On zero-order consistency residue and background pressure for the conservative SPH fluid dynamics

Feng Wang; Xiangyu Hu

REVIEW 4 major objections 6 minor 36 references

On zero-order consistency residue and background pressure for the conservative SPH fluid dynamics

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read One residual term in conservative SPH pressure gradients is proposed as the common cause of spurious damping in waves and channels.

desk verdict The channel-flow part is a solid, reproducible finding, but the free-surface attribution is underdetermined until the residue power is computed directly and the limiter dissipation is isolated. read the letter →

arxiv 2507.18210 v1 pith:MSI2NSPQ submitted 2025-07-24 physics.flu-dyn cs.CEphysics.comp-ph

Feng Wang , Xiangyu Hu This is my paper · ORCID

classification physics.flu-dyncs.CEphysics.comp-ph

keywords smoothedparticlehydrodynamicszero-orderconsistencykernelgradientcorrectionbackgroundpressurenumericaldissipationfree-surfaceflowchannelnozzlebenchmark

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 that a single quantity, the zero-order consistency residue, is the common root cause of the non-physical numerical damping seen in conservative smoothed particle hydrodynamics (SPH) simulations of both gravity-driven free-surface flows and pressure-driven channel flows. The residue is the term $2p_i \sum_j \nabla W_{ij} V_j$ left over when the conservative pairwise-average pressure gradient fails to reproduce a constant pressure field, and it acts as a spurious dissipative force whose strength grows with local pressure. In numerical tests, the damping grows with water depth and initial wave amplitude in an inviscid standing wave, and with channel length and outlet pressure in long channels, while remaining insensitive to spatial resolution. The paper also shows that the reverse kernel gradient correction reduces but cannot fully remove the effect, leaving measurable velocity loss in the engineering nozzle benchmark. If correct, the result reframes a known accuracy complaint as a consistency defect with a specific mechanism, and it warns that grid refinement alone will not cure the problem.

What carries the argument

The load-bearing object is the zero-order consistency residue, the second term on the right of Eq. (13), $2p_i \sum_j \nabla W_{ij} V_j$. It arises because a pairwise-average pressure gradient cannot vanish on a constant field when the particle distribution is irregular; the kernel-gradient sum is the zero-order consistency error, and its multiplication by the local pressure $p_i$ makes deep water, long channels, and high outlet pressure the dangerous regimes. The paper uses this identity in two arguments: a dimensional analysis of the residue-induced acceleration that predicts the dimensionless energy-dissipation rate is independent of water depth, and a balance condition at the channel mid-section, $1/L - \sum_j \nabla W_{ij} V_j = 0$, which separates regimes where the residue dominates the driving pressure gradient. The reverse kernel gradient correction (RKGC), a momentum-conserving rebuild of kernel gradients, is the counter-mechanism used to test the identification.

What would settle it

Run the inviscid standing wave case with a pressure-gradient discretization that is exactly zero-order consistent, for example a difference-based or fully corrected gradient, while keeping the same Riemann solver, particle distribution, and wave parameters; if the exponential energy decay remains at the same rate, the decay is not caused by the zero-order consistency residue.

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Extended reading notes

Core claim

The central claim is that the zero-order gradient inconsistency of the strictly conservative SPH formulation is not just an accuracy defect but a source of physical-looking damping. When the conservative gradient is used to reproduce a constant pressure field, an irreducible residual term appears, $(\nabla p)_i = \sum_j W_{ij} V_j (\nabla p)_i + 2p_i \sum_j \nabla W_{ij} V_j$, and the second term, named the zero-order consistency residue, dissipates the flow because the kernel-gradient summation runs opposite to the flow direction. In gravity-driven free surfaces, hydrostatic pressure makes the residue grow with water depth; in pressure-driven channels, the inlet pressure required by long domains does the same. The paper reports exponential energy decay in the standing wave, velocity loss that grows with channel length and outlet pressure but not with resolution, and a normalized energy-decay rate that is independent of water depth, as the depth scaling of pressure cancels in the dimensional analysis. The reverse kernel gradient correction (RKGC) mitigates the residue but leaves about 3.5% velocity loss in the longest tested laminar channel and 2.1% in the 3D nozzle benchmark, where the uncorrected simulation loses 33.7% of the axis velocity.

Load-bearing premise

The attribution of all the measured standing-wave energy decay to the zero-order consistency residue requires that the dissipation produced by the Riemann solver's pressure limiter, which is active in these inviscid flows, is negligible compared with the residue's effect.

