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A Shifted Boundary Method for Thermal Flows

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Pith's one-line read The Shifted Boundary Method on octree meshes accurately enforces Dirichlet and Neumann thermal boundary conditions, producing correct Nusselt numbers across Rayleigh 10^3–10^9 and Reynolds 1–5×10^4 without body-fitted meshes.

desk verdict A solid, well-validated extension of SBM to thermal flows that overclaims in the abstract and lacks convergence evidence at the top of its Rayleigh range. read the letter →

arxiv 2501.00143 v2 pith:N22LC7RU submitted 2024-12-30 physics.flu-dyn cs.NAmath.NA

Cheng-Hau Yang , Guglielmo Scovazzi , Adarsh Krishnamurthy , Baskar Ganapathysubramanian This is my paper · ORCID
classification physics.flu-dyncs.NAmath.NA MSC 65M6076D0576R1080A19
keywords ShiftedBoundaryMethodImmersedComputationalfluiddynamicsIncompleteOctreeOptimalsurrogateWeakconditionsBuoyancy-drivenconvectionResidual-basedvariationalmultiscale
topics Navier-Stokes Regularity
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 aims to establish that the Shifted Boundary Method (SBM), implemented on incomplete octree meshes, delivers quantitatively accurate coupled flow-and-heat simulations without body-fitted meshing. It claims that both Dirichlet (temperature) and Neumann (heat-flux) boundary conditions can be imposed precisely on a surrogate boundary via a Taylor-expansion shift, with a correction factor $\tilde{n}\cdot n$ that fixes the area mismatch between the surrogate and true boundaries. The authors validate the method on two- and three-dimensional tests—lid-driven cavities with obstacles, heated cylinders with constant wall temperature and uniform heat flux, natural convection around a sphere, and a gyroid—across Rayleigh numbers $10^3$–$10^9$ and Reynolds numbers $1$–$5\times 10^4$. If the claim holds, thermal engineering simulations on complex geometries can rely on automatically generated octree meshes while still extracting accurate Nusselt numbers and boundary fluxes.

What carries the argument

The load-bearing object is the shifted boundary condition of Eq. (42), $S_{D,h}u = E_{u_D}$ on the surrogate boundary $\tilde{\Gamma}_{D,h}$, where $S_{D,h}u(\tilde{x}) = u(\tilde{x}) + \nabla u(\tilde{x})\cdot d(\tilde{x})$ and $d$ is the distance vector from the surrogate boundary point $\tilde{x}$ to its closest-point projection on the true boundary. Together with the area correction factor $\tilde{n}\cdot n$ in the Neumann term of Eq. (46), this shift converts a cut-cell boundary into a surrogate boundary where derivatives can be evaluated directly from nodal values of linear shape functions. The formulation is Nitsche-based, with consistency, adjoint-consistency, and penalty terms, and it collapses to standard Nitsche's method when the mesh is body-fitted ($d\to 0$).

What would settle it

A mesh refinement study at Ra=$10^{8}$ for the natural-convection sphere and at Ra=$10^{3}$ for the gyroid, if it showed the Nusselt numbers not converging or behaving erratically, would falsify the smoothness premise in the very regimes the paper claims to cover. A geometry with a sharp corner where two true-boundary points are equidistant from the same surrogate point would also break the closest-point projection assumption.

Watch

Extended reading notes

Core claim

The central claim is that the SBM with octree meshes yields accurate thermal fluxes (Nusselt numbers) for both Dirichlet and Neumann boundary conditions, provided the surrogate boundary includes all cut elements ($\lambda=1$) and the Neumann term contains the area correction factor $\tilde{n}\cdot n$. The method uses a Taylor expansion to shift boundary conditions from the true boundary to the surrogate boundary, discarding the remainder, and an Nitsche-based variational formulation to enforce them weakly. The authors show that local and global Nusselt numbers converge at rates near 1 to 1.44 even with linear elements, and that omitting the area correction produces $O(1)$ errors in the global Nusselt number because the surrogate boundary area can overestimate the true boundary by a factor like $\pi/4$ for a cylinder.

