Görtler vortices in Falkner–Skan flows with suction and blowing

Abstract

In this paper, we use nonlinear calculations to study curved boundary-layer flows with pressure gradients and self-similar suction or blowing. For an accelerated outer flow, stabilization occurs in the linear region while the saturation amplitude of vortices is larger than for flows with a decelerating outer flow. The combined effects of boundary-layer suction and a favourable pressure gradient can give a significant stabilization of the flow. Streamwise vortices can be amplified on both concave and convex walls for decelerated Falkner-Skan flow with an overshoot in the velocity profile. The disturbance amplitude is generally lower far downstream compared with profiles without overshoot. the wall jet without freestream as described by Glauert [4]. Libby and Liu have shown that there exists at least four branches for adverse pressure gradients and Zaturska and Banks [5] found a new branch related to favourable pressure gradients. Solutions to the Falkner-Skan equation in the form of overshoot velocity profiles have also been included in the textbooks by White [6] and Sobey [7].Overshoot profiles are unstable on both concave and convex walls and streamwise vortices can develop. On curved walls the Görtler number is the appropriate parameter for the stability problem. The Görtler number is defined as Go = Re √ , the curvature parameter = /R, where R is the radius of curvature of the wall and = √ x/U ∞ is the boundary-layer thickness, where x is the streamwise coordinate and is the kinematic viscosity. The Reynolds number is defined as Re = U ∞ / , where U ∞ is the freestream velocity. Streamwise vortices in curved boundary layers have been studied frequently both numerically and experimentally, for a review see Hall [8], Saric [9], and Floryan [10]. However, studies of streamwise vortices in wall jet flows are more sparse. The first parallel neutral stability calculations were made by Kahawita [11] for the Glauert wall jet without freestream. Non-parallel linear stability theory was used by Floryan [12,13], followed by Matsson [14] who studied the influence of system rotation and self-similar suction or blowing on wall jets. Wadey [15] used asymptotic methods for linear theory to study Görtler vortices in wall jet flows for large spanwise wavenumbers. It was found that curved wall jets on both concave and convex walls were more stable to streamwise vortices than boundary-layer flows.An experimental study of streamwise vortices appeared for wall jet flow on a concave wall, see Matsson [16], followed by nonlinear simulations of the same flow by Le Cunff and Zebib [17]. The simulations were able to capture the primary instability of streamwise vortices in the experiments. They also found that for low Görtler numbers the disturbance amplitude initially could be increased but the amplitude further downstream attained an amplitude which was lower than the starting level. However, at higher Görtler numbers the streamwise vortices increased exponentially in amplitude followed by a maximum and an almost constant level furt...

Görtler vortices and streaks in boundary layer subject to pressure gradient: excitation by free stream vortical disturbances, nonlinear evolution and secondary instability

“…For a Falkner–Skan boundary layer, the local linear stability analysis of Ragab & Nayfeh (1980) indicates that an adverse pressure gradient is destabilising. This conclusion was confirmed by the marching method calculations with the upstream condition being taken to be an eigenmode or a more or less arbitrary disturbance velocity distribution (Goulpié et al 1996; Matsson 2008). Rogenski, de Souza & Floryan (2016) performed simulations using the linearised Navier–Stokes (N–S) equations, which require an artificial ‘buffer region’.…”

Effect of Pressure Gradient on the Development of Görtler Vortices

“…The results for the nonlinear development and saturation of Görtler vortices obtained by Matsson (2008) can be compared with the results obtained by Souza and his group (Rogenski et al 2013;2016a;2016b), who have been using direct numerical simulation (DNS) to study the development of Görtler vortices in boundary layers with pressure gradient. In a first paper Rogenski et al (2013) studied the effect of curvature variation on the nonlinear development of Görtler vortices as a way to control their growth and consequent transition to turbulence.…”

“…xx/xx 03/14 smaller wavenumber (β = 0.22) chosen by Matsson (2008) is more unstable than their higher wavenumber (β = 0.52), showing that the former is closer to the wavenumber associated with the maximum amplification. That pattern was the same for either suction/ blowing and pressure gradient, in a sense that suction and favorable pressure gradient are stabilizing.…”

“…Both concave and convex walls can be unstable for adverse pressure gradient if a base flow has an overshoot in the velocity profile. The base flow considered by Matsson (2008) was of the Falkner-Skan family including cases with overshoot similar to wall jet flows. Pressure gradient and suction and blowing change not only the boundary layer thickness, but also the lateral spreading of the characteristic mushroom-like structures of the Görtler vortices.…”

“…In other words, the vortices strength increases, but not their instability, as they remain coherent further downstream. Matsson (2008) associated this higher kinetic energy with instability, but in reality, maintaining the vortical structure without saturation or breakdown for a longer streamwise extent is actually associated with higher stability of the mushroom structure (secondary instability). Adverse pressure gradient results are more unstable on the linear region, saturate earlier and at a lower amplitude compared to the Blasius results.…”