Identifying invariant solutions with Nektar++

Stanisław Gepner (Warsaw University of Technology)

Co-authors: G. Kawahara

Several canonical flows are known to remain stable for arbitrary high values of the Reynolds numbers. In the case of those flows transition to turbulence is the result of the onset of equilibrium states, populating the phase space, rather than a cycle of bifurcations that takes the flow away from its laminar form. Those equilibria solutions influence dynamics of the flow by attracting the solution from the sufficiently perturbed laminar state and along their stable manifolds and then govern the dynamics of the flow evolution by ejecting the solution along the respective unstable trajectories. An important group of invariant states are those that lie on the boundary that separates the laminar and turbulent attraction basins. Such states form relative attractors on the boundary and manifest as simple or relative periodic orbits or travelling waves. Identification of such states is a worthwhile endeavour as it sheds light on the path the flow needs to take in order to cross the laminar-turbulent edge and in principle should allow for the formulation of control strategies with the goal of either accelerating or postponing the transition. In this work we present several Nektar++ based tools designed to aid in the identification of such states.

Suppressing buoyant instability using forcing based on receptivity

Abhishek Kumar (Coventry University)

By using receptivity as an external force, this study seeks to suppress the buoyant instability that arises in mixed baroclinic convection in a cavity. We consider a nearly semi-cylindrical cavity with an upper free surface, where fluid can be fed in, and porous lower boundaries, where fluid can escape. This cavity comprises a semicircular lower boundary, two adiabatic sidewalls and is infinitely extended in the third direction. This configuration describes problems involving melted solid materials, for example, in metallurgical casting processes. Our earlier linear stability analysis and direct numerical simulations show that the unstable modes are three-dimensional [1]. Next, we propose a mechanism to suppress these unstable modes. Our results demonstrate how the adjoint modes (receptivity) can be used as an external force to suppress the instability. There is a crucial role played by the amplitude and phase of the adjoint modes. We implement this time-dependent receptivity forcing in the spectral-element code NEKTAR++ [2].

References: [1] Kumar and Pothérat, J. Fluid Mech. 885, A40 (2020). [2] Moxey et al., Comput. Phys. Commun. 249, 107110 (2020).

Stability of flow in a 90-degree bend

Alexander Proskurin (Altai state technical university)

The presentation describes stability of two-dimensional flow in a channel that consists of straight inlet and outlet branches and a circular 90-degree curved bend. An incompressible viscous fluid flows through the elbow under the action of a constant pressure gradient between the inlet and outlet. Navier-Stokes equations were solved numerically using a high-fidelity spectral/hp element method. In a range of Reynolds numbers, an adaptive selective frequency damping method was used to get steady-state flow. It was found that three separation bubbles and vortex shedding can exist in the bend. The modal stability of two- and three-dimensional perturbations was investigated. The critical Reynolds number of the two-dimensional disturbances was found by extrapolation from lower Reynolds number results. It is much greater than the three-dimensional one, but the two-dimensional flow could be subcritically unstable with respect to the externally imposed small-amplitude white noise. For three-dimensional perturbations, the dependence of the critical Reynolds numbers on the bending radius was obtained. For the case of a moderate bending radius, a neutral curve is provided and eigenfunctions are studied in detail. Three-dimensional instability can be caused by a periodic or monotonically growing mode, and these unstable modes relate to the recirculation bubbles that occur after the bend.

