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arxiv: 2606.17795 · v1 · pith:QU3CHC2Fnew · submitted 2026-06-16 · ⚛️ physics.flu-dyn

Dynamics of a vortex column of supercritical fluid across the pseudo-boiling line

Jordi Poblador-Ibanez , Fazle Hussain This is my paper

Pith reviewed 2026-06-26 22:54 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords supercritical fluidvortex columnpseudo-boiling linevorticity transportviscous mechanismscarbon dioxidelow-Mach flow
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The pith

In a supercritical vortex column, three extra viscous mechanisms beyond diffusion alter vorticity when crossing the pseudo-boiling line, including one that generates reverse vorticity.

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

The paper analyzes an axisymmetric vortex column in supercritical carbon dioxide with an imposed radial thermal layer that forces the fluid across the pseudo-boiling line. Vorticity evolution deviates markedly from the classical Oseen vortex because viscosity and density vary sharply with temperature and pressure near the critical point. Viscous diffusion is joined by a stretching term, an alignment effect between vorticity and property gradients, and a source term arising from swirl interacting with those gradients. The source term can create regions of reverse vorticity that locally increase circulation. This matters for understanding vortex persistence and mixing in systems that use supercritical fluids at conditions where property changes are extreme.

Core claim

Vorticity evolution in the axisymmetric vortex depends strongly on core temperature and ambient pressure near the critical point, differing substantially from the Oseen solution during thermal mixing. Beyond diffusion, three additional viscous mechanisms become significant across the pseudo-boiling line: a vorticity stretching term, alignment of vorticity with viscosity and density gradients, and a vorticity source from the interaction of fluid swirl with those gradients. The third mechanism can generate reverse vorticity, locally increasing circulation and substantially modifying the temporal evolution of the vortex.

What carries the argument

The vorticity transport equation under the low-Mach approximation with strongly varying properties, where the three additional viscous terms (stretching, alignment, and swirl-gradient source) act across the pseudo-boiling line.

If this is right

  • Reverse vorticity appears only when the core is hotter or colder than the surroundings in a way that crosses the pseudo-boiling line.
  • The magnitude of the new vorticity source grows as ambient pressure approaches the critical pressure because property gradients become steeper.
  • Local circulation can increase temporarily instead of monotonically decreasing as in constant-property flows.
  • The temporal history of vortex radius and peak vorticity is altered during the thermal mixing phase.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The identified source term could be added to reduced-order vortex models used in engineering simulations of supercritical heat exchangers or turbines.
  • Similar mechanisms may appear in other axisymmetric swirling flows with strong radial property stratification, such as in combustion chambers.
  • If the axisymmetric assumption breaks down, the new vorticity source might seed three-dimensional instabilities that further modify mixing.

Load-bearing premise

The low-Mach approximation together with the imposed radial thermal layer and axisymmetric assumption capture the dominant physics of property variation without needing full compressibility or three-dimensional effects.

What would settle it

A direct numerical simulation or experiment at the same low Reynolds number and near-critical pressures that shows no reverse vorticity generation and no measurable deviation from the Oseen decay law when the vortex core crosses the pseudo-boiling line.

Figures

Figures reproduced from arXiv: 2606.17795 by Fazle Hussain, Jordi Poblador-Ibanez.

