Lipo Wang

The length-scale distribution function of the distance between extremal points in passive scalar turbulence

In order to extract small-scale statistical information from passive scalar fields obtained by direct numerical simulation (DNS) a new method of analysis is introduced. It consists of determining local minimum and maximum points of the fluctuating scalar field via gradient trajectories starting from every grid point in the directions of ascending and descending scalar gradients. The ensemble of grid cells from which the same pair of extremal points is reached determines a spatial region which is called a ‘dissipation element’. This region may be highly convoluted but on average it has an elongated shape with, on average, a nearly constant diameter of a few Kolmogorov scales and a variable length that has the mean of a Taylor scale. We parameterize the geometry of these elements by the linear distance between their extremal points and their scalar structure by the absolute value of the scalar difference at these points. The joint p.d.f. of these two parameters contains most of the information needed to reconstruct the statistics of the scalar field. It is decomposed into a marginal p.d.f. of the linear distance and a conditional p.d.f. of the scalar difference. It is found that the conditional mean of the scalar difference follows the 1/3 inertial-range Kolmogorov scaling over a large range of length-scales even for the relatively small Reynolds number of the present simulations. This surprising result is explained by the additional conditioning on minima and maxima points. A stochastic evolution equation for the marginal p.d.f. of the linear distance is derived and solved numerically. The stochastic problem that we consider consists of a Poisson process for the cutting of linear elements and a reconnection process due to molecular diffusion. The resulting length-scale distribution compares well with those obtained from the DNS.

Length-scale distribution functions and conditional means for various fields in turbulence

Dissipation elements are identified for various direct numerical simulation (DNS) fields of homogeneous shear turbulence. The fields are those of the fluctuations of a passive scalar, of the three components of velocity and vorticity, of the second invariant of the velocity gradient tensor, turbulent kinetic energy and viscous dissipation. In each of these fields trajectories starting from every grid point are calculated in the direction of ascending and descending gradients, reaching a local maximum and minimum point, respectively. Dissipation elements are defined as spatial regions containing all the grid points from which the same pair of minimum and maximum points is reached. They are parameterized by the linear length between these points and the difference of the field variable at these points. In analysing the changes that occur during one time step in the linear length as well as in the number of grid points contained in the elements, it is found that rapid splitting and attachment processes occur between elements. These processes are much more frequent than the previously identified processes of cutting and reconnection. The model for the length-scale distribution function that had previously been proposed is modified to include these additional processes. Comparisons of the length-scale distribution function for the various fields with the proposed model show satisfactory agreement. The conditional mean difference of the field variable at the minimum and maximum points of dissipation elements is calculated for the passive scalar field and the three components of velocity. While the conditional mean difference follows the 1/3 inertial-range Kolmogorov scaling for the passive scalar field, the scaling exponent differs from the 1/3 law for each of the three components of velocity. This is thought to be due to the relatively high shear rate of the DNS calculations. The conditional mean viscous dissipation shows, differently from all other field variables analysed, a pronounced dependence on the linear length of elements. This is explained by intermittency. This finding is used to evaluate the production and the dissipation term of the empirically derived e-equation that is often used in engineering calculations.

Gradient trajectory analysis of a scalar field with external intermittency

The passive scalar field of a temporally evolving shear layer is investigated using gradient trajectories as a means to analyse the scalar probability density function and the conditional scalar dissipation rate in the presence of external intermittency. These results are of significance for turbulent combustion, where improved predictions of the statistics of the conditional dissipation rate are needed in several models. First, the variation of the conventional first and second moments of the conditional dissipation rate across the layer is quantitatively documented in detail. A strong dependence of the conditional dissipation rate on the lateral position and on the conditioning value of the scalar is observed. The dependence on the transverse distance to the centre-plane partially explains the double-hump profile usually reported when this dependence is ignored. The variation with the scalar observed in the ratio between the second and first moments would invalidate certain assumptions commonly done in turbulent combustion. It is also seen that conditioning on the scalar does not reduce the fluctuation of the dissipation rate with respect to unconditional values. Next, the role of external intermittency in these results is investigated. For that purpose, the flow is partitioned into different zones based on different types of gradient trajectories passing through each point, thereby introducing non-local information in comparison with the standard turbulent/non-turbulent separation based on the conventional intermittency function. In addition to the homogeneous outer regions, three zones are identified: a turbulent zone, a turbulence interface and quasi-laminar diffusion layers. The relative contribution from each of these zones to the conventional intermittency factor is reported. The statistics are then conditioned on each of these zones, and the spatial variation of the scalar distribution and of the conditional scalar dissipation rate is explained in terms of the observed zonal statistics. For the Reynolds numbers of the present simulation, between 1500 and 3000 based on the vorticity thickness and the velocity difference, and a Schmidt number equal to 1, it results that the major contribution to both statistics is due to the turbulence interfaces. At the same time, the turbulent zone shows a distinct behaviour, being approximately homogeneous but anisotropic.

On properties of fluid turbulence along streamlines

Geometrical and dynamical properties of turbulent flows have been investigated by streamline segment analysis. Starting from each grid point, a streamline segment is defined as the part of its streamline bounded by the two adjacent extremal points of the velocity magnitude. Physically the streamline segments can be extended into a more meaningful concept, namely the streamtube segments, which are non-overlapping and space filling. This decomposition of the flow allows for new insights into vector-related statistics in turbulence. According to the variation of velocity, the streamline segments can be sorted into positive and negative segments. The overall properties of turbulent flows can be newly understood and explained from the statistics of these segments with simple structures; for instance, the negative skewness of the velocity derivative becomes naturally a kinematic outcome. Furthermore, from direct numerical simulations conditional statistics of pressure and kinetic energy dissipation along the streamline segments are evaluated and discussed.

