Interscale energy transfer in the merger of wakes of a multiscale array of rectangular cylinders

The near wake of a flow past a multiscale array of bars is studied by means of particle image velocimetry (PIV). The aim of this research is to understand the nature of multiscale flows, where multiple coherent motions of nonuniform sizes and characteristic frequencies (i.e., sheddings of particular bars in our considered case) interact with each other. The velocity fields acquired from the experiments are triple decomposed into their mean, a number of coherent fluctuations, and their stochastic part according to a triple decomposition technique introduced recently by Baj et al., Phys. Fluids 27, 075104 (2015). This nonstandard approach allows us to monitor the interactions between different coherent fluctuations representative of sheddings of the particular bars. Further, additional equations governing the kinetic energy of the recognized velocity components are derived to provide better insight into the dynamics of these interactions. Interestingly, apart from the coherent fluctuations associated with sheddings, some additional, secondary coherent fluctuations are also recognized. These seem to appear as a result of nonlinear triadic interactions between the primary shedding modes when the two shedding structures of different characteristic frequencies are in close proximity to one another. The secondary coherent motions are almost exclusively supplied with energy by the primary coherent motions, whereas the latter are driven by the mean flow. It is also found that the coherent fluctuations play an important role in exciting the stochastic fluctuations, as the energy is not fed to the stochastic fluctuations directly from the mean flow but rather through the coherent modes.

Figure 1

Characterization of the free stream at the center plane: (a) the mean streamwise velocity and the turbulence intensity profiles, (b) PSDs of the streamwise and transverse velocity fluctuations evaluated in the bulk flow, and (c) streamwise correlation functions of the transverse velocity fluctuations evaluated in the bulk flow and at the midheight of the boundary layer.

Figure 2

(a) Multiscale grid, (b) reference uniform grid, and (c) the outer frame.

Figure 3

Schematic of the experimental setup.

Figure 4

Locations of the fields of view of particular measurements.

Figure 5

Transverse profiles and stitched fields of view for experiment 1: (a) mean streamwise velocity and (b) fluctuations intensity (wakes' intersection points are marked with crosses, based on experiment 1).

Figure 6

PSDs of streamwise and transverse velocity components measured at the center line of (a) wake I $\left(x/H=0.57\right)$, (b) wake II $\left(x/H=0.07\right)$, and (c) wake III $(x/H=0.07$, based on experiment 2).

Figure 7

Shift of the shedding peaks of the transverse velocity PSDs measured in different experiments at the center line of (a) wake II $\left(x/H=0.07\right)$ and (b) wake III $\left(x/H=0.07\right)$.

Figure 8

Compensated PSDs of the transverse velocity component in the flow past the multiscale array at $x/H=$ (a) 0.07, (b) 0.24, (c) 0.41, and (d) 0.57. (e) Compensated PSD of the transverse velocity component in the reference flow past an uniform array at $x/H=0.07$ (based on experiment 2).

Figure 9

Appearance of secondary spectral peaks associated with frequencies $f_{\mathrm{sh}}^{\mathrm{II}}-f_{\mathrm{sh}}^{\mathrm{I}}$ and $f_{\mathrm{sh}}^{\mathrm{II}}+f_{\mathrm{sh}}^{\mathrm{I}}$ at $(x/H,y/H)=(0.75,-0.28)$ (based on experiment 2).

Figure 10

Characteristic frequencies of the secondary coherent fluctuations at stations: $x/H=$ (a) 0.57 and (b) 0.23 (based on experiment 3).

Figure 11

Energies (i.e., ${\mathrm{\Phi }}_{\mathrm{OMD}}·{\underline{\mathrm{\Phi }}}_{\mathrm{OMD}}^T$, left-hand side) and curls (i.e., $|\nabla \times {\mathrm{\Phi }}_{\mathrm{OMD}}|$, right-hand side) of the OMD modes associated with frequencies: (a) $f^{\mathrm{I}}$, (b) $f^{\mathrm{II}}$, and (c) $f^{\mathrm{III}}$; lines denote points of equal curls' phase values (values normalized with their respective maxima, based on experiment 1).

Figure 12

An example of the Fourier modes extraction: (a) a function $a(y,\tau )$ to be phase averaged, (b) the associated phase reference $\varphi \left(\tau \right)$, (c) the phase average $\langle a\rangle (y,\varphi )$, and (d) the resultant second Fourier mode ${\mathrm{\Phi }}_{\mathrm{PA}}\left(y\right)$.

