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REVIEW 4 major objections 7 minor 1 cited by

The efficient implementation of transport velocity formulation

T0 review · 4 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read SPH clumping fixed by smoothing-length transport correction

desk verdict Useful SPH limiter and broad benchmarks, but Eq. (14)'s coefficient is off by a factor of 2.19 and the central correction is effectively calibrated, not derived. read the letter →

arxiv 2411.13992 v1 pith:TXP4ZJKZ submitted 2024-11-21 physics.flu-dyn

Zhentong Wang , Oskar J. Haidn , Xiangyu Hu This is my paper · ORCID
classification physics.flu-dyn MSC 76M2865M75 PACS 47.11.-j
keywords weakly-compressibleSPHtensileinstabilitytransportvelocitysmoothinglengthvariable-resolutionflowslimiterparticleclumpingvoidregions
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

The paper proposes a modified transport-velocity correction for weakly-compressible SPH that is scaled to the smoothing length instead of the background pressure, making the correction suitable for variable-resolution simulations. To stop overcorrection in low-velocity flows, the correction is multiplied by a limiter that switches on only when a local disorder measure indicates a non-uniform particle distribution. The resulting displacement correction is $\Delta \mathbf{e}_r = -0.2\, \beta\, h_{\min}^2 \sum_j \nabla W_{ij} V_j$ with $\beta$ given by the limiter. Numerical tests on the Taylor-Green vortex, lid-driven cavity, fluid-structure interaction, multi-resolution cylinder flow, and a 3D FDA nozzle show the method avoids overcorrection, prevents clumping and void regions, and matches reference solutions.

What carries the argument

The central object is the limiter $\beta$ of Eq. (15), a switch built on the scalar disorder measure $h^2 |\sum_j \nabla W_{ij} V_j|^2$, where $h=h_{\min}$ in variable-resolution regions. When the measure is above the threshold $5\times10^{-4}$, $\beta = \min(C \cdot \text{measure}, 1)$ with $C=10^3$; otherwise $\beta=0$. This multiplier sits in front of the smoothing-length-scaled transport displacement $\Delta \mathbf{e}_r = -0.2\, \beta\, h_{\min}^2 \sum_j \nabla W_{ij} V_j$, so the same formula handles both uniform and disordered particle clouds without needing a background pressure.

What would settle it

Run a low-velocity variable-resolution simulation, such as a lid-driven cavity with a fine region embedded in a coarse region and maximum velocity below $10^{-3}$, and check whether the limiter stays off in the uniform coarse region while particles still develop clumps or an energy plateau appears. If overcorrection or clumping persists outside the tested parameter range, the threshold and gain are not general.

Watch

Extended reading notes

Core claim

On the paper's own terms, the transport-velocity correction need not depend on a global background pressure. The authors derive, from the background-pressure formulation of Adami et al. under dual-criteria time stepping, that the correction per advection step is a displacement proportional to $h^2 \sum_j \nabla W_{ij} V_j$ with coefficient $-0.2$. They then bound this correction with a limiter $\beta$ that is zero unless $h^2|\sum_j \nabla W_{ij} V_j|^2$ exceeds $5\times10^{-4}$, and saturates at 1 with gain $C=10^3$. The central discovery is that this disorder-triggered, smoothing-length-scaled limiter removes the overcorrection visible in low-velocity Taylor-Green flow while preserving the stabilising effect in disordered or variable-resolution regions.

Load-bearing premise

The limiter's fixed threshold and gain constants, tuned on the test cases, are assumed to separate reliably the particle distributions that need transport correction from those that do not, across all velocities and resolutions.

Editorial extensions

If this is right

  • Low-velocity flows no longer show the energy decay plateau caused by the original background-pressure formulation.
  • The correction scales with smoothing length, so variable-resolution SPH uses the same formula with $h=h_{\min}$ in refined regions.
  • The limiter allows transport velocity to be used with dual-criteria time stepping without the sound-speed-dependent background pressure.
  • Accuracy benchmarks for cavity flow, FSI, and the FDA nozzle match reference data, so the change does not sacrifice solution quality.
  • Two- and three-dimensional tests show the same stabilising effect, so the method generalises beyond planar flows.

