Benjamin Sanderse

Accuracy analysis of explicit Runge-Kutta methods applied to the incompressible Navier-Stokes equations

This paper investigates the temporal accuracy of the velocity and pressure when explicit Runge-Kutta methods are applied to the incompressible Navier-Stokes equations. It is shown that, at least up to and including fourth order, the velocity attains the classical order of accuracy without further constraints. However, in case of a time-dependent gradient operator, which can appear in case of time-varying meshes, additional order conditions need to be satisfied to ensure the correct order of accuracy. Furthermore, the pressure is only first-order accurate unless additional order conditions are satisfied. Two new methods that lead to a second-order accurate pressure are proposed, which are applicable to a certain class of three- and four-stage methods. A special case appears when the boundary conditions for the continuity equation are independent of time, since in that case the pressure can be computed to the same accuracy as the velocity field, without additional cost. Relevant computations of decaying vortices and of an actuator disk in a time-dependent inflow support the analysis and the proposed methods.

Energy-conserving Runge-Kutta methods for the incompressible Navier-Stokes equations

Energy-conserving methods have recently gained popularity for the spatial discretization of the incompressible Navier-Stokes equations. In this paper implicit Runge-Kutta methods are investigated which keep this property when integrating in time. Firstly, a number of energy-conserving Runge-Kutta methods based on Gauss, Radau and Lobatto quadrature are constructed. These methods are suitable for convection-dominated problems (such as turbulent flows), because they do not introduce artificial diffusion and are stable for any time step. Secondly, to obtain robust time-integration methods that work also for stiff problems, the energy-conserving methods are extended to a new class of additive Runge-Kutta methods, which combine energy conservation with L-stability. In this class, the Radau IIA/B method has the best properties. Results for a number of test cases on two-stage methods indicate that for pure convection problems the additive Radau IIA/B method is competitive with the Gauss methods. However, for stiff problems, such as convection-dominated flows with thin boundary layers, both the higher order Gauss and Radau IIA/B method suffer from order reduction. Overall, the Gauss methods are the preferred method for energy-conserving time integration of the incompressible Navier-Stokes equations.

Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier-Stokes equations

Harlow and Welch [Phys. Fluids 8 (1965) 2182-2189] introduced a discretization method for the incompressible Navier-Stokes equations conserving the secondary quantities kinetic energy and vorticity, besides the primary quantities mass and momentum. This method was extended to fourth order accuracy by several researchers [25,14,21]. In this paper we propose a new consistent boundary treatment for this method, which is such that continuous integration-by-parts identities (including boundary contributions) are mimicked in a discrete sense. In this way kinetic energy is exactly conserved even in case of non-zero tangential boundary conditions. We show that requiring energy conservation at the boundary conflicts with order of accuracy conditions, and that the global accuracy of the fourth order method is limited to second order in the presence of boundaries. We indicate how non-uniform grids can be employed to obtain full fourth order accuracy.

Runge-Kutta methods for the incompressible Navier-Stokes equations

Time integration of the incompressible Navier-Stokes equations with Runge-Kutta methods is not straightforward due to the differential-algebraic nature of the equations. In this work we investigate the temporal order of accuracy of velocity and pressure for both explicit and implicit methods. This is done by applying existing theory on Runge-Kutta methods for differential-algebraic equations to the incompressible Navier-Stokes equations. We focus on a specific class of Runge-Kutta methods, namely symplectic Runge-Kutta methods, which in the case of the incompressible Navier-Stokes equations are energy-conserving.

Efficient ultimate load estimation for offshore wind turbines using interpolating surrogate models

During the design phase of an offshore wind turbine, it is required to assess the impact of loads on the turbine life time. Due to the varying environmental conditions, the effect of various uncertain parameters has to be studied to provide meaningful conclusions. Incorporating such uncertain parameters in this regard is often done by applying binning, where the probability density function under consideration is binned and in each bin random simulations are run to estimate the loads. A different methodology for quantifying uncertainties proposed in this work is polynomial interpolation, a more efficient technique that allows to more accurately predict the loads on the turbine for specific load cases. This efficiency is demonstrated by applying the technique to a power production test problem and to IEC Design Load Case 1.1, where the ultimate loads are determined using BLADED. The results show that the interpolating polynomial is capable of representing the load model. Our proposed surrogate modeling approach therefore has the potential to significantly speed up the design and analysis of offshore wind turbines by reducing the time required for load case assessment.

Simulation of Elongated Bubbles in a Channel Using the Two-Fluid Model

This paper investigates the capability of the two-fluid model to predict the bubble drift velocity of elongated bubbles in channels. The two-fluid model is widely used in the oil and gas industry for dynamic multiphase pipeline simulations. The bubble drift velocity is an important quantity in predicting pipeline flushing and slug flow. In this paper, it is shown that the two-fluid model in its standard form predicts a bubble drift velocity of (similar to the shallow water equations), instead of the exact value of as derived by Benjamin[1]. Modifying the two-fluid model with the commonly employed momentum correction parameter leads to a steady solution (in a moving reference frame), but still predicts an erroneous bubble drift velocity. To get the correct bubble drift velocity, it is necessary to include the pressure variation along the channel height due to both the hydrostatic component and the vertical momentum flux. GRAPHICAL ABSTRACT