Figure 13

Energies (i.e., ${\mathrm{\Phi }}_{\mathrm{PA}}·{\underline{\mathrm{\Phi }}}_{\mathrm{PA}}^T$, left-hand side) and curls (i.e., $|\mathbf{\nabla }\times {\mathrm{\Phi }}_{\mathrm{PA}}|$, right-hand side) of the second Fourier modes of the phase-averaged velocity fluctuations associated with frequencies: (a) $f^{\mathrm{II}-\mathrm{I}}$, (b) $f^{\mathrm{II}+\mathrm{I}}$, (c) $f^{\mathrm{III}-\mathrm{II}}$, and (d) $f^{\mathrm{III}+\mathrm{II}}$; continuous lines denote points of equal curls' phase value; dashed lines indicate locations of maxima of amplitude of [(a), (b)] ${\tilde{\mathbf{u}}}^{\mathrm{II}}$ and [(c), (d)] ${\tilde{\mathbf{u}}}^{\mathrm{III}}$; crosses designate the wakes' intersection points inferred from the fluctuations intensity field [see Fig. 5]; while circles mark the maximum of the mode's energy field (values normalized with their respective maxima, based on experiment 1).

Figure 14

Terms of the mean energy budget (8a) and the affiliated fluctuation intensity profiles at $x/H=$ (a) 0.07, (b) 0.24, (c) 0.41, and (d) 0.57 (lines are based on experiment 2 and markers are based on experiment 3). A reproduction of Fig. 5 is attached at the very bottom.

Figure 15

Terms of the mean energy budget (8a) averaged across the transverse direction at $x/H=$ (a) 0.07, (b) 0.24, (c) 0.41, and (d) 0.57 (values are expressed as percentages of the respective total gains, based on experiment 2).

Figure 16

Terms of the coherent energy budget (9a) evaluated for the primary coherent fluctuations $({\tilde{\mathbf{u}}}^{\mathrm{II}}$, upper plots; ${\tilde{\mathbf{u}}}^{\mathrm{III}}$, lower plots) and the affiliated fluctuations intensity profiles at $x/H=$ (a) 0.07 and (b) 0.24 (lines are based on experiment 2 and markers are based on experiment 3). A reproduction of Fig. 5 is attached at the very bottom.

Figure 17

Terms of the coherent energy budget (9a) evaluated for the primary coherent fluctuations $({\tilde{\mathbf{u}}}^{\mathrm{I}}$, upper plots; ${\tilde{\mathbf{u}}}^{\mathrm{II}}$, lower plots) and the affiliated fluctuation intensity profiles at $x/H=$ (a) 0.41 and (b) 0.57 (lines are based on experiment 2 and markers are based on experiment 3). A reproduction of Fig. 5 is attached at the very bottom.

Figure 18

Terms of the coherent energy budget (9a) averaged across the transverse direction evaluated for ${\tilde{\mathbf{u}}}^{\mathrm{I}}$ at $x/H=$ (a) 0.41 and (b) 0.57; for ${\tilde{\mathbf{u}}}^{\mathrm{II}}$ at $x/H=$ (c) 0.07, (d) 0.24, (e) 0.41, and (f) 0.57; and for ${\tilde{\mathbf{u}}}^{\mathrm{III}}$ at $x/H=$ (g) 0.07 and (h) 0.24 (values are expressed as percentages of the respective total gains, based on experiment 2).

Figure 19

Terms of the coherent energy budget (9a) evaluated for the secondary coherent fluctuations and the affiliated fluctuations intensity profiles at (a) $x/H=0.24$ $({\tilde{\mathbf{u}}}^{\mathrm{III}-\mathrm{II}}$, the upper plot; ${\tilde{\mathbf{u}}}^{\mathrm{III}+\mathrm{II}}$, the lower plot) and (b) $x/H=0.57$ $({\tilde{\mathbf{u}}}^{\mathrm{II}-\mathrm{I}}$, the upper plot; ${\tilde{\mathbf{u}}}^{\mathrm{II}+\mathrm{I}}$, the lower plot, lines are based on experiment 2 and markers are based on experiment 3). A reproduction of Fig. 5 is attached at the very bottom.

Figure 20

Terms of the coherent energy budget (9a) averaged across the transverse direction evaluated at (a) $x/H=0.24$ for ${\tilde{\mathbf{u}}}^{\mathrm{II}-\mathrm{I}}$, (b) $x/H=0.57$ for ${\tilde{\mathbf{u}}}^{\mathrm{III}-\mathrm{II}}$, (c) $x/H=0.24$ for ${\tilde{\mathbf{u}}}^{\mathrm{II}+\mathrm{I}}$, and (d) $x/H=0.57$ for ${\tilde{\mathbf{u}}}^{\mathrm{III}+\mathrm{II}}$ (values are expressed as percentages of the respective total gains, based on experiment 2).

