Flow Statistics in the Transitional Regime of Plane Channel Flow

Abstract

The transitional regime of plane channel flow is investigated above the transitional point below which turbulence is not sustained, using direct numerical simulation in large domains. Statistics of laminar-turbulent spatio-temporal intermittency are reported. The geometry of the pattern is first characterized, including statistics for the angles of the laminar-turbulent stripes observed in this regime, with a comparison to experiments. High-order statistics of the local and instantaneous bulk velocity, wall shear stress and turbulent kinetic energy are then provided. The distributions of the two former quantities have non-trivial shapes, characterized by a large kurtosis and/or skewness. Interestingly, we observe a strong linear correlation between their kurtosis and their skewness squared, which is usually reported at much higher Reynolds number in the fully turbulent regime.

Transitional Channel Flow: A Minimal Stochastic Model

“…An additional criterion, the synchronous or asynchronous nature of the dynamics, has been found relevant to distinguish among them [27]. In the context of the present model, as soon as probability p 4 grows from zero, fully one-sided configurations previously reached after the termination of a (11,12) after [14]. The total turbulent fraction is F t (B) + F t (R) and the order parameter characterising the transition is abs(F t (B)−F t (R)).…”

“…where X ± represents what are now the densities B and R . The coefficients in (11) and (12) are then related to the probabilities introduced in the model as a ∝…”

“…The results of the mean-field approach, system (11) and (12), rephrased from [14], are depicted in Figure 14 (left). Upon variation of parameter c representing p 4 up to an unknown rescaling factor, all along the one-sided regime (c < c cr ), the total turbulent fraction is seen to decrease while the order parameter measuring the lack of symmetry similarly decreases to zero according to the usual Landau square-root law.…”

“…Of interest here, the transitional range of plane channel flow displays a remarkable series of steps at decreasing from large values where a regime of featureless turbulence prevails. It has been the subject of numerous studies and references to them can be found in the article by Kashyap, Duguet, and Dauchot in this special issue [ 12 ]; see also [ 13 ]. Our own observations based on numerical simulations are described in [ 14 , 15 ] and summarized in Figure 1 .…”

“…Other definitions involve the friction velocity , or the stream-wise speed averaged over the gap . They are related either empirically, vis., vs. , or theoretically, vis., to be used in particular for connecting to the work presented in [ 12 ], and some other articles. See [ 14 ] for details.…”