Daniele De Grazia | Imperial College London (original) (raw)
Papers by Daniele De Grazia
Journal of Computational Physics, 2018
This study focuses on the dispersion and diffusion characteristics of high-order energystable flu... more This study focuses on the dispersion and diffusion characteristics of high-order energystable flux reconstruction (ESFR) schemes via the spatial eigensolution analysis framework proposed in [1]. The analysis is performed for five ESFR schemes, where the parameter 'c' dictating the properties of the specific scheme recovered is chosen such that it spans the entire class of ESFR methods, also referred to as VCJH schemes, proposed in [2]. In particular, we used five values of 'c', two that correspond to its lower and upper bounds and the others that identify three schemes that are linked to common high-order methods, namely the ESFR recovering two versions of discontinuous Galerkin methods and one recovering the spectral difference scheme. The performance of each scheme is assessed when using different numerical intercell fluxes (e.g. different levels of upwinding), ranging from "under-" to "over-upwinding". In contrast to the more common temporal analysis, the spatial eigensolution analysis framework adopted here allows one to grasp crucial insights into the diffusion and dispersion properties of FR schemes for problems involving nonperiodic boundary conditions, typically found in open-flow problems, including turbulence, unsteady aerodynamics and aeroacoustics.
Journal of Scientific Computing, 2015
This paper investigates the connections between many popular variants of the well-established dis... more This paper investigates the connections between many popular variants of the well-established discontinuous Galerkin method and the recently developed high-order flux reconstruction approach on irregular tensor-product grids. We explore these connections by analysing three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and three types of flux reconstruction schemes for solving systems of conservation laws on irregular tensor-product meshes. We demonstrate that the existing connections established on regular grids are also valid on deformed and curved meshes for both linear and nonlinear problems, provided that the metric terms are accounted for appropriately. We also find that the aliasing issues arising from nonlinearities either due to a deformed/curved elements or due to the nonlinearity of the equations are equivalent and can be addressed using the same strategies both in the discontinuous Galerkin method and in the flux reconstruction approach. In particular, we show that the discontinuous Galerkin and the flux reconstruction approach are equivalent also when using higher-order quadrature rules that are commonly employed in the context of over-or consistent-integration-based dealiasing methods. The connections found in this work help to complete the picture regarding the relations between these two numerical approaches and show the possibility of using over-or consistent-integration in an equivalent manner for both the approaches. Keywords Spectral/hp methods • Discontinuous Galerkin • Flux reconstruction • Aliasing • Irregular grids B G. Mengaldo
Journal of Computational Physics, 2015
High-order methods are becoming increasingly attractive in both academia and industry, especially... more High-order methods are becoming increasingly attractive in both academia and industry, especially in the context of computational fluid dynamics. However, before they can be more widely adopted, issues such as lack of robustness in terms of numerical stability need to be addressed, particularly when treating industrial-type problems where challenging geometries and a wide range of physical scales, typically due to high Reynolds numbers, need to be taken into account. One source of instability is aliasing effects which arise from the nonlinearity of the underlying problem. In this work we detail two dealiasing strategies based on the concept of consistent integration. The first uses a localised approach, which is useful when the nonlinearities only arise in parts of the problem. The second is based on the more traditional approach of using a higher quadrature. The main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods, thereby reducing aliasing-driven instabilities. We demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations, where, in the latter, both volumetric and interface approximations must be considered. We show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations.
Computer Physics Communications, 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.
International Journal for Numerical Methods in Fluids, 2014
With high-order methods becoming more widely adopted throughout the field of computational fluid ... more With high-order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. The flux reconstruction approach allows various well-known high order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of the Lagrange basis representing the solution and from the quadrature approximation used to integrate the mass matrix and the other terms of the discretisation. By considering both a linear and nonlinear advection equation on a regular grid, we examine the mathematical properties which connect these discretisations. These arguments are further confirmed by the results of an empirical numerical study.
