Session BB: Turbulent Mixing

Turbulent transport of a high Schmidt number passive scalar below an air--water interface with zero mean shear

Laboratory measurements in a stirred tank reveal the turbulent flux on the water side of an air--water interface. The tank is stirred from below by random jets, providing turbulence that is homogeneous and isotropic in the horizontal direction. This flow is an interesting counterpoint to cases where the turbulence at the surface is driven by shear there. The Taylor microscale Reynolds number $R_\lambda \approx 300$, giving a wide range of scales. The contribution of these various scales to turbulent scalar flux is investigated by simultaneously measuring the velocity field and the concentration field of dissolved CO$_2$. This is accomplished via quantitative imaging, and allows cospectra, structure functions, and coherent structures to be investigated. These are compared to theory for sheared interfaces, as well as to similar results obtained by other researchers at lower Schmidt and Reynolds numbers via CFD.   

Effect of the initial conditions on scalar dispersion in uniform-density grid turbulence

According to the equilibrium similarity analysis, self-preservation solutions are valid at all scales of motion for many `classical' flows. In grid-turbulence for example, the energy decay rate is of a power-law form and can depend on initial conditions, hence decay-rate constants cannot be universal, except possibly in the limit of ``infinite'' Reynolds number. Most of the earlier attempts to validate this theory do not cover a sufficiently wide range of initial conditions or decay times. In the present work, a passive scalar is released from a point source in uniform-density grid turbulence (downstream of a towed grid) using three different momentum-flux conditions: 1) a momentum-less-wake condition, 2) a wake condition, and 3) a jet condition. Using laser-induced fluorescence techniques, data obtained from stream-wise cuts of the scalar field are recorded for the complete turbulence decay range, with varying initial Reynolds numbers based on the mesh length of the grid reaching Re$_{M}$ = 5.6$\times $10$^{4}$. Using fitting parameters to extract the power-law decay range, the scaling for the different inflow conditions is computed and differences are discussed. It is shown that scalar dispersion in grid turbulence retains the influence of the initial conditions over the entire decay range.   

Effect of scalar-field initial conditions on the internal intermittency of turbulent passive scalars

Lepore and Mydlarski\footnote{Lepore and Mydlarski, 2009, {\it Phys. Rev. Lett.}, {\bf 103}, 034501.} recently measured the inertial-convective-range structure function scaling exponents of a turbulent passive scalar field ($\xi_n$) in hydrodynamically identical flows for which the passive scalar field was injected using two different techniques. The flow under consideration was the turbulent wake of a circular cylinder and the scalar fields were injected by: (i) heating the cylinder, and (ii) use of a mandoline. Lepore and Mydlarski showed that the higher-order structure function scaling exponents were dependent on the passive scalar field boundary conditions -- a result that (i) explained the previously observed variations in $\xi_n$ reported by different research groups (who generated their passive scalar fields using different methods), and (ii) is, ostensibly, in violation of Kolmogorov-Oboukhov-Corrsin theory. The present work further explores the role of the passive scalar field boundary conditions in turbulent mixing by means of additional statistical analyses (conditional expectations, inverse structure functions of passive scalar increments, etc.) in an attempt to elucidate the relationship between the boundary conditions and the internal intermittency of the turbulent passive scalar field.   

On the universality of dissipation mechanisms in mixing

We consider partitioning of energy in incompressible miscible variable density flows and in particular choose to distinguish between energy that is dissipated due to viscosity, and that which performs mixing. Variable density fluids exhibit a change in potential energy exclusively associated with mixing, as well as an overall change due to the system relaxation. The relationship between these energies provides a global measure of mixing efficiency that can be easily diagnosed from experiments and comparable simulations. Rayleigh-Taylor instability is one of the most efficient mechanisms for mixing: typically half of the initially available potential energy is used in doing so, an efficiency of 50\%. Unfortunately initial and final states in a Rayleigh-Taylor unstable flow are such that, by construction, mixing efficiency is bounded at 50\%. Our new configuration has an energetically admissible end-state that could be achieved if the mixing efficiency were 75\%, but our experiments show that despite the possibility of achieving a higher mixing efficiency, the system once more relaxes to 50\%. We conclude that the processes governing the distribution of energy are micro-scale, and not pre-determined by initial conditions.   

On the diapycnal diffusivity in homogeneous stably stratified turbulence

Quantifying the irreversible diapycnal mixing that occurs in stably stratified turbulence is fundamental to the understanding and modeling of geophysical flows, and for predicting dispersion in these flows. In this study, data of diapycnal mixing from direct numerical simulations of homogeneous stably stratified turbulence, both with and without shear, and from grid turbulence experiments, are reviewed and analyzed to investigate the scaling of the diapycnal diffusivity. In these homogeneous flows the instantaneous diapycnal diffusivity is given exactly by $K_d = \epsilon_\rho/\left(\partial\overline{\rho}/\partial z\right)^2$ and may be expressed in terms of the large scale properties of the turbulence as $K_d = \gamma L_E^2/T_L$, where $L_E = (\overline{{\rho'}^2})^\frac12/|\partial\overline{\rho}/\partial z|$ is the Ellison overturning length-scale, $T_L = k/\epsilon$ is the turbulence decay time-scale, and $\gamma$ is half the mechanical to scalar time-scale ratio. Our results show that $L_E$ and $T_L$ can explain all the variations in $K_d$ over a wide range of shear and stratification strengths (including shear-free and neutrally stratified cases) while $\gamma$ remains approximately constant. This result is also found to be independent of Prandtl (or Schmidt) number.   

