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A weakly compressible SPH method for RANS simulation of wall-bounded turbulent flows

T0 review · 6 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper claims that a particle-based SPH solver for the k–epsilon RANS equations achieves, for the first time, resolution-converged mean velocity and turbulent kinetic energy in wall-bounded turbulent flows.

desk verdict A serious SPH-RANS method paper with genuine benchmark progress; the convergence claim is plausible but rests on hand-calibrated switches that are not shown to vanish, so it needs revision rather than rejection. read the letter →

arxiv 2501.18397 v1 pith:R7APA2HM submitted 2025-01-30 physics.flu-dyn cs.CE

Feng Wang , Zhongguo Sun , Xiangyu Hu This is my paper · ORCID
classification physics.flu-dyncs.CE
keywords SmoothedparticlehydrodynamicsRANSturbulencemodelk-epsilonWall-boundedturbulentflowWallLagrangianmethodConvergencestudykineticenergyover-prediction
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 sets out to make Smoothed Particle Hydrodynamics (SPH) usable for wall-bounded turbulent flow by solving the two-equation k–epsilon RANS equations in a Lagrangian particle framework. Its central assertion is that, with a package of corrections, the method converges with resolution: both mean velocity and turbulent kinetic energy settle toward reference values as particles are refined, in straight, curved, and mildly separated channels. If true, this removes a known obstacle, as particle-based RANS has typically over-predicted turbulent kinetic energy and lacked benchmark validation in wall-bounded cases. The authors identify the root causes as over-damping in the core flow, spurious production in the plug-flow region, and the kinematically sharp shear discontinuity at the wall, and they propose targeted fixes for each.

What carries the argument

The load-bearing objects are four interacting corrections. ARD uses $\mu_c = \max(\mu_R, \tilde{\mu}_{ij})$ to pick, per pairwise interaction, whichever dissipation is larger, the Riemann numerical viscosity or the eddy viscosity, so stability is supplied where eddy viscosity is low and removed where it would over-damp. LTVF multiplies the transport-velocity position correction by a limiter $\beta_{tvf} = \min(m h^2 \|R^0_{\nabla\phi}\|_2, 1)$, so particles are re-centered only when the consistency residue is actually large. The weighted near-wall gradient compensation replaces the SPH velocity gradient with a blend of the wall-function gradient and the SPH gradient for neighbors in the wall layer, with weight $w_s$ set to 0.1, or 0.5 when ARD is active. The constant $y_p$ strategy sets the wall-adjacent layer height as a model parameter, and the level-set boundary-offset technique shifts the dummy-particle wall inward by $\delta = y_p - d_p^r/2$ so that $y^+$ stays in the wall-function's effective range as resolution rises.

What would settle it

Repeat the straight-channel convergence sweep beyond Nf = 80 with ARD active and w_s fixed at 0.5, then compute the difference of k and U from the FDM and DNS references; if the difference stops shrinking or the wall-nearest k bias reappears, the claimed convergence owes to the tuned blending rather than to the resolved physical balances.

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Extended reading notes

Core claim

The authors claim that the inconsistency between the Lagrangian character of SPH and the RANS wall-model setup is the reason SPH-RANS has not previously converged for wall-bounded turbulence, and that it can be fixed. In the main stream, an Adaptive Riemann-eddy Dissipation (ARD) switches dissipation between the Riemann solver's numerical viscosity and the eddy viscosity, preventing both particle clustering in high-shear layers and over-damping in the core; a Limited Transport Velocity Formulation (LTVF) suppresses spurious turbulent kinetic energy production caused by position corrections in plug-flow regions. Near the wall, a particle-based wall model evaluates shear stress and velocity gradients from tangential velocities, a weighted compensation scheme restores the under-estimated velocity gradient in the layer just above the wall, and a constant y_p strategy with a level-set boundary offset keeps the non-dimensional wall distance fixed as resolution increases. The result, the paper claims, is the first SPH-RANS convergence study in which both velocity and k converge and agree with DNS, finite-volume, and experimental references at engineering-acceptable resolutions.

Load-bearing premise

The hand-set numerical dissipation and the hand-calibrated blending weight w_s are assumed not to corrupt the RANS mean-flow and turbulence balances as the resolution increases.

