The effect of a local mesh refinement on hydraulic modelling of river meanders

Large‐scale river models are generally discretized by relatively large mesh cells resulting in bathymetry discretization errors and numerical effects. These hydraulic models are generally calibrated by altering the bed roughness to compensate for these errors and effects. Consequently, the calibrated roughness values are mesh‐dependent while generally local mesh refinements are executed after model calibration to study the effects of river interventions. This study shows both the errors caused by bathymetry discretization and numerical effects for locally refined meshes. First, schematised river meanders with a flat bed in the transverse flow direction are analysed to isolate the induced numerical effects by the mesh. Afterwards, a case study is considered to verify if similar mesh influences are found in natural river meanders. Curvilinear, triangular and hybrid (combination of curvilinear and triangular cells) meshes are used with different resolutions. The analysis shows that in the schematised river meanders lower depth‐averaged flow velocities and larger water depths are simulated with coarser meshes. In the case study, substantial differences in hydrodynamics between the meshes are obtained suggesting that the bathymetry discretization is more influential than the numerical effects. Finally, it was found that triangular meshes, and rivers with narrow meander bends, are most sensitive to mesh resolution. Especially in these cases, it is desirable to refine the mesh at the desired locations before model calibration.


| INTRODUCTION
A detailed insight into flow patterns in rivers is crucial as it helps analysing the impacts of river interventions regarding, for example, sediment management and flood protection. A common approach to investigate such processes is by making use of two-dimensional depth-averaged (2DH) hydraulic models since vertical motions are assumed to be insignificant and the water depths in rivers are relatively shallow compared to the width (Altaie & Dreyfuss, 2018;Lai, 2010;Hardy, Bates, & Anderson, 1999). Therefore, the Shallow Water Equations (SWEs) are generally used, which are given by the depth-averaged continuity equation and the momentum equations (Section 2.1). In order to simulate flow pattern predictions with 2DH hydraulic models, study areas are discretized with meshes. Generally, fully curvilinear or fully triangular meshes are used to solve the governing equations in hydraulic river models. A combination of curvilinear and triangular mesh cells is also possible and is known as a hybrid mesh.
In the field of computational fluid dynamics (CFD), the optimum mesh resolution is generally found by performing a mesh convergence analysis (Tezdogan, Incecik, & Turan, 2016). In such an analysis, multiple model runs are performed with an increasing mesh resolution until a further increase in resolution does not result in a significant change in model output. However, due to computational time restrictions for large-scale river applications, hydraulic models commonly have mesh cell sizes in the order of +10 m. Consequently, the model output of such coarse 2DH river models is largely influenced by the mesh setup in terms of mesh resolution and mesh shape (Bomers, Schielen, & Hulscher, 2019). Mesh coarsening and a poor alignment between the mesh and the direction of the flow lead to a smoothed hydrograph, resulting in lower depth-averaged flow velocities and higher water depths (Caviedes-Voullième, Garcia-Navarro, & Murillo, 2012). This mesh-generated effect has a diffusion-like appearance and is known as "numerical diffusion". This effect is a result of a truncation error and is induced when numerical algorithms, such as the upwind scheme, are used to solve the advection problem in the SWEs.
Another mesh-generated effect is the accuracy of the bathymetry discretization reflecting how well the digital elevation model (DEM) is schematised by the mesh. Typically, the bathymetry discretization accuracy is higher for high-resolution meshes. A mesh should be fine enough to capture important flow features and geometrical structures. Caviedes-Voullième et al. (2012) showed that small details in river bathymetry can sometimes affect large-scale flow behaviour.
The outcomes of hydraulic river models largely depend on the bathymetry discretization since the river cross-sectional area can be overestimated and underestimated by low-resolution meshes, influencing the discharge capacity and consequently the simulated water levels (Bomers et al., 2019;Caviedes-Voullième et al., 2012).
Numerical diffusion and bathymetry discretization influence model outcomes in large-scale river models since relatively low mesh resolutions are used. Therefore, hydraulic models are generally calibrated by altering the channel roughness until the model output is close to measurements. As such, the channel roughness compensates for the mesh-related errors (Bomers et al., 2019) and these calibrated roughness values are mesh-dependent. However, to assess the impact of river interventions, meshes are commonly locally refined after model calibration at the intervention locations to accurately schematise the bathymetry, for example, to accurately simulate the discharge partitioning at the bifurcation point of a side channel (Ji & Zhang, 2019) or to simulate the effect of in-stream structures for river restoration to improve physical habitats (Theodoropoulos et al., 2020). However, a local mesh refinement results in a change of the numerical effects and re-calibration might be required. The effects of a local increase in mesh resolution are not yet fully understood as it is currently unknown to what extent a locally refined mesh in narrow and wide river bends influences model outcomes.
In Caviedes-Voullième et al. (2012) and Bomers et al. (2019), the importance of mesh resolution is discussed. In these studies, the influences of numerical diffusion and bathymetry discretization effects were interrelated as both studies considered case studies. Consequently, it is unclear to what extent numerical effects or the bathymetry accuracy impact hydraulic river modelling outcomes separately.
Therefore, this study aims to understand how numerical effects and the accuracy of the bathymetry discretization affect model outcomes to identify if a local refinement of the mesh is appropriate after model calibration. We express model outcomes in terms of simulated water depths, water levels and depth-averaged flow velocity profiles in meander bends. To assess the model outcomes, we construct various meshes for both (1) schematised river meanders with a flat bed in the transverse flow direction to exclude the effects of bathymetry accuracy and (2) a case study where both numerical effects and bathymetry discretization influence model outcomes. A local mesh refinement is performed to determine its effects on local water depths and flow velocities.
The paper is organised as follows. Section 2 introduces the set-up of the hydraulic models of the case study and the schematised river meanders. Section 3 firstly presents the results of the schematised river meanders to solely consider the numerical effects. Next, the results of the case study are presented to also consider bathymetry discretization effects. Section 4 discusses the results of the considered cases. Section 5 presents the conclusions of the paper.
In the SWEs, t represents the time (s), u and v are the depth-averaged flow velocities (m=s) in respectively x À and y À direction, g is the gravitational acceleration (m=s 2 ), ζ is the water level (m), ν is the kinematic viscosity (m 2 =s). ρ represents the density of the water (kg=m 3 ), which is assumed to be incompressible. f is the Coriolis frequency (rad=s). τ b u and τ b v are respectively bottom friction (N=m 2 ) in x-and y-direction. τ w u and τ w v are the wind friction acting at the free surface (N=m 2 ) in x À and y À direction respectively. Furthermore, it is important to recognise that the inertia, advection, hydrostatic pressure, diffusion, Coriolis force, wind and bottom friction terms are represented with, respectively, terms 1 until 7 in Equations (2) and (3).