Editorial extensions

If this is right

In the inviscid standing wave, residue-induced energy decay follows a clear exponential law whose rate increases with initial wave amplitude and drops to nearly zero when the amplitude is zero.
In the normalized standing-wave variables, the energy-decay rate is essentially the same across water depths, which the dimensional analysis explains by the cancellation of depth in the hydrostatic and residue pressure scalings.
For velocity-inlet and pressure-outlet channels, maximum centerline velocity loss grows with channel length and outlet pressure, is absent under periodic boundary conditions, and gets slightly worse rather than better with refinement (14.67% to 17.47% as the number of particles across the channel rises from 20 to 80).
Kernel gradient correction reduces but does not eliminate the residue: about 3.5% velocity loss remains at length-to-height 60, turbulent kinetic energy under-prediction near the inlet is mitigated, and the 3D nozzle benchmark drops from a 33.7% uncorrected axis-velocity loss to 2.1% with correction.
In high-background-pressure engineering geometries, the residue cannot be neglected, so the paper advises caution with conservative SPH and suggests simplified physical models when the background pressure is unavoidable.

Reading between the lines

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

A direct testable extension would recompute the inviscid standing wave with a strictly zero-order-consistent pressure gradient and the same Riemann solver; if the exponential decay persists, the attribution to the residue would need revision.
The dimensional result implies a scaling prediction not tested in the paper: at fixed wave shape, the normalized decay rate should scale with the dynamic pressure and the smoothing-length-normalized consistency error, so varying the sound speed or smoothing length should move the decay rate in a predictable way.
Because the velocity loss is resolution-insensitive, standard convergence studies can look satisfactory while the inlet-region momentum balance is wrong; reporting the residue term as a routine diagnostic would expose the defect in future SPH work.
The same mechanism should appear in other conservative particle methods built on pairwise-average gradients and run with large background pressure, making cross-method comparison a way to see whether this is an SPH-specific issue or a general particle-approximation property.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

What you should know: the channel-flow half of this paper is strong and genuinely new. Long VIPO channels lose centerline velocity to a resolution-insensitive damping that periodic BCs eliminate, outlet pressure modulates, and RKGC only partly fixes. I know of no prior report of this channel-length-dependent loss, and the turbulence case shows the same mechanism eating TKE near the inlet. The FDA nozzle numbers (33.7% velocity loss without correction, 2.1% with) are a practical punchline. The dimensional argument for water-depth-independent dimensionless dissipation rates is neat and testable. Credit where due: the paper ships code, reports measured trends rather than fits to the model, and is honest about RKGC's limits.

The soft spot is the free-surface leg. The standing-wave energy decay is measured with the Eq. (8) pressure-limiter active, and the paper itself says that limiter operates only in inviscid flows. So the measured dE/dt mixes residue contribution with Riemann-solver dissipation. The residue's dissipative direction is imported from Colagrossi et al. and never verified in the oscillating flow, where velocity reverses each half-period; net decay needs a sustained correlation, not just a fixed kernel-sum bias. The paper does not compute residue power directly from the simulation. That is the load-bearing gap. Separately, the Hw = 3.0 fit has R2 = 0.30, yet the paper claims close agreement across depths; that claim needs qualification.

These are addressable, not fatal. The channel-flow evidence stands on its own: periodic BC removes the effect, outlet pressure trends match the hypothesis, resolution insensitivity shows it is not a discretization error, and RKGC partial recovery is consistent. The paper's central mechanism is plausible and the engineering relevance is clear.

Who should read it: SPH method developers and CFD practitioners using conservative SPH in internal flows or wave tanks. General CFD readers can skip the method detail but should know the long-channel caveat.

Recommendation: yes, send to peer review. The channel-flow result alone justifies referee time. Ask for direct residue-work measurement and a limiter-free run in the standing-wave tests, or at least a bound on the limiter's contribution. That is a revision request, not a rejection.

Referee Report

4 major / 6 minor

Summary. The paper identifies the "zero-order consistency residue" (Eq. 13: 2 p_i Σ_j ∇W_ij V_j) in the conservative SPH pressure-gradient discretization as the common root cause of non-physical numerical damping in two flow classes: gravity-driven free-surface flows (studied via inviscid standing waves) and pressure-driven internal flows (laminar and turbulent channels, plus the FDA nozzle). It argues that background pressure amplifies the residue, presents sensitivity studies of energy decay or velocity loss to wave amplitude, water depth, channel length, outlet pressure, and resolution, and evaluates the reverse kernel gradient correction (RKGC) as a partial remedy. The paper claims the damping is resolution-insensitive and only partially removable by RKGC, with residual velocity losses up to 17.47% in the long channel and 33.7% in the 3D FDA nozzle without correction.

