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Assessment of subgrid scale mixing models used in LES at high pressures

Neelakantan Padmanabhan, Richard S. Miller

arXiv:2609.37758v1physics.flu-dyncs.NAmath.NA

TL;DR

LES-FMDF molecular-mixing closures need evaluation for high-pressure flows, where multicomponent diffusion effects may become more significant. This study compares three closures with DNS-derived CSD from turbulent slot jets across a broad pressure range and finds deviations between the predictions and DNS. The comparison therefore provides evidence about closure applicability in the assessed flows.

  • Problem

    The applicability of LES-FMDF molecular-mixing closures for CSD at high pressure remains to be established, where multicomponent differential and cross-diffusion effects may become more significant.

    Source: §1 · p. 2

  • Method

    The study evaluates IEM, MC, and MAPPING against DNS-derived FMDF and CSD from spatially evolving turbulent slot jets spanning 1 atm to 100 atm.

    Source: §1 · p. 2

  • Results

    CSD predictions from the evaluated mixing models deviate from DNS in the pressure-dependent comparison.

    Source: §6 · p. 35

  • Takeaways & Limitations

    The comparison shows that applicability of the evaluated closures to large-pressure flows cannot be assumed from their use in other conditions.

    Source: §1 · p. 2, §6 · p. 35

    The mapping-closure assessment is limited to a single scalar; its multiple-scalar EMST extension is not studied.

    Source: §3.1.4 · p. 14

Abstract

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Molecular mixing models for the conditional scalar diffusion (CSD) term in the filtered mass density function (FMDF) transport equation used in Large Eddy Simulation (LES-FMDF) approach are evaluated against direct numerical simulation (DNS) of spatially evolving jets. A database of DNS of turbulent mixing slot jets of $C_{7}H_{16} / O_2$ and $C_{7}H_{16} / Air$ is developed. The formulation includes the compressible form of the governing equations, a generalized multicomponent diffusion model, a cubic real gas equation of state and realistic property models. Simulations are conducted over a wide range of initial pressures ($1 atm < P_0 < 100 atm$) and jet width based Reynolds numbers of 850 and 1300. FMDF of mixture fraction at various filter widths are obtained from the simulation at several spatial locations within the flow. The CSD term in the exact transport equation of FMDF is calculated from the DNS. An \textit{a priori} analysis of Interaction by Exchange of Mean (IEM), Modified Curl (MC) and Mapping Closure (MAPPING) mixing models for CSD is conducted. At high pressures, significant deviations are observed between the CSD measured from the DNS and those predicted by the models. The source of the deviations is determined to be the mixing frequency used in the models. The significance of the mixing frequency and its variation with change in pressure and diffusion models is studied. The primary objective of this work is to assess the applicability of aforementioned mixing models to flows at large pressures and determine the causes for deviation in prediction.

1. Introduction

High-pressure combustion simulations need accurate turbulent-mixing models, but the high-pressure performance of molecular mixing models used to close the FMDF conditional scalar diffusion term is uncertain. This study evaluates those models using DNS of real-gas turbulent slot jets across a broad pressure range.

Introduction

  • Combustion devices such as diesel engines and gas turbines operate at P ≥10atm, and rocket engines exceed 100atm, motivating accurate turbulent-mixing models for simulation.The paper identifies RANS and LES as simulation approaches used in the design and development of these devices.Source: §1 · p. 1
  • PDF, FDF, and FMDF methods represent flow statistics through joint distributions, closing the chemical source term while leaving SGS convection and conditional scalar diffusion unclosed.The introduction associates these methods with Pope, Colucci et.al., and Jaberi et.al., respectively.Source: §1 · p. 2
  • IEM, MC, MAPPING, and other CSD models were developed mainly from atmospheric-pressure, ideal-gas data and had been observed to perform well at atmospheric pressure.The introduction states that their efficacy at high pressures was unknown.Source: §1 · p. 2
  • The study evaluates molecular mixing models for conditional scalar diffusion using DNS of spatially evolving turbulent slot jets from 1atm to 100atm.The database uses real-gas effects, realistic property models, and a generalized diffusion model; FMDF and CSD are measured at multiple spatial locations.Source: §1 · p. 2
  • At high pressures, multicomponent differential and cross-diffusion effects may become more significant, motivating evaluation of models developed under atmospheric, ideal-gas conditions.The efficacy of these models at high pressures was not established in the introduction.Source: §1 · p. 2

2. DNS of a turbulent slot jet

The DNS combines compressible governing equations with real-gas and multicomponent-diffusion modeling, and uses specified domain, boundary, and inflow treatments to simulate turbulent slot jets.