7th AIAA Theoretical Fluid Mechanics Conference, 2014
ABSTRACT The nature of boundary conditions, and how they are implemented, can have a sig- nifican... more ABSTRACT The nature of boundary conditions, and how they are implemented, can have a sig- nificant impact on the stability and accuracy of a Computational Fluid Dynamics (CFD) solver. The objective of this paper is to assess how different boundary conditions impact the performance of compact discontinuous high-order spectral element methods (such as the discontinuous Galerkin method and the Flux Reconstruction approach), when these schemes are used to solve the Euler and compressible Navier-Stokes equations on unstruc- tured grids. Specifically, the paper will investigate inflow/outflow and wall boundary con- ditions. In all studies the boundary conditions were enforced by modifying the boundary flux. For Riemann invariant (characteristic), slip and no-slip conditions we have consid- ered a direct and an indirect enforcement of the boundary conditions, the first obtained by calculating the flux using the known solution at the given boundary while the second achieved by using a ghost state and by solving a Riemann problem. All computations were performed using the open-source software Nektar++ (www.nektar.info)
Physical Review Fluids, 2018
In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary l... more In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary layer by a small wall roughness. Specifically, we present a direct numerical simulation (DNS) of a two-dimensional boundary-layer flow over a flat plate encountering a three-dimensional Gaussian-shaped hump. This work was motivated by the lack of DNS data of boundary-layer flows past roughness elements in a similar regime which is typical of civil aviation. The Mach and Reynolds numbers are chosen to be relevant for aeronautical applications when considering small imperfections at the leading edge of wings. We analyze different heights of the hump: The smaller heights result in a weakly nonlinear regime, while the larger result in a fully nonlinear regime with an increasing laminar separation bubble arising downstream of the roughness element and the formation of a pair of streamwise counterrotating vortices which appear to support themselves.
With high-order methods becoming more widely adopted throughout the field of computational fluid ... more With high-order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. The flux reconstruction approach allows various well-known high order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of t...
This is to certify that the work presented in this thesis has been carried out at Imperial Colleg... more This is to certify that the work presented in this thesis has been carried out at Imperial College London and has not been previously submitted to any other university or technical institution for a degree or award. The thesis comprises only my original work, except where due acknowledgement is made in the text. The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work.
This study focuses on the dispersion and diffusion characteristics of high-order energy-stable fl... more This study focuses on the dispersion and diffusion characteristics of high-order energy-stable flux reconstruction (ESFR) schemes via the spatial eigensolution analysis framework proposed in [1]. The analysis is performed for five ESFR schemes, where the parameter 'c' dictating the properties of the specific scheme recovered is chosen such that it spans the entire class of ESFR methods, also referred to as VCJH schemes, proposed in [2]. In particular, we used five values of 'c', two that correspond to its lower and upper bounds and the others that identify three schemes that are linked to common high-order methods, namely the ESFR recovering two versions of discontinuous Galerkin methods and one recovering the spectral difference scheme. The performance of each scheme is assessed when using different numerical intercell fluxes (e.g. different levels of upwinding), ranging from " under-" to " over-upwinding ". In contrast to the more common temporal analysis, the spatial eigensolution analysis framework adopted here allows one to grasp crucial insights into the diffusion and dispersion properties of FR schemes for problems involving non-periodic boundary conditions, typically found in open-flow problems, including turbulence, unsteady aerodynamics and aeroacoustics.
In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary l... more In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary layer by a small wall roughness. Specifically, we present a direct numerical simulation (DNS) of a two-dimensional boundary-layer flow over a flat plate encountering a three-dimensional Gaussian-shaped hump. This work was motivated by the lack of DNS data of boundary-layer flows past roughness elements in a similar regime which is typical of civil aviation. The Mach and Reynolds numbers are chosen to be relevant for aeronautical applications when considering small imperfections at the leading edge of wings. We analyze different heights of the hump: The smaller heights result in a weakly nonlinear regime, while the larger result in a fully nonlinear regime with an increasing laminar separation bubble arising downstream of the roughness element and the formation of a pair of streamwise counterrotating vortices which appear to support themselves.
This paper investigates the connections between many popular variants of the well-established dis... more This paper investigates the connections between many popular variants of the
well-established discontinuous Galerkin method and the recently developed high-order flux reconstruction approach on irregular tensor-product grids. We explore these connections by analysing three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and three types of flux reconstruction schemes for solving systems of conservation laws on irregular tensor-productmeshes.We demonstrate that the existing connections established on regular grids are also valid on deformed and curved meshes for both linear and nonlinear problems, provided that the metric terms are accounted for appropriately. We also find that the aliasing issues arising from nonlinearities either due to a deformed/curved elements or due to the nonlinearity of the equations are equivalent and can be addressed using the same strategies both in the discontinuous Galerkin method and in the flux reconstruction approach. In particular, we show that the discontinuous Galerkin and the flux reconstruction
approach are equivalent also when using higher-order quadrature rules that are commonly employed in the context of over- or consistent-integration-based dealiasing methods.The connections found in this work help to complete the picture regarding the relations between these two numerical approaches and show the possibility of using over- or consistent-integration in an equivalent manner for both the approaches.
