Numerical prediction of isothermally reacting mixing layer using vortex‐in‐cell and filtered density function

A large eddy simulation based on the filtered vorticity transport equation and the filtered density function (FDF) transport equation developed in an earlier study is extended to predict a chemically reacting flow with no heat release. The filtered vorticity transport equation is solved using the vortex‐in‐cell scheme in conjunction with the dynamic eddy viscosity subgrid‐scale models. The transport equation for FDF is solved using the Lagrangian Monte‐Carlo method. The methodology is tested on a chemically reacting spatially growing mixing layer with no heat release. The effects of Damköhler number (Da) on the concentration structure of the reacting mixing layer, the mean reactant and product concentrations and on the reactant FDF are investigated. It is shown that mixing has a greater effect on scalar field within the vortex structure as compared with the braid regions. Also for high Da, the reaction zones are mainly limited to the thin reacting interfacial zones, i.e. the contact zone between the reactants, whereas for low Da, the reacting zones are spread as reacting pockets within the vortex structure. The effects of Da on mean reactant and product concentrations, root‐mean‐square concentration fluctuations and probability density are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Vortex-In-Cell and Probability Density Function Approach for a Passive Scalar Field in a Mixing Layer

A Reynolds-averaged simulation based on the vortex-in-cell (VIC) and the transport equation for the probability density function (PDF) of a scalar has been developed to predict the passive scalar field in a two-dimensional spatially growing mixing layer. The VIC computes the instantaneous velocity and vorticity fields. Then the mean-flow properties, i.e. the mean velocity, the root-mean-square (rms) longitudinal and lateral velocity fluctuations, the Reynolds shear stress, and the rms vorticity fluctuations are computed and used as input to the PDF equation. The PDF transport equation is solved using the Monte Carlo technique. The convection term uses the mean velocities from the VIC. The turbulent diffusion term is modeled using the gradient transport model, in which the eddy diffusivity, computed via the Boussinesq's postulate, uses the Reynolds shear stress and gradients of mean velocities from the VIC. The molecular mixing term is closed by the modified Curl model.

The computational results were compared with two-dimensional experimental results for passive scalar. The predicted turbulent flow characteristics, i.e. mean velocity and rms longitudinal fluctuations in the self-preserving region, show good agreement with the experimental measurements. The profiles of the mean scalar and the rms scalar fluctuations are also in reasonable agreement with the experimental measurements. Comparison between the mean scalar and the mean velocity profiles shows that the scalar mixing region extends further into the free stream than does the momentum mixing region, indicating enhanced transport of scalar over momentum. The rms scalar profiles exhibit an asymmetry relative to the concentration centerline, and indicate that the fluid on the high-speed side mixes at a faster rate than the fluid on the low-speed side. The asymmetry is due to the asymmetry in the mixing frequency cross-stream profiles. Also, the PDFs have peaks biased toward the high-speed side.     

LES of Premixed Methane Flame Impinging on the Wall Using Non-adiabatic Flamelet Generated Manifold (FGM) Approach

In this paper LES of flame wall interaction using a non-adiabatic FGM approach is reported for a premixed methane fuel jet impinging on a spherical disk. Nitrogen is used as co-flow in order to avoid the interaction with the surrounding air and combustion. The flow field is described by means of the Smagorinsky model with Germano procedure for SGS stresses. The SGS scalar flux in scalar transport equations is modeled by the linear eddy diffusivity model. Two aspects are especially addressed in this paper. First focus is on the grid resolution required near the wall without including a special wall-adapted SGS modeling in reacting configurations. The second aspect is devoted to the integration of the near wall kinetic effects into the FGM framework. The results for the flow field, mixing and combustion properties are presented and analyzed in terms of grid resolution, Reynolds number (in reacting and non-reacting case) and adiabaticity. Comparisons with available experimental data show satisfactory agreement. An outline of the thermal and flow boundary layer analysis is subsequently provided.     