2. DNS of a turbulent slot jet

  • The DNS solves compressible continuity, momentum, energy, and species equations with ideal- and real-gas equations of state, realistic properties, and multicomponent diffusion.High-pressure real-gas effects use the Peng–Robinson equation of state; heat and mass fluxes include multicomponent differential and cross-diffusion effects.Source: §2 · pp. 2–3

2.1. Problem geometry and numerical approach

  • The domain spans Reynolds-number-dependent streamwise and cross-stream extents, and mesh refinement tests establish resolution-independent statistics before the smallest tested mesh is used.Spatial derivatives use eighth-order explicit central differences, with fourth-order Runge–Kutta time integration and filtering at each stage.Source: §2.1 · pp. 4–6, Table 1 · p. 6

2.2. Initial and Boundary conditions

  • All simulations use a convective Mach number of 0.35; nonreflecting inlet and outlet conditions, transverse sponge zones, and spanwise periodicity define the boundaries.Jet and co-flow velocities are 86.8 m/s and 17.4 m/s, and simulations run for nondimensional time 200–240 to obtain converged statistics.Source: §2.2 · pp. 6, 9

2.3. Inflow condition

  • A smooth top-hat inflow profile and combined sinusoidal and circular spatial forcing induce three-dimensional instability and transition within 10Djet of the inlet.Forcing amplitude is 0.001–0.002 and the selected frequency is 0.25–0.4 for the specified jet and co-flow velocities.Source: §2.3 · pp. 9–10

3. FMDF method

The FMDF method uses a Favre-filtered joint PDF to describe scalars and turbulent flow variables in compressible LES. Its chemical reaction source term is closed, but conditional scalar diffusion (CSD) remains unclosed and requires a mixing model.

3. FMDF method

  • FMDF derives a transport equation for a Favre-filtered joint PDF of scalars and flow variables, following Jaberi et al. and the PDF method of Pope.The joint statistical description allows flow moments to be determined.Source: §3 · pp. 10–11
  • The reaction source term is closed, while subgrid scalar convection and conditional scalar diffusion remain unclosed in the FMDF transport equation.CSD represents molecular transport in physical space and molecular mixing in composition space; the section motivates mixing models for its treatment.Source: §3 · p. 11
  • The transport equation is solved with Lagrangian Monte Carlo, using stochastic differential equations for spatial transport and a mesh-free Lagrangian setup for composition transport.The composition evolution of a Lagrangian fluid element is formulated alongside this transport procedure.Source: §3 · p. 11

3.1. Molecular mixing models

The section outlines four molecular-mixing closures and a shared mixing frequency, distinguishing their particle-interaction mechanisms, locality, and scope. Their applicability depends on the scalar representation and, for several models, the mixing frequency.

3.1.1. Interaction by Exchange of Mean

  • IEM, also called LMSE, relaxes particle composition toward the filter mean at a rate proportional to ε/k, but is local in composition space.Its locality may make it unsuitable for dispersed flows with large spatial concentration variation.Source: §3.1.1 · pp. 11–12

3.1.2. Interaction by Exchange of Conditional Mean

  • IECM uses velocity-conditioned averages so particles in the same eddy interact, addressing IEM’s dispersion inconsistency and potentially improving SGS mixing predictions in dispersed flows.IECM is nevertheless non-local in composition space.Source: §3.1.2 · p. 12

3.1.3. Modified Curl

  • Modified Curl, proposed by Janicka et al. from Curl’s model, stochastically updates particle pairs using a random factor and shares IEM’s mixing frequency.It is local in physical space but not composition space; the unequal-weight variant depends on particle-weight distribution.Source: §3.1.3 · p. 12

3.1.4. Mapping Closure

  • Mapping Closure maps sorted scalar compositions to a Gaussian reference field, preserving mean and variance decay while remaining local in composition space.Pope’s unequal-weight formulation applies only to one scalar; the EMST multiple-scalar extension is not studied.Source: §3.1.4 · pp. 13–14

3.1.5. Mixing frequency

  • The models’ mixing frequency has units of 1/s and can be determined from simulation parameters by dimensional analysis; Jaberi et al. proposed an expression for it.The standard constants for atmospheric-pressure flows are drawn from Jaberi et al..Source: §3.1.5 · p. 14

3.2. Measurement of subgrid scale terms from DNS

DNS subgrid quantities are obtained by filtering and volume-averaging flow properties, then comparing the resulting FMDF and conditional scalar diffusion with SGS models.

