ . For Fr = ∞, 𝐶 𝑦 ≈ −0.05 and for Fr = 6, 𝐶 𝑦 ≈ 0.05.
This value is approximately 12% of the streamwise drag force

Coefficient of Friction 𝐶 𝑓
Figure 3(a-h) show the contours of scaled shear stress
Re0.5 |𝜏𝑥 | on the leeside (left) and windside (right) for all cases
at 𝛼 = 10◦ . In figure 3(i,j), we present the variation of 𝐶 𝑓 on thespheroid surface and in the 𝑦 = 0 plane on both sides. Similar
to the pressure contours, a distinct asymmetry is present on
the leeside of Fr = ∞ and 6 flows as shown figure 3(a,c), to be
discussed in more detail in the next subsection.
The shear stress contours show that flow separation primarily happens on the leeside at 𝛼 = 10◦ across all Fr. The
region of separated flow can be identified by the region of|𝜏𝑥 | → 0, marked in green in figure 3 contours. For Fr = ∞and 6, flow separates primarily from the two lateral sides on3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 2: Pressure contours on the leeside (a,c,e) and windside (b,d,f) of the spheroid shown for all simulated Fr at 𝛼 = 10◦ .

Dashed lines in (a) and (c) correspond to 𝑦 = 0. Also shown is the variation of 𝐶 𝑝 (i, j) on the leeside and windside of the
body surface in the 𝑦 = 0 plane. Potential solution for 𝐶 𝑝 is also shown in dashed line, obtained from Piquet & Queutey(1992).

in both cases. Intuitively, 𝐶 𝑦 should be equal to zero due
to reflectional symmetry in the configuration about the 𝑦 = 0
plane. Non-zero 𝐶 𝑦 implies lateral asymmetry in the flow on
the body. Pressure and friction contours in figure 2(a,c) and3(a,c), respectively, show that this is indeed the case and theasymmetry originates on the leeside of the body at 𝑥/𝐷 ' 1(the 𝑦 = 0 intersection of the body surface is shown to better
identify this lateral asymmetry). No asymmetry is present on
the windside flow over the spheroid. When the value of Fr
is decreased to 1.9, 𝐶 𝑦 → 0. Further decrease to Fr = 1 also
results in 𝐶 𝑦 = 0. This indicates that the flow asymmetry is
suppressed as the stratification is increased. Figure 2(e,g) andfigure 3(e,g) also confirm that for Fr = 1.9 and 1, no asymmetryis visually evident on the leeside of the spheroid. Hence, twoimportant findings from the analysis of 𝐶 𝑦 are: (i) the weakly
stratified (Fr = 6) and unstratified (Fr = ∞) flow exhibit lateralasymmetry and (ii) this asymmetry is suppressed as the strengthof stratification is increased, i.e., at Fr = 1.9 and 1.

fied case is in accord with previous studies on flow past slenderbodies (Tezuka & Suzuki, 2006; Ashok et al., 2015a,b; Jiang
et al., 2015). It is also interesting to note that the 𝐶 𝑦 of Fr = ∞and Fr = 6 are similar in magnitude but flipped in sign. We
hypothesize that the Fr = ∞ and Fr = 6 flows might be locked in
two different reflectional-symmetry-breaking states, with eachstate being equally probable. There can be a switching between these two states at a very large timescale. The existenceof a long time scale for switching between different reflectionsymmetry-breaking states have been extensively researched inflow past three-dimensional blunt bodies (Grandemange et al.,2013; Rigas et al., 2014; Dalla Longa et al., 2019). Interestingly, Jiang et al. (2015), who also found lateral asymmetry intheir flow (6:1 spheroid at 𝛼 = 45◦ ), did not find a switch evenafter 600𝐷/𝑈∞ . We aim to investigate the characteristics of
the intermediate to far wake, besides looking at near-body flow.
Hence, running the simulations for 𝑇 ∼ 1000𝐷/𝑈∞ would beprohibitively expensive and out of scope of the current work.

Our finding regarding lateral asymmetry in the unstrati-

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 3: Contours of Re0.5 |𝜏𝑥 | (a-h) on the leeside and windside of the spheroid for all Fr at 𝛼 = 10◦ . Dashed lines in (a)and (c) correspond to 𝑦 = 0. Variation of 𝐶 𝑓 (i, j) on the leeside and windside of the body at 𝑦 = 0 plane.
CONCLUSIONS∞, the flow separates from the lateral sides on the lee of the
body while at Fr = 1, the flow separates predominantly near
the vertical center plane 𝑦 = 0. In the future studies, we plan toinvestigate flow separation and topology of shed vortices andtheir potential link to the flow asymmetry in 𝛼 = 10◦ cases aswell as wake dynamics.

Large eddy simulations (LES) are performed to study thecharacteristics of flow over a 6:1 spheroid placed at an angle
of incidence 𝛼 = 10◦ . We present results at Re = 5 × 103 andFr = ∞, 6, 1.9, and 1. Additionally, flows past a 6:1 spheroid
at the same (Re, Fr) combinations but at 𝛼 = 0◦ are simulated
to provide a basis for comparison. We find that the buoyancy effect introduced by stratification strongly modulates thepressure and friction forces on the body. The critical Froudenumber (Ortiz-Tarin et al., 2019) given by Fr 𝑐 = 𝐿/𝜋𝐷 is defined kinematically by equating the body length to the halfwavelength of the generated lee wave. For the 6:1 spheroid, theeffect of stratification on 𝐶 𝑝 , 𝐶 𝑓 , and 𝐶𝑑 is more pronounced
at Fr = 1 than at the critical Froude number Fr 𝑐 ≈ 1.9. For
both 𝛼 = 0◦ and 10◦ cases, 𝐶𝑑 monotonically increases withdecreasing Fr. At 𝛼 = 10◦ , a distinct lateral asymmetry is visible in the 𝐶 𝑝 and 𝐶 𝑓 contours for Fr = 6 and ∞ cases. This
gives rise to a non-zero lateral force whose magnitude is approximately 12% of the streamwise drag at 𝛼 = 10◦ . Further
increasing the stratification kills the asymmetry at Fr = 1.9 and
1. At 𝛼 = 10◦ , we also find that the flow separation over the
body is strongly dependent on the value of Fr. At Fr = 6 andACKNOWLEDGEMENTS
We gratefully acknowledge the support of the Office ofNaval Research grant N0014-20-1-2253.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 4: Force coefficients decomposed between pressure and shear contribution: (a) coefficient of drag 𝐶𝑑 , (b) coefficientof lift 𝐶𝑙 , (c) lateral force 𝐶 𝑦 at 𝛼 = 10◦ , and (d) 𝐶𝑑 at 𝛼 = 0◦ .

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