Effects of upstream perturbations on the solution of the laminar and fully turbulent boundary layer equations with pressure gradients

Abstract

The aim of this work is to contribute to the understanding of sensitivity of boundary layers to the upstream boundary condition and history effects for both laminar and fully turbulent states in equilibrium conditions as well as some non-equilibrium turbulent boundary layers. Solutions of the two-dimensional boundary layer equations are obtained numerically for this study together with the RANS (Reynolds-averaged Navier-Stokes) approach for turbulence modeling. The external pressure gradient is imposed through an evolution of the external velocity of the form U e ∝ (x − x 0) m , and boundary layers are initialized from a profile giving a perturbed shape factor. It is found that laminar boundary layers require very long distances for convergence towards the non-disturbed profiles in terms of the initial boundary layer thickness (∼ 10 4 δ in) and that this distance is dependent on m. In turbulent boundary layers much shorter distances, although still large (∼ 10 2 δ in), are observed and they are also dependent on m. The maximum adverse pressure gradient for which convergence to a reference solution is possible is also studied finding that there is no limit for attached laminar boundary layers, whereas turbulent boundary layers do not converge once they are out of equilibrium. The convergence distances in turbulent boundary layers are also studied in terms of the turnover length (δU + e) because it has been shown to be more appropriate to refer the convergence distance to this length rather than the boundary layer thickness. The values for convergence using this criterion are extended to pressure gradient boundary layers. Moreover an equivalent criterion is proposed and studied for laminar boundary layers based on the viscous characteristic time.

Assessment of ZDES for WMLES of turbulent boundary layers with pressure gradient and mild flow separation

“…The behavior of SEM for studies of zero-pressure-gradient turbulent boundary layers is very satisfactory Deck et al [19]. Despite the numerous studies of the effects of upstream perturbations on turbulent boundary layers, some discrepancies where found when it comes to adverse pressure gradient conditions that led the authors to a prior study in order to identify whether disturbances of the boundary layer in such conditions would eventually disappear or, contrary to that, increase [20]. The results show that convergence to a reference state requires important distances of the order of 10 4 and 10 2 times the initial boundary layer thickness for laminar and turbulent boundary layers respectively.…”

Control of flow separation over an aerofoil by external acoustic excitation at a high Reynolds number

“…The shape factor is considered to be an indicator of the state of the boundary layer 33,34 . A strong adverse pressure gradient in the uncontrolled flow results in a rise in the shape factor, in agreement with the findings by Vaquero et al 33 .…”

“…The shape factor is considered to be an indicator of the state of the boundary layer 33,34 . A strong adverse pressure gradient in the uncontrolled flow results in a rise in the shape factor, in agreement with the findings by Vaquero et al 33 . The reason behind this is the B/L thickening due to an adverse pressure gradient, as illustrated by the significant rise in displacement thickness in Figure 17 downstream of the onset of flow separation.…”

Outer layer turbulence dynamics in a high-Reynolds-number boundary layer up to recovering from mild separation

“…An external pressure gradient may also lead to non-equilibrium conditions in the boundary layer. However, this is not necessarily the case since near-equilibrium conditions may be obtained for a given family of streamwise pressure gradient distributions (Rotta 1962;Mellor & Gibson 1966;Townsend 1976;Bobke et al 2017;Vaquero, Renard & Deck 2019b).…”