The structure and dynamics of reacting plane mixing layers

The reacting two-dimensional plane mixing layer has been studied in two configurations: a rearward facing step and a two-stream mixing layer. Observations have been made of the steady state behavior, and the unsteady behavior when the flow was forced by a specific acoustic frequency. The steady behavior of the mean properties of the reacting flows is similar to that of non-reacting free shear flows except for the global effects of thermodynamic property changes. The structure of these flows is qualitatively similar to that of non-reacting flows. Vortices form by the two-dimensional Kelvin-Helmholtz instability and grow by subharmonic combination until the mixing layer interacts with the walls. Entrainment is dominated by the two-dimensional vortex motion. Three-dimensional instabilities give rise to secondary vortices which are coherent over several Kelvin-Helmholtz structures and dominate the fine scale mixing process. The mixing transition corresponds to a loss of coherence in the layer. Unsteady behavior occurs when there are resonant interactions with the Kelvin-Helmholtz instability or the instability associated with the recirculation vortex in the rearward facing step flow. Modeling efforts are reported which show promise of simulating the essential features of plane mixing layers.A version of this paper was presented at the ASME Winter Annual Meeting of 1984 and printed in AMD-Vol. 66     

A Comprehensive Study of Improved Delayed Detached Eddy Simulation with Wall Functions

The mechanisms for nonlinear saturation of a bluff-body stabilised turbulent premixed flame are investigated using LES with the transported flame surface density (TFSD) approach to combustion modelling. The numerical simulation is based on a previous detailed experimental investigation. Results for both the unforced non-reacting and reacting flows are validated against experiment, demonstrating that the fundamental flow features and predicted flame structure are well captured. Key terms in the FSD transport equation are then used to describe the generation and destruction of flame surface area for the unforced reacting flow. In order to investigate the non-linear response of the unsteady heat release rate to acoustic forcing, four harmonically forced flames are considered having the same forcing frequency (160 Hz) but different amplitudes of 10 %, 25 %, 50 % and 64 % of the mean inlet velocity. The flame response is characterised using the Flame Describing Function (FDF). Accurate prediction of the FDF is obtained using the current approach. The computed forced flame structure matches well with the experiment, where effects of shear layer rollup and growth of the vortices on the flame can be clearly observed. Transition to nonlinearity is also observed in the computed FDF. The mechanisms leading to the saturation of the flame response in the higher amplitude case are characterised by inspecting the terms in the FSD transport equation at conditions when the integrated heat release is at its maximum and minimum, respectively. Pinch-off and flame rollup can be seen in snapshots taken at the phase angle of maximum integrated heat release. Conversely, intense vortex shedding and flame-sheet collapse around the shear-layer, as well as small-scale destruction of flame elements in the wake, can be seen in snapshots taken at the phase angle of minimum integrated heat release. The pivotal role of FSD destruction on nonlinear saturation of the flame response is confirmed through the analysis of phase-averaged terms in the FSD transport equation taken at different locations. The phase-averaged subgrid curvature term is found to concentrate in the cusps and downstream regions where flame annihilation is dominant.     

Large eddy simulation (2D) of spatially developing mixing layer using vortex‐in‐cell for flow field and filtered probability density function for scalar field

A large eddy simulation based on filtered vorticity transport equation has been coupled with filtered probability density function transport equation for scalar field, to predict the velocity and passive scalar fields. The filtered vorticity transport has been formulated using diffusion‐velocity method and then solved using the vortex method. The methodology has been tested on a spatially growing mixing layer using the two‐dimensional vortex‐in‐cell method in conjunction with both Smagorinsky and dynamic eddy viscosity subgrid scale models for an anisotropic flow. The transport equation for filtered probability density function is solved using the Lagrangian Monte‐Carlo method. The unresolved subgrid scale convective term in filtered density function transport is modelled using the gradient diffusion model. The unresolved subgrid scale mixing term is modelled using the modified Curl model. The effects of subgrid scale models on the vorticity contours, mean streamwise velocity profiles, root‐mean‐square velocity and vorticity fluctuations profiles and negative cross‐stream correlations are discussed. Also the characteristics of the passive scalar, i.e. mean concentration profiles, root‐mean‐square concentration fluctuations profiles and filtered probability density function are presented and compared with previous experimental and numerical works. The sensitivity of the results to the Schmidt number, constant in mixing frequency and inflow boundary conditions are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

