Nora Okong'o

Consistent large-eddy simulation of a temporal mixing layer laden with evaporating drops. Part 1. Direct numerical simulation, formulation and a priori analysis

Large-eddy simulation (LES) models are presented and evaluated on a database obtained from direct numerical simulation (DNS) of a three-dimensional temporal mixing layer with evaporating drops. The gas-phase equations are written in an Eulerian frame for two perfect gas species (carrier gas and vapour emanating from the drops), while the liquid-phase equations are written in a Lagrangian frame. The effect of drop evaporation on the gas phase is considered through mass, momentum and energy source terms. The DNS database consists of transitional states attained by layers with different initial Reynolds numbers and initial liquid-phase mass loadings. Budgets of the LES equations at the transitional states show that, for the mass loadings considered, the filtered source terms (FSTs) are smaller than the resolved inviscid terms and some subgrid scale (SGS) terms, but larger than the resolved viscous stress, heat flux and mass flux terms. The irreversible entropy production (i.e. the dissipation) expression for a two-phase flow with phase change is derived, showing that the dissipation contains contributions due to viscous stresses, heat and species-mass fluxes, and source terms. For both the DNS and filtered flow fields at transition, the two leading contributions are found to be the dissipation due to the energy source term and that due to the chemical potential of the mass source. Therefore, the modelling effort is focused on both the SGS fluxes and the FSTs in the LES equations. The FST models considered are applicable to LES in which the grid is coarser than the DNS grid and, consistently, ‘computational’ drops represent the DNS physical drops. Because the unfiltered flow field is required for the computation of the source terms, but would not be available in LES, it was approximated using the filtered flow field or the filtered flow field augmented by corrections based on the SGS variances. All of the FST models were found to overestimate DNS-field FSTs, with the relative error of modelling the unfiltered flow field compared to the error of using computational drops showing a complex dependence on filter width and number of computational drops. For modelling the SGS fluxes and (where possible) SGS variances, constant-coefficient Smagorinsky, gradient and scale-similarity models were assessed on the DNS database, and calibrated coefficients were statistically equivalent when computed on single-phase or two-phase flows. The gradient and scale-similarity models showed excellent correlation with the SGS quantities. An a posteriori study is proposed to evaluate the impact of the studied models on the flow-field development, so as to definitively assess their suitability for LES with evaporating drops.

Direct numerical simulation of a transitional supercritical binary mixing layer: Heptane and nitrogen

Direct numerical simulations (DNS) of a supercritical temporal mixing layer are conducted for the purpose of exploring the characteristics of high-pressure transitional mixing behaviour. The conservation equations are formulated according to fluctuation-dissipation (FD) theory, which is consistent with non-equilibrium thermodynamics and converges to kinetic theory in the low-pressure limit. According to FD theory, complementing the low-pressure typical transport properties (viscosity, diffusivity and thermal conductivity), the thermal diffusion factor is an additional transport property which may play an increasingly important role with increasing pressure. The Peng–Robinson equation of state with appropriate mixing rules is coupled to the dynamic conservation equations to obtain a closed system. The boundary conditions are periodic in the streamwise and spanwise directions, and of non-reflecting outflow type in the cross-stream direction. Due to the strong density stratification, the layer is considerably more difficult to entrain than equivalent gaseous or droplet-laden layers, and exhibits regions of high density gradient magnitude that become very convoluted at the transitional state. Conditional averages demonstrate that these regions contain predominantly the higher-density, entrained fluid, with small amounts of the lighter, entraining fluid, and that in these regions the mixing is hindered by the thermodynamic properties of the fluids. During the entire evolution of the layer, the dissipation is overwhelmingly due to species mass flux followed by heat flux effects with minimal viscous contribution, and there is a considerable amount of backscatter in the flow. Most of the species mass flux dissipation is due to the molecular diffusion term with significant contributions from the cross-term proportional to molecular and thermal diffusion. These results indicate that turbulence models for supercritical fluids should primarily focus on duplicating the species mass flux rather than the typical momentum flux, which constitutes the governing dissipation in atmospheric mixing layers. Examination of the passive-scalar probability density functions (PDFs) indicates that neither the Gaussian, nor the beta PDFs are able to approximate the evolution of the DNS-extracted PDF from its inception through transition. Furthermore, the temperature–species PDFs are well correlated, meaning that their joint PDF is not properly approximated by the product of their marginal PDFs; this indicates that the traditional reactive flow modelling based on replacing the joint PDF representing the reaction rate by the product of the marginal PDFs is not appropriate. Finally, the subgrid-scale temperature–species PDFs are also well correlated, and the species PDF exhibits important departures from the Gaussian. These results suggest that classic PDFs used in atmospheric pressure flows would not capture the physics of this supercritical mixing layer, either in an assumed PDF model at the larger scale, or at the subgrid scale.

