Nektar++: An open-source spectral/<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.gif" display="inline" overflow="scroll"><mml:mi>h</mml:mi><mml:mi>p</mml:mi></mml:math> element framework

Cantwell, Chris D.; Moxey, David; Comerford, Andrew; Bolis, A.; Rocco, G.; Mengaldo, Gianmarco; Grazia, Daniele De; Yakovlev, Sergiy; Lombard, J.-E.; Ekelschot, Dirk; Jordi, Bastien; Xu, Hanlin; Mohamied, Yumnah; Eskilsson, Claes; Nelson, Brent; Vos, Peter; Biotto, C.; Kirby, Robert M.; Sherwin, Spencer J.

doi:10.1016/j.cpc.2015.02.008

Cited by 469 publications

(345 citation statements)

References 39 publications

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“…A stiffly stable splitting scheme is adopted which decouples the velocity and pressure fields; time integration is achieved by a second-order accurate implicit-explicit scheme (Karniadakis, Israeli & Orszag 1991;Cantwell et al 2015).…”

Section: Numerical Strategy and Flow Configurationsmentioning

confidence: 99%

Destabilisation and modification of Tollmien–Schlichting disturbances by a three-dimensional surface indentation

Mughal

Gowree

et al. 2017

J. Fluid Mech.

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We consider the influence of a smooth three-dimensional (3-D) indentation on the instability of an incompressible boundary layer by linear and nonlinear analyses. The numerical work was complemented by an experimental study to investigate indentations of approximately 11δ 99 and 22δ 99 width at depths of 45 %, 52 % and 60 % of δ 99 , where δ 99 indicates 99% boundary layer thickness. For these indentations a separation bubble confined within the indentation arises. Upstream of the indentation, spanwise-uniform Tollmien-Schlichting (TS) waves are assumed to exist, with the objective to investigate how the 3-D surface indentation modifies the 2-D TS disturbance. Numerical corroboration against experimental data reveals good quantitative agreement. Comparing the structure of the 3-D separation bubble to that created by a purely 2-D indentation, there are a number of topological changes particularly in the case of the widest indentation; more rapid amplification and modification of the upstream TS waves along the symmetry plane of the indentation is observed. For the shortest indentations, beyond a certain depth there are then no distinct topological changes of the separation bubbles and hence on flow instability. The destabilising mechanism is found to be due to the confined separation bubble and is attributed to the inflectional instability of the separated shear layer. Finally for the widest width indentation investigated (22δ 99 ), results of the linear analysis are compared with direct numerical simulations. A comparison with the traditional criteria of using N-factors to assess instability of properly 3-D disturbances reveals that a general indication of flow destabilisation and development of strongly nonlinear behaviour is indicated as N = 6 values are attained. However N-factors, based on linear models, can only be used to provide indications and severity of the destabilisation, since the process of disturbance breakdown to turbulence is inherently nonlinear and dependent on the magnitude and scope of the initial forcing.

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Section: Numerical Strategy and Flow Configurationsmentioning

confidence: 99%

Destabilisation and modification of Tollmien–Schlichting disturbances by a three-dimensional surface indentation

Mughal

Gowree

et al. 2017

J. Fluid Mech.

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“…Eigenspectra obtained by the LPPE closure are compared with those delivered by two well-validated incompressible DNS codes, nek5000 [16] and nektar++ [10], both of which feature spectral element spatial discretization and time-stepping procedures for global instability analysis. In the case of the two-dimensional lid-driven cavity, the matrix-forming finite-element code FreeFEM++ is also employed to solve the BiGlobal EVP.…”

Section: Resultsmentioning

confidence: 99%

“…The issue of boundary conditions for a pressure perturbation that ensures flow incompressibility is typically absent in time-stepping methods, since it has been dealt with in the direct numerical simulation code used, e.g., by time-advancing velocity fields that satisfy the continuity equation, as done in the nektar++ [10,30], nek5000 [13,16] or Semtex [7] spectral element codes. The analogous procedure of weak formulation of the equations of motion, followed in the finite-element package FreeFEM++ [25], ensures that no issues exist regarding appropriate boundary conditions for the pressure perturbation, when this package is adapted to perform global linear instability analysis [12,17,32,53].…”

mentioning

confidence: 99%

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

Theofilis

2017

Theor. Comput. Fluid Dyn.

