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REVIEW 3 major objections 4 minor 19 references

Vortex topology in the lee of a 6:1 prolate spheroid

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Across 48 large-eddy simulations of a 6:1 prolate spheroid, the lee-side boundary layer always separates symmetrically, and the recirculation is always one of three states: a proto-vortex, a coherent vortex, or a recirculating wake.

desk verdict A genuinely useful 48-case LES taxonomy of leeward flow on the 6:1 spheroid, undermined by an algebraic error in the threshold that defines its headline scalings. read the letter →

arxiv 2507.03187 v1 pith:QUMQQ3XN submitted 2025-07-03 physics.flu-dyn

Marc Plasseraud , Krishnan Mahesh This is my paper · ORCID
classification physics.flu-dyn MSC 76D1076D1776F6576M12 PACS 47.32.C47.27.Ep47.27.nb
keywords prolatespheroidlarge-eddysimulationboundarylayerseparationvortextopologyrecirculatingwakeReynoldsnumbereffectsangleofattacknormalforce
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper uses 48 large-eddy simulations to map the flow behind a 6:1 prolate spheroid over Reynolds numbers from 150,000 to 4 million and angles of attack from 10 to 90 degrees. It claims that in every case the boundary layer separates symmetrically and the leeward recirculation appears in just one of three states: a proto-vortex with no distinct rotation center, a coherent vortex aligned with the body axis, or an incoherent recirculating wake. In the coherent state the separated shear layer rolls into a vortex with a pressure-minimum core, and along the body the recirculation area grows quadratically while the total circulation grows roughly linearly. These axial evolutions, not the total size of the recirculation, set the normal force and pitching moment. The result matters because it reduces a complex, three-dimensional separated flow to a small set of topology types whose growth laws connect directly to vehicle loads.

What carries the argument

The central object is the coherent vortex, identified by the largest closed iso-surface of stagnation pressure; this definition supplies a stable boundary for measuring radius, circulation, and area. The argument's engine is an axial control-volume balance in which the constant-stagnation-pressure vortex boundary acts as a material surface, so the mass added across the separated shear layer per unit length drives the axial mass flow to grow linearly while the separation-length geometry ($L_s \sim x$) makes the area grow quadratically. The load connection comes from a Riabouchinsky-style balance between shear-layer stress and wall suction, and from Crocco's equation expressing the in-plane vortex force as the Lamb-vector integral, which isolates the left/right turbulent-stress asymmetry as the only contributor.

What would settle it

A wind-tunnel test on a 6:1 prolate spheroid at Re=1.5M and 2.5M, alpha=20 and 40 degrees, measuring the azimuthal separation line and the presence or absence of a coherent lee vortex: the paper predicts a sharp boundary-layer-state jump between those Reynolds numbers, so if the separation-line shift or vortex state is identical at both, the central Re-dependence claim is wrong.

Watch

Extended reading notes

Core claim

For all 48 combinations of Reynolds number and angle of attack considered, the boundary layer of the inclined 6:1 prolate spheroid separates symmetrically and its leeward recirculation belongs to one of three categories: proto-vortex, coherent vortex, or recirculating wake. In the coherent-vortex state, the separated shear layer rolls into a three-dimensional vortex aligned with the spheroid axis, whose center is a pressure minimum and which converts azimuthal momentum into axial momentum. Along the axis, the total recirculation area grows as $A_t \sim x^2$ while the total circulation grows roughly as $\Gamma_t \sim x$, with the exponent depending on angle of attack. The paper further shows that the normal force and pitching moment follow from this axial evolution: suction is highest where swirl and vortex stretching are greatest, not where the recirculation is largest, and the overturning moment comes from the first half of the body.

Load-bearing premise

The boundary-layer state at separation is the paper's explanatory variable for topology and Re-dependence, but that state is produced by dynamic-Smagorinsky LES with a trip inherited from prior work, not by resolving or modeling laminar-to-turbulent transition; if the LES misplaces transition, the separation lines, state boundaries, and Re-dependence of loads are all affected.

