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Sven Westerbeek

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DeHNSSo

The Delft Harmonic Navier-Stokes Solver for Nonlinear Stability Problems with Complex Geometric Features

Journal article (2024) - S.H.J. Westerbeek, S.J. Hulshoff, H.M. Schuttelaars, M. Kotsonis

A nonlinear Harmonic Navier-Stokes (HNS) framework is introduced for simulating instabilities in laminar spanwise-invariant shear layers, featuring sharp and smooth wall surface protuberances. While such cases play a critical role in the process of laminar-to-turbulent transition, classical stability theory analyses such as parabolized or local stability methods fail to provide (accurate) results, due to their underlying assumptions. The generalized incompressible Navier-Stokes (NS) equations are expanded in perturbed form, using a spanwise and temporal Fourier ansatz for flow perturbations. The resulting equations are discretized using spectral collocation in the wall-normal direction and finite-difference methods in the streamwise direction. The equations are then solved using a direct sparse-matrix solver. The nonlinear mode interaction terms are converged iteratively. The solution implementation makes use of a generalized domain transformation to account for geometrical smooth surface features, such as humps. No-slip conditions can be embedded in the interior domain to account for the presence of sharp surface features such as forward- or backward-facing steps. Common difficulties with Navier-Stokes solvers, such as the treatment of the outflow boundary and convergence of nonlinear terms, are considered in detail. The performance of the developed solver is evaluated against several cases of representative boundary layer instability growth, including linear and nonlinear growth of Tollmien-Schlichting waves in a Blasius boundary layer and stationary crossflow instabilities in a swept flat-plate boundary layer. The latter problem is also treated in the presence of a geometrical smooth hump and a sharp forward-facing step at the wall. HNS simulation results, such as perturbation amplitudes, growth rates, and shape functions, are compared to benchmark flow stability analysis methods such as Parabolized Stability Equations (PSE), Adaptive Harmonic Linearized Navier-Stokes (AHLNS), or Direct Numerical Simulations (DNS). Good agreement is observed in all cases. The HNS solver is subjected to a grid convergence study and a simple performance benchmark, namely memory usage and computational cost. The computational cost is found to be considerably lower than high-fidelity DNS at comparable grid resolutions.@en

Linear and Nonlinear Stability Analysis of a Three-Dimensional Boundary Layer over a Hump

Conference paper (2023) - S.H.J. Westerbeek, Juan Alberto Franco Sumariva, Theodorus Michelis, Stefan Hein, M. Kotsonis

The Parabolized Stability Equations (PSE), Adaptive Harmonic Linearized Navier-Stokes (AHLNS) and Harmonic Navier-Stokes (HNS) solvers are used to analyze the linear and nonlinear stability of swept-wing boundary layers under the influence of smooth wall deformations of varying s ...

Direct numerical simulation of interaction between a stationary crossflow instability and forward-facing steps

Journal article (2022) - J. Casacuberta Puig, S. Hickel, S.H.J. Westerbeek, M. Kotsonis

The interaction between forward-facing steps of several heights and a pre-existing critical stationary crossflow instability of a swept-wing boundary layer is analysed. Direct numerical simulations (DNS) are performed of the incompressible three-dimensional laminar base flow and the stationary distorted flow that arise from the interaction between an imposed primary stationary crossflow perturbation and the steps. These DNS are complemented with solutions of the linear and the nonlinear parabolised stability equations, used towards identifying the influence of linearity and non-parallelism near the step. A fully stationary solution of the Navier-Stokes equations is enforced numerically, in order to isolate the mechanisms pertaining to the interaction of the stationary disturbance with the step. Results provide insight into the salient modifications of the base laminar boundary layer due to the step, and the response of the incoming crossflow instability to these changes. The fundamental spanwise Fourier mode of the disturbance field gradually lifts up as it approaches the step and passes over it. The flow environment around the step is characterised by a sudden spanwise modulation of the base-flow streamlines. Additional stationary perturbation structures are induced at the step, which manifest in the form of spanwise-aligned velocity streaks near the wall. Shortly downstream of the step, the fundamental component of the crossflow perturbation maintains a rather constant amplification for the smallest steps studied. For the largest step, however, the fundamental crossflow perturbation is stabilised significantly shortly downstream of the largest step. This surprising result is ascribed to a modulation of the kinetic energy transfer between the base flow and the fundamental perturbation field, which is brought forward as a new step interaction mechanism. Possible non-modal growth effects at the step are discussed. Furthermore, the results from DNS indicate significant amplification of the high-order harmonic crossflow components downstream of the step.