Significance. If the root-cause attribution is correct, the paper offers a unified explanation for a long-observed excessive-dissipation problem in conservative SPH and gives practical guidance on background-pressure selection and correction schemes. The study is systematic, releases its code through the SPHinXsys repository, and its empirical trends (periodic BC removes damping, lower outlet pressure reduces loss, deeper water and larger amplitude increase decay, RKGC mitigates but does not eliminate) are internally consistent. However, the central causal claim hinges on isolating the residue contribution from other numerical dissipation, and that isolation is incomplete for the free-surface leg of the study.

major comments (4)

[§2.1.1, Eq. (8)] The standing-wave energy-decay experiments in §4.2 are performed while the dissipation limiter β_ij = min(η max(v_ij · e_ij, 0), c_0) is active, because the paper states that this term is activated only in inviscid flow cases such as the standing wave. The measured dE*/dt is therefore the sum of the residue contribution and the Riemann-solver limiter dissipation, and no run with β_ij = 0 or separate estimate of the limiter's power is reported. Since both contributions grow with wave amplitude, the observed exponential decay cannot be uniquely attributed to the zero-order consistency residue. Please add a control simulation with the limiter disabled or directly compute the residue power and show that it accounts for the measured energy loss.
[§3, after Eq. (13)] The dissipative character of the residue is justified by the statement, imported from Ref. [29], that 'the direction of kernel gradient summation is generally opposite to the flow direction.' This directional bias is not verified for the oscillating standing-wave flow, where the velocity changes sign every half-period; a fixed bias cannot by itself produce net decay unless it is correlated with the velocity over the oscillation cycle. The paper never computes the work done by the residue term in the simulations, so the free-surface attribution remains underdetermined and the claim that a single residue explains both the free-surface and channel-flow observations is not fully supported.
[§4.2.2, Eqs. (16)-(18)] The dimensional analysis concluding that the dimensionless energy dissipation rate is independent of water depth relies on the hydrostatic scaling p_i ∝ ρ g H_w. In the standing-wave test the pressure also contains a dynamic component proportional to ρ g A, so p_i/(ρ g H_w) = 1 + O(A/H_w), which is not exactly constant across the tested depths. The fitted slopes in Table 3 vary by roughly 10%, so the claim that the similarity is 'well explained' by Eq. (18) needs either a more careful pressure decomposition or an explicit demonstration that the dynamic-pressure correction is negligible in the fitted time interval.
[§5.2.4, Fig. 19] The abstract's phrase 'resolution-insensitive' is not what Fig. 19 shows: the maximum centerline velocity loss increases from 14.67% at Nf = 20 to 17.47% at Nf = 80. The defensible statement is that the damping is not reduced by spatial refinement, but the paper should explain why the loss slightly grows with resolution, since an increasing error with Nf is not the behavior one would naively expect from a pure discretization error and is itself informative about the nature of the residue.

minor comments (6)

[§5] In the opening sentence of Section 5, 'we firstly analysis this problem' should read 'we first analyze this problem.'
[§4.2.2, after Eq. (17)] The phrase 'we need to time the non-dimensionalized velocity' should be 'multiply by the non-dimensionalized velocity.'
[§7] In the Conclusion, 'The non-dimensionless energy dissipation rate' should be 'the dimensionless energy dissipation rate.'
[Fig. 11] The label 'Inlet buffer' in Fig. 11 is not defined or discussed in the text; please clarify what region this refers to and why it is shown.
[Table 3] For Hw = 3.0 the reported R² is 0.2951, which is too low to support the claim in §4.2.2 that the energy-decay curves are 'well consistent with the logarithmic law.' Please discuss this outlier explicitly rather than attributing it only to numerical oscillation.
[Abstract and §1] The paper says the RKGC technique is 'introduced'; since it is adopted from Ref. [34], 'applied' or 'evaluated' would be more accurate.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the residue is a defined mathematical term, the energy-decay and velocity-loss data are measured, and the water-depth independence prediction is a dimensional consequence tested against independently fitted slopes.