Figure 1
Figure 1. Figure 1: Map of cp for CO2 as a function of pr and Tr . The real-fluid model described in section 2 is used to obtain cp. (a) contour plot of ln(cp/cp o ) showing the saturation line, pseudo-boiling line and critical point. cp o = 846 J/(kgK); and (b) cp distributions at different pr . The (pseudo-)boiling temperatures Tb and Tpb are shown. numbers and the fluid can show features of phase transition, i.e., pseudo-b… view at source ↗
Figure 2
Figure 2. Figure 2: Validation of the RFM against NIST data for sCO [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of ρ/µ and Re for sCO2 at different reduced pressures pr across the pseudo-boiling line between Tmin and Tmax, with Remax = 500. the velocity fluctuation in the flow of O(1 − 10 uτ) and l is a vortex diameter of O(1 − 10 µm), ω0 falls within O(104 − 107 s −1 ). When calculating Γ0 based on a vortex size of l, one obtains Γ0 of O(10−8 − 10−3 m2/s), or Re of O(1 − 104 ). This broad range of Re impl… view at source ↗
Figure 4
Figure 4. Figure 4: Initial profiles of the sCO2 vortex column with Remax = 500 and pr = 2. The dashed vertical line represents rc. (a) ωz and T; (b) ν; and (c) α. an incompressible Oseen vortex at Remax = 1000, for which no numerical destabilization and good agreement with the analytical solution was found. Regardless of the numerical origin of this destabilisation in our simulations, the observation highlights how easily vo… view at source ↗
Figure 5
Figure 5. Figure 5: Grid convergence of relevant terms in the vorticity equation ( [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Collapse of the numerical solution of the vortex with a cold core obtained with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution with t + of the vorticity ωz/ω0 of the hot core (left) and the cold core (right) at different pressures. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a) pr = 2; (b) pr = 1.5; and (c) pr = 1.3. 3.3. Vortex Evolution Based on Core Temperature and Thermodynamic Pressure This section compares the evolution of the vortex column between the cold and the hot vortex cores o… view at source ↗
Figure 8
Figure 8. Figure 8: Evolution with t + of the azimuthal velocity and circulation of the hot core (left) and the cold core (right) at different pressures. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a)-(b) uθ/(ω0rc) and Γ/Γ0 for pr = 2; (c)-(d) uθ/(ω0rc) and Γ/Γ0 for pr = 1.5; and (e)-(f) uθ/(ω0rc) and Γ/Γ0 for pr = 1.3. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution with t + of the vortex core radius rcore at different pressures compared against the incompressible Oseen’s model. (a) hot core; and (b) cold core. across the vortex during the early stages, where its evolution is affected by heat transfer. Therefore, the aim is to compare the evolution of rcore against Oseen’s model in the asymptotic state as the temperature becomes uniform. That is, νTmin is se… view at source ↗
Figure 10
Figure 10. Figure 10: Evolution with t + of relevant variables in the compressible vortex for the hot core (left) and the cold core (right) at different pressures. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a)-(c) ν; (d)-(f) α; and (g)-(i) T. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Evolution with t + of relevant variables in the compressible vortex for the hot core (left) and the cold core (right) at different pressures. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a)-(c) ur ; and (d)-(f) ∇ · u. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Evolution with t + of the terms in the vorticity equation non-dimensionalized with ω0 for the hot core (left) and the cold core (right) at pr = 2. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a) t + = 0; (b) t + = 0.0562; (c) t + = 0.1685; (d) t + = 0.3370; (e) t + = 0.6739; and (f) t + = 1.2917. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Viscous stresses acting on a two-dimensional fluid element in cylindrical coordinates under axisymmetric conditions. [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution with t + of the various viscous terms in (17) non-dimensionalised with ω0 for the hot core (left) and the cold core (right) at pr = 2. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a) t + = 0; (b) t + = 0.0562; (c) t + = 0.1685; (d) t + = 0.3370; (e) t + = 0.6739; and (f) t + = 1.2917. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Evolution with t + of the partial derivatives of viscosity and density found in (17) for the hot core (left) and the cold core (right) at pr = 2. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a) ∂µ ∂r ; (b) ∂ρ ∂r ; and (c) 1 r ∂ ∂r [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: Evolution with t + of the various sub-terms from term (B) given by (18) and non-dimensionalised with ω0 for the hot core (left) and the cold core (right) at pr = 2. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a) t + = 0; (b) t + = 0.1685; and (c) t + = 0.6739. stretching in the hot core configuration and the opposite in the cold core configuration. In contrast, first-order … view at source ↗
Figure 18
Figure 18. Figure 18: Sketch of viscous mechanism represented by term (C1) for redistributing vorticity based on the alignment between the viscosity and [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Evolution with t + of the various sub-terms from term (C) given by (19) and non-dimensionalised with ω0 for the hot core (left) and the cold core (right) at pr = 2. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a) t + = 0; (b) t + = 0.1685; and (c) t + = 0.6739. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Evolution with t + of the various sub-terms from term (D) given by (20) and non-dimensionalized with ω0 for the hot core (left) and the cold core (right) at pr = 2. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a) t + = 0; (b) t + = 0.1685; and (c) t + = 0.6739. Lastly, the physical mechanisms for vorticity generation or decay described by terms (D1) and (D2) based on differe… view at source ↗
Figure 21
Figure 21. Figure 21: Evolution with t + of the various partial derivatives of viscosity and density found in (17) for the hot core (left) and the cold core (right). The dashed vertical lines represent the vortex centre (red) and rc (grey). At pr = 1.5: (a) ∂µ ∂r ; (b) ∂ρ ∂r ; (c) 1 r ∂ ∂r [PITH_FULL_IMAGE:figures/full_fig_p023_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Evolution with t + of the various viscous terms in (17) non-dimensionalised with ω0 for the hot core (left) and the cold core (right) at pr = 1.5. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a) t + = 0; (b) t + = 0.0589; (c) t + = 0.1768; (d) t + = 0.3241; (e) t + = 0.6777; and (f) t + = 1.2964. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Evolution with t + of the various viscous terms in (17) non-dimensionalised with ω0 for the hot core (left) and the cold core (right) at pr = 1.3. The dashed vertical lines represent the vortex centre (red) and rc (grey). (a) t + = 0; (b) t + = 0.0618; (c) t + = 0.1545; (d) t + = 0.3399; (e) t + = 0.6799; and (f) t + = 1.2980. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Distributions at t + = 0.0618 and t + = 0.1545 of ωz/ω0 and ur for the hot core at pr = 1.3. The dotted vertical lines represent the location of the pseudo-boiling line given by Tpb ≈ 316 K. (a) ωz/ω0; and (b) ur . As t + increases, the evolution of the viscous term in the cold core highlights the stronger fluid properties gradients across the pseudo-boiling line. Viscous diffusion dominates the overall t… view at source ↗
Figure 25
Figure 25. Figure 25: Distribution at t + = 0.0618 and t + = 0.1545 of the terms in the vorticity equation given by (17) non-dimensionalised with ω0 for the hot core at pr = 1.3. The dotted vertical lines represent the location of the pseudo-boiling line given by Tpb ≈ 316 K. (a) global distributions; (b) distribution in the gas-like side of the vortex at t + = 0.0618; and (c) distribution in the gas-like side of the vortex at… view at source ↗
Figure 26
Figure 26. Figure 26: Distributions in the gas-like side of the vortex of the various viscous terms in ( [PITH_FULL_IMAGE:figures/full_fig_p027_26.png] view at source ↗
read the original abstract

