Turbulent Mixing Model Research Articles

Assumed β-pdf Model for Turbulent Mixing: Validation and Extension to Multiple Scalar Mixing

Abstract Many engineering calculations of turbulent reacting flows employ the assumed-pdf approach. On the form of the assumed pdf, however, there has been little consensus and the choice has often been ad hoc. The objective of this work is to investigate the validity of the assumed β-pdf model for the case of the inert mixing of two scalars and extend the applicability of the model to multiple scalar mixing. By comparing the β-pdf model favorably with the two-scalar mixing data obtained from direct numerical simulations (DNS), it is demonstrated that the use of this model is justified for all stages of the mixing process in statistically-stationary, isotropic turbulence. It is also shown analytically that during the final stages of mixing the model β-pdf reduces to a Gaussian-pdf, consistent with the observations from experiments and DNS, The suggested multivariate β-pdf model for multiple-scalar mixing relates algebraically, The mean scalar concentrations and the turbulent scalar-energy—to the joint pdf of the scalar concentrations. The model requires that only one extra transport equation—for the turbulent scalar-energy—be solved, apart from the mean equations. The model can be used in conjunction with any model for the turbulence velocity field and is expected to provide a simple method for calculating scalar mixing in turbulent flows.

The Early Phase of Spark Ignition

Abstract Experimental evidence relating to the early phase of the spark ignition process is reviewed. This suggests that the evolution of the spark kernel after the initial shockwave formation is governed by well defined fluid mechanical structures. In the case of an axial spark this takes the form of a toroidal vortex pair which, depending on the discharge conditions, can be either a laminar or turbulent structure, with a rapid transitions between the two states. Plasma jet and surface discharge igniters on the other hand produce an axially propagating spherical turbulent kernel. It is proposed that igniters can be classifed into two major groups on the basis of their underlying fluid mechanical structures, either radially or axially propagating mixing elements. A mathematical model of the turbulent toroidal mixing element is developed and compared with some of the available experimental data on kernel growth rates. The consequences of the internal vorticity of the kernels for ignition in turbulent mixtures are examined and some of the relevant published data is re-interpreted in terms of the turbulent mixing model.

A model of turbulent mixing across a density interface including the effect of rotation

A new theoretical approach is presented, which determines the entrainment laws with and without background rotation in a two-layer stratified fluid with one layer stirred by turbulent motions. The model gives the entrainment coefficient E as a function of the local turbulent Richardson number Ri = g ′ l / u 2 , the Rossby number Ro = u/fl and the Peelet number Pe = ul /κ. The following entrainment laws are obtained: (i) without rotation and for a high Peclet number: $E \propto Ri^{-\frac{3}{2}}$ ; (ii) without rotation and for a low to moderate Peclet number: $E \propto Pe^{-\frac{1}{3}}Ri^{-1}$ (iii) with rotation and for a high Peclet number: E ∝ RoRi −l . These entrainment laws are consistent with experiments for the three different cases. The model relies to a great extent on the spectral distribution of interface oscillations measured in experiments. Comparison is made with experiments and with earlier models of entrainment.

Turbulent mixing model based on ordered pairing

In Turbulent reactive flows the fluid composition at a point changes due to convection, reaction, and mixing (i.e., molecular transport). Modeling approaches based on the transport equation for the joint probability density function (jpdf) of velocity and composition have proved successful, largely because convection and reaction can be treated exactly, without modeling assumptions. Mixing has to be modeled, however, and current mixing models are deficient in several respects. The first contribution of this article is to define a model problem pertaining to turbulent diffusion flames in the flamelet regime. When applied to this problem, existing models incorrectly yield composition jpdfs corresponding to distributed combustion. The second contribution is to develop and demonstrate a new class of mixing models that performs satisfactorily for the model problem, and also for the case of the decaying field of a conserved scalar.