Figure 21

Components of the triadic production term of the coherent energy budget (9a) governing energy transfer between primary and secondary coherent fluctuations and the affiliated fluctuations intensity profiles evaluated at $x/H=$ (a) 0.24 (triads $\{{\tilde{\mathbf{u}}}^{\mathrm{II}},{\tilde{\mathbf{u}}}^{\mathrm{III}},{\tilde{\mathbf{u}}}^{\mathrm{III}-\mathrm{II}}\}$ and $\{{\tilde{\mathbf{u}}}^{\mathrm{II}},{\tilde{\mathbf{u}}}^{\mathrm{III}},{\tilde{\mathbf{u}}}^{\mathrm{III}+\mathrm{II}}\})$ and (b) 0.57 (triads $\{{\tilde{\mathbf{u}}}^{\mathrm{I}},{\tilde{\mathbf{u}}}^{\mathrm{II}},{\tilde{\mathbf{u}}}^{\mathrm{II}-\mathrm{I}}\}$ and $\{{\tilde{\mathbf{u}}}^{\mathrm{I}},{\tilde{\mathbf{u}}}^{\mathrm{II}},{\tilde{\mathbf{u}}}^{\mathrm{II}+\mathrm{I}}\})$. The integrals of the transverse profiles of the particular triadic production term's components shown in diagrammatic form for $x/H=$ (c) 0.24 and (d) 0.57 (based on experiment 2).

Figure 22

Comparison of velocity gradients' correlations calculated explicitly from the data and evaluated based on the local axisymmetry model (A1) (based on experiment 3).

Figure 23

OMD eigenvalues evaluated for different fields of view of experiment 1 on top of the enveloped PSD (i.e., maximum of PSD taken over the respective field of view).

Figure 24

Example of the triple decomposition: (a) total, coherent, and stochastic velocity fluctuations signals and (b) PDFs of the respective fluctuations' components (based on experiment 2).

Figure 25

PSDs of total $u_2^{\prime }$ (left-hand side) and residual $u_2^{\prime \prime }$ (right-hand side) transverse velocity fluctuations after (a) OMD-based coherent fluctuations ${\tilde{\mathbf{u}}}^{\mathrm{II}}$ and ${\tilde{\mathbf{u}}}^{\mathrm{III}}$ are removed at $x/H=0.07$, (b) phase-averaging-based coherent fluctuations ${\tilde{\mathbf{u}}}^{\mathrm{III}-\mathrm{II}}$ and ${\tilde{\mathbf{u}}}^{\mathrm{III}+\mathrm{II}}$ are removed at $x/H=0.24$ (based on experiment 1).

Figure 26

Mean profiles of absolute values of coherent vorticity fluctuations, resolved from different experiments, associated to (a) ${\tilde{\mathbf{u}}}^{\mathrm{II}}$ at $x/H=0.23,0.24$, (b) ${\tilde{\mathbf{u}}}^{\mathrm{III}}$ at $x/H=0.23,0.24$, (c) ${\tilde{\mathbf{u}}}^{\mathrm{III}-\mathrm{II}}$ at $x/H=0.23,0.24$, and (d) ${\tilde{\mathbf{u}}}^{\mathrm{III}+\mathrm{II}}$ at $x/H=0.22,0.24$.

Figure 27

(a) PSD of an amplitude signal $A^l$ on top of PSDs of associated total and coherent transverse velocity fluctuations and (b) PDF of an amplitude signal (based on experiment 2).

Figure 28

Measurements the of correlations between different components of the velocity fluctuations, see (D1), evaluated based on data acquired from experiment 2 at $x/H=$ (a) 0.07, (b) 0.24, (c) 0.41, and (d) 0.57 and from experiment 3 at $(x/H,y/H)=$ (e) $(0.22,-0.34)$, (f) $(0.23,-0.26)$, (g) $(0.23,-0.35)$, and (h) $(0.36,-0.37)$.

Figure 29

PDF and the cumulative PDF of a ratio between the convergence interval and the confidence interval evaluated for a normally distributed random variable.

Figure 30

Demonstration of convergence check: (a) evolution of $95\%$ convergence intervals around converging convection terms of energy budgets and (b) the final convergence interval around a production term (based on experiment 2).