Secondary flow in spanwise-periodic in-phase sinusoidal channels

Authors: Vidal, A., Nagib, H.M., Schlatter, P., Vinuesa, R.
Document Type: Article
Pubstate: Published
Journal: Journal of Fluid Mechancis
Volume: 851   288-316
Year: 2018


Direct numerical simulations (DNSs) are performed to analyze the secondary flow of Prandtl's second kind in fully developed spanwise-periodic channels with in-plane sinu-soidal walls. The secondary flow is characterized for different combinations of wave parameters defining the wall geometry at Re h = 2500 and 5000, where h is the half-height of the channel. The total cross-flow rate in the channel Q yz is defined along with a theoretical model to predict its behavior. Interaction between the secondary flow from opposite walls is observed if λ h A, where A and λ are the amplitude and wavelength of the sinusoidal function defining the wall geometry. As the outer-scaled wavelength (λ/h) is reduced the secondary vortices become smaller and faster, increasing the total cross-flow rate per wall. However, if the inner-scaled wavelength (λ +) is below 130 viscous units the cross-flow decays for smaller wavelengths. By analyzing cases in which the wavelength of the wall is much smaller than the half-height of the channel λ << h, we show that the cross-flow distribution depends almost entirely on the separation between the scales of the instantaneous vortices, where the upper and lower bounds are determined by λ/h and λ + , respectively. Therefore, the distribution of the secondary flow relative to the size of the wave at a given Re h can be replicated at higher Re h by decreasing λ/h and keeping λ + constant. The mechanisms that contribute to the mean cross-flow are analyzed in terms of the Reynolds stresses and using quadrant analysis to evaluate the probability density function of the bursting events. These events are further classified with respect to the sign of their instantaneous spanwise velocities. Sweeping events and ejections are preferentially located in the valleys and peaks of the wall, respectively. The sweeps direct the instantaneous cross-flow from the core of the channel towards the wall, turning in the wall-tangent direction towards the peaks. The ejections drive the instantaneous cross-flow from the near-wall region towards the core. This preferential behavior is identified as one of the main contributors to the secondary flow.