Development of a Combustion Module within the Nektar++ Framework

Aiden Forknall (Loughborough University)

Co-authors: Jialin Su, Andrew Garmory

In this project combustion modelling methods will be implemented into the framework of Nektar++, a spectral/hp element solver. The aim is to create a reactive multiphase flow solver capable of accurately modelling combustion flow regimes at low Mach numbers. To date, a pseudo-compressible multiphase flow solver has been developed based on the Incompressible Navier-Stokes code provided by Nektar++. The solver uses a low Mach variable density approach, able to simulate mixing between fuel and oxidizer
phases. Validations have been performed with a canonical turbulent flow case and combustor relevant geometry such as cold flow of Sandia Piloted CH4/Air Flames. Simulations are LES, polynomial order 4, and have been completed using Lovelace and Sulis. For modelling of reactive flow, a 1D flamelet generated manifold (FGM) formulation dependant on mixture fraction, Z, is currently in development. Further FGM expansion is expected to include dependences on the variance of mixture fraction, Z’’, and if time permits, the progress variable, c, and its variance, c’’. The completed solver will be validated with canonical flame datasets and assessed against relatively ‘low’ and ‘high’ order reacting flow codes. Subsequently, the developed program will be applied to setups that are more representative of future combustors.

High-Order Mesh Generation with NekMesh + Demo

Kaloyan Kirilov (Imperial College London)

High-fidelity spectral/hp element simulations require high-order, geometry conforming curvilinear meshes. This poses a significant challenge when arbitrary geometries are considered. Therefore, we will present the main capabilities of our high-order mesh generator, NekMesh. First, we will show our main mesh generation pipeline – from CAD to valid high-quality curvilinear mesh. This will be followed by an alternative industrial pipeline suitable for very complex geometries. Last, a demonstration of this pipeline will be performed and the current work on NekMesh will be outlined.

Project NEPTUNE: Towards exascale fusion simulations with Nektar++

Bin Liu (Imperial College London) / Mashy Green (Kings College London)

The study of the anisotropic heat transport in magnetized plasma is fundamental to the stability and control of thermonuclear fusion process in tokamak reactors. Currently, tokamak is the leading candidate for a practical fusion reactor, using the magnetic confinement approach to produce controlled thermonuclear fusion power. The high-order spectral element method with highly accurate high-order elements is proven to be a promising numerical approach to accurately approximate the strong thermal anisotropy in the magnetized plasma. In this talk, extensions to NekMesh, the mesh generation tool offered as part of the Naktar++ framework, are discussed. These included methods to generate accurate high-order anisotropic meshes that incorporate domains with internal structures defined in CAD are discussed. Additional mesh control features are introduced to offer fine tuning of refinement and anisotropy directed at the internal CAD structures. These are done by utilizing techniques originally developed for aeronautical problems such as r-adaption and iso-parametric boundary layer generation. Furthermore, in order to advance Nektar++ towards the next generation of HPC and the exascale era required for feasible fusion simulations, a highly efficient matrix-free Helmholtz operator with single-instruction multiple-data (SIMD) vectorization is implemented in Nektar++. The details of the mathematical derivation and their performance are presented and compared with the matrix-based operators.

Direct numerical simulation of the turbulence characteristics in a cylinder wake

Hongyi Jiang (University of Western Australia)

The primary vortex street, turbulent kinetic energy and energy dissipation rate in the wake of a circular cylinder are examined at a Reynolds number of 1000 and up to 120 cylinder diameters downstream of the cylinder. The turbulence characteristics are quantified using direct numerical simulation (based on the framework of Nektar++), which provides a comprehensive dataset that is almost impossible to acquire from physical experiments. The energy dissipation rate is decomposed into the components due to the mean flow, the coherent primary vortices and the remainder. It is found that the remainder component, which develops only in a three-dimensional turbulent wake, accounts for the majority of the total dissipation rate for almost the entire wake. The turbulence dissipation in the wake is largely locally homogeneous, but not locally isotropic or axisymmetric, even after the annihilation of the primary vortex street. The spatial distribution of the turbulence characteristics relative to the spatial distribution of the primary and streamwise vortices is also examined.