Passive Scalar Transport in a Turbulent Mixing Layer

This paper describes a combined experimental and numerical study of scalar transport in spatially developing, two-stream, turbulent mixing layers with velocity ratios of approximately 2:1. The experimental mixing layer was created by an S-shaped splitter plate mounted in a wind tunnel, and the concentration field was realized by releasing incense smoke into the high-speed side boundary layer above the splitter plate. Simultaneous measurements of the velocity and concentration fields were performed. A 12-sensor hot-wire probe was used to measure the velocity field and its gradients, while the concentration field was recorded with digital photographs of the laser-illuminated smoke. In parallel, a large-eddy simulation (LES) of the spatially developing mixing layer was carried out. Auxiliary turbulent boundary layer LES were used to provide high quality inflow boundary conditions for the velocity and concentration fields. By synchronizing the velocity and concentration measurements, concentration fluxes were also determined. Octant analysis based on the sign combinations of the velocity and concentration fuctuations was performed on the flux data to investigate the scalar transport processes. It was found that octants compatible with mean gradient transport of the scalar contribute most to the scalar fluxes. Conditional planar averages of scalar and momentum fluxes were obtained to determine their spatial distribution with respect to the organized roller and rib vortices of the mixing layer, and distinct patterns were observed. The simulation provided additional insight about the flow and scalar flux distribution topology. This topology was found to be partially compatible with simple models of roller and rib vortices that transport the scalar in a mean gradient sense.     

On Fokker–Planck Equations for Turbulent Reacting Flows. Part 2. Filter Density Function for Large Eddy Simulation

The application of large eddy simulation (LES) to turbulent reacting flow calculations is faced with several closure problems. Suitable parametrizations for filtered reaction rates for instance are hardly available in general. A way to overcome these problems is investigated here. This is done by extending LES equations for filtered velocities and scalars (mass fractions of species and temperature) to equations that involve subgrid scale (SGS) fluctuations. Such equations are called filter density function (FDF) methods because they determine the FDF, which is essentially the probability density function of SGS variables. The FDF model considered involves only three parameters: C ₀ that controls the generation of velocity fluctuations and two parameters which determine the relaxation of velocity and scalar fluctuations. The consideration of this model may be seen as the analysis of a limiting case: the implications of the most simple equations for the dynamics of SGS fluctuations are investigated in this way. These equations were proved recently by various simulations. Here, the FDF model is used analytically to improve simpler methods. Existing models for the SGS stress tensor in velocity LES equations and the diffusion coefficient in scalar FDF equations are generalized in this way. The advantages of these models compared to existing ones are pointed out. These investigations provide further evidence for the suitability of the FDF model considered and they provide its parameters. A theoretical value C ₀ = 19/12 is derived, which agrees very well with the results of direct numerical simulation. This estimate implies the same value for the universal Kolmogorov constant of the energy spectrum, which is consistent with the results of many measurements. The other two model parameters can be obtained then by dynamic procedures. Therefore, the closure problems of LES equations are overcome in this way such that adjustable parameters are not involved.     

New Molecular Transport Model for FDF/LES of Turbulence with Passive Scalar

The paper addresses the issue of modelling and computation of wall-bounded turbulent flows with passive scalars. In the present approach, the large eddy simulation (LES) method is used to compute the velocity field in the near-wall zone. The LES is coupled with the Lagrangian filtered density function (FDF) model for the transport of a passive scalar. In the paper, we propose two models to account for the molecular transport near the wall and investigate their behaviour in the limit case of small filter widths. One of the models is tested numerically, and computational results for a heated channel flow are compared with the available DNS data.     

