Papers by Spencer J. Sherwin
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 : Selected Papers from the ICOSAHOM Conference, London, UK, July 9-13, 2018
AIAA SCITECH 2023 Forum, Jan 19, 2023
A recently developed computational framework for jet noise is used to compute the noise generated... more A recently developed computational framework for jet noise is used to compute the noise generated by an isolated and installed jet. The framework consists of two parts. In the first part, the spectral/hp element framework Nektar++ (www.nektar.info) is used to compute the near-field flow. Nektar++ solves the unfiltered Navier-Stokes equations on unstructured grids using the high-order discontinuous Galerkin method. The discrete equations are integrated in time using an implicit scheme based on the matrix-free Newton-GMRES method. In the second part, the Antares library (www.cerfacs.fr/antares/) is used to compute the far-field noise. Antares solves the Ffowcs Williams -Hawkings equation for a permeable integration surface in the time domain using a source-time dominant algorithm. The simulations are validated against experimental data obtained in the Doak Laboratory Flight Jet Rig, located at the University of Southampton. For the isolated jet, good agreement is achieved, both in terms of the flow statistics and the far-field noise. The discrepancies observed for the isolated jet are believed to be caused by an under-resolved boundary layer in the simulations. For the installed jet, the flow statistics are also well predicted. In the far-field, very good agreement is achieved for downstream observers. For upstream observers, some discrepancies are observed for very high and very low frequencies.
Computer Aided Design, Mar 1, 2016
With high-order methods becoming increasingly popular in both academia and industry, generating c... more With high-order methods becoming increasingly popular in both academia and industry, generating curvilinear meshes that align with the boundaries of complex geometries continues to present a significant challenge. Whereas traditional low-order methods use planar-faced elements, high-order methods introduce curvature into elements that may, if added naively, cause the element to self-intersect. Over the last few years, several curvilinear mesh generation techniques have been designed to tackle this issue, utilizing mesh deformation to move the interior nodes of the mesh in order to accommodate curvature at the boundary. Many of these are based on elastic models, where the mesh is treated as a solid body and deformed according to a linear or non-linear stress tensor. However, such methods typically have no explicit control over the validity of the elements in the resulting mesh. In this article, we present an extension of this elastic formulation, whereby a thermal stress term is introduced to 'heat' or 'cool' elements as they deform. We outline a proof-of-concept implementation and show that the adoption of a thermo-elastic analogy leads to an additional degree of robustness, by considering examples in both two and three dimensions.
Journal of Computational Physics, Feb 1, 2016
This study addresses linear dispersion-diffusion analysis for the spectral/hp continuous Galerkin... more This study addresses linear dispersion-diffusion analysis for the spectral/hp continuous Galerkin (CG) formulation in one dimension. First, numerical dispersion and diffusion curves are obtained for the advection-diffusion problem and the role of multiple eigencurves peculiar to spectral/hp methods is discussed. From the eigencurves' behaviour, we observe that CG might feature potentially undesirable non-smooth dispersion/diffusion characteristics for under-resolved simulations of problems strongly dominated by either convection or diffusion. Subsequently, the linear advection equation augmented with spectral vanishing viscosity (SVV) is analysed. Dispersion and diffusion characteristics of CG with SVV-based stabilization are verified to display similar non-smooth features in flow regions where convection is much stronger than dissipation or vice-versa, owing to a dependency of the standard SVV operator on a local Péclet number. First a modification is proposed to the traditional SVV scaling that enforces a globally constant Péclet number so as to avoid the previous issues. In addition, a new SVV kernel function is suggested and shown to provide a more regular behaviour for the eigencurves along with a consistent increase in resolution power for higher-order discretizations, as measured by the extent of the wavenumber range where numerical errors are negligible. The dissipation characteristics of CG with the SVV modifications suggested are then verified to be broadly equivalent to those obtained through upwinding in the discontinuous Galerkin (DG) scheme. Nevertheless, for the kernel function proposed, the full upwind DG scheme is found to have a slightly higher resolution power for the same dissipation levels. These results show that improved CG-SVV characteristics can be pursued via different kernel functions with the aid of optimization algorithms.