Modelling of subgrid-scale phenomena in supercritical transitional mixing layers: an a priori study

A database of transitional direct numerical simulation (DNS) realizations of a supercritical mixing layer is analysed for understanding small-scale behaviour and examining subgrid-scale (SGS) models duplicating that behaviour. Initially, the mixing layer contains a single chemical species in each of the two streams, and a perturbation promotes roll-up and a double pairing of the four spanwise vortices initially present. The database encompasses three combinations of chemical species, several perturbation wavelengths and amplitudes, and several initial Reynolds numbers specifically chosen for the sole purpose of achieving transition. The DNS equations are the Navier-Stokes, total energy and species equations coupled to a real-gas equation of state; the fluxes of species and heat include the Soret and Dufour effects. The large-eddy simulation (LES) equations are derived from the DNS ones through filtering. Compared to the DNS equations, two types of additional terms are identified in the LES equations: SGS fluxes and other terms for which either assumptions or models are necessary. The magnitude of all terms in the LES conservation equations is analysed on the DNS database, with special attention to terms that could possibly be neglected. It is shown that in contrast to atmospheric-pressure gaseous flows, there are two new terms that must be modelled: one in each of the momentum and the energy equations. These new terms can be thought to result from the filtering of the nonlinear equation of state, and are associated with regions of high density-gradient magnitude both found in DNS and observed experimentally in fully turbulent high-pressure flows. A model is derived for the momentum-equation additional term that performs well at small filter size but deteriorates as the filter size increases, highlighting the necessity of ensuring appropriate grid resolution in LES. Modelling approaches for the energy-equation additional term are proposed, all of which may be too computationally intensive in LES. Several SGS flux models are tested on an a priori basis. The Smagorinsky (SM) model has a poor correlation with the data, while the gradient (GR) and scale-similarity (SS) models have high correlations. Calibrated model coefficients for the GR and SS models yield good agreement with the SGS fluxes, although statistically, the coefficients are not valid over all realizations. The GR model is also tested for the variances entering the calculation of the new terms in the momentum and energy equations; high correlations are obtained, although the calibrated coefficients are not statistically significant over the entire database at fixed filter size. As a manifestation of the small-scale supercritical mixing peculiarities, both scalar-dissipation visualizations and the scalar-dissipation probability density functions (PDF) are examined. The PDF is shown to exhibit minor peaks, with particular significance for those at larger scalar dissipation values than the mean, thus significantly departing from the Gaussian behaviour.

Direct numerical simulations of O2/H2 temporal mixing layers under supercritical conditions

Direct numerical simulations of a supercritical oxygen/hydrogen temporal three-dimensional mixing layer are conducted to explore the features of high-pressure transitional mixing behavior. The conservation equations are formulated according to fluctuation–dissipation theory and are coupled to a modified Peng–Robinson equation of state. The boundary conditions are periodic in the streamwise and spanwise directions and of nonreflecting outflow type in the cross-stream direction. Simulations are conducted with initial Reynolds numbers of 6 x 10^2 and 7.5 x 10^2, initial pressure of 100 atm, and temperatures of 400 K in the O_2 and 600 K in the H_2 stream. Each simulation encompasses the rollup and pairing of four initial spanwise vortices into a single vortex. The layer eventually exhibits distorted regions of high density-gradient-magnitude similar to the experimentally observed wisps of fluid at the boundary of supercritical jets. Analysis of the data reveals that the higher-Reynolds-number layer reaches transition, whereas the other one does not. The transitional layer is analyzed to elucidate its characteristics.

Consistent large-eddy simulation of a temporal mixing layer laden with evaporating drops. Part 2. A posteriori modelling

Large-eddy simulation (LES) is conducted of a three-dimensional temporal mixing layer whose lower stream is initially laden with liquid drops which may evaporate during the simulation. The gas-phase equations are written in an Eulerian frame for two perfect gas species (carrier gas and vapour emanating from the drops), while the liquid-phase equations are written in a Lagrangian frame. The effect of drop evaporation on the gas phase is considered through mass, species, momentum and energy source terms. The drop evolution is modelled using physical drops, or using computational drops to represent the physical drops. Simulations are performed using various LES models previously assessed on a database obtained from direct numerical simulations (DNS). These LES models are for: (i) the subgrid-scale (SGS) fluxes and (ii) the filtered source terms (FSTs) based on computational drops. The LES, which are compared to filtered-and-coarsened (FC) DNS results at the coarser LES grid, are conducted with 64 times fewer grid points than the DNS, and up to 64 times fewer computational than physical drops. It is found that both constant-coefficient and dynamic Smagorinsky SGS-flux models, though numerically stable, are overly dissipative and damp generated small-resolved-scale (SRS) turbulent structures. Although the global growth and mixing predictions of LES using Smagorinsky models are in good agreement with the FC-DNS, the spatial distributions of the drops differ significantly. In contrast, the constant-coefficient scale-similarity model and the dynamic gradient model perform well in predicting most flow features, with the latter model having the advantage of not requiring a priori calibration of the model coefficient. The ability of the dynamic models to determine the model coefficient during LES is found to be essential since the constant-coefficient gradient model, although more accurate than the Smagorinsky model, is not consistently numerically stable despite using DNS-calibrated coefficients. With accurate SGS-flux models, namely scale-similarity and dynamic gradient, the FST model allows up to a 32-fold reduction in computational drops compared to the number of physical drops, without degradation of accuracy; a 64-fold reduction leads to a slight decrease in accuracy.