Editorial extensions

If this is right

  • SPH-RANS can be validated on standard wall-bounded benchmarks such as channels, curved ducts, and converging–diverging ducts, reaching the same mean-flow and k answers as mesh-based RANS at comparable coarse resolutions.
  • The k over-prediction known from earlier particle RANS implementations is attributed to TVF position corrections in plug-flow regions and to particle-vortex production in high-shear regions, and the LTVF and ARD fixes address those mechanisms directly.
  • Because wall dummy particles are retained, the framework can be extended to turbulent fluid–structure interaction without redesigning the boundary treatment.
  • The convergence machinery, namely constant y_p, boundary offset, and y^+-consistent refinement, provides a template for convergence studies of other RANS models in particle methods.
  • The level-set boundary offset makes the approach applicable to complex geometries, not just straight channels.

Reading between the lines

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

  • If the claimed convergence survives un-tuned resolution sweeps with the same w_s and no recalibration, the ARD switch effectively behaves like an automated sub-particle dissipation model, which would make ARD worth testing as an implicit LES-type closure for non-RANS SPH.
  • The constant-y_p device forgoes the boundary offset once FSI is involved; the paper's own limitation note suggests a wall model independent of y^+ would be needed for rigorous FSI convergence, and that is the natural next test.
  • The same diagnosed inconsistency, namely kinematic shear discontinuity plus particle migration near walls, should appear in other Lagrangian solvers such as MPS, so the LTVF limiter and weighted gradient compensation may transfer there.
  • The weight w_s being recalibrated from 0.1 to 0.5 when ARD is active hints that the compensation is coupled to the dissipation balance; a resolution- or flow-dependent w_s would be the decisive check of the mechanism.
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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

6 major / 6 minor

Summary. The manuscript proposes a weakly compressible SPH method for solving the k-epsilon RANS equations in wall-bounded turbulent flows. The central methodological contributions are: an adaptive Riemann-eddy dissipation (ARD) switch, a limited transport velocity formulation (LTVF), a particle-based Lagrangian wall model with a weighted near-wall gradient compensation, and a constant-y_p strategy combined with a level-set boundary-offset technique for resolution-independent wall-adjacent placement. The method is validated on a straight channel, mildly- and strongly-curved channels, and a half converging-diverging (HCD) channel, with comparisons against FDM, FVM, DNS, and experimental data. The headline claim is that this is the first SPH-RANS method to achieve good resolution convergence for both mean velocity and turbulent kinetic energy in wall-bounded flows.

Significance. If the convergence claim is substantiated, this is a meaningful advance for particle-based RANS simulation of engineering wall-bounded flows. The paper identifies a real inconsistency between the Lagrangian particle dynamics and the RANS/wall-model representation, and it proposes several systematic remedies rather than a purely ad hoc fix. The external validation base is genuinely useful: the straight channel is compared with DNS, the curved channels with experiments and FVM, and the HCD channel with two independent FVM codes. The release of the implementation in the open-source SPHinXsys repository is also a concrete strength. However, the paper's central claim currently rests on numerical mechanisms whose consistency with the RANS equations at vanishing particle spacing is not demonstrated: the ARD switch and the weighted near-wall gradient compensation contain hand-set parameters and are not subjected to a quantitative consistency or sensitivity analysis. The convergence evidence is also weaker than stated for parts of the validation matrix. The contribution is therefore promising but not yet fully supported.

