Direct Numerical Simulation Database Research Articles

Direct Numerical Simulation Database of High-Speed Flow over Parameterized Curved Walls

This study presents a direct numerical simulation (DNS) database of high-speed turbulent boundary layers (TBLs) subject to pressure gradients due to parametrically varied backward-facing and forward-facing wall curvatures, with an inflow Mach number of 4.9 and a friction Reynolds number of Reτ≃1000 immediately before the onset of wall curvature. The Mach and Reynolds numbers are significantly higher than those reported in the literature for the DNS of pressure-gradient TBLs. The flow conditions and baseline wall geometries are representative of experiments in the high-speed blowdown wind tunnel at the National Aerothermochemistry Laboratory at Texas A&M University. The wall steepness of the baseline geometry for both the backward-facing and forward-facing walls was systematically varied to cause attached, incipiently separated, and fully separated flows. Precomputed flow statistics, including turbulent kinetic energy budgets, are available on the website of the Turbulence Modeling Resource of the NASA Langley Research Center, allowing other investigators to query any property of interest.

Physics-informed data-driven unsteady Reynolds-averaged Navier–Stokes turbulence modeling for particle-laden jet flows

With the assistance of deep learning (DL), we present a framework for predicting the turbulent eddy viscosity in unsteady Reynolds-averaged Navier–Stokes (URANS) simulations for particle-laden jet flows. We report a complete workflow from identifying the input flow and particle quantities in the training phase to predicting the flow and particle fields in the testing phase. The framework incorporates a deep neural network model, also known as multi-layer perceptrons, into the momentum equations of the Euler–Lagrangian gas–solid flow system. A data-driven, physics-informed DL approach was employed to predict the modeled turbulent eddy viscosity field, formulated as a function of the instantaneous flow and particle quantities. In the training phase, these regression functions were trained with an existing high-fidelity direct numerical simulation database. In the testing phase, the trained model was then used to predict the instantaneous local eddy viscosity to update the closure term and to solve the URANS equations iteratively. A series of round, turbulent particle-laden jets in a co-flow with various Stokes numbers were assessed, including those beyond the range of conditions employed for training. The proposed DL–URANS model was found to provide enhanced accuracy for predicting both flow and particle quantities when compared with the baseline URANS simulation.

On the generation of near-wall dilatational motions in hypersonic turbulent boundary layers

Dilatational motions in the shape of travelling wave packets have been identified recently to be dynamically significant in hypersonic turbulent boundary layers. The present study investigates the mechanisms of their generation and their association with the solenoidal motions, especially the well-recognized near-wall self-sustaining process of the regeneration cycle between the velocity streaks and quasi-streamwise vortices. By exploiting the direct numerical simulation databases and orchestrating numerical experiments, we explore systematically the near-wall flow dynamics in the processes of the formation and transient growth of low-speed streaks. We conclude via theoretical ansatz that the nonlinearity related to the parallel density and pressure gradients close to the wall due to the restriction of the isothermal boundary condition is the primary cause of the generation of the dilatational structures at small scales. In fully developed turbulence, the formation and the existence of healthy dilatational travelling wave packets require the participation of the turbulence at scales larger than those of the near-wall regeneration cycles, especially the occurrence of the bursting events that generate vortex clusters. This is proven by the less intensified dilatational motions in the numerical experiments in which the Orr mechanism is alleviated and the vortical structures and turbulent bursts are weakened.

