On determining characteristic length scales in pressure-gradient turbulent boundary layers

Abstract

In the present work, we analyze three commonly used methods to determine the edge of pressure gradient turbulent boundary layers: two based on composite profiles, the one by Chauhan et al. [“Criteria for assessing experiments in zero pressure gradient boundary layers,” Fluid Dyn. Res. 41, 021404 (2009)] and the one by Nickels [“Inner scaling for wall-bounded flows subject to large pressure gradients,” J. Fluid Mech. 521, 217–239 (2004)], and the other one based on the condition of vanishing mean velocity gradient. Additionally, a new method is introduced based on the diagnostic plot concept by Alfredsson et al. [“A new scaling for the streamwise turbulence intensity in wall-bounded turbulent flows and what it tells us about the ‘outer’ peak,” Phys. Fluids 23, 041702 (2011)]. The boundary layers developing over the suction and pressure sides of a NACA4412 wing section, extracted from a direct numerical simulation at chord Reynolds number Rec = 400 000, are used as the test case, besides other numerical and experimental data from favorable, zero, and adverse pressure-gradient flat-plate turbulent boundary layers. We find that all the methods produce robust results with mild or moderate pressure gradients, although the composite-profile techniques require data preparation, including initial estimations of fitting parameters and data truncation. Stronger pressure gradients (with a Rotta–Clauser pressure-gradient parameter β larger than around 7) lead to inconsistent results in all the techniques except the diagnostic plot. This method also has the advantage of providing an objective way of defining the point where the mean streamwise velocity is 99% of the edge velocity and shows consistent results in a wide range of pressure gradient conditions, as well as flow histories. Collapse of intermittency factors obtained from a wide range of pressure-gradient and Re conditions on the wing further highlights the robustness of the diagnostic plot method to determine the boundary layer thickness (equivalent to δ99) and the edge velocity in pressure gradient turbulent boundary layers.

Effect of adverse pressure gradients on turbulent wing boundary layers

“…Note that in this study we employ the edge velocity U e for outer scaling given the fact that in PG TBLs there is not a clear definition of the outer velocity scale (unlike in ZPGs, where the free-stream velocity is constant), since the streamwise pressure gradient leads to a mean velocity gradient for y n > δ 99 . Furthermore, Vinuesa et al (2016) argued that U e , obtained using the method reported in their study, constitutes a robust outer scale over a wide range of Reynolds numbers and pressure gradients.…”

“…First, the integral properties of the boundary layer such as pressure-gradient magnitude, friction Reynolds number (Re τ ), momentum-thickness Reynolds number (Re θ ) and others will be presented and assessed both on the suction side of the wing (denoted by the subscript ss) and the pressure side (denoted by the subscript ps). Note that Re τ = δ 99 u τ /ν and Re θ = U e θ /ν, where δ 99 and U e are the 99 % boundary-layer thickness and the mean velocity at the boundary-layer edge (both obtained using the method outlined by Vinuesa et al (2016)), and θ is the momentum thickness. Second, the turbulence statistics gathered from the simulation are shown.…”

“…First, note that the computation of β is started at, approximately, x/c = 0.2. The reason is that, as mentioned above, the calculation of δ 99 is based on the diagnostic scaling (Vinuesa et al 2016), which requires the turbulent intensity. Thus, it can only be used after the boundary layer becomes fully turbulent, i.e.…”

Revisiting History Effects in Adverse-Pressure-Gradient Turbulent Boundary Layers

“…Note that the linear region extends with increasing R e towards lower U / U e -values [23]. Regarding PG TBLs, Drózdz et al [48] and Vinuesa et al [39] showed that the diagnostic scaling also collapsed boundary layers subjected to a wide range of pressure-gradient conditions when introducing the shape factor on the left-hand-side of relation (10) as: . In Fig.…”

POD Analysis of Entropy Generation in a Laminar Separation Boundary Layer

“…Figure 7 shows the variation of boundary layer thickness, displacement boundary layer thickness, momentum boundary layer thickness and the shape factor along the streamwise. What should be noted that strong pressure gradient may lead to an inconsistent boundary layer edge by the common techniques to define the boundary layer edge [29]. The shape factor maintains about 2 to 3 in W1-3, where the boundary layer keeps laminar.…”