major comments (6)
  1. [§5.1, Figs. 10-12] The resolution-convergence study that supports the paper's headline claim is performed with the ARD switch deliberately deactivated, as stated in the text: 'Since there is no strong flow separation in this case, the ARD technique is not activated for the convergence test.' Yet the separated HCD case is only convergent when ARD is active. The paper therefore demonstrates convergence for the no-ARD variant and stable, resolution-improving behavior for the ARD variant, but it does not demonstrate that the ARD variant is itself a consistent discretization. Please add a quantitative budget showing the ARD contribution to the momentum and k/epsilon equations (Eqs. (5), (7), (8) with Eq. (32)) decreasing relative to the physical production and dissipation terms as dp tends to zero, or run a straight-channel convergence study with ARD active.
  2. [§3.1, Eq. (32)] The modulation function mu_c = max(mu_R, e_mu_ij) replaces the physical eddy viscosity by the Riemann-based numerical viscosity whenever the latter is larger. This is an ad hoc switch, not a scale-separated correction: mu_R contains rho0 c0 h and the limiter beta_ij, and the text gives no order-of-convergence estimate for the error introduced when mu_R exceeds e_mu_ij. Because the same switch is essential for HCD stability, the convergence claim requires either a proof that the switch does not alter the RANS balance in the continuum limit or a convergence test with the switch active and its contribution explicitly monitored.
  3. [§4.3, Eq. (40)] The weighted near-wall gradient compensation uses ws=0.1 without ARD and ws=0.5 with ARD. The weight is a free parameter that is explicitly recalibrated when ARD is switched on, and no sensitivity study or error estimate is provided. Since P-layer particles exist at every resolution, this term is active in the continuum limit. The paper must show that the compensated gradient asymptotically approaches the wall-model gradient (Eq. (38)) as dp decreases and that the results are not sensitive to ws in a range independent of ARD. Without this, the observed convergence may reflect the chosen weight rather than the consistency of the discretization.
  4. [§5.2, Fig. 18] For the mildly-curved channel the two resolutions use different yp values (yp=1.59e-3 for Nf=20 and yp=7e-4 for Nf=40), unlike the constant-yp strategy used for the straight and HCD channels. A convergence test with a changing wall-model parameter is not a resolution-convergence test in the same sense. Please either repeat the mildly-curved study at fixed yp or explicitly label the comparison as a combined resolution/wall-model sensitivity study.
  5. [§5.1, first paragraph] The straight-channel SPH simulation is initialized from the converged FDM solution to accelerate flow development. This weakens the convergence evidence: with the same equations, parameters, and a nearly converged initial condition, the SPH integration might preserve the FDM profile without independently generating it. Please quantify the evolution from the initial state, for example by starting from a uniform or coarse initial condition and showing that the SPH solution reaches the same converged profile, or by reporting the decay of residuals from the FDM initialization, to rule out initialization bias.
  6. [§5.3, final paragraph] The statement that 'occasionally, with different initial particle relaxation distribution, a different vortex shape would appear' indicates that the reported HCD solution is not unique for a fixed resolution and parameter set. Since the HCD case is one of the central convergence demonstrations, please quantify the run-to-run variability (number of relaxation seeds, range of velocity and turbulent-kinetic-energy errors) and show that the convergence trend is larger than the seed-to-seed scatter.
minor comments (6)
  1. [Abstract] 'an level-set-based' should be 'a level-set-based'.
  2. [§2.5 heading] 'Duel-criteria time stepping' should be 'Dual-criteria time stepping'.
  3. [§5.1, Figs. 10-12] The dashed line at y/H=0.05 in the legends is not explained in the text; please define it in the caption or in the text.
  4. [§3.2, Eq. (33)] The notation ||R^0_nabla_phi||_2 should be explicitly defined; Eq. (26) defines R^0_nabla_phi as a vector, but the limiter in Eq. (33) uses it as a scalar magnitude without stating the norm.
  5. [§4.2] The relation R=2h-dp and the statement that yp=dp/2 is applied to all particles in the Pext layer would benefit from a one-sentence derivation, since the figure alone does not make clear why a single yp is applied to a layer of finite thickness.
  6. [§5.1-§5.3] The time-averaging procedure for the SPH results is not specified: the averaging window, number of samples, and stationarity checks should be reported for all quantitative profiles, since the comparisons in Figs. 10-12, 18, 20-21, and 29-30 rely on time-averaged data.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central convergence claim is validated against external DNS, experiments, and independent FVM codes, and the new schemes are defined by stated equations rather than by the quantities they predict.

full rationale

The paper's derivation chain does not reduce its reported predictions to its inputs. The new ingredients (ARD, LTVF, the particle-based wall model, the weighted near-wall compensation, and the constant-yp/boundary-offset strategy) are introduced as explicit formulas appended to the standard k-epsilon RANS discretization, and none of the reported velocity or turbulent-kinetic-energy profiles is algebraically equal to a tuned constant or to a fitted quantity. Convergence is checked against external references: DNS of Lee and Moser for the straight channel; experiments of Hunt and Joubert and Eskinazi and Yeh plus FVM results for the curved channels; and two independent FVM codes (OpenFOAM and FLUENT) for the HCD channel. The self-citations to the Riemann solver, TVF, RKGC, and SPHinXsys refer to base methods published elsewhere, not to the present convergence claim, so they are independent support rather than a self-justifying chain. The hand-set parameters (ws, m, yp) are calibration choices and the paper openly reports recalibrating ws when ARD is active; however, no equation shows a predicted benchmark quantity being equal to one of these constants, so this is a calibration and robustness concern, not circularity. The ARD indicator decreasing with refinement is a diagnostic that partly follows from the h-scaling of mu_R, but the convergence claim is carried by the profile comparisons themselves, not by the indicator. Thus no circular step is established.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The method relies on several hand-set coefficients (ws, m, alpha, eta, yp, start-up viscosity) and the standard k-epsilon wall-function closure. No new physical entities are introduced; the P/Pext layers are discretization constructs, not invented physics.