Turbulent activity in the near-wall region of adverse pressure gradient turbulent boundary layers

Two direct numerical simulation (DNS) databases are investigated to understand the effect of the outer-layer turbulence on the inner layer’s structures and energy transfer mechanisms. The first DNS database is the non-equilibrium adverse-pressure-gradient (APG) turbulence boundary layer (TBL) of Gungor et al. [1]. Its Reynolds number and the inner-layer pressure gradient parameter reach above 8000 and 10, respectively. The shape factor spans between 1.4 and 3.3, which indicates the flow has various velocity defect situations. The second database is the same flow as the first one but the outer layer turbulence is artificially eliminated in this flow. Turbulence is removed above 0.15 local boundary layer thickness. For the analysis, we chose four streamwise positions with small, moderate, large, and very-large velocity defect. We compare the wall-normal distribution of Reynolds stresses, two-point correlations and spectral distributions of energy, production and pressure strain. The results show that the inner layer turbulence can sustain itself when the outer-layer turbulence does not exist regardless of the velocity defect or the pressure gradient. The two-point correlations of both cases show that outer large-scale structures affect the inner layer structures significantly. The streamwise extent of the correlation contours scales with pressure-viscous units. This shows the importance of the pressure gradient’s effect on the inner-layer structures. The spectral distributions demonstrate that the energy transfer mechanisms are probably the same in the inner layer regardless of the velocity defect, which suggests the near-wall cycle may exist even in very-large defect APG TBLs where the mean shear in the inner layer is considerably lower than small-defect APG TBLs.

Assessment of Laws of the Wall During Flame–Wall Interaction of Premixed Flames Within Turbulent Boundary Layers

The validity of the usual laws of the wall for Favre mean values of the streamwise velocity component and temperature for non-reacting flows has been assessed for turbulent premixed flame-wall interaction using Direct Numerical Simulation (DNS) data. Two different DNS databases corresponding to friction velocity-based Reynolds number of 110 and 180 representing unsteady head-on quenching of statistically planar flames within turbulent boundary layers have been considered. The usual log-law based expressions for the Favre mean values of the streamwise velocity and temperature for the inertial layer have been found to be inadequate at capturing the corresponding variations obtained from DNS data. The underlying assumptions of constant shear stress and the equilibrium of production and dissipation of turbulent kinetic energy underpinning the derivation of the usual log-law for the mean streamwise velocity have been found to be rendered invalid within the usual inertial layer during flame-wall interaction for both cases considered here. The heat flux does not remain constant within the usual inertial layer, and the turbulent flux of temperature exhibits counter-gradient transport within the so-called inertial layer for the cases considered in this work. These render the assumptions behind the derivation of the usual log-law for temperature to be invalid for application to turbulent flame-wall interaction. It has been found that previously proposed empirical modifications to the existing laws of the wall, which account for density and kinematic viscosity variations with temperature, do not significantly improve the agreement with the corresponding DNS data in the inertial layer and the inaccurate approximations for the kinematic viscosity compensated wall normal distance and the density compensated streamwise velocity component contribute to this disagreement. The DNS data has been utilised here to propose new expressions for the kinematic viscosity compensated wall normal distance and the density compensated streamwise velocity component, which upon using in the empirically modified law of wall expressions have been demonstrated to provide reasonable agreement with DNS data.

Relations between Reynolds stresses and their dissipation rates during premixed flame–wall interaction within turbulent boundary layers

A direct numerical simulation (DNS) database for head-on quenching of premixed flames propagating across turbulent boundary layers representative of friction Reynolds numbers, Reτ, of 110 and 180 has been utilized to analyze the interrelation between Reynolds stresses and their dissipation rates during flame–wall interaction. The Reynolds stresses and their dissipation rates exhibit significant deviations from the corresponding non-reacting flow profiles within the flame brush and in the burned gas region. This behavior is prominent for the components in the wall-normal direction because the mean direction of flame normal acceleration due to thermal expansion aligns with the wall-normal direction in this configuration. The anisotropy of Reynolds stresses and their dissipation rate tensors have been found to be qualitatively similar, but the anisotropic behavior weakens with increasing Reτ. However, the components of the anisotropy tensors of Reynolds stresses and viscous dissipation rate are not related according to a linear scaling, and thus, the models based on this assumption do not successfully capture the viscous dissipation rate components obtained from the DNS data. By contrast, a model, which includes the invariants of the anisotropy tensor of Reynolds stresses and satisfies the limiting conditions, has been found to capture especially the diagonal components of the viscous dissipation rate tensor more successfully for both non-reacting and reacting cases considered in this work. However, the quantitative prediction of this model suffers for the components in the wall-normal direction for lower values of Reτ, but the performance of this model improves with an increase in Reτ.