Fast and slow changes of the length of gradient trajectories in homogeneous shear turbulence

Gradient trajectories in scalar fields have recently received attention in the context of dissipation elements [1], [2] which in turn are of interest for the flamelet concept in nonpremixed combustion [3]. Dissipation elements are space filling regions in a scalar field defined such that gradient trajectories starting from any point within the element in ascending and descending directions reach the same minimum and maximum points. Gradient trajectories advance preferentially through regions of the scalar field that have been smoothed by the combined action of diffusion and extensive strain. Since the extensive strain in these regions is of the order of the inverse of the integral time scale T, dimensional analysis predicts the mean length l m of dissipation elements to be of the order of the Taylor length [4].

A new view of flow topology and conditional statistics in turbulence.

By partitioning a turbulent flow field into relative simple units, the original complex system may be better understood from studying decomposed structures. In this paper, some general principles for identifying geometrical decomposition are discussed. Logically, to make analysis more objective and quantitative, the decomposed units need to be non-arbitrarily defined and space filling. Following this vein, we introduced two topological approaches satisfying these prerequisites and the relevant work is reviewed. For a given scalar variable, dissipation elements are defined as the spatial regions that the gradient trajectories of this scalar can share the same pair of scalar extremums (one maximum and one minimum), whereas for the general vector variables, vector tube segments are the part of vector tubes bounded by adjacent extremums of the magnitude of the given vector. Both structures can be characterized by representative shape parameters: the length scale and the extremum difference. On the basis of direct numerical simulation data, the statistics of the shape parameters have been studied. Physically, those structures reveal the ‘nature’ topology of turbulence, and thus their characteristic parameters reflect the flow properties. For instance, when the vector tube segment approach is applied to the velocity case, the negative skewness of the velocity derivative can be explained by the asymmetry of the joint probability density function of the shape parameters of streamtube segments. Conditional statistics based on these newly defined structures identify finer flow physics and are believed helpful for modelling improvement. Application examples illustrate that, in principle, these methods can generally be applied to different flow cases under different situations.

Structures of the vorticity tube segment in turbulence

To address the geometrical properties of the turbulent velocity vector field, a new concept named streamtube segment has been developed recently [L. Wang, “On properties of fluid turbulence along streamlines,” J. Fluid Mech. 648, 183–203 (2010)10.1017/S0022112009993041]. According to the vectorial topology, the entire velocity field can be partitioned into the so-called streamtube segments, which are organized in a non-overlapping and space-filling manner. In principle, properties of turbulent fields can be reproduced from those of the decomposed geometrical units with relatively simple structures. A similar idea is implemented to study the turbulent vorticity vector field using the vorticity tube segment structure. Differently from the conventional vortex tubes, vorticity tube segments are space-filling and can be characterized by non-arbitrary parameters, which enables a more quantitative description rather than just an illustrative explanation of turbulence behaviors. From analyzing the direct numerica...

Application of Riblets on Turbine Blade Endwall Secondary Flow Control

Within the past 10 years, significant improvements have been achieved in the laser manufacturing process. It is feasible now to design various small-scale surface features (such as dimples, riblets, grooves, etc.) in gas turbine applications with the current manufacturing readiness level of laser surface texturing techniques. In this paper, the potential of adding riblets on a turbine endwall has been investigated through combined computational fluid dynamics and experimental studies in a low-speed linear cascade environment. Detailed comparisons of the flow structures have been made for cases with and without riblets on the endwall. The numerical results show that endwall riblets can effectively reduce the strength of the pressure side leg of the horseshoe vortex, lower the cross-passage pressure gradient, and alleviate the lift up of the passage vortex. Oil film flow visualization and exit aerodynamic loss survey in experiments support the computational fluid dynamics observations: The passage vortex lo...

A compensation-defect model for the joint probability density function of the scalar difference and the length scale of dissipation elements

Dissipation element analysis is a new approach to study turbulent scalar fields. Gradient trajectories starting from each material point in a fluctuating scalar field ϕ′(x,t) in ascending and descending directions will inevitably reach a maximal and a minimal point. The ensemble of material points sharing the same pair ending points is named a dissipation element. Dissipation elements can be parametrized by the length scale l and the scalar difference Δϕ′, which are defined as the straight line connecting the two extremal points and the scalar difference at these points, respectively. The decomposition of a turbulent field into dissipation elements is space filling. This allows us to reconstruct certain statistical quantities of fine scale turbulence which cannot be obtained otherwise. The marginal probability density function (PDF) of the length scale distribution had been modeled in the previous work based on a Poisson random cutting-reconnection process and had been compared to data from direct numeri...

Mean velocity increment conditioned on gradient trajectories of various scalar variables in turbulence

This paper focuses on the generality of the velocity structure along the gradients of several field variables in scalar form, such as the passive scalar, the Cartesian projections of the velocity vector, kinetic energy and energy dissipation. These scalar variables behave differently, which can be explained mathematically by the fact that the source terms in their governing equations are different. Consequently, it is reasonable to expect some generalities among different scalars if the respective source terms are instantaneously small. The validity of this expectation has been discussed for individual cases, especially at high Reynolds numbers. It is found that the pointwise mean strain rates along gradients of various scalars are always negative. By analyzing the scalar gradient orientations, it is seen that statistically the gradient vectors with large magnitudes align with each other, whereas those with small magnitudes tend to be randomly organized. In terms of the two-point velocity increment, for the passive scalar it has recently been derived that there exists a nearly linear scaling of the velocity difference with respect to the arc length of gradient trajectories (Wang 2009 Phys. Rev. E 79 046325). Theoretically, this scaling can also be extended to other scalars with satisfactory agreement with numerical results. PACS number: 47.27. i (Some figures in this article are in colour only in the electronic version.)