Linearly implicit Rosenbrock-type Runge–Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows

Bassi, Francesco; Botti, Lorenzo Alessio; Colombo, Alessandro; Ghidoni, Antonio; Massa, Francesco Carlo

doi:10.1016/j.compfluid.2015.06.007

Cited by 96 publications

(86 citation statements)

References 44 publications

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“…The free stream flow is of

false(ρ, u, v, M a false) = false(1.0, \sqrt{2} false/ 2, \sqrt{2} false/ 2, 0.05 false)

and the gas constant R =1.0 for this case. The perturbation is defined as

({, \begin{matrix} array \\ array δ u = - \frac{α}{2 π} (y - y_{0}) e^{ϕ (1 - r^{2})} \\ array δ v = \frac{α}{2 π} (x - x_{0}) e^{ϕ (1 - r^{2})} \\ array δ T = - \frac{α^{2} (γ - 1)}{16 ϕ γ π^{2}} e^{2 ϕ (1 - r^{2})}, \end{matrix})

where

ϕ = \frac{1}{2}

and α =5 are parameters that define the vortex strength. r =( x − x 0 ) 2 +( y − y 0 ) 2 is the distance to the center of the vortex ( x 0 , y 0 ).…”

Section: Numerical Resultsmentioning

confidence: 99%

An implicit high‐order preconditioned flux reconstruction method for low‐Mach‐number flow simulation with dynamic meshes

Wang

2019

Numerical Methods in Fluids

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Summary A fully implicit high‐order preconditioned flux reconstruction/correction procedure via reconstruction (FR/CPR) method is developed to solve the compressible Navier‐Stokes equations at low Mach numbers. A dual‐time stepping approach with the second‐order backward differentiation formula (BDF2) is employed to ensure temporal accuracy for unsteady flow simulation. When dynamic meshes are used to handle moving/deforming domains, the geometric conservation law is implicitly enforced to eliminate errors due to the resolution discrepancy between BDF2 and the spatial FR/CPR discretization. The large linear system resulted from the spatial and temporal discretizations is tackled with the restarted generalized minimal residual solver in the PETSc (portable, extensible toolkit for scientific computation) library. Through several benchmark steady and unsteady numerical tests, the preconditioned FR/CPR methods have demonstrated good convergence and accuracy for simulating flows at low Mach numbers. The new flow solver is then used to study the effects of Mach number on unsteady force generation over a plunging airfoil when operating in low‐Mach‐number flows. It is observed that weak compressibility has a significant impact on thrust generation but has a negligible effect on lift generation of an oscillating airfoil.

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“…The free stream flow is of

false(ρ, u, v, M a false) = false(1.0, \sqrt{2} false/ 2, \sqrt{2} false/ 2, 0.05 false)

and the gas constant R =1.0 for this case. The perturbation is defined as

({, \begin{matrix} array \\ array δ u = - \frac{α}{2 π} (y - y_{0}) e^{ϕ (1 - r^{2})} \\ array δ v = \frac{α}{2 π} (x - x_{0}) e^{ϕ (1 - r^{2})} \\ array δ T = - \frac{α^{2} (γ - 1)}{16 ϕ γ π^{2}} e^{2 ϕ (1 - r^{2})}, \end{matrix})

where

ϕ = \frac{1}{2}

and α =5 are parameters that define the vortex strength. r =( x − x 0 ) 2 +( y − y 0 ) 2 is the distance to the center of the vortex ( x 0 , y 0 ).…”

Section: Numerical Resultsmentioning

confidence: 99%

An implicit high‐order preconditioned flux reconstruction method for low‐Mach‐number flow simulation with dynamic meshes