3.2. Measurement of subgrid scale terms from DNS

  • DNS subgrid characteristics, including FMDF and conditional scalar diffusion, are directly computed and compared with SGS models to evaluate closure at high pressures.The DNS domain is divided into smaller domains, where flow properties are volume-averaged.Source: §3.2 · pp. 14–15

3.2.1. Measurement of FMDF

  • The FMDF is obtained by spatially filtering a DNS scalar, sampling its instantaneous subgrid fluctuations, and constructing their distribution.Unlike a RANS PDF, which characterizes fluctuations across flow realizations, the FMDF describes instantaneous subgrid fluctuations.Source: §3.2.1 · p. 15

3.2.2. Measurement of CSD

  • CSD is measured by conditioning DNS mixture-fraction diffusion rates on mixture fraction within cubic filters at filter-width ratios r = 4 and 16.The analysis focuses on mixture fraction; conditional averages are formed from grid points within the filter volume.Source: §3.2.2 · pp. 15–16

4. Results · 4.1. DNS flow evolution

The DNS captures the jet’s transition from a laminar flow through vortex formation and breakup to turbulence, while its mean-velocity and fluctuation statistics show agreement with experimental or DNS benchmarks. The flow evolution is compared with slot- and plane-jet experiments because the slot jet evolves structurally like a plane jet.

4.1.1. Structural Evolution of Jet

  • The jet transitions from laminar flow to Kelvin–Helmholtz vortex rings, downstream vortex pairing and breakup, and turbulence beyond the end of the potential core.The laminar region extends to x1 < 10Djet; vortex rings appear at 10Djet < x1 < 15Djet, and the potential core ends around x1 > Lx/2.Source: §4.1.1 · p. 17

4.1.2. Mean Velocity of Flow

  • The normalized streamwise mean-velocity excess collapses to a semi-self-similar profile for x1 > 7.5Djet and agrees with slot- and plane-jet experiments.The cited comparisons are with Shestakov et al., Gutmark and Wygnanski, and Ramaprian and Chandrasekhara.Source: §4.1.2 · p. 17, Figure 6 · p. 19
  • The slot-jet evolution is validated against both slot- and plane-jet experiments because its flow structure resembles that of plane jets.The authors state that comparisons are appropriate when Reynolds numbers are in a similar range, despite differences in physical models.Source: §4.1.2 · pp. 17–18
  • The cross-stream mean-velocity profile also agrees well with experimental results.The cross-stream velocity itself is used as the mean-velocity excess because Uco,x2 = 0.Source: §4.1.2 · p. 18, Figure 6 · p. 19
  • The downstream centerline velocity-excess decay compares well with experiments and varies inversely with x1, as in Stanley et al.’s DNS.Experimental comparisons include Shestakov et al., Thomas and Prakash, and Hussain and Clark.Source: §4.1.2 · p. 18, Figure 7 · p. 20
  • DNS-measured r.m.s. velocity fluctuations were compared with round-jet DNS by Ries et al., slot-jet experiments by Zhe and Modi, and plane-jet experiments by Khayrullina et al.; they show good agreement with experimental data.Source: §4.1.2 · p. 18

4.2. FMDF

The jet-centerline FMDF changes strongly with generalized diffusion as pressure and Reynolds number rise, while it remains similar across pressure under standard Fickian diffusion.

4.2. FMDF

  • Bimodal FMDF shapes indicate highly segregated mixture fractions, whereas unimodal shapes indicate high concentrations of either fuel or oxidizer.Near the nozzle along the jet centerline and in the freestream, fuel and oxidizer concentrations are typically high, respectively.Source: §4.2 · p. 22
  • With generalized diffusion, the jet-centerline FMDF is bimodal at 1 atm, more segregated at higher Reynolds number, and multimodal as pressure rises, indicating intensified mixing.The generalized model accounts for diffusion due to concentration and pressure gradients, which become significant at large pressures.Source: §4.2 · p. 22
  • With standard Fickian diffusion, the FMDF is predominantly unimodal at atmospheric pressure and remains similar between 1 atm and 100 atm.Source: §4.2 · p. 22

4.3. CSD

The DNS-based CSD magnitude increases non-linearly with ambient pressure, while its shape and magnitude vary relatively little over time. The section compares DNS CSD against IEM, MC, and MAPPING models and notes that conditioning discrete scalar-diffusion values produces a discontinuous CSD shape.

