Supporting computer code for the paper 'Energy stable and momentum conserving hybrid finite element method for the incompressible Navier--Stokes equations'

Supporting computer code for the paper 'Energy stable and momentum conserving hybrid finite element method for the incompressible Navier-Stokes equations' in SIAM Journal on Scientific Computing

An embedded--hybridized discontinuous Galerkin finite element method for the Stokes equations

We present and analyze a new embedded--hybridized discontinuous Galerkin finite element method for the Stokes problem. The method has the attractive properties of full hybridized methods, namely an $H({\rm div})$-conforming velocity field, pointwise satisfaction of the continuity equation and \emph{a priori} error estimates for the velocity that are independent of the pressure. The embedded--hybridized formulation has advantages over a full hybridized formulation in that it has fewer global degrees-of-freedom for a given mesh and the algebraic structure of the resulting linear system is better suited to fast iterative solvers. The analysis results are supported by a range of numerical examples that demonstrate rates of convergence, and which show computational efficiency gains over a full hybridized formulation

A locally conservative and energy-stable finite element for the Navier--Stokes problem on time-dependent domains

We present a finite element method for the incompressible Navier--Stokes problem that is locally conservative, energy-stable and pressure-robust on time-dependent domains. To achieve this, the space--time formulation of the Navier--Stokes problem is considered. The space--time domain is partitioned into space--time slabs which in turn are partitioned into space--time simplices. A combined discontinuous Galerkin method across space--time slabs, and space--time hybridized discontinuous Galerkin method within a space--time slab, results in an approximate velocity field that is $H({\rm div})$-conforming and exactly divergence-free, even on time-dependent domains. Numerical examples demonstrate the convergence properties and performance of the method

A pressure-robust embedded discontinuous Galerkin method for the Stokes problem by reconstruction operators

The embedded discontinuous Galerkin (EDG) finite element method for the Stokes problem results in a point-wise divergence-free approximate velocity on cells. However, the approximate velocity is not H(div)-conforming and it can be shown that this is the reason that the EDG method is not pressure-robust, i.e., the error in the velocity depends on the continuous pressure. In this paper we present a local reconstruction operator that maps discretely divergence-free test functions to exactly divergence-free test functions. This local reconstruction operator restores pressure-robustness by only changing the right hand side of the discretization, similar to the reconstruction operator recently introduced for the Taylor--Hood and mini elements by Lederer et al. (SIAM J. Numer. Anal., 55 (2017), pp. 1291--1314). We present an a priori error analysis of the discretization showing optimal convergence rates and pressure-robustness of the velocity error. These results are verified by numerical examples. The motivation for this research is that the resulting EDG method combines the versatility of discontinuous Galerkin methods with the computational efficiency of continuous Galerkin methods and accuracy of pressure-robust finite element methods

An exactly mass conserving space-time embedded-hybridized discontinuous Galerkin method for the Navier-Stokes equations on moving domains

This paper presents a space-time embedded-hybridized discontinuous Galerkin (EHDG) method for the Navier--Stokes equations on moving domains. This method uses a different hybridization compared to the space-time hybridized discontinuous Galerkin (HDG) method we presented previously in (Int. J. Numer. Meth. Fluids 89: 519--532, 2019). In the space-time EHDG method the velocity trace unknown is continuous while the pressure trace unknown is discontinuous across facets. In the space-time HDG method, all trace unknowns are discontinuous across facets. Alternatively, we present also a space-time embedded discontinuous Galerkin (EDG) method in which all trace unknowns are continuous across facets. The advantage of continuous trace unknowns is that the formulation has fewer global degrees-of-freedom for a given mesh than when using discontinuous trace unknowns. Nevertheless, the discrete velocity field obtained by the space-time EHDG and EDG methods, like the space-time HDG method, is exactly divergence-free, even on moving domains. However, only the space-time EHDG and HDG methods result in divergence-conforming velocity fields. An immediate consequence of this is that the space-time EHDG and HDG discretizations of the conservative form of the Navier--Stokes equations are energy stable. The space-time EDG method, on the other hand, requires a skew-symmetric formulation of the momentum advection term to be energy-stable. Numerical examples will demonstrate the differences in solution obtained by the space-time EHDG, EDG, and HDG methods

Energy stable and momentum conserving hybrid finite element method for the incompressible Navier-Stokes equations

A hybrid method for the incompressible Navier-Stokes equations is presented. The method inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equal-order interpolations can be used for the velocity and pressure fields, and mass can be conserved locally. Using continuous Lagrange multiplier spaces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same as for a continuous Galerkin method on the same mesh. Different from our earlier investigations on the approach for the Navier-Stokes equations, the pressure field in this work is discontinuous across cell boundaries. It is shown that this leads to very good local mass conservation and, for an appropriate choice of finite element spaces, momentum conservation. Also, a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed, even in the absence of a pointwise solenoidal velocity field. Mass conservation, momentum conservation, and global energy stability are proved for the time-continuous case and for a fully discrete scheme. The presented analysis results are supported by a range of numerical simulations. © 2012 Society for Industrial and Applied Mathematics

Energy Stable and Momentum Conserving Hybrid Finite Element Method for the Incompressible Navier–Stokes Equations

