Jens Henrik Goebbert

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...

Decomposition of the turbulent kinetic energy field into regions of compressive and extensive strain

Based on direct numerical simulations of homogeneous shear turbulence, homogeneous isotropic decaying turbulence and a turbulent channel flow, the scaling of the two-point velocity difference along gradient trajectories 〈Δun|s〉 as well as between the extreme points of the instantaneous turbulent kinetic energy field k is studied. In the first step, we examine the linear 〈Δun|s〉∝s·a∞ scaling, where s denotes the separation arclength along a gradient trajectory and a∞ is the asymptotic value of the conditional mean strain rate of large dissipation elements. Then, we investigate the scaling of the velocity difference between scalar extreme points 〈Δun|l〉 as well as the probability density function of the Euclidean distance P(l) between them, conditioned on compressive and extensive strain regions. We observe that while the overall velocity difference along gradient trajectories and between diffusively connected scalar extreme points exhibits linear scaling behaviour 〈Δun|l〉∝l, the conditional velocity differences of extensive 〈Δun|l+〉 and compressive regions 〈Δun|l−〉 are in contrast to the K41 theory proportional to l2/3. The scaling exponent of the overall comes to one part from the purely extensive (compressive) 〈Δun|l+〉 (〈Δun|l−〉), while the second contribution is due to the difference in weighting the different regions, thus involving the conditioned pdfs P(l+) and P(l−). We find that the latter relation scales with l1/3. The decomposition of 〈Δun|l〉 into two contributions scaling with l2/3 and l1/3, respectively, hence yields an alternative explanation for the observed linear regime.

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.

Dissipation Element Analysis of Inhomogenous Turbulence

Gradient trajectories recently received attention in the context of dissipation elements, which are space-filling regions whose theoretical description has successfully been tested via Direct Numerical Simulations (DNS) of homogenous shear flows by Wang and Peters (J. Fluid Mech. 554: 457–475, 2006; J. Fluid Mech. 608: 113–138, 2008). In the present work, the DNS of a Kolmogorov flow is evaluated via a parallelized gradient trajectory search code to verify the theory of dissipation elements in inhomogenous turbulence and to analyze a new k-\(\sqrt{k}L\) model derived by Menter et al. (Turbulence, Heat and Mass Transfer, 5, 2006).

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.