| Case study: Grensmaas stretch
The Grensmaas, a stretch of the Meuse River, is used as a case study. The Grensmaas consists of both moderate and sharp mean- An initial water level was set throughout the entire spatial domain which corresponds to a discharge of 250m 3 =s. At km-2, the system was forced with a semi-stationary discharge, which can be divided into two categories: (i) low-and (ii) high-range semi-stationary discharges.
The model domain was initially forced with a low-range of 250m 3 =s and was eventually increased to a high range of 3430m 3 =s after three days. A predefined rating curve based on measurements and hydraulic model simulations was used to set up the downstream boundary condition at km-64.

| Schematised river meanders
The river characteristics of the schematised river meanders were chosen such that these correspond with those of the Grensmaas.
In this way, we can relate the outcomes of the schematised river meanders with those of the case study. For the development of the schematised river meanders, we used the following river characteristics of the Grensmaas: (i) main channel width; (ii) floodplain width, which is half of the average total floodplains width; (iii) floodplain height with respect to the bed level of the main channel; (iv) river slope and (v) bottom friction in the main channel and floodplains.
We set up schematised river meanders based on the intrinsic function (Langbein & Leopold, 1966). The so-called sine-generated curve (Langbein & Leopold, 1966) describes the rate of change in direction along its path by a sinusoidal function. The angle between the meander curve and the horizontal is defined as the "direction angle" (θ). We adopted this theoretical approach since such curves are formulated to have the least average curvature per unit length. This results in the least total work needed for a fluid particle to accelerate (by changing its direction) through the river meander. Consequently, this leads to flow patterns that are comparable to those in a natural river meander (Langbein & Leopold, 1966).
To capture the extremes of these geometrical characteristics in the Grensmaas, we analysed three meander curves: (i) a moderate curve without floodplains; (ii) a sharp curve without floodplains and (iii) a sharp curve with floodplains ( Figure 2). All schematised river meanders consisted of a total meander length of 30 km. Furthermore, a so-called "valley slope" of 4:49e À 04m=m was applied, which is a slope along a straight line through the meander bends. In this way, identical bed levels were obtained over a specified distance in the horizontal direction for both the moderate and sharp meanders.
A constant initial water level of 0m with respect to the bed level of the main channel at x ¼ 0 was set throughout the entire spatial domains of the three schematised river meanders. At x ¼ 0, all three schematised river meanders were forced with a unique semistationary discharge until similar water levels are obtained between the schematised river meanders. We divided the forcing into two low and three high-range semi-stationary discharges. To obtain similar water levels, we used a low discharge range of 946 m 3 /s for the moderate meander with no floodplains and 844 m 3 /s for both the sharp meander with and without floodplains. For the high-discharge range, we used discharges of 3,002, 2,679 and 3,801 m 3 /s for the moderate meander without floodplains, the sharp meander without floodplains and the sharp meander with floodplains, respectively. Each schematised case was initially forced with the low-range and was eventually increased to the high range after ten days. We used predefined rating curves based on steady uniform flow considerations for the downstream boundary conditions. For the schematised river meanders, we reviewed the hydrodynamics in the second river bends simulated by various mesh structures

