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arxiv: 2607.06322 · v1 · pith:L4KMLEGZ · submitted 2026-07-07 · physics.flu-dyn

Strain-Rate-Consistent varepsilon-Based Non-Premixed Flamelet Model

Sylvain L. Walsh , Yalu Zhu , Feng Liu , William A. Sirignano This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 reserved 2026-07-08 09:47 UTCglm-5.2pith:L4KMLEGZrecord.jsonopen to challenge →

Figure 1
Figure 1. Figure 1: : Flow configuration and computational grid. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] reproduced from arXiv: 2607.06322
classification physics.flu-dyn
keywords flamelet modelnon-premixed combustionstrain-rate consistencyturbulence dissipation rateflamelet progress variableRANScombustion modelingturbulence-chemistry interaction
0
0 comments X

The pith

Swapping the progress variable for turbulence dissipation fixes strain-rate

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

This paper identifies a structural inconsistency in the widely used flamelet/progress-variable (FPV) model for non-premixed combustion: the progress variable that selects the local flamelet state evolves through advection, diffusion, and chemical production, with no explicit dependence on the local strain rate. The authors show that this decoupling causes the FPV model to preferentially sample near-equilibrium flamelet states even in high-strain regions, weakening the intended connection between the computed flow field and the strain-rate-controlled flamelet response. They propose an alternative, the epsilon-Z flamelet model, in which the turbulence kinetic energy dissipation rate (epsilon) is used to directly infer the imposed flamelet strain rate. This restores a one-to-one link between the local turbulent state and the flamelet lookup, so that strain-induced effects like flame standoff, local quenching, and pressure-dependent extinction emerge naturally from the coupling. To handle the discontinuities that strain-rate-based selection reintroduces, the model explicitly transports a reduced set of species, allowing combustion products to persist through locally quenched regions without a progress variable. A reactant-availability scaling corrects tabulated source terms when the transported composition drifts off the flamelet manifold. Two-dimensional RANS simulations of a transonic accelerating reacting mixing layer demonstrate that the epsilon-Z model produces more physically localized heat-release fields, concentrates combustion in a narrower range of turbulent dissipation rates, and achieves better consistency with the pressure-dependent flammability limit than the FPV model.

Core claim

The central finding is that the FPV model's progress variable, driven by chemical production and transport rather than by local strain rate, collapses toward equilibrium flamelet states throughout the reacting layer regardless of the actual strain-rate field. This is shown to be a consequence of the source-term structure: the mixture-fraction-integrated progress-variable source term has a locally attracting region at high flamelet parameter values, causing the transported solution to relax toward near-equilibrium states. Replacing the progress variable with epsilon as the coupling variable restores direct strain-rate consistency, and explicit transport of selected species preserves composi!t

What carries the argument

The epsilon-to-strain-rate mapping S* = (1/2) sqrt(C_vd * epsilon / (nu * [S1^2 + 1 - S1])) (Eq. 26), which links the RANS-computed turbulence dissipation rate to the flamelet compressive strain rate; the reactant-availability scaling alpha = alpha_f * alpha_ox (Eq. 37), which limits tabulated source terms when transported species mass fractions fall below flamelet-manifold values; and the explicit transport of a reduced species set (6 major species plus 1 lumped residual) to preserve product continuity across quenched regions.

If this is right

  • Simulations of combustion chambers with strong strain gradients (e.g., turbine burners, rocket injectors) would see different predictions for flame anchoring, extinction, and reignition when using the epsilon-Z model instead of FPV, with direct consequences for combustor design and stability analysis.
  • The finding that FPV preferentially samples near-equilibrium states suggests that published FPV-based predictions of heat-release spatial distribution may be systematically broader and less physically localized than the actual strain-rate field would imply.
  • The reactant-availability scaling for off-manifold states provides a template for handling composition drift in other tabulated-chemistry approaches where transported species diverge from precomputed manifolds.
  • Extension to three-dimensional LES would require quantifying the cost of explicit species transport against FPV's lower equation count, and assessing whether the epsilon-to-S* relation holds under subgrid-scale turbulence modeling.