Editorial extensions

If this is right

  • At fixed Reynolds number and incidence, the lee-side suction and lift are larger on the forward half of the spheroid, which produces the overturning pitching moment despite the body's nose-tail symmetry.
  • Increasing angle of attack raises normal force up to about 70 degrees; beyond that the vortex loses coherence and suction drops.
  • Increasing Reynolds number delays separation and shrinks the recirculation, lowering lift at a given incidence.
  • The quadratic area growth with near-linear circulation growth implies a decreasing mean axial velocity and swirl along the vortex, consistent with the observed decay of vortex strength toward the tail.
  • The three-state classification (proto-vortex, coherent vortex, recirculating wake) is exhaustive for the studied range, so a simulation or experiment at any new point in this Re-alpha range should land in one of these states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same three-state classification and area/circulation scaling may apply to other slender axisymmetric bodies at incidence; a quick test would be to compute $A_t(x)$ and $\Gamma_t(x)$ for a 3:1 ellipsoid or a submarine-like hull.
  • Editorial inference: the finding that suction tracks swirl rather than recirculation size suggests a control-surface design rule: to maximize side force, promote early vortex inception and high swirl, not a large recirculation bubble.
  • Editorial inference: because the paper's boundary-layer state is produced by an inherited trip, a transition-resolving simulation at Re=1.5M-2.5M would either confirm or shift the state boundaries and the inferred critical Reynolds number.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper presents a parametric large-eddy simulation study of flow over a 6:1 prolate spheroid at six Reynolds numbers (Re = 0.15M, 1M, 1.5M, 2M, 3M, 4M) and eight angles of attack (alpha = 10 to 90 deg). The authors classify the leeward recirculation into three states (proto-vortex, coherent vortex, recirculating wake) and describe how the axial evolution of vortex radius, circulation, stagnation pressure, and swirl changes with Reynolds number and incidence. They connect these flow-topology features to the normal force and pitching moment using a control-volume mass balance and a Riabouchinsky-type cavity model, and report power-law scalings A_t ~ x^2 for recirculation area and Gamma_t ~ x^alpha for total circulation. The central claims are that separation is always symmetric in the investigated parameter range, that the coherent-vortex state converts azimuthal momentum into axial momentum, and that maximum suction occurs where swirl is high rather than where recirculation is largest.

Significance. If the quantitative results hold, the paper provides a valuable taxonomy and a large, openly described LES dataset for a canonical separated flow, building on a novel stagnation-pressure-based vortex definition (Plasseraud & Mahesh 2024a). The grid-convergence check (Table 1, 218M vs 470M: 0.11% difference in normal force) and the use of multiple consistent observables (vorticity, stagnation pressure, skin friction, helicity) lend qualitative support to the three-state classification. However, the quantitative layer--quadratic area growth, circulation scaling, and the load-topology link--is built on a threshold derived in Section 3.6 that appears to contain an algebraic error, and the Reynolds-number dependence rests on an assumed transition location that the LES does not resolve. These concerns are central to the paper's main claims, not peripheral. The qualitative taxonomy and the dataset itself are likely to be useful to the community even if the specific scalings require revision.

