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The relative advantages and disadvantages of Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) are discussed. The topics are presented in viewgraph form.
The basic objective of this research is to extend the capabilities of Large Eddy Simulations (LES) and Direct Numerical Simulations (DNS) for the computational analyses of high speed reacting flows. In the efforts related to LES, we were primarily involved with assessing the performance of the various modern methods based on the Probability Density Function (PDF) methods for providing closures for treating the subgrid fluctuation correlations of scalar quantities in reacting turbulent flows. In the work on DNS, we concentrated on understanding some of the relevant physics of compressible reacting flows by means of statistical analysis of the data generated by DNS of such flows. In the research conducted in the second year of this program, our efforts focused on the modeling of homogeneous compressible turbulent flows by PDF methods, and on DNS of non-equilibrium reacting high speed mixing layers. Some preliminary work is also in progress on PDF modeling of shear flows, and also on LES of such flows.
Closed form analytical expressions are obtained for predicting the limited rate of reactant conversion in a binary reaction of the type F + rO yields (1 + r) Product in unpremixed homogeneous turbulence. These relations are obtained by means of a single point Probability Density Function (PDF) method based on the Amplitude Mapping Closure. It is demonstrated that with this model, the maximum rate of the reactants' decay can be conveniently expressed in terms of definite integrals of the Parabolic Cylinder Functions. For the cases with complete initial segregation, it is shown that the results agree very closely with those predicted by employing a Beta density of the first kind for an appropriately defined Shvab-Zeldovich scalar variable. With this assumption, the final results can also be expressed in terms of closed form analytical expressions which are based on the Incomplete Beta Functions. With both models, the dependence of the results on the stoichiometric coefficient and the equivalence ratio can be expressed in an explicit manner. For a stoichiometric mixture, the analytical results simplify significantly. In the mapping closure, these results are expressed in terms of simple trigonometric functions. For the Beta density model, they are in the form of Gamma Functions. In all the cases considered, the results are shown to agree well with data generated by Direct Numerical Simulations (DNS). Due to the simplicity of these expressions and because of nice mathematical features of the Parabolic Cylinder and the Incomplete Beta Functions, these models are recommended for estimating the limiting rate of reactant conversion in homogeneous reacting flows. These results also provide useful insights in assessing the extent of validity of turbulence closures in the modeling of unpremixed reacting flows. Some discussions are provided on the extension of the model for treating more complicated reacting systems including realistic kinetics schemes and multi-scalar mixing with finite rate chemical reactions in more complex configurations.
Results are presented of direct numerical simulations (DNS) of spatially developing shear flows under the influence of infinitely fast chemical reactions of the type A + B yields Products. The simulation results are used to construct the compositional structure of the scalar field in a statistical manner. The results of this statistical analysis indicate that the use of a Beta density for the probability density function (PDF) of an appropriate Shvab-Zeldovich mixture fraction provides a very good estimate of the limiting bounds of the reactant conversion rate within the shear layer. This provides a strong justification for the implementation of this density in practical modeling of non-homogeneous turbulent reacting flows. However, the validity of the model cannot be generalized for predictions of higher order statistical quantities. A closed form analytical expression is presented for predicting the maximum rate of reactant conversion in non-homogeneous reacting turbulence.
Results are presented of direct numerical simulations (DNS) of a randomly perturbed compressible, spatially developing planar jet under the influence of a finite rate Arrhenius chemical reaction of the type F + O yields Product + Heat. The objectives of the simulations are to assess the compositional structure of the flame and to determine the influence of reaction exothermicity by means of statistical sampling of the data generated by DNS. It is shown that even with this idealized kinetics model, the simulated results exhibit features in accordance with experimental data. These results indicate that the Damkoehler number is an important parameter in determining the statistical composition of the reacting field, and that the results are insensitive to the mechanism by which this parameter is varied. It is demonstrated that as the intensity of mixing is increased and the effect of finite rate chemistry is more pronounced, the magnitudes of the ensemble mean and rms of the product mass fraction decrease, and those of the reactants mass fraction increase. Also, at higher mixing rates the joint probability density functions of the reactants' mass fractions shift towards higher values within the composition domain indicating a lower reactedness. These trends are consistent with those observed experimentally and are very useful in portraying the statistical structure of non-equilibrium diffusion flames. The DNS generated data are also utilized to examine the applicability of the 'laminar diffusion flamelet model' in predicting the rate of the reactant conversion with finite rate chemistry. This examination indicates that the bounds of the product formation scale reasonably well with those obtained by the flamelet model. However, in the range of the relatively low values of the Damkoehler numbers considered, the scatter of the results is substantially more than that to be completely determined by the closure.