free parameters (6)
  • ws (weighted compensation) = 0.1 without ARD, 0.5 with ARD
    Blending weight between wall-model gradient and inner-particle velocity gradient in Eq. (40); chosen by hand and recalibrated when ARD is toggled.
  • m (LTVF limiter slope) = 1000
    Decay slope in Eq. (33) that limits TVF correction; selected empirically, no sensitivity study reported.
  • alpha (TVF coefficient) = 0.2
    General TVF correction coefficient in Eq. (25), carried over from prior literature and used in LTVF.
  • eta (Riemann dissipation limiter) = 3
    Dissipation limiter parameter in Eq. (18), described as a generally effective empirical parameter.
  • yp (wall-adjacent distance) = 0.025 for HCD; 1.59e-3 and 7.0e-4 for mildly-curved at Nf=20/40; fixed by constant-yp strategy elsewhere
    First-layer distance controls wall function y+; chosen per case and per resolution except when the constant-yp strategy is applied.
  • initial eddy viscosity for delayed activation = 1e-3
    Fixed high eddy viscosity used during laminar start-up in the strongly-curved case (Section 5.2); ad hoc numerical device.
assumptions (4)
  • domain assumption The standard k-epsilon RANS model with constants C_mu, C1, C2, sigma_k, sigma_eps is valid for the benchmark flows.
    Transport equations (7)-(8) and Boussinesq closure (4) are assumed to model the flows without modification.
  • domain assumption The step-wise (standard) wall function is valid and requires y+ in the effective range (roughly 30-100).
    Eqs. (10)-(14) and Section 2.2; the constant-yp strategy is designed to keep y+ inside this range.
  • ad hoc to paper The ARD switching rule max(mu_R, e_mu_ij) preserves the RANS solution.
    Eq. (32) adds Riemann dissipation where eddy viscosity is low; no analysis proves this dissipation is confined to numerical scales or consistent with the RANS equations.
  • ad hoc to paper The weighted near-wall gradient compensation with ws=0.1/0.5 produces the correct production term Gk.
    Eq. (40) blends wall-model and SPH velocity gradients with a hand-set weight; no error estimate or sensitivity study is provided.

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Pith. "Pith review of A weakly compressible SPH method for RANS simulation of wall-bounded turbulent flows." pith.science (2026). https://pith.science/paper/R7APA2HM

@misc{pith2026250118397,
  author       = {Pith},
  title        = {Pith review of: A weakly compressible SPH method for RANS simulation of wall-bounded turbulent flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7APA2HM}},
  note         = {Machine review of arXiv:2501.18397}
}
abstract

This paper presents a Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method for solving the two-equation Reynolds-Averaged Navier-Stokes (RANS) model. The turbulent wall-bounded flow with or without mild flow separation, a crucial flow pattern in engineering applications, yet rarely explored in the SPH community, is simulated. The inconsistency between the Lagrangian characteristic and RANS model, mainly due to the intense particle shear and near-wall discontinuity, is firstly revealed and addressed by the mainstream and nearwall improvements, respectively. The mainstream improvements, including Adaptive Riemann-eddy Dissipation (ARD) and Limited Transport Velocity Formulation (LTVF), address dissipation incompatibility and turbulent kinetic energy over-prediction issues. The nearwall improvements, such as the particle-based wall model realization, weighted near-wall compensation scheme, and constant $y_p$ strategy, improve the accuracy and stability of the adopted wall model, where the wall dummy particles are still used for future coupling of solid dynamics. Besides, to perform rigorous convergence tests, an level-set-based boundary-offset technique is developed to ensure consistent $y^+$ across different resolutions. The benchmark wall-bounded turbulent cases, including straight, mildly- and strongly-curved, and Half Converging and Diverging (HCD) channels are calculated. Good convergence is, to our best knowledge, firstly achieved for both velocity and turbulent kinetic energy for the SPH-RANS method. All the results agree well with the data from the experiments or simulated by the Eulerian methods at engineering-acceptable resolutions. The proposed method bridges particle-based and mesh-based RANS models, providing adaptability for other turbulence models and potential for turbulent fluid-structure interaction (FSI) simulations.