Where Pith is reading between the lines

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

  • The sensitivity of the model to the coefficient C_vd (set to 1) and the assumed strain-rate distribution S1 (set to 1/2 for axisymmetric counterflow) is not assessed. If DNS data reveals C_vd varies significantly with flow geometry or Reynolds number, the predicted flamelet strain rates and hence the heat-release distribution could shift substantially.
  • The use of a Dirac delta-PDF for epsilon ignores turbulent fluctuations in dissipation rate; in highly turbulent regions, the nonlinear S*(epsilon) mapping could produce systematically biased mean strain rates, analogous to the known issue with delta-PDFs for the progress variable.
  • The off-manifold reactant-availability scaling is a local correction that does not account for the history of compositional departure; in flows with repeated quench-reignition cycles, accumulated off-manifold drift could make the tabulated source terms increasingly unrepresentative of the actual chemistry.

Load-bearing premise

The entire model output depends on the relation between the turbulence dissipation rate epsilon and the flamelet strain rate S*, which requires a coefficient C_vd and an assumed strain-rate distribution. The paper sets C_vd to 1 and assumes axisymmetric counterflow geometry, but states that C_vd remains to be definitively determined from DNS data. If this single coefficient does not accurately represent the turbulent cascade across different flow conditions, the inferred fl!t

What would settle it

Run a DNS of a turbulent non-premixed flame with known strain-rate statistics and compare the DNS-inferred flamelet strain rates against the epsilon-based S* prediction. If the epsilon-to-S* mapping systematically mispredicts the local flamelet state (e.g., by more than the difference between the FPV and epsilon-Z models shown here), the central advantage of the proposed model over FPV would not hold.

Figures

Figures reproduced from arXiv: 2607.06322 by Feng Liu, Sylvain L. Walsh, William A. Sirignano, Yalu Zhu.

Figure 2
Figure 2. Figure 2: : Left: Solutions to the flamelet equations presented as S-shaped curves for [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: : Mapping of the flamelet maximum temperature and the heat-release rate [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: : Strain-rate parametrized flamelet solutions at different background pres [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: : Schematic of on- and off- manifold states. [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: : Profiles of temperature and velocity at three different streamwise locations [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: : Temperature contours for OSK (top), FPV (center) end [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: : Left a): contours of the mean strain-rate magnitude, [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: : (a) Spatial distribution of the normalized flamelet parameter and (b) [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: : Quantities of interest for the FPV results at the streamwise location x = [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 3
Figure 3. Figure 3: The upper stable branch is used in this comparison, since [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 11
Figure 11. Figure 11: Heat-release rate, Q e˙ , predicted by the OSK, FPV, and ϵ–Z combustion models. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: : Heat-release-rate weighted joint statistics of [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Mixture compositions and YeCO field. different local thermochemical states, not merely different mixing-layer thicknesses. In the same way that the distribution of λ affects the FPV heat-release field, it also affects the retrieved species mass fractions and the corresponding implied chemical state represented by the table. By contrast, the OSK and ϵ–Z models solve explicit species transport equations. Mo… view at source ↗
read the original abstract

This numerical study examines a strain-rate inconsistency in the conventional flamelet/progress-variable (FPV) formulation for non-premixed combustion and proposes an alternative coupling based on the turbulence kinetic energy dissipation rate, $\varepsilon$. Two-dimensional Reynolds-averaged Navier-Stokes (RANS) simulations of a transonic accelerating reacting mixing layer are performed using one-step kinetics, a conventional FPV model, and the proposed $\varepsilon$-$Z$ flamelet model. The analysis focuses on the relation between the RANS-computed mean strain-rate field and the local strain rate imposed on the flamelet through the coupling between the flow computation and the flamelet library. In the FPV formulation, the flamelet state is selected through a transported progress variable, whose evolution is governed by advection, diffusion, and chemical production rather than by the local strain-rate environment. The present results show that this can lead to preferential sampling of near-equilibrium flamelet states in high-strain regions, thereby weakening the intended connection between the computed flow field and the strain-rate-controlled flamelet response. In the $\varepsilon$-$Z$ formulation, $\varepsilon$ is used to infer the imposed flamelet strain rate, $S^*$, so that the local flamelet state is directly constrained by the modeled turbulence field and the pressure-dependent flammability limit. Selected species are transported explicitly, allowing products to persist through locally quenched regions, while a reactant-availability scaling limits tabulated source terms when the transported composition departs from the flamelet manifold.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