A k-ε model for turbulent mixing in shock-tube flows induced by Rayleigh–Taylor instability

A k-ε model for turbulent mixing induced by Rayleigh–Taylor instability is described. The classical linear closure relations are supplemented with algebraic relations in order to be valid under strong gradients. Calibrations were made against two shock-tube experiments (Andronov et al. [Sov. Phys. JETP 44, 424 (1976); Sov. Phys. Dokl. 27, 393 (1982)] and Houas et al. [Proceedings of the 15th International Symposium on Shock Waves and Shock Tubes (Stanford U.P., Stanford, CA, 1986)]) using the same set of constants. The new interpretation of the experimental data of Brouillette and Sturtevant [Physica D 37, 248 (1989)], where the mixing length is discriminated from the wall jet, requires a different numerical value for the Rayleigh–Taylor source term coefficient. A detailed physical study is given in both cases. It turns out that the spectrum is narrower in the Brouillette and Sturtevant case than in the Andronov et al. case but the small length scales are of the same magnitude.

A binomial sampling model for scalar turbulent mixing

The closure problem generated by the molecular mixing term in the turbulent convection of scalars is studied. The statistical average of this term both in moment formulations and in the probability density function (pdf) approach implicitly encloses the turbulence straining action on scalar gradients leading to a significant enhancement of the molecular dissipative effects. Previous pdf model equations are examined in terms of cumulants evolution and reasons for their failure are diagnosed. A new noninteractive model is proposed, combining a linear mean square estimation (LMSE) deterministic subprocess affecting all the Monte Carlo particles, used to represent the pdf, and a binomial sampling acting on a fraction of them. The scalar lower and/or upper bounds are naturally considered in the formulation. For unbounded scalars, or when the scalar standard deviation is much smaller than the absolute value of the difference between the bounds and the scalar mean, the binomial sampling tends to a Gaussian one. The extension of this model to investigate the joint statistics of velocity and scalars is also considered. Numerical results for the homogeneous turbulent mixing of one scalar with initial values of either 0 or 1 are presented. The evolution of the first four moments and the pdf is qualitatively reasonable. The correct asymptotic Gaussian state is predicted.

Linear-eddy modelling of turbulent transport. Part 3. Mixing and differential molecular diffusion in round jets

The linear-eddy model of turbulent mixing represents a spatially developing flow by simulating the time development along a comoving transverse line. Along this line, scalar quantities evolve by molecular diffusion and by randomly occurring spatial rearrangements, representing turbulent convection. The modelling approach, previously applied to homogeneous turbulence and to planar shear layers, is generalized to axisymmetric flows. This formulation captures many features of jet mixing, including differential molecular diffusion effects. A novel experiment involving two unmixed species in the nozzle fluid is proposed and analysed.

Closure models for turbulent reacting flows

AbstractIn this paper, a simple procedure based on fast and slow reaction asymptotics has been employed to derive first‐order closure models for the nonlinear reaction terms in turbulent mass balances from mechanistic models of turbulent mixing and reaction. The coalescence‐redispersion (CRD) model, the interaction by exchange with the mean (IEM) model, the three‐environment (3E) model, and the four‐environment (4E) model have been used to develop closure equations. The closure models have been tested extensively against experimental data for both single and multiple reactions. The closures based on slow asymptotics for the CRD, 3E and 4E models provide very good predictions of all of the experimental data, while other models available either in the literature or derived here are not adequate. The simple new closure equations developed in this paper may be useful in modeling systems involving turbulent mixing and complex chemical reactions.

Modeling of inhomogeneities in the toroidal jet-stirred reactor

The ideal stirred reactor, where mixing and molecular diffusion are fast compared to the chemical reactions being studied and where backmixing is sufficient to eliminate all composition gradients within the reactor, is a modeling tool of classic and continuing interest in “reaction engineerig”. Some reactions, however, can be faster than the mixing process and interpretation of data taken under these conditions may require modeling of turbulent mixing and its interaction with the chemical processes of interest. In this article, some progress on application of the Curl coalescence-redispersion model, to description of the mixing chemistry interaction is reported. Comparison with experimental results from fuel-lean CO H 2 combustion indicates that for fast reactions, such as CO H 2air , it is important to take the jet mixing nature of the flow into account. For the slower rich C 2H 2air reaction the exit gas composition can be quite well modeled by assuming a simple stirred reactor. Progress on cold flow tracer studies and two dimensional modeling of the jet mixing is also reported.