Reading between the lines

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

  • The disorder measure could be made dimensionless and adaptive, replacing the fixed $5\times10^{-4}$ threshold with a value tied to kernel shape, which would remove the case-dependent tuning.
  • The limiter is likely sensitive to the Wendland kernel and $h=1.3dp$ used here; kernels with different derivative behaviour may need recalibration of $C$ and the threshold.
  • If the measure is local and cheap, it could also gate particle-shifting or artificial-viscosity corrections in other SPH variants, not just transport velocity.
  • A direct comparison against the particle-shifting technique on the same variable-resolution FSI cases would clarify which strategy better preserves free-surface volume.
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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

4 major / 7 minor

Summary. The manuscript proposes an improved transport-velocity formulation for weakly-compressible SPH. In Section 3, the authors replace the background-pressure-dependent correction of Adami et al. with a smoothing-length-scaled correction, Eq. (16), and introduce a limiter, Eq. (15), that suppresses the correction for well-ordered particle distributions to avoid overcorrection in low-velocity flows. The formulation is claimed to extend the transport-velocity approach to variable-resolution simulations and to prevent particle clumping and void formation without the overcorrection of the original method. Section 4 validates the method on a Taylor-Green vortex, 2D and 3D lid-driven cavity flows, flow-induced vibration of an elastic beam, multi-resolution flow around a cylinder, and the 3D FDA nozzle, with comparisons to references and experiments.

Significance. The practical importance of a transport-velocity correction that scales with smoothing length rather than background pressure is clear: it could make the method applicable to multi-resolution SPH and address the known overcorrection at low velocities. The paper has several strengths: the benchmark suite is broad, including 2D and 3D cases, fluid-structure interaction, variable resolution, and an experimental device; the SPHinXsys code is publicly accessible, which supports reproducibility; and the proposed limiter targets a real limitation of the original formulation. However, the central coefficient in Eq. (16) is not derived from the stated constants in Eq. (14), and the limiter constants are calibrated from numerical cases without a sensitivity study. These issues place the core quantitative claim in need of revision before the result can be relied upon.

major comments (4)
  1. [Section 3, Eq. (14)] The coefficient ξ = -0.2 is not obtained from the stated formulas. Substituting Eq. (13) with α = 7.0 and Eq. (9) with CFL_ad = 0.25 into Eq. (12), and using m_j = ρ_j V_j with ρ_i ≈ ρ_0, gives Δe_r = -α CFL_ad^2 h^2 Σ_j ∇W_ij V_j = -7 × (0.25)^2 h^2 Σ_j ∇W_ij V_j = -0.4375 h^2 Σ_j ∇W_ij V_j. The printed value -0.2 is smaller by a factor of 2.1875. Since all validations in Section 4 use the coefficient -0.2 of Eq. (16), the derivation in Eq. (14) does not support the implemented correction; an independent re-implementation following the printed constants would apply a much larger correction and would not reproduce the reported results. The authors should either correct the derivation with consistent constants or explicitly state that ξ is a calibrated parameter. This issue is load-bearing because Eq. (16) is the central claim of the paper.
  2. [Section 3, Eq. (15)] The limiter constants C = 10^3 and the threshold 5 × 10^-4 are introduced 'according to the numerical cases'. Since the limiter is added specifically to remove the overcorrection observed with the original formulation, part of the reported improvement is built into these fitted parameters. The manuscript provides no sensitivity study with respect to C or the threshold, and no scale analysis showing that these values are robust across resolutions, Reynolds numbers, or variable-resolution ratios. Without such evidence, the general-accuracy claim for low-velocity and multi-resolution flows is not fully supported. At minimum, the authors should report results for nearby values of C and the threshold, or derive the constants from dimensional or error considerations.
  3. [Section 4.2] The statement that the lid-driven cavity results show 'approximate second-order convergence' is not backed by a convergence table or error norms. The figure shows velocity profiles at three resolutions, but no quantitative errors are reported. Please add a table of L1/L2 errors and observed orders for the velocity profiles, or soften the convergence claim.
  4. [Sections 3 and 4.4] The formulation uses h_min in Eqs. (14)-(16) for variable-resolution flows, but the manuscript does not justify why the minimum smoothing length is the correct length scale for the correction across a resolution interface, nor does it test sensitivity to the resolution ratio or to the location of the interface. Since the central novelty includes variable-resolution applicability, this assumption should be either derived or tested, for example by varying the refinement ratio and measuring differences in the solution.
minor comments (7)
  1. [Section 1] The word 'stablity' in the introduction should be 'stability'.
  2. [Section 2.3, Eq. (8)] The text refers to ρ* as the density prior to reinitialization, but Eq. (8) does not define ρ* or show how it enters the summation; please clarify the notation.
  3. [Section 2.2, Eqs. (4) and (5)] The notation p_L/p_R in Eq. (5) is inconsistent with p_l/p_r in Eq. (4); please unify.
  4. [Section 4.5] The phrase 'as depicted in the left panel of Figure 12 (left panel)' repeats 'left panel'; remove the duplicate.
  5. [Section 5] The word 'sceneries' in the conclusion should be 'scenarios'.
  6. [Table 1] The column header 'Amplitude in y direction/(D)' is unclear; please state explicitly that the amplitude is normalized by D.
  7. [Figure 13] The caption refers to panels (a) and (b), but the panels are not labeled in the figure; please add labels.