major comments (3)
  1. [Section 3.6] The derivation of the recirculation threshold P_s ≈ 0.49 P_inf_s is algebraically inconsistent. From P_s = P + 1/2 ρ u^2 and u^2/u_p^2 = 0.992, the exact expression is P_s = 0.008 P + 0.992 P_inf_s. The stated approximation P_s ≈ 0.49 P_inf_s would require P ≈ −63 P_inf_s, which is unphysical. Moreover, the factor 0.992 does not follow from the 0.01 tolerance in Section 3.5; the consistent value would be u^2/u_p^2 = (0.99)^2 = 0.9801. This threshold is load-bearing: it defines A_t (used for the x^2 scaling in Section 5.7 and Figures 39–40), the total circulation Γ_t (Figures 41–42), the primary-vortex boundary in Section 3.7, and the load integral f_y^recirculation(x) = ∫(0.49 P_inf_s − p_wall) dφ in Section 5.9.2. The authors should either correct the derivation or demonstrate that the reported scalings and load conclusions are insensitive to the threshold choice, for example by recomputing A_t, Γ_t, and the load integral for thresholds of 0.4 and 0.6 P_inf_s.
  2. [Section 5.2.2] The attribution of the jump in δ99 and Re_θ between Re = 1M and 1.5M to a laminar-to-turbulent transition at Re ≈ 1.5M conflicts with the cited experimental critical regime of Re = 2–3M (Ahn 1992) and with the authors' own earlier finding that trips do not guarantee a turbulent boundary layer at high incidence (Plasseraud et al. 2023). The LES uses a dynamic-Smagorinsky SGS model with a trip inherited from prior work, so the transition location is not resolved or independently modeled. The separation lines (Figures 6 and 9), the boundary-layer state (Figures 10–11), and the load trends (Figures 48–49) all depend on this assumed transition. Please provide a sensitivity check, such as a comparison against experimental separation locations in the critical regime, or discuss how the main conclusions would change if transition occurs at Re = 2–3M rather than at 1.5M.
  3. [Section 5.7] The control-volume mass balance that leads to the quadratic growth of A_t assumes that the surface S_Ps, an iso-surface of constant stagnation pressure, is a material surface with zero mass flux. In the turbulent, time-averaged flow considered here, the mean stagnation pressure is not constant along mean streamlines because of Reynolds stresses, and the mean mass flux across the iso-surface need not vanish. The authors state that this holds in the inviscid limit, but the LES flow is viscous and turbulent. Please quantify the neglected turbulent flux across S_Ps using the LES data (for example, by computing the actual mean normal velocity on the iso-surface), or provide a justification that the error is small enough to support the derived scaling.
minor comments (4)
  1. [Introduction / Section 5.4] The Introduction states that 'the primary vortex is attached and coherent at low angles of attack (10°, 20°)', but Section 5.4 classifies the 20° case as a proto-vortex without a distinct center of rotation; please reconcile this terminology.
  2. [Section 5.5.2] The Burgers vortex profile is written as (u_x, u_theta, u_r) = (−a r, ..., 2 a z); the radial and axial components appear to be interchanged, since a standard Burgers vortex has u_r = −a r and u_z = 2 a z.
  3. [Section 3.6] The notation for the freestream stagnation pressure appears both as P∞s and P_inf_s in different places; please unify the notation.
  4. [Figure 10] The y-axis label of Figure 10 is ambiguous; please state the units and whether the value is normalized by L.

Circularity Check

2 steps flagged · score 4.0 of 10

The 0.49 P∞_s recirculation threshold is asserted to follow from the δ99 definition but is algebraically unjustified; this threshold then reappears as the 'constant stagnation pressure' shear layer and enters the load/topology force integral, making part of the load–topology link definitional rather than a derived result.

  1. self definitional [Section 3.6 and Section 5.9.2]
    "the separated region is defined as the area where 𝑃𝑠 < 0.49𝑃∞𝑠 and 𝑟 >𝛿99. ... In the proto–vortex and 3D vortex states, the stagnation pressure is constant at 𝑃𝑠 = 0.49𝑃∞𝑠 along the separated shear layer but has higher values in the vicinity of the meridian plane."

