Philip Schaefer

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.

The length distribution of streamline segments in homogeneous isotropic decaying turbulence

Based on the profile of the absolute value u of the velocity field ui along streamlines, the latter are partitioned into segments at their extreme points as proposed by Wang [J. Fluid Mech. 648, 183–203 (2010)]10.1017/S0022112009993041. It is found that the boundaries of all streamline segments, i.e., points where the gradient projected in streamline direction ∂u/∂s vanishes, define a surface in space. This surface also contains all local extreme points of the scalar u-field, i.e., points where the gradient in all directions of the field of the absolute value of the velocity and thereby those of the turbulent kinetic energy (k = u2/2, where k is the instantaneous turbulent kinetic energy) vanishes. Such points also include stagnation points of the flow field, which are absolute minimum points of the turbulent kinetic energy. As local extreme points are the ending points of dissipation elements, an approach for space-filling geometries in turbulent scalar fields, such elements in the turbulent kinetic ener...

Gradient trajectory analysis in a Jet flow for turbulent combustion modelling

Based on planar high-speed Rayleigh scattering measurements of the mixture fraction Z of propane discharging from a turbulent round jet into co-flowing carbon dioxide at nozzle-based Reynolds numbers Re 0 = 3000–8600, we use scalar gradient trajectories to investigate the local structure of the turbulent scalar field with a focus on the scalar turbulent/non-turbulent interface. The latter is located between the fully turbulent part of the jet and the outer flow. Using scalar gradient trajectories, we partition the turbulent scalar field into these three regions according to an approach developed by Mellado et al. (J.P. Mellado, L. Wang, and N. Peters, Gradient trajectory analysis of a scalar field with external intermittency, J. Fluid Mech. 626 (2009), pp. 333–365.). Based on these different regions, we investigate in a next step zonal statistics of the scalar probability density function (pdf) P(Z) as well as the scalar difference along the trajectory ΔZ and its mean scalar value Zm , where the latter two quantities are used to parameterize the scalar profile along gradient trajectories. We show that the scalar pdf P(Z) can be reconstructed from zonal gradient trajectory statistics of the joint pdf P(Zm , ΔZ). Furthermore, on the one hand we relate our results for the scalar turbulent/non-turbulent interface to the findings made in other experimental and numerical studies of the turbulent/non-turbulent interface, and on the other hand discuss them in the context of the flamelet approach and the modelling of pdfs in turbulent non-premixed combustion. Finally, we compare the zonal statistics for P(Z) with the composite model of Effelsberg and Peters (E. Effelsberg and N. Peters, A composite model for the conserved scalar pdf, Combust. Flame 50 (1983), pp. 351–360) and observe a very good qualitative and quantitative agreement.

Asymptotic analysis of homogeneous isotropic decaying turbulence with unknown initial conditions

In decaying grid turbulence there is a transition from the initial state immediately behind the grid to the state of fully developed turbulence downstream, which is believed to be self-similar. This state is characterized by a power law decay of the turbulent kinetic energy with a time-independent decay exponent n. The value of this exponent, however, depends on the initial distribution of the velocity, about which we have only general information at the very best. For homogeneous isotropic decaying turbulence, the evolution of the two-point velocity correlation is described by the von Kármán–Howarth equation. In the non-dimensionalized form of this equation a decay exponent dependent term occurs, whose coefficient will be called δ. We exploit the fact that δ vanishes for n→2, which is shown to correspond to the limit d→∞, where d denotes the dimensionality of space to formulate a singular perturbation problem. It is shown that a distinguished limit exists for d→∞ and δ→0. We obtain in the limit of infinitely large Reynolds numbers an outer layer of limited, but a priori unknown extension, as well as an inner layer of the thickness of the order , where the Kolmogorov scaling is valid. To leading order, we obtain an algebraic balance in the outer layer between the two-point correlation and the third-order structure function. In the inner layer the analysis yields the emergence of higher order terms to the classical K41 scaling. All leading order solutions are shown to be subject to a band of uncertainty of the order , which is argued to be due to intrinsically unknown initial conditions.

A model equation for the joint distribution of the length and velocity difference of streamline segments in turbulent flows

Streamlines recently received attention as natural geometries of turbulent flow fields. Similar to dissipation elements in scalar fields, streamlines are segmented into smaller subunits based on local extreme points of the absolute value of the velocity field u along the streamline coordinate s, i.e., points where the projected gradient in streamline direction us = 0. Then, streamline segments are parameterized using their arclength l between two neighboring extrema and the velocity difference Δ at the extrema. Both parameters are statistical variables and streamline segments are characterized by the joint probability density function (jpdf) P(l, Δ). Based on a previously formulated model for the marginal pdf of the arclength, P(l), which contains terms that account for slow changes as well as fast changes of streamline segments, a model for the jpdf is formulated. The jpdfs, when normalized with the mean length, lm, and the standard deviation of the velocity difference σ, obtained from two different dir...

