SCIENCE CHINA Technological Sciences

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*Corresponding author (email: prao@ams.sunysb.edu)

• RESEARCH PAPER • October 2013 Vol.56 No.10: 2355–2360

doi: 10.1007/s11431-013-5340-0

Macro and micro issues in turbulent mixing

MELVIN J, KAUFMAN R, LIM H, KAMAN T, RAO P* & GLIMM J

Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA

Received June 29, 2013; accepted August 16, 2013; published online September 9, 2013

Numerical prediction of turbulent mixing can be divided into two subproblems: to predict the geometrical extent of a mixing region and to predict the mixing properties on an atomic or molecular scale, within the mixing region. The former goal suffices for some purposes, while important problems of chemical reactions (e.g. flames) and nuclear reactions depend critically on the second goal in addition to the first one. Here we review recent progress in establishing a conceptual reformulation of conver-gence, and we illustrate these concepts with a review of recent numerical studies addressing turbulence and mixing in the high Reynolds number limit. We review significant progress on the first goal, regarding the mixing region, and initial progress on the second goal, regarding atomic level mixing properties. New results concerning non-uniqueness of the infinite Reynolds number solutions and other consequences of a renormalization group point of view, to be published in detail elsewhere, are summarized here.

The notion of stochastic convergence (of probability measures and probability distribution functions) replaces traditional pointwise convergence. The primary benefit of this idea is its increased stability relative to the statistical “noise” which char- acterizes turbulent flow. Our results also show that this modification of convergence, with sufficient mesh refinement, may not be needed. However, in practice, mesh refinement is seldom sufficient and the stochastic convergence concepts have a role. Related to this circle of ideas is the observation that turbulent mixing, in the limit of high Reynolds number, appears to be non-unique. Not only have multiple solutions been observed (and published) for identical problems, but simple physics based arguments and more refined arguments based on the renormalization group come to the same conclusion.

Because of the non-uniqueness inherent in numerical models of high Reynolds number turbulence and mixing, we also in-clude here numerical examples of validation. The algorithm we use here has two essential components. We depend on Front Tracking to allow accurate resolution of flows with sharp interfaces or steep gradients (concentration or thermal), as are com-mon in turbulent mixing problems. The higher order and enhanced algorithms for interface tracking, both those already devel-oped, and those proposed here, allow a high resolution and uniquely accurate description of sample mixing problems. Addi-tionally, we depend on the use of dynamic subgrid scale models to set otherwise missing values for turbulent transport coeffi-cients, a step that breaks the non-uniqueness.

stochastic convergence, turbulent mixing, renormalization group, dynamic subgrid scale models

Citation: Melvin J, Kaufman R, Lim H, et al. Macro and micro issues in turbulent mixing. Sci China Tech Sci, 2013, 56: 23552360, doi: 10.1007/s11431-013-5340-0

1 Theory for turbulent mixing

1.1 Stochastic convergence and Young measures

Our ideas for stochastic simulation are intuitively appeal-

ing and they are elementary to implement in a Large Eddy Simulation (LES) as a post-processing step. Nonetheless, the ideas represent an important conceptual reorganiza-tion for LES. The basic idea is to partition the available information in a simulation into space-time localization information and solution state space stochastic fluctuation information [1, 2]. The partition is achieved through in-troduction of a coarse grid, coarser than the simulation

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grid, whose cells we call supercells. The supercells (with their reduced space-time localization information) provide the reduced level of space-time localization. Within a single supercell, we do not distinguish space-time information. Rather, we regard all solution values in the supercell as samples from a space-time dependent probability distribu-tion (PDF). In this way, the supercell allows enhanced lo-calization of state space fluctuations, in the form of a dis-crete approximation to a full probability measure. Such an object is called a Young measure, and through an expanded notion of weak solutions, it can be differentiated. Hence, the Young measure can be considered as a solution of the gov-erning partial differential equation, which for high Reynolds number (Re) LES, is more or less the Euler equation. We regard the supercell PDF interpretation of the numerical solution as a numerical approximation to a Young measure. The weak convergence (with integration over the solution state space as well as over space-time) is known as w* con-vergence, as it occurs in the dual of a Banach space. The Banach space, as far as the solution state values are con-cerned, is the space of continuous functions on state space, and the dual is the space of measures on state space. This dual is the origin of the Young measures.

