In flow-by capacitive deionization (CDI) brackish water flows between two electrodes that capacitively remove salt. We assume low inlet concentrations so “salt shocks” appear in the electrodes and the process becomes diffusion-limited. For unit charge efficiency, a simplified model is derived consisting of two coupled partial differential equations. We obtain approximations, and exact solutions in terms of the Lambert W function, for the salt concentration as a function of time and space and for the equilibrium charge-voltage relation. These surprisingly simple solutions compare well with the results from comprehensive two-dimensional simulations. Useful analytical expressions are obtained for optimal geometrical and operational parameters that maximize the productivity and minimize the specific energy losses. By making cells much thinner the productivity can be increased an order of magnitude compared to typical values in the literature. The optimal electrode is found to be roughly six times thinner than the spacer. The associated pressure drop is around 0.4 bar per 1 mM of inlet salt concentration, making our recommendations practically feasible only for relatively low concentrations. The obtained model and analytical expressions provide useful guidance to strongly improve the design process.

A novel hyper-reduction method is proposed that conserves kinetic energy and momentum for reduced order models of the incompressible Navier-Stokes equations. The main advantage of conservation of kinetic energy is that it endows the hyper-reduced order model (hROM) with a nonlinear stability property. The new method poses the discrete empirical interpolation method (DEIM) as a minimization problem and subsequently imposes constraints to conserve kinetic energy. Two methods are proposed to improve the robustness of the new method against error accumulation: oversampling and Mahalanobis regularization. Mahalanobis regularization has the benefit of not requiring additional measurement points. Furthermore, a novel method is proposed to perform energy- and momentum-conserving temporal localization with the principle interval decomposition: new interface conditions are derived such that energy and momentum are conserved for a full time-integration instead of only during separate intervals. The performance of the new energy- and momentum-conserving hyper-reduction methods and the energy- and momentum-conserving temporal localization method is analysed using three convection-dominated test cases; a shear-layer roll-up, two-dimensional homogeneous isotropic turbulence and a time-periodic inviscid flow consisting of a vortex in a uniform background flow. Our main finding is that energy conservation in combination with oversampling or regularization leads to a robust method with excellent long time stability properties. When any of these two ingredients is missing, accuracy and/or stability is significantly impaired.

In this paper we present a complete framework for the energy-stable simulation of stratified incompressible flow in channels, using the one-dimensional two-fluid model. Building on earlier energy-conserving work on the basic two-fluid model, our new framework includes diffusion, friction, and surface tension. We show that surface tension can be added in an energy-conserving manner, and that diffusion and friction have a strictly dissipative effect on the energy. We then propose spatial discretizations for these terms such that a semi-discrete model is obtained that has the same conservation properties as the continuous model. Additionally, we propose a new energy-stable advective flux scheme that is energy-conserving in smooth regions of the flow and strictly dissipative where sharp gradients appear. This is obtained by combining, using flux limiters, a previously developed energy-conserving advective flux with a novel first-order upwind scheme that is shown to be strictly dissipative. The complete framework, with diffusion, surface tension, and a bounded energy, is linearly stable to short wavelength perturbations, and exhibits nonlinear damping near shocks. The model yields smoothly converging numerical solutions, even under conditions for which the basic two-fluid model is ill-posed. With our explicit expressions for the dissipation rates, we are able to attribute the nonlinear damping to the different dissipation mechanisms, and compare their effects.

The pressure-free two-fluid model (PFTFM) is a recent reformulation of the one-dimensional two-fluid model (TFM) for stratified incompressible flow in ducts (including pipes and channels), in which the pressure is eliminated through intricate use of the volume constraint. The disadvantage of the PFTFM was that the volumetric flow rate had to be specified as an input, even though it is an unknown quantity in case of periodic boundary conditions. In this work, we derive an expression for the volumetric flow rate that is based on the demand for energy (and momentum) conservation. This leads to PFTFM solutions that match those of the TFM, justifying the validity and necessity of the derived choice of volumetric flow rate. Furthermore, we extend an energy-conserving spatial discretization of the TFM, in the form of a finite volume scheme, to the PFTFM. We propose a discretization of the volumetric flow rate that yields discrete momentum and energy conservation. The discretization is extended with an energy-conserving discretization of the source terms related to gravity acting in the streamwise direction. Our numerical experiments confirm that the discrete energy is conserved for different problem settings, including sloshing in an inclined closed tank, and a traveling wave in a periodic domain. The PFTFM solutions and the volumetric flow rates match the TFM solutions, with reduced computation time, and with exact momentum and energy conservation.

