Esteban Ferrer

A high order Discontinuous Galerkin - Fourier incompressible 3D Navier-Stokes solver with rotating sliding meshes

We present the development of a sliding mesh capability for an unsteady high order (order?3) h/p Discontinuous Galerkin solver for the three-dimensional incompressible Navier-Stokes equations. A high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh, through the use of curved boundary elements and mixed triangular-quadrilateral meshes.A second order stiffly stable method is used to discretise in time the Arbitrary Lagrangian-Eulerian form of the incompressible Navier-Stokes equations. Spatial discretisation is provided by the Symmetric Interior Penalty Galerkin formulation with modal basis functions in the x-y plane, allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques. Spatial discretisation in the z-direction is provided by a purely spectral method that uses Fourier series and allows computation of spanwise periodic three-dimensional flows. The developed solver is shown to provide high order solutions, second order in time convergence rates and spectral convergence when solving the incompressible Navier-Stokes equations on meshes where fixed and rotating elements coexist.In addition, an exact implementation of the no-slip boundary condition is included for curved edges; circular arcs and NACA 4-digit airfoils, where analytic expressions for the geometry are used to compute the required metrics.The solver capabilities are tested for a number of two dimensional problems governed by the incompressible Navier-Stokes equations on static and rotating meshes: the Taylor vortex problem, a static and rotating symmetric NACA0015 airfoil and flows through three bladed cross-flow turbines. In addition, three dimensional flow solutions are demonstrated for a three bladed cross-flow turbine and a circular cylinder shadowed by a pitching NACA0012 airfoil.

Wind turbine blade tip comparison using CFD

The effect of wind turbine blade tip geometry is numerically analysed using Computational Fluid Dynamics (CFD). Three different rotating blade tips are compared for attached flow conditions and the flow physics around the geometries are analysed. To this end, the pressure coefficient (Cp) is defined based on the stagnation pressure rather than on the inflow dynamic pressure. The tip geometry locally modifies the angles of attack (AOA) and the inflow dynamic pressure at each of the studied sections. However not all 3D effects could be reduced to a change of these two variables. An increase in loadings (particularly the normal force) towards the tip seem to be associated to a spanwise flow component present for the swept-back analysed tip. Integrated loads are ranked to asses wind turbine tip overall performance. It results from the comparison that a better tip shape that produced better torque to thrust ratios in both forces and moments is a geometry that has the end tip at the pitch axis. The work here presented shows that CFD may prove to be useful to complement 2D based methods on the design of new wind turbine blade tips.

CFD predictions of transition and distributed roughness over a wind turbine airfoil

ows and distributed roughness over a wind turbine airfoil to account for variable life cycle operational regimes are presented. The panel method code Xfoil 6.96 and the CFD commercial code Fluent 12.0.3 Beta are evaluated for the prediction of 2D wind turbine airfoil aerodynamic performance. The aim of the work is to asses the accuracy of the various methods in the determination of integrated loads (i.e. Cl, Cd, L/D and Cm1=4) for airfoils with both clean and rough surfaces that simulate contamination arising from wind turbine operational life. Numerical data are compared to experimental results for the NREL S814 wind turbine airfoil. Xfoil and Fluent are used to predict the three states present in a wind turbine blade life cycle. The aerodynamic characteristics are calculated with transition and no roughness (also called clean conguration), with fully turbulent ow and no roughness and nally applying roughness under fully turbulent ow conditions. Validation cases are presented where computations are compared to experimental data for cases with locally distributed roughness at the leading edge of the airfoil. Roughness of this type simulates airfoil contamination by bugs, dirt or debris. Results for smooth clean surfaces and fully turbulent ows (e.g. tripped boundary layer) are very similar for the two codes for attached or mildly separated ows. The transitional model (k ! Re ) of Fluent gives reasonable results when compared to transition free experimental data and Xfoil computed with free transition. To account for roughness eects, the SST k ! model modied to take into account surface roughness has been used in its Fluent version. CFD calculations, where roughness is modelled, are in good agreement with experimental data. It is shown that roughness originated from contamination has a more damaging eect on aerodynamics than a boundary layer tripping. It is concluded that CFD can simulate all the variety of ows that an operational wind turbine airfoil can encounter throughout its life cycle.

