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arxiv: 1907.01173 · v2 · pith:LFZGJNYRnew · submitted 2019-07-02 · ⚛️ physics.comp-ph · physics.flu-dyn

Numerical modelling of shock-bubble interactions using a pressure-based algorithm without Riemann solvers

Fabian Denner , Berend van Wachem This is my paper

Pith reviewed 2026-05-25 10:51 UTC · model grok-4.3

classification ⚛️ physics.comp-ph physics.flu-dyn
keywords shock-bubble interactionpressure-based algorithmcompressible interfacial flowmesh resolutiongas-liquid interfacenumerical modellingshock wave
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The pith

A pressure-based algorithm without Riemann solvers predicts shock-bubble interactions with mesh resolution mattering critically only in liquid-gas cases.

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

The paper tests a fully-coupled pressure-based algorithm without Riemann solvers on shock waves hitting one- and two-dimensional bubbles. It reports that mesh resolution and advection schemes exert only minor influence on bubble shape, position and dominant waves when the bubble is gas in gas. When the bubble is gas in liquid, however, mesh resolution controls the predicted shape, position, post-shock evolution and the pressure and temperature fields. This distinction is examined because shock-bubble interactions occur in many engineering flows and the algorithm offers a potential simplification over classical Riemann-solver methods.

Core claim

The fully-coupled pressure-based algorithm without Riemann solvers models the interaction of shock waves with bubbles such that, for a gas bubble in gas, mesh resolution and advection schemes have only minor influence on bubble shape, position and the behaviour of dominant shock waves and rarefaction fans, whereas for a gas bubble in liquid the mesh resolution has a critical influence on shape, position, post-shock evolution, pressure and temperature distribution.

What carries the argument

The fully-coupled pressure-based algorithm without Riemann solvers applied to compressible interfacial flows.

If this is right

  • Lower mesh resolutions suffice for reliable predictions of bubble dynamics in gas-gas shock interactions.
  • High mesh resolution is required to obtain consistent post-shock pressure, temperature and bubble evolution in liquid-gas cases.
  • The algorithm supplies an alternative to density-based Riemann-solver methods for these interfacial flows.
  • Advection-scheme choice has limited impact on the dominant flow features in the gas-gas configuration.

Where Pith is reading between the lines

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

  • The same mesh-resolution sensitivity may appear in other liquid-gas compressible problems simulated with pressure-based methods.
  • Direct comparison with laboratory shock-tube experiments on gas bubbles in liquids could test whether the sensitivity reflects physical behaviour.
  • Extension of the study to three-dimensional geometries would indicate whether additional resolution demands arise beyond the two-dimensional results.

Load-bearing premise

The numerical implementation of the fully-coupled pressure-based algorithm contains no coding or discretization errors that would create the reported difference in mesh sensitivity between the gas-gas and liquid-gas cases.

What would settle it

Running the liquid-gas cases at successively doubled mesh resolutions and comparing the resulting bubble shape, position and pressure fields against independent experimental data or established high-resolution reference solutions would show whether the predictions converge to a unique result.

Figures

Figures reproduced from arXiv: 1907.01173 by Berend van Wachem, Fabian Denner.

