Pre-multiplied spanwise power spectra of velocity fluctuations for CP- and CPR-flows: (a,b) kzΦuu/UτS2, (c,d) kzΦvv/UτS2 and (e,f) kzФww/UτS2. Time evolution of instantaneous negative u structures (u/UτS) for CP flow in the xy plane. Contours of negative u structures in (left) xz plane (y+ = 7) and (right) yz plane for CP flow.
Distributions of the local convection velocity (Uc) at the tail of the DS during the coalescence process for the CP (solid lines) and CPR flows (dashed lines). The induced roll cell-like behavior by the POD mode of the artificially constructed VLSM patterns (a) in the trimetric view and (b) on the yz plane at x/h = 20πh: (i) CP- and (ii) CPR- flow (I). Linear estimates of the conditional velocity fields conditioned by a spanwise eddy strength event in the channel centerline (yr/h = 1) on the xy plane: (a) CP and (b) CPR flow.
List of Tables
Explanation of terms and abbreviations
Introduction
Similar to the results found in TBL with bar roughness, previous studies in fully developed turbulent channel flows (i.e., pure Poiseuille flows) with uniaxial bar roughness have shown failure of wall similarity in the outer layer (Leonardi et al. The failure of wall similarity in the outer layer for TBLs and unidirectional turbulent channel flows with bar roughness has been explained through the modification of turbulent structures by surface roughness. They reported that the reduction of Reynolds stresses in the outer part of the layer is attributed to weakened very large-scale motions (VLSM) and motions of rotating cells near the centerline.
In the present study, we investigate the modification of the overall process for the maintenance (or formation) of VLSMs and roll cell motions near the centerline in a CP flow by the surface roughness based on the previous approaches (especially from Toh & Itano 2005). ;Hwang et al. 2016; Lee et al. 2019).
Computational details
The role of large-scale ejection in the formation of VLSM shows that the two different approaches of the LSM chaining process and the co-support cycle are interrelated. In addition, Lee et al. based on synthetically fabricated VLSM patterns using a hairpin packet model. 2019) showed that the roll-cell patterns are a pure kinematic consequence of velocity induction within the VLSM and move to organize small-scale near-wall motions beneath the VLSM patterns, consistent with the previous observation of Toh & Itano (2005). Using data from DoS of a smooth-walled CP flow (Lee et al. 2018), we show that bottom-up and top-down interactions similar to the self-supporting cycle in turbulent Poiseuille flow are important for the spatial organization of VLSM and roll-cells near the centerline .
However, the dynamics related to the generation of the VLSMs and roll cell motions, both in the inner and outer layers in a CP flow, are affected by the surface roughness, leading to the weakened VLSMs and roll cells with associated reduction of the Reynolds. emphasizes. Below, a brief description of the computational approach for DoSs of turbulent Couette-Poiseuille flows over smooth and rough walls is given in § 2. After introducing an idealized scheme for the formation process of the VLSMs and roll cells, we investigate systematic modification of the inside-outside interactions for the formation of the VLSMs and roll cells by the surface roughness in § 3.
The two DNSs are performed independently of each other in a very long computational domain in the flow direction with a domain size (Lx, Ly, Lz) = (40πh, 2h, 6πh), which is sufficient to capture the longest structures for a CP flow ( Tsukahara et al. 2006). The transverse 2-D bars are periodically arranged in the flow direction only on the bottom wall, and the roughness height (k) is k/h = 0.12, similar to previous studies by Orlandi et al. 2008) in one-sided turbulent channel flows with bar roughness. The streamwise pitch (p) between the roughness elements is p/k = 8 to introduce strong inner and outer layer interaction with a maximum shape resistance (Leonardi et al. 2003; Lee & Sung 2007).
The friction velocity (Uτ) is directly estimated from the total drag, which is the sum of the averaged skin friction drags and the shapes averaged in the horizontal plane (Leonardi et al. 2003; Lee & Sung 2007). Further numerical details and validation of our DoS data are found in Lee et al.
Results and discussion
- Weakened VLSMs and roll-cell motions
- Top-down and bottom-up interactions
- Spanwise congregation motion by roll-cells
- Formation of a VLSM
- Formation of roll-cell motions
The inset in the upper left corner shows the Reynolds shear stress profiles in the outer coordinates for the CP and CPR. To relate the congregational motions for the negative u stripes (Figures 5 and 6) with a roll cell motion, the conditional correlation coefficient (R) with the streamwise velocity fluctuations (u) is defined as. For the CPR flow in Figure 7(b), the spatial characteristics of the correlation contours for R[wroll-cells, u] are similar to those for the CP flow.