full rationale

The central derivation is not circular. Equation (13) defines the zero-order consistency residue as a component of the conservative pressure-gradient discretization, following the external mathematical derivation of Colagrossi et al. [29]; giving the term a name does not insert the conclusion. The paper's quantitative evidence, namely the standing-wave energy-decay slopes in Tables 2 and 3 and the channel centerline-velocity losses in Table 5, is measured from simulations rather than generated by fitting the residue model. Equation (18) is a dimensional-analysis identity for the residue power; its prediction that the dimensionless decay rate is independent of water depth follows from hydrostatic scaling and is then compared with independently fitted slopes, so the comparison is not forced by construction. The channel analysis uses the self-cited TVF boundedness result [31] as a premise, and the RKGC technique [34] is also from the same group, but these citations support method components rather than the root-cause attribution. The attribution is independently supported by the periodic-versus-VIPO comparison, the outlet-pressure sweep, the resolution sweep, and the RKGC on/off tests. The main legitimate concerns are correctness and identification issues, not circularity: the standing-wave runs keep the Eq. (8) dissipation limiter active, and the sign and direction of the residue force are imported from [29] rather than computed from the oscillating flow, so the free-surface energy decay is not uniquely isolated as residue-induced. These concerns do not make the derivation equivalent to its inputs, and no equation or fitted parameter is renamed as a prediction. The low score reflects only normal self-citation in the method stack.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

The paper rests on three main pillars: the residue definition from prior literature, the background-pressure-driven mechanism, and the numerical parametric evidence. The free parameters are limited: eta=3 is the only tuned constant in the dissipation limiter. The axioms are mostly domain assumptions about the sign of the kernel-gradient sum and the subdominance of Riemann-solver dissipation, both load-bearing for the causal story. No new physical entities are introduced.

free parameters (1)

Dissipation limiter eta = 3
Appears in Eq. (8) as beta_ij = min(eta max(v_ij dot e_ij, 0), c0). The paper states 'eta = 3 is determined by numerical tests and is used throughout this work' (Section 2.1.1). This is a hand-tuned constant that directly controls the Riemann-solver dissipation in inviscid standing wave tests, the very cases where energy decay is attributed to the residue.

assumptions (4)

domain assumption The transport velocity formulation strictly bounds the zero-order consistency error (sum_j grad W_ij V_j).
Invoked in Section 3 and Section 5.1 to argue that the remaining driver of the residue is the background pressure term p_i, not the unbounded consistency error. The paper cites its own Ref. [31] for this bound.
domain assumption The direction of the kernel-gradient summation is generally opposite to the flow direction, making the residue dissipative.
Used in Section 3 ('the residual term dissipates the flow because the direction of kernel gradient summation is generally opposite to the flow direction[29]'). This sign property is not derived in the paper and is essential for the claim that the residue causes damping.
domain assumption The numerical dissipation from the low-dissipation Riemann solver is negligible in the inviscid standing wave tests.
In Section 2.1.1 the paper notes the intermediate-pressure dissipation term in Eq. (8) 'is activated only in inviscid flow cases, such as the standing wave'. The energy decay analysis in Section 4 attributes all observed decay to the residue, implicitly assuming the solver dissipation is subdominant.
standard math Hydrostatic pressure scaling p_i ~ rho g Hw holds for the nondimensionalization of the residue in free-surface flows.
Used in Section 4.2.2 to derive Eq. (18) and conclude the dimensionless dissipation rate is independent of water depth. This is a standard physics assumption for the standing wave configuration.

invented entities (1)

Zero-order consistency residue (2 p_i sum_j grad W_ij V_j) independent evidence
purpose: A named diagnostic to quantify the residual force from the conservative pressure gradient when the kernel-gradient sum is nonzero; used as the presumed root cause of numerical damping.
The term itself appears in the prior literature (Eq. 13 cites Colagrossi et al. [29]), and the paper does not claim a new force or physical field. The parametric tests (water depth, wave amplitude, channel length, outlet pressure, RKGC) provide falsifiable handles, so this is a diagnostic quantity rather than a postulated physical entity.

