Here we shall explain the proposed method of temporal tracking of the educed vortices, which is primarily based on the centroid location of each structure, but additionally on their overlapped volumes with the structures of the subsequent snapshot. Note that similar time-tracking criteria have been employed independently, for instance, by Lehew et al. (2013) and Lozano-Durán and Jiménez (2014) respectively.
The proposed hybrid criterion was developed in order to achieve a compromise between the com- putationally efficient nature of the centroid-based method, and the robustness of the overlapped- volume method that is computationally more expensive. However, the attractive capability of the volumetric technique of identifying occasional splitting and merging events of the vortices had to be compromised. For the sake of clarity, detailed explanations on the limitation of the cur- rent time-tracking method will be offered with appropriate examples in the following part of this section.
Before proceeding any further, let us define some key terminologies necessary to explain the track- ing method. We shall indicate any field values φ (x, y, z,t) in an arbitrary snapshot with the super-
script n, i.e. φn= φ (x, y, z,tn), where tn is a discrete point in time corresponding to the snapshot
of concern. Consequently, the field value in the subsequent snapshot at tn+1= tn+ ∆tsis denoted
as φn+1.
Prior to the time-tracking, the centroid locations of every educed structure i are computed as mentioned earlier. Additionally, the mean velocity components on the computed centroid locations (hu(yc, zc)ii, hv(yc, zc)ii and hw(yc, zc)ii) need to be computed by interpolation from the simulation
grid. Furthermore, the signed threshold values that have been used to educe those streamwise vortices are also stored to identify their sense of rotation. In the following, therefore, only the vortices with the same sense of rotation will be connected.
Upon the initiation of the time-tracking procedure, every structure at tn+1, which is denoted by the
index j = 1 . . . Nvn+1, is translated backwards in space using the local mean velocity, viz.
[ ˜xc, ˜yc, ˜zc]j= [xc, yc, zc]j− ∆ts[hu(yc, zc)ji, hv(yc, zc)ji, hw(yc, zc)ji] (5.3)
where [ ˜xc, ˜yc, ˜zc]jdenotes the shifted centroid coordinates of every structure j. This shifting proce-
dure is mostly consistent with the one applied in Lozano-Durán and Jiménez (2014), and based on the earlier findings in the turbulent plane channel flow that the advection velocities of the small- scale turbulent fluctuations are approximately equal to the local mean velocities (cf. Kim and Hussain (1993), del Álamo and Jiménez (2009)). Note the method proposed by Lozano-Durán and Jiménez (2014) translates every vertex that defines the skeletons of their turbulent structures independently, based on the local mean velocity fields interpolated onto the vertices. In this way, the morphing of the structures resulted from the local velocity gradient in the wall-normal direc- tion can be taken into account, in contrast to the current method that simply translates the whole structures around their centroids without any deformations. Although this part of their procedure
seems promising, it had to be omitted for the current implementation again due to the associated high computational effort and storage requirements.
Some exceptions should be mentioned for the advection velocities of the near-wall small-scale
structures below y+≈ 15, as well as the large-scale structures that both exist in the plane channel
turbulence. Whilst the small-scale near-wall structures travel with an approximately constant ve-
locity at ≈ 11uτ, those large-scale structures were found to travel with a constant velocity that is
approximately equivalent to the bulk flow velocity throughout the entire channel domain (cf. del Álamo and Jiménez (2009)). Despite the physical importance of those quantities in the current flow configuration, we need to proceed without any further discussion on this topic at this point, since it is simply out of the scopes of the current study and technique. However, further investiga- tions on this topic should be carried out separately in the near future, preferably by developing a new spectral-based method similar to the one employed by del Álamo and Jiménez (2009).
Let us now consider a particular vortex at tn, denoted by the index ˜i. The ultimate purpose of
this part of the tracking procedure is to find its successor in the subsequent snapshot at tn+1 (the
succeeding vortex is denoted by ˜j).
After shifting every vortex structure at tn+1 backwards in space based on the mean velocity field
at the centroids, we construct a set of the position difference vectors r˜i, j, which are defined with
respect to the centroid location of the vortex ˜i at tnand its distance to the centroid locations of every
vortex j. As a result of the backward shifting, the position difference vectors should now be solely
the product of the geometrical morphing of those vortices over the time interval ∆ts. Subsequently,
the position difference vectors are sorted according to their magnitude in ascending order, and the vortex associated with the first element of the sorted vector is selected as a succeeding candidate
˜j. Furthermore, the magnitude of r˜i, ˜jis compared against the predefined searching radius rth, and
if |r˜i, ˜j| < rth, then a connection between ˜i and ˜j is established (cf. Figure 5.7(a,b) for illustrative
examples). Note that the value of rthis designed to be function of Vest, based on a consideration
that the larger structures are subjected to the stronger velocity shear and should therefore result a
larger displacement of the centroid location by higher degree of morphing. The definition of rthis
as follows:
rth= αr(Vest)1/3 (5.4)
where αris a user-specified constant parameter that was set at 0.5 throughout the current study.
