Orr-Sommerfeld Stability Equation Research Articles

On closures for reduced order models—A spectrum of first-principle to machine-learned avenues

For over a century, reduced order models (ROMs) have been a fundamental discipline of theoretical fluid mechanics. Early examples include Galerkin models inspired by the Orr–Sommerfeld stability equation and numerous vortex models, of which the von Kármán vortex street is one of the most prominent. Subsequent ROMs typically relied on first principles, like mathematical Galerkin models, weakly nonlinear stability theory, and two- and three-dimensional vortex models. Aubry et al. [J. Fluid Mech. 192, 115–173 (1988)] pioneered the data-driven proper orthogonal decomposition (POD) modeling. In early POD modeling, available data were used to build an optimal basis, which was then utilized in a classical Galerkin procedure to construct the ROM, but data have made a profound impact on ROMs beyond the Galerkin expansion. In this paper, we take a modest step and illustrate the impact of data-driven modeling on one significant ROM area. Specifically, we focus on ROM closures, which are correction terms that are added to the classical ROMs in order to model the effect of the discarded ROM modes in under-resolved simulations. Through simple examples, we illustrate the main modeling principles used to construct the classical ROMs, motivate and introduce modern ROM closures, and show how data-driven modeling, artificial intelligence, and machine learning have changed the standard ROM methodology over the last two decades. Finally, we outline our vision on how the state-of-the-art data-driven modeling can continue to reshape the field of reduced order modeling.

Calculation of the Orr-Sommerfeld stability equation for the plane Poiseuille flow

The stability of plane Poiseuille flow depends on eigenvalues and solutions which are generated by solving Orr-Sommerfeld equation with input parameters including real wavenumber and Reynolds number . In the reseach of this paper, the Orr-Sommerfeld equation for the plane Poiseuille flow was solved numerically by improving the Chebyshev collocation method so that the solution of the Orr-Sommerfeld equation could be approximated even and odd polynomial by relying on results of proposition 3.1 that is proved in detail in section 2. The results obtained by this method were more economical than the modified Chebyshev collocation if the comparison could be done in the same accuracy, the same collocation points to find the most unstable eigenvalue. Specifically, the present method needs 49 nodes and only takes 0.0011s to create eigenvalue while the modified Chebyshev collocation also uses 49 nodes but takes 0.0045s to generate eigenvalue with the same accuracy to eight digits after the decimal point in the comparison with , see [4], exact to eleven digits after the decimal point.

Stability in channel flow with fiber suspensions

The constitutive equation of fiber suspensions is established on the basis of fiber orientation tensors. The modified Orr-Sommerfeld stability equation is obtained further and numerically solved by aid of spectral method and finite difference method. The computational results of channel flow without fibers agree well with the experimental data with a higher degree of accuracy than previous numerical results. The results of the channel flow with fiber suspensions indicate that the presence of fibers attenuates the instability of flow, increases the critical Reynolds number, reduces the growth rate of perturbations and narrows the range of unstable waves. The extent of the effect of fibers on the flow stability is in direct proportion to the volume fraction and aspect-ratio of the fibers.

Transition Prediction for Three-Dimensional Boundary Layers in Computational Fluid Dynamics Applications

A method is presented for estimating the laminar/turbulent transition location in three-dimensional boundary layers for computational fluid dynamics (CFD) applications requiring numerous transition estimates with no user intervention. Given the Reynolds number and the Cp distribution, the location of transition is estimated based on the attachment-line state, the potential for relaminarization, the occurrence of laminar separation, and the growth of instabilities. Transition caused by instability is estimated based on N factors calculated for Tollmien-Schlichting waves and for stationary crossflow instabilities. A neural network is used (in place of solving the Orr-Sommerfeld stability equation) for determining the instability growth rates. The current version of the method assumes incompressible flow. The boundary-layer flow and instabilities are calculated based on an infinite-sweep (strip boundary layer) approximation; the instability calculations also employ the parallel-flow approximation

Computational aspects of the spectral Galerkin FEM for the Orr-Sommerfeld equation

Since Orszag's paper [‘Accurate solution of the Orr–Sommerfeld stability equation’, J. Fluid Mech., 50, 689–703 (1971)], most of the subsequent spectral techniques for solving the Orr–Sommerfeld equation (OSE) employed the Tau discretization and Chebyshev polynomials. The use of the Tau discretization appears to be accompanied by so-called spurious eigenvalues not related to the OSE and a singular matrix B in the generalized eigenvalue problem. Starting from a variational formulation of the OSE, a spectral discretization is performed using a Galerkin method. By adopting integrated Legendre polynomials as basis functions, the boundary conditions can be satisfied exactly for any spectral order and the non-singular matrices A and B are obtained in Ax=λBx. For plane Poiseuille flow, the stiffness and the mass matrices are sparse with bandwidths 7 and 5 respectively, and the entries can be calculated explicitly (thus avoiding quadrature errors) for any polynomial flow profile U. According to the convergence results [Hancke, ‘Calculating large spectra in hydrodynamic stability: a p FEM approach to solve the Orr–Sommerfeld equation’, Diploma Thesis, Swiss Federal Institute of Technology Zürich, Seminar for Applied Mathematics, 1998; Hancke, Melenk and Schwab, ‘A spectral Galerkin method for hydrodynamic stability problems’, Research Report No. 98-06, Seminar for Applied Mathematics, Swiss Federal Institute of Technology, Zürich], no spurious eigenvalue has been found. Numerical experiments with spectral orders up to p=600 illustrate the results. Copyright © 2000 John Wiley & Sons, Ltd.

