TY - JOUR

T1 - Accurate numerical approximation of the absolute stability of unbounded flows

AU - Groot, Koen J.

AU - Schuttelaars, Henk M.

PY - 2020/1/15

Y1 - 2020/1/15

N2 - The initial stage of the laminar–turbulent transition of semi-infinite flows can be characterized as either an absolute or convective instability, naturally associated with localized wave packets. A convective instability is directly linked to an absolute instability in a different reference frame. Therefore, our aim is to determine the absolute stability of a flow in a given but arbitrary reference frame, which can only be directly inferred from the absolute eigenvalue spectrum. If advective processes are present, the associated absolute eigenfunctions grow exponentially in space in the advective direction. The eigenvalue spectrum is usually computed numerically, which requires truncating the domain and prescribing artificial boundary conditions at these truncation boundaries. For separated boundary conditions, the resulting spectrum approaches the absolute spectrum as the domain length tends to infinity. Since advective processes result in spatially exponentially growing eigenfunctions, it becomes increasingly difficult to represent these functions numerically as the domain length increases. Hence, a naive numerical implementation of the eigenvalue problem may result in a computed spectrum that strongly deviates from the (mathematically correct) absolute spectrum due to numerical errors. To overcome these numerical inaccuracies, we employ a weighted method ensuring the convergence to the absolute spectrum. From a physical point of view, this method removes the advection-induced spatial exponential growth from the eigenfunctions. The resulting (absolute) spectrum allows for a direct interpretation of the character of the pertinent perturbations and the eigensolutions can be used to construct and analyse the evolution of localized wave packets in an efficient way.

AB - The initial stage of the laminar–turbulent transition of semi-infinite flows can be characterized as either an absolute or convective instability, naturally associated with localized wave packets. A convective instability is directly linked to an absolute instability in a different reference frame. Therefore, our aim is to determine the absolute stability of a flow in a given but arbitrary reference frame, which can only be directly inferred from the absolute eigenvalue spectrum. If advective processes are present, the associated absolute eigenfunctions grow exponentially in space in the advective direction. The eigenvalue spectrum is usually computed numerically, which requires truncating the domain and prescribing artificial boundary conditions at these truncation boundaries. For separated boundary conditions, the resulting spectrum approaches the absolute spectrum as the domain length tends to infinity. Since advective processes result in spatially exponentially growing eigenfunctions, it becomes increasingly difficult to represent these functions numerically as the domain length increases. Hence, a naive numerical implementation of the eigenvalue problem may result in a computed spectrum that strongly deviates from the (mathematically correct) absolute spectrum due to numerical errors. To overcome these numerical inaccuracies, we employ a weighted method ensuring the convergence to the absolute spectrum. From a physical point of view, this method removes the advection-induced spatial exponential growth from the eigenfunctions. The resulting (absolute) spectrum allows for a direct interpretation of the character of the pertinent perturbations and the eigensolutions can be used to construct and analyse the evolution of localized wave packets in an efficient way.

KW - Absolute spectrum

KW - Advection-diffusion equation

KW - Numerical method

KW - Streamwise BiGlobal stability problem

KW - Symmetrization

UR - http://www.scopus.com/inward/record.url?scp=85076607439&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2019.132224

DO - 10.1016/j.physd.2019.132224

M3 - Article

AN - SCOPUS:85076607439

VL - 402

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

M1 - 132224

ER -