A low Reynolds number multiple-time-scale turbulence model (LMS) and its application to fully developed turbulent channel flows and pulsating pipe flows are presented. The LMS can describe the inequilibrium turbulence phenomena down to the viscous sublayer. The calculated fluid flow and turbulence fields for the channel flows are in better agreement with the direct numerical simulation (DNS) results than those obtained using a Reynolds stress turbulence model, and the calculated near-wall dissipation rates are in qualitatively correct agreement with the DNS results. The LMS also successfully predicts the rapidly varying phase-lead of the wall shearing stress that occurs in a narrow range of the dimensionless frequency omega (+) = (omega x nu)/(upsilon x tau(exp2)) for the pulsating pipe flows while various other turbulence models fail to predict this phenomenon, and the LMS yields significantly improved numerical results for a wide range of the dimensionless frequency compared with those obtained using a rapid distortion theory (RDT).
The constant C(sub mu) and the near-wall damping function f(sub mu) in the eddy-viscosity relation of the kappa-epsilon model are evaluated from direct numerical simulation (DNS) data for developed channel and boundary layer flow at two Reynolds numbers each. Various existing f(sub mu) model functions are compared with the DNS data, and a new function is fitted to the high-Reynolds-number channel flow data. The epsilon-budget is computed for the fully developed channel flow. The relative magnitude of the terms in the epsilon-equation is analyzed with the aid of scaling arguments, and the parameter governing this magnitude is established. Models for the sum of all source and sink terms in the epsilon-equation are tested against the DNS data, and an improved model is proposed.
In an effort towards understanding turbulent scalar mixing, we study the effect of molecular mixing, first in isolation and then by accounting for the effects of the velocity field. The chief motivation for this approach stems from the strong resemblance of the scalar probability density function (PDF) obtained from the scalar field evolving from the heat conduction equation that arises in a turbulent velocity field. However, the evolution of the scalar dissipation is different for the two cases. We attempt to account for these differences, which are due to the velocity field, using a Lagrangian frame analysis. After establishing the usefulness of this approach, we use the heat-conduction simulations (HCS), in lieu of the more expensive direct numerical simulations (DNS), to study many of the less understood aspects of turbulent mixing. Comparison between the HCS data and available models are made whenever possible. It is established that the beta PDF characterizes the evolution of the scalar PDF during mixing from all types of non-premixed initial conditions.
This paper reports results from the numerical implementation and testing of the compressible large eddy-simulation (LES). Relevant quantities from 32-cubed coarse grid LES solutions are compared with results generated from direct numerical simulations (DNS) of three-dimensional compressible turbulence that have been run both with sufficient resolution, at 96 cubed. The 32 cubed LES results overall agree well with their 96 cubed DNS counterparts. Moreover, the new DNS results confirm several recent conclusions about compressible turbulence that have been based primarily on two-dimensional simulations.
The probability density functions (pdf's) of the fluctuating velocity components, as well as their first and second derivatives, are calculated using data from the direct numerical simulations (DNS) of fully developed turbulent channel flow. It is observed that, beyond the buffer region, the pdf of each of these quantities is independent of the distance from the channel wall. It is further observed that, beyond the buffer region, the pdf's for all the first derivatives collapse onto a single universal curve and those of the second derivatives also collapse onto another universal curve, irrespective of the distance from the wall. The kinetic-energy dissipation rate exhibits log normal behavior.
The progress is reported of the Center for Modeling of Turbulence and Transition (CMOTT). The main objective of the CMOTT is to develop, validate and implement the turbulence and transition models for practical engineering flows. The flows of interest are three-dimensional, incompressible and compressible flows with chemical reaction. The research covers two-equation (e.g., k-e) and algebraic Reynolds-stress models, second moment closure models, probability density function (pdf) models, Renormalization Group Theory (RNG), Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS).
An analysis of the near-wall behavior of the proper orthogonal decomposition (POD) eigenfunctions derived from direct numerical simulation (DNS) of channel flow is performed. Consistent with previous studies, a low order multi-mode reconstruction of the kinetic energy and Reynolds shear stress suffices. A similar reconstruction of the isotropic dissipation rate is shown to be insufficient, however. An analysis is performed of the multi-mode composition of the dissipation rate in the near-wall region, and it is shown that a significant number of higher-order modes are required to achieve the correct asymptotic consistency in the near-wall region. In an attempt to avoid this problem, a length scale definition is proposed in terms of an integration of the correlation tensor which factors in the presence of the wall. The wall is accounted for by only integrating out to 2y(+) and not over the entire domain. Viscous and inviscid estimates for the dissipation were used in the near-wall and core regions respectively, in conjunction with this length scale representation to obtain an estimate of the dissipation throughout the domain. The resulting dissipation exhibits the proper behavior near the wall and in the inertial layer. A 1 POD mode estimate of the length scale is computed and found to agree quite well with the length scale obtained when the entire correlation tensor is used.