Developments in Formulation and Application of the Filtered Density Function

An overview is presented of the state of progress in large eddy simulation of turbulent combustion via the filtered density function (FDF). This includes the formulations based on both the joint velocity-scalar FDF, and the marginal scalar FDF. In the former, the most up-to-date and comprehensive form of the model is presented along with its applications for LES of some relatively simple flows. In the latter, results are presented of the most recent LES of a complex turbulent flame. Both of the models are described in the context of a variable density flow via consideration of the filtered mass density function (FMDF).Presented at the Workshop on Quality Assessment of Unsteady Methods for Turbulent Combustion, Darmstadt, Germany, June 16–17, 2005.     

Conditional Moment Closure for Large Eddy Simulations

A conditional moment closure (CMC) based combustion model for large-eddy simulations (LES) of turbulent reacting flow is proposed and evaluated. Transport equations for the conditionally filtered species are derived that are consistent with the LES formulation and closures are suggested for the modelling of the conditional velocity, conditional scalar dissipation and the fluctuations around the conditional mean. A conventional β-probability density distribution of the scalar is used together with dynamic modelling for the sub-grid fluxes. The model is validated by comparison of simulations with measurements of a piloted, turbulent methane-air jet diffusion flame.     

Large Eddy Simulation of Scalar Mixing in a Coaxial Confined Jet

The highly turbulent flow occurring inside gas-turbine combustors requires accurate simulation of scalar mixing if CFD methods are to be used with confidence in design. This has motivated the present paper, which describes the implementation of a passive scalar transport equation into an LES code, including assessment/testing of alternative discretisation schemes to avoid over/undershoots and excessive smoothing. Both second order accurate TVD and higher order accurate DRP schemes are assessed. The best performance is displayed by a DRP method, but this is only true on fine meshes; it produces similar (or larger) errors to a TVD scheme on coarser meshes, and the TVD approach has been retained for LES applications. The unsteady scalar mixing performance of the LES code is validated against published DNS data for a slightly heated channel flow. Excellent agreement between the current LES predictions and DNS data is obtained, for both velocity and scalar statistics. Finally, the developed methodology is applied to scalar transport in a confined co-axial jet mixing flow, for which experimental data are available. Agreement with statistically averaged fields for both velocity and scalar, is demonstrated to be very good, and a considerable improvement over the standard eddy viscosity RANS approach. Illustrations are presented of predicted time-resolved information e.g. time histories, and scalar pdf predictions. The LES results are shown, even using a simple Smagorinsky SGS model, to predict (correctly) lower values of the turbulent Prandtl number in the free shear regions of the flow, compared to higher values in the wall-affected regions. The ability to predict turbulent Prandtl number variations (rather than input these as in combustor RANS CFD models) is an important and promising feature of the LES approach for combustor flow simulation since it is known to be important in determining combustor exit temperature traverse.     

Experimental and Numerical Study of the Scalar Turbulent/Non-Turbulent Interface Layer in a Jet Flow

Based on two large-eddy simulations (LES) of a non-reacting turbulent round jet with a nozzle based Reynolds number of 8,610 with the same configuration as the one that has recently been investigated experimentally (Gampert et al., 2012; J Fluid Mech, 2012; J Fluid Mech 724:337, 2013), we examine the scalar turbulent/non-turbulent (T/NT) interface layer in the mixture fraction field of the jet flow between ten and thirty nozzle diameters downstream. To this end, the LES—one with a coarse grid and one with a fine grid—are in a first step validated against the experimental data using the axial decay of the mean velocity and the mean mixture fraction as well as based on radial self-similar profiles of mean and root mean square values of these two quantities. Then, probability density functions (pdf) of the mixture fraction at various axial and radial positions are compared and the quality of the LES is discussed. In general, the LES results are consistent with the experimental data. However, in the flow region where the imprint of the T/NT interface layer is dominant in the mixture fraction pdf, discrepancies are observed. In a next step, statistics of the T/NT interface layer are studied, where a satisfactory agreement for the pdf of the location of the interface layer from the higher resolved LES with the experimental data is observed, while the one with the coarse grid exhibits considerable deviations. Finally, the mixture fraction profile across the interface is investigated where the same trend as for the pdf of the location is present. In particular, it is found that the sharp interface that is present in experimental studies (Gampert et al., J Fluid Mech, 2013; Westerweel et al., J Fluid Mech 631:199, 2009) is less distinct in the LES results and rather diffused in radial direction outside of the T/NT interface layer.     