Convergence of nonlinear terms is normally a problem, precisely because there is no control over the convergence of the fluctuations. For w* convergence, in the sense of Young measures, with Banach spaces and duals as indicated, this difficulty is removed. A series of papers [1−3] have developed this point of view, with convergence [4−6] doc-umented (see Figure 1). Something important has happened, with w* convergence of approximate Young measures to a limit, the nonlinear terms converge. For example, chemi-cally reactive flow (turbulent combustion) does not need to rely on reaction flame structure models that accept average temperature and concentration values as input and then

compensate for these average values, i.e., the fluctuations normally missing in a conventional LES or Reynolds- averaged Navier-Stokes (RANS) framework. With stochas-tic convergence, there is no need for such a chemistry re-lated flame structure model, with its assumptions of a steady state flame structure, questionable at least during flame ini-tiation and extinction. What is known as LES with finite rate chemistry, free of flame models, is enabled. The mesh requirement for finite rate chemistry is not Direct Numerical Simulation (DNS), full resolution of all turbulent length scales, but rather the resolution of all internal layers within the flame, and resolution or numerical convergence of the solution fluctuations, i.e., the Young measures. We actually find a convergence stronger than w*, namely L1 conver-gence of the cumulative distribution functions (CDFs).

The famous k−5/3 law [7] for the velocity fluctuations can be considered as a Sobolev norm bound for the solutions of turbulent flow. This point of view was taken as a hypothesis [8], and turned into a convergence proof for the zero viscos-ity limit of the Navier-Stokes equation. The convergence is in an Lp norm, to a weak solution which lies in an Lp space. This proof is in contrast to the Young measure w* conver-gence of refs. [9, 10], which does not require Kolmogorov’s 1941 (K41) hypothesis, but offers a weaker type of conver-gence to a weaker limit, not known even to be a solution of the limiting Euler equations. Taken together, these results suggest that the w* convergence to a Young measure is an intermediate result, and not the optimal or final convergence result.

In spite of this theoretical support for Lp convergence to a classical weak solution, we regard this type of convergence as not available in practice for typical grids, and thus find the notion of w* convergence to a Young measure to be useful for analysis of numerical solutions. To state this fact more boldly, numerical convergence for LES solutions (al-

Figure 1 Mixing data taken from the midplane of a Rayleigh-Taylor (RT) two-fluid mixing problem. Left: Raw simulation data of mixture fractions with three levels of mesh refinement; Right: L1 norm comparison of CDFs (coarse grid to fine and medium grid to fine) in individual supercells.

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ways under resolved) is not the study of the solution as x→0. It is the study of the solution as x varies within an affordable band always within the range of inertial length scales.

1.2 Renormalization group and subgrid scale terms

LES aims to compute some but not all of the turbulent vor-tices, the larger but not the smaller ones. An averaging pro-cess, introduced at the grid level, cuts off the subgrid level turbulence, but it introduces new (unclosed) terms, such as the difference between the average of a product and the product of an average, which must be approximated (mod-eled) for the equations to close. This cutoff occurs within a strongly interacting region and is thus a central complica-tion to LES simulations. These unclosed terms, called sub-grid scale (SGS) terms, are approximated by a model times a coefficient. The model for a flux term has a specified functional form, usually a solution gradient (a Smagorinsky, or gradient diffusion model). A theory [11−13] determines the otherwise missing coefficients of the postulated models, and allows reliable computations to proceed. In ref. [14], we have developed a renormalization group analysis of the generation of subgrid scale terms, with higher order correc-tions and a systematic expansion for the unclosed terms. The expansion terms would seem to allow higher accuracy in the modeling of the subgrid terms, but the expansion terms fail to be definite in sign and give rise to unstable algorithms.

1.3 Non-uniqueness of apparently numerically con-verged solutions

A wide range of simulation results have been proposed as solutions for an identical high Re turbulent mixing problem [15]. Additionally, systematic variation of the SGS coeffi-cient has been observed to change the atomic mixing prop-erties of nominally converged solutions [16]. On this basis, we have observed that the selection of the high Re limit is influenced by numerical considerations. The renormaliza-tion group framework, or simple theoretical arguments based on a count on the number of dimensionless variables (only one of which is eliminated by setting Re to infinity) suggest non-unique solutions in the high Re limit, with the distinct solutions labeled by the dimensionless ratios of transport coefficients taken in the limit process. For LES, there are turbulent transport coefficients in addition to the molecular ones, and these also influence the limit. Here the dynamic SGS theory is very helpful and gives a unique value for them. Numerical non-uniqueness results in part from alternate choices for the transport coefficients, and in part from algorithms, which in effect introduce the numeri-cal equivalent of a turbulent closure term (Implicit Large Eddy Simulation, ILES). In Table 1, we summarize the ob-served numerical and experimental variation in efforts to

Table 1 Variabilities in from a variety of experimental and numerical sources

Experimental variabilities Experimental variability Due to experimental initial conditions

20% 5%‒30%

Numerical issues ILES to experiment discrepancy [15] ILES to ILES simulation discrepancy [15] Numerical variation from transport coefficients [17, 18] FT/LES/SGS to experiment discrepancy [18]

100% 50% 5% 5%

determine the overall growth rate of the Rayleigh-Taylor instability, known as .