Figure 1
Figure 1. Figure 1: Profiles of the velocity u, pressure ∆p = p − pII and temperature T of the interaction of a shock wave with Ms = 1.1 with a one-dimensional helium-bubble in air on meshes with different mesh spacings ∆x at time t = 6.5 × 10−4 s. −50 0 50 100 150 200 250 1.1 1.2 1.3 1.4 1.5 1.6 1.7 u [m/s] x [m] 400 ∆x 2000 ∆x 10000 ∆x (a) Velocity u 0 200 400 600 800 1000 1.1 1.2 1.3 1.4 1.5 1.6 1.7 ρ [kg/m 3 ] x [m] (b) D… view at source ↗
Figure 2
Figure 2. Figure 2: Profiles of the velocity u, density ρ and temperature T of the interaction of a shock wave with Ms = 1.1 with a one-dimensional air-bubble in water on meshes with different mesh spacings ∆x at time t = 4.0 × 10−4 s. Air is taken to have a heat capacity ratio of γ0,Air = 1.4 and a specific gas constant of R0,Air = 288.0 J kg−1 K −1 , and helium is assumed to have a heat capacity ratio of γ0,He = 1.648 and a… view at source ↗
Figure 3
Figure 3. Figure 3: Profiles of the pressure ∆p = p − pII of the interaction of a shock wave with Ms = 1.1 with a one-dimensional air-bubble in water on meshes with different mesh spacings ∆x at (a) time t = 4.0 × 10−4 s and (b) time t = 6.5 × 10−4 s. The theoretical Riemann solution is shown as a reference. uI , pI , TI uII, pII, TII 170 mm 220 mm 445 mm 44.5 mm us 50 mm Inlet Outlet Wall Symmetry x y [PITH_FULL_IMAGE:figur… view at source ↗
Figure 4
Figure 4. Figure 4: Schematic illustration of the computational setup of the two-dimensional R22 bubble in air interacting with a shock wave [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Contours of the density gradient (1 − 0.75ψ)|∇ρ| (upper half) and the pressure p (lower half) of the two-dimensional shock-bubble interaction of the R22 bubble in air on a Cartesian mesh with different mesh resolutions ∆x at τ = t aR22,II/d0 = 0.68, using the Minmod scheme. (a) ∆x = d0/200 (b) ∆x = d0/300 (c) ∆x = d0/500 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Contours of the density gradient (1 − 0.75ψ)|∇ρ| (upper half) and the pressure p (lower half) of the two-dimensional shock-bubble interaction of the R22 bubble in air on a Cartesian mesh with different mesh resolutions ∆x at τ = t aR22,II/d0 = 1.15, using the Minmod scheme. 1 2 3 4 5 6 7 0.1 0.15 0.2 0.25 0.3 0.35 ρ [kg/m 3 ] x [m] 200 ∆x 300 ∆x 500 ∆x (a) τ = 0.68 1 2 3 4 5 6 0.1 0.15 0.2 0.25 0.3 0.35 ρ … view at source ↗
Figure 7
Figure 7. Figure 7: Profiles of the density ρ along the x-axis at y = 0.005 m of the two-dimensional shock-bubble interaction of the R22 bubble in air on Cartesian meshes with different mesh spacings ∆x at different dimensionless times τ = t aR22,II/d0, using the Minmod scheme. resolution increases. As mentioned in the introduction, this is to be expected, yet a coherent and sufficiently accurate description of the magnitude … view at source ↗
Figure 8
Figure 8. Figure 8: Contours of the density gradient (1 − 0.75ψ)|∇ρ| (upper half) and the pressure p (lower half) of the two-dimensional shock-bubble interaction of the R22 bubble in air on a Cartesian mesh with ∆x = d0/500 at τ = t aR22,II/d0 = 1.15, using the first-order upwind scheme, the Minmod scheme and Superbee scheme. for the three considered mesh resolutions. 1 2 3 4 5 6 7 0.1 0.15 0.2 0.25 0.3 0.35 ρ [kg/m 3 ] x [m]… view at source ↗
Figure 9
Figure 9. Figure 9: Profiles of the density ρ along the x-axis at y = 0.005 m of the two-dimensional shock-bubble interaction of the R22 bubble in air on Cartesian meshes with mesh spacing ∆x = d0/500 at different dimensionless times τ = t aR22,II/d0, using the first-order upwind scheme, the Minmod scheme and the Superbee scheme. Considering different TVD differencing schemes for the discretisation of advected variables (see … view at source ↗
Figure 10
Figure 10. Figure 10: Schematic illustration of the computational setup of the air bubble in water interacting with a shock wave with Mach [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Contours of the pressure p (upper half) and the temperature T (lower half) of the two-dimensional shock-bubble interaction of the air bubble in water on a Cartesian mesh with ∆x = d0/200 at different times t. Both the pressure scale and the temperature scale are logarithmic. γ0,Air = 1.4, a pressure constant of Π0,Air = 0 Pa and a specific gas constant of R0,Air = 288.0 J kg−1 K −1 . The applied computati… view at source ↗
Figure 12
Figure 12. Figure 12: Contours of the pressure p (upper half) and the temperature T (lower half) of the two-dimensional shock-bubble interaction of the air bubble in water on a Cartesian mesh with ∆x = d0/400 at different times t. Both the pressure scale and the temperature scale are logarithmic. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Contours of the pressure p (upper half) and the temperature T (lower half) of the two-dimensional shock-bubble interaction of the air bubble in water on a Cartesian mesh with ∆x = d0/600 at different times t. Both the pressure scale and the temperature scale are logarithmic. Figures 11-13 show the contours of the pressure and temperature distribution of the water-air system at different times t for equidi… view at source ↗
Figure 14
Figure 14. Figure 14: Profiles of pressure p, density ρ and temperature T and along the x-axis at y = 6 × 10−4 m of the two-dimensional shock-bubble interaction of the air bubble in water on Cartesian meshes with different mesh spacings ∆x at time t = 3.0 × 10−6 s. 5. Conclusions In the current paper, the numerical modelling of shock-bubble interactions using the pressure-based algorithm proposed by Denner et al. (2018), where… view at source ↗
Figure 15
Figure 15. Figure 15: Profiles of pressure p, density ρ and temperature T and along the x-axis at y = 6 × 10−4 m of the two-dimensional shock-bubble interaction of the air bubble in water on Cartesian meshes with different mesh spacings ∆x at time t = 3.8 × 10−6 s. 0 0.5 1 1.5 2 2.5 3 3.5 0 4 8 12 16 20 24 p [GPa] x [mm] 200 ∆x 400 ∆x 600 ∆x (a) Pressure p 0 5000 10000 15000 20000 13 14 15 16 17 18 ρ [kg/m 3 ] x [mm] (b) Densi… view at source ↗
Figure 16
Figure 16. Figure 16: Profiles of pressure p, density ρ and temperature T and along the x-axis at y = 6 × 10−4 m of the two-dimensional shock-bubble interaction of the air bubble in water on Cartesian meshes with different mesh spacings ∆x at time t = 4.5 × 10−6 s. course, an accurate prediction of shock-bubble interactions is, therefore, a prerequisite for numerical methods to be utilised in research and process development p… view at source ↗
read the original abstract