| Computation scheme of the numerical meshes
To discretize the governing equations, we applied a finite volume method on a staggered scheme. Water levels are stored at the cell centres whereas velocity variables are found at the cell faces (Harlow & Welch, 1965;Tu, Yeoh, & Liu, 2013). This approach differs from a collated arrangement, in which all SWEs variables are discretized at the same positions (Meier, Alves, & Mori, 1999 et al., 2013). According to Stelling (1983), a staggered scheme is a more effective discretization method for the SWEs as the number of mesh points is reduced by a factor four in comparison to a collocated scheme reducing the computation time.
Triangular and hybrid meshes have gained much attention over the recent years due to the flexibility they provide in complex geometries.
For computational efficiency, we used orthogonal grids such that the pressure gradients only depend on two pressure points, which reduces computation time and results in higher model accuracy (Kleptsova, Pietrzak, & Stelling, 2009). The orthogonality principle enforces the criteria that the corners of two adjacent grid cells are placed on a common circle and that the line segment, which connects the circumcenter of two neighbouring cells, intersects orthogonally with the interface between them (Bomers et al., 2019;Casulli & Walters, 2000;Kernkamp, Dam, Stelling, & Goede, 2011;Kleptsova et al., 2009). In this study, we aimed for a maximum deviation of 2 from perfectly orthogonal mesh cells (90 )

| Meshes
Several meshes were constructed to analyse the influence of different mesh shapes and mesh resolutions on hydraulic modelling outcomes. Here, we present the meshes for the schematised river meanders and the Grensmaas respectively followed by the locally refined meshes for the Grensmaas.   • Triangular low-resolution mesh: the set-up is similar to the highresolution variant, but now with 4 mesh cells in the main channel.
• Hybrid high-resolution mesh: a curvilinear mesh was used in the main channel with 20 mesh cells in the transverse flow direction.
The floodplains were discretized with a triangular mesh such that each triangular mesh cell at the main channel-floodplain boundary is connected to one curvilinear mesh cell.
• Hybrid low-resolution mesh: the set-up is similar to the highresolution variant, but now with 10 mesh cells in the main channel.

| Meshes for the Grensmaas
Similar mesh structures were constructed for the Grensmaas as for the schematised case with floodplains (Table 1 and Figure 4d

| Meshes for the Grensmaas with local refinements
To gain a better understanding of how a local increase in mesh resolution influences hydraulic river modelling outcomes in the case study, we considered two local mesh refinements for each constructed lowresolution mesh (Table 1). A local mesh refinement was implemented in an area with narrow floodplains (Figure 1: around CS 1) and wide floodplains (Figure 1: around CS 2). For each mesh shape, the local refinements had the same resolution as the higher resolution variants.
In order to prevent mesh cells being connected to more mesh cells than the number of its net links, we applied triangular mesh cells at the transition between the low resolution and the locally refined regions. However, applying triangular mesh cells at the transition is at the expense of the orthogonality and smoothness criteria (ratio of the areas of the two adjacent mesh cells) of the locally refined curvilinear mesh due to the exponentially increasing mesh cell dimensions towards the outer edges of the floodplains. Therefore, the cell dimensions were gradually increased using quadrilateral mesh cells ensuring that the smoothness criteria were met between the low-resolution mesh and the locally refined regions.