Interference and ground effects on flow past two inclined flat plates in tandem arrangement

Yifeng Al (Hong Kong University of Science and Technology)

Co-authors: Lei Zhou, Kam Tim Tse, Hongfu Zhang

Direct Numerical Simulation (DNS) was conducted on the flow past two tandem inclined flat plates at a Reynolds number of 150. With an inclined angle θ=25°, the high-order spectral-element method is employed for higher accuracy. Fluid force, power spectral density analysis, flow structures, and instability have been investigated focusing on both interference and ground effects. The gap spacing ratio was selected as X/L=1 to 6 at intervals of 1 and the plate height was chosen as G/L=0.2 to 1.6 at intervals of 0.2 in the vicinity of the ground. It was found that two asymmetrical vortices formed behind the upstream plate (UP), and then directly reattached to the downstream plate (DP), where larger ratios of X/L and G/L enhanced the vortex shedding behind the plates and magnify the fluid forces. A small gap spacing X/L was found not sufficient to develop the gap flow, where tandem plates were considered as a single body of vortex shedding. With G/L lower than 0.5, the vorticity generated at the trailing edge was gradually neutralized by the boundary shear layer which contributes to lower fluid force. Moreover, the instability of tandem plates was governed by the gap spacing ratio.

Flow transition of low Reynolds number flows past a plunging airfoil

Ankang Gao (University of Science and Technology of China)

Co-authors: C. D. Cantwell, S. J. Sherwin

The two-dimensional to three-dimensional transition of a flow past a plunging NACA 0012 airfoil at Reynolds numbers from Re=400 to 10 000 and an angle of attack of 15 degrees was investigated using global linear stability analysis and spanwise-homogeneous direct numerical simulation (DNS). Below a Strouhal number of 0.5, the transition Reynolds number is higher than the static airfoil which indicates the plunging motion stabilises the two-dimensional baseflow. For higher frequencies, an period-doubling mode first becomes unstable, which has a peak Floquet multiplier around a spanwise wavelength of 2c. This unstable mode also dominates in three-dimensional direct numerical simulations (DNS). Finally, a short-wave mode becomes unstable at St_c=0.95, which generates more small-scale vorticies in the DNS result.

Using Nektar++ to Predict the Noise from Isolated and Installed Jets

Daniel Lindblad (Imperial College London)

Ever since jet airliners were introduced in the 1950s, jet noise has been a major concern for the airline industry. Today, the amount of noise produced by the turbulent jet is considerably lower than it was 70 years ago. Despite this, jet noise is still the dominant source of aircraft noise during take-off. Therefore, further reductions in jet noise are necessary to meet increasingly stringent noise certification tests, and thereby reduce the negative environmental impacts of aircraft noise. To date, most reductions in jet noise have been realized by increasing the diameter of the engine. This makes it possible to reduce the velocity of the jet, while at the same time keeping the thrust constant. A lower velocity does in turn lead to lower noise levels and better fuel efficiency. However, larger engines must also be placed closer to the wing in order to provide enough ground clearance. This leads to a new source of aircraft noise as the turbulent jet impinges on the wing: installed jet noise. In this talk, I will present some recent efforts on modeling isolated and installed jet noise using the high-order Nektar++ code.

High fidelity compressible flow simulations of a turbo-machinery geometrical configuration using nektar++

João Isler (Imperial College London)

High fidelity numerical simulations at high Reynolds number, although extremely expensive, are nowadays possible due to the efficiency of new algorithms on highly parallel systems. Among the computational fluid dynamics methodologies employed, high-order methods are becoming increasingly popular since the industrial sector strives to increase fidelity and accuracy for design purposes while retaining computational efficiency. To this end, the well-known highly scalable spectral/hp element framework Nektar++ is used. This approach therefore brings the possibility to have access to a wide range of spatial and temporal scales in turbulent flows and also study instability mechanisms of shear flows with a lower computational cost. In this work, we investigated compressible flow simulations in an engine intake at high Reynolds numbers in absence of transonic effects and under strong adverse pressure gradients by means of computational simulations. Computational resource savings were, in this study, achieved by reducing the computational domain, so that, although only a small part of the domain was computed, it was possible to reproduce the physical behaviour expected in the whole computational domain. This methodology combined with the high-order methods demonstrated to be highly efficient and drastically reduced the computational time to perform the three-dimensional compressible flow simulations.