Revisiting the modelling framework for the unresolved scalar variance

The unresolved scalar variance in large-eddy simulations of turbulent flows is a fundamental physical and modelling parameter. Despite its importance, relatively few algebraic models have been developed for this important variable with the most prominent models to date being the classic scale-similarity and gradient models. In this work a new generalized modelling framework based on reconstruction has been developed, which in contrast to classic modelling approaches allows the construction of base static variance models of arbitrary accuracy. It is demonstrated that higher-order reconstructions naturally lead to base static variance models of increased accuracy, and that the classic scale-similarity and gradient models are subsets of more general and higher-order models. The classic scale-similarity assumption for developing dynamic models is also revisited, and it is demonstrated that this can essentially be reinterpreted as a two-level reconstruction approach. Based on this result, a new general methodology is proposed that allows the construction of dynamic models for any given base static model, and a corresponding general reconstruction operator, algebraic or iterative. Consequently, improved static and dynamic models for the scalar variance are developed. The newly developed models are then thoroughly tested a priori using two high-fidelity direct numerical simulation databases corresponding to two substantially different flame and flow configurations, and are shown to outperform classic algebraic models for the variance.

Timescales Associated with the Evolution of Reactive Scalar Gradient in Premixed Turbulent Combustion: A Direct Numerical Simulation Analysis

The fractional change in the reaction progress variable gradient depends on the flow normal straining within the flame and also upon the corresponding normal gradients of the reaction rate and its molecular diffusion transport. The statistical behaviours of the normal strain rate and the contributions arising from the normal gradients of the reaction rate and molecular diffusion rate within the flame were analysed by means of a Direct Numerical Simulation (DNS) database of statistically planar turbulent premixed flames ranging from the wrinkled/corrugated flamelets regime to the thin reaction zones regime. The interaction of flame-normal straining with the flame-normal gradient of molecular diffusion rate was found to govern the reactive scalar gradient transport in the preheat zone, where comparable timescales for turbulent straining and molecular diffusion are obtained for small values of Karlovitz numbers. However, the molecular diffusion timescale turns out to be smaller than the turbulent straining timescale for high values of Karlovitz numbers. By contrast, the reaction and hot product zones of the flame remain mostly unaffected by turbulence, and the reactive scalar gradient transport in this zone is determined by the interaction between the flame-normal gradients of molecular diffusion and chemical reaction rates.

Calibrating sub-grid scale models for high-order wall-modeled large eddy simulation

AbstractHigh-order methods have demonstrated orders of magnitude reduction in computational cost for large eddy simulation (LES) over low-order methods in the past decade. Most such simulations are wall-resolved implicit LES (ILES) without an explicit sub-grid scale (SGS) model. The use of high-order ILES for severely under-resolved LES such as wall-modeled LES (WMLES) often runs into robustness and accuracy issues due to the low dissipation embedded in these methods. In the present study, we investigate the performance of several popular SGS models, the static Smagorinsky model, the wall-adapting local eddy-viscosity (WALE) model and the Vreman model, to improve the robustness and accuracy of under-resolved LES using high-order methods. The models are implemented in the high-order unstructured grid LES solver called hpMusic based on the discontinuous flux reconstruction method. The length scales in these SGS models are calibrated using the direct numerical simulation (DNS) database for the turbulent channel flow problem. The Vreman model has been found to produce the most accurate and consistent results with a proper choice of the length scale for WMLES.