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The linearized pressure Poisson equation (LPPE) is used in two and three spatial dimensions in the respective matrix-forming solution of the BiGlobal and TriGlobal eigenvalue problem in primitive variables on collocated grids. It provides a disturbance pressure boundary condition which is compatible with the recovery of perturbation velocity components that satisfy exactly the linearized continuity equation. The LPPE is employed to analyze instability in wall-bounded flows and in the prototype open Blasius boundary layer flow. In the closed flows, excellent agreement is shown between results of the LPPE and those of global linear instability analyses based on the time-stepping nektar++, Semtex and nek5000 codes, as well as with those obtained from the FreeFEM++ matrix-forming code. In the flat plate boundary layer, solutions extracted from the two-dimensional LPPE eigenvector at constant streamwise locations are found to be in very good agreement with profiles delivered by the NOLOT/PSE space marching code. Benchmark eigenvalue data are provided in all flows analyzed. The performance of the LPPE is seen to be superior to that of the commonly used pressure compatibility (PC) boundary condition: at any given resolution, the discrete part of the LPPE eigenspectrum contains converged and not converged, but physically correct, eigenvalues. By contrast, the PC boundary closure delivers some of the LPPE eigenvalues and, in addition, physically wrong eigenmodes. It is concluded that the LPPE should be used in place of the PC pressure boundary closure, when BiGlobal or TriGlobal eigenvalue problems are solved in primitive variables by the matrix-forming approach on collocated grids.Keywords Global linear instability · Matrix-forming · Collocated grids · Pressure boundary conditions IntroductionGlobal linear instability theory is enjoying increasing acceptance in the fluid mechanics community, as also witnessed by the contents of this volume. The theory has permitted revealing previously unknown mechanisms responsible for laminar-turbulent flow transition in complex three-dimensional geometries with maximally one homogeneous spatial direction. Two main approaches are followed for the numerical work associated with modal and non-modal global linear instability analysis, namely matrix-forming vs. time-stepping, both of which have been discussed in a recent review [61]. The matrix-forming approach is interesting when dealing Communicated by Ati

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“…Numerical strategy A spectral/hp element discretisation, implemented in the Nektar++ package (Cantwell et al 2015), is used in this work to solve the linear as well as nonlinear NavierStokes equations. A stiffly stable splitting scheme is adopted, which decouples the velocity and pressure fields, and time integration is achieved by a second-order accurate implicit-explicit scheme (Karniadakis, Israeli & Orszag 1991).…”

Section: 3mentioning

confidence: 99%

Influence of localised smooth steps on the instability of a boundary layer

Lombard

Sherwin

2017

J. Fluid Mech.

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We consider a smooth, spanwise-uniform forward-facing step defined by a Gauss error function of height 4 %-30 % and four times the width of the local boundary layer thickness δ 99 . The boundary layer flow over a smooth forward-facing stepped plate is studied with particular emphasis on stabilisation and destabilisation of the twodimensional Tollmien-Schlichting (TS) waves and subsequently on three-dimensional disturbances at transition. The interaction between TS waves at a range of frequencies and a base flow over a single or two forward-facing smooth steps is conducted by linear analysis. The results indicate that for a TS wave with a frequency F ∈ [140, 160] (F = ων/U 2 ∞ × 10 6 , where ω and U ∞ denote the perturbation angle frequency and free-stream velocity magnitude, respectively, and ν denotes kinematic viscosity), the amplitude of the TS wave is attenuated in the unstable regime of the neutral stability curve corresponding to a flat plate boundary layer. Furthermore, it is observed that two smooth forward-facing steps lead to a more acute reduction of the amplitude of the TS wave. When the height of a step is increased to more than 20 % of the local boundary layer thickness for a fixed width parameter, the TS wave is amplified, and thereby a destabilisation effect is introduced. Therefore, the stabilisation or destabilisation effect of a smooth step is typically dependent on its shape parameters. To validate the results of the linear stability analysis, where a TS wave is damped by the forward-facing smooth steps direct numerical simulation (DNS) is performed. The results of the DNS correlate favourably with the linear analysis and show that for the investigated frequency of the TS wave, the K-type transition process is altered whereas the onset of the H-type transition is delayed. The results of the DNS suggest that for the perturbation with the non-dimensional frequency parameter F = 150 and in the absence of other external perturbations, two forward-facing smooth steps of height 5 % and 12 % of the boundary layer thickness delayed the H-type transition scenario and completely suppressed for the K-type transition. By considering Gaussian white noise with both fixed and random phase shifts, it is demonstrated by DNS that transition is postponed in time and space by two forward-facing smooth steps.

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Nektar++: An open-source spectral/hp element framework

Cited by 469 publications

References 39 publications

Destabilisation and modification of Tollmien–Schlichting disturbances by a three-dimensional surface indentation

Destabilisation and modification of Tollmien–Schlichting disturbances by a three-dimensional surface indentation

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

Influence of localised smooth steps on the instability of a boundary layer

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