The interaction of a shock wave with a bubble features in many engineering and emerging technological applications, and has been used widely to test new numerical methods for compressible interfacial flows. Recently, density-based algorithms with pressure-correction methods as well as fully-coupled pressure-based algorithms have been established as promising alternatives to classical density-based algorithms based on Riemann solvers. The current paper investigates the predictive accuracy of fully-coupled pressure-based algorithms without Riemann solvers in modelling the interaction of shock waves with one-dimensional and two-dimensional bubbles in gas-gas and liquid-gas flows. For a gas bubble suspended in another gas, the mesh resolution and the applied advection schemes are found to only have a minor influence on the bubble shape and position, as well as the behaviour of the dominant shock waves and rarefaction fans. For a gas bubble suspended in a liquid, however, the mesh resolution has a critical influence on the shape, the position and the post-shock evolution of the bubble, as well as the pressure and temperature distribution.

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 / 1 minor

Summary. The paper investigates the predictive accuracy of a fully-coupled pressure-based algorithm without Riemann solvers for shock-bubble interactions in one- and two-dimensional gas-gas and liquid-gas flows. It claims that mesh resolution and advection schemes exert only minor influence on bubble shape, position, and dominant wave behavior in gas-gas cases, but that mesh resolution exerts critical influence on these quantities plus post-shock pressure and temperature fields in liquid-gas cases.