Circularity Check

2 steps flagged · score 6.0 of 10

The limiter constants and the -0.2 coefficient are fitted inputs presented as derived; the claimed derivation in Eq. (14) is arithmetically inconsistent with the printed alpha and CFL_ad values.

  1. fitted input called prediction [Section 3, Eq. (15) and Eq. (16)]
    "beta = { min(C h^2 |Σ_j ∇W_ij V_j|^2, 1), h^2 |Σ_j ∇W_ij V_j|^2 > 5 × 10^-4; 0, otherwise } (15) with C = 10^3 according to the numerical cases."

    The limiter's threshold (5×10^-4) and gain (C=10^3) are explicitly chosen 'according to the numerical cases', i.e. fitted to the low-velocity and variable-resolution behaviors that the paper then reports as validation. The claim that Eq. (16) avoids overcorrection in the Taylor-Green case is therefore a restatement of the tuning target rather than an independent prediction; the constants encode the desired outcome. Independent benchmark cases (cavity, FSI, FDA) keep the rest of the validation meaningful, but this specific improvement is built into the parameters.

  2. other [Section 3, Eq. (14); Eq. (16); constants in Sec. 2.4 and Eq. (13)]
    "Upon inserting Eq. (13) into Eq. (12), the displacement correction ... is formulated as ∆e_r = 1/2 a_t(∆t_ad)^2 = ... = ξ h^2 Σ_j ∇W_ij V_j, (14) with the coefficient ξ = −0.2 accordingly."

    The stated inputs are α=7.0 (Eq. 13) and CFL_ad=0.25 (Sec. 2.4). Direct substitution gives ξ = -α CFL_ad^2 = -7 × 0.25^2 = -0.4375, not -0.2. The printed 'ξ = -0.2 accordingly' does not follow from the displayed equations, so the central coefficient in Eq. (16) is an additional fitted value rather than a derived consequence. The central correction formula thus presupposes the number it claims to obtain; an outside implementation using the stated constants would apply a correction 2.19 times larger and could not reproduce the results.

full rationale

The paper is not globally circular: the lid-driven cavity, FSI, flow-around-cylinder, and FDA-nozzle results are checked against external references (Ghia et al., Turek-Hron, experiment/PIV), and the transport-velocity idea is taken from the independent Adami et al. formulation. However, two load-bearing pieces of the claimed improvement are inputs in disguise. First, the limiter constants in Eq. (15) are selected 'according to the numerical cases' and then the same low-velocity/variable-resolution behavior is presented as validation, which is fitted-input-called-prediction circularity for the overcorrection claim. Second, the derivation of Eq. (14) is arithmetically inconsistent with the printed alpha=7.0 and CFL_ad=0.25: the algebra yields xi=-0.4375, so the xi=-0.2 in Eq. (16) is not 'accordingly' derived but is a calibrated/ansatz constant. These issues concern the central correction formula, but because the method still passes external benchmarks and the code is open-source, the paper is only partially circular rather than wholly reducible to its inputs. Score 6.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The formulation depends on four numerical parameters. xi is presented as derived, but the paper's own constants imply about -0.44, not -0.2. C and the threshold are explicitly fitted to test cases. alpha is inherited from prior work. No new physical entities are introduced; the limiter is a numerical device. The central claim therefore rests on calibrated constants plus standard WCSPH assumptions.