Load-bearing premise

The method's accuracy rests on the premise that the flow and temperature fields are smooth enough across the gap between the surrogate octree boundary and the true boundary that the discarded Taylor remainder in the shifted boundary condition is negligible, and that the closest-point projection is uniquely defined.

Editorial extensions

If this is right

  • Thermal engineering simulations on complex geometries (heat exchangers, building ventilation, urban heat islands) can use automatically generated octree meshes, bypassing labor-intensive body-fitted meshing.
  • The linear semi-implicit Navier-Stokes solver runs about 60% faster than the fully implicit version at the same accuracy for the benchmark cylinder case, making high-Rayleigh or high-Reynolds simulations more affordable.
  • Neumann (heat-flux) boundary conditions, often a weak point of immersed boundary methods, are computable on the true boundary from octree meshes, as demonstrated by mesh-converged global fluxes at about order 1.
  • The area correction term is essential: without it, global Nusselt numbers are overestimated by roughly the ratio of surrogate to true boundary areas ($\pi/4$ for a circle in a square grid), so the SBM shift is a qualitative prerequisite, not a refinement.

Reading between the lines

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

  • The smoothness assumption in the Taylor expansion suggests that at very high Rayleigh numbers, where thermal boundary layers become extremely thin, the octree must resolve the boundary layer; the absence of a mesh convergence study at Ra=10^8 and for the gyroid leaves this as an open practical question.
  • Closest-point projection may fail to be unique near sharp corners or non-Lipschitz boundaries; level-set alternatives are mentioned but not tested, so robustness on non-smooth geometry is a plausible boundary of the method's applicability.
  • The observed $\pi/4$ area-correction ratio implies a general principle for immersed methods on Cartesian grids: boundary flux errors scale with the mismatch between surrogate and true boundary measure, and correcting that measure may be as important as the boundary-condition shift itself.
  • One could test the method's generality by applying it to a conjugate heat-transfer problem (solid and fluid coupled at an interface), where the Neumann condition on the fluid side is driven by the solid's flux; the SBM's ability to compute fluxes on both sides of a cut cell would be directly exercised.
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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

3 major / 5 minor

Summary. This paper presents an implementation of the Shifted Boundary Method (SBM) on incomplete octree meshes for coupled thermal incompressible flow. The authors combine a linear semi-implicit BDF2 Navier-Stokes discretization with a fully implicit energy equation, VMS stabilization, SUPG and backflow stabilization, and Nitsche/SBM weak boundary conditions. They validate the method with manufactured solutions, cylinder-flow benchmarks, lid-driven cavities, flow past heated cylinders with constant wall temperature and constant wall heat flux, natural convection around a sphere in a cube (Ra=1e3--1e8), Rayleigh-Benard convection up to Ra=1e9, and a gyroid/sphere demonstrator. The main claims are that Octree-SBM enforces Dirichlet and Neumann thermal boundary conditions accurately on non-boundary-fitted meshes and that coupled thermal-flow statistics are captured across laminar, transitional, and turbulent regimes without any additional numerical treatment beyond RB-VMS.

Significance. The potential contribution is substantial if the claims are fully supported. Avoiding boundary-fitted meshes while retaining accurate boundary-flux evaluation is practically valuable for complex-geometry thermal flows, and the speedup measured in Section 4.1 (roughly factor 2--2.4) is a concrete efficiency gain. The paper's strengths include its broad independent benchmark base (Chen, Khanafer, Yoon, Bharti, Hsu, Xu), the manufactured-solution verification in Appendix A.1, and the reported mesh-convergence orders for Nusselt numbers in Figures 13 and 16. The use of an open-source framework and detailed benchmark tables makes the results reproducible in principle. However, the highest-Rayleigh-number validation is thinner than the abstract suggests, and the 'no additional numerical treatments' statement is inaccurate as written.