Flow and thermal fields modeling in jet impingement configurations using a Reynolds stress turbulence closure within the RANS and Sensitized-RANS framework

The flow and thermal field characteristics underlying turbulent convective heat transfer in two reference configurations, representing a jet exiting a planar slot and a circular tube and impinging on a heated wall, are studied computationally. The performance of a conventional Reynolds Stress Model (RSM) and its scale-resolving variant, employed in the context of a time-accurate Reynolds-Averaged Navier–Stokes (RANS) simulation, is investigated in conjunction with different modeling approaches for the turbulent heat flux uj′θ′¯, whose gradient represents the turbulence-induced heat source in the temperature transport equation. In this so-called Sensitized-RANS modeling strategy, employing an improved eddy-resolving instability-sensitive Reynolds Stress Model (IISRSM, Jakirlić and Maduta (2015)), the unresolved ‘sub-scale’ turbulent correlations represent the solutions of a set of corresponding transport equations for the entire stress tensor ui′uj′¯. The turbulent scalar-flux models follow from the classical gradient-diffusion approach with the diffusion-like coefficients formulated as a function of both the scalar eddy-viscosity and the anisotropic Reynolds stress tensor. The latter model expressions also take into account various non-linear relationships in terms of turbulent stress components. The computational results of the fully-developed channel flow, featured only by the mean shear and constant wall heat flux, representing a preliminary investigated flow configuration, agree very well with the corresponding DNS (Direct Numerical Simulation) database for both RSM versions in conjunction with all the turbulent heat flux models adopted. However, the study of the two flow impingement cases with the different structure of the impinging jet and nozzle-to-wall distance variations, characterized by a much more complex flow straining, using the considered conventional Reynolds stress model shows significantly different results for the Reynolds stress and thermal fields compared to the available DNS data, especially in the immediate impact region manifested by a significant overprediction of the turbulence intensity. In contrast, the IISRSM-relevant results for the mean flow and temperature fields, the associated integral characteristics, and the corresponding turbulence quantities are in very good agreement with the reference DNS data, due to the much higher predictive capabilities of this eddy-resolving turbulence model compared to its baseline counterpart.

A priori tests of turbulence models for compressible flows

Purpose The purpose of the paper is to analyse the performances of closures and compressibility corrections classically used in turbulence models when applied to highly-compressible turbulent boundary layers (TBLs) over flat plates. Design/methodology/approach A direct numerical simulation (DNS) database of TBLs, covering a wide range of thermodynamic conditions, is presented and exploited to perform a priori analyses of classical and recent closures for turbulent models. The results are systematically compared to the “exact” terms computed from DNS. Findings The few compressibility corrections available in the literature are not found to capture DNS data much better than the uncorrected original models, especially at the highest Mach numbers. Turbulent mass and heat fluxes are shown not to follow the classical gradient diffusion model, which was shown instead to provide acceptable results for modelling the vibrational turbulent heat flux. Originality/value The main originality of the present paper resides in the DNS database on which the a priori tests are conducted. The database contains some high-enthalpy simulations at large Mach numbers, allowing to test the performances of the turbulence models in the presence of both chemical dissociation and vibrational relaxation processes.

An augmented invariant-based model of the pressure Hessian tensor using a combination of physics-assisted neural networks

Modeling the velocity gradient dynamics in incompressible turbulence requires modeling two unclosed quantities: the pressure Hessian tensor and the viscous Laplacian tensor. In this work, we model the pressure Hessian tensor using a combination of two different physics-embedded deep neural networks. The first neural network is trained specifically to predict the alignment tendencies of the eigen-vectors of the pressure Hessian tensor, whereas the second neural network is trained only to predict the magnitude of the tensor. This separation of tasks allows us to define mathematically optimal and physics-informed customized loss functions separately for the two aspects (alignment and magnitude) of the tensor. Both neural networks take invariants of the velocity gradient tensor as inputs. Even though the training of the two networks is performed using direct numerical simulation database of an incompressible stationary isotropic turbulence at a particular Reynolds number, we extensively evaluate the model at different Reynolds numbers and in different kinds of flow fields. In incompressible flows, the proposed model shows significant improvements over the existing phenomenological model (the recent fluid deformation closure model or the RFD model) of the pressure Hessian tensor. While the improvements in the alignment tendencies are convincingly evident in the shapes of the probability density functions of the cosines of various angles between eigenvectors, the improvements in the prediction of the magnitude of the pressure Hessian tensor using the new model are quantifiable in the range of 28%–89% (depending on the type of the flow field) compared to the RFD model.