free parameters (4)
  • Transport correction coefficient xi = -0.2
    Eq. (14); claimed to follow from alpha=7 and CFLad=0.25, but substitution gives about -0.44, so xi appears to be an independently set magnitude.
  • Limiter gain C = 10^3
    Eq. (15); set according to the numerical cases rather than derived.
  • Limiter threshold = 5x10^-4
    Eq. (15); hand-set cutoff below which the transport correction is set to zero.
  • Background-pressure coefficient alpha = 7.0
    Eq. (13), taken from Ref. [32]; inherited parameter that controls the original transport-velocity magnitude.
assumptions (4)
  • domain assumption Weakly-compressible approximation with artificial EOS p = c^2(rho - rho_0) and c = 10Umax keeps density variation below 1%.
    Section 2.1; the method and all benchmarks rely on this standard WCSPH closure, not on a more exact incompressible treatment.
  • domain assumption The Riemann-based SPH discretization with the dissipation limiter beta (eta=3) is a faithful discretization of the governing equations.
    Section 2.2; accuracy and stability of the base solver are inherited from Refs. [24,27,29], and the paper does not re-derive them.
  • domain assumption Density reinitialization via Eq. (8) removes density errors without biasing the transport-velocity result.
    Section 2.3; the correction interacts with the reinitialized density field, and no sensitivity to reinitialization is reported.
  • ad hoc to paper Using the minimum smoothing length h_min in the transport correction is valid for variable-resolution flows.
    Section 3, after Eq. (15); the paper asserts h in Eqs. (14)-(15) is h_min but provides no derivation or sensitivity study for mixed-resolution configurations.

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Cite this review

Pith. "Pith review of The efficient implementation of transport velocity formulation." pith.science (2026). https://pith.science/paper/TXP4ZJKZ

@misc{pith2026241113992,
  author       = {Pith},
  title        = {Pith review of: The efficient implementation of transport velocity formulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TXP4ZJKZ}},
  note         = {Machine review of arXiv:2411.13992}
}
read the original abstract

The standard smoothed particle hydrodynamics (SPH) method suffers from tensile instability, resulting in particle clumping and void regions under negative pressure conditions. In this study, we extend the transport-velocity formulation of Adami et al. (2013) \cite{adami2013transport} in the weakly-compressible SPH (WCSPH) framework to address this long-standing issue. Rather than relying on background pressure, our modified and improved transport-velocity correction scales directly to the smoothing length, making it suitable for variable-resolution flows. Additionally, we introduce a limiter to the new formulation to prevent overcorrection, especially for flow with small velocities. These modifications enhance the general applicability of the transport velocity in fluid dynamics. Numerical tests involving low-velocity and variable-resolution cases demonstrate that the new formulation offers a general and accurate solution for multi-physics SPH simulations.

Figures

Figures reproduced from arXiv: 2411.13992 by the authors.

Figure 1
Figure 1. The maximum velocity (left panel) and energy decay (right panel) in the Taylor-Green [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Taylor–Green vortex (Re = 100): Particle snapshots with a resolution of 50×50 particles at different time instants. boundary conditions along both the x− and y− axes is specified as follows:    u(x, y, t) = − expbt cos(2πx)sin(2πy) v(x, y, t) = expbt sin(2πx)cos(2πy) . (17) Here, the decay rate is given by b = −8π 2/Re, with the Reynolds number Re = 100. The total kinetic energy decay rate is −16π 2/Re, and the f… view at source ↗
Figure 3
Figure 3. Taylor–Green vortex (Re = 100): Instantaneous velocity field and velocity vectors with a resolution of dp = 1/100 (left panel) and decay of kinetic energy (right panel). trast, the proposed formulation is observed to align with the decay of kinetic energy with the analytical solution, validating that it effectively avoids the overcorrection issue. 4.2. Lid-driven cavity flow Although the primary goal of the proposed… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Lid-driven square cavity flow (Re = 1000): The geometry and boundary conditions setup and the velocity contour ranging from 0 to 1. x v 0 0.2 0.4 0.6 0.8 1 -0.6 -0.4 -0.2 0 0.2 0.4 Reference Original dp=1/50 Present dp=1/50 Present dp=1/100 Present dp=1/200 u y -0.5 0 …
Figure 5
Figure 5. Figure 5: Lid-driven square cavity flow (Re = 1000): The horizontal velocity component u along x = 0.5L (left panel) and the vertical velocity component v along y = 0.5L (right panel) with the spatial resolutions as dp = 1/50, 1/100 and 1/200, and the comparisons with the refere…
Figure 6
Figure 6. Figure 6: Sketch of the two-dimensional flow-induced oscillation of an elastic beam attached to a [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: The fluid velocity magnitude contour at different instances, with the corresponding beam [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Flow-induced vibration of an elastic beam attached to a cylinder: [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Multi-resolution flow around cylinder (Re = 100): Geometry, boundary conditions and the refinement region setup (top panel), as well as particle volume, where the reference particle volume is 0.01 and the minimal particle volume is 0.0025, with the resolution of D/dp =…
Figure 10
Figure 10. Figure 10: Multi-resolution flow around cylinder (Re = 100): velocity contour ranging from 1.1 × 10−3 to 1.4 with the spatial resolution of D/dp = 20 at different time instants. t C D 0 50 100 150 200 0 0.6 1.2 1.8 2.4 3 D/dp=10 D/dp=20 D/dp=40 t C L 0 50 100 150 200 -1 -0.5 0 0…
Figure 11
Figure 11. Figure 11: Multi-resolution flow around cylinder (Re = 100): Drag (left panel) and lift (right panel) coefficients with the time using the proposed formulation with Re = 100. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Three-dimensional lid-driven square cavity flow ( [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: Three-dimensional lid-driven square cavity flow ( [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: Three-dimensional FDA nozzle: Geometry and boundary conditions. [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: Three-dimensional FDA nozzle (Re=500): Velocity contour using the proposed formu [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]
Figure 16
Figure 16. Figure 16: Three-dimensional FDA nozzle (Re=500): Axial velocity using the present algorithm [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]
Figure 17
Figure 17. Figure 17: Three-dimensional FDA nozzle (Re=500): Axial velocity profiles from cross-section [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]

