Sharath S. Girimaji

Partially-averaged navier-stokes model for turbulence : A reynolds-averaged navier-stokes to direct numerical simulation bridging method

A turbulence bridging method purported for any filter-width or scale resolution-fully averaged to completely resolved-is developed. The method is given the name partially averaged Navier-Stokes (PANS) method. In PANS, the model filter width (extent of partial averaging) is controlled through two parameters: the unresolved-to-total ratios of kinetic energy (f k ) and dissipation (f e ). The PANS closure model is derived formally from the Reynolds-averaged Navier-Stokes (RANS) model equations by addressing the following question: if RANS represents the closure for fully averaged statistics, what is the corresponding closure for partially averaged statistics? The PANS equations vary smoothly from RANS equations to Navier-Stokes (direct numerical simulation) equations, depending on the values of the filter-width control parameters. Preliminary results are very encouraging.

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...

FULLY-EXPLICIT AND SELF-CONSISTENT ALGEBRAIC REYNOLDS STRESS MODELS

A fully explicit, self-consistent algebraic expression (for Reynolds stress) which is the exact solution to the Reynolds stress transport equation in the “weak-equilibrium” limit for two-dimensional mean flows for all linear and some quasi-linear pressure-strain models, is derived. Current explicit algebraic Reynolds stress models derived by employing the “weak-equilibrium” assumption treat the production-to-dissipation (P/ɛ) ratio as a constant, resulting in an effective viscosity that can be singular away from the equilibrium limit. In this paper the set of simultaneous algebraic Reynolds stress equations in the weak-equilibrium limit are solved in the full nonlinear form and the eddy viscosity is found to be nonsingular. Preliminary tests indicate that the model performs adequately, even for three-dimensional mean-flow cases. Due to the explicit and nonsingular nature of the effective viscosity, this model should mitigate many of the difficulties encountered in computing complex turbulent flows with the algebraic Reynolds stress models.

Straining and scalar dissipation on material surfaces in turbulence : implications for flamelets

Direct numerical simulations of turbulence are used to examine the straining on material surfaces, and the behavior of thin diffusive layers. The results are related to questions arising in the study of turbulent premixed and diffusion flames in the flamelet regime. The simulations are of constant-density, homogeneous, isotropic turbulence, with artificial forcing of the velocity field to maintain statistical stationarity. Taylor-scale Reynolds numbers (Rλ) up to 93 are achieved. It is found that the total rate-of-strain a in the tangent plane of a material surface is positive (i.e., extensive) with 80% probability. This straining causes the area of the surface to double every 2.5 Kolmogorov time scales (τη). A premixed flamelet can be viewed as a surface that propagates at a speed w (i.e., the local laminar flame speed) relative to the fluid ahead. It is shown that the distance z between such a propagating surface and an initially coincident material surface remains small if w is small compared to the Kolmogorov velocity scale. For this case, the statistics of z are characterized. Subject to certain assumptions, the thin diffusive layers between blobs of fluid of different concentration adopt a self-similar form (at least for small times). It is found that the scalar dissipation χ0 in the center of these layers is approximately log-normally distributed. The mean thickness of these layers is approximately 2 Batchelor scales, and is less than 5 Batchelor scales with 98% probability. The joint probability density of χ0 and a shows that χ0 fluctuates significantly about its quasi-static value based on a (for a > 0). The integral time scales of a and χ0 are found to be approximately τη and 4τη, respectively. None of the results obtained shows significant Reynolds-number dependence when normalized by the Kolmogorov scales.

Material-element deformation in isotropic turbulence

The evolution of infinitesimal material line and surface elements in homogeneous isotropic turbulence is studied using velocity-gradient data generated by direct numerical simulations (DNS). The mean growth rates of length ratio (I) and area ratio (A) of material elements are much smaller than previously estimated by Batchelor (1952) owing to the effects of vorticity and of non-persistent straining. The probability density functions (p.d.f.’s) of 1/(1) and A/(A) do not attain stationarity as hypothesized by Batchelor (1952). It is shown analytically that the random variable 1/(1) cannot be stationary if the variance and integral timescale of the strain rate along a material line are non-zero and DNS data confirm that this is indeed the case. The application of the central limit theorem to t!e material element evolution equations suggests that the standardized variables Z( = (In 1- (In l))/(var Z);) and d( = (1nA - (In A))/(varA)i) should attain stationary distributions that are Gaussian for all Reynolds numbers. The p.d.f.s of 1 and d calculated from DNS data appear to attain stationary shapes that are independent of Reynplds number. The stationary values of the flatness factor and super-skewness of both 1 and d are in close agreement with those of a Gaussian distribution. Moreover, the mean and variance of In I (and 1nA) grow linearly in time (normalized by the Kolmogorov timescale, T~), at rates that are nearly independent of Reynolds number. The statistics of material volume-element deformation are also studied and are found to be nearly independent of Reynolds number. An initially spherical infinitesimal volume of fluid deforms into an ellipsoid. It is found that the largest and the smallest of the principal axes grow and shrink respectively, exponentially in time at comparable rates. Consequently, to conserve volume, the intermediate principal axis remains approximately constant. The performance of the stochastic model of Girimaji & Pope (1990) for the velocity gradients is also studied. The model estimates of the growth rates of (lnl) and (1nA) are close to the DNS values. The growth rate of the xariances are estimated by the model to within 17 %. The stationary distributions of I and d obtained from the model agree very well with those calculated from DNS data. The model also performs well in calculating the statistics of material volume-element deformation.