Evolution of Flame Displacement Speed Within Flame Front in Different Regimes of Premixed Turbulent Combustion

A transport equation for the flame displacement speed evolution in premixed flames is derived from first principles, and the mean behaviours of the terms of this equation are analysed based on a Direct Numerical Simulation database of statistically planar turbulent premixed flames with a range of different Karlovitz numbers. It is found that the regime of combustion (or Karlovitz number) affects the statistical behaviour of the mean contributions of the terms of the displacement speed transport equation which are associated with the normal strain rate and curvature dependence of displacement speed. The contributions arising from molecular diffusion and flame curvature play leading order roles in all combustion regimes, whereas the terms arising from the flame normal straining and reactive scalar gradient become leading order contributors only for the flames with high Karlovitz number values representing the thin reaction zones regime. The mean behaviours of the terms of the displacement speed transport equation indicate that the effects arising from fluid-dynamic normal straining, reactive scalar gradient and flame curvature play key roles in the evolution of displacement speed. The mean characteristics of the various terms of the displacement speed transport equation are explained in detail and their qualitative behaviours can be expounded based on the behaviours of the corresponding terms in the case of 1D steady laminar premixed flames. This implies that the flamelet assumption has the potential to be utilised for the purpose of any future modelling of the unclosed terms of the displacement speed transport equation even in the thin reaction zones regime for moderate values of Karlovitz number.

Direct Numerical Simulation of a High-Pressure Turbine Stage: Unsteady Boundary Layer Transition and the Resulting Flow Structures

Abstract In the present study, we investigate the unsteady boundary layer transition based on the direct numerical simulation database of a high-pressure turbine (HPT) stage (Zhao and Sandberg, 2021, High-Fidelity Simulations of a High-Pressure Turbine Stage: Effects of Reynolds Number and Inlet Turbulence, ASME Turbo Expo 2021, Paper No. GT2021-58995), focusing on the transition mechanisms on the rotor blade, affected by the incoming periodic wakes and the background freestream turbulence (FST). On the basis of the fully resolved flow fields, we provide detailed analysis of the flow structures responsible for the transition, and two distinctive transition paths have been identified. The first path is the typical bypass transition via the instability of Klebanoff streaks, which happens when the transition region is not directly affected by the wake. The suction-side boundary layer is disturbed at the leading edge, resulting in the formation of streamwise streaks. These streaky structures endure varicose instability in the region with adverse pressure gradient (APG), then quickly break down into turbulent spots, which then evolve into fully turbulent flow. The other transition path is a consequence of the direct interaction between the wake structures and the blade boundary layer, when the wake apex starts to affect the transitional region. To be specific, the wake structures directly interact with the separation bubble in the APG region, causing sudden breakdown into turbulence. A calmed region is found to follow the wake-induced turbulent boundary layer. It is observed that the recovery to a calmed region can be impacted by the FST, as the calmed region in case with no FST is much longer compared to cases with stronger FST.

Compressibility effects in supersonic and hypersonic turbulent boundary layers subject to wall disturbances

