Pressure-Gradient Turbulent Boundary Layers Developing Around a Wing Section

Abstract

A direct numerical simulation database of the flow around a NACA4412 wing section at R e c = 400,000 and 5∘ angle of attack (Hosseini et al. Int. J. Heat Fluid Flow 61, 117–128, 2016), obtained with the spectral-element code Nek5000, is analyzed. The Clauser pressure-gradient parameter β ranges from ≃ 0 and 85 on the suction side, and from 0 to − 0.25 on the pressure side of the wing. The maximum R e 𝜃 and R e τ values are around 2,800 and 373 on the suction side, respectively, whereas on the pressure side these values are 818 and 346. Comparisons between the suction side with zero-pressure-gradient turbulent boundary layer data show larger values of the shape factor and a lower skin friction, both connected with the fact that the adverse pressure gradient present on the suction side of the wing increases the wall-normal convection. The adverse-pressure-gradient boundary layer also exhibits a more prominent wake region, the development of an outer peak in the Reynolds-stress tensor components, and increased production and dissipation across the boundary layer. All these effects are connected with the fact that the large-scale motions of the flow become relatively more intense due to the adverse pressure gradient, as apparent from spanwise premultiplied power-spectral density maps. The emergence of an outer spectral peak is observed at β values of around 4 for λ z ≃ 0.65δ 99, closer to the wall than the spectral outer peak observed in zero-pressure-gradient turbulent boundary layers at higher R e 𝜃. The effect of the slight favorable pressure gradient present on the pressure side of the wing is opposite the one of the adverse pressure gradient, leading to less energetic outer-layer structures.

On determining characteristic length scales in pressure gradient turbulent boundary layers

“…The velocity profiles shown in Figure 3 are expressed in the direction tangential to the wing surface at each location, and the y coordinate is measured in the direction normal to the wall. Profiles are shown between x/c = 0.15 and 0.98, which according to Vinuesa et al [9] corresponds to a Reynolds number range 500 < Re θ < 2, 800. The pressure gradient can be quantified in terms of the so-called Rotta-Clauser pressure-gradient parameter β = −Δ/u τ dU e /dx t (where dU e /dx t is the derivative of the tangential edge velocity with respect to the tangential direction x t ), which is in the range 0 < β < 85 in this particular dataset.…”

“…All the methods are tested on turbulent boundary-layer profiles developing over the suction side of a NACA4412 wing profile, extracted from a recent DNS at Reynolds number 400,000 [9,10]. While the four methods provide robust results when the pressure gradient is mild or moderate, stronger pressure gradients (β > 7) lead to inconsistent results in all the techniques except the diagnostic plot one.…”

“…This database was obtained through direct numerical simulation (DNS), using the spectral element code Nek5000 [15], and a full description of the setup is given by Vinuesa et al [9]. This flow is a good test case to assess the merits and deficiencies of the various methods, due to the fact that the boundary layers developing over the wing are subjected to the effect of wall curvature, as well as pressure gradients.…”

“…One reason for the fallacy is attributed to the velocity gradient that is not necessarily zero for y > δ, i.e., U (y > δ) = constant as can be observed in Figure 1, which shows TBL profiles along the suction side of a wing [9,10]. Due to the relatively low Reynolds numbers Re and the strong pressure gradient conditions (where similarity laws usually fail) the edge of the boundary layer, its velocity U e (which is identical to the freestream velocity U ∞ in the case of ZPG TBLs), the boundary layer thickness δ, and the integral parameters (for which δ is needed to truncate the integrals), are all ill-defined.…”

Lossy Data Compression Effects on Wall-bounded Turbulence: Bounds on Data Reduction

“…6 (right), consists of 28,000 spectral elements with N = 11, which amount to a total of 48.4 million grid points. Additional simulation details, including the implementation in Nek5000, can be found in [15,27].…”