Adaptation strategies for high order discontinuous Galerkin methods based on Tau-estimation

In this paper three p-adaptation strategies based on the minimization of the truncation error are presented for high order discontinuous Galerkin methods. The truncation error is approximated by means of a ?-estimation procedure and enables the identification of mesh regions that require adaptation. Three adaptation strategies are developed and termed a posteriori, quasi-a priori and quasi-a priori corrected. All strategies require fine solutions, which are obtained by enriching the polynomial order, but while the former needs time converged solutions, the last two rely on non-converged solutions, which lead to faster computations. In addition, the high order method permits the spatial decoupling for the estimated errors and enables anisotropic p-adaptation.These strategies are verified and compared in terms of accuracy and computational cost for the Euler and the compressible Navier-Stokes equations. It is shown that the two quasi-a priori methods achieve a significant reduction in computational cost when compared to a uniform polynomial enrichment. Namely, for a viscous boundary layer flow, we obtain a speedup of 6.6 and 7.6 for the quasi-a priori and quasi-a priori corrected approaches, respectively.

An interior penalty stabilised incompressible discontinuous Galerkin–Fourier solver for implicit large eddy simulations

Abstract We present an implicit Large Eddy Simulation (iLES) h / p high order (≥2) unstructured Discontinuous Galerkin–Fourier solver with sliding meshes. The solver extends the laminar version of Ferrer and Willden, 2012 [34] , to enable the simulation of turbulent flows at moderately high Reynolds numbers in the incompressible regime. This solver allows accurate flow solutions of the laminar and turbulent 3D incompressible Navier–Stokes equations on moving and static regions coupled through a high order sliding interface. The spatial discretisation is provided by the Symmetric Interior Penalty Discontinuous Galerkin (IP-DG) method in the x – y plane coupled with a purely spectral method that uses Fourier series and allows efficient computation of spanwise periodic three-dimensional flows. Since high order methods (e.g. discontinuous Galerkin and Fourier) are unable to provide enough numerical dissipation to enable under-resolved high Reynolds computations (i.e. as necessary in the iLES approach), we adapt the laminar version of the solver to increase (controllably) the dissipation and enhance the stability in under-resolved simulations. The novel stabilisation relies on increasing the penalty parameter included in the DG interior penalty (IP) formulation. The latter penalty term is included when discretising the linear viscous terms in the incompressible Navier–Stokes equations. These viscous penalty fluxes substitute the stabilising effect of non-linear fluxes, which has been the main trend in implicit LES discontinuous Galerkin approaches. The IP-DG penalty term provides energy dissipation, which is controlled by the numerical jumps at element interfaces (e.g. large in under-resolved regions) such as to stabilise under-resolved high Reynolds number flows. This dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps, thus providing an efficient stabilization mechanism for iLES. The IP-DG stabilisation is complemented with a Spectral Vanishing Viscosity (SVV) method, in the z -direction, to enhance stability in the continuous Fourier space. The coupling between the numerical viscosity in the DG plane and the SVV damping, provides an efficient approach to stabilise high order methods at moderately high Reynolds numbers. We validate the formulation for three turbulent flow cases: a circular cylinder at Re = 3900 , a static and pitch oscillating NACA 0012 airfoil at Re = 10000 and finally a rotating vertical-axis turbine at Re = 40000 , with Reynolds based on the circular diameter, airfoil chord and turbine diameter, respectively. All our results compare favourably with published direct numerical simulations, large eddy simulations or experimental data. We conclude that the DG-Fourier high order solver, with IP-SVV stabilisation, proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery.

Insights on Aliasing Driven Instabilities for Advection Equations with Application to Gauss–Lobatto Discontinuous Galerkin Methods

We analyse instabilities due to aliasing errors when solving one dimensional non-constant advection speed equations and discuss means to alleviate these types of errors when using high order discontinuous Galerkin (DG) schemes. First, we compare analytical bounds for the continuous and discrete version of the PDEs. Whilst traditional

Onset of three-dimensional flow instabilities in lid-driven circular cavities

L^2

CFD for Wind and Tidal Offshore Turbines

L2 norm energy bounds applied to the discrete PDE do not always predict the physical behaviour of the continuous version of the equation, more strict elliptic norm bounds correctly bound the behaviour of the continuous PDE. Having derived consistent bounds, we analyse the effectiveness of two stabilising techniques: over-integration and split form variations (conservative, non-conservative and skew-symmetric). Whilst the former is shown to not alleviate aliasing in general, the latter ensures an aliasing-free solution if the splitting form of the discrete PDE is consistent with the continuous equation. The success of the split form de-aliasing is restricted to DG schemes with the summation-by-parts simultaneous-approximation-term properties (e.g. DG with Gauss–Lobatto points). Numerical experiments are included to illustrate the theoretical findings.