Figures

Figures reproduced from arXiv: 2501.18397 by the authors.

Figure 1
Figure 1. The profiles (from left to right) of the mean flow velocity U, turbulent kinetic energy k, turbulent dissipation rate ϵ and eddy viscosity µt , obtained from k − ϵ RANS model in a fully-developed turbulent straight channel. Note the main stream and wall-adjacent region is separated by the dash lines indicating the first fluid layer thickness yp. Riemann dissipation may lead to instability in the side stream where hi… view at source ↗
Figure 2
Figure 2. The typical flow field in a turbulent expanding channel obtained with￾out Riemann dissipation. unsteady vortex-like pattern, which is inconsistent with RANS model char￾acterized with smooth velocity distribution except at wall-adjacent location. These numerical vortexes may introduce extra production of the turbu￾lent kinetic energy, and break the fundamental eddy-viscosity assumption, eventually reducing the accura… view at source ↗
Figure 3
Figure 3. In a fully-developed turbulent straight channel, the results with the original and limited TVF on velocity gradient and turbulent kinetic energy. Nf gives the number of fluid particles on the cross-section. gradient, even at the high resolution, disappears and the over-prediction problem is well addressed. 4. Near wall treatment and boundary condition 4.1. Near-wall Lagrangian characteristics In RANS model, as shown… view at source ↗
Figures from the paper (27 more)
Figure 4
Figure 4. Figure 4: Division of the near wall regions for the particle-based method. For all the fluid particles that are near the wall, the distance to wall r n w is calculated. The particles that satisfy r n w < R, are defined in the Pext layer, where R = 2h − dp. For those particle, wh…
Figure 5
Figure 5. Figure 5: The particle-based wall model for complex geometry when b = 1, nb are the unit normal vectors evaluated at wall particles. 4.3. Wall boundary conditions and weighted compensation scheme In this work, for the fluid particles near wall or within the Pext layer, 4 types o…
Figure 6
Figure 6. Figure 6: Concept of boundary-offset technique: (a) at the coarse resolution setup, the P layer coalesces with the space between the first layer particles and the wall surface (boundary), and the dummy particles are used to represent the original solid wall; (b) at a refinement …
Figure 7
Figure 7. Figure 7: Boundary-offset technique for complex geometry. ϕw = 0 represents the wall, ϕd = −yp +dpf /2 refers to the boundary (interface between the fluid and wall dummy particles), and dpf is the fluid particle diameter. ϕouter = ϕd + Ld is the outer bound of dummy particles, w…
Figure 8
Figure 8. Figure 8: The initial fluid domain for straight channel under different resolutions when use the offset model. 0 P y dp = 0.05 dp = 0.016 Inflow Outflow Physical wall Concave Position of the 1st layer fluid particles Zoom in [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: The particle distribution for wavy channel under the two different resolutions when use the offset model. is constrained by the wall model. To achieve a rigorous convergence study for the entire FSI system, one possible solution is using more advanced wall models which…
Figure 10
Figure 10. Figure 10: Velocity convergence profiles in a linear scale for the FDM and SPH method. ARD and weighted velocity gradient compensation, on the velocity and k profiles. Activation of the ARD technique makes the particle distribution very lattice and hence the degree of kernel tru…
Figure 11
Figure 11. Figure 11: Velocity convergence profiles in a logarithmic scale for the FDM and SPH method. resolution increases to Nf = 40. As for the weighted compensation for the velocity gradient, without this scheme, the sub-wall-nearest k is significantly smaller although with the Binner …
Figure 12
Figure 12. Figure 12 [PITH_FULL_IMAGE:figures/full_fig_p032_12.png]
Figure 13
Figure 13. Figure 13: Comparison with the FVM result on mainstream velocity and the effect of the proposed technique. and 148400, respectively, the same as the experiments[55, 56]. For the mildly-curved channel, the contours of the velocity and the three turbulent variables are represented…
Figure 14
Figure 14. Figure 14: Comparison with the FVM result on turbulent kinetic energy and the effect of the proposed technique. be a critical problem since the WCSPH method is sensitive to the start-up impulse from the velocity inlet. The common technique includes using a gradually increasing i…
Figure 15