@en

Receptivity of crossflow instability to discrete roughness amplitude and location

Journal article (2022) - G. Zoppini, S.H.J. Westerbeek, D. Ragni, M. Kotsonis

The effect of discrete roughness elements on the development and breakdown of stationary crossflow instability on a swept wing is explored. Receptivity to various element heights and chordwise locations is explored using a combination of experimental and theoretical tools. Forcing configurations, determined based on linear stability predictions, are manufactured and applied on the wing in a low turbulence facility. Measurements are performed using infrared thermography, quantifying the transition front location, and planar particle image velocimetry, providing a reconstruction of stationary crossflow instabilities and their associated growth. Measurements are corroborated with simulations based on nonlinear parabolised stability equations. Results confirm the efficacy of discrete roughness elements in introducing and conditioning stationary crossflow instabilities. Primary instability amplitudes and resulting laminar-turbulent transition location are found to strongly depend on both roughness amplitude and chordwise location. The Reynolds number based on element height is found to satisfactorily approximate the initial forcing amplitude, revealing the importance of local velocity effects in non-zero-pressure gradient flows. Direct estimation of initial perturbation amplitudes from nonlinear simulations suggests the existence of pertinent flow mechanisms in the element vicinity, active in conditioning the onset of modal instabilities. Dedicated velocimetry planes, elucidate the development of a momentum deficit wake which rapidly decays downstream of the element followed by mild growth, representing the first experimental evidence of transient behaviour in swept wing boundary layers. The outcome of this work identifies a strong scalability of the transition dynamics to roughness amplitude and location, warranting the upscaling of roughness elements to more accessible, measurable and spatially resolved configurations in future experiments.

@en

Effect of streamwise surface undulations on the nonlinear stability of crossflow instabilities

Conference paper (2022) - S.H.J. Westerbeek, M. Kotsonis

The nonlinear stability of three-dimensional boundary layers over various undulated surfaces was calculated using the generalized Nonlinear Parabolized Stability Equations (NPSE). The results are compared with a flat plate configuration to assess the effect of the undulation s ...

Contributed

Maritime Drone Swarm

Bachelor thesis (2023) - D. Çelebi, M.N.L. de Bruijn, J. de Wit, P. Garcia de Vinuesa Garcia, O.P. Heukelom, J. Louro Fuentes, M.A. Rombouts, M.H. van Opstal, J.I.J. Verhoeff, M.J. Wolf, B.D.W. Remes, Bianca Giovanardi, S.H.J. Westerbeek

Non-Modal Growth Induced by Discrete Roughness Elements in a Swept Wing Boundary Layer

Modelling and Diagnostics

Master thesis (2022) - A.D. Sequeira, M. Kotsonis, G. Zoppini, S.H.J. Westerbeek

Laminar-to-turbulent boundary layer transition on the wings and stabilizers of aircraft leads to a large increase in the skin-friction drag they experience during flight. This has motivated prolific research into understanding and delaying the transition process, in an effort to improve the economic and ecological impact of air travel. The swept wing poses a particular challenge to this, as the transition scenario in flight is typically governed by the growth and breakdown of stationary crossflow instabilities (CFI). These are destabilized by flow conditions that would suppress instabilities on straight wings, such as Tollmien–Schlichting waves, complicating the use of laminar flow control techniques. Previous research has shown stationary CFI to be highly receptive to surface roughness, with the use of discrete roughness elements (DRE) having emerged as a forcing/control strategy. However, the mechanisms behind which DRE condition the onset of CFI in a swept wing boundary layer are still a topic of ongoing investigations.

The research presented in this Master thesis aims to further characterize the relation between the DRE forcing configuration and the onset of swept wing boundary layer instabilities, through an investigation of its wake flowfield. The DRE wake has previously been observed to be highly non-modal in nature, limiting the use of modal linear stability theory in predicting stationary disturbance evolution in the element vicinity. To address this, a linear, non-modal, parabolized stability framework is derived, capable of simulating transient growth within the DRE wake region. The numerical framework is initialized with experimental DRE wake data from a previous study, for a DRE array of critical forcing height. The predicted stability characteristics qualitatively agree with experiment, with the numerical solver successfully evolving the stationary DRE wake structure into stationary CFI downstream of the array. However, a quantitative match with experiment could not be attained. The results were also noted to be sensitive to the choice of initial conditions, as well as to the numerical discretization.

In addition to the stationary numerical simulations, hot-wire anemometry measurements are conducted at the TU Delft low turbulence wind tunnel, providing a first-time experimental characterization of the evolution of unsteady disturbances in the DRE wake for a swept wing boundary layer. Regions of high fluctuation intensity are observed to be localized around the top and sides of each element in the DRE array, with their spectral characteristics associated to circular cylinder shedding. For a critical DRE forcing, the evolution of the wake flow is largely governed by the stationary structures, although unsteady disturbance energy is observed to undergo a brief period of transient energy amplification. A super-critical DRE forcing introduces strongly tonal fluctuations into the DRE wake, whose amplification is enhanced by interaction between the stationary structures. The high fluctuation levels are sustained and grow to non-linear amplitudes, inhibiting the relaxation of steady disturbance energy. This leads to rapid, local turbulent breakdown in the vicinity of the elements.