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Reference graph

Works this paper leans on

50 extracted references · 48 canonical work pages · cited by 1 Pith paper

  1. [1]

    Adami, X

    S. Adami, X. Hu, N. A. Adams, A transport-velocity formulation for smoothed particle hydrodynamics, Journal of Computational Physics 241 (2013) 292–307

    2013

  2. [2]

    A numerical approach to the testing of the fission hypothesis, author=Lucy, Leon B, The astronomical journal 82 (1977) 1013–1024

    work page 1977

  3. [3]

    R. A. Gingold, J. J. Monaghan, Smoothed particle hydrodynamics: theory and application to non-spherical stars, Monthly notices of the royal astronomical society 181 (3) (1977) 375–389

    work page 1977

  4. [4]

    J. J. Monaghan, SPH compressible turbulence, Monthly Notices of the Royal Astronomical Society 335 (3) (2002) 843–852. 26

    work page 2002

  5. [5]

    W. C. Welton, Two-dimensional PDF/SPH simulations of compressible turbu- lent flows, Journal of Computational Physics 139 (2) (1998) 410–443

    work page 1998

  6. [6]

    J. J. Monaghan, Simulating free surface flows with SPH, Journal of computa- tional physics 110 (2) (1994) 399–406

    work page 1994

  7. [7]

    L. D. Libersky, A. G. Petschek, T. C. Carney, J. R. Hipp, F. A. Allahdadi, High strain Lagrangian hydrodynamics: a three-dimensional SPH code for dynamic material response, Journal of computational physics 109 (1) (1993) 67–75

    work page 1993

  8. [8]

    Randles, L

    P. Randles, L. D. Libersky, Smoothed particle hydrodynamics: some recent improvements and applications, Computer methods in applied mechanics and engineering 139 (1-4) (1996) 375–408

    work page 1996

Show all 50 references
  1. [9]

    S. M. Longshaw, B. D. Rogers, Automotive fuel cell sloshing under temporally and spatially varying high acceleration using GPU-based Smoothed Particle Hydrodynamics (SPH), Advances in Engineering Software 83 (2015) 31–44

    2015

  2. [10]

    Z. Wang, C. Zhang, O. J. Haidn, X. Hu, An Eulerian SPH method with WENO reconstruction for compressible and incompressible flows, Journal of Hydrody- namics 35 (2) (2023) 210–221

    2023

  3. [11]

    Khayyer, H

    A. Khayyer, H. Gotoh, Y. Shimizu, On systematic development of FSI solvers in the context of particle methods, Journal of Hydrodynamics 34 (3) (2022) 395–407

    2022

  4. [12]