An Experimental Investigation of the Turbulence Structure of a Lifted H2/N2 Jet Flame in a Vitiated Co-Flow

Measurements of mean velocity components, turbulent intensity, and Reynolds shear stress are presented in a turbulent lifted H₂/N₂ jet flame as well as non-reacting air jet issuing into a vitiated co-flow by laser doppler velocimetry (LDV) technique. The objectives of this paper are to obtain a velocity data base missing in the previous experiment data of the Dibble burner and so provide initial and flow field data for evaluating the validity of various numerical codes describing the turbulent partially premixed flames on this burner. It is found that the potential core is shortened due to the high ratio of jet density to co-flow density in the non-reacting cases. However, the existence of flame suppressed turbulence in the upstream region of the jet dominates the length of potential core in the reacting cases. At the centreline, the normalized axial velocities in the reacting cases are higher than the non-reacting cases, and the relative turbulent intensities of the reacting flow are smaller than in the non-reacting flow, where a self-preserving behaviour for the relative turbulent intensities exists at the downstream region. The profiles of mean axial velocity in the lifted flame distribute between the non-reacting jet and non-premixed flame both in the axial and radial distributions. The radial distributions of turbulent kinetic energy in the lifted flames exhibit a change in distributions indicating the difference of stabilisation mechanisms of the two lifted flame. The experimental results presented will guide the development of an improved modelling for such flames.     

A Length-Scale Equation

We derive an equation for the average length-scale in a turbulent flow from a simple physical model. This is a tensorial length-scale. We use as a model the evolution of a blob of turbulent kinetic energy under the influence of production, dissipation, and transport, as well as distortion by the mean motion. A single length-scale is defined which is biased toward the smallest of the scales in the various directions. Constants are estimated by consideration of homogeneous decay. Preliminary computations are carried out in a mixing layer and a two-dimensional jet, using the new length-scale equation and the equation for the turbulent kinetic energy. The results are compared with data and with the predictions of the classical k-epsilon equations; the new results are quite satisfactory. In particular, the plane jet/round jet anomaly is approximately resolved. This revised version was published online in July 2006 with corrections to the Cover Date.     

The effect of inflow conditions on the transition to turbulence in large eddy simulations of spatially developing mixing layers

Large Eddy Simulations (LES) of spatially developing turbulent mixing layers have been performed for flows of uniform density and Reynolds numbers of up to 50,000 based on the visual thickness of the layer and the velocity difference across it. On a fine LES grid, a validation simulation performed with a hyperbolic tangent inflow profile produces flow statistics that compare extremely well with reference Direct Numerical Simulation (DNS) data. An inflow profile derived from laminar Blasius profiles produces a flow that is significantly different to the reference DNS, particularly with respect to the initial development of the flow. When compared with experimental data, however, it is the boundary layer-type inflow simulation produces the better prediction of the flow statistics, including the mean transition location. It is found that the boundary layer inflow condition is more unstable than the hyperbolic tangent inlet profile. A suitably designed coarse LES grid produces good predictions of the mean transition location with boundary layer inflow conditions at a low computational cost. The results suggest that hyperbolic tangent functions may produce unreliable DNS data when used as the initial condition for studies of the transition in the mixing layer flow.     