Recently, higher-order compact schemes have seen increasing use in the DNS (Direct Numerical Simulations) of the Navier-Stokes equations. Although they do not have the spatial resolution of spectral methods, they offer significant increases in accuracy over conventional second order methods. They can be used on any smooth grid, and do not have an overly restrictive CFL dependence as compared with the O(N(exp -2)) CFL dependence observed in Chebyshev spectral methods on finite domains. In addition, they are generally more robust and less costly than spectral methods. The issue of the relative cost of higher-order schemes (accuracy weighted against physical and numerical cost) is a far more complex issue, depending ultimately on what features of the solution are sought and how accurately they must be resolved. In any event, the further development of the underlying stability theory of these schemes is important. The approach of devising suitable boundary clusters and then testing them with various stability techniques (such as finding the norm) is entirely the wrong approach when dealing with high-order methods. Very seldom are high-order boundary closures stable, making them difficult to isolate. An alternative approach is to begin with a norm which satisfies all the stability criteria for the hyperbolic system, and look for the boundary closure forms which will match the norm exactly. This method was used recently by Strand to isolate stable boundary closure schemes for the explicit central fourth- and sixth-order schemes. The norm used was an energy norm mimicking the norm for the differential equations. Further research should be devoted to BC for high order schemes in order to make sure that the results obtained are reliable. The compact fourth order and sixth order finite difference scheme had been incorporated into a code to simulate flow past circular cylinders. This code will serve as a verification of the full spectral codes. A detailed stability analysis by Carpenter (from the fluid Mechanics Division) and Gottlieb gave analytic conditions for stability as well as asymptotic stability. This had been incorporated in the code in form of stable boundary conditions. Effects of the cylinder rotations had been studied. The results differ from the known theoretical results. We are in the middle of analyzing the results. A detailed analysis of the effects of the heating of the cylinder on the shedding frequency had been studied using the above schemes. It has been found that the shedding frequency decreases when the wire was heated. Experimental work is being carried out to affirm this result.
The primary objective of this research is to extend current capabilities of Large Eddy Simulations (LES) and Direct Numerical Simulations (DNS) for the computational analyses of high speed reacting flows. Our efforts in the first two years of this research have been concentrated on a priori investigations of single-point Probability Density Function (PDF) methods for providing subgrid closures in reacting turbulent flows. In the efforts initiated in the third year, our primary focus has been on performing actual LES by means of PDF methods. The approach is based on assumed PDF methods and we have performed extensive analysis of turbulent reacting flows by means of LES. This includes simulations of both three-dimensional (3D) isotropic compressible flows and two-dimensional reacting planar mixing layers. In addition to these LES analyses, some work is in progress to assess the extent of validity of our assumed PDF methods. This assessment is done by making detailed companions with recent laboratory data in predicting the rate of reactant conversion in parallel reacting shear flows. This report provides a summary of our achievements for the first six months of the third year of this program.
The objectives of this research are to employ direct numerical simulation (DNS) to study the phenomenon of mixing (or lack thereof) in compressible free shear flows and to suggest new means of enhancing mixing in such flows. The shear flow configurations under investigation are those of parallel mixing layers and planar jets under both non-reacting and reacting nonpremixed conditions. During the three-years of this research program, several important issues regarding mixing and chemical reactions in compressible shear flows were investigated.
Results of a numerical investigation of the dispersion of solid particles in decaying isotropic turbulence are presented. The 3D time-dependent velocity field of a homogeneous nonstationary turbulence is computed using the method of direct numerical simulation (DNS). The dispersion characteristics of three different solid particles (corn, copper, and glass) injected in the flow are obtained by integrating the complete equation of particle motion along the instantaneous trajectories of 22-cubed particles for each particle type, and then by performing ensemble averaging. Good agreement was achieved between the present DNS results and the measured time development of the mean-square displacement of the particles. Questions of how and why the dispersion statistics of a solid particle differ from those of its corresponding fluid point and surrounding fluid and what influences inertia and gravity have on these statistics are also discussed.
We apply the dynamic subgrid-scale model (DSGSM) to large-eddy simulation (LES) of turbulent channel flow with a square rib mounted on one wall. The Reynolds number (Re) is 3210 based on the mean velocity above the obstacle and the obstacle height. Near-wall structures are resolved with the no-slip boundary condition. The results show better agreement with direct numerical simulation (DNS) than LES with a fixed model constant, verifying the value of the DSGSM for simulating complex turbulent flows.
The constant C sub mu and the near-wall damping function f sub mu in the eddy-viscosity relation of the k-epsilon model are evaluated from direct numerical simulation (DNS) data for developed channel and boundary layer flow at two Reynolds numbers each. Various existing f sub mu model functions are compared with the DNS data, and a new function is fitted to the high-Reynolds-number channel flow data. The epsilon-budget is computed for the fully developed channel flow. The relative magnitude of the terms in the epsilon-equation is analyzed with the aid of scaling arguments, and the parameter governing this magnitude is established. Models for the sum of all source and sink terms in the epsilon-equation are tested against the DNS data, and an improved model is proposed.