Computation of compressible, laminar boundary layers on swept, tapered wings

Abstract

Nomenclature a = speed of sound C = normalized density-viscosity product, C p = pressure coefficient, 2(p -p x )/p x Ul, c = chord length c fx = local skin friction coefficient for circumferential flow, 2(/jLdu/dz) w /p e ul f = transformed circumferential component of vector potential, i^(0, r, z)/(2£r/r 0 ) 1/2 G = transformed radial component of vector potential (alternative), (v e /q e )g g = transformed radial component of vector potential, w> 2 (0, r, z)/r(2gr/r 0 ) l/2 v e H = total enthalpy M x = Mach number of undisturbed stream, UJa p = static pressure q e = resultant external velocity, r = radial distance along generator, measured from cone apex, O r () = radius of sphere centered at O, intersecting leading edge of streamwise section S = enthalpy parameter, HIH e -1 T = static temperature U x = velocity of undisturbed stream (w, v, w) = velocity components in (0, r, z) directions v = spanwise velocity, -v x = arc distance around surface, r 0 0 y = spanwise distance, r () -r z = distance normal to surface y = ratio of principal specific heat capacities of gas A = transformed boundary-layer thickness 8 * = displacement thickness of circumferential flow, S'o (1 -pulp e u e ) dz 77 = similarity variable used in present work, u e /f, p dz/V2fr/r 0 VK-C = similarity variable used by Kaups and Cebeci, Vv e /p e /z e r 0 Jg p dz 0 = polar angle in developed plane, measured from stagnation line A H A 2 = sweep angles of leading and trailing edges fji = coefficient of viscosity of gas f = transformed circumferential coordinate, So Pet*> e u e r () d0 p = mass density of gas o-= Prandtl number of gas

Automatic Transition Prediction for High-Lift Systems Using a Hybrid Flow Solver

“…The transition prediction module consists of a laminar boundarylayer code for swept, tapered wings 5 and an e N -database method for Tollmien-Schlichting (TS) instabilities (see Ref. 6).…”

Automatic Transition Prediction and Application to Three-Dimensional High-Lift Configurations

“…The first steps towards the setup of a RANS-based CFD tool with automatic transition prediction were made, for example, in [5], where a RANS solver and an e N method were applied and in [6], where a RANS solver, a laminar boundary-layer method [7], and an e N method were coupled. There, the boundary-layer method was used to produce highly accurate laminar, viscous layer data to be analyzed by a linear stability code.…”

Modeling of Natural Transition in Properly Three-Dimensional Flows

“…In addition to the high numerical accuracy needed for the second derivatives, stability in the numerical scheme for solving the boundary layer equations is also very important. Some well-established boundary layer calculation methods 12 have been found to suffer from unsuspected instabilities that cause severe spurious oscillations in second derivatives 13 . Nonetheless, a method for calculating three-dimensional, laminar boundary layers, developed by Horton 14 , has overcome the difficulties and has been designed specifically for the purpose of providing profile data for use in three-dimensional PSE computations on wings of arbitrary planform; it operates on data extracted from computed solutions of the RANS equations.…”