Figure 15. Figure 15: The y + on the wall-adjacent particles at different resolutions. a delayed activation scheme, specifically for the Lagrangian turbulence sim￾ulation. That means initially the turbulent module is not activated except from the calculation of the turbulent viscous force.…
Figure 16
Figure 16. Figure 16: Geometry of the (a) mildly-curved and (b) strongly-curved channels. and from the experiment[56]. The SPH simulations are conducted at the two different resolutions, and a very good convergence is observed for both velocity and k. The FVM results are obtained by using …
Figure 17
Figure 17. Figure 17: Contours of the mildly-curved channel near outlet. 0.0 0.2 0.4 0.6 0.8 1.0  SPH Nf=20 SPH Nf=40 Exp Hunt & Joubert (1979) FVM Farzad et al. (1983) 0.9 0.6 0.5 0.4 0.7 0.3 1.0 0.8 [PITH_FULL_IMAGE:figures/full_fig_p037_17.png]
Figure 18
Figure 18. Figure 18: Outlet velocity comparison for the mildly-curved channel. with the experiment, all of them under-predict the velocity profile near the outer curved wall because of the secondary flow. As for the k profile, near the inner curve, the SPH results agree well with the FVM …
Figure 19
Figure 19. Figure 19: Contours of the strongly-curved channel near outlet. outer curve, the standard k − ϵ model obtained a smaller value while the SPH method yielded a slightly bigger one. This difference is properly due to the low resolution that was used in the FVM simulation. But in th…
Figure 20
Figure 20. Figure 20: Comparison of the outlet velocity for the strongly-curved channel. The Reynolds number is 40000, and the velocity inlet and zero pressure outlet boundary conditions are used. The simulations are conducted by the FVM and SPH methods. The mesh-based computations are bas…
Figure 21
Figure 21. Figure 21: Comparison of the turbulent kinetic energy profiles for the strongly￾curved channel. y x 3 1.5 1 1.5 5 5 30° Monitoring line Platform Bottom wall Up wall [PITH_FULL_IMAGE:figures/full_fig_p040_21.png]
Figure 22
Figure 22. Figure 22: Geometry of the HCD channel and the monitoring position. including the front and rear edges of the platform. The velocity field, shown in [PITH_FULL_IMAGE:figures/full_fig_p040_22.png]
Figure 23
Figure 23. Figure 23: y + value of the HCD channel from the FVM when yp = 0.025. appear and wall-adjacent particles become no longer body-fitted, which not only deteriorates the stability but also reduces the accuracy. The contour of the turbulent kinetic energy exhibits a stronger disturb…
Figure 24
Figure 24. Figure 24: Velocity contours of the HCD channel calculated by the SPH method with or without the ARD technique. wall treatment, although largely reduce the computational effort, results in a discontinuity between the inner fluid domain and the unresolved near wall domain. For th…
Figure 25
Figure 25. Figure 25: Turbulent kinetic energy contours of the HCD channel calculated by the SPH method with or without the ARD technique. (d) With ARD, Nf=120 k 0.0 5.0e-2 1.1e-1 (a) Without ARD, Nf=80 (b) With ARD, Nf=80 Particle vortex (c) Without ARD, Nf=120 Particle vortex Smooth and …
Figure 26
Figure 26. Figure 26: Turbulent kinetic energy in the HCD channel with or without the ARD technique at two additional resolutions 43 [PITH_FULL_IMAGE:figures/full_fig_p043_26.png]
Figure 27
Figure 27. Figure 27: demonstrates the imposing area and degree of the adaptive dissipation under different resolutions. The brighter color indicates a stronger degree of the dissipation applied. And the regions with strong shear, such as the converging segment, back edge of the platform a…
Figure 28
Figure 28. Figure 28: Velocity contour comparison between the FVM and SPH methods. 6. Conclusion In this paper, we propose a WCSPH method for solving the wall-bounded turbulent flow without or with gentle flow separation. The k-ϵ RANS equa￾tions with the wall model are discretized within t…
Figure 29
Figure 29. Figure 29: Velocity comparison between the FVM and SPH methods on the monitoring line. the Lagrangian challenges of the RANS model, such as intense particle shear and near-wall discontinuities, we introduce specialized techniques for both the main stream and near-wall regions. T…
Figure 30
Figure 30. Figure 30: Turbulent kinetic energy comparison between the FVM and SPH methods on the monitoring line. particle vortex caused by the inconsistency and the over-correction in the turbulent plug flow region. For the near-wall treatments, we propose a Lagrangian-meshless imple￾ment…

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