Parabolized Stability Equations Code with Automatic Inflow for Swept Wing Transition Analysis

Abstract

This paper aims at developing a parabolized stability equation transition analysis code coupled to the industrial inhouse accurate Reynolds-averaged Navier-Stokes code developed at Israel Aerospace Industries for stability analysis of three-dimensional boundary layers over swept wings. In the first step, the parabolized stability equation derivation in a body-fitted coordinate system is exposed, as well as its classical initialization, using the solution procedure of the global Chebyshev eigenvalue problem for the linear stability equations. This initialization procedure demands large user interactions, and it prevents the parabolized stability equation method to be of practical use in an industrial environment. Therefore, one exposes in the second step a new way to automatically generate the inflow solution of the linear stability equations needed by the parabolized stability equations. This new robust approach paves the way for innovative automatic industrial use of the parabolized stability equation technology without the need to involve the user. This approach supplies the inflow conditions required for the parabolized stability equation marching procedure without the need to solve the standalone linear stability global eigenvalue problem. The solution of the linear stability equations at initialization is now obtained by iteratively converging the local linear stability solver starting from the new inflow empirical approximation obtained in two special coordinate systems for crossflow and Tollmien-Schlichting, respectively. These formulas represent some invariant functions of the stability characteristics of the exact Navier-Stokes laminar boundary-layer profiles instead of self-similar boundary-layer profiles like the Falkner-Skan-Cooke. Standard academic validation test cases and industrial results of the parabolized stability equation/Reynolds-average Navier-Stokes codes on a generic unmanned aerial vehicle swept wing are exposed. Nomenclature c jk = entries of Jacobi transformation matrix, J c jk = entries of inverse Jacobi matrix, J −1 h = first Lamé coefficient K 1 = local curvature of the airfoil M, M ∞ = Mach number N = amplification factor (N-factor) Q ∞ = velocity vector at infinity R = Reynolds number U, V, W = velocity components along x, y, and z axes, respectively u k = velocity components in the Cartesian coordinate system w k = velocity components in the local curvilinear coordinate system x i = general curvilinear coordinates (where i is equal to 1, 2, or 3) x n , y n , z n = orthogonal normal to the leading-edge coordinate systemx s , y s , z s = orthogonal streamline-oriented coordinate system x,ỹ,z = orthogonal "tilde" coordinate system y i = Cartesian coordinate system (where i is equal to 1, 2, or 3) α = complex wave number in the longitudinal x direction α i = spatial amplification rate in the x direction α r = real wave number in the longitudinal x direction β = real wave number in the spanwise z direction β H = Hartree parameter Δ 0 = displacement thickness of the boundary layer in t...

Application of surrogate models to stability analysis and transition prediction in hypersonic flows

“…To cover the condition above, flow parameters for calculating training samples are selected at Mach numbers M∞=4.3: 0.2: 5.5, unit Reynolds number Re∞=1.4×10 7 : 0.3×10 7 : 3.5×10 7 and wall temperatures Tw=500: 50: 600, as listed in Table 3 and Fig. 11, which add up to 280 base flows.…”

“…In order to take into account flow transition in CFD simulations, a lot of transition prediction methods have been proposed. These methods include: 1) stability-theory-based e N method, which is a physics-based method, modeling the evolution of perturbations inside boundary layers, such as the local LST (linear stability theory) [2]- [5], Bi-global LST [6], PSE (parabolized stability equations) method [7] [8] and direct numerical simulation method [9] [10]. 2) Transition criteria, which utilize empirical criteria derived from theoretical and experiment study, such as AHD (Arnal-Habiballah-Delcourt) criteria [11], Gleyzes criteria [12] and C1 criteria [13].…”

Application of surrogate models to stability analysis and transition prediction in hypersonic flows

“…These methods include: 1) stability-theory-based e N method, which is a physics-based method modeling the evolution of perturbations inside boundary layers, e.g. the local linear stability theory (LST) [2][3][4][5], Bi-global LST [6], parabolized stability equations (PSE) method [7,8] and direct numerical simulation method [9,10]. 2) Transition criteria, which utilize empirical criteria derived from theoretical and experimental study, e.g.…”

Automatic Transition Prediction in a Navier–Stokes Solver Using Linear Stability Theory

“…Furthermore, Campbell and Lynde [10] showed a coupling of an unstructured code with different stability solvers for a laminar wing design framework. Kosarev et al [11] demonstrated the application of a coupling between RANS and a parabolized stability equation solver with automated input generation using an LST code.…”