major comments (3)
  1. [Abstract; Sections 2.3.1 and 2.4, Eqs. (29), (34), (35)] The abstract's claim that results are obtained 'without any additional numerical treatments, beyond RB-VMS' is contradicted by the formulation itself. Equation (29) contains an explicit SUPG term for the energy equation, and Eqs. (34)--(35) add backflow stabilization to both momentum and energy. These are additional stabilization treatments beyond the RB-VMS fine-scale terms. The sentence should be reworded to say what is actually meant (for example, 'without an additional turbulence model') or the SUPG and backflow terms should be acknowledged as part of the proposed framework.
  2. [Section 4.4, Tables 11 and 12; Appendix A.2] The central claim of quantitative accuracy up to Ra=1e8 is not yet supported at the top of the range. Table 12 lists single-resolution results for Ra=1e8 (NuT=18.75, NuSp=54.82) with no mesh-convergence study, while the nearby convergence study at Ra=1e7 (Table 11) shows a 4.5% change in NuSp between levels 8 and 9 (31.63 to 33.05) and a 2.8% change in NuT (11.25 to 11.57). Since the Ra=1e8 case uses the same refinement strategy, its discretization error could be comparable to the agreement level claimed for lower Ra. The same single-resolution limitation applies to the Ra=1e9 Rayleigh-Benard comparison in Appendix A.2, which rests on one 512x512 mesh. The authors should either add convergence evidence at these regimes or temper the abstract's range claim to the values where convergence is demonstrated.
  3. [Section 2.8, Remark after Eq. (46); Section 4.3, Fig. 16] The claim that SBM ensures 'precise enforcement' of Neumann boundary conditions should be qualified by the acknowledged simplification in the SBM Neumann term. The remark after Eq. (46) states that the Hessian shift is dropped and that this can reduce the L2 convergence order by one. The UHF tests show first-order convergence in the global flux (order 1.02 in Fig. 16b), which is adequate for the low-Reynolds-number cases shown but does not establish precise Neumann enforcement at high Rayleigh or Reynolds numbers. The authors should state this limitation explicitly in the abstract or conclusions, or provide additional high-regime Neumann-boundary validation.
minor comments (5)
  1. [Section 4.3, first paragraph] The sentence beginning 'We first consider a a forced convection problem' contains a duplicated article; please proofread for similar typographical artifacts that appear elsewhere in the text.
  2. [Section 4.4, Table 13 caption] The caption says 'for different Reynolds numbers,' but the parameter being varied is the Rayleigh number; correct the caption.
  3. [Section 2.3.1, after Eq. (21)] The text says 'CM and CE are chosen as 36,' but CE does not appear in the displayed equations (only CM appears in the momentum stabilization parameter); clarify where CE is used or remove the reference to it.
  4. [Section 4.5] The gyroid results are presented qualitatively through streamlines and temperature contours, with no quantitative accuracy check; please state explicitly that this is a demonstration case rather than a validation case.
  5. [Figure 19 and surrounding text] The text lists mesh refinement levels separately for Ra=1e3/1e4, Ra=1e5/1e6, and Ra=1e7/1e8; adding a small table that maps each Rayleigh number to its base and local refinement levels would improve readability and reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found: the SBM formulation is derived in-paper from a Taylor expansion and validated against independent literature benchmarks and manufactured solutions, so the central accuracy claim does not reduce to its own inputs.

full rationale

The paper's central claim is that Octree-SBM accurately enforces Dirichlet and Neumann thermal boundary conditions on non-boundary-fitted meshes. The shifted boundary condition (Eq. 42) is directly derived from the Taylor expansion in Eq. 39, with the discarded remainder explicitly identified as o(||d||^2); it is not defined in terms of the Nusselt numbers later reported. The SBM variational forms (Eqs. 45 and 46) are stated to collapse to standard Nitsche formulations when d -> 0, and the Neumann area-correction factor (n . nT) is derived through the integration-by-parts calculation in Eq. 47 rather than tuned to match benchmark outputs. Validation is anchored to independent external studies (Chen, Khanafer, Yoon, Bharti, Dennis, Ahmad and Qureshi, Hsu, Zukauskas and Ziugzda, Pachpute, and others) and to manufactured-solution convergence tests in Appendix A.1, so the agreement with literature is not produced by construction. The only self-referential element is the mesh-convergence study in Section 4.2.2, where the finest-mesh simulation is labeled 'ground truth'; the authors explicitly acknowledge that 'the "ground truth" solution is not the exact solution but rather the result of the highest refinement level simulation,' and this is a standard relative-convergence check, not a fitted parameter renamed as a prediction. Self-citations to prior SBM work (e.g., refs. [37,40,41,47,65]) are used for implementation context and algorithmic details, but the load-bearing shifted-boundary formulas are re-derived in the present paper and externally benchmarked, so those citations do not carry the argument. The absence of a mesh-convergence study at Ra = 1e8 and in the gyroid case is a support gap at the top of the claimed Rayleigh-number range, but that is an evidence-completeness concern, not circularity: no input quantity is identical by construction to the claimed output quantity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim relies on standard stabilized-FEM machinery (VMS, SUPG, Nitsche-type penalty constants), a Boussinesq incompressible flow model, and two SBM-specific premises: smoothness of the solution across the shift distance, and uniqueness of the closest-point projection for the test geometries. No new physical entities are introduced. The penalty and stabilization constants are hand-chosen algorithmic parameters, not fitted to the validation data.