Computer Physics Communications, Jul 1, 2015
Nektar++ is an open-source software framework designed to support the development of highperforma... more Nektar++ is an open-source software framework designed to support the development of highperformance scalable solvers for partial differential equations using the spectral/hp element method. High-order methods are gaining prominence in several engineering and biomedical applications due to their improved accuracy over low-order techniques at reduced computational cost for a given number of degrees of freedom. However, their proliferation is often limited by their complexity, which makes these methods challenging to implement and use. Nektar++ is an initiative to overcome this limitation by encapsulating the mathematical complexities of the underlying method within an efficient C++ framework, making the techniques more accessible to the broader scientific and industrial communities. The software supports a variety of discretisation techniques and implementation strategies, supporting methods research as well as application-focused computation, and the multi-layered structure of the framework allows the user to embrace as much or as little of the complexity as they need. The libraries capture the mathematical constructs of spectral/hp element methods, while the associated collection of pre-written PDE solvers provides out-of-the-box application-level functionality and a template for users who wish to develop solutions for addressing questions in their own scientific domains.
Springer eBooks, 2005
There are three important steps in the computational modelling of any physical process: (i) probl... more There are three important steps in the computational modelling of any physical process: (i) problem definition, (ii) mathematical model, and (iii) computer simulation. The first natural step is to define an idealization of our problem of interest in terms of a set of relevant quantities which we would like to measure. In defining this idealization we expect to obtain a well-posed problem, this is one that has a unique solution for a given set of parameters. It might not always be possible to guarantee the fidelity of the idealization since, in some instances, the physical process is not totally understood. An example is the complex environment within a nuclear reactor where obtaining measurements is difficult. The second step of the modeling process is to represent our idealization of the physical reality by a mathematical model: the governing equations of the problem. These are available for many physical phenomena. For example, in fluid dynamics the Navier-Stokes equations are considered to be an accurate representation of the fluid motion. Analogously, the equations of elasticity in structural mechanics govern the deformation of a solid object due to applied external forces. These are complex general equations that are very difficult to solve both analytically and computationally. Therefore, we need to introduce simplifying assumptions to reduce the complexity of the mathematical model and make it amenable to either exact or numerical solution. For example, the irrotational (without vorticity) flow of an incompressible fluid is accurately represented by the Navier-Stokes equations but, if the effects of fluid viscosity are small, then Laplace's equation of potential flow is a far more efficient description of the problem.
Computers & Fluids, Oct 1, 2015
Boundary perturbations are considered as flow control forcing and their distributions are optimis... more Boundary perturbations are considered as flow control forcing and their distributions are optimised to suppress transient energy growth induced by the most energetic disturbances in the domain. For a given control cost (square integration of the control forcing), the optimal control calculated from the proposed
Journal of Fluid Mechanics, Jun 1, 2005
A straight tube with a smooth axisymmetric constriction is an idealized representation of a steno... more A straight tube with a smooth axisymmetric constriction is an idealized representation of a stenosed artery. We examine the three-dimensional instabilities and transition to turbulence of steady flow, steady flow plus an oscillatory component, and an idealized vascular pulsatile flow in a tube with a smooth 75 % stenosis using both linear stability analysis and direct numerical simulation. Steady flow undergoes a weak Coanda-type wall attachment and turbulent transition through a subcritical bifurcation, leading to hysteretic behaviour with respect to changes in Reynolds number. The pulsatile flows become unstable through a subcritical period-doubling bifurcation involving alternating tilting of the vortex rings that are ejected from the throat with each pulse. These tilted vortex rings rapidly break down through a self-induction mechanism within the confines of the tube. While the linear instability modes for pulsatile flow have maximum energy well downstream of the stenosis, we have established using direct numerical simulation that breakdown can gradually propagate upstream until it occurs within a few tube diameters of the constriction, in agreement with previous experimental observations. At the Reynolds numbers employed in the present study, transition is localized, with relaminarization occurring further downstream. A non-exhaustive investigation has also been undertaken into the receptivity of the axisymmetric shear layer in the idealized physiological pulsatile flow, with the results suggesting it has localized convective instability over part of the pulse cycle.
