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Turbulent Duct Flow Controlled with Spanwise Wall Oscillations

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Abstract

The spanwise oscillation of channel walls is known to substantially reduce the skin-friction drag in turbulent channel flows. In order to understand the limitations of this flow control approach when applied in ducts, direct numerical simulations of controlled turbulent duct flows with an aspect ratio of A R = 3 are performed. In contrast to channel flows, the spanwise extension of the duct is limited. Therefore, the spanwise wall oscillation either directly interacts with the duct side walls or its spatial extent is limited to a certain region of the duct. The present results show that this spanwise limitation of the oscillating region strongly diminishes the drag reduction potential of the control technique. We propose a simple model that allows estimating the achievable drag reduction rates in duct flows as a function of the width of the duct and the spanwise extent of the controlled region.

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Acknowledgements

This research was conducted using the resources provided by the Swedish National Infrastructure for Computing (SNIC) at High Performance Computing Center North (HPC2N) and at National Supercomputer Center (NSC) at Linköping University. Steffen Straub is grateful for the financial support of the Prof. Dr.-Ing. Erich Müller-Stiftung and the Dr.-Ing. Willy-Höfler-Stiftung.

Ricardo Vinuesa and Phlipp Schlatter acknowledge the support by the Knut and Alice Wallenberg Foundation and the Swedish Research Council (VR).

Bettina Frohnapfel acknowledges the support by the Alexander von Humboldt Foundation in form of a Feodor-Lynen-Fellowship.

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Authors and Affiliations

  1. Institute of Fluid Mechanics, Karlsruhe Institute of Technology, 76131, Karlsruhe, Germany

    Steffen Straub, Bettina Frohnapfel & Davide Gatti

  2. Linné FLOW Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Ricardo Vinuesa & Philipp Schlatter

Authors
  1. Steffen Straub
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  2. Ricardo Vinuesa
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  4. Bettina Frohnapfel
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  5. Davide Gatti
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Corresponding author

Correspondence to Steffen Straub.

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Appendix

Appendix

1.1 Validation

In order to validate the implementation of the oscillating wall, the DNS results of one uncontrolled and one controlled turbulent channel flow simulated in Nek5000 are compared against literature data from Moser, Kim and Mansour (MKM) [39] and results obtained with a a classic pseudo-spectral DNS code (abbreviated in the following with PSC) [30]. The profile of the mean streamwise velocity \(\overline {u}^{+}\) is shown in Fig. 13a. Very good agreement between the present results (c:0), data from MKM and results generated with the second code (PSC:0), is found for the uncontrolled case (lower curves). Also the controlled cases show very good agreement. Note that the results are scaled with the actual friction velocity of each flow [2].

Fig. 13
figure 13

Inner-scaled a mean streamwise velocity \(\overline {u}^{+}\), b RMS of streamwise velocity fluctuations, c RMS of wall-normal velocity fluctuations and d RMS of spanwise velocity fluctuations in the channel compared to literature references [39] (MKM) and results generated with a pseudo-spectral code [30] (PSC). Blue lines correspond to uncontrolled flows and purple lines to controlled flows with A + = 12

Full size image

The RMS values of the streamwise velocity fluctuations are presented in Fig. 13b. Here, the applied control technique causes a reduction of the inner-scaled streamwise velocity fluctuations close to the wall up to y + ≈ 30. A slight undershoot of the peak value for c:0 is visible for both controlled and uncontrolled cases which might be caused by a slightly insufficient spatial resolution [40].

Figure 13c shows the RMS of wall-normal velocity fluctuations. The agreement between uncontrolled channel cases c:0, MKM and CPL:0 is very good, as is the agreement between the two controlled cases c:12 and CPL:12. Excellent agreement is also found for the spanwise velocity fluctuations shown in Fig. 13d. The large fluctuations at the wall for the controlled cases are a result of the applied decomposition into mean and fluctuating values (see Section 2) since the value at the wall is determined by the prescribed oscillation amplitude. Finally, we compare the Reynolds stresses \(-\overline {u^{\prime } v^{\prime }}^{+}\) in Fig. 14 which shows reasonable agreement for all considered cases.

Fig. 14
figure 14

Inner-scaled Reynolds shear stress in the channel. See Fig. 13 for different line styles

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Straub, S., Vinuesa, R., Schlatter, P. et al. Turbulent Duct Flow Controlled with Spanwise Wall Oscillations. Flow Turbulence Combust 99, 787–806 (2017). https://doi.org/10.1007/s10494-017-9846-6

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