free parameters (3)
  • Nitsche/SBM penalty parameters C_B^M (momentum) and C_B^E (energy) = 200 and 400; increased by factor 10 upon non-convergence
    Hand-chosen stabilization constants in Eqs. 45-46. These are algorithmic tuning parameters, not fitted to reference data, but the escalation procedure is ad hoc.
  • Backflow stabilization coefficients beta_0 and beta_theta = 0.5
    Hand-chosen in Section 2.4 to damp outflow backflow; not fitted to benchmarks.
  • VMS constant C_M = 36
    Standard variational multiscale stabilization constant in Eq. 21 for the momentum equation.
assumptions (5)
  • domain assumption Incompressible Newtonian fluid with Boussinesq-type buoyancy forcing f_i = theta * delta_im; non-dimensional parameters chosen per convection type (Table 1).
    Underlies the governing equations Eqs. 1-7 and restricts applicability to moderate temperature differences where density variations are linear.
  • domain assumption The shifted boundary condition (Eq. 42) is an accurate approximation because the discarded Taylor remainder R_D is o(||d||^2) (Eq. 39), requiring solution smoothness across the surrogate-to-true boundary gap.
    At high Rayleigh numbers with thin thermal boundary layers, errors depend on mesh resolution; this is acknowledged but only partially tested (no Ra=1e8 convergence study).
  • domain assumption The closest-point projection M_h (Eq. 37a) yields a uniquely defined, accurate distance vector d for the sphere, cylinder, and gyroid surfaces used in the tests.
    Ambiguous projections at edges or high curvature would introduce errors; the paper cites earlier corner analysis [41,42] but does not verify this for the gyroid.
  • standard math VMS and SUPG stabilization with the stated parameters produce a stable, convergent discrete solution for the tested regimes.
    Standard stabilized FEM framework; the MMS results (Appendix A.1) support this for the base solver.
  • domain assumption Reference benchmark values cited from the literature are sufficiently accurate to serve as ground truth for validation.
    Validation comparisons in Tables 8, 9, 12 and Figures 7, 8, 10, 11, 14, 18 assume these references are correct.

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Pith. "Pith review of A Shifted Boundary Method for Thermal Flows." pith.science (2026). https://pith.science/paper/N22LC7RU

@misc{pith2026250100143,
  author       = {Pith},
  title        = {Pith review of: A Shifted Boundary Method for Thermal Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N22LC7RU}},
  note         = {Machine review of arXiv:2501.00143}
}
abstract

This paper presents an incomplete Octree mesh implementation of the Shifted Boundary Method (Octree-SBM) for multiphysics simulations of coupled flow and heat transfer. Specifically, a semi-implicit formulation of the thermal Navier-Stokes equations is used to accelerate the simulations while maintaining accuracy. The SBM enables precise enforcement of field and derivative boundary conditions on cut (intercepted) elements, allowing for accurate flux calculations near complex geometries, when using non-boundary fitted meshes. Both Dirichlet and Neumann boundary conditions are implemented within the SBM framework, with results demonstrating that the SBM ensures precise enforcement of Neumann boundary conditions on Octree-based meshes. We illustrate this approach by simulating flows across different regimes, spanning several orders of magnitude in both the Rayleigh number ($Ra \sim 10^3$--$10^9$) and the Reynolds number ($Re \sim 10^0$--$10^4$), and covering the laminar, transitional, and turbulent flow regimes. Coupled thermal-flow phenomena and their statistics across all these regimes are accurately captured without any additional numerical treatments, beyond a Residual-based Variational Multiscale formulation (RB-VMS). This approach offers a reliable and efficient solution for complex geometries, boundary conditions and flow regimes in computational multiphysics simulations.