High-order methods are becoming increasingly attractive in both academia and industry, especially... more High-order methods are becoming increasingly attractive in both academia and industry, especially in the context of computational fluid dynamics. However, before they can be more widely adopted, issues such as lack of robustness in terms of numerical stability need to be addressed, particularly when treating industrial-type problems where challenging geometries and a wide range of physical scales, typically due to high Reynolds numbers, need to be taken into account. One source of instability is aliasing effects which arise from the nonlinearity of the underlying problem. In this work we detail two dealiasing strategies based on the concept of consistent integration. The first uses a localised approach, which is useful when the nonlinearities only arise in parts of the problem. The second is based on the more traditional approach of using a higher quadrature. The main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods, thereby reducing aliasing-driven instabilities. We demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations, where, in the latter, both volumetric and interface approximations must be considered. We show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations.
Nektar++ is an open-source software framework designed to support the development of high perform... more Nektar++ is an open-source software framework designed to support the development of high performance 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.
The nature of boundary conditions, and how they are implemented, can have a significant impact on... more The nature of boundary conditions, and how they are implemented, can have a significant impact on the stability and accuracy of a Computational Fluid Dynamics (CFD) solver. The objective of this paper is to assess how di erent boundary conditions impact the performance of compact discontinuous high-order spectral element methods (such as the discontinuous Galerkin method and the Flux Reconstruction approach), when these schemes are used to solve the Euler and compressible Navier-Stokes equations on unstructured grids. Speci cally, the paper will investigate inflow/outflow and wall boundary conditions. In all studies the boundary conditions were enforced by modifying the boundary flux. For Riemann invariant (characteristic), slip and no-slip conditions we have considered a direct and an indirect enforcement of the boundary conditions, the first obtained by calculating the flux using the known solution at the given boundary while the second achieved by using a ghost state and by solving a Riemann problem. All computations were performed using the open-source software Nektar++ (www.nektar.info).
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
With high-order methods becoming more widely adopted throughout the field of computational fluid ... more With high-order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. One of the most recent methods to be developed is the flux reconstruction approach, which allows various well-known high-order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the more widely adopted discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of the Lagrange basis representing the solution and from the quadrature approximation used to integrate the mass matrix and the other terms of the discretization. By considering both linear and nonlinear advection equations on a regular grid, we examine the mathematical properties that connect these discretizations. These arguments are further confirmed by the results of an empirical numerical study.
Journal of Computational Physics, 2018
This study focuses on the dispersion and diffusion characteristics of high-order energystable flu... more This study focuses on the dispersion and diffusion characteristics of high-order energystable flux reconstruction (ESFR) schemes via the spatial eigensolution analysis framework proposed in [1]. The analysis is performed for five ESFR schemes, where the parameter 'c' dictating the properties of the specific scheme recovered is chosen such that it spans the entire class of ESFR methods, also referred to as VCJH schemes, proposed in [2]. In particular, we used five values of 'c', two that correspond to its lower and upper bounds and the others that identify three schemes that are linked to common high-order methods, namely the ESFR recovering two versions of discontinuous Galerkin methods and one recovering the spectral difference scheme. The performance of each scheme is assessed when using different numerical intercell fluxes (e.g. different levels of upwinding), ranging from "under-" to "over-upwinding". In contrast to the more common temporal analysis, the spatial eigensolution analysis framework adopted here allows one to grasp crucial insights into the diffusion and dispersion properties of FR schemes for problems involving nonperiodic boundary conditions, typically found in open-flow problems, including turbulence, unsteady aerodynamics and aeroacoustics.
Journal of Scientific Computing, 2015
This paper investigates the connections between many popular variants of the well-established dis... more This paper investigates the connections between many popular variants of the well-established discontinuous Galerkin method and the recently developed high-order flux reconstruction approach on irregular tensor-product grids. We explore these connections by analysing three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and three types of flux reconstruction schemes for solving systems of conservation laws on irregular tensor-product meshes. We demonstrate that the existing connections established on regular grids are also valid on deformed and curved meshes for both linear and nonlinear problems, provided that the metric terms are accounted for appropriately. We also find that the aliasing issues arising from nonlinearities either due to a deformed/curved elements or due to the nonlinearity of the equations are equivalent and can be addressed using the same strategies both in the discontinuous Galerkin method and in the flux reconstruction approach. In particular, we show that the discontinuous Galerkin and the flux reconstruction approach are equivalent also when using higher-order quadrature rules that are commonly employed in the context of over-or consistent-integration-based dealiasing methods. The connections found in this work help to complete the picture regarding the relations between these two numerical approaches and show the possibility of using over-or consistent-integration in an equivalent manner for both the approaches. Keywords Spectral/hp methods • Discontinuous Galerkin • Flux reconstruction • Aliasing • Irregular grids B G. Mengaldo
Journal of Computational Physics, 2015
High-order methods are becoming increasingly attractive in both academia and industry, especially... more High-order methods are becoming increasingly attractive in both academia and industry, especially in the context of computational fluid dynamics. However, before they can be more widely adopted, issues such as lack of robustness in terms of numerical stability need to be addressed, particularly when treating industrial-type problems where challenging geometries and a wide range of physical scales, typically due to high Reynolds numbers, need to be taken into account. One source of instability is aliasing effects which arise from the nonlinearity of the underlying problem. In this work we detail two dealiasing strategies based on the concept of consistent integration. The first uses a localised approach, which is useful when the nonlinearities only arise in parts of the problem. The second is based on the more traditional approach of using a higher quadrature. The main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods, thereby reducing aliasing-driven instabilities. We demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations, where, in the latter, both volumetric and interface approximations must be considered. We show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations.