Journal of Fluid Mechanics, Jun 23, 2015
This study is focused on two-and three-dimensional incompressible flow past a circular cylinder f... more This study is focused on two-and three-dimensional incompressible flow past a circular cylinder for Reynolds number Re ≤ 1000. To gain insight into the mechanisms underlying the suppression of unsteadiness for this flow we determine the nonlinear optimal open-loop control driven by surface-normal wall transpiration. The spanwise-constant wall transpiration is allowed to oscillate in time although steady forcing is determined to be most effective. At low levels of control cost, defined as the square integration of the control, the sensitivity of unsteadiness with respect to wall transpiration is a good approximation of the optimal control. The distribution of this sensitivity suggests that the optimal control at small magnitude is achieved by applying suction upstream of the upper and lower separation points and blowing at the trailing edge. At high levels of wall transpiration, the assumptions underlying the linearised sensitivity calculation become invalid since the base flow is eventually altered by the size of the control forcing. The large-magnitude optimal control is observed to spread downstream of the separation point and draw the shear layer separation towards the rear of the cylinder through suction, while blowing along the centreline eliminates the recirculation bubble in the wake. We further demonstrate that it is possible to completely suppress vortex shedding in twoand three-dimensional flow past a circular cylinder up to Re = 1000, accompanied by 70% drag reduction when a nonlinear optimal control of moderate magnitude (with root mean square value 8% of the free stream velocity) is applied. This is confirmed through linearised stability analysis about the steady-state solution when the nonlinear optimal wall transpiration is applied. While continuously distributed wall transpiration is not physically realisable, the study highlights localised regions where discrete control strategies could be further developed. It also highlights the appropriate range of application of linear and nonlinear optimal control to this type of flow problem.
Journal of Biomechanics, Aug 1, 2011
The accuracy of the nonlinear one-dimensional (1-D) equations of pressure and flow wave propagati... more The accuracy of the nonlinear one-dimensional (1-D) equations of pressure and flow wave propagation in Voigt-type visco-elastic arteries was tested against measurements in a well-defined experimental 1:1 replica of the 37 largest conduit arteries in the human systemic circulation. The parameters required by the numerical algorithm were directly measured in the in vitro setup and no data fitting was involved. The inclusion of wall visco-elasticity in the numerical model reduced the underdamped high-frequency oscillations obtained using a purely elastic tube law, especially in peripheral vessels, which was previously reported in this paper Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J. Biomech. 40,[3476][3477][3478][3479][3480][3481][3482][3483][3484][3485][3486]. In comparison to the purely elastic model, visco-elasticity significantly reduced the average relative root-mean-square errors between numerical and experimental waveforms over the 70 locations measured in the in vitro model: from 3.0% to 2.5% ðp o 0:012Þ for pressure and from 15.7% to 10.8% ðp o 0:002Þ for the flow rate. In the frequency domain, average relative errors between numerical and experimental amplitudes from the 5th to the 20th harmonic decreased from 0.7% to 0.5% ðp o 0:107Þ for pressure and from 7.0% to 3.3% ðp o 10 À6 Þ for the flow rate. These results provide additional support for the use of 1-D reduced modelling to accurately simulate clinically relevant problems at a reasonable computational cost.