    The later statement that stagnation pressure is constant along the separated shear layer restates the threshold used to define the recirculation boundary: the outer edge of the region 𝑃𝑠 < 0.49𝑃∞𝑠 is necessarily the 0.49 isosurface. This definitional constancy is then used in the Riabouchinsky-style load balance, with f_y^recirculation(x) = ∫(0.49𝑃∞𝑠 − 𝑝_wall)d𝜙. Thus the claim that suction from the recirculation scales with its width, and the explanation of where suction peaks, are partly built into the chosen boundary definition rather than being an independent physical finding. The wall pressure distribution itself is independent LES data, so the circularity is partial.

  2. other [Section 3.6]
    "since we have ®𝑢·®𝑢/𝑢2 𝑝 =®𝑢·®𝑢/(2(𝑃∞𝑠 −𝑃)) = 0.992, the following threshold on 𝑃𝑠 could be taken: 𝑃𝑠 =𝑃(1.0− 0.992)+ 0.992𝑃∞𝑠 ≈ 0.49𝑃∞𝑠"

    The algebra does not support the 0.49 value: with the factor 0.992, the expression gives 0.008𝑃 + 0.992𝑃∞𝑠, and equating this to 0.49𝑃∞𝑠 would require 𝑃 ≈ −63𝑃∞𝑠, an unphysical condition not stated or derived. The 0.49 threshold is therefore an arbitrary input presented as consistent with the δ99 boundary-layer definition. Since the same threshold defines the recirculation area A_t, the closed stagnation-pressure vortex boundary, and the load integral in Section 5.9.2, the reported scalings (A_t ~ x^2, Γ_t ~ x) and the load–topology interpretation characterize a chosen iso-surface, not an independently constrained physical boundary.

full rationale

The paper's central parametric results — symmetric separation, the three topologies, separation-line trends, and the computed normal force and pitching moment — come directly from the time-averaged LES flow fields and grid convergence, not from the threshold definition. Those parts are not circular. The circularity is concentrated in the diagnostic framework: Section 3.6 defines the recirculation as the region below 0.49P∞_s, and Section 5.9.2 then treats the 0.49 isosurface as the physical shear-layer boundary, using it in the force decomposition. The threshold's claimed derivation is algebraically invalid, so the subsequent 'constant stagnation pressure' and the resulting Riabouchinsky load relation are partly self-definitional. The self-citations to Plasseraud & Mahesh (2024a,b) supply the vortex-boundary and material-surface method; these are method citations rather than uniqueness theorems, and the material-surface property is a standard Bernoulli/Crocco consequence. The moderate score of 4 reflects that the load–topology link is partially built into the definition, while the bulk of the empirical Reynolds-number/angle-of-attack mapping remains independent LES content.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The quantitative layer rests on the hand-chosen 0.49P_infinitys recirculation threshold (whose stated derivation is inconsistent), the stagnation-pressure vortex boundary from the authors' prior paper, the helicity-root separation detector, and several physical idealizations: a material constant-P_s surface in the control-volume balance, a Riabouchinsky cavity balance, and a tripped boundary layer whose state LES does not explicitly transition. The central taxonomy is grounded in direct field observations and does not depend on new entities; no particles, forces, or conserved quantities are invented.