Dynamic induction control is a wind farm flow control strategy that utilises wind turbine thrust variations to accelerate breakdown of the aerodynamic wake and improve downstream turbine performance. However, when floating wind turbines are considered, additional dynamics and challenges appear that make optimal control difficult. In this work, we propose an adjoint optimisation framework for non-linear economic model-predictive control, which utilises a novel coupling of an existing aerodynamic wake model to floating platform hydrodynamics. Analysis of the frequency response for the coupled model shows that it is possible to achieve wind turbine thrust variations without inducing large motion of the rotor. Using economic model-predictive control, we find dynamic induction results that lead to an improvement of 7 % over static induction control, where the dynamic controller stimulates wake breakdown with only small variations in rotor displacement. This novel model formulation provides a starting point for the adaptation of dynamic wind farm flow control strategies for floating wind turbines.

Control-oriented models provide a basis for wind farm control to improve power production and reduce structural loading. Wake steering is considered to be one of the most promising techniques to achieve this. Wind turbine wakes under yaw misalignment are deflected downstream and have been shown to produce a curled or kidney-shaped structure. A Navier-Stokes based code called FRED was developed to model wind farm flow in 2D to perform yaw control. To tackle the differences between 2D and 3D flow, this work introduces a generalised continuity correction and wind turbine force scaling terms to the FRED framework. The effectiveness of approximating 3D results is tested by comparison with 3D simulations in the same framework. The continuity correction is now applicable to general wind directions and effective in reducing wake width and speed-up effects. The magnitude of wake deflection can be tuned using a force scaling term. However, we show that there remains a qualitative difference in the deflection profile downstream, as well as a difference in the propagation of yaw effects over time. From this study we can conclude that there is a fundamental difference between 2D and 3D flow physics in spatial and temporal dynamics which makes the 2D modelling approach challenging for control without further empirical adjustments. The necessary corrections are likely to be complex and non-physical, leading to a departure from the first principles foundation that FRED is developed from.

FLORIDyn is a parametric control-oriented dynamic model suitable to predict the dynamic wake interactions between wind turbines in a wind farm. In order to improve the accuracy of FLORIDyn, this study proposes to calibrate the tuning parameters present in the model by employing a probabilistic setting using the UQ4WIND framework. The strategy relies on constructing a surrogate model (based on polynomial chaos expansion), which is then used to perform both global sensitivity analysis and Bayesian calibration. For our analysis, a nine wind turbine configuration in a yawed setting constitutes the test case. The results of sensitivity analysis offer valuable insight into the time-dependent influence of the model parameters onto the model output. The model parameter tied to the turbine efficiency appear to be the most sensitive parameter affecting the model output. The calibrated FLORIDyn model using the Bayesian approach yield predictions much closer to the measurement data, which is equipped with an uncertainty estimate.

Wind farm flow control aims to improve wind turbine performance by reducing aerodynamic wake interaction between turbines. Dynamic, physics-based models of wind farm flows have been essential for exploring control strategies such as wake redirection and dynamic induction control. Free-vortex methods can provide a computationally efficient way to model wind turbine wake dynamics for control optimisation. We present a control-oriented free-vortex wake model of a 2D and 3D actuator disc to represent wind turbine wakes. The novel derivation of the discrete adjoint equations allows efficient gradient evaluation for gradient-based optimisation in an economic model-predictive control algorithm. Initial results are presented for mean power maximisation in a two-turbine case study. An induction control signal is found using the 2D model that is roughly periodic and supports previous results on dynamic induction control to stimulate wake mixing. The 3D model formulation effectively models a curled wake under yaw misalignment. Under time-varying wind direction, the optimisation finds solutions demonstrating both wake steering and a smooth transition to greedy control. The free-vortex wake model with gradient information shows potential for efficient optimisation and provides a promising way to further explore dynamic wind farm flow control.

A novel pressure-free two-fluid model formulation is proposed for the simulation of one-dimensional incompressible multiphase flow in pipelines and channels. The model is obtained by simultaneously eliminating the volume constraint and the pressure from the widely used two-fluid model (TFM). The resulting ‘pressure-free two-fluid model’ (PF-TFM) has a number of attractive features: (i) it features four evolution equations (without additional constraints) that can be solved very quickly with explicit time integration methods; (ii) it keeps the conservation properties of the original two-fluid model, and therefore the correct shock relations in case of discontinuities; (iii) its solutions satisfy the two TFM constraints exactly: the volume constraint and the volumetric flow constraint; (iv) it offers a convenient form to analytically analyse the equation system, since the constraint has been removed. A staggered-grid spatial discretization and an explicit Runge-Kutta time integration method are proposed, which keep the constraints exactly satisfied when numerically solving the PF-TFM. Furthermore, for the case of strongly imposed boundary conditions, a novel adapted Runge-Kutta formulation is proposed that keeps the volumetric flow exact in time while retaining high order accuracy. Several test cases confirm the theoretical properties and show the efficiency of the new pressure-free model.