The evolution of an axisymmetric vortex column in a weakly compressible supercritical fluid is analysed. A thermal layer is imposed to radially stratify the fluid and uncover effects of the large fluid property variations across the pseudo-boiling line. A multi-dimensional flow solver based on a low-Mach approximation is employed. Using supercritical carbon dioxide as the fluid, we examine axisymmetric configurations at low Reynolds number with the vortex core hotter or colder than the surrounding fluid and for different thermodynamic pressures close to the critical pressure. Vorticity evolution depends strongly on the core temperature and ambient pressure, differing substantially from the classical Oseen solution during the thermal mixing process under highly varying fluid properties. Viscous effects dominate the vorticity evolution. Beyond diffusion, three additional viscous mechanisms are identified, which become significant across the pseudo-boiling line: (1) a vorticity stretching term, (2) an alignment of vorticity and viscosity/density gradients, and (3) a vorticity source due to the interaction between the fluid swirl and the viscosity and density gradients. The first two mechanisms alter existing vorticity, while the latter injects new vorticity. In fact, the third mechanism can generate reverse vorticity, locally increasing circulation and substantially modifying the temporal evolution of the vortex.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes the evolution of an axisymmetric vortex column in weakly compressible supercritical CO2 using a low-Mach approximation, with an imposed radial thermal layer to induce property variations across the pseudo-boiling line. At low Reynolds number, vorticity evolution deviates from the classical Oseen vortex; beyond viscous diffusion, three additional mechanisms are reported from the simulations: a vorticity stretching term, alignment of vorticity with viscosity/density gradients, and a source term arising from swirl interacting with those gradients (the last of which can generate reverse vorticity and locally increase circulation).