Significance. If the mesh-sensitivity contrast is confirmed to be physical rather than numerical, the work would establish practical resolution guidelines for pressure-based methods in high-density-ratio compressible flows and position the algorithm as a viable alternative to Riemann-solver approaches. The manuscript supplies no quantitative error norms, convergence rates, or cross-validation against experiments or established Riemann-solver codes, which limits the strength of this contribution.

major comments (2)
  1. [Abstract and §4] Abstract and §4 (results): the assertion that mesh resolution has 'critical influence' on bubble shape, position, and post-shock evolution in liquid-gas flows is stated qualitatively only; no L1 or L2 error norms, grid-convergence rates, or quantitative comparison of interface position versus resolution are reported, leaving the magnitude of the sensitivity unquantified.
  2. [§3 and §4.2] §3 (numerical method) and §4.2 (liquid-gas results): the interface advection and pressure-velocity coupling at density ratios ~1000 are not shown to be free of resolution-dependent truncation or artificial-viscosity errors; without a side-by-side comparison against a Riemann-solver reference solution on the same meshes, the observed sensitivity cannot be distinguished from method-specific artifacts.
minor comments (1)
  1. [§4 figures] Figure captions in §4 could explicitly state the exact mesh resolutions (cells per bubble diameter) used for each panel to allow direct replication of the reported sensitivity trends.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript examining the predictive accuracy of a fully-coupled pressure-based algorithm for shock-bubble interactions. We address each major comment below, indicating revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (results): the assertion that mesh resolution has 'critical influence' on bubble shape, position, and post-shock evolution in liquid-gas flows is stated qualitatively only; no L1 or L2 error norms, grid-convergence rates, or quantitative comparison of interface position versus resolution are reported, leaving the magnitude of the sensitivity unquantified.

    Authors: We agree the description relies on qualitative visual comparisons of results at successive resolutions (e.g., 100–400 cells per diameter). The critical influence is shown by large changes in bubble deformation, translation, and post-shock pressure/temperature fields at density ratios ~1000. In revision we will add quantitative support by reporting the variation in bubble centroid position, interface area, and peak post-shock pressure as functions of mesh size, together with simple L2 differences between successive grids. revision: yes

  2. Referee: [§3 and §4.2] §3 (numerical method) and §4.2 (liquid-gas results): the interface advection and pressure-velocity coupling at density ratios ~1000 are not shown to be free of resolution-dependent truncation or artificial-viscosity errors; without a side-by-side comparison against a Riemann-solver reference solution on the same meshes, the observed sensitivity cannot be distinguished from method-specific artifacts.

    Authors: The study characterises the behaviour of the pressure-based method itself rather than benchmarking it against Riemann-solver codes. The pronounced sensitivity at high density ratios is consistent with the physical stiffness introduced by the density jump, which amplifies small interface errors regardless of solver type. A direct mesh-matched comparison to a Riemann-solver implementation lies outside the present scope and would require separate code development and validation. revision: no

Circularity Check

0 steps flagged

No circularity: direct numerical outcomes from standard discretization

full rationale

The paper reports mesh-resolution and advection-scheme effects on shock-bubble interactions obtained from direct numerical simulations with a pressure-based algorithm. No load-bearing step reduces a claimed prediction or result to a fitted parameter, self-citation, or definitional equivalence; the gas-gas versus liquid-gas contrast is presented as an observed computational outcome compared against expected physical wave propagation, without any self-referential closure or renaming of known results. The work is therefore self-contained against external physical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the study relies on standard assumptions of continuum mechanics and numerical stability in compressible flow solvers; no free parameters, ad-hoc axioms, or invented entities are introduced or fitted.

axioms (1)
  • domain assumption The governing equations for compressible interfacial flows are correctly discretized by the pressure-based algorithm
    Invoked when claiming predictive accuracy of the method.

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