free parameters (1)
  • Recirculation stagnation-pressure threshold coefficient = 0.49 P_infinitys
    Hand-chosen threshold defining the recirculation area A_t, hence Gamma_t and the x^2/x scaling claims; the section 3.6 derivation is internally inconsistent (P_s = 0.008P + 0.992P_infinitys does not equal 0.49P_infinitys for physical pressures).
assumptions (7)
  • domain assumption Dynamic-Smagorinsky wall-resolved LES on the 470M-cell overset grid produces the correct boundary-layer state at separation at Re up to 4M without resolving or modeling natural transition.
    Invoked throughout sections 2 and 5; section 5.2.2 attributes the delta_99/Re_theta jump to a laminar-to-turbulent change at Re approximately 1.5M, but the simulation neither resolves nor controls transition, and Ahn (1992) places the experimental critical regime at Re = 2M to 3M.
  • domain assumption The trip inherited from Plasseraud et al. (2023) gives a repeatable boundary-layer state and negligible port/starboard asymmetry, even at alpha = 90 deg.
    Used in section 5.2.1 where the alpha = 90 deg separation slope is taken as zero considering a negligible effect of the asymmetry introduced by the trip; trip geometry is not summarized in this paper.
  • domain assumption Helicity-density sign change marks the primary separation line for all 48 cases.
    Section 3.4 adopts the helicity-root criterion (Moffatt and Tsinober 1992), validated by Chesnakas and Simpson only at 10 deg and 20 deg; it is assumed valid up to alpha = 90 deg, where the paper notes skin-friction minima can give false positives.
  • domain assumption The vortex boundary is the largest closed stagnation-pressure iso-surface (Plasseraud and Mahesh 2024a).
    Section 3.7; this definition underlies r_0, Gamma_v, Delta_P_s, swirl, and the Crocco-based zero-vortex-force result in section 5.9.2; no sensitivity analysis is reported.
  • domain assumption Three flow-throughs of time averaging give converged mean statistics in all topologies, including the unsteady recirculating wake.
    Section 3.3 sets the averaging protocol; no statistical convergence check or error bars are given, although the wake state is unsteady.
  • domain assumption In the control-volume balance, the constant-stagnation-pressure surface is material, with zero mean mass flux.
    Section 5.7 applies steady inviscid Bernoulli to a turbulent mean flow to set the integral of u dot n over S(P_s) to zero; turbulence and viscous effects on that surface are neglected. This yields the quadratic area growth.
  • domain assumption The Riabouchinsky force balance f_y(x) = integral over S of tau_s, proportional to L_s times tau_s, applies to the spheroid lee cavity.
    Equation (5.1) in section 5.7 adapts Roshko's cylinder-cavity model; the paper acknowledges constant-velocity and pressure assumptions are not exactly met in the spheroid cavity but retains the balance for the load-topology link.

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Cite this review

Pith. "Pith review of Vortex topology in the lee of a 6:1 prolate spheroid." pith.science (2026). https://pith.science/paper/QUMQQ3XN

@misc{pith2026250703187,
  author       = {Pith},
  title        = {Pith review of: Vortex topology in the lee of a 6:1 prolate spheroid},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QUMQQ3XN}},
  note         = {Machine review of arXiv:2507.03187}
}
read the original abstract

A large scale parametric study of the flow over the prolate spheroid is presented to understand the effect of Reynolds number and angle of attack on the separation, the wake formation and the loads. Large-Eddy Simulation is performed for six Reynolds numbers ranging from Re = 0.15M to Re = 4M and for eight angles of attack ranging from 10 degrees to 90 degrees. For all the cases considered, the boundary layer separates symmetrically and forms a recirculation region. Several distinct flow topologies are observed that can be grouped into three categories: proto-vortex, coherent vortex and recirculating wake. In the proto-vortex state, the recirculation does not have a distinct center of rotation, instead, a two-layer detached flow structure is formed. In the coherent vortex state, the separated shear layer rolls into a three-dimensional vortex that is aligned with the axis of the spheroid. This vortex has a clear center of rotation corresponding to a minimum of pressure and transforms the azimuthal momentum from the separated shear layer into axial momentum. In the recirculating wake regime, the recirculation is incoherent and the primary separation forms a dissipative shear layer that is convected in the direction of the free-stream. This symmetric pair of shear layers bounds a low-momentum recirculating cavity on the leeward side of the spheroid. The properties of these states are not constant, but evolve along the axis of the spheroid and are dictated by the characteristics of the boundary layer at separation. The variation of the flow with Reynolds number and angle of attack is described, and its connection to the loads on the spheroid are discussed.

Figures

Figures reproduced from arXiv: 2507.03187 by the authors.