Journal of Computational Physics, Nov 1, 2018
This work focuses on the accuracy and stability of high-order nodal discontinuous Galerkin (DG) m... more This work focuses on the accuracy and stability of high-order nodal discontinuous Galerkin (DG) methods for under-resolved turbulence computations. In particular we consider the inviscid Taylor-Green vortex (TGV) flow to analyse the implicit large eddy simulation (iLES) capabilities of DG methods at very high Reynolds numbers. The governing equations are discretised in two ways in order to suppress aliasing errors introduced into the discrete variational forms due to the under-integration of non-linear terms. The first, more straightforward way relies on consistent/over-integration, where quadrature accuracy is improved by using a larger number of integration points, consistent with the degree of the non-linearities. The second strategy, originally applied in the high-order finite difference community, relies on a split (or skew-symmetric) form of the governing equations. Different split forms are available depending on how the variables in the non-linear terms are grouped. The desired split form is then built by averaging conservative and non-conservative forms of the governing equations, although conservativity of the DG scheme is fully preserved. A preliminary analysis based on Burgers' turbulence in one spatial dimension is conducted and shows the potential of split forms in keeping the energy of higher-order polynomial modes close to the expected levels. This indicates that the favourable dealiasing properties observed from split-form approaches in more classical schemes seem to hold for DG. The remainder of the study considers a comprehensive set of (under-resolved) computations of the inviscid TGV flow and compares the accuracy and robustness of consistent/over-integration and split form discretisations based on the local Lax-Friedrichs and Roe-type Riemann solvers. Recent works showed that relevant split forms can stabilize higher-order inviscid TGV test cases otherwise unstable even with consistent integration. Here we show that stable high-order cases achievable with both strategies have comparable accuracy, further supporting the good dealiasing properties of split form DG. The higher-order cases achieved only with split form schemes also displayed all the main features expected from consistent/over-integration. Among test cases with the same number of degrees of freedom, best solution quality is obtained with Roe-type fluxes at moderately high orders (around sixth order). Solutions obtained with very high polynomial orders displayed spurious features attributed to a sharper dissipation in wavenumber space. Accuracy differences between the two dealiasing strategies considered were, however, observed for the low-order cases, which also yielded reduced solution quality compared to high-order results.
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018
Springer eBooks, 2020
Modified Equation Analysis for the Discontinuous Galerkin Formulation
Lecture notes in computational science and engineering, 2015
ABSTRACT In this paper we present an assessment of the discontinuous Galerkin (DG) formulation th... more ABSTRACT In this paper we present an assessment of the discontinuous Galerkin (DG) formulation through modified equation analysis (MEA). When applied to linear advection, MEA can help to clarify wave-propagation properties previously observed in DG. In particular, a connection between MEA and dispersion-diffusion (eigensolution) analysis is highlighted. To the authors&#39; knowledge this is the first application of MEA to DG schemes, and as such this study focuses only on element-wise constant and linear discretizations in one dimension. For the linear discretization, we found that the physical mode&#39;s accuracy can be increased via upwinding. MEA&#39;s application to higher order solutions and non-linear problems is also briefly discussed. In special, we point out that MEA&#39;s applicability in the analysis of DG-based implicit large eddy simulations seems infeasible due to convergence issues.
International Journal for Numerical Methods in Engineering, Sep 17, 2020
We propose a rp-adaptation method for the simulation of compressible flows with shocks. The flow ... more We propose a rp-adaptation method for the simulation of compressible flows with shocks. The flow solver used is the inviscid version of the compressible module of the high-order hp/spectral framework Nektar++ [1].
Eigenspectral Analysis of Preconditioners in an Adaptive Compressible Flow Solver
Lecture notes in computational science and engineering, Nov 29, 2022
Numerical study on the effect of geometric approximation error in the numerical solution of PDEs using a high-order curvilinear mesh
arXiv (Cornell University), Aug 23, 2019
When time-dependent partial differential equations (PDEs) are solved numerically in a domain with... more When time-dependent partial differential equations (PDEs) are solved numerically in a domain with curved boundary or on a curved surface, mesh error and geometric approximation error caused by the inaccurate location of vertices and other interior grid points, respectively, could be the main source of the inaccuracy and instability of the numerical solutions of PDEs. The role of these geometric errors in deteriorating the stability and particularly the conservation properties are largely unknown, which seems to necessitate very fine meshes especially to remove geometric approximation error. This paper aims to investigate the effect of geometric approximation error by using a high-order mesh with negligible geometric approximation error, even for high order polynomial of order p. To achieve this goal, the high-order mesh generator from CAD geometry called NekMesh is adapted for surface mesh generation in comparison to traditional meshes with non-negligible geometric approximation error. Two types of numerical tests are considered. Firstly, the accuracy of differential operators is compared for various p on a curved element of the sphere. Secondly, by applying the method of moving frames, four different time-dependent PDEs on the sphere are numerically solved to investigate the impact of geometric approximation error on the accuracy and conservation properties of high-order numerical schemes for PDEs on the sphere.
