The extended GrtlerHmmerlin model for linear instability of three-dimensional incompressible swept attachment-line boundary layer flow

Abstract

A simple extension of the classic Görtler–Hämmerlin (1955) (GH) model, essential for three-dimensional linear instability analysis, is presented. The extended Görtler–Hämmerlin model classifies all three-dimensional disturbances in this flow by means of symmetric and antisymmetric polynomials of the chordwise coordinate. It results in one-dimensional linear eigenvalue problems, a temporal or spatial solution of which, presented herein, is demonstrated to recover results otherwise only accessible to the temporal or spatial partial-derivative eigenvalue problem (the former also solved here) or to spatial direct numerical simulation (DNS). From a numerical point of view, the significance of the extended GH model is that it delivers the three-dimensional linear instability characteristics of this flow, discovered by solution of the partial-derivative eigenvalue problem by Lin & Malik (1996a), at a negligible fraction of the computing effort required by either of the aforementioned alternative numerical methodologies. More significant, however, is the physical insight which the model offers into the stability of this technologically interesting flow. On the one hand, the dependence of three-dimensional linear disturbances on the chordwise spatial direction is unravelled analytically. On the other hand, numerical results obtained demonstrate that all linear three-dimensional instability modes possess the same (scaled) dependence on the wall-normal coordinate, that of the well-known GH mode. The latter result may explain why the three-dimensional linear modes have not been detected in past experiments; criteria for experimental identification of three-dimensional disturbances are discussed. Asymptotic analysis based on a multiple-scales method confirms the results of the extended GH model and provides an alternative algorithm for the recovery of three-dimensional linear instability characteristics, also based on solution of one-dimensional eigenvalue problems. Finally, the polynomial structure of individual three-dimensional extended GH eigenmodes is demonstrated using three-dimensional DNS, performed here under linear conditions.

Structural changes of laminar separation bubbles induced by global linear instability

“…Numerical method The results of the BiGlobal instability analysis have been verified by linear and nonlinear three-dimensional DNS monitoring the linear development of the disturbance velocity components associated with the stationary leading global mode. The DNS code is a modification of that of Lundbladh et al (1994) and has also been used by Theofilis et al (2003) in the verification of the polynomial modes of swept Hiemenz flow. It considers a velocity-vorticity formulation of the incompressible continuity and Navier-Stokes equations.…”

Linear global instability of non-orthogonal incompressible swept attachment-line boundary-layer flow

“…Lin & Malik (1997) went on to analyse by global linear stability theory the effect of streamwise curvature and concluded that it stabilizes the flow, thus offering additional motivation to analyse the stability of plane stagnation flows first. Theofilis et al (2003) also performed global instability analysis and DNS of the incompressible swept Hiemenz flow, fully confirming the existence of the sequence of the global modes predicted by Lin & Malik (1996), and proposed a polynomial model to describe the chordwise dependence of the amplitude functions of these modes. The polynomial model converts the partial differential equation (PDE)-based global linear instability analysis EVP into an ordinary differential equation (ODE)-based one, without loss of physical information in the linear regime.…”

“…However, instead of embarking upon parametric studies of instability by numerical solution of the partial-derivative EVP or by DNS, the question is addressed next whether it is possible to simplify the full system of equations describing linear stability by a polynomial model analogous with that proposed by Theofilis et al (2003) for the orthogonal case; attention is turned to this issue next.…”

“…Subsequently, a simple extension of the extended Görtler-Hämmerlin ordinary differential equation (ODE)-based polynomial model proposed by Theofilis et al (2003) for orthogonal flow, which includes previous models as special cases and recovers global instability analysis results, is presented for non-orthogonal flow. Direct numerical simulations have been used to verify the analysis results and unravel the limits of validity of the basic flow model analysed.…”

The linearized pressure Poisson equation for global instability analysis of incompressible flows

“…It should, however, be stressed that in the present matrix-forming context the issues arising from the splitting formulation of the incompressible equations of motion, where inappropriate use of pressure boundary conditions in the time-integration algorithm may violate mass conservation, are not directly related to the boundary conditions to close the linearized Navier-Stokes system. Pressure compatibility (PC) boundary conditions, derived from the momentum equations and collocated at the wall, have been shown to perform well in classic linear stability analysis [41] and their extension in two spatial dimensions is the natural candidate to provide the sought closure of the system of linearized equations of motion; indeed, these conditions have been used successfully in the global instability analysis of a number of well-studied incompressible flows [35,36,57,63,64].…”