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

Helicity and Physical Fidelity in Turbulence Modeling

This thesis is a study of physical fidelity in turbulence modeling. We first consider conservation laws in several popular turbulence models and find that of the Leray, Leray-deconvolution, Bardina and Stolz-Adams approximate deconvolution model (ADM), all but the Bardina model conserve a model energy. Only the ADM conserves a model helicity. Since the ADM conserves a model energy and helicity, we then investigate a joint helicity-energy spectrum in the ADM. We find that up to a filter-dependent length scale, the ADM cascades energy and helicity jointly in the same manner as the Navier-Stokes equations.We also investigate helicity treatment in discretizations of turbulence models. For inviscid, periodic flow, we implement energy conserving discretizations of the ADM, Leray, and Leray-deconvolution models as well as the Navier-Stokes equations (NSE) and observe helicity treatments. We find that of none of the models conserve helicity (or model helicity) in the discretizations. Since the Leray-deconvolution model of turbulence is newly developed, our implementation is new and thus we analyze the trapezoidal Galerkin scheme that we implement and compare it to the usual Leray model.Lastly, we develop an energy and helicity conserving trapezoidal Galerkin scheme for the Navier-Stokes equations. We prove conservation properties for the scheme, stability, and show the scheme does not lose asymptotic accuracy compared to the usual trapezoidal Galerkin scheme. We also present numerical experiments that compare the energy and helicity conserving scheme to more typical schemes

Improved Accuracy for Fluid Flow Problems Via Enhanced Physics

This thesis is an investigation of numerical methods for approximating solutions to fluid flow problems, specifically the Navier-Stokes equations (NSE) and magnetohydrodynamic equations (MHD), with an overriding theme of enforcing more physical behavior in discrete solutions. It is well documented that numerical methods with more physical accuracy exhibit better long-time behavior than comparable methods that enforce less physics in their solutions. This work develops, analyzes and tests finite element methods that better enforce mass conservation in discrete velocity solutions to the NSE and MHD, helicity conservation for NSE, cross-helicity conservation in MHD, and magnetic field incompressibility in MHD

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

The EMAC scheme for Navier-Stokes simulations, and application to flow past bluff bodies

The Navier-Stokes equations model the evolution of water, oil, and air flow (air under 220 m.p.h.), and therefore the ability to solve them is important in a wide array of engineering design problems. However, analytic solution of these equations is generally not possible, except for a few trivial cases, and therefore numerical methods must be employed to obtain solutions. In the present dissertation we address several important issues in the area of computational fluid dynamics. The first issue is that in typical discretizations of the Navier-Stokes equations such as the mixed finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier-Stokes equations dictate that they do. It is widely believed in the computational fluid dynamics community that the more physics is built into the discretization, the more accurate and stable the discrete solutions are, especially over longer time intervals. In chapter 3 we study conservation properties of Galerkin methods for the incompressible Navier-Stokes equations, without the divergence constraint strongly enforced. We show that none of the commonly used formulations (convective, conservative, rotational, and skew-symmetric) conserve each of energy, momentum, and angular momentum (for a general finite element choice). We aim to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convective formulation does not conserve most of these quantities). In chapter 3 we also perform a number of numerical experiments, which verify the theory and test the new formulation. To study the performance of our novel formulation of the Navier-Stokes equations, we need reliable reference solutions/statistics. However, there is not a significant amount of reliable reference solutions for the Navier-Stokes equations in the literature. Accurate reference solutions/statistics are difficult to obtain due to a number of reasons. First, one has to use several millions of degrees of freedom even for a two-dimensional simulation (for 3D one needs at least tens of millions of degrees of freedom). Second, it usually takes a long time before the flow becomes fully periodic and/or stationary. Third, in order to obtain reliable solutions, the time step must be very small. This results in a very large number of time steps. All of this results in weeks of computational time, even with the highly parallel code and efficient linear solvers (and in months for a single-threaded code). Finally, one has to run a simulation for multiple meshes and time steps in order to show the convergence of solutions. In the second chapter we perform a careful, very fine discretization simulations for a channel flow past a flat plate. We derive new, more precise reference values for the averaged drag coefficient, recirculation length, and the Strouhal number from the computational results. We verify these statistics by numerical computations with the three time stepping schemes (BDF2, BDF3 and Crank-Nicolson). We carry out the same numerical simulations independently using deal.II and Freefem++ software. In addition both deal.II/Q2Q1 and Freefem/P2P1 element types were used to verify the results. We also verify results by numerical simulations with multiple meshes, and different time step sizes. Finally, in chapter 4 we compute reference values for the three-dimensional channel flow past a circular cylinder obstacle, with both time-dependent inflow and with constant inflow using up to 70.5 million degrees of freedom. In contrast to the linearization approach used in chapter 2, in chapter 4 we numerically study fully nonlinear schemes, which we linearize using Newton\u27s method. In chapter 4 we also compare the performance of our novel EMAC scheme with the four most commonly used formulations of the Navier-Stokes equations (rotational, skew-symmetric, convective and conservative) for the three-dimensional channel flow past circular cylinder both with the time-dependent inflow and with constant inflow

Stable computing with an enhanced physics based scheme for the 3D Navier--Stokes equations

We study extensions of an earlier developed energy and helicity preserving scheme for the 3D Navier-Stokes equations and apply them to a more general class of problems. The scheme is studied together with stabilizations of grad-div type in order to mitigate the effect of the Bernoulli pressure error on the velocity error. We prove stability, convergence, discuss conservation properties, and present numerical experiments that demonstrate the advantages of the scheme

Stable computing with an enhanced physics based scheme for the 3d Navier-Stokes equations

We study extensions of an earlier developed energy and helicity preserving scheme for the 3D Navier-Stokes equations and apply them to a more general class of problems. The scheme is studied together with stabilizations of grad-div type in order to mitigate the effect of the Bernoulli pressure error on the velocity error. We prove stability, convergence, discuss conservation properties, and present numerical experiments that demonstrate the advantages of the schem