4 major / 10 minor

Summary. This manuscript examines a strain-rate inconsistency in the conventional flamelet/progress-variable (FPV) formulation for non-premixed combustion and proposes an alternative coupling based on the turbulence kinetic energy dissipation rate, epsilon. Two-dimensional RANS simulations of a transonic accelerating reacting mixing layer are performed using one-step kinetics (OSK), a conventional FPV model, and the proposed epsilon-Z flamelet model. The central diagnosis is that the FPV formulation decouples the selected flamelet state from the local strain-rate field, leading to preferential sampling of near-equilibrium states in high-strain regions. The proposed epsilon-Z model restores direct strain-rate consistency by inferring the flamelet strain rate S* from epsilon and produces more physically localized heat-release fields. Explicit transport of selected species is used to preserve composition transport across locally extinguished regions.

Significance. The FPV inconsistency analysis (Sec. 4.3) is well-argued and self-contained; it identifies a genuine structural issue in how the transported progress variable selects flamelet states independently of the local strain-rate environment. The heat-release-weighted statistics (Fig. 12) provide a quantitative and falsifiable comparison between FPV and epsilon-Z, showing that 11.9% of FPV heat release occurs in locally extinguished regions versus 1.7% for epsilon-Z. The epsilon-S* relation (Eq. 26) is drawn from a parameter-free derivation by Sirignano et al. [30] for a 3D rotating counterflow, not a fitted result. The off-manifold reactant-availability scaling (Eqs. 35-38) is a pragmatic and clearly documented correction. The model produces falsifiable predictions (flame standoff at ~2 mm, downstream quenching at x~110 mm) that are directly tied to the strain-rate coupling. However, the significance of the epsilon-Z predictions is tempered by the absence of validation against DNS or experimental data and the lack of sensitivity analysis for the coefficient C_vd.