    J. J. Monaghan, Smoothed particle hydrodynamics, Reports on progress in physics 68 (8) (2005) 1703. 27

    2005

  5. [13]

    Schuessler, D

    I. Schuessler, D. Schmitt, Comments on smoothed particle hydrodynamics, As- tronomy and Astrophysics, vol. 97, no. 2, Apr. 1981, p. 373-379. 97 (1981) 373–379

    1981

  6. [14]

    G. R. Johnson, S. R. Beissel, Normalized smoothing functions for SPH im- pact computations, International Journal for Numerical Methods in Engineering 39 (16) (1996) 2725–2741

    1996

  7. [15]

    Colagrossi, M

    A. Colagrossi, M. Landrini, Numerical simulation of interfacial flows by smoothed particle hydrodynamics, Journal of computational physics 191 (2) (2003) 448–475

    2003

  8. [16]

    Monaghan, On the problem of penetration in particle methods, Journal of Computational physics 82 (1) (1989) 1–15

    J. Monaghan, On the problem of penetration in particle methods, Journal of Computational physics 82 (1) (1989) 1–15

    1989

  9. [17]

    R. Xu, P. Stansby, D. Laurence, Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach, Journal of com- putational Physics 228 (18) (2009) 6703–6725

    2009

  10. [18]

    X. Xu, P. Yu, A technique to remove the tensile instability in weakly compress- ible SPH, Computational Mechanics 62 (5) (2018) 963–990

    2018

  11. [19]

    S. J. Lind, R. Xu, P. K. Stansby, B. D. Rogers, Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves, Journal of Computational Physics 231 (4) (201...

    2012

  12. [20]

    Antuono, A

    M. Antuono, A. Colagrossi, S. Marrone, D. Molteni, Free-surface flows solved by means of SPH schemes with numerical diffusive terms, Computer Physics Communications 181 (3) (2010) 532–549. 28

    2010

  13. [21]

    P.-N. Sun, A. Colagrossi, S. Marrone, A.-M. Zhang, The δplus-SPH model: Simple procedures for a further improvement of the SPH scheme, Computer Methods in Applied Mechanics and Engineering 315 (2017) 25–49

    2017

  14. [22]

    Vacondio, B

    R. Vacondio, B. D. Rogers, P. K. Stansby, P. Mignosa, J. Feldman, Variable res- olution for SPH: a dynamic particle coalescing and splitting scheme, Computer Methods in Applied Mechanics and Engineering 256 (2013) 132–148

    2013

  15. [23]

    Lyu, P.-N

    H.-G. Lyu, P.-N. Sun, Further enhancement of the particle shifting technique: Towards better volume conservation and particle distribution in SPH simula- tions of violent free-surface flows, Applied Mathematical Modelling 101 (2022) 214–238

    2022

  16. [24]

    Zhang, X

    C. Zhang, X. Hu, N. A. Adams, A weakly compressible SPH method based on a low-dissipation Riemann solver, Journal of Computational Physics 335 (2017) 605–620

    2017

  17. [25]

    Litvinov, X

    S. Litvinov, X. Hu, N. A. Adams, Towards consistence and convergence of con- servative SPH approximations, Journal of Computational Physics 301 (2015) 394–401

    2015

  18. [26]

    Zhang, X

    C. Zhang, X. Y. Hu, N. A. Adams, A generalized transport-velocity formulation for smoothed particle hydrodynamics, Journal of Computational Physics 337 (2017) 216–232

    2017

  19. [27]

    Zhang, M

    C. Zhang, M. Rezavand, X. Hu, Dual-criteria time stepping for weakly com- pressible smoothed particle hydrodynamics, Journal of Computational Physics 404 (2020) 109135. 29

    2020

  20. [28]

    Zhang, M

    C. Zhang, M. Rezavand, Y. Zhu, Y. Yu, D. Wu, W. Zhang, J. Wang, X. Hu, SPHinXsys: An open-source multi-physics and multi-resolution library based on smoothed particle hydrodynamics, Computer Physics Communications 267 (2021) 108066

    2021

  21. [29]

    E. F. Toro, Riemann solvers and numerical methods for fluid dynamics: a prac- tical introduction, Springer Science & Business Media, 2013

    2013

  22. [30]

    X. Y. Hu, N. A. Adams, A multi-phase SPH method for macroscopic and meso- scopic flows, Journal of Computational Physics 213 (2) (2006) 844–861

    2006

  23. [31]