Computer Physics Communications, 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.
International Journal for Numerical Methods in Fluids, 2014
With high-order methods becoming more widely adopted throughout the field of computational fluid ... more With high-order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. The flux reconstruction approach allows various well-known high order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of the Lagrange basis representing the solution and from the quadrature approximation used to integrate the mass matrix and the other terms of the discretisation. By considering both a linear and nonlinear advection equation on a regular grid, we examine the mathematical properties which connect these discretisations. These arguments are further confirmed by the results of an empirical numerical study.
7th AIAA Theoretical Fluid Mechanics Conference, 2014
ABSTRACT The nature of boundary conditions, and how they are implemented, can have a sig- nifican... more ABSTRACT The nature of boundary conditions, and how they are implemented, can have a sig- nificant impact on the stability and accuracy of a Computational Fluid Dynamics (CFD) solver. The objective of this paper is to assess how different boundary conditions impact the performance of compact discontinuous high-order spectral element methods (such as the discontinuous Galerkin method and the Flux Reconstruction approach), when these schemes are used to solve the Euler and compressible Navier-Stokes equations on unstruc- tured grids. Specifically, the paper will investigate inflow/outflow and wall boundary con- ditions. In all studies the boundary conditions were enforced by modifying the boundary flux. For Riemann invariant (characteristic), slip and no-slip conditions we have consid- ered a direct and an indirect enforcement of the boundary conditions, the first obtained by calculating the flux using the known solution at the given boundary while the second achieved by using a ghost state and by solving a Riemann problem. All computations were performed using the open-source software Nektar++ (www.nektar.info)
Physical Review Fluids, 2018
In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary l... more In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary layer by a small wall roughness. Specifically, we present a direct numerical simulation (DNS) of a two-dimensional boundary-layer flow over a flat plate encountering a three-dimensional Gaussian-shaped hump. This work was motivated by the lack of DNS data of boundary-layer flows past roughness elements in a similar regime which is typical of civil aviation. The Mach and Reynolds numbers are chosen to be relevant for aeronautical applications when considering small imperfections at the leading edge of wings. We analyze different heights of the hump: The smaller heights result in a weakly nonlinear regime, while the larger result in a fully nonlinear regime with an increasing laminar separation bubble arising downstream of the roughness element and the formation of a pair of streamwise counterrotating vortices which appear to support themselves.
With high-order methods becoming more widely adopted throughout the field of computational fluid ... more With high-order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. The flux reconstruction approach allows various well-known high order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of t...
This is to certify that the work presented in this thesis has been carried out at Imperial Colleg... more This is to certify that the work presented in this thesis has been carried out at Imperial College London and has not been previously submitted to any other university or technical institution for a degree or award. The thesis comprises only my original work, except where due acknowledgement is made in the text. The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work.
This study focuses on the dispersion and diffusion characteristics of high-order energy-stable fl... more This study focuses on the dispersion and diffusion characteristics of high-order energy-stable flux reconstruction (ESFR) schemes via the spatial eigensolution analysis framework proposed in [1]. The analysis is performed for five ESFR schemes, where the parameter 'c' dictating the properties of the specific scheme recovered is chosen such that it spans the entire class of ESFR methods, also referred to as VCJH schemes, proposed in [2]. In particular, we used five values of 'c', two that correspond to its lower and upper bounds and the others that identify three schemes that are linked to common high-order methods, namely the ESFR recovering two versions of discontinuous Galerkin methods and one recovering the spectral difference scheme. The performance of each scheme is assessed when using different numerical intercell fluxes (e.g. different levels of upwinding), ranging from " under-" to " over-upwinding ". In contrast to the more common temporal analysis, the spatial eigensolution analysis framework adopted here allows one to grasp crucial insights into the diffusion and dispersion properties of FR schemes for problems involving non-periodic boundary conditions, typically found in open-flow problems, including turbulence, unsteady aerodynamics and aeroacoustics.