major comments (4)
  1. Sec. 3.4, Eq. (26): The mapping from epsilon to S* depends on C_vd, which is set to 1 without sensitivity analysis. The paper itself states this coefficient 'remains to be definitively determined from DNS data.' Since S* is the sole variable selecting the flamelet state from the library (Eq. 28-29), C_vd directly controls every model output: flame standoff distance, downstream quenching location (x~110 mm), and the heat-release-weighted statistics in Fig. 12. A factor-of-two change in C_vd would shift S* by sqrt(2), potentially moving the flame standoff and quenching locations substantially and altering the comparison with FPV that constitutes the paper's main evidence. The claim that epsilon-Z produces 'more physically localized' heat-release fields is not established without either (a) showing insensitivity to C_vd or (b) validation against DNS/experiment. The FPV inconsistency诊断 (Sec.
  2. 4.3) is self-contained and does not depend on C_vd, so that part stands independently. But the positive claims about epsilon-Z require a sensitivity analysis or at minimum a bounded discussion of how robust the qualitative findings are to C_vd variation.
  3. Sec. 3.4, Eq. (29): The Dirac delta-PDF for S* (or epsilon) is acknowledged as needing further investigation but its impact is not assessed. Since S* varies substantially across the mixing layer (Fig. 8b), the assumption of zero variance in S* at the flamelet scale may significantly bias the retrieved heat-release rates, particularly near the flammability limit where the S-curve is steep. A brief discussion of the expected direction of bias (e.g., Jensen's inequality applied to the nonlinear heat-release response) would strengthen the assessment.
  4. Sec. 4.1, Fig. 7 and Sec. 4.4, Fig. 11: The epsilon-Z model predicts a flame standoff of ~2 mm and downstream quenching at x~110 mm, both attributed to strain-rate-induced extinction. However, without an independent reference (DNS or experiment), it is unclear whether these are physically correct predictions or artifacts of the epsilon-S* mapping. The OSK model predicts zero standoff, but OSK uses one-step kinetics without strain-rate coupling, so it is not an independent validation of the standoff mechanism. The paper should clarify what evidence supports the physical correctness of these specific predictions beyond internal consistency with the model's own assumptions.
minor comments (10)
  1. Sec. 2.1: 'a a non-premixed flame' — duplicate article.
  2. Sec. 3.2: 'momentum equations momentum equations' — duplicated phrase.
  3. Sec. 3.3: 'has ben ensured' — typo for 'has been.'
  4. Sec. 3.3: 'we compare compare Eq. (24)' — duplicated word.
  5. Sec. 3.4, Fig. 4b caption: 'shoes surfaces' — typo for 'shows.'
  6. Sec. 4.3, Fig. 10 caption: The caption references panels a), b), c) but the figure sub-labels should be verified for consistency with the text description.
  7. Sec. 3.4.3, Eq. (38): The reactant-availability scaling assumes linear dependence on local reactant concentration (justified via elementary reaction analysis, Eqs. 39-41). For chain-branching steps or reactions with non-unity stoichiometric coefficients, this linearity may not hold. A brief note on the range of validity would be helpful.
  8. Table 1: The footnote markers (**, ***, ****) are somewhat hard to track. Consider integrating key footnotes into the table cells or using a more standard footnote format.
  9. Sec. 4.4, Fig. 12: The nonlinear color mapping for the KDE is mentioned but the specific mapping function is not specified. A brief note on the transformation would aid reproducibility.
  10. Sec. 3.4.2: The constraint C_vd < 1 is stated as necessary for physical validity, but C_vd = 1 is adopted. This apparent tension should be explicitly addressed — is C_vd = 1 a limiting case or an approximation that slightly violates the constraint?

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for a careful and substantive review. The referee correctly identifies that the FPV inconsistency analysis (Sec. 4.3) stands independently of the epsilon-Z model's assumptions, and that the positive claims about epsilon-Z require either sensitivity analysis or validation. We address each major comment below.

read point-by-point responses

  1. Referee: Sec. 3.4, Eq. (26): C_vd set to 1 without sensitivity analysis; S* directly controls every model output; factor-of-two change in C_vd would shift S* by sqrt(2), potentially moving flame standoff and quenching locations; claim of 'more physically localized' heat-release not established without sensitivity or validation.

    Authors: The referee is correct that C_vd directly controls S* and therefore every model output. We will add a sensitivity analysis varying C_vd over a physically admissible range (the constraint C_vd < 1 from Sirignano et al. [30] provides an upper bound; we will test C_vd = 0.5, 0.75, and 1.0). We will report the resulting shifts in flame standoff distance, downstream quenching location, and the heat-release-weighted statistics of Fig. 12. We expect the qualitative findings—localization of heat release relative to FPV, and the reduced fraction of heat release in locally extinguished regions—to be robust because they arise from the structural difference in coupling (epsilon-driven vs. progress-variable-driven), not from the precise value of C_vd. However, we agree that this must be demonstrated rather than assumed. We will also add a bounded discussion of how the quantitative predictions (standoff distance, quenching location) scale with C_vd, noting that S* ~ sqrt(C_vd * epsilon / nu), so the standoff location is determined by where S*(x) crosses S*_fl(p(x)), and the sensitivity of that crossing point depends on the streamwise gradient of epsilon and the pressure variation. We will temper the claim of 'more physically localized' heat-release fields to explicitly state that this is relative to FPV and contingent on the assumed C_vd range. revision: yes

  2. Referee: Sec. 3.4, Eq. (29): Dirac delta-PDF for S* (or epsilon) not assessed; zero variance in S* may bias retrieved heat-release rates, particularly near flammability limit where S-curve is steep; brief discussion of expected bias direction (Jensen's inequality) requested.