Scalar transport in plane mixing layers

This paper describes the application of the Eulerian, single-point, single-time joint-scalar probability density function (PDF) equation for predicting the scalar transport in mixing layer with a high-speed and a low-speed stream. A finite-volume procedure is applied to obtain the velocity field with the k-ε closure being used to describe turbulent transport. The scalar field is represented through the modelled evolution equation for the scalar PDF and is solved using a Monte Carlo simulation. The PDF equation employs gradient transport modelling to represent the turbulent diffusion, and the molecular mixing term is modelled by the LMSE closure. There is no source term for chemical reaction as only an inert mixing layer is considered here. The experimental shear layer data published by Batt is used to validate the computational results despite the fact that comparisons between experiments and computational results are difficult because of the high sensitivity of the shear layer to initial conditions and free stream turbulence phenomena. However, the bimodal shape of the RMS scalar fluctuation as was measured by Batt can be reproduced with this model, whereas standard gradient diffusion calculations do not predict the dip in this profile. In this work for the first time an explanation is given for this phenomenon and the importance of a micromixing model is stressed. Also it is shown that the prediction of the PDF shape by the LMSE model is very satisfactory. Received on 27 October 1998     

External Intermittency Simulation in Turbulent Round Jets

Direct numerical and large eddy simulation (DNS and LES) are applied to study passive scalar mixing and intermittency in turbulent round jets. Both simulation techniques are applied to the case of a low Reynolds number jet with Re = 2,400, whilst LES is also used to predict a high Re = 68,000 flow. Comparison between time-averaged results for the scalar field of the low Re case demonstrate reasonable agreement between the DNS and LES, and with experimental data and the predictions of other authors. Scalar probability density functions (pdfs) for this jet derived from the simulations are also in reasonable accord, although the DNS results demonstrate the more rapid influence of scalar intermittency with radial distance in the jet. This is reflected in derived intermittency profiles, with LES generally giving profiles that are too broad compared to equivalent DNS results, with too low a rate of decay with radial distance. In contrast, good agreement is in general found between LES predictions and experimental data for the mixing field, scalar pdfs and external intermittency in the high Reynolds number jet. Overall, the work described indicates that improved sub-grid scale modelling for use with LES may be beneficial in improving the accuracy of external intermittency predictions by this technique over the wide range of Reynolds numbers of practical interest.     

Investigation of scalar measurement error in diffusion and mixing processes

In variable density, multi-fluid and reacting flows, the degree of molecular mixing is a critical component of turbulent transfer and mixing models. Also, in many microflows and low Reynolds number flows, scalar diffusion length- and time-scales play a significant role in the mixing dynamics. Characterization of such molecular mixing processes requires scalar measurement devices with a small probe volume size. Spatial averaging, which occurs due to finite probe volume size, can lead to errors in resolving the density or scalar gradients between pockets of unmixed fluids. Given a probe volume size and a priori knowledge of the functional profile of the diffusion layer being measured, we obtain an estimate for the measurement error due to spatial averaging and make the corrections accordingly. An analytical model for the measure of scalar mixing is developed as a predictor for the growth of scalar gradients in a variable scalar flow. The model is applied to a buoyancy-driven mixing layer with a Prandtl number of 7. Measurements within the mixing layer have shown that initial entrainment of unmixed fluid causes a decrease in the measured amount of molecular mixing at the centerplane. Following this period of initial entrainment, the fluids within the mixing layer exhibit an increase in the degree of molecular mixing.     

Direct numerical simulation of turbulent mixing of a passive scalar in pipe flow

Turbulent mixing of a passive scalar in fully developed turbulent pipe flow has been investigated by means of a Direct Numerical Simulation (DNS). The scalar is released from a point source located on the centreline of the pipe. The domain size of the concentration field has been chosen large enough to capture the different stages of turbulent mixing. Results are presented for mean concentration profiles, turbulent fluxes, concentration fluctuations, probability density functions and higher-order moments. To validate the numerical simulations the results are compared with experimental data on mixing in grid-turbulence that have been reported in the literature. The agreement between the experimental measurements and the computations is satisfactory. We have also considered the Probability Density Function (PDF). For small diffusion times and positions not on the plume centreline, our results lead to a PDF of an exponential form with a large peak at zero concentration. When the diffusion time increases, the PDF shifts from a exponential to a more Gaussian form.