Vertical fluxes of nitrate associated with salt fingers in the world's oceans

The primary means by which the oceans serve as a sink for atmospheric carbon dioxide is through the vertical flux of sinking organic carbon. This organic material is derived from primary photosynthetic production in the upper ocean. Its rate of loss to the deep sea, the “new production”, is rigorously constrained by the upward flux of the limiting nutrient, nitrate. New production estimates have been made by using velocity microstructure measurements to infer nitrate fluxes, where it is assumed that turbulent production (mixing) balances viscous energy dissipation. However, estimates based on a salt finger convection model lead to nitrate fluxes which are as much as an order of magnitude larger than a turbulent mixing model predicts and agree much better with fluxes inferred from biological uptake measurements and tracer studies. Heat flux estimates are also higher, but by less than a factor of 3. Observations of the “scaled dissipation ratio” Γ, a quantity that is based on both velocity and temperature microstructure measurements, may provide a means of distinguishing between the two mixing hypotheses. Where both turbulent production and salt fingering contribute to the mixing this distinction is more difficult, but by using measurements of Γ the potential error in the vertical flux estimate due to an improper choice of mixing models is greatly reduced.

Turbulent mixer model with application to homogeneous, instantaneous chemical reactions

A method to evaluate the decay of concentration variance in turbulent mixers is presented. The turbulent mixer model links together macromixing (large-scale convective motions and turbulent dispersion) with successive stages of micromixing process (inertial-convective disintegration of large eddies, formation of laminated structures within energy dissipating vortices and molecular diffusion within deforming laminated structures). A beta probability density function is used for an inert scalar type composition variable formed with reagents concentrations difference in order to estimate conversion of instantaneous reactions in the system. The results calculated from the model for the multijet mixers and reactors are compared with experimental results available in the literature.

A Linear- Eddy Model of Turbulent Scalar Transport and Mixing

Abstract Transport and mixing of diffusive scalars in turbulent flows are simulated computationally based on a novel representation of the temporal evolution along a transverse line moving with the mean fluid velocity. The scalar field along this line evolves by Fickian diffusion, representing molecular processes, and by randomly occurring events called block inversions. Block inversion, representing the effect of turbulent convection, consists of the random selection of an interval (Y0 − 1/2, Y0 + 1/2) of the line, where the interval size l may he either fixed or randomly selected, and replacement of the scalar field θ(y) within that interval by θ(2y0 For fixed l, the model requires a single input parameter, the Peclet number. To demonstrate the performance of the model, this formulation is used to compute the spatial development of diffusive scalar fields downstream of several source configurations in homogeneous turbulence. Generalization to inhomogeneous turbulence is discussed, as well as a formulati...

Statistical model of turbulent mixing of relaxing gases behind an array of small-scale nozzles

An approach to the description of mixing of the relaxing gases behind an array of small-scale nozzles is proposed, on the basis of a spectral model of turbulence and the density function of the probability distribution for the concentration.

Modeling of turbulent molecular mixing

The general coalescence-dispersion (C/D) closure provides phenomenological modeling of turbulent molecular mixing. The models of Curl ( AIChEJ, Vol. 9) and Dopazo and O'Brien ( Combust. Sci. Tech., Vol. 13) appear as two limiting C/D models that “bracket” the range of results one obtains by any other model. This finding is used to investigate the sensitivity of the results to the choice of the model. Inert scalar mixing is found to be less model-sensitive than mixing accompanied by chemical reaction. Infinitely fast chemistry approximation is used to relate the C/D approach to Toor's earlier results ( Turbulence in Mixing Reactions, R. S. Brodkey, Ed.). Pure mixing and infinite rate chemistry calculations are compared to study further a recent result of Hsieh and O'Brien (Combust. Sci. Tech., Vol. 46), who found that higher concentration moments are not sensitive to chemistry.