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Cite this review

Pith. "Pith review of On zero-order consistency residue and background pressure for the conservative SPH fluid dynamics." pith.science (2026). https://pith.science/paper/MSI2NSPQ

@misc{pith2026250718210,
  author       = {Pith},
  title        = {Pith review of: On zero-order consistency residue and background pressure for the conservative SPH fluid dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSI2NSPQ}},
  note         = {Machine review of arXiv:2507.18210}
}

read the original abstract

As one of the major challenges for the conservative smoothed particle hydrodynamics (SPH) method, the zero-order consistency issue, although thought to be mitigated by the particle regularization scheme, such as the transport velocity formulation, significantly damps the flow in a long channel for both laminar and turbulent simulations. Building on this finding, this paper not only thoroughly analyzes the damping reason in this pressure-driven channel flow, but also relates this problem with the excessive numerical dissipation in the gravity-driven free-surface flow. The common root cause of the non-physical numerical damping in the two typical flow scenarios, the zero-order gradient consistency residue, is exposed. The adverse influence of the background pressure on the residue for the two scenarios is revealed and discussed. To comprehensively understand the behavior of the residue and mitigate its potential adverse effects, we conduct both theoretical analysis and numerical experiments focusing on the key sensitive factors. For studying the residue-induced non-physical energy dissipation in the gravity-driven free-surface flow, the water depth and input dynamic pressure in the inviscid standing wave case are tested. To investigate the velocity loss in the pressure-driven channel flow, we examine the effects of the channel length, resolution, and outlet pressure. The state-of-the-art reverse kernel gradient correction technique is introduced for the two typical flows, and proved to be effective in reducing the residue effect, but we find its correction capability is fundamentally limited. Finally, the FDA nozzle, an engineering benchmark, is tested to demonstrate the residue influence in a complex geometry, highlighting the necessity of correction schemes in scenarios with unavoidable high background pressure.

Figures

Figures reproduced from arXiv: 2507.18210 by the authors.

**Figure 1.** The velocity contour and cross-sectional velocity profiles in a long laminar channel(L/H = 60) computed by the conservative weakly compressible SPH method, where L and H is channel length and height, respectively, and U is the axis velocity. Second, a similar unexpected numerical dissipation has been observed in the free surface flows for decades. The dissipation leads to a significant under-prediction of the wave h… view at source ↗

**Figure 2.** The flow characteristic of the gravity-driven free-surface flow, and with the increase of the water depth Hw, the background pressure pb increases, leading to the increment of the residue. 4.2. Sensitivity analysis In the previous analysis, we identified the background pressure as the primary source of the residue. This background pressure can be decomposed into two components: a dynamic part (dynamic pressure) and … view at source ↗

**Figure 3.** Standing wave: the initial fluid particle distributions with 4 different initial wave amplitudes, (a) 0.0 (b) 0.1 (c) 0.2 (d) 0.3. And the black dot lines denote the average water depth [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (25 more)

**Figure 8.** Figure 8: 17 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 4.** Figure 4: Standing wave: time evolution of dimensionless total mechanical energy for varying initial wave amplitudes (linear time axis), the region enclosed by the dashed lines, t ∗ ∈ [20, 70], is used for fitting. The linear fitting results are summarized in table 3. The R2 and…

**Figure 5.** Figure 5: Standing wave: time evolution of dimensionless total mechanical energy for varying initial wave amplitudes (logarithmic time axis) and the linear fitting results, where the data is obtained for t ∗ ∈ [20, 70]. (a) (b) (c) Hw/m 2.1 0.0 [PITH_FULL_IMAGE:figures/full_fig…

**Figure 6.** Figure 6: Standing wave: the initial fluid particle distributions for testing the water depths, Hw = (a) 1.0 (b) 1.5 (c) 2.0. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

**Figure 8.** Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

**Figure 9.** Figure 9: Standing wave: time evolution of the dimensionless total mechanical energy for varying water depths with or without the RKGC technique. this problem from a theoretical perspective and then conduct numerical tests for validation. 5.1. Theoretical analysis Similar to the…

**Figure 10.** Figure 10: Standing wave: time evolution of the dimensionless total mechanical energy for varying initial wave amplitudes with or without the RKGC technique. VIPO boundary conditions but different channel lengths (L1 ≫ L2), since the pressure gradients are the same, the inlet pr…

**Figure 11.** Figure 11: Theoretical pressure distribution along the main-stream(x) direction in the two channels with different lengths. gradient does the overall flow begin to exhibit noticeable dissipative behavior. When these two terms reach a balance, we obtain: (∇p)i − 2pi X j ∇WijVj = …

**Figure 12.** Figure 12: Laminar channel flow with periodic boundary condition: (a) the velocity contour and (b) cross-sectional velocity profiles. The five cross-sectional velocity profiles are monitored at positions x/L = 0, 0.25, 0.5, 0.75. Some labels are omitted for simplicity. The segme…