Figure 1
Figure 1. Schematic of the body–oriented coordinate systems. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. a): Time–averaged skin–friction coefficient [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Time–averaged axial vorticity in a transverse section at [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (46 more)
Figure 4
Figure 4. Figure 4: Skin friction coefficient versus 𝑥/𝐿 and 𝜙 at 𝛼 = 20◦ , 𝑅𝑒 = 4𝑀 for the (a) coarse grid, (b) medium grid and (c) fine grid. leeward with increasing Reynolds number. This results in a closing of the separation angle and a shrinking of the primary separation. This effect…
Figure 5
Figure 5. Figure 5: Time–averaged axial vorticity in a transverse slice at [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Azimuthal location of primary separation versus x/L from a) to h): [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Separation line versus angle of attack at [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Slope of separation line regression versus angle of attack. Each curve is a [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Average azimuth of separation line versus Reynolds number. [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: 𝛿99 averaged over the primary separation versus 𝛼. Each curve is a different Reynolds number [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: 𝑅𝑒 𝜃 averaged over the primary separation versus 𝛼. 𝛼 ∥ 𝜕𝑃𝜕𝑥 ∥ [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: Norm of the pressure gradient averaged over the primary separation versus [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 13
Figure 13. Figure 13: Spheroid surface colored by normalized dot product of skin friction vector and [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]
Figure 14
Figure 14. Figure 14: Averaged ratio of axial to azimuthal velocity at primary separation versus [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]
Figure 15
Figure 15. Figure 15: Time–averaged stagnation pressure in a transverse slice at (a): [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]
Figure 16
Figure 16. Figure 16: Time–averaged stagnation pressure in a transverse slice at [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]
Figure 17
Figure 17. Figure 17: Time–averaged a) axial velocity, b) pressure, c) vorticity magnitude and d) [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]
Figure 18
Figure 18. Figure 18: Time–averaged stagnation pressure in a transverse slice at [PITH_FULL_IMAGE:figures/full_fig_p016_18.png]
Figure 19
Figure 19. Figure 19: Time–averaged a) axial velocity, b) pressure, c) vorticity magnitude and d) [PITH_FULL_IMAGE:figures/full_fig_p017_19.png]
Figure 20
Figure 20. Figure 20: Location of the primary vortex center vs [PITH_FULL_IMAGE:figures/full_fig_p018_20.png]
Figure 21
