Explicit Runge–Kutta schemes for incompressible flow with improved energy-conservation properties

The application of pseudo-symplectic Runge–Kutta methods to the incompressible Navier–Stokes equations is discussed in this work. In contrast to fully energy-conserving, implicit methods, these are explicit schemes of order p that preserve kinetic energy to order q, with q>p. Use of explicit methods with improved energy-conservation properties is appealing for convection-dominated problems, especially in case of direct and large-eddy simulation of turbulent flows. A number of pseudo-symplectic methods are constructed for application to the incompressible Navier–Stokes equations and compared in terms of accuracy and efficiency by means of numerical simulations.Peer ReviewedPostprint (author's final draft

Effects of discrete energy and helicity conservation in numerical simulations of helical turbulence

Helicity is the scalar product between velocity and vorticity and, just like energy, its integral is an in-viscid invariant of the three-dimensional incompressible Navier-Stokes equations. However, space-and time-discretization methods typically corrupt this property, leading to violation of the inviscid conservation principles. This work investigates the discrete helicity conservation properties of spectral and finite-differencing methods, in relation to the form employed for the convective term. Effects due to Runge-Kutta time-advancement schemes are also taken into consideration in the analysis. The theoretical results are proved against inviscid numerical simulations, while a scale-dependent analysis of energy, helicity and their non-linear transfers is performed to further characterize the discretization errors of the different forms in forced helical turbulence simulations

Effects of discrete energy and helicity conservation in numerical simulations of helical turbulence

Helicity is the scalar product between velocity and vorticity and, just like energy, its integral is an in-viscid invariant of the three-dimensional incompressible Navier-Stokes equations. However, space-and time-discretization methods typically corrupt this property, leading to violation of the inviscid conservation principles. This work investigates the discrete helicity conservation properties of spectral and finite-differencing methods, in relation to the form employed for the convective term. Effects due to Runge-Kutta time-advancement schemes are also taken into consideration in the analysis. The theoretical results are proved against inviscid numerical simulations, while a scale-dependent analysis of energy, helicity and their non-linear transfers is performed to further characterize the discretization errors of the different forms in forced helical turbulence simulations

A minimum-dissipation time-integration strategy for large-eddy simulation of incompressible turbulent flows

Adaptive time stepping can significantly enhance the accuracy and the efficiency of computational methods. In this work, a time-integration strategy with adaptive time step control is proposed for large-eddy simulation of turbulent flows. The algorithm is based on Runge-Kutta methods and consists in adjusting the time-step size dynamically to ensure that the numerical dissipation rate due to the temporal scheme is smaller than the molecular and subgrid-scale ones within a desired tolerance. The effectiveness of the method, as compared to standard CFL-like criteria, is assessed by large-eddy simulations of the three-dimensional Taylor-Green Vortex

An efficient time advancing strategy for energy-preserving simulations

Energy-conserving numerical methods are widely employed within the broad area of convection-dominated systems. Semi-discrete conservation of energy is usually obtained by adopting the so-called skew-symmetric splitting of the non-linear convective term, defined as a suitable average of the divergence and advective forms. Although generally allowing global conservation of kinetic energy, it has the drawback of being roughly twice as expensive as standard divergence or advective forms alone. In this paper, a general theoretical framework has been developed to derive an efficient time-advancement strategy in the context of explicit Runge–Kutta schemes. The novel technique retains the conservation properties of skew-symmetric-based discretizations at a reduced computational cost. It is found that optimal energy conservation can be achieved by properly constructed Runge–Kutta methods in which only divergence and advective forms for the convective term are used. As a consequence, a considerable improvement in computational efficiency over existing practices is achieved. The overall procedure has proved to be able to produce new schemes with a specified order of accuracy on both solution and energy. The effectiveness of the method as well as the asymptotic behavior of the schemes is demonstrated by numerical simulation of Burgers' equation.Postprint (published version

Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods

A novel reduced-order model (ROM) formulation for incompressible flows is presented with the key property that it exhibits non-linearly stability, independent of the mesh (of the full order model), the time step, the viscosity, and the number of modes. The two essential elements to non-linear stability are: (1) first discretise the full order model, and then project the discretised equations, and (2) use spatial and temporal discretisation schemes for the full order model that are globally energy-conserving (in the limit of vanishing viscosity). For this purpose, as full order model a staggered-grid finite volume method in conjunction with an implicit Runge-Kutta method is employed. In addition, a constrained singular value decomposition is employed which enforces global momentum conservation. The resulting `velocity-only' ROM is thus globally conserving mass, momentum and kinetic energy. For non-homogeneous boundary conditions, a (one-time) Poisson equation is solved that accounts for the boundary contribution. The stability of the proposed ROM is demonstrated in several test cases. Furthermore, it is shown that explicit Runge-Kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation

Effects of discrete energy and helicity conservation in numerical simulations of helical turbulence

Helicity is the scalar product between velocity and vorticity and, just like energy, its integral is an inviscid invariant of the three-dimensional incompressible Navier-Stokes equations. However, space- and time-discretization methods typically corrupt this prop- erty, leading to violation of the inviscid conservation principles. This work investigates the discrete helicity conservation properties of spectral and finite-differencing methods, in relation to the form employed for the convective term. Effects due to Runge-Kutta time- advancement schemes are also taken into consideration in the analysis. The theoretical results are proved against inviscid numerical simulations, while a scale-dependent analy- sis of energy, helicity and their non-linear transfers is performed to further characterize the discretization errors of the different forms in forced helical turbulence simulations.Postprint (published version

Energy preserving turbulent simulations at a reduced computational cost

Energy-conserving discretizations are widely regarded as a fundamental requirement for high-fidelity simulations of turbulent flows. The skew-symmetric splitting of the nonlinear term is a well-known approach to obtain semi-discrete conservation of energy in the inviscid limit. However, its computation is roughly twice as expensive as that of the divergence or advective forms alone. A novel time-advancement strategy that retains the conservation properties of skew-symmetric-based schemes at a reduced computational cost has been developed. This method is based on properly constructed Runge–Kutta schemes in which a different form (advective or divergence) for the convective term is adopted at each stage. A general framework is presented to derive schemes with prescribed accuracy on both solution and energy conservation. Simulations of homogeneous isotropic turbulence show that the new procedure is effective and can be considerably faster than skew-symmetric-based techniques.Postprint (published version

Approximate projection method for the incompressible Navier–Stokes equations

Postprint (published version