A novel mathematical framework is derived for the addition of nodes to univariate and interpolatory quadrature rules. The framework is based on the geometrical interpretation of the Vandermonde matrix describing the relation between the nodes and the weights and can be used to determine all nodes that can be added to an interpolatory quadrature rule with positive weights such that the positive weights are preserved. In the case of addition of a single node, the derived inequalities that describe the regions where nodes can be added are explicit. Besides addition of nodes these inequalities also yield an algorithmic description of the replacement and removal of nodes. It is shown that it is not always possible to add a single node while preserving positive weights. On the other hand, addition of multiple nodes and preservation of positive weights is always possible, although the minimum number of nodes that need to be added can be as large as the number of nodes of the quadrature rule. In case of addition of multiple nodes the inequalities describing the regions where nodes can be added become implicit. It is shown that the well-known Patterson extension of quadrature rules is a special case that forms the boundary of these regions and various examples of the applicability of the framework are discussed. By exploiting the framework, two new sets of quadrature rules are proposed. Their performance is compared with the well-known Gaussian and Clenshaw–Curtis quadrature rules, demonstrating the advantages of our proposed nested quadrature rules with positive weights and fine granularity.

For the purpose of uncertainty propagation a new quadrature rule technique is proposed that has positive weights, has high degree, and is constructed using only samples that describe the probability distribution of the uncertain parameters. Moreover, nodes can be added to the quadrature rule, resulting in a sequence of nested rules. The rule is constructed by iterating over the samples of the distribution and exploiting the null space of the Vandermonde system that describes the nodes and weights, in order to select which samples will be used as nodes in the quadrature rule. The main novelty of the quadrature rule is that it can be constructed using any number of dimensions, using any basis, in any space, and using any distribution. It is demonstrated both theoretically and numerically that the rule always has positive weights and therefore has high convergence rates for sufficiently smooth functions. The convergence properties are demonstrated by approximating the integral of the Genz test functions. The applicability of the quadrature rule to complex uncertainty propagation cases is demonstrated by determining the statistics of the flow over an airfoil governed by the Euler equations, including the case of dependent uncertain input parameters. The new quadrature rule significantly outperforms classical sparse grid methods.

A novel method is proposed to infer Bayesian predictions of computationally expensive models. The method is based on the construction of quadrature rules, which are well-suited for approximating the weighted integrals occurring in Bayesian prediction. The novel idea is to construct a sequence of nested quadrature rules with positive weights that converge to a quadrature rule that is weighted with respect to the posterior. The quadrature rules are constructed using a proposal distribution that is determined by means of nearest neighbor interpolation of all available evaluations of the posterior. It is demonstrated both theoretically and numerically that this approach yields accurate estimates of the integrals involved in Bayesian prediction. The applicability of the approach for a fluid dynamics test case is demonstrated by inferring accurate predictions of the transonic flow over the RAE2822 airfoil with a small number of model evaluations. Here, the closure coefficients of the Spalart–Allmaras turbulence model are considered to be uncertain and are calibrated using wind tunnel measurements.

An efficient algorithm is proposed for Bayesian model calibration, which is commonly used to estimate the model parameters of non-linear, computationally expensive models using measurement data. The approach is based on Bayesian statistics: using a prior distribution and a likelihood, the posterior distribution is obtained through application of Bayes' law. Our novel algorithm to accurately determine this posterior requires significantly fewer discrete model evaluations than traditional Monte Carlo methods. The key idea is to replace the expensive model by an interpolating surrogate model and to construct the interpolating nodal set maximizing the accuracy of the posterior. To determine such a nodal set an extension to weighted Leja nodes is introduced, based on a new weighting function. We prove that the convergence of the posterior has the same rate as the convergence of the model. If the convergence of the posterior is measured in the Kullback-Leibler divergence, the rate doubles. The algorithm and its theoretical properties are verified in three different test cases: analytical cases that confirm the correctness of the theoretical findings, Burgers' equation to show its applicability in implicit problems, and finally the calibration of the closure parameters of a turbulence model to show the effectiveness for computationally expensive problems.

A novel approach is proposed to reduce, compared with the conventional binning approach, the large number of aeroelastic code evaluations that are necessary to obtain equivalent loads acting on wind turbines. These loads describe the effect of long-term environmental variability on the fatigue loads of a horizontal-axis wind turbine. In particular, Design Load Case 1.2, as standardized by IEC, is considered. The approach is based on numerical integration techniques and, more specifically, quadrature rules. The quadrature rule used in this work is a recently proposed “implicit” quadrature rule, which has the main advantage that it can be constructed directly using measurements of the environment. It is demonstrated that the proposed approach yields accurate estimations of the equivalent loads using a significantly reduced number of aeroelastic model evaluations (compared with binning). Moreover, the error introduced by the seeds (introduced by averaging over random wind fields and sea states) is incorporated in the quadrature framework, yielding an even further reduction in the number of aeroelastic code evaluations. The reduction in computational time is demonstrated by assessing the fatigue loads on the NREL 5 MW reference offshore wind turbine in conjunction with measurement data obtained at the North Sea, for both a simplified and a full load case.