This is another indication of the common nature of mergers leading to longer structures for the CPR stream. The time-evolving current view in Fig. 11 suggests that the weakened large-scale ejection for the CPR-flow is attributed to the small number of negative u-structures involved in the congregation motion (here there are five lines for the CP-flow and two steaks for the CPR-flow). As expected, the DS tail convective velocities for the CPR flow (dashed lines) are higher than those for the CP flow (solid lines) due to the weak large ejection for the CPR flow.
Specifically, the merger frequency of the LSMs (ρm) is estimated to be 0.317 and 0.223 for the CP and CPR currents. The conditionally averaged flow fields show that streamwise merging of the LSMs to form a VLSM is a robust feature in the CP flow. In Figure 14, the force of the roller cell movement for the CPR flow also decreases compared to that for the CP flow.
In the next section, we will show that the reduction in roll-cell power for CPR flow is attributed to a weakened VLSM. The estimated eigenvalues of the n = 1 modes for the CP- and CPR- flows are shown in Fig. 15. A direct comparison of VLSMs and roll-cells between the CP- and CPR- flows shows a weakened state of the structures for the CPR- flow, similar to our observation in Figs. 3 and 14.
In other words, the generation of the weakened VLSMs by the surface roughness for the CPR flow leads to the weakened rolling cells or very large-scale circulations.
Summary and conclusions
For the CP flow, the collective behavior of negative u structures in the near-wall region was found due to the circulation of rotating cells centered in the channel core. The introduction of surface roughness into the flow led to the suppression of the near-wall wide pile motions, and it was observed that strong pile motion occurs in the outer layer (y/h = 0.25) for the CPR flow. The formation process for VLSMs for CP flow was very similar to that observed in previous studies for turbulent channel/funnel and boundary layer flows: the VLSM is generated by the streamwise coupling of LSMs due to the change of convection velocity between upstream and downstream LSMs.
The slow speed of the tail of the downstream LSM was closely associated with the generation of a strong large-scale ejection, which was formed by the spanwise merging motions of the near-wall streaks. Although similar process for the formation of the VLSMs was observed for the CPR flow, the presence of a weak large-scale ejecta motion reduced the merger frequency between adjacent LSMs, resulting in the weakened VLSMs for the CPR flow. The weak large-scale ejection for the CPR flow was explained by the reduction in radius of the roll cell motion by the surface roughness.
Since the reduced roll-cell radius of motion caused the overall spanwise motion of a small number of negative u-structures in the outer layer, the strength of the large-scale ejection for the CPR flow became lower than that for the CP flow, resulting in a smaller difference in the convection velocity between the upstream and downstream LSM. POD analysis to extract the most dominant VLSM patterns for CP and CPR currents showed that the most energetic POD mode for both currents occurs at a range wavenumber of = 4, which is very similar to that between negative (or positive) VLSMs for CP- and CPR-flows in time-averaged flow fields. In addition, the reconstructed velocity fluctuation fields for the most energetic POD mode for both currents showed that wavy cell patterns are the most dominant.
The rotation cells were simply passive dynamic kinematic manifestations of aligned LSMs projecting to POD modes. Direct induction of rolling cell patterns from VLSMs showed that the weakened rolling cell-like patterns for CPR flow are a direct consequence of the weakened VLSM patterns.
2020 Direct numerical simulation of a turbulent Couette-Poiseuille flow, part 2: large and very large scale motions. 2005 Experimental investigation of the average velocity and turbulence characteristics of the planar Couette flow: effects with a low Reynolds number and a large longitudinal vortex structure. 2016 Comparison of turbulent boundary layers over smooth and rough surfaces up to high Reynolds numbers.
2006 DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region. 2012 Direct numerical simulation of a 30R long turbulent pipe flow at R+ = 685: large- and very large-scale motions.
APPENDIX
Open circle symbols indicate the wall normal reference locations and red cross symbols indicate vortex cores located in the upstream and downstream locations with respect to x/h = 0. Linear estimates of the conditional velocity fields conditioned by a positively signed eddy strength event in the channel centerline (yr/h = 1) on the xy plane: (a) CP and (b) CPR currents. The normal wall velocity (v) along the line y/h = 0.0 is shown in the 1D plot (red lines), and an estimated vortex pattern with its core radius (r0) is shown in the contours.
The increased eddy pattern in size for the CPR flow compared to the CP flow is consistent with the previous finding of Volino et al. on turbulent boundary layer flow with rod roughness.
ACKNOWLEDGEMENTS