The dispersion of patterns written in turbulent air

We study the mixing of passive objects in turbulence by writing structures in turbulent air and following their deformation in time. The writing is done by fusing ${\rm O}_2$ and ${\rm N}_2$ molecules into NO in the focus of a strong ultraviolet laser beam. By crossing several of these laser beams, patterns that have both small and large scales can be painted. The patterns are visualized a while later by inducing fluorescence of the NO molecules with a second UV laser and registering the image. The width of the lines that make the pattern is approximately $50 \:\mu$m, a few times the Kolmogorov length, the overall size of the patterns ($\approx 4$mm) is inside the inertial range of the used turbulent jet flow. Thus, we are able to study turbulent dispersion both at micro- and macroscales in a frame of reference that moves with the flow. In this way we have measured the spreading of clouds whose size is a few times the Kolmogorov length and the Batchelor dispersion of objects whose size is inside the inertial range. Patterns are compressible objects and spontaneously develop concentration fluctuations. We show for the first time the remarkable statistical properties of these fluctuations.   

Enthalpy Diffusion in Multicomponent Flows

The enthalpy diffusion flux in the multicomponent energy equation is a well known yet frequently neglected term. It accounts for energy changes associated with compositional changes resulting from species diffusion. The term prevents local violations of the entropy condition in flows where significant mixing occurs between species of dissimilar molecular weight. In simulations of nonpremixed combustion, omission of the enthalpy flux can lead to anomalous temperature gradients, which may cause mixing regions to exceed ignition conditions. The term can also play a role in generating acoustic noise in turbulent mixing layers. Euler solvers that rely on numerical diffusion to blend fluids at the grid scale cannot reliably predict temperatures in mixing regions. On the other hand, Navier-Stokes solvers that incorporate enthalpy diffusion can provide much more accurate results. In constructing turbulence closures for high Reynolds number mixing, the same turbulent diffusion model that appears in the species mass transport equation should also appear in the energy equation as part of a ``turbulent enthalpy diffusion''; otherwise the energy and species transport equations will not be consistent.   

A bivariate beta distribution as a presumed pdf for two mixture fractions

In turbulent reacting flows, an assumed pdf of a conserved scalar is often used to characterize the composition of the fluid. If more than a single fuel stream is present multiple mixture fractions are necessary, in which case a representation of the joint statistics is required. Here, several joint distributions proposed in the literature and a new bivariate beta distribution are investigated with regard to their suitability as a presumed pdf for the mixing of two conserved scalars. The bivariate beta distribution has the advantage that the marginal distributions reduce to univariate beta distributions, which have been shown to perform well for the mixing of a single conserved scalar. The presumed distributions are compared to DNS data over a range of initial scalar fields, including variations in the state of mixing and proportions of each scalar.   

Numerical Implementation of Molecular Transport and Mixing in LES/FDF of Turbulent Flows

In large-eddy simulations of turbulent flames, the effect of molecular diffusion on scalar transport is significant and therefore needs to be modeled correctly in Lagrangian Filtered Density Function (FDF) methods. McDermott \textit{et al.} (2007) show that in FDF methods, modeling molecular diffusion as a mean drift term in the scalar equation avoids the spurious production of scalar variance. Following McDermott \textit{et al.}, we incorporate the effects of molecular diffusion in the Interaction by Exchange with the Mean (IEM) mixing model. In this study, we evaluate two second-order-accurate numerical methods \textit{viz.} Particle-In-Cell (PIC) and Cloud-In-Cell (CIC) for implementing mixing. Given the nominal particle number density and problem geometry in a typical FDF calculation of a jet flame, the estimated mean field needs to be smoothed both for variance reduction and for proper treatment of empty cells near the jet axis. Our numerical studies show that while both implementations achieve detailed conservation and guarantee boundedness of the scalar field, CIC is computationally expensive and PIC is dependent on the scalar bounds. But for a given accuracy, PIC with smoothing incurs typically half the cost of CIC, for an appropriate choice of the smoothing parameter. This research is supported by the Department of Energy under Grant No. DE-FG02-90ER.   

Large-Eddy simulation and measurements of turbulent mixing in a confined rectangular jet

Large-eddy simulations (LES) of a passive scalar were performed for a confined rectangular liquid jet (Re = 20,000) and compared with the simultaneous particle image velocimetry (PIV) and planar laser induced fluorescence (PLIF) measurements. A finite-difference LES code was used to obtain velocity data, which was already proved to provide a very good agreement with PIV experiment data in the previous study. Both finite-difference and finite-volume formulation were used to discretize and solve the filtered scalar transport equation. The effects of numerical schemes and subgrid models on the LES results were investigated. Model validation was performed by comparing LES data for one-point statistics such as the passive scalar mean and variance, turbulence flux and probability distribution function with the PLIF data. In addition, LES data for the two-point spatial auto-correlations of passive scalar fluctuations and cross-corelations of passive scalar fluctuation and velocity fluctuations were also computed and compared with the simultaneous PIV and PLIF data.