    Authors: We agree that the Dirac delta assumption for S* warrants discussion of the expected bias. We will add a paragraph analyzing the bias direction. The key consideration is the curvature of the heat-release rate as a function of S*. On the stable burning branch, the volumetric heat-release rate increases with S* (as shown in Fig. 4b) up to a peak near the flammability limit, then drops to zero. Where the response is convex (d^2 Q/dS*^2 > 0), Jensen's inequality implies that neglecting variance underestimates the mean heat-release rate; where it is concave, the mean is overestimated. Near the flammability limit, the response is steeply decreasing, so the curvature is negative and the delta-PDF assumption would overestimate heat release. This bias would act to sharpen the extinction transition relative to a more realistic distribution. We will add this discussion and note it as a limitation that should be addressed in future work with a presumed PDF for epsilon or S*. revision: yes

  3. Referee: Sec. 4.1, Fig. 7 and Sec. 4.4, Fig. 11: epsilon-Z predicts flame standoff ~2 mm and downstream quenching at x~110 mm attributed to strain-rate-induced extinction; without independent reference (DNS or experiment), unclear whether physically correct or artifacts of epsilon-S* mapping; OSK not independent validation of standoff mechanism; paper should clarify what evidence supports physical correctness beyond internal consistency.

    Authors: The referee is correct that without DNS or experimental validation, we cannot claim that the specific predictions (2 mm standoff, quenching at x~110 mm) are quantitatively correct. We will revise the manuscript to make this limitation explicit. We can offer the following indirect evidence, which we will state as supporting but not conclusive: (1) The flame standoff mechanism—high shear at the splitter plate producing local strain rates exceeding the flammability limit—is physically plausible and consistent with the known behavior of strained diffusion flames near bluff-body or splitter-plate trailing edges. (2) The downstream quenching at x~110 mm is driven by the pressure-dependent reduction of S*_fl, which is a property of the flamelet library (computed with detailed kinetics) rather than an artifact of the epsilon-S* mapping. (3) The OSK model's zero standoff is not independent validation, as the referee notes, because OSK lacks strain-rate coupling; we will clarify this explicitly. We agree that DNS or experimental comparison is needed to establish physical correctness and will state this as a required next step. We will reframe the predictions as falsifiable hypotheses of the model rather than validated results. revision: partial

standing simulated objections not resolved
  • We cannot provide DNS or experimental validation in the revised manuscript. The configuration (transonic accelerating mixing layer at 30 bar with methane/vitiated-air) does not have a corresponding DNS or experimental dataset that we are aware of. The referee's request for validation is legitimate, but we cannot satisfy it within the scope of this revision. We will explicitly acknowledge this as a limitation and frame the epsilon-Z predictions as falsifiable model predictions awaiting validation.

Circularity Check

0 steps flagged

No significant circularity; the ε–S* relation is a self-contained algebraic derivation from the viscous dissipation definition, and the FPV inconsistency diagnosis is an independent diagnostic finding.