In ref. [16], we outlined a validation/verification program for LES in the high Re regime, based on the code FronTier (FT)/LES/SGS. In contrast to RANS, the LES/SGS frame-work has no adjustable parameters. Our validation is in the regime of classical turbulent mixing experiments, conducted at Re~3.5×104. Beyond the experimental Re range of 3.5× 104, we employed a mild extrapolation, i.e., a mathematical verification step, to Re values of 6×105 and higher, relevant to inertial confinement fusion (ICF). We have observed that the transport coefficients and the atomic level mixing CDFs display only a mild norm dependence on change of Re (about 8%, just marginally above the mesh convergence related errors of about 6% for the same problem) and are also norm convergent under mesh refinement, in a purely hydro study.

The truncation error associated with the advection of the nonlinear terms in the governing equations affects the tur-bulent mixing rate. The error arises in the numerical con-struction of solution or flux values located on the grid cell faces, and in their extrapolation to time dependent values for time integration over the current time step from cell av-eraged values of the primitive variables taken at the begin-ning of a time step. Numerical algorithms are typically high order, but more or less universally degrade to first order near a sharp interface or steep solution gradient. In this re-gime of large first order errors, the numerical truncation errors are comparable to the SGS models. This fact is often cited as a basis for ignoring the SGS terms and substituting numerical truncation errors (ILES). Of course, the result has a somewhat random dependence on the taste of the pro-grammer, and is not compatible with scientifically based Verification and Validation (V&V).

2 Numerical modeling of complex mixing flows

2.1 Macro observables: the mixing zone

2.1.1 Rayleigh-Taylor instabilities

FT/LES/SGS has achieved systematic agreement with a wide range of Rayleigh-Taylor experiments, including mix-ing of both concentration and thermal fields. Uncertainty over initial conditions (not recorded) has been resolved, by

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backward solution of the fluid equations from an early time (recorded) to an initial time (not recorded), and with uncer-tainty quantification for the possible errors in the recon-struction [19] (see Figure 2). Agreement is precise enough to distinguish between distinct experiments and their dis-tinct values of α (see Table 2).

2.1.2 Richtmyer-Meshkov

FronTier was used in a three-way code comparison (with RAGE and PROMETHEUS), with experiment and with theory [23] for single mode Richtmyer-Meshkov instabili-ties. Extensive comparison of FT/LES/SGS with the already validated RAGE code for multimode Richtmyer-Meshkov instabilities, before and after reshock, gave generally excel-lent results [24], with exceptions related to the thermal field.

2.1.3 Two-phase Taylor-Couette flow

We studied two-phase (oil based and aqueous phases) Tay-lor-Couette flow in a high-speed turbulent regime. We found as a late time statistical steady state a flow regime with segregation of two continuous phases, with the heavy fluid (aqueous) on the outside, in what could be called a

Figure 2 Three simulations of a RT experiment, with reconstructed ini-tial data long wavelength modes. The three simulations are set to (a) 0 × the best estimate of the long wavelength modes, (b) the best estimate and (c) 2 × the best estimate. From this plot, we see a ±5% effect from the long wavelength perturbations.

Table 2 Experimental and FT/LES/SGS simulation values of com-pared for a series of experiments, with error bars if provided by either the simulation or the experiment, and the discrepancy if any

Experiment αexp αsim Discrepancy (%)

Smeeton-Youngs [20] #112 0.052 0.055 6

Smeeton-Youngs [20] #105 0.072 0.072−0.080 0

Smeeton-Youngs 10 exp. 0.055−0.077 0.066 0

Andrews-Banerjee [21] 0.065−0.07 0.069 0

Andrews-Mueschke [22] 0.059−0.081 0.075 0

Andrews-Mueschke [22] 0.08−0.09 0.084 0

centrifuge regime (see Figure 3). In this flow, in view of the high Reynolds and Taylor numbers, there is a turbulent Taylor-Couette flow in each of the two continuous phases. The present levels of resolution appear to describe the bulk mass distribution, but the degree to which minority phases are entrained in each of the two distinct continuous phase regions and the role of subgrid scale macrostructure to de-termine the phase analysis are yet to be determined. The purpose of this simulation is to assess the influence of the fluid mixing on a chemical reaction, which occurs at the interface between the two phases.

2.2 Micro observables: molecular scale mixing proper-ties

2.2.1 Validation studies

Some of our mesh convergence studies (verification) are mentioned above. We have also carried out, to a limited extent, validation studies (see Figure 4).