Figure 21. Figure 21: Time–averaged a) axial, b) tangential, c) radial velocity along a [PITH_FULL_IMAGE:figures/full_fig_p018_21.png]
Figure 22
Figure 22. Figure 22: Time–averaged a) axial, b) tangential, c) radial velocity along circle of radius [PITH_FULL_IMAGE:figures/full_fig_p019_22.png]
Figure 23
Figure 23. Figure 23: Slope at origin of 𝑢 𝜃 versus average axial vorticity in the vortex 𝜔 𝑣 𝑥 . velocity decreases with 𝑥 such that at 𝑥/𝐿 = 0.9, it is a local minimum, with 𝑢𝑥 increasing linearly with 𝑟. The aximuthal velocity increases linearly from the center to around half of the rad…
Figure 24
Figure 24. Figure 24: Vortex stretching 𝜔𝑗 𝜕𝑢𝑥 𝜕𝑥𝑗 vs 𝑥/𝐿 for 𝑅𝑒 = 2𝑀. 𝑥/𝐿 𝑢𝑟=0 [PITH_FULL_IMAGE:figures/full_fig_p020_24.png]
Figure 25
Figure 25. Figure 25: Vortex center velocity vs 𝑥/𝐿 for 𝑅𝑒 = 2𝑀. 𝑥/𝐿 𝑥/𝐿 a) b) 𝑟 ✛ double vortex [PITH_FULL_IMAGE:figures/full_fig_p020_25.png]
Figure 26
Figure 26. Figure 26: Primary vortex radius 𝑟 vs 𝑥/𝐿 for 𝛼 ∈ [20◦ , 70◦ ], 𝑅𝑒 = 4𝑀 (a) and for 𝛼 = 40◦ , 𝑅𝑒 ∈ [0.15𝑀, 4𝑀] (b). 𝑙𝑜𝑔(𝑥) 𝑙𝑜𝑔(𝑟) [PITH_FULL_IMAGE:figures/full_fig_p020_26.png]
Figure 27
Figure 27. Figure 27: Logarithm of the primary vortex radius vs [PITH_FULL_IMAGE:figures/full_fig_p020_27.png]
Figure 28
Figure 28. Figure 28: Visualization of the spheroid at a) 𝛼 = 40◦ and 𝑅𝑒 = 1𝑀; b) 𝛼 = 40◦ and 𝑅𝑒 = 4𝑀. The wall of the spheroid is shaded by instantaneous skin friction coefficient; one half of the flow (𝜙 ∈ [0 ◦ , 180◦ ]) is shown in transverse slices spaced by Δ𝑥 = 0.01 and shaded by tim…
Figure 29
Figure 29. Figure 29: Primary vortex circulation vs 𝑥/𝐿 for 𝛼 ∈ [20◦ , 70◦ ], 𝑅𝑒 = 4𝑀 (a) and for 𝛼 = 40◦ , 𝑅𝑒 ∈ [0.15𝑀, 4𝑀] (b). 𝑥/𝐿 Γ𝑣/Γ𝑡 [PITH_FULL_IMAGE:figures/full_fig_p022_29.png]
Figure 30
Figure 30. Figure 30: Ratio of primary vortex circulation over total circulation vs [PITH_FULL_IMAGE:figures/full_fig_p022_30.png]
Figure 31
Figure 31. Figure 31: Pressure at the center of the primary vortex vs [PITH_FULL_IMAGE:figures/full_fig_p022_31.png]
Figure 32
Figure 32. Figure 32: Stagnation pressure differential in the primary vortex vs [PITH_FULL_IMAGE:figures/full_fig_p023_32.png]
Figure 33
Figure 33. Figure 33: Swirl number vs x for 𝛼 ∈ [20◦ , 60◦ ], 𝑅𝑒 = 2𝑀. Figures 31 and 32 show the pressure in the center of the vortex and the stagnation pressure differential Δ𝑃𝑠 = 𝑃 𝑠𝑎𝑑𝑑𝑙𝑒 𝑠 − 𝑃 0 𝑠 where 𝑃 𝑠𝑎𝑑𝑑𝑙𝑒 𝑠 is the stagnation pressure at the saddle point and 𝑃 0 𝑠 is the stagnati…
Figure 34
Figure 34. Figure 34: Time–averaged a) axial velocity, b) secondary velocity and c) axial vorticity in [PITH_FULL_IMAGE:figures/full_fig_p024_34.png]
Figure 35
Figure 35. Figure 35: Time–averaged stagnation pressure in a transverse slice for [PITH_FULL_IMAGE:figures/full_fig_p025_35.png]
Figure 36
Figure 36. Figure 36: Time–averaged a) axial velocity, b) pressure and c) vorticity magnitude for [PITH_FULL_IMAGE:figures/full_fig_p025_36.png]
Figure 37