Note on the Two-Point Boundary Value Numerical Solution of the Orr-Sommerfeld Stability Equation

Note on the Two-Point Boundary Value Numerical Solution of the Orr-Sommerfeld Stability Equation

The discrete temporal eigenvalue spectrum of the generalised Hiemenz flow as solution of the Orr-Sommerfeld equation

A spectral collocation method is used to obtain the solution to the Orr-Sommerfeld stability equation. The accuracy of the method is established by comparing against well documented flows, such as the plane Poiseuille and the Blasius Boundary layers. The focus is then placed on the generalised Hiemenz flow, an exact solution to the Navier-Stokes equations constituting the base flow at the leading edge of swept cylinders and aerofoils. The spanwise profile of this flow is very similar to that of Blasius but, unlike the latter case, there is no rational approximation leading to the Orr-Sommerfeld equation. We will show that if, based on experimentally obtained intuition, a nonrational reduction of the full system of linear stability equations is attempted and the resulting Orr-Sommerfeld equation is solved, the linear stability critical Reynolds number is overestimated, as has indeed been done in the past. However, as shown by recent Direct Numerical Simulation results, the frequency eigenspectrum of instability waves may still be obtained through solution of the Orr-Sommerfeld equation. This fact lends some credibility to the assumption under which the Orr-Sommerfeld equation is obtained insofar as the identification of the frequency regime responsible for linear growth is concerned. Finally, an argument is presented pointing towards potential directions in the ongoing research for explanation of subcriticality in the leading edge boundary layer.

Effect of Finite Thickness Flexible Boundary Upon the Stability of a Poiseuille Flow

The stability behavior, the stress, and velocity distributions for a plane Poiseuille flow bounded by a finite thickness elastic layer is studied. The analysis is performed by utilizing the coupled relationships between the Orr-Sommerfeld stability equation for the fluid and the Navier equations for the solid. The numerical instabilities experienced in the solution of the Orr-Sommerfeld equation have been overcome with the use of Davey’s orthonormalization technique. This study focuses only on the Tollimen-Schlichting instabilities. This mode is the most unstable of the three different types of instabilities. The results show that certain combinations of parameters can lead to improved stability conditions. Under these conditions the normal and shear stress distributions may behave completely different in certain regions of the fluid.

A modified tau spectral method that eliminates spurious eigenvalues

A modified tau spectral method is presented which eliminates the spurious eigenvalues produced by the usual tau method. The modified tau method essentially involves an appropriate factorization of the differential operators in the eigenvalue problem. It is developed for eigenvalue problems posed as single differential equations or as systems of such equations. Several eigenvalue problems are solved using both the usual and modified Chebyshev-tau methods, including the Orr-Sommerfeld stability equation for plane Poiseuille flow. The convergence of the modified tau method is shown to be at least as rapid as that of the usual tau method. The use of the tau coefficients in indicating convergence is also discussed.

Tau method approximate solution of highorder differential eigenvalue problems defined in the complex plane, with an application to orr‐sommerfeld stability equation

AbstractRecent new formulations of the Tau method have made it possible to apply it to a variety of new problems. In this paper we consider the numerical estimation of complex eigenvalues of high‐order differential equations by using the Tau method. The example considered is Orr–Sommerfeld's stability equation, for which there is a considerable literature on high‐accuracy numerical estimations. Our numerical results compare favourably with the most accurate ones reported in the recent literature.

On singular ordinary linear differential operators related to Orr-Sommerfeld stability equation

A usable, not technically complicated theory unlike that of Wasow of linear, singular, even order differential operators with general boundary conditions is proposed and the behaviour of eigenvalues and eigenfunctions is discussed in the limit as ε → 0, ε > 0. Completeness of the eigenvalue spectrum when the operator is formally self-adjoint is discussed and formal solutions are constructed and finally justification of the formal approximations is shown by proving the existence of actual solutions of the differential equation approximated by formal solutions. Also, modifications necessary for the consideration of the critical point are suggested. This theory is unlike that of Reid since it provides rigorous justification of the approximations and moreover, this theory is applicable to a large class of physical problems rewritten in such a way that they contain a small parameter.

Accurate solution of the Orr–Sommerfeld stability equation

The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772·22. It is explained why expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions.

On the divergent growth of molecular fluctuations in classical shear flow

A two-point hydrodynamical equation governing the fluctuation evolution in classical shear flows is derived via the B-B-G-K-Y procedure and a double series moment expansion. The resulting equation has a close similarity to Orr-Sommerfeld's stability equation.