In case of more radical changes of the centroid locations that results |r˜i, ˜j| > rth, which is often
associated with vortex splitting/merging events, the overlap volume between the vortex ˜i and ˜j needs to be assessed. In order to compute the overlapped volume, their vertices are interpolated onto a dedicated separate three-dimensional grid with no variation in the grid spacing in each direction (in contrast to the simulation grid which has variations in two cross-stream directions).
In this way, every cubical cell of the dedicated grid has an identical volume Vc that one simply
needs to count those cells whose centroid locations are shared by the two structures in order to compute the overlapped volume. Furthermore, the spatial resolution of the dedicated grid, which
is solely depend on the number of the grid points in each direction (i.e. Nx, Ny and Nz), can be
5.4 Temporal evolution of educed vortices
study, we selected the numbers of the grid points in each direction as a half of the numbers of the Fourier/Chebyshev-Gauss-Lobatto grid points utilised in the simulation.
Consequently, the overlapped volume V˜i∧ ˜jis defined as follows:
V˜i∧ ˜j=
∑
Vc, Vc∈ {Vc|Vc∈ ˜i∧Vc∈ ˜j} (5.5)where Vc is the volume of one uniform grid cell. Finally, the ratio of V˜i∧ ˜jto the volume of the
structure ˜j is compared against another user-specified threshold ratio αv, that was set at 0.4 in this
case. A connection between ˜i and ˜j is established if the condition V˜i∧ ˜j/V˜j> αvis met. Note that
in case of splitting, only the large-enough child-structure that is closest to the parent structure in their centroid locations are considered as the successor, whilst the remaining child structures are recognised as independent newly-born structures to be tracked from that point in time (cf. Figure 5.7(b,c)). This limited capability of identifying the splitting events is one of the compromises made for this current method that was mentioned earlier.
If either the centroid distance or the overlapped volume criterion is met, then the time is advanced
(tn+1→ tn) with ˜j becoming new ˜i (i.e. the two candidates are identified as one), and the tracking
process continues. If none of the two conditions are met, on the other hand, the time-tracking of ˜i is terminated at that point in time and ˜j becomes a new track. Please refer to Figure 5.7(d) for an illustration of a case of loosing a track artificially.
Streamwise domain periodicity is taken into account continuously by checking the gap between
(xc)˜iand the streamwise domain limit Lx. In case of the gap is smaller than 0.5∆tshu(yc, zc)˜ii, the
centroid streamwise location is simply translated by Lxbackwards before the successor searching
routine is initiated.
To this end, it is important to note that the temporal tracking is performed in an iterative manner, by starting from all the vortices that are not marked as initial/succeeding structures previously. Similarly, those structures that have been tracked before are not considered to be candidates of the successors. In other words, all the educed vortex structures are marked as initial points or successors strictly only once, which implies to the incapability of detecting the merging events within the current framework. The limitation is considered to be acceptable for the current purpose of investigating the dynamical aspects of the vortices in the mixed-boundary corners and free-slip plane, since the merging events are usually associated with the inverse cascade to the turbulent kinetic energy that is not our interest here. In any case, incorporating the sophisticated graph- based structural identity management performed in Lozano-Durán and Jiménez (2014) should be considered in the future, in order to achieve more complete picture of the splitting/merging events in the open duct flow.
(a) tn tn+1 backward- shifting |r˜i, ˜j| < rth (b) tn tn+1 |r˜i, ˜j| < rth (c) tn tn+1 |r˜i, ˜j| > rth V˜i∧ ˜j/V˜j> αv (d) tn tn+1 |r˜i, ˜j| > rth V˜i∧ ˜j/V˜j< αv
Figure 5.7: Schematics of four representative scenarios in the vortex tracking procedure, where: (a) case of successful vortex connection by the condition |r˜i, ˜j| < rthwithout splitting; (b) same as (a) with splitting; (c) case of successful vortex connection by the condition V˜i∧ ˜j/V˜j> αv; (d) case of unsuccessful vortex connection. Diagrams on the left and right sides are with and without backward shifting of the structures at tn+1respectively. Red structures, the vortex ˜i at tn; half-transparent red structures, the candidates (left) or the connected successor (right) ˜j at tn+1; grey structures, independent structures without being connected. Only the closest candidates of ˜j have their centre of gravity indicated. The length of the dual-head arrows indicates |r˜i, ˜j|.
5.4 Temporal evolution of educed vortices
Figure 5.8: Vortex temporal-tracking result in open duct flow with A = 1 and Reb= 2205; flow direction is from left to right. The image shows several arbitrarily selected structures over a time sequence covering approximately 8 bulk time units (≈ 82.5∆tν), connected by solid lines. The colour of the structures indicates positive (red) and negative (blue) values of the streamwise vorticity thresholds employed to educe those structures.