Streamline segment statistics propagation in inhomogeneous turbulence

The streamline segmentation method proposed by Wang is applied to a turbulent wavy channel flow to investigate the impact of pressure-driven strain on the streamline segment probability distribution. While the asymmetrical shape of the probability distribution is known to react to streamwise strains, in regions of detached flow this impact is severely less pronounced. Using tomographic particle-image velocimetry measurements and numerical data in attached and detached flow, the origin for this observation is found in the strong compressive strain during attached flow. The responsible mechanism is determined by introducing a concept of streamline segment locality, adapting the propagation equation for streamline segment statistics to inhomogeneous flow and applying it to the numerical data. A locally increased segment cutting rate is identified to cause the reduced sensitivity of the segment probability distribution to the local strain.

Figure 1

Difference between positive and negative mean segment lengths depending on local flow acceleration for various positions in the wavy channel flow [19].

Figure 2

Test section and tomographic PIV setup.

Figure 3

Field of view (FOV) of the tomographic PIV setup. (ES) defines the FOV in Ref. [19].

Figure 4

Profiles of first- and second-order statistics in multiple streamwise locations. The wall position is indicated by the vertical solid line.

Figure 5

Artificially generated reference flow for a fully attached flow field.

Figure 6

Selection criterion and attached and detached flow in the TPIV data (a) Selection criterion, (b) Mean attached flow field, (c) Mean detached flow field.

Figure 7

Strain and statistical asymmetry in the expansion slope region (ES) based on the TPIV data (a) Normalized mean streamwise strain, (b) Ratio between the mean segment length of positive and negative segments (thick) and empirical prediction (fine).

Figure 8

Strain and statistical asymmetry in the expansion slope region (ES) based on the DNS data (a) Normalized mean streamwise strain, (b) Ratio between the mean segment length of positive and negative segments (thick) and empirical prediction (fine).

Figure 9

Strain and statistical asymmetry in the trough region (T) based on the DNS data (a) Normalized mean streamwise strain, (b) Ratio between the mean segment length of positive and negative segments (thick) and empirical prediction (fine).

Figure 10

Strain and statistical asymmetry in the contraction slope region (CS) based on the DNS data (a) Normalized mean streamwise strain, (b) Ratio between the mean segment length of positive and negative segments (thick) and empirical prediction (fine).

Figure 11

Overall relation between local normalized acceleration and streamline segment length ratio based on the DNS data.

Figure 12

Schematic of streamline segment selection based on subvolumes.

Figure 13

Association of streamline segments with grid points.

Figure 14

Detection and cutting location for a single streamline segment.

Figure 15

Number of segments detected per cell.

Figure 16

Ratio between positive and negative segment numbers.

Figure 17

Logarithm to base 10 of the magnitude of convective time scales (a) Logarithm of $\tilde{T}{l}_m$ (b) Logarithm of $\tilde{T}\sigma$.

Figure 18

Logarithm to base 10 of the parameter fitting residuals. (a) Parameters for homogeneous turbulence, (b) fully free fit, (c) $\widetilde{c_{l,\nu }}$ fixed at 0.5, and (d) $\widetilde{c_{\mathrm{\Delta },\nu }}$ fixed at 1.

Figure 19

Exemplary jPDF fluxes. (a) Fluxes based on statistics, (b) parameters for homogeneous turbulence, (c) fully free fit, (d) $\widetilde{c_{l,\nu }}$ fixed at 0.5, and (e) $\widetilde{c_{\mathrm{\Delta },\nu }}$ fixed at 1.

Figure 20

Comparison of inner kinematics and background strain distributions (a) $\widetilde{C_{\mathrm{\Delta }}}$-distribution (b) Background strain.

Figure 21

Dimensionless timescales of inner kinematics (a) ${\tilde{T}}^{+}$-distribution (b) ${\tilde{T}}^{-}$-distribution.

Figure 22

Dimensionless occurrence rates of cutting and reattachment processes (a) ${\mathrm{\Lambda }}_c$-distribution (b) ${\mathrm{\Lambda }}_a$-distribution.

Figure 23

Dimensionless source terms for the velocity difference within the segments (a) $\tilde{K}$-distribution (b) ${\tilde{K}}^{*}$-distribution.

Figure 24

Original (a) and synthetically generated jPDF (b) at $x/h=1.5$ and $y/h=0.8$.