In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary l... more In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary layer by a small wall roughness. Specifically, we present a direct numerical simulation (DNS) of a two-dimensional boundary-layer flow over a flat plate encountering a three-dimensional Gaussian-shaped hump. This work was motivated by the lack of DNS data of boundary-layer flows past roughness elements in a similar regime which is typical of civil aviation. The Mach and Reynolds numbers are chosen to be relevant for aeronautical applications when considering small imperfections at the leading edge of wings. We analyze different heights of the hump: The smaller heights result in a weakly nonlinear regime, while the larger result in a fully nonlinear regime with an increasing laminar separation bubble arising downstream of the roughness element and the formation of a pair of streamwise counterrotating vortices which appear to support themselves.
This paper investigates the connections between many popular variants of the well-established dis... more This paper investigates the connections between many popular variants of the
well-established discontinuous Galerkin method and the recently developed high-order flux reconstruction approach on irregular tensor-product grids. We explore these connections by analysing three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and three types of flux reconstruction schemes for solving systems of conservation laws on irregular tensor-productmeshes.We demonstrate that the existing connections established on regular grids are also valid on deformed and curved meshes for both linear and nonlinear problems, provided that the metric terms are accounted for appropriately. We also find that the aliasing issues arising from nonlinearities either due to a deformed/curved elements or due to the nonlinearity of the equations are equivalent and can be addressed using the same strategies both in the discontinuous Galerkin method and in the flux reconstruction approach. In particular, we show that the discontinuous Galerkin and the flux reconstruction
approach are equivalent also when using higher-order quadrature rules that are commonly employed in the context of over- or consistent-integration-based dealiasing methods.The connections found in this work help to complete the picture regarding the relations between these two numerical approaches and show the possibility of using over- or consistent-integration in an equivalent manner for both the approaches.
High-order methods are becoming increasingly attractive in both academia and industry, especially... more High-order methods are becoming increasingly attractive in both academia and industry, especially in the context of computational fluid dynamics. However, before they can be more widely adopted, issues such as lack of robustness in terms of numerical stability need to be addressed, particularly when treating industrial-type problems where challenging geometries and a wide range of physical scales, typically due to high Reynolds numbers, need to be taken into account. One source of instability is aliasing effects which arise from the nonlinearity of the underlying problem. In this work we detail two dealiasing strategies based on the concept of consistent integration. The first uses a localised approach, which is useful when the nonlinearities only arise in parts of the problem. The second is based on the more traditional approach of using a higher quadrature. The main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods, thereby reducing aliasing-driven instabilities. We demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations, where, in the latter, both volumetric and interface approximations must be considered. We show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations.
Nektar++ is an open-source software framework designed to support the development of high perform... more Nektar++ is an open-source software framework designed to support the development of high performance 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.
The nature of boundary conditions, and how they are implemented, can have a significant impact on... more The nature of boundary conditions, and how they are implemented, can have a significant impact on the stability and accuracy of a Computational Fluid Dynamics (CFD) solver. The objective of this paper is to assess how di erent boundary conditions impact the performance of compact discontinuous high-order spectral element methods (such as the discontinuous Galerkin method and the Flux Reconstruction approach), when these schemes are used to solve the Euler and compressible Navier-Stokes equations on unstructured grids. Speci cally, the paper will investigate inflow/outflow and wall boundary conditions. In all studies the boundary conditions were enforced by modifying the boundary flux. For Riemann invariant (characteristic), slip and no-slip conditions we have considered a direct and an indirect enforcement of the boundary conditions, the first obtained by calculating the flux using the known solution at the given boundary while the second achieved by using a ghost state and by solving a Riemann problem. All computations were performed using the open-source software Nektar++ (www.nektar.info).
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
With high-order methods becoming more widely adopted throughout the field of computational fluid ... more With high-order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. One of the most recent methods to be developed is the flux reconstruction approach, which allows various well-known high-order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the more widely adopted discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of the Lagrange basis representing the solution and from the quadrature approximation used to integrate the mass matrix and the other terms of the discretization. By considering both linear and nonlinear advection equations on a regular grid, we examine the mathematical properties that connect these discretizations. These arguments are further confirmed by the results of an empirical numerical study.