This study presents a numerical investigation of vortex rings generated at low Reynolds numbers within radially confined domains. The motivation stems from the lack of previous studies addressing these combined conditions, which are relevant to wall-bounded microjet flows. We solve the governing equations using the entropy-damped artificial-compressibility (EDAC) formulation with compact finite-difference schemes (CS-FD) and a third-order total variation diminishing Runge–Kutta (TVD-RK3) method; moreover, an immersed-boundary method (IBM) resolves curvilinear walls on a Cartesian grid. Validation against high-Reynolds-number confined and low-Reynolds-number unconfined experiments, and against direct numerical simulations, confirms the solver’s accuracy. We examine vortex rings with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L/D_0=4.0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">/</mo> <msub> <mi>D</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>4.0</mn> </mrow> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$150\le Re\le 1000$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>150</mn> <mo>≤</mo> <mi>R</mi> <mi>e</mi> <mo>≤</mo> <mn>1000</mn> </mrow> </math></EquationSource> </InlineEquation> and confinement ratios <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$D_C/D_0=1.75,\,2.0,\,2.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mi>C</mi> </msub> <mo stretchy="false">/</mo> <msub> <mi>D</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.75</mn> <mo>,</mo> <mspace width="0.166667em" /> <mn>2.0</mn> <mo>,</mo> <mspace width="0.166667em" /> <mn>2.5</mn> </mrow> </math></EquationSource> </InlineEquation>, plus an unconfined reference. In all cases, two dissipation zones appear: one within the ring and another near the wall. The dominant site shifts with Re: for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Re\le 250$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>e</mi> <mo>≤</mo> <mn>250</mn> </mrow> </math></EquationSource> </InlineEquation> dissipation concentrates in the ring, whereas for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Re\ge 500$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>e</mi> <mo>≥</mo> <mn>500</mn> </mrow> </math></EquationSource> </InlineEquation> it localizes near the wall. Furthermore, confinement reduces streamwise displacement and circulation by narrowing the effective cross section and intensifying the wall-attached vorticity layer. Decreasing Re suppresses roll-up of this layer, thereby allowing longer travel before viscous losses dominate. Lower Re also stabilizes the ring: no three-dimensionality is observed for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Re\le 500$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>e</mi> <mo>≤</mo> <mn>500</mn> </mrow> </math></EquationSource> </InlineEquation> at any confinement, while at <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$Re=1000$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>e</mi> <mo>=</mo> <mn>1000</mn> </mrow> </math></EquationSource> </InlineEquation> azimuthal undulations arise under tighter confinement (<InlineEquation ID="IEq8"> <EquationSource Format="TEX">$D_C/D_0\le 2.0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mi>C</mi> </msub> <mo stretchy="false">/</mo> <msub> <mi>D</mi> <mn>0</mn> </msub> <mo>≤</mo> <mn>2.0</mn> </mrow> </math></EquationSource> </InlineEquation>) but remain absent for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$D_C/D_0=2.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mi>C</mi> </msub> <mo stretchy="false">/</mo> <msub> <mi>D</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2.5</mn> </mrow> </math></EquationSource> </InlineEquation>, indicating that strong confinement lowers the Reynolds number threshold for breakdown.

Vortex rings at low Reynolds numbers in confined domains

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2026年5月8日

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Diego Silva-Soto,
Martín Salinas-Vázquez,
Carlos Palacios-Morales,
Rafael Chávez-Martínez,
William Vicente

摘要

This study presents a numerical investigation of vortex rings generated at low Reynolds numbers within radially confined domains. The motivation stems from the lack of previous studies addressing these combined conditions, which are relevant to wall-bounded microjet flows. We solve the governing equations using the entropy-damped artificial-compressibility (EDAC) formulation with compact finite-difference schemes (CS-FD) and a third-order total variation diminishing Runge–Kutta (TVD-RK3) method; moreover, an immersed-boundary method (IBM) resolves curvilinear walls on a Cartesian grid. Validation against high-Reynolds-number confined and low-Reynolds-number unconfined experiments, and against direct numerical simulations, confirms the solver’s accuracy. We examine vortex rings with $L/D_0=4.0$ $L / D_{0} = 4.0$ over $150\le Re\le 1000$ $150 \leq R e \leq 1000$ and confinement ratios $D_C/D_0=1.75,\,2.0,\,2.5$ $D_{C} / D_{0} = 1.75, 2.0, 2.5$ , plus an unconfined reference. In all cases, two dissipation zones appear: one within the ring and another near the wall. The dominant site shifts with Re: for $Re\le 250$ $R e \leq 250$ dissipation concentrates in the ring, whereas for $Re\ge 500$ $R e \geq 500$ it localizes near the wall. Furthermore, confinement reduces streamwise displacement and circulation by narrowing the effective cross section and intensifying the wall-attached vorticity layer. Decreasing Re suppresses roll-up of this layer, thereby allowing longer travel before viscous losses dominate. Lower Re also stabilizes the ring: no three-dimensionality is observed for $Re\le 500$ $R e \leq 500$ at any confinement, while at $Re=1000$ $R e = 1000$ azimuthal undulations arise under tighter confinement ( $D_C/D_0\le 2.0$ $D_{C} / D_{0} \leq 2.0$ ) but remain absent for $D_C/D_0=2.5$ $D_{C} / D_{0} = 2.5$ , indicating that strong confinement lowers the Reynolds number threshold for breakdown.