    J. J. Monaghan, Smoothed particle hydrodynamics, Annual review of astronomy and astrophysics 30 (1) (1992) 543–574

    1992

  24. [32]

    Zhang, W

    S. Zhang, W. Zhang, C. Zhang, X. Hu, A Lagrangian free-stream boundary condition for weakly compressible smoothed particle hydrodynamics, Journal of Computational Physics 490 (2023) 112303

    2023

  25. [33]

    Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Advances in computational Mathematics 4 (1995) 389–396

    H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Advances in computational Mathematics 4 (1995) 389–396

    1995

  26. [34]

    G. I. Taylor, A. E. Green, Mechanism of the production of small eddies from large ones, Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences 158 (895) (1937) 499–521

    1937

  27. [35]

    U. Ghia, K. N. Ghia, C. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, Journal of computational physics 48 (3) (1982) 387–411. 30

    1982

  28. [36]

    Turek, J

    S. Turek, J. Hron, Proposal for numerical benchmarking of fluid-structure in- teraction between an elastic object and laminar incompressible flow, Springer, 2006

    2006

  29. [37]

    L. Han, X. Hu, SPH modeling of fluid-structure interaction, Journal of Hydro- dynamics 30 (2018) 62–69

    2018

  30. [38]

    Zhang, M

    C. Zhang, M. Rezavand, X. Hu, A multi-resolution SPH method for fluid- structure interactions, Journal of Computational Physics 429 (2021) 110028

    2021

  31. [39]

    Bhardwaj, R

    R. Bhardwaj, R. Mittal, Benchmarking a coupled immersed-boundary-finite- element solver for large-scale flow-induced deformation, AIAA journal 50 (7) (2012) 1638–1642

    2012

  32. [40]

    F.-B. Tian, H. Dai, H. Luo, J. F. Doyle, B. Rousseau, Fluid–structure interac- tion involving large deformations: 3D simulations and applications to biological systems, Journal of computational physics 258 (2014) 451–469

    2014

  33. [41]

    F. M. White, J. Majdalani, Viscous fluid flow, Vol. 3, McGraw-Hill New York, 2006

    2006

  34. [42]

    Chiu, R.-K

    P.-H. Chiu, R.-K. Lin, T. W. Sheu, A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier–Stokes equa- tions in time-varying complex geometries, Journal of Computational Physics 229 (12) (2010) 4476–4500

    2010

  35. [43]

    D.-V. Le, B. C. Khoo, J. Peraire, An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries, Journal of Compu- tational Physics 220 (1) (2006) 109–138. 31

    2006

  36. [44]

    Brehm, C

    C. Brehm, C. Hader, H. F. Fasel, A locally stabilized immersed boundary method for the compressible Navier–Stokes equations, Journal of Computational Physics 295 (2015) 475–504

    2015

  37. [45]

    Russell, Z

    D. Russell, Z. J. Wang, A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, Journal of Computational Physics 191 (1) (2003) 177–205

    2003

  38. [46]

    H. C. Ku, R. S. Hirsh, T. D. Taylor, A pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations, Journal of Com- putational Physics 70 (2) (1987) 439–462

    1987

  39. [47]

    Komaizi, A

    D. Komaizi, A. Niknam, Hybrid finite-volume–finite-element scheme for 3D sim- ulation of thermal plasma arc configuration, AIP Advances 14 (1) (2024)

    2024

  40. [48]

    S. F. Stewart, E. G. Paterson, G. W. Burgreen, P. Hariharan, M. Giarra, V. Reddy, S. W. Day, K. B. Manning, S. Deutsch, M. R. Berman, et al., Assess- ment of CFD performance in simulations of an idealized medical device: results of FDA’s first computational interlaboratory stu...

    2012

  41. [49]

    Simulation of the FDA nozzle benchmark: A lattice Boltzmann study, au- thor=Huang, Feng and No¨ el, Romain and Berg, Philipp and Hosseini, Seyed Ali, Computer Methods and Programs in Biomedicine 221 (2022) 106863

    2022

  42. [50]

    Hariharan, M

    P. Hariharan, M. Giarra, V. Reddy, S. Day, K. Manning, S. Deutsch, S. Stew- art, M. Myers, M. Berman, G. Burgreen, et al., Experimental particle image velocimetry protocol and results database for validating computational fluid dy- 32 namic simulations of the fda benchmark noz...

    2011

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