TBC

Hari Sundar (University of Utah)

TBC

Implementation of the finite in time integration scheme

Allen Sanderson (University of Utah)

In this short talk will describe the implementation of the Finite in Time (FIT) Integration Scheme into Nektar++. The scheme is a fast convolution algorithm for computing solutions to (Caputo) time-fractional differential equations. It is an explicit solver that expresses the solution as an integral over a Talbot curve, which is discretized with quadrature. First-order quadrature is currently implemented.

An Efficient Implicit Solver for Incompressible flow

Henrik Wustenberg (Imperial College London)

Under-resolved Direct Numerical Simulation (uDNS) offers highly resolved insight into complex and unsteady flow phenomena such as vortices. The numerical algorihtms of uDNS however are difficult to stabilise whilst maintaining a high computational efficiency. In particular for industrial applications, complex geometries provide challenging test cases for a uDNS solver and significantly decrease the overall efficiency through numerical instabilities. Implicit time- stepping increases the stability of the algorithm, however, at larger computational costs. This work explores the potential of implicit time-stepping using Velocity-Correction schemes. While the cost for each time step increase, implicit schemes lift the CFL restriction for the time step and allow balancing temporal and spatial discretisation errors to achieve an efficient solution. At this stage of the project, a linearly implicit Velocity-Correction scheme based on Dong and Shen (2010) is presented and its stability margin compared to the widely used Semi-Implicit Velocity-Correction scheme.

High-fidelity vortex-induced vibration of a wind turbine blade under extreme conditions

Mohsen Lahooti (Imperial College London)

Modern wind turbines are put to the stand-still state under certain conditions such as above-design wind speeds or scheduled maintenance. In such situations the blade is prone to experiencing the incoming wind flow with a very high angle of attack that could lead to massive separation and development and shedding vortices from the blade, resulting in an unsteady loading on the blade structure. Further, due to the large size and hence enhanced flexibility of modern wind turbine blades, this unsteady loading of the flow together with the aeroelastic response of the blade could result in vibration of the blade in such conditions which is known as the stand-still vibration. In this talk, the simulation results of the stand-still vibration of the NREL 5MW blade under high wind speed condition will be presented. Simulations are performed using a recently developed fluid-structure solver that has been implemented in the Nektar++ framework and is based on the coupling with the uDNS incompressible Navier-Stokes solver of Nektar++ and Geometrically-Exact composite beam (GECB) beam solver of SHARPy library.

Direct Numerical Simulation study of the NACA0012 at high angle-of-attack by means of the Nektar++ incompressible solver

Guglielmo Vivarelli (Imperial College London)

Stall and post-stall conditions occur in different modern engineering settings, such as compressors and turbines in gas engines and wind turbines to name a few. To be able to design such components a detailed understanding of the complex flow physics generally encountered in these cases is required. Separation, turbulent transition, shear layers are an example of some of the intricate flow behaviour that may appear.

In this work, an extensive Direct Numerical Simulation study of the solution at angles of attack 9, 10 and 12 degrees will be carried out. Similar flow features will be seen, but their location and intensity differs as the aerofoil is tilted more and more. Boundary layer quantities will be computed to assess the situation and the difference between the cases. Following this, an in-depth analysis of the effects of the mesh resolution, but in particular, of the spatial discretisation scheme will be carried out. The evidence suggests a very strong dependency on the small scale behaviour that mainly effects the leading edge bubble separation. Finally, a separate study will be carried out to understand the effects of the spanwise extrusion on the overall solution. The 10 degrees angle of attack case displayed a non-negligible dependency at both the leading edge and downstream of the 50% chord position. It is not conclusive as to whether the widest case (Lz=8) was sufficient to capture the entire flow physics. The 12 degree case, on the other hand, demonstrated that there is a much smaller dependency on the spanwise length.

