The Discrete Temporal Eigenvalue Spectrum of the Generalised Hiemenz Boundary Layer 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

On Linear And Nonlinear Instability in the Infinte Swept Attachment-Line Boundary Layer

On linear and nonlinear instability of the incompressible swept attachment-line boundary layer VASSILIOS THEOFILIS a1 a1 DLR, Institute for Fluid Mechanics, Division Transition and Turbulence, Bunsenstraße 10, D-37073 Göttingen, Germany Abstract The stability of an incompressible swept attachment-line boundary layer flow is studied numerically, within the Görtler–Hämmerlin framework, in both the linear and nonlinear two-dimensional regimes in a self-consistent manner. The initial-boundary-value problem resulting from substitution of small-amplitude excitation into the incompressible Navier–Stokes equations and linearization about the generalized Hiemenz profile is solved. A comprehensive comparison of all linear approaches utilized to date is presented and it is demonstrated that the linear initial-boundary-value problem formulation delivers results in excellent agreement with those obtained by solution of either the temporal or the spatial linear stability theory eigenvalue problem for both zero suction and a layer in which blowing is applied. In the latter boundary layer recent experiments have documented the growth of instability waves with frequencies in a range encompassed by that of the unstable Görtler–Hämmerlin linear modes found in our simulations. In order to enable further comparisons with experiment and, thus, assess the validity of the Görtler–Hämmerlin theoretical model, we make available the spatial structure of the eigenfunctions at maximum growth conditions. The condition on smallness of the imposed excitation is subsequently relaxed and the resulting nonlinear initial-boundary-value problem is solved. Extensive numerical experimentation has been performed which has verified theoretical predictions on the way in which the solution is expected to bifurcate from the linear neutral loop. However, it is demonstrated that the two-dimensional model equations considered do not deliver subcritical instability of this flow; this strengthens the conjecture that three-dimensionality is, at least partly, responsible for the observed discrepancy between the linear theory critical Reynolds number and the subcritical turbulence observed either experimentally or in three-dimensional numerical simulations. Further, the present nonlinear computations demonstrate that the unstable flow has its line of maximum amplification in the neighbourhood of the experimentally observed instability waves, in a manner analogous to the Blasius boundary layer. In line with previous eigenvalue problem and direct simulation work, suction is observed to be a powerful stabilization mechanism for naturally occurring instabilities of small amplitude

On the Spatial Structure of Global Linear Instabilities And Their Experimental Identification

The purpose of the present paper is to discuss the spatial structure of global instabilities, solutions of the partial derivative eigenvalue problem resulting from a nonparallel linear instability analysis of the incompressible Navier–Stokes and continuity equations, as developing upon four prototype essentially two-dimensional steady laminar flows. Theoretical knowledge of these eigendisturbances is instrumental to devising measurement techniques appropriate for their experimental recovery and ultimate control of laminar-turbulent transition mechanisms. Raising the awareness of the global linear flow eigenmodes contributes to redefining the boundaries between experimental observations which may be attributed to linear global as opposed to nonlinear mechanism

Boundary Layer Growth on a Rotating And Accelerating Sphere

Boundary-layer growth on a sphere is studied when it is set into motion with constant acceleration and constant angular velocity, the latter being normal to the former. Analytic expressions are derived for the velocity components of the incompressible fluid in terms of a power series of the time of motion as well as for the skin frictio

Massively Parallel Solution of the BiGlobal Eigenvalue Problem Using Dense Linear Algebra

Linear instability of complex flows may be analyzed by numerical solutions of partial-derivative-based eigenvalue problems; the concepts are, respectively, referred to as BiGlobal or TriGlobal instability, depending on whether two or three spatial directions are resolved simultaneously. Numerical solutions of the BiGlobal eigenvalue problems in flows of engineering significance, such as the laminar separation bubble in which global eigenmodes have been identified, reveal that recovery of (two-dimensional) amplitude functions of globally stable but convectively unstable flows (i.e., flows which sustain spatially amplifying disturbances in a local instability analysis context) requires resolutions well beyond the capabilities of serial, in-core solutions of the BiGlobal eigenvalue problems. The present contribution presents a methodology capable of overcoming this bottleneck via massive parallel solution of the problem at hand; the approach discussed is especially useful when a large window of the eigenspectrum is sought. Two separated flow applications, one in the boundary-layer on a flat plate and one in the wake of a stalled airfoil, are briefly discussed as demonstrators of the class of problems in which the present enabling technology permits the study of global instability in an accurate manner