A velocity-biased turbulent mixing model for passive scalars in homogeneous turbulence

In applications to turbulent reactive flows, the probability density function approach requires a model for molecular diffusivity. The objective of the present work is to develop a consistent model for passive scalars in turbulent flows, improving the calculation of the velocity-scalar correlation coefficient. This is done by using the Langevin model for the velocity and by adjusting two model parameters of a velocity-biased mixing model for the scalar, which are determined by reference to homogeneous isotropic turbulent flows with a uniform mean cross-stream scalar gradient. Calculated and measured profiles are in good agreement.

A long-range transport model for ammonia and ammonium for Europe

This paper reviews available data on ammonia (NH 3) emission and other parameters used in the model, viz. dry deposition velocities, scavenging coefficients and the pseudo-first order reaction rate. There are some striking differences in emission and subsequent behaviour of NH 3 and sulphur dioxide (SO 2). NH 3 is predominantly emitted near or at ground level, whereas SO 2 is mainly emitted from higher sources. Moreover, the conversion rate of NH 3 to ammonium (NH 4 +) is higher than for SO 2 to sulphate (SO 4 2−). The approximate treatment in the Lagrangian model of turbulent mixing (instantaneous homogeneous mixing over the whole mixing layer) had to be modified by the introduction of several correction factors derived from detailed diffusion calculations. Contrary to SO 2, the concentration at ground-level of NH 3 will be determined to a large extent by local sources, which hinder model verification. For this purpose only measurements in the Netherlands could be used. The model has been run for the year 1980. The computed ammonium concentrations in air and precipitation are in agreement with the available measured data. The model results show that the dry deposition of NH 3 in any country is mainly caused by inland sources, whereas the dry and wet deposition of ammonium is to a considerable extent caused by foreign sources. Moreover some numerical aspects of the model are discussed and the results of a sensitivity analysis are shown.

Theoretical Remarks on a Phenomenological Model of Turbulent Mixing

Abstract In the pdf approach to turbulent combustion the molecular diffusion is to be modeled. In recent years, several authors have employed the Curl model (Curl, 1963) for closure. A modified version of the Curl model has been introduced by Pope (1982b). The modified model contains an unspecified function A∥α) [cf. Eq. (la)] which has been defined in the context of Monte-Carlo simulation. The paper Clarifies the physical meaning of this function by relating it to the physics of the initial phase of mixing of an inert scalar in homogeneous turbulence. A relationship is established between the modified Curl model and the linear mean square estimation closure introduced by Dopazo and O'Brien (1976). The asymptotic behavior of the pdf is discussed.

Instantaneous planar measurement of the complete three-dimensional scalar gradient in a turbulent jet

A new technique is described that permits calculation of all three instantaneous components of the concentration gradient vector within a cross section of a turbulent gas jet. This is made possible by the simultaneous measurement of gas concentrations from two closely spaced parallel cross sections using laser Rayleigh scattering from two illumination sheets of different wavelength. The use of a second sheet gives access to an extra dimension of information, which is valuable in testing models of turbulent mixing.

A 'flip' experiment in a chemically reacting turbulent mixing layer

An experimental investigation of entrainment and mixing in a reacting, uniform density, liquid plane shear layer has been carried out using laser-induced fluorescence diagnostics. Results indicate that the reactants mix on a molecular level and react at a composition that is nearly uniform across the transverse extent of the layer. The composition of the mixed fluid is found to be asymmetric with an excess of high-speed fluid, suggesting that entrainment into the shear layer is also asymmetric. These results are at variance with predictions of conventional models of turbulent transport and mixing.

A convective model for turbulent mixing in rotating convection zones

The effects of rotation are included in an analytical model for the convective motions in a plane-parallel layer of an ideal fluid. The turbulent stress tensor, formed by taking products and averages of the various velocity components, is calculated for an arbitrary eddy size and shape. Heuristic formulae presented for determining the size and shape of the dominant eddy then give a fully specified stress tensor. Applications for this stress tensor in problems of stellar internal dynamics, heat flow, scalar diffusion, and dynamo theory are suggested. The resultant stresses tend to produce differential rotation profiles with rapidly rotating equators and interiors. The dynamo activity associated with these convective motions tends to occur near the lower boundary of the convection zone.