In the present study, we investigate the compressibility effects in supersonic and hypersonic turbulent boundary layers under the influence of wall disturbances by exploiting direct numerical simulation databases at Mach numbers up to 6. Such wall disturbances enforce extra Reynolds shear stress on the wall and induce mean streamline curvature in rough wall turbulence that leads to the intensification of turbulent motions in the outer region. The turbulent and fluctuating Mach numbers, the density and the velocity divergence fluctuation intensities suggest that the compressibility effects are enhanced by the increment of the free-stream Mach number and the implementation of the wall disturbances. The differences between the Reynolds and Favre average due to the density fluctuations constitute approximately $9\,\%$ of the mean velocity close to the wall and $30\,\%$ of the Reynolds stress near the edge of the boundary layer, indicating their non-negligibility in turbulent modelling strategies. The comparatively strong compressive events behaving as eddy shocklets are observed at the free-stream Mach number of $6$ only in the cases with wall disturbances. By further splitting the velocity into the solenoidal and dilatational components with the Helmholtz decomposition, we found that the dilatational motions are organized as travelling wave packets in the wall-parallel planes close to the wall and as forward inclined structures in the form of radiated waves in the vertical planes. Despite their increased magnitudes and higher portion in the Reynolds normal and shear stresses, the dilatational motions show no tendency of contributing significantly to the skin friction and the production of turbulent kinetic energy due to their mitigation by the cross-correlation between the solenoidal and dilatational velocity components.

Coherent structures and turbulent model refinement in oblique shock/hypersonic turbulent boundary layer interactions

In the present study, we investigate the evolution of turbulent statistics and coherent structures in hypersonic turbulent boundary layers at the Mach number of 5 impinged by oblique shock waves generated by the wedge with the angles of 14°, 10°, and 6°, inducing strong, mild, and incipient flow separation, by exploiting direct numerical simulation databases, for the purpose of revealing the underlying flow physics that are of significance to turbulent modeling. We found that the large-scale structures are amplified within the interaction zone, manifested in the form of large-scale low- and high-speed streaks with the spanwise length scale of boundary layer thickness, and gradually decay downstream, the process of which is extremely long. The abrupt variation in the characteristic length, time, and velocity scales as well as the incompatible viscous dissipation of the mean and turbulent kinetic energy results in the incorrect predictions by the Reynolds-Averaged Navier–Stokes (RANS) equation simulations, provided the models are established based on solving the transport equations of the turbulent kinetic equation and its viscous dissipation (k−ε or k−ω models, for instance). To amend this issue, we propose to refine the parameters in the model as the functions of wall pressure, the flow quantities related to multiple flow features. The RANS simulations with the k−ω SST model utilizing the proposed refinement improve greatly the accuracy of the skin friction, wall heat flux, and Reynolds shear stress downstream of the interaction zone, and the wall pressure distributions in hypersonic turbulence over compression ramp, suggesting its promising prospect in engineering applications.

Forecasting small-scale dynamics of fluid turbulence using deep neural networks

Turbulence in fluid flows is characterized by a wide range of interacting scales. Since the scale range increases as some power of the flow Reynolds number, a faithful simulation of the entire scale range is prohibitively expensive at high Reynolds numbers. The most expensive aspect concerns the small-scale motions; thus, major emphasis is placed on understanding and modeling them, taking advantage of their putative universality. In this work, using physics-informed deep learning methods, we present a modeling framework to capture and predict the small-scale dynamics of turbulence, via the velocity gradient tensor. The model is based on obtaining functional closures for the pressure Hessian and viscous Laplacian contributions as functions of velocity gradient tensor. This task is accomplished using deep neural networks that are consistent with physical constraints and explicitly incorporate Reynolds number dependence to account for small-scale intermittency. We then utilize a massive direct numerical simulation database, spanning two orders of magnitude in the large-scale Reynolds number, for training and validation. The model learns from low to moderate Reynolds numbers and successfully predicts velocity gradient statistics at both seen and higher (unseen) Reynolds numbers. The success of our present approach demonstrates the viability of deep learning over traditional modeling approaches in capturing and predicting small-scale features of turbulence.