Real-Gas Effects on Mean Flow and Temporal Stability of Binary-Species Mixing Layers

Real-gas effects on the mean flow and inviscid stability of temporal mixing layers are examined for supercritical heptane/nitrogen and oxygen/hydrogen mixtures. The analysis is based on the compressible Navier–Stokes equations for conservation of mass, momentum, total energy, and species mass, with heat and species-mass fluxes derived from fluctuation-dissipation theory and incorporating Soret and Dufour effects. An approximate form of the equations is used to obtain a system of similarity equations for the streamwise velocity, the temperature, and the mass fraction. The similarity profiles show important real-gas nonideal-mixture effects, particularly for the temperature, in departing from the incompressible error-function similarity solution. Realistic Schmidt and Prandtl numbers were found to be important to the similarity profiles. A linear, inviscid stability analysis is then performed using the similarity profile, as well as analytical error-function profiles, as its basic flow. The stability analysis shows that the similarity profile has larger growth rates at a given wavelength and a shorter more unstable wavelength than the error-function profiles and than an incompressible flow. The similarity profile also has a larger range of unstable wavelengths than the error-function profiles.

Detailed characteristics of drop-laden mixing layers: Large eddy simulation predictions compared to direct numerical simulation

Results are compared from direct numerical simulation (DNS) and large eddy simulation (LES) of a temporal mixing layer laden with evaporating drops to assess the ability of LES to reproduce detailed characteristics of DNS. The LES used computational drops, each of which represented eight physical drops, and a reduced flow field resolution using a grid spacing four times larger than that of the DNS. The LES also used models for the filtered source terms, which express the coupling of the drops with the flow, and for the unresolved subgrid-scale (SGS) fluxes of species mass, momentum, and enthalpy. The LESs were conducted using one of three different SGS-flux models: dynamic-coefficient gradient (GRD), dynamic-coefficient Smagorinsky (SMD), and constant-coefficient scale similarity (SSC). The comparison of the LES with the filtered-and-coarsened (FC) DNS considered detailed aspects of the flow that are of interest in ignition or full combustion. All LESs captured the largest-scale vortex, the global amount of vapor emanating from the drops, and the overall size distribution of the drops. All LESs tended to underpredict the global amount of irreversible entropy production (dissipation). The SMD model was found unable to capture either the global or local vorticity variation and had minimal small-scale activity in dynamic and thermodynamic variables compared to the FC-DNS. The SMD model was also deficient in predicting the spatial distribution of drops and of the dissipation. In contrast, the GRD and SSC models did mimic the small-scale activity of the FC-DNS and the spatial distribution of drops and of the dissipation. Therefore, the GRD and SSC models are recommended, while the SMD model seems inappropriate for combustion or other problems where the local activity must be predicted.

Detailed characteristics of drop-laden mixing layers: LES predictions compared to DNS

Results have been compared from Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) of a temporal mixing layer laden with evaporating drops, to assess the ability of LES to reproduce detailed characteristics of DNS.

A Priori Analysis of Subgrid-Scale Models for Large Eddy Simulations of Supercritical Binary-Species Mixing Layers

Models for large eddy simulation (LES) are assessed on a database obtained from direct numerical simulations (DNS) of supercritical binary-species temporal mixing layers. The analysis is performed at the DNS transitional states for heptane/nitrogen, oxygen/hydrogen and oxygen/helium mixing layers. The incorporation of simplifying assumptions that are validated on the DNS database leads to a set of LES equations that requires only models for the subgrid scale (SGS) fluxes, which arise from filtering the convective terms in the DNS equations. Constant-coefficient versions of three different models for the SGS fluxes are assessed and calibrated. The Smagorinsky SGS-flux model shows poor correlations with the SGS fluxes, while the Gradient and Similarity models have high correlations, as well as good quantitative agreement with the SGS fluxes when the calibrated coefficients are used.

Helicity in supercritical temporal mixing layers

Databases of transitional states obtained from Direct Numerical Simulations (DNS) of temporal, supercritical mixing layers for two species systems, 02/H2 and C7Hle/N2, are analyzed to elucidate species-specific turbulence aspects.