Drag reducing riblets on nektar++

Chi Hin Chan (Imperial College London)

Riblets, inspired from the ribbed texture of shark-skins have the potential to reduce turbulent skin-friction drag. Riblets are attractive due to its nature as a passive flow control device which requires no energy input to alter the flow favourably. An optimally designed riblet can achieve a skin-friction drag reduction of up to ∼ 10%, which can be exploited in skin-friction dominant engineering applications (i.e planes, submarines). The performance of a riblet strongly depends on its size and cross-sectional shape. An optimal riblet has a groove-cross sectional length of l_g^+ ∼ 10.7 (l_g^+ = A_g h/ν, where, A g is the riblet’s groove cross-sectional area), while its dependence on its cross-sectional shape is not known and is worth exploring. Preliminary studies of drag reducing riblets are performed using direct numerical simulations on nektar++ due to its ability to discretise complex geometries while achieving spectral accuracy. In particular, a turbulent channel flow filled with riblets is discretised by a combination of Fourier modes and spectral/hp elements. A study on mesh dependency and the assosciated challenges will be presented in the workshop.

Scale Resolved Simulations of an Inverted Multi-Element Wing in Ground Effect

James Slaughter (Imperial College London)

Presented is a computational study of a spanwise homogeneous inverted multi-element wing in ground effect. At a ride height of 0.36h/c, a slice of a F1 front wing has been taken and extruded 0.16c in the z direction. The study has examined the flow physics and salient flow features using iLES through a Spectral/hp Element type discretization. Force coefficients of -8.33 and 0.17 have been found for lift and drag, respectively. Through examination of the lift and surface pressure fluctuation through techniques such as Power Spectral Density and Cross Spectral Phase, we present relationships between key modes on all aerofoils at St=40,60,140,200. An in-phase relationship can be seen at St=40 whilst a distinct out-of-phase relationship can be seen at higher Strouhal Number. Transition mechanisms on the suction surfaces of all flaps have been characterized with the first two elements transitioning via Laminar Separation Bubble, with the third transitioning via a bypass mechanism. This study is presented as to create a benchmark for the testing and review of current and new models and processes as applied to motorsports and Formula 1 aerodynamics.

Studies of disturbance growth in transonic boundary layers over complex geometries using embedded DG simulations

Ganlin Lyu (Imperial College London)

Laminar boundary layer natural transition for external flows is of particular interest in both the aeronautical industry and academia. The transitional process is dominated by the linear growth of disturbances, e.g., Tollmien–Schlichting waves and crossflow waves, and therefore a correct prediction on the development of the disturbances is necessary for a successful transitional analysis. Most conventional studies focused on the disturbances developing based on incompressible boundary layer flows over ideal, clean geometries. However, the physical settings for wider real applications are different for the flow compressibility and geometrical complexity. The compressibility stems from the transonic operational conditions, and for the real geometries the main source of the complexity is the existence of surface imperfections, which typically take the form of steps and gaps whose sizes are comparable with the boundary layer thickness. In the current study we therefore further extend the physical settings to transonic laminar boundary layer at realistic Reynolds numbers, and over wing sections with surface imperfections.

Exploring the implementation of the ALE method in Nektar++

Edward Laughton (Kings College London)

The arbitrary Lagrangian-Eulerian (ALE) method is a finite element formulation where the computational mesh is not fixed in space and is allowed to vary arbitrarily with time. In this talk we examine the ALE method with respect to Nektar++, looking at the formulation of the method itself in high-order schemes using a DG spatial discretisation and the current code implementation. This follows on from previous work focused on non-conformal interfaces which when combined with the ALE method now allows for arbitrary rotating and sliding computational domains in Nektar++.