Michael Gauding

Extensive strain along gradient trajectories in the turbulent kinetic energy field

Based on direct numerical simulations of forced turbulence, shear turbulence, decaying turbulence, a turbulent channel flow as well as a Kolmogorov flow with Taylor-based Reynolds numbers Re? between 69 and 295, the normalized probability density function of the length distribution of dissipation elements, the conditional mean scalar difference ?kl at the extreme points as well as the scaling of the two-point velocity difference along gradient trajectories ?un are studied. Using the field of the instantaneous turbulent kinetic energy k as a scalar, we find good agreement between the model equation for as proposed by Wang and Peters (2008 J. Fluid Mech. 608 113?38) and the results obtained in the different direct numerical simulation cases. This confirms the independence of the model solution from both the Reynolds number and the type of turbulent flow, so that it can be considered universally valid. In addition, we show a 2/3 scaling for the mean conditional scalar difference. In the second part of the paper, we examine the scaling of the conditional two-point velocity difference along gradient trajectories. In particular, we compare the linear s/? scaling, where ? denotes an integral time scale and s the separation arclength along a gradient trajectory in the inertial range as derived by Wang (2009 Phys. Rev. E 79 046325) with the s?a? scaling, where a? denotes the asymptotic value of the conditional mean strain rate of large dissipation elements.

Line segments in homogeneous scalar turbulence

The local structure of a turbulent scalar field in homogeneous isotropic turbulence is analyzed by direct numerical simulations (DNS) with different Taylor micro-scale based Reynolds numbers between 119 and 529. A novel signal decomposition approach is introduced where the signal of the scalar along a straight line is partitioned into segments based on the local extremal points of the scalar field. These segments are then parameterized by the distance l between adjacent extremal points and the scalar difference Δϕ at the extrema. Both variables are statistical quantities and a joint distribution function of these quantities contains most information to statistically describe the scalar field. It is highlighted that the marginal distribution function of the length becomes independent of Reynolds number when normalized by the mean length lm. From a statistical approach, it is further shown that the mean length scales with the Kolmogorov length, which is also confirmed by DNS. For turbulent mixing, the scala...

Generalised scale-by-scale energy-budget equations and large-eddy simulations of anisotropic scalar turbulence at various Schmidt numbers

A necessary condition for the accurate prediction of turbulent flows using large-eddy simulation (LES) is the correct representation of energy transfer between the different scales of turbulence in the LES. For scalar turbulence, transfer of energy between turbulent length scales is described by a transport equation for the second moment of the scalar increment. For homogeneous isotropic turbulence, the underlying equation is the well-known Yaglom equation. In the present work, we study the turbulent mixing of a passive scalar with an imposed mean gradient by homogeneous isotropic turbulence. Both direct numerical simulations (DNS) and LES are performed for this configuration at various Schmidt numbers, ranging from 0.11 to 5.56. As the assumptions made in the derivation of the Yaglom equation are violated for the case considered here, a generalised Yaglom equation accounting for anisotropic effects, induced by the mean gradient, is derived in this work. This equation can be interpreted as a scale-by-scale energy-budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. The equation is evaluated for the conducted DNS, followed by a discussion of physical effects present at different scales for various Schmidt numbers. For an analysis of the energy transfer in LES, a generalised Yaglom equation for the second moment of the filtered scalar increment is derived. In this equation, new terms appear due to the interaction between resolved and unresolved scales. In an a-priori test, this filtered energy-budget equation is evaluated by means of explicitly filtered DNS data. In addition, LES calculations of the same configuration are performed, and the energy budget as well as the different terms are thereby analysed in an a-posteriori test. It is shown that LES using an eddy viscosity model is able to fulfil the generalised filtered Yaglom equation for the present configuration. Further, the dependence of the terms appearing in the filtered energy-budget equation on varying Schmidt numbers is discussed.

High-order structure functions for passive scalar fed by a mean gradient

Abstract Transport equations for even-order structure functions are written for a passive scalar mixing fed by a mean scalar gradient, with a Schmidt number Sc = 1 . Direct numerical simulations (DNS), in a range of Reynolds numbers R λ ∈ [88, 529] are used to assess the validity of these equations, for the particular cases of second-and fourth-order moments. The involved terms pertain to molecular diffusion, transport, production, and dissipative-fluxes. The latter term, present at all scales, is equal to: i) the mean scalar variance dissipation rate, ⟨ χ ⟩, for the second-order moments transport equation; ii) non-linear correlations between χ and second-order moments of the scalar increment, for the fourth-order moments transport equation. The equations are further analysed to show that the similarity scales (i.e., variables which allow for perfect collapse of the normalised terms in the equations) are, for second-order moments, fully consistent with Kolmogorov–Obukhov–Corrsin (KOC) theory. However, for higher-order moments, adequate similarity scales are built from ⟨ χ n ⟩. The similarity is tenable for the dissipative range, and the beginning of the scaling range.