Figures

Figures reproduced from arXiv: 2501.00143 by the authors.

Figure 1
Figure 1. The surrogate domain, its boundary, and the distance vector d. The mapping sketched in Figure 1c is defined as follows: Mh : Γ˜ h → Γ , (37a) x˜ 7→ x , (37b) where Mh maps any point x˜ ∈ Γ˜ h on the surrogate boundary to a point x = Mh(x˜) on the physical boundary Γ. In this study, Mh is defined as the closest-point projection of x˜ onto Γ, as illustrated in Figure 1c. Using this mapping, a distance vector function … view at source ↗
Figure 2
Figure 2. Diagram illustrating the block iteration technique used to perform multiphysics simulations of thermal incompressible flow (NSHT). 3. Implementation details. 3.1. Numerical implementations Our computational framework is built on two core components: Dendro-KT [10, 74] and PETSc, both of which play essential roles in enabling high-performance, large-scale scientific simulations across various domains. These tools wor… view at source ↗
Figure 3
Figure 3. Schematic graph of various NSHT simulations performed in the paper. Additionally, efficient distance function calculations play an essential role in the SBM. To address this, we use the k-d tree nanoflann library [81]. For further details on the implementation of these components, readers are referred to Yang et al. [47]. 3.2. Block-iterative strategy The block-iterative strategy we adopt has proven effective in add… view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Two-dimensional flow past a cylinder (Section 4.1): plot of the various levels of mesh refinement. An increase of mesh resolution by 1 represents a reduction in element size by a factor of 2. Here, the finest element and the coarsest element vary by factor of 64 in len…
Figure 5
Figure 5. Figure 5: Thermal lid-driven cavity test with one circular obstacle (Section 4.2.1): contour lines of the non-dimensional temperature at steady state are plotted from 0 to 1 at regular intervals of 0.1. 4.2. Dirichlet boundary condition for heat transfer in two dimensions 4.2.1.…
Figure 6
Figure 6. Figure 6: Thermal lid-driven cavity test with one circular obstacle (Section 4.2.1): close-up of the temperature contour at steady state: the red line indicates the geometric boundary, while the white line represents the zero-temperature contour. The two closely match. Nulocal …
Figure 7
Figure 7. Figure 7: Thermal lid-driven cavity test with one circular obstacle (Section 4.2.1): comparison of the local Nusselt number along the bottom wall (∇θ · n, i.e., the non-dimensional thermal flux), at steady state. Here, n is the inward-facing normal vector directed toward the cyl…
Figure 8
Figure 8. Figure 8: Thermal lid-driven cavity test with one circular obstacle (Section 4.2.1): comparison of the steady-state temperature distributions against the results of Chen et al. [91] at various Richardson numbers. The temperature profiles are shown along (a) x = 0.15, (b) x = 0.8…
Figure 9
Figure 9. Figure 9: Thermal lid-driven cavity test with two circular obstacles (Section 4.2.1): steady-state temperature contours and local mesh refinement. The contour lines represent non-dimensional temperatures ranging from 0 to 1, plotted at regular intervals of 0.1. parameters are Re…
Figure 10
Figure 10. Figure 10: Thermal lid-driven cavity test with two circular obstacles (Section 4.2.1): distribution at steady state of the local Nusselt number (Nulocal = ∇θ · n, i.e., a non-dimensional thermal flux) and comparison against the simulations of Khanafer et al. [92]. (a) Local Nu o…
Figure 11
Figure 11. Figure 11: Thermal lid-driven cavity test with two circular obstacles (Section 4.2.1): steady-state local Nusselt number distribution (Nulocal = ∇θ·n, non-dimensional thermal flux) along the cavity walls, compared with Khanafer et al. [92]. 4.2.2. Flow past a heated cylinder wit…
Figure 12