2.2.2 Parameters for w* convergence

We use L1 norm convergence of the Young measure CDFs. Other choices are to apply an Lp or L∞ norm. In addition, these norms can be applied to the PDFs rather than to the CDFs. Any of these changes will make the convergence

Figure 3 Simulation of a small slice from a fluid mixing contactor, with high speed Taylor-Couette flow. The flow is initialized in a maximum unstable configuration, with all the heavy fluid on the inside. The left frame shows the chaotic transition at an intermediate time to a stable con-figuration, and the right frame shows a late time stable configuration.

Figure 4 Second moment of the concentration distribution (θ) for hot-cold and for fresh-salt water mixing RT experiments. Shown are ex-perimental data points, our simulations and the results of a DNS simulation [22, 25].

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more difficult to demonstrate, but also more useful once it is demonstrated. For the present (L1 convergence of CDFs), the convergence is slow, and so the use of a more difficult norm appears to be ill advised. There remains the question of the utility of this weaker notion of convergence. In many problems, the final output is an integrated measure of reac-tion (total fuel burned, energy released), with limited spatial resolution. For comparison to such experimental data, it would seem that the present formulation, with integrated measures of convergence and reduced space-time resolution, might be sufficient. Of course, any remaining shortfall in accuracy can be addressed by the obvious expedient of ad-ditional mesh resolution.

Within the framework of L1 convergence for the CDFs, we have two parameters to set. The first is the bin size for the discrete data bins for the Young measure in solution state space. We found this to be an insensitive parameter, and we have typically used 10 bins per variable [16]. The second parameter is the supercell size. The results are very sensitive to this parameter. We studied the relative im-portance of mesh errors, statistical sampling errors and physical parameter variation (Reynolds number and varia-tion of turbulent concentration diffusion) [16]. Using a study based in part on the beta distribution as an approxi-mation or model for the concentration, we found the ex-pected slow (n1/2) rate of convergence for the statistical errors. We also found, but did not explain, a slow conver-gence regarding the mesh errors. It is possible that this phenomenon is related to K41 statistics and the lack of smoothness of the solution. In spite of these difficulties, we were able to show that the mesh and statistical errors and the Reynolds number effects were two to three times small-er than those due to variation of turbulent transport coeffi-cients, thereby showing that the turbulence transport de-pendence was a quantitatively verified phenomenon.

2.2.3 A software tool

A tool to allow ready support for w* convergence studies is available at http://www. ams.sunysb.edu/wstar.

3 Conclusions

Mesh convergence, in LES, is not possible in the normal sense of pointwise convergence, as new solution phenome-na emerge as the mesh is refined. This is a common view based on computational experience. We have shown math-ematically, on the basis of the assumed Kolmogorov 1941 velocity statistics and for an incompressible flow, that strong Lp convergence to a solution of the incompressible (infinite Reynolds number) Euler equation does occur. This mathematical result is not consistent with common experi-ence. For practical levels of grid refinement, such conver-gence is not achievable. We focus on observational length scales to circumvent this problem. If the observation has a

length scale which is large when compared to the grid, for example comparable to the overall flow and its geometry, we call it macro. If it is small relative to the grid spacing, as with chemical reactions, which depend on atomic collisions, we call it micro.

In this paper we summarize substantial progress of the authors and co-workers in the study of turbulent mixing generated by Rayleigh-Taylor and Richtmyer-Meshkov (acceleration driven) turbulent mixing flows, at the level of macro observables. These results include verification and validation, a goal which has eluded most workers using LES. We outline some recent theoretical observations that cast light on the difficulty others have had with LES validation. Foremost among these is the notion that solutions of the compressible, multi-fluid Euler equation, appear to be non- unique, with the non-unique solutions parameterized by turbulent transport terms and even by the choice of numeri-cal algorithm.

Concerning LES convergence for micro observables, the results are of a more initial nature. We present a theoretical framework, of a statistical nature, which appears to allow LES convergence; we show partial results that support this claim. The LES micro observable convergence framework is related to the Young measures familiar to conservation law theorists. The use of these ideas in the study of LES convergence is novel; it appears to be a significant concep-tual advance. It allows convergence of nonlinear functionals of the solution, including chemical reaction rates, or what is known as finite rate chemistry, without the use of flame structure models. Technical details have been investigated with this method and are summarized here.

This work was supported in part by the Nuclear Energy University Pro-gram of the Department of Energy, project NEUP-09-349, Battelle Energy Alliance LLC 00088495 (subaward with DOE as prime sponsor), Leland Stanford Junior University 2175022040367A (subaward with DOE as prime sponsor), Army Research Office W911NF0910306. Computational resources were provided by the Stony Brook Galaxy cluster and the Stony Brook/BNL New York Blue Gene/L IBM machine. This research used re-sources of the Argonne Leadership Computing Facility at Argonne Nation-al Laboratory, which is supported by the Office of Science of the U.S. De-partment of Energy under contract DE-AC02-06CH11357. Stony Brook University Preprint number SUNYSB-AMS-12-04.

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