**Figure 13.** Figure 13: Laminar channel flow: the velocity contours with different length-toheight ratios (δ = (a) 15, (b) 30, (c) 45 and (d) 60 from up to bottom). Each case has five cross-sectional velocity profiles monitored at positions x/L = 0, 0.25, 0.5, 0.75. Some labels are omitted …

**Figure 14.** Figure 14: Laminar channel flow: The velocity contour computed by the finite volume method under the same simulation settings as the WCSPH method. Reynolds number is 20000 and the analytical data obtained[21] from the finite difference method are imposed as the inflow boundary c…

**Figure 15.** Figure 15: Turbulent channel flow: the contours of the turbulent kinetic energy and velocity together with the cross-sectional data in the long turbulent channel (δ = 60, Re = 20000). The cross-sectional profiles are monitored at positions x/L = 0, 0.25, 0.5, 0.75. The dot lines…

**Figure 16.** Figure 16: Turbulent channel flow: the velocity contours and the cross-sectional profiles of the laminar channels at positions x/L = 0, 0.25, 0.5, 0.75 calculated by the finite volume method. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_16.png]

**Figure 17.** Figure 17: Laminar channel flow: the centerline velocity profiles of the long laminar channel (δ = 60) calculated by the WCSPH method at the four outlet pressure (pout = 1500, −100, −500, −1000). (a) (b) (c) (d) P/Pa -1500 2000 0 [PITH_FULL_IMAGE:figures/full_fig_p034_17.png]

**Figure 18.** Figure 18: Laminar channel flow: The pressure contours near outlet [0.92L, L] of the long laminar channel (δ = 60) calculated by the WCSPH method at the four outlet pressure (pout = (a)1500,(b) − 500,(c) − 1000,(d) − 1500). 5.2.4. Resolution Aiming at reducing the above-mentione…

**Figure 19.** Figure 19: Laminar channel flow: The centerline velocity of the long laminar channels (δ = 60) calculated by the conservative WCSPH method at the three resolutions (Nf = 20, 40, 80, where Nf is the number of fluid particles across the cross-section) 35 [PITH_FULL_IMAGE:figures/…

**Figure 20.** Figure 20: Laminar channel flow: the centerline velocity profiles of the laminar channels of different lengths(δ = 60, 45, 30, 15) calculated by the WCSPH method with or without RKGC. 6. The residue in the FDA nozzle To reveal the significance of the zero-order residue issue in …

**Figure 21.** Figure 21: The results of a very long channel (δ = 60) calculated by the WCSPH method with the RKGC at the laminar and turbulent situations, and the crosssectional profiles are obtained at positions x/L = 0, 0.25, 0.5, 0.75. (a) The laminar case regarding the velocity; (b) the …

**Figure 22.** Figure 22: The geometry of the (a) FDA nozzle (b) reversed FDA nozzle. First, we conduct the 2D simulation to study the influence of the complex geometry on the residue, and the two geometries are tested, (1) the FDA nozzle shape; (2) the reversed FDA nozzle shape (as shown in …

**Figure 23.** Figure 23: Two-dimensional FDA nozzle flow: the axis velocity profiles of the 2D FDA and reversed FDA nozzle cases calculated by the WCSPH method with or without the RKGC, the maximum velocity losses are denoted. Next, we simulate the full-size 3D FDA nozzle, and the flow parame…

**Figure 24.** Figure 24: Two-dimensional FDA nozzle flow: the axis pressure profiles of the 2D FDA and reversed FDA nozzle cases calculated by the WCSPH method with or without the RKGC. gradient correction technique, the max axis velocity loss could reach 33.7% compared with the experiment re…

**Figure 25.** Figure 25: Three-dimensional FDA nozzle flow: the velocity contours sliced at the x-plane calculated by the WCSPH method (a) with RKGC (b) without RKGC. 7. Conclusion This study investigates the long-standing zero-order consistency issue in the conservative smoothed particle hyd…

**Figure 26.** Figure 26: Three-dimensional FDA nozzle flow: the comparison of the axis velocity profiles calculated by the WCSPH method with RKGC or without RKGC, and the corresponding experiment data. The dotted lines present the monitoring positions for the cross-sectional data. In the cas…

**Figure 27.** Figure 27: Three-dimensional FDA nozzle flow: the comparison of the crosssectional velocity profiles, at L/dthr = (a) 3 (b) 9, calculated by the WCSPH method with RKGC or without RKGC, and the corresponding experiment data. To address zero-order residue issue in the high backgr…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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