full rationale

The paper's central derivation chain proceeds as follows: (1) Eq. 25 writes the exact viscous dissipation rate Φ for a 3D rotating counterflow in terms of the imposed strain rates S*, S*S1, S*S2; (2) Eq. 26 algebraically solves for S* in terms of ε/ν, yielding S* = (1/2)√(C_vd ε/(ν[S1² + 1 − S1])). This is a parameter-free derivation given the stated assumptions (C_vd=1, S1=1/2 for axisymmetric counterflow). The coefficient C_vd is not fitted to the present simulation data—it is set to 1 based on the theoretical framework of Sirignano et al. [30], and the paper explicitly acknowledges it 'remains to be definitively determined from DNS data.' The self-citation to [30] (which shares authors with the present paper) provides the origin of the framework, but the derivation is reproduced self-contained in Eqs. 25–26 and does not depend on any unverified result from [30] to be load-bearing. The FPV inconsistency analysis (Sec. 4.3) is an independent diagnostic: it computes λ/λ_max from the FPV CFD solution and shows it collapses to ~1 regardless of local strain rate—this is a simulation output, not a restatement of an input. The ε-Z model's predictions (flame standoff, quenching at x≈110 mm, heat-release localization) follow from applying Eq. 26 in the CFD solver and are not equivalent to the inputs by construction. The self-citations to [23, 30, 32, 33] are normal continuity references and do not form a chain where the present paper's central claim reduces to an unverified prior claim. The only minor concern is that C_vd=1 is adopted 'consistent with the assumptions outlined in [30]' without independent validation, but this is a modeling assumption (acknowledged as such), not a circularity. Score 1 reflects the minor self-citation to [30] for the theoretical framework, which is not load-bearing for circularity since the derivation is reproduced in the present paper and the assumption is explicitly flagged as unvalidated. The absence of sensitivity analysis to C_vd is a correctness risk, not a circularity issue.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 1 invented entities

The model introduces two key free parameters (C_vd, S1) without sensitivity analysis. The off-manifold scaling is justified by an ad hoc linear-scaling argument. The lumped residual species is a pragmatic construct. The core ε–S* relation is parameter-free in derivation but its application requires the assumed values.

free parameters (5)
  • C_vd = 1.0
    Dimensionless coefficient accounting for distribution of viscous dissipation across smallest turbulent scales. Set to 1 without calibration or sensitivity analysis (Sec. 3.4).
  • S1 = 0.5
    Normalized major tensile strain rate, set to 1/2 for axisymmetric counterflow (Sec. 3.4).
  • C_chi = 2.0
    Proportionality constant for scalar dissipation rate closure in FPV (Eq. 12), used only in the FPV baseline, not in the ε–Z model.
  • Sc_T, Pr_T = 0.7
    Turbulent Schmidt and Prandtl numbers, standard model constants.
  • Progress variable definition C = Y_H2O + Y_CO2 + Y_CO = N/A
    Ad hoc choice of product species for the FPV model; the paper claims findings are insensitive to this choice but does not prove it.
axioms (5)
  • domain assumption Flamelet equations are quasi-laminar and quasi-steady (Eq. 18)
    Standard flamelet assumption invoked in Sec. 3.2; all flamelet-based models depend on this.
  • domain assumption The scalar dissipation rate χ(Z) follows the erfc-based profile (Eq. 19)
    Standard assumed profile used in flamelet library generation; invoked in Sec. 3.2.
  • domain assumption β-PDF for mixture fraction and Dirac δ-PDF for ε/λ
    Presumed PDF approach in Eq. 23; the δ-PDF for ε is acknowledged as needing further investigation (Sec. 3.4).
  • ad hoc to paper Reactant consumption rates scale linearly with local reactant concentration ratio Y_X/Y_X,TAB
    Justification for the off-manifold scaling (Eq. 37-38) in Sec. 3.4.3; based on bimolecular elementary reaction structure but applied as a bounded limiter.
  • domain assumption k-ω SST turbulence model accurately represents the turbulent dissipation rate ε in reacting mixing layers
    The entire ε–Z coupling depends on ε from the turbulence model (Eq. 27); standard RANS closure assumption.
invented entities (1)
  • Lumped residual species Y_N (Eq. 33) no independent evidence
    purpose: Represents all non-transported species (H2, H, O, OH, HO2, CH3, CH2O) as a single composite species to reduce computational cost.
    Practical modeling construct; no independent evidence needed but it is an invented entity in the model framework. Validated only by the claim that ξ > 0.95 throughout the flamelet solution space.

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