Turbulent drag reduction by spanwise wall forcing. Part 1. Large-eddy simulations

Abstract

Turbulent drag reduction (DR) through streamwise travelling waves of the spanwise wall oscillation is investigated over a wide range of Reynolds numbers. Here, in Part 1, wall-resolved large-eddy simulations in a channel flow are conducted to examine how the frequency and wavenumber of the travelling wave influence the DR at friction Reynolds numbers $Re_\tau = 951$ and $4000$ . The actuation parameter space is restricted to the inner-scaled actuation (ISA) pathway, where DR is achieved through direct attenuation of the near-wall scales. The level of turbulence attenuation, hence DR, is found to change with the near-wall Stokes layer protrusion height $\ell _{0.01}$ . A range of frequencies is identified where the Stokes layer attenuates turbulence, lifting up the cycle of turbulence generation and thickening the viscous sublayer; in this range, the DR increases as $\ell _{0.01}$ increases up to $30$ viscous units. Outside this range, the strong Stokes shear strain enhances near-wall turbulence generation leading to a drop in DR with increasing $\ell _{0.01}$ . We further find that, within our parameter and Reynolds number space, the ISA pathway has a power cost that always exceeds any DR savings. This motivates the study of the outer-scaled actuation pathway in Part 2, where DR is achieved through actuating the outer-scaled motions.

On the relationship between manipulated inter-scale phase and energy-efficient turbulent drag reduction

“…Ai is Airy function of the first kind, and R {• • • } is the real part of the argument. Marusic et al (2021) and Rouhi et al (2023) have previously compared the P + in /P + o computed using w + (from their LES data) with that estimated from the GSL theory (A3), and obtained good agreement. Notably, the maximum difference was found to be less than 1 % for their OSA cases, while it was less than 3 % for the ISA cases.…”

“…Here, T osc = 2π/ω osc and A are the time period and amplitude of the spanwise oscillation, respectively; κ x = 2π/λ is the streamwise wavenumber of the travelling wave where λ is the wavelength; t denotes time; and u, v and w denote the velocity fluctuations along the streamwise (x), wall-normal (y) and spanwise (z) directions, respectively. This approach of imposing spanwise oscillations on the wall has been investigated extensively (refer to Ricco et al (2021) and references therein), predominantly for its ability to achieve significant drag reduction (DR) through actuating the near-wall inner-scaled motions having a characteristic viscous-scaled time scale of T + ≈ 100, which we refer to here as the inner-scaled actuation (ISA) strategy (Rouhi et al 2023). These investigations have mostly been limited to low-Reynolds-number flows, for which the inner scales are the dominant (if not sole) contributors to the total drag (Schoppa & Hussain 2002;Kim 2011).…”

“…These investigations have mostly been limited to low-Reynolds-number flows, for which the inner scales are the dominant (if not sole) contributors to the total drag (Schoppa & Hussain 2002;Kim 2011). Oscillating these wall elements at the associated small time scales (T + osc 100), however, incurs a large power cost (∝(T osc ) −(5/2) ; Quadrio & Ricco 2011), making net power savings from the ISA strategy less likely (Rouhi et al 2023). Considering the inverse relationship between power cost and oscillation time period, previous studies performed at low Reynolds numbers (Gatti & Quadrio 2016; some cases in Marusic et al 2021) have attempted to reduce skin-friction drag by targeting the outer scales, associated with large T + osc .…”