Impact of Deep Gaps on Laminar–Turbulent Transition in Compressible Boundary-Layer Flow

Abstract

Two-dimensional direct numerical simulations are used to study the impact of deep gaps on laminar-turbulent transition in compressible boundary-layer flow. For these, the gap depth-to-width ratio is always larger than five. They are located on a flat plate without pressure gradient. A steady base flow is used with a Mach number of 0.6, freestream temperature of 288 K, and free-stream pressure of 1 bar. Subsequently, Tollmien-Schlichting waves are introduced by suction and blowing at the wall, and their growth over the gap is evaluated by N factors. The influence of the gap on laminar-turbulent transition is quantified by the difference ΔN compared with the N factor obtained for a flat plate without gap. A periodic influence of the gap depth on ΔN is observed. In the direct numerical simulations, acoustic waves enter the gap and form a standing wave due to reflections, similar as occurring in organ pipes. The feedback of the standing wave on the boundary-layer flow above is essential for the observed ΔN variations. In a second case, the influence of a specific gap placed in front of a forward-facing step is studied as well. Here, a reduction of the N factor and hence a delay of transition, relative to the flow with step alone, are reached due to the presence of the gap. Nomenclature A = dimensionless amplitude A m = dimensionless maximum amplitude in vertical direction a = normalized amplitudẽ c = speed of sound, m∕s c p = heat capacity at constant pressure, J∕K c v = heat capacity at constant volume, J∕K d = dimensionless gap depth h = dimensionless step height L ref = reference length, m Ma = Mach number N = factor describing amplification of TS waves n = factor describing the amplification of a TS wave for one frequency Pr = Prandtl number p = dimensionless pressure p w = dimensionless wall pressure Re d = Reynolds number based on the gap depth r = receptivity factor t = time, s u = dimensionless horizontal velocity v = dimensionless vertical velocity w = dimensionless gap width x = dimensionless length in horizontal direction y = dimensionless length in vertical direction α = phase shift between waves γ = heat capacity ratio ΔN = deviation of an N factor from the N factor of the flat plate Δα = constant phase shift between standing wave and new TS wave δ = boundary-layer thickness, m δ 1 = displacement thickness, m λ = dimensionless wave length μ = dynamic viscosity, kg∕s · mρ = density, kg∕m 3 τ w = dimensionless wall shear stress ω = dimensionless circular frequency ∼ = dimensional quantities Subscripts an = antinode amplitude of standing wave dis = disturbances (fluctuations) of flow quantities do = values of downward traveling waves in the gap op = values at the gap opening res = amplitude of superposition of TS waves up = values of upward traveling waves in the gap ∞ = free-stream quantities

Effect of Small Surface Deformations on the Stability of Tollmien–Schlichting Disturbances

“…Gap deformations of variable depth and width were modeled for several freestream Reynolds numbers and a constant transonic Mach number. Deeper and wider gap indentations were found to amplify the growth of TS disturbances, agreeing with previous observations concerning the effects of surface deformations [14,21,42,43]. A correlation was established that related the variation in the N factor with the gap dimensions; this particular relationship was also found to give a reasonable estimate for ΔN variations for the realistic gap deformations considered by Ashworth et al [44].…”

“…where the subscript notations of gap and clean again refers to solutions over the deformed and nondeformed geometries. (A similar analysis was undertaken in the studies of other surface deformations [14,21,23,35,37,38,42].) Figure 9 displays N-factor variations for all of the deformations modeled in this study and for 10 −7 R ∞ 1.223.…”

“…Further studies on surface deformations have examined the effect of humps [29][30][31][32], dimples [33], corrugation [34], steps [35][36][37][38][39], roughness distributions [40,41], and deep gaps [42]. More recently, Xu et al [43] studied the impact of three-dimensional surface indentations on the destabilization of TS disturbances, where separation bubbles could form within sufficiently deformed indentations.…”

Experimental analysis of suction on step-induced boundary-layer transition

“…This refs. [18,19]. For cases with suction (starboard), however, 375 transition was shifted towards a more downstream location.…”

“…slits) downstream of a backward facing step at low Mach numbers (M < 0.1); numerical studies have been 45 conducted to examine suction through a gap upstream of a forward-facing step in refs. [17], [18] and [19]…”