4.3. CSD

  • Although the CSD is stochastic, its shape and magnitude vary relatively little with time.Source: §4.3 · p. 22
  • DNS CSD is compared with IEM, MC, and MAPPING predictions; one case uses heptane-air at 1 atm, Re0 = 850, Le = 1, ideal-gas EOS, and Fickian diffusion.For that case, the calculated mixing frequency is Ωm = 1.26 × 10^4.Source: §4.3 · pp. 22, 24
  • The absolute CSD magnitude increases non-linearly with ambient pressure across the reported pressure conditions.The comparison covers DNS at 1, 35, and 100 atm with generalized and simplified Fickian diffusion models.Source: §4.3 · pp. 22, 24
  • The CSD shape is discontinuous because discrete scalar-diffusion values within each filter volume are conditioned on mixture fraction.Source: §4.3 · p. 24

4.4. Comparison of CSD and Mixing models

The IEM, Modified Curl, and Mapping Closure models show distinct CSD mismatches, with the IEM discrepancy linked to mixing frequency and the other models’ predictions affected by their mixing procedures.

4.4.1. Interaction by Exchange of Mean

  • At high pressures, IEM mixing frequency is overpredicted by at least 10× with generalized diffusion and underpredicted by 10× with Fickian diffusion; IEM overpredicts CSD by 10×.Least-squares curve fitting calibrated model parameters to DNS data and determined corrected mixing frequencies. With corrected IEM mixing frequencies, IECM CSD trends were qualitatively similar, and model and DNS residual norms were comparable.Source: §4.4.1 · p. 30

4.4.2. Modified Curl

  • Modified Curl generally follows the CSD trend but overpredicts DNS; at high pressures its slope is smaller, and random particle selection introduces nonphysical fluctuations.The model constant is CMC = 1 at high pressures and CMC = 2 for atmospheric flow with generalized diffusion.Source: §4.4.2 · pp. 30–31

4.4.3. Mapping Closure

  • Mapping Closure’s sorted-scalar mixing underpredicts average conditional diffusion, while its unsorted-scalar fit resembles Modified Curl but is not physical.The model preserves localness in composition space by sorting scalars before mixing; evaluation of the multiple-scalar EMST extension is not included.Source: §4.4.3 · p. 31

5. Discussion

The discussion identifies mixing frequency as a major source of model uncertainty and attributes high-pressure discrepancies to diffusion behavior. It describes corrections for generalized diffusion and a DNS-based alternative for estimating mixing frequency.

5.1. Mixing frequency

  • At P0 = 100 atm, the calculated mixing frequency differs from its calibrated value by a factor of 10 with Fickian diffusion, and the discrepancy is larger with generalized diffusion.Mixing frequency is common to the molecular mixing models and is identified as their largest source of uncertainty when fitting DNS data.Source: §5.1 · p. 33
  • At atmospheric pressure, γdev is approximately unity; at high pressure it differs from 1 because standard Fickian diffusion overpredicts diffusion, so γdev replaces γ in calculating Ωm.The passage attributes the high-pressure discrepancy to diffusivity variation and increasingly important cross-diffusion effects.Source: §5.1 · p. 33
  • Because γ significantly affects Ωm, accurate generalized-diffusion modeling requires an effective diffusion coefficient; Ma derived a species-specific coefficient.The discussion refers readers to Ma for analysis of molecular mixing models using this effective coefficient at high pressures.Source: §5.1 · pp. 33–34
  • DNS-derived mean turbulent kinetic energy and dissipation rates yield mixing frequencies for IEM; calibrated Ck−ϵ is approximately 1 in most cases, suggesting an alternative for generalized-diffusion LES.The calibrated constants are presented in Table 6.Source: §5.1 · p. 34, Table 6 · p. 34

6. Conclusion

DNS-based CSD comparisons show that commonly used mixing models deviate at high pressures because their mixing-frequency diffusivity is misestimated. An effective diffusivity accounting for cross-diffusion or a dissipation-to-turbulent-kinetic-energy ratio is proposed as an alternative.

6. Conclusion

  • The DNS uses a real-gas equation of state, generalized diffusion, and temperature- and pressure-dependent properties; atmospheric-pressure flow statistics agree with prior studies.The flow's structural evolution at larger pressures is assumed to be qualitatively similar.Source: §6 · pp. 34–35
  • At large pressures, mixing-model CSD predictions deviate from DNS because constant diffusivity under-predicts mixing frequency while a deviation factor over-predicts it.An effective diffusivity accounting for cross-diffusion effects is identified as the ideal term for estimating mixing frequency.Source: §6 · p. 35
  • Alternatively, mixing frequency may be estimated from the ratio of mean dissipation rate to mean turbulent kinetic energy.Deriving an effective diffusivity that accounts for high-pressure cross-diffusion effects is beyond this work's scope.Source: §6 · p. 35
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