Applying an Extended Flamelet Model for a Multiple Injection Operating Strategy in a Common-Rail DI Diesel Engine

Subject of this work is the recently introduced extended Representative Interactive Flamelet (RIF) model for multiple injections. First, the two-dimensional laminar flamelet equations, which can describe the transfer of heat and mass between two-interacting mixture fields, are presented. This is followed by a description of the various mixture fraction and mixture fraction variance equations that are required for the RIF model extension accounting for multiple injection events. Finally, the modeling strategy for multiple injection events is described: Different phases of combustion and interaction between the mixture fields resulting from different injections are identified. Based on this, the extension of the RIF model to describe any number of injections is explained. Simulation results using the extended RIF model are compared against experimental data for a Common-Rail DI Diesel engine that was operated with three injection pulses. Simulated pressure curves, heat release rates, and pollutant emissions are found to be in good agreement with corresponding experimental data. For the pilot injection and the main or post injection, respectively, different ignition phenomena are pointed out and the influence of the scalar dissipation rate on these ignition phenomena is detailly investigated. 2009 SAE International.

Evaluation of Modeling Approaches for NOx Formation in a Common-Rail DI Diesel Engine within the Framework of RepresentativeInteractive Flamelets (RIF)

Representative Interactive Flamelets (RIF) have proven successful in predicting diesel engine combustion. The RIF concept is based on the assumption that chemistry is fast compared to the smallest turbulent time scales, associated with the turnover time of a Kolmogorov eddy. The assumption of fast chemistry may become questionable with respect to the prediction of pollutant formation; the formation of NOx, for example, is a rather slow process. For this reason, three different approaches to account for NOx emissions within the flamelet approach are presented and discussed in this study. This includes taking the pollutant mass fractions directly from the flamelet equations, a technique based on a three-dimensional transport equation as well as the extended Zeldovich mechanism. Combustion and pollutant emissions in a Common-Rail DI diesel engine are numerically investigated using the RIF concept. Special emphasis is put on NOx emissions. A surrogate fuel for diesel consisting of a mixture of n-decane (70% liquid volume fraction) and alpha-methylnaphthalene (30% liquid volume fraction) is applied in the simulations. One engine operating point is considered with a variation of start of injection. The simulation results are discussed and compared to experimental data.

Dissipation element analysis of a turbulent non-premixed jet flame

The objective of the present work is to examine the interaction between turbulent mixing and chemistry by employing the method of dissipation elements in a non-premixed turbulent jet flame. The method of dissipation elements [L. Wang and N. Peters, J. Fluid Mech. 554, 457–475 (2006)] is used to perform a space-filling decomposition of the turbulent jet flow into different regimes conditioned on their location with respect to the reaction zone. Based on the non-local structure of dissipation elements, this decomposition allows us to discern whether points away from stoichiometry are connected through a diffusive layer with the reaction zone. In a next step, a regime based statistical analysis of dissipation elements is carried out by means of data obtained from a direct numerical simulation. Turbulent mixing and chemical reactions depend strongly on the mixture fraction gradient. From a budget between strain and dissipation, the mechanism for the formation and destruction of mean gradients along dissipation elements is inspected. This budget reveals that large gradients in the mixture fraction field occur at a small but finite length scale. Finally, the inner structure of dissipation elements is examined by computing statistics along gradient trajectories of the mixture fraction field. Thereby, the method of dissipation elements provides a statistical characterization of flamelets and novel insight into the interaction between chemistry and turbulence.

Generalized Energy Budget Equations for Large-Eddy Simulations of Scalar Turbulence

The energy transfer between different scales of a passive scalar advected by homogeneous isotropic turbulence is studied by an exact generalized transport equation for the second moment of the scalar increment. This equation can be interpreted as a scale-by-scale energy budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. These effects are analyzed by means of direct numerical simulation where each term is directly accessible. To this end, a variation of the Taylor micro-scale based Reynolds number between 88 and 754 is performed. Understanding the energy transport between scales is crucial for Large-Eddy Simulation (LES). For an analysis of the energy transfer in LES, a transport equation for the second moment of the filtered scalar increment is introduced. In this equation new terms appear due to the interaction between resolved and unresolved scales, which are analyzed in the context of an a priori and an a posteriori test. It is further shown that LES using an eddy viscosity approach is able to fulfill the correct inter-scale energy transport for the present configuration.

Compressive and extensive strain along gradient trajectories

Based on direct numerical simulations of forced turbulence, shear turbulence, decaying turbulence, a turbulent channel flow as well as a Kolmogorov flow with Taylor based Reynolds numbers Reλ between 69 and 295, the normalized probability density function of the length distribution () of dissipation elements, the conditional mean scalar difference at the extreme points as well as the scaling of the two-point velocity difference along gradient trajectories are studied. Using the field of the instantanous turbulent kinetic energy k as a scalar, we find a good agreement between the model equation for () as proposed by Wang and Peters (2008) and the results obtained in the different DNS cases. This confirms the independance of the model solution from both, the Reynolds number and the type of turbulent flow, so that it can be considered universally valid. In addition, we show a 2/3 scaling for the mean conditional scalar difference. In the second part of the paper, we examine the scaling of the conditional two-point velocity difference along gradient trajectories. In particular, we compare the linear s/τ scaling, where τ denotes an integral time scale and s the separation arclength along a gradient trajectory in the inertial range as derived by Wang (2009) with the s · a∞ scaling, where a∞ denotes the asymtotic value of the conditional mean strain rate of large dissipation elements.