Figure 37. Figure 37: Three components of time–averaged velocity in the body frame of reference [PITH_FULL_IMAGE:figures/full_fig_p026_37.png]
Figure 38
Figure 38. Figure 38: Total recirculation area versus 𝑥/𝐿 for 𝛼 ∈ [20◦ , 70◦ ], 𝑅𝑒 = 4𝑀 (a) and for 𝛼 = 40◦ , 𝑅𝑒 ∈ [0.15𝑀, 4𝑀] (b). 𝑥/𝐿 𝐴 𝑡 [PITH_FULL_IMAGE:figures/full_fig_p026_38.png]
Figure 39
Figure 39. Figure 39: Total area of recirculation 𝐴 𝑡 versus x/L (black) at 𝛼 = 30, 𝑅𝑒 = 0.15𝑀, compared with a quadratic regression (red). a regime in which the recirculation is decoherent. The slope of the 𝐴 𝑡 versus 𝑥 curve is increasingly steeper with increasing angle of attack. This i…
Figure 40
Figure 40. Figure 40: Logarithm of the recirculation area vs 𝑙𝑜𝑔(𝑥) for 𝛼 ∈ [20◦ , 60◦ ]. 𝑥/𝐿 𝑥/𝐿 Γ𝑡 a) b) [PITH_FULL_IMAGE:figures/full_fig_p027_40.png]
Figure 41
Figure 41. Figure 41: Recirculation circulation vs 𝑥/𝐿 for a): 𝛼 ∈ [20◦ , 60◦ ], 𝑅𝑒 = 4𝑀; b): 𝛼 = 40◦ , 𝑅𝑒 ∈ [0.15𝑀, 4𝑀]. [20◦ , 60◦ ]. The slope of the curves is close to 2, showing that the quadratic scaling of 𝐴 𝑡 with 𝑥 discussed in figure 39 can be extended to a large portion of the c…
Figure 42
Figure 42. Figure 42: Logarithm of the total circulation vs 𝑙𝑜𝑔(𝑥) for 𝛼 ∈ [20◦ , 60◦ ]. 𝐴 𝑡 (𝑥) 𝐴 𝑡 (𝑥 + Δ𝑥) S𝑖 S𝑃𝑠 ≡ S (𝑃𝑠 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) [PITH_FULL_IMAGE:figures/full_fig_p028_42.png]
Figure 43
Figure 43. Figure 43: Schematic of the control volume of the recirculation area, bounded by [PITH_FULL_IMAGE:figures/full_fig_p028_43.png]
Figure 44
Figure 44. Figure 44: Schematic of the Riabouchinsky model for the prolate spheroid recirculation [PITH_FULL_IMAGE:figures/full_fig_p029_44.png]
Figure 45
Figure 45. Figure 45: Local separation length 𝐿𝑠 vs 𝑥/𝐿. 𝐴 𝑡 ∝ 𝐿𝑠 · (𝑟𝑥 + 𝐿𝑠𝜋/2). Since 𝐿𝑠 ∝ 𝑥, 𝐴 𝑡 ∝ 𝑥 2 in the leading order, as previously observed. This plateau observed at 𝛼 = 90◦ is understood to be similar to the previously commented vortex burst. 5.8. Evolution of the skin friction…
Figure 46
Figure 46. Figure 46: Instantaneous skin–friction coefficient versus [PITH_FULL_IMAGE:figures/full_fig_p031_46.png]
Figure 47
Figure 47. Figure 47: Leeward side of the prolate spheroid showing the time–averaged skin friction [PITH_FULL_IMAGE:figures/full_fig_p032_47.png]
Figure 48
Figure 48. Figure 48: Normal force coefficient 𝐹𝑦 versus angle of attack 𝛼 for all six Reynolds numbers. formed in the lee of the spheroid because constant velocity and pressure implies constant stagnation pressure, which is not the case in this study. In the proto–vortex and 3D vortex sta…
Figure 49
Figure 49. Figure 49: Pitching moment coefficient 𝑀𝑧 versus angle of attack 𝛼 for all six Reynolds numbers. the in–plane vortex–induced force is zero by Crocco’s equation: 𝑓 𝑣𝑜𝑟 𝑡𝑒𝑥 𝑦 (𝑥) = ∫ A ⟨®𝑢 × ®𝜔⟩ = ∫ A (∇𝑃𝑠 +∇𝜏) = 𝑃 0 𝑠 ∫ S 𝑛ˆ + ∫ S 𝜏𝑠 ·𝑛ˆ = ∫ S 𝜏𝑠 ·𝑛ˆ where A is the 2D area of the…

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