The local topology of stream- and vortex lines in turbulent flows

Tangent lines to a given vector field, such as streamlines or vortex lines, define a local unit vector t that points everywhere in the lines direction. The local behavior of the lines is characterized by the eigenvalues of the tensor T=∇·t. In case of real eigenvalues, t can be interpreted as a normal vector to a surface element, whose shape is defined by the eigenvalues of T. These eigenvalues can be used to define the mean curvature −H and the Gaussian curvature K of the surface. The mean curvature −H describes the relative change of the area of the surface element along the field line and is a measure for the local relative convergence or divergence of the lines. Different values of (H, K) determine whether field lines converge or diverge (elliptic concave or elliptic convex surface element, stable/unstable nodes), converge in one principal direction and diverge in another (saddle) or spiral inwards or outwards (stable/unstable focus). In turbulent flows, a plethora of local field line topologies are ...

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.

Experimental Investigation of Dissipation Element Statistics in a Jet Flow

We present a detailed experimental investigation of conditional statistics related to dissipation elements based on the scalar field \( \theta \) and its scalar dissipation rate \( \chi \). Based on high frequency two-dimensional measurements of the mass fraction of propane in a turbulent round jet discharging into surrounding air, we acquire data resolving the Kolmogorov scale in every spatial direction using a high-speed Rayleigh scattering technique and Taylor’s hypothesis. The Reynolds number (based on nozzle diameter and jet exit velocity) varies between 2,000 and 6,700 and the Schmidt number between \( 1.2 \) and \( 1.7 \). The experimental results for the normalized marginal pdf \( \tilde{P}(\tilde{l}) \) of the length of dissipation elements follow closely the theoretical model derived by Wang and Peters (J. Fluid. Mech. 608:113–138, 2008). We also find that the mean linear distance between two extreme points of an element is of the order of the Taylor microscale \( \lambda \). Furthermore, the conditional mean \( \langle\varDelta \theta |l\rangle \) scales with Kolmogorov’s 1/3 power.

Local Dynamics and Statistics of Streamline Segments in Fluid Turbulence

Based on local extreme points of the absolute value \( u \) of the velocity field \( u_{i} \), streamlines are partitioned into segments as proposed by Wang (J. Fluid. Mech. 648:183–203, 2010). The temporal evolution of the arc length l of streamline segments is analyzed and associated with the motion of the isosurface defined by all points on which the gradient in streamline direction \( \partial u/\partial s \) vanishes. This motion is diffusion controlled for small segments, while large segments are mainly subject to strain and pressure influences. Due to the non-locality of streamline segments, their temporal evolution is not only a result of slow but also of fast changes, which differ by the magnitude of the jump \( \varDelta l \) that occurs within a small time step \( \varDelta t \). The separation of the dynamics into slow and fast changes allows the derivation of a transport equation for the probability density function (pdf) P(l) of the arc length l of streamline segments. While slow changes in the pdf transport equation translate into a convection and a diffusion term when terms up to second order are included, the dynamics of the fast changes yield integral terms. The convection velocity corresponds to the first order jump moment, while the diffusion term includes the second order jump moment. It is theoretically and from DNS data of homogeneous isotropic decaying turbulence at two different Reynolds numbers concluded that the normalized first order jump moment is quasi-universal, while the second order one is proportional to the inverse of the square root of the Taylor based Reynolds number \( Re_{\lambda }^{ - 1/2} \). It’s inclusion thus represents a small correction in the limit of large Reynolds numbers. Numerical solutions of the pdf equation yield a good agreement with the pdf obtained from the DNS data. It is also concluded on theoretical grounds that the mean length of streamline segments scales with the Taylor microscale rather than with any other turbulent length scale, a finding that can be confirmed from the DNS.

Geometrical Features of Streamlines and Streamline Segments in Turbulent Flows

Streamlines constitute natural geometries in turbulent flows. In this work streamlines are segmented based on local extrema of the field of the absolute value of the velocity along the streamline coordinate. Streamline segments are parameterized based on their arclength and a theoretical scaling of the mean length with the geometric mean of the Kolmogorov length and the Taylor microscale is derived and found to be in good agreement with direct numerical simulations.