A hybrid method for the incompressible Navier–Stokes equations is presented. The ethod inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods hen momentum advection becomes significant, equal-order interpolations can be used for the velocity nd pressure fields, and mass can be conserved locally. Using continuous Lagrange multiplier paces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same s for a continuous Galerkin method on the same mesh. Different from our earlier investigations on he approach for the Navier–Stokes equations, the pressure field in this work is discontinuous across ell boundaries. It is hown that this leads to very good local mass conservation and, for an appropriate hoice of finite element spaces, momentum conservation. Also, a new form of the momentum ransport terms for the method is constructed such that global energy stability is guaranteed, even n the absence of a pointwise solenoidal velocity field. Mass conservation, momentum conservation, nd global energy stability are proved for the time-continuous case and for a fully discrete scheme. he presented analysis results are supported by a range of numerical simulations.Hydraulic EngineeringCivil Engineering and Geoscience

Rivers running deep : complex flow and morphology in the Mahakam River, Indonesia

Rivers in tropical regions often challenge our geomorphological understanding of fluvial systems. Hairpin bends, natural scours, bifurcate meander bends, tie channels and embayments in the river bank are a few examples of features ubiquitous in tropical rivers. Existing observation techniques fall short to grasp the complex governing processes of flow and morphology. In this thesis new observational techniques are introduced and applied to study the Mahakam River, East Kalimantan, Indonesia. The observations reveal a new type of morphological regime, characterized by non-harmonic meanders, scour and strong variation of the cross-sectional area. The anomalous geometry induces a complex three-dimensional flow pattern causing longitudinal flow to be concentrated near the bed of the river. In Chapter 2 a wavelet based technique is introduced to characterize meander shape in a quantitative, objective manner. A scale space forest composed of a set of rooted trees represents the meandering planform. Based on the rooted trees, the locally dominant meander wavelengths are defined along the river. Sub-meander scale spectral density in the wavelet transform is used to determine a set of metrics quantifying bend skewness and fattening. Negative fattening parameterizes the so-called non-harmonic or hairpin bend character of meanders. The super-meander scale tree represents the embedding of meanders into larger-scale fluctuations, spanning from double-headed meander scales until the scale of the valley thalweg. The new approach is used to quantify the anomalous planform geometry of the Mahakam River in a comparison with the Red River and the Purus River. The geometry of the Mahakam River is analyzed into more detail in Chapter 3, where the highly curved non-harmonic meanders are related to deep scours in the river bed. A total of 35 scours is identified which exceed three times the average river depth, and four scours exceed the river depth over four times. The maximum scour depth strongly correlates with channel curvature and systematically occurs half a river width upstream of the bend apex. Most scours occur in a freely meandering zone of the river. A systematic reconnaissance of the river banks reveals a switch of erosion-deposition patterns at high curvature. Advancing banks normally observed at the inner side of a bend are mostly found at the outer side of high-curvature reaches, while eroding banks switch from the outer side for mildly curved bends to the inner side for bends with high curvature. The overall lateral migration rate of the river is low. These results indicate a switch of morphological regime at high curvatures, which requires detailed flow measurements to unravel the underlying physical processes. Taking flow measurements in the deep scours of the Mahakam River presents a challenge to contemporary methods in hydrography. Acoustic Doppler Current Profilers (ADCPs) are capable of profiling flow velocity over large distances from a research vessel, but the existent data processing techniques assume homogeneity of the flow between the divergent acoustic beams. This assumption fails for complex three dimensional flows as found in the scours. In Chapter 4 a new ADCP data processing technique is developed that strongly reduces the extent over which the flow needs to be assumed homogeneous. The new method is applied to flow measurements collected in a river bend with a scour exceeding 40 m depth. Results based on the new approach reveal secondary flow patterns which remain invisible adopting the conventional method. Chapter 5 aims to better understand flow in sharp bends, by combining analyses of the flow measurements from a deep scour with Large Eddy Simulations of the flow. The three-dimensional flow field is strongly dominated by horizontal circulations at both sides of the scour. The dramatic increase in cross-sectional area (from 2200 m2 to 7000 m2 ) plays a crucial role in the generation of the two horizontal recirculation cells. An existing formulation to predict water surface gradients in bends is extended to include the effect of cross-sectional area variations, next to the effect of curvature changes. Variation in the cross-sectional area develops adverse water surface gradients explaining the flow recirculation. The depth increase toward the scour causes a strong downward flow (up to 12 cm s − 1 ) creating a non-hydrostatic pressure distribution, steering the core of the flow toward the bed. The latter aspect is poorly reproduced by the Large Eddy Simulations, which may relate to the representation of turbulent shear stresses. In Chapter 6 a novel technique is introduced to better monitor turbulence properties in complex river flows from ADCP measurements, exploiting what is discarded in observations of the mean flow. It extends the so-called variance method, using two ADCPs instead of one. The availability of eight acoustic beams, four from each ADCP, changes an otherwise unsolvable set of equations with six unknowns into an overdetermined system of eight equations with six unknowns. This allows to solve for the complete Reynolds stress tensor, yielding profiles of Reynolds stresses over almost the entire water column. Widely applied assumptions on turbulence anisotropy ratios are shown to be incorrect, which reveals a knowledge gap in open channel turbulence. Chapter 7 uses the technique developed in Chapter 6 to investigate the degree in which bed shear stress can be monitored continuously from an ADCP mounted horizontally at the river bank (HADCP). A calibrated boundary layer model is applied to estimate time-series of cross-river bed-shear stress profiles from HADCP velocity measurements. It is concluded the HADCP measurement can represent the regional bed shear stresses, as inferred from a logarithmic velocity profile, reasonably well. These regional bed-shear stresses, in turn, poorly represent the local estimates obtained from coupled ADCP measurements, which are more directly related to processes of sediment transport and complex river morphology. Detailed observations of turbulence properties may be the key to improve our understanding of complex river flow and morphology. </p