Significance. If the mechanisms are shown to be robust, the work isolates how strong property gradients in near-critical fluids can produce non-classical vorticity sources and sinks, offering a concrete starting point for modeling vortex mixing and circulation changes in supercritical applications. The low-Mach axisymmetric setup with controlled thermal stratification is a reasonable choice for isolating these variable-property effects.

major comments (2)
  1. [Results / vorticity analysis] The central claim that the three mechanisms 'become significant' and alter circulation rests on simulation outcomes, yet the manuscript supplies no quantitative support such as term-by-term budgets from the vorticity equation, relative magnitudes across the pseudo-boiling line, or circulation histories isolating each contribution. This absence is load-bearing for the reported departure from Oseen behavior.
  2. [Numerical methods] No grid-convergence or discretization-error data are referenced for the reported mechanisms or circulation evolution, leaving open whether the identified terms are numerically converged under the low-Mach axisymmetric discretization.
minor comments (2)
  1. [Governing equations] Explicitly state the low-Mach vorticity equation (including all variable-property viscous terms) so that the origin of the three additional mechanisms can be traced directly to specific terms.
  2. [Parameter space] Clarify the thermodynamic conditions (reduced pressures and core-to-ambient temperature differences) at which each mechanism is stated to dominate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments identify areas where additional quantitative evidence and verification would strengthen the manuscript. We address each point below and will incorporate the requested material in a revised version.

read point-by-point responses

  1. Referee: [Results / vorticity analysis] The central claim that the three mechanisms 'become significant' and alter circulation rests on simulation outcomes, yet the manuscript supplies no quantitative support such as term-by-term budgets from the vorticity equation, relative magnitudes across the pseudo-boiling line, or circulation histories isolating each contribution. This absence is load-bearing for the reported departure from Oseen behavior.

    Authors: We agree that explicit term-by-term budgets and relative-magnitude comparisons would make the significance of the three mechanisms more transparent. The mechanisms were identified by direct inspection of the variable-property vorticity equation under the low-Mach formulation, and the simulations show clear departures from Oseen evolution when property gradients are large. Nevertheless, the manuscript does not presently contain the requested budgets or isolated circulation histories. In the revision we will add these diagnostics, including (i) instantaneous and time-integrated budgets of each term evaluated across the pseudo-boiling line and (ii) circulation histories obtained by successively enabling or disabling the three additional terms. This will provide the quantitative support needed to substantiate the claim. revision: yes

  2. Referee: [Numerical methods] No grid-convergence or discretization-error data are referenced for the reported mechanisms or circulation evolution, leaving open whether the identified terms are numerically converged under the low-Mach axisymmetric discretization.

    Authors: The referee is correct that no grid-convergence or discretization-error estimates are reported. The computations were performed on a sequence of successively refined meshes, but these checks were not documented. In the revised manuscript we will include a dedicated convergence subsection that reports (i) the grid resolutions employed, (ii) the observed order of accuracy for the vorticity field and circulation, and (iii) the sensitivity of the three additional viscous terms to further refinement. This will confirm that the reported mechanisms are numerically converged. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives its central claims from direct numerical simulation of the low-Mach axisymmetric Navier-Stokes equations with variable density and viscosity across the pseudo-boiling line. The three viscous mechanisms are obtained by inspecting the resulting vorticity transport equation terms under the imposed radial thermal stratification; they are not fitted parameters, not renamed inputs, and not justified solely by self-citation. The abstract and setup make clear that the mechanisms emerge as simulation outcomes rather than by construction from the model assumptions themselves. No load-bearing self-citation chain or ansatz smuggling is indicated.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the low-Mach approximation and the axisymmetric idealization being sufficient to isolate the property-gradient effects. No free parameters are explicitly fitted in the abstract; the thermodynamic pressures and Reynolds number are chosen inputs.

axioms (2)
  • domain assumption Low-Mach approximation is valid for the weakly compressible supercritical fluid under the imposed thermal stratification.
    Stated directly in the abstract as the basis for the flow solver.
  • domain assumption Axisymmetric configuration with an imposed radial thermal layer captures the dominant vorticity mechanisms across the pseudo-boiling line.
    The setup is presented without discussion of three-dimensional or compressibility corrections.

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Reference graph

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