On the Birth of Stall Cells on Airfoils

Critical point theory asserts that two-dimensional topologies are defined as degeneracies and any three-dimensional disturbance of a two-dimensional flow will lead to a new three-dimensional flowfield topology, regardless of the disturbance amplitude. Here, the topology of the composite flowfields reconstructed by linear superposition of the two-dimensional flow around a stalled airfoil and the leading stationary three-dimensional global eigenmode has been studied. In the conditions monitored the two-dimensional flow is steady and laminar and is separated over a fraction of the suction side, while the amplitudes considered in the linear superposition are small enough for the linearization assumption to be valid. The multiple topological bifurcations resulting have been analysed in detail; the surface streamlines generated by the leading stationary global mode of the separated flow have been found to be strongly reminiscent of the characteristic stall cells, observed experimentally on airfoils just beyond stall in both laminar and turbulent flow

Accurate Parabolic Navier-Stokes solutions of the supersonic flow around and elliptic cone

Flows of relevance to new generation aerospace vehicles exist, which are weakly dependent on the streamwise direction and strongly dependent on the other two spatial directions, such as the flow around the (flattened) nose of the vehicle and the associated elliptic cone model. Exploiting these characteristics, a parabolic integration of the Navier-Stokes equations is more appropriate than solution of the full equations, resulting in the so-called Parabolic Navier-Stokes (PNS). This approach not only is the best candidate, in terms of computational efficiency and accuracy, for the computation of steady base flows with the appointed properties, but also permits performing instability analysis and laminar-turbulent transition studies a-posteriori to the base flow computation. This is to be contrasted with the alternative approach of using order-of-magnitude more expensive spatial Direct Numerical Simulations (DNS) for the description of the transition process. The PNS equations used here have been formulated for an arbitrary coordinate transformation and the spatial discretization is performed using a novel stable high-order finite-difference-based numerical scheme, ensuring the recovery of highly accurate solutions using modest computing resources. For verification purposes, the boundary layer solution around a circular cone at zero angle of attack is compared in the incompressible limit with theoretical profiles. Also, the recovered shock wave angle at supersonic conditions is compared with theoretical predictions in the same circular-base cone geometry. Finally, the entire flow field, including shock position and compressible boundary layer around a 2:1 elliptic cone is recovered at Mach numbers 3 and

Numerical considerations in spectral multidomain methods for BiGlobal instability analysis of open cavity configurations

A novel approach for the solution of the viscous incompresible and/or compressible BiGlobal eigenvalue problems (EVP) in complex open cavity domains is discussed. The algorithm is based on spectral multidomain spatial discretization, decomposing space into rectangular subdomains which are resolved by spectral collocation based on Chebyshev polynomials. The eigenvalue problem is solved by Krylov subspace iteration. Here particular emphasis is placed on aspects of the parallel developments that have been necessary, on account of the high computing demands placed on the solver, as ever more complex “T-store” configurations are addressed

Linear instability analysis of incompressible flow over a cuboid cavity

Direct numerical simulations are performed to analyze the three-dimensional instability of flows over three-dimensional cavities. The flow structures at different Reynolds numbers are investigated by using the spectral-element solver nek5000. As the Reynolds number increasing, the lateral wall effects become more important, the recirculation zone shrinks, the front vortex increases and the flow structure inside of the cavity becomes more complex. Results show that the flow bifurcates from a steady state to an oscillatory regime beyond a value of Reynolds number Re = 1100

Molecular Dynamics Simulations of Couette flow

The first steps towards developing a continuum-molecular coupled simulations techniques are presented, for the purpose of computing macroscopic systems of confined fluids. The idea is to compute the interface wall-fluid by Molecular Dynamics simulations, where Lennard-Jones potential (and others) have been employed for the molecular interactions, so the usual non slip boundary condition is not specified. Instead, a shear rate can be imposed at the wall, which allows to obtain the properties of the wall material by means of an iterative method. The remaining fluid region will be computed by a spectral hp method. We present MD simulations of a Couette flow, and the results of the developed boundary conditions from the wall fluid interaction