ANALYSIS AND MODELING OF SUBGRID SCALAR MIXING USING NUMERICAL DATA

Direct numerical simulations (DNS) of passive scalar mixing in isotropic turbulence is used to study, analyze and, subsequently, model the role of small (subgrid) scales in the mixing process. In particular, we attempt to model the dissipation of the large scale (supergrid) scalar fluctuations caused by the subgrid scales by decomposing it into two parts: (i) the effect due to the interaction among the subgrid scales, E_phi^ . Model comparison with DNS data shows good agreement. This model is expected to be useful in the large eddy simulations of scalar mixing and reaction.

Partially-averaged Navier Stokes Model for Turbulence: Implementation and Validation

Partially-averaged Navier Stokes (PANS) is a suite of turbulence closure models of various modeled-to-resolved scale ratios ranging from Reynolds-averaged Navier Stokes (RANS) to Navier-Stokes (direct numerical simulations). The objective of PANS, like hybrid models, is to resolve large scale structures at reasonable computational expense. The modeled-to-resolved scale ratio or the level of physical resolution in PANS is quantified by two parameters: the unresolved-to-total ratios of kinetic energy (fk) and dissipation (fe). The unresolved-scale stress is modeled with the Boussinesq approximation and modeled transport equations are solved for the unresolved kinetic energy and dissipation. In this paper, we first present a brief discussion of the PANS philosophy followed by a description of the implementation procedure and finally perform preliminary evaluation in benchmark problems.

A diffusion model for velocity gradients in turbulence

In this paper a stochastic model for velocity gradients following fluid particles in incompressible, homogeneous, and isotropic turbulence is presented and demonstrated. The model is constructed so that the velocity gradients satisfy the incompressibility and isotropy requirements exactly. It is further constrained to yield the first few moments of the velocity gradient distribution similar to those computed from full turbulence simulations (FTS) data. The performance of the model is then compared with other computations from FTS data. The model gives good agreement of one‐time statistics. While the two‐time statistics of strain rate are well replicated, the two‐time vorticity statistics are not as good, reflecting perhaps a certain lack of embodiment of physics in the model. The performance of the model when used to compute material element deformation is qualitatively good, with the material line‐element growth rate being correct to within 5% and that of surface element correct to within 20% for the low...

On the modeling of scalar diffusion in isotropic turbulence

The objectives of this paper are (i) to establish analytically some important features of the scalar diffusion process in turbulence and (ii) to derive a closure model for this process in terms of the scalar probability density function (pdf). The present analysis shows that in isotropic turbulence the conditional scalar dissipation [χ(ψ)=D<∂φ/∂xi∂φ/∂xi‖φ=ψ≳], its derivative (∂χ/∂ψ), and the conditional scalar diffusion [Θ(ψ)=D<∂2φ/∂xi ∂xi‖φ=ψ≳] are zero at the extreme values of scalar concentration. Models for conditional‐dissipation ratio (χ/es) and conditional‐diffusion ratio (Θ/es) are derived from the observation [Girimaji, Combust. Sci. Technol. 78, 177 (1991); NASA Contract. Rep. CR 4446 (1992)] that the scalar pdf can be characterized by the β pdf at all stages of non‐premixed mixing. The conditional‐dissipation model is compared with the DNS data of Eswaran and Pope [Phys. Fluids 31, 506 (1988)] and the mapping‐closure‐based model [O’Brien and Jiang, Phys. Fluids A 3, 3121 (1991)]. The applicatio...

Assumed PDF turbulence-chemistry closure with temperature-composition correlations

Abstract An assumed probability density function (PDF) approach that accounts for temperature and composition fluctuations, including effects of temperature-composition correlations, has been developed. Parametric studies were performed to determine the gross behavior of the assumed PDF and provide insight into the expected trends of the model for a given kinetic mechanism. The model was implemented into an existing Navier-Stokes solver and applied to a supersonic, co-axial H 2 /air burner. Proper prediction of the turbulence effects on the chain initiating step of the kinetic process is a first-order concern in supersonic combustion applications because of small flow residence times. Temperature-composition correlations were extracted and compared with results obtained by direct integration of a PDF evolution equation. The results of this comparison showed that the model properly predicted both the magnitude and sign of the cross-correlation terms associated with the chain initiating step of the H 2 /air kinetic mechanism. Correlations for all but one of the remaining kinetic steps qualitatively reproduced those obtained by direct integration.