On wall pressure fluctuations in conical shock wave/turbulent boundary layer interaction

The structure and the frequency spectra of wall pressure fluctuations beneath a planar turbulent boundary layer interacting with a conical shock wave at Mach number $M_\infty =2.05$ and Reynolds number $\textit {Re}_\theta \approx 630$ (based on the upstream boundary layer momentum thickness) are examined to elucidate the effects of pressure gradient and flow separation on the characteristics of the wall pressure fluctuations, by exploiting a direct numerical simulation database. Upstream of the interaction, in the zero pressure gradient region, wall pressure statistics compare well with canonical compressible boundary layers in terms of fluctuation intensities and frequency spectra. Across the main interaction zone (APG1), the root-mean-square of wall pressure fluctuations becomes very large (corresponding to approximately 173.3 dB), with maximum increase approximately 12.7 dB from the incoming level. In the second adverse pressure gradient zone (APG2), the root-mean-square of wall pressure fluctuations attains a second peak (corresponding to $164.7$ dB), with an increase of 8.4 dB from the upstream level. Both the APG1 and APG2 regions feature a substantial fraction of flow reversal events, which are, however, scattered and interspersed with regions of attached flow. The wall pressure power spectral density exhibits a broadband and energetic low-frequency component associated with the global unsteadiness of the separation bubble/conical shock system. Analysis of the two-point correlations and wavenumber/frequency spectra of wall pressure fluctuations further suggests that the typical eddies become more elongated along the spanwise direction, as the flow in the separated region tends to escape the centreline, and the convection velocity is significantly reduced.

Unsteady evolution of distributed roughness-induced vortices under re-entry conditions

AbstractThis study investigates a re-entry scenario of an Apollo-like space capsule with Direct Numerical Simulations (DNS). The simulation includes the chemical equilibrium gas model. Cross-flow-like vortices are induced through random distributed roughness patches on the capsule surface. Two different machine learning methods are used to predict the maximum vorticity magnitude downstream of a pseudo-random roughness patch, the wall-normal location of the vortex core and spanwise and wall-normal gradient maxima of u. A large DNS database is formed for training and testing of the neural networks. In order to understand the influence of the vorticity magnitude on the transition process, local one-dimensional inviscid (LODI) relations are used to describe perturbations at the inflow. The disturbance evolution in the streamwise direction is analysed with a two-dimensional Fourier transformation in time and space. We show how the vorticity magnitudes of the cross-flow-like vortices, spanwise and wall-normal derivatives of the streamwise velocity influence the transition location.

Turbulence Effects on the Statistical Behaviour and Modelling of Flame Surface Density and the Terms of Its Transport Equation in Turbulent Premixed Flames

The influence of the ratio of integral length scale to flame thickness on the statistical behaviours of flame surface density (FSD) and its transport has been analysed using a Direct Numerical Simulation database of three-dimensional statistically planar turbulent premixed flames for different turbulence intensities. It has been found that turbulent burning velocity based on volume-integration of reaction rate and flame surface area increase but the peak magnitudes of the FSD and the terms of the FSD transport term decrease with an increase in length scale ratio for a given turbulence intensity. The flame brush thickness and flame wrinkling increase with an increase in length scale ratio for all turbulence intensities. However, the qualitative behaviours of the unclosed terms in the FSD transport equation remain unaltered by the length scale ratio and in all cases the tangential strain rate term and the curvature term act as leading order source and sink, respectively. A decrease in length scale ratio for a given turbulence intensity leads to a decrease in Damköhler number and an increase in Karlovitz number. This has an implication on the alignment of reactive scalar gradient with local strain rate eigenvectors, which in turn increases positive contribution of the tangential strain rate term with a decrease in length scale ratio. Moreover, an increase in Karlovitz number increases the likelihood of negative contribution of the curvature term. Thus, the magnitude of the negative contribution of the FSD curvature term increases with a decrease in length scale ratio for a given turbulence intensity. The model for the tangential strain rate term, which explicitly considers the scalar gradient alignment with local principal strain rate eigenvectors, has been shown to be more successful than the models that do not account for the scalar gradient alignment characteristics. Moreover, the existing model for the curvature and propagation term needed modification to account for greater likelihood of negative values for higher Karlovitz number. However, the models for the unclosed flux of FSD and the mean reaction rate closure are not significantly affected by the length scale ratio.