Reducing errors caused by geometrical inaccuracy to solve partial differential equations with moving frames on curvilinear domain
Computer Methods in Applied Mechanics and Engineering, Aug 1, 2022
28th AIAA/CEAS Aeroacoustics 2022 Conference, Jun 13, 2022
In this work, the open-source spectral/hp element framework Nektar++ (www.nektar.info) is coupled... more In this work, the open-source spectral/hp element framework Nektar++ (www.nektar.info) is coupled with the Antares library (www.cerfacs.fr/antares/) to predict noise from a subsonic jet. Nektar++ uses the high-order discontinuous Galerkin method to solve the compressible Navier-Stokes equations on unstructured grids. Unresolved turbulent scales are modeled using an implicit Large Eddy Simulation approach. In this approach, the favourable dissipation properties of the discontinuous Galerkin method are used to remove the highest resolved wavenumbers from the solution. For time-integration, an implicit, matrix-free, Newton-Krylov method is used. To compute the far-field noise, Antares solves the Ffowcs Williams -Hawkings equation for a permeable integration surface in the time-domain using a source-time dominant algorithm. The simulation results are validated against experimental data obtained in the Doak Laboratory Flight Jet Rig, located at the University of Southampton.
Journal of Fluid Mechanics, Apr 11, 2022
The vortex dynamics of leading-edge vortices on plunging high-aspect-ratio (AR = 10) wings and ai... more The vortex dynamics of leading-edge vortices on plunging high-aspect-ratio (AR = 10) wings and airfoils were investigated by means of volumetric velocity measurements, numerical simulations, and stability analysis in order to understand the deformation of the leading-edge vortex filament and spanwise instabilities. The vortex filaments on both the wing and airfoil exhibit spanwise waves, but with different origins. The presence of a wing tip causes the leg of the vortex to remain attached to the wing upper surface, while the initial deformation of the filament near the wing-tip resembles a helical vortex. The essential features can be modelled as the deformation of initially L-shaped semi-infinite vortex column. In contrast, the instability of the vortices is well captured by the instability of counter-rotating vortex pairs, which are formed either by the trailing-edge vortices or the secondary vortices rolled-up from the wing surface. The wavelengths observed in the experiments and simulations are in agreement with the stability analysis of counter-rotating vortex pairs of unequal strength.
International Journal of Turbomachinery, Propulsion and Power, 2022
Reynolds-Averaged Navier–Stokes (RANS) methods continue to be the backbone of CFD-based design; h... more Reynolds-Averaged Navier–Stokes (RANS) methods continue to be the backbone of CFD-based design; however, the recent development of high-order unstructured solvers and meshing algorithms, combined with the lowering cost of HPC infrastructures, has the potential to allow for the introduction of high-fidelity simulations in the design loop, taking the role of a virtual wind tunnel. Extensive validation and verification is required over a broad design space. This is challenging for a number of reasons, including the range of operating conditions, the complexity of industrial geometries and their relative motion. A representative industrial low pressure turbine (LPT) cascade subject to wake passing interactions is analysed, adopting the incompressible Navier–Stokes solver implemented in the spectral/hp element framework Nektar++. The bar passing effect is modelled by leveraging a spectral-element/Fourier Smoothed Profile Method. The Reynolds sensitivity is analysed, focusing in detail on...
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Papers by Spencer J. Sherwin