Wang

2019

Numerical Methods in Fluids

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“…With this choice, the matrix

boldP false(boldw false) \in R^{m} \otimes R^{m}

does not depend on w and reduces to identity matrix P = I . Alternatives to the set of conservative variables for compressible flows were investigated by several authors; in these cases, the transformation matrix P ( w ) can differ from I , see Bassi et al and Bassi et al The Cartesian components f c ( w ) and g c ( w ) of the convective F c ( w ) flux function equal

f_{c} false(boldw false) = ([], \begin{matrix} array ρ u \\ array ρ h_{0} u \\ array ρ u u + p \\ array ρ v u \end{matrix}), g_{c} false(boldw false) = ([], \begin{matrix} array ρ v \\ array ρ h_{0} v \\ array ρ u v \\ array ρ v v + p \end{matrix}),

where ρ is the fluid density, p is the pressure, u and v are the velocity components, respectively, e 0 and h 0 are the total energy and enthalpy per unit mass, respectively. The total enthalpy per unit mass is defined as h 0 = e 0 + p / ρ .…”

Section: Basic Concepts For Dg On Agglomerated Meshesmentioning

confidence: 99%

An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

Zenoni

Leicht

Colombo

et al. 2017

Numerical Methods in Fluids

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Summary In this work, we exploit the possibility to devise discontinuous Galerkin discretizations over polytopic grids to perform grid adaptation strategies on the basis of agglomeration coarsening of a fine grid obtained via standard unstructured mesh generators. The adaptive agglomeration process is here driven by an adjoint‐based error estimator. We investigate several strategies for converting the error field estimated solving the adjoint problem into an agglomeration factor field that is an indication of the number of elements of the fine grid that should be clustered together to form an agglomerated element. As a result the size of agglomerated elements is optimized for the achievement of the best accuracy for given grid size with respect to the target quantities. To demonstrate the potential of this strategy we consider problem‐specific outputs of interest typical of aerodynamics, eg, the lift and drag coefficients in the context of inviscid and viscous flows test cases.

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“…The time discretization of Eq. (23) is efficiently performed by means of a linearly implicit Rosenbrock-type Runge-Kutta schemes, following the implementation presented in [27], that can be written as:…”

Section: Time Integrationmentioning

confidence: 99%

Discontinuous Galerkin Computation of Gaseous Mixture Coaxial Jets

Franchina¹,

Savini²,

Bassi³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. A novel approach based upon Discontinuous Galerkin (DG) discretization, applied to the divergence form of the multicomponent Navier-Stokes equations, is here presented and used to compute non reactive turbulent axisymmetric gaseous jets. The original key feature is the use of L 2 -projection form of the (perfect gas) equation of state. This choice mitigates problems typically encountered by the front-capturing schemes in computing multicomponent flow fields, i.e. spurious oscillations across material and contact surfaces where the mixture composition is changing. The solver makes also use of a shock-capturing technique based on artificial dissipation selectively added into the equations and tuned in connection with the magnitude of inviscid residuals of the equations and on suitable coefficients accounting for the variation of the unknown variables within and across grid elements. A simple limiting procedure is introduced in order to avoid the occurrence of unphysical gas properties due to negative and/or greater that one mass fractions values within the domain. The DG code based on the proposed novel technique for multicomponent flow computation is here employed to study the mixing mode and the preferential diffusion mechanism of a mixture jet of helium and carbon dioxide in a surrounding flow of air, both in laminar and turbulent flow regimes. Mass diffusion is modelled by means of Fick's first law and use is made of constant Prandtl and Schmidt numbers in Wilcox's (2008) k − ω model. Third-order accurate results are presented, discussed and compared with the available experimental data. They confirm the possible existence in coaxial jets of different periodic flow structures, greatly affecting mixing rates, and different species diffusive mass fluxes. The relative importance of both phenomena depends on the flow regime and its characteristics. The tests carried out give at the same time indications about the accuracy of the proposed method and its effectiveness in computing complex unsteady flow fields.

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Linearly implicit Rosenbrock-type Runge–Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows

Cited by 96 publications

References 44 publications

An implicit high‐order preconditioned flux reconstruction method for low‐Mach‐number flow simulation with dynamic meshes

An implicit high‐order preconditioned flux reconstruction method for low‐Mach‐number flow simulation with dynamic meshes

An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

Discontinuous Galerkin Computation of Gaseous Mixture Coaxial Jets

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