Figure 12. Figure 12: Flow past a heated cylinder with constant wall temperature (CWT, Section 4.2.2): distribution of the levels of mesh refinement. Note that the mesh is highly refined near the cylinder boundary, where the thermal boundary layer forms. size (0.01) used in the boundary-fi…
Figure 13
Figure 13. Figure 13: Mesh convergence study for the time-averaged global Nusselt number (Nu = P −1 R Γ ∇θ · ndΓ, where P represents the perimeter of the circular cylinder) in the flow past a heated cylinder with constant wall temperature at Re = 100 (CWT, see Section 4.2.2). with an order…
Figure 14
Figure 14. Figure 14: Flow past a heated cylinder with constant wall temperature (CWT, Section 4.2.2): Comparison with Hsu [98] of the (time-averaged) global Nusselt number (Nu = P −1 R Γ ∇θ · ndΓ, where P represents the perimeter of the circular cylinder) and the local Nusselt number (∇θ …
Figure 15
Figure 15. Figure 15: Flow past a heated cylinder with constant wall heat (UHF, Section 4.3): mesh refinement levels, with finer grids near the cylinder boundary. (a) Mesh convergence study for the global Nusselt number (Nu = P −1 R θ −1 dΓ), where P represents the perimeter of the circula…
Figure 16
Figure 16. Figure 16: Mesh convergence study for the global Nusselt number (Nu = P −1 R θ −1 dΓ) and the global non-dimensional flux (Nu = P −1 R ∇θ·n dΓ) in the flow past a heated cylinder with uniform heat flux at Re = 10 (UHF, see Section 4.3) [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]
Figure 17
Figure 17. Figure 17: Flow past a heated cylinder with constant wall heat (UHF, Section 4.3): temperature contours at steady-state for various Reynolds numbers. Nulocal (a) Re = 10 Nulocal (b) Re = 20 [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]
Figure 18
Figure 18. Figure 18: Flow past a heated cylinder with constant wall heat flux (UHF, Section 4.3): Comparison against Bharti et al. [66] of the local Nusselt number (θ −1 ) distribution at steady state, as a function of the angular position. 4.4. Natural convection around a sphere in a cub…
Figure 19
Figure 19. Figure 19: Natural convection around a sphere in a cubic enclosure (Section 4.4): element sizes for various Rayleigh numbers [PITH_FULL_IMAGE:figures/full_fig_p025_19.png]
Figure 20
Figure 20. Figure 20: Natural convection around a sphere in a cubic enclosure (Section 4.4): instantaneous (non-dimensional) temperature contours for various Rayleigh numbers. Contours are plotted from 0 to 1 at regular intervals of 0.1 [PITH_FULL_IMAGE:figures/full_fig_p026_20.png]
Figure 21
Figure 21. Figure 21: Natural convection around a sphere in a cubic enclosure (Section 4.4): instantaneous velocity Line Integral Convolution (LIC) visual￾izations at various Rayleigh numbers [PITH_FULL_IMAGE:figures/full_fig_p027_21.png]
Figure 22
Figure 22. Figure 22: Natural convection around complex obstacles (Section 4.5): illustration of the problem setup and mesh refinement strategy. while on the surface of the sphere, it is set as θ = 1. θ = −1 (non-dimensional temperature) represents a relatively lower temperature than other…
Figure 23
Figure 23. Figure 23: Natural convection around complex obstacles (Section 4.5): streamlines and temperature contours after the flow reached steady-state. Temperature contour values are set to -1, -0.6, -0.2, 0.2, 0.6, and 1. Streamlines are colored by the y-direction velocity [PITH_FULL_…
Figure 24
Figure 24. Figure 24: Natural convection around complex obstacles (Section 4.5): streamlines of the flow at steady-state passing through the gyroid (front view). This image highlights the intricate streamline pattern as they navigate through the gyroid obstacle in a natural convection sett…
Figure 25
Figure 25. Figure 25: Scaling performance of the Octree-SBM computation for two-dimensional flow past a cylinder with a uniform heat flux, as evaluated on TACC’s Frontera supercomputer (Section 4.6). linear semi-implicit Navier-Stokes and heat transfer equations, which significantly enhanc…

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