| Schematised river meanders
The following section presents the results of the schematised moderate river meander without floodplains followed by the sharp river meander without and with floodplains respectively as this order helps to isolate the induced numerical effects.

| Moderate river meander (main channel)
In the moderate river meander, it was found that the four meshes pre- However, larger water depths were predicted by the coarser curvilinear and triangular meshes (Figure 5a). Furthermore, we observed greater differences between the minimum and maximum depthaveraged velocities through the river bend with the high-resolution variants of both mesh shapes (Figure 6a-d). A more diffused (uniform) depth-averaged flow velocity profile was predicted by the coarser meshes, which illustrates the influence of the numerical diffusion.
With the lower discharge range, we found the same differences between the meshes, but to a lesser extent.
In terms of the differences between the mesh shapes, larger water depths and lower depth-averaged flow velocities were obtained with the high-resolution triangular mesh than with the high-resolution curvilinear mesh (Figures 5a and 6a-d). The opposite occurred for the low-resolution variants as the lowest resolution triangular mesh simulated lower water depths and higher maximum depth-averaged flow velocities (Figure 5a and 6a-d).
When applying coarser meshes, larger water depth and depthaveraged flow velocity differences were obtained than when using a different mesh shape indicating that mesh resolution had a larger effect on model outcomes than mesh shape.

| Sharp river meander (main channel)
In comparison to the more moderate river meander, we obtained similar flood patterns using all four meshes in the sharp river meander (Figure 5b and 6e-h). In terms of the differences between the meshes, larger water depths and lower depth-averaged flow velocities were simulated with the coarser meshes (Figure 6e-h). However, in the sharper bend greater differences in depth-averaged flow velocities were predicted between the inner and outer bend using the four meshes compared to the more moderate bend.

| Numerical and bathymetry discretization effects
Similar to the schematised river meanders, higher water levels were simulated by the coarser variants of the curvilinear, hybrid and triangular meshes for the highest discharge range in the narrow river bend ( Figure 8a). Nonetheless, in comparison to the schematised river meanders, substantially larger water level and depth-averaged flow velocity differences were obtained between the meshes in the case study ( Figure 8). In addition, significant differences in water levels and depth-averaged flow velocities between the meshes were found at the lower discharge range (Table 2).
With respect to the mesh shapes, the curvilinear and hybrid meshes predicted comparable depth-averaged flow velocities and water levels under the same level of mesh resolution. Under the same mesh resolution, water level differences between the curvilinear and hybrid meshes ranged 0 À 5cm when considering both the low and high discharge scenarios. The triangular meshes predicted substantially higher water levels, where the differences between the curvilinear meshes with comparable mesh resolutions ranged 9 À 87cm. This indicates that a higher resolution was required with the triangular meshes in order to achieve the same level of accuracy as the curvilinear and hybrid meshes.
Less depth-averaged flow velocity differences were found between the meshes in the floodplain areas of the wide river bend (Figure 8c,d). Consequently, throughout the river bend with wide floodplains, relatively smaller differences in depth-averaged flow velocities were obtained between the meshes compared to the narrow river bend. Therefore, due to the large floodplain effect, bathymetry discretization and numerical effects which were predominantly induced in the main channel were distributed throughout the model domain and hence dampened. These findings were in line with the results of the schematised sharp river meander with floodplains.

| Impact of local mesh refinements
When considering the local mesh refinements, the simulated water levels at the bend apex CS 1 converged towards those of the higher F I G U R E 5 Cross-sectional view of the simulated water depth for the moderate and sharp river meander (main channel cases) with the highest discharge range at CS 1 and CS 2 by the four considered meshes for each river meander. However, the simulated water levels at CS 1 by the former deviated significantly compared to the latter mesh. This indicated that curvilinear and hybrid meshes were more responsive to local mesh refinements than triangular meshes.
Similar results were obtained in the wide river bend (Table 2).
However, in comparison to the narrow river bend, water levels simulated by the Grensmaas_Tri_LR_Loc_Ref became closer to those of the higher resolution variant. This can be explained by the dampening effect of wide floodplains, which led to overall closer water level predictions by all six meshes.
The outcomes in both the wide and narrow river bends showed that a local mesh refinement was an effective technique to reduce the effects of the bathymetry discretization locally since the water levels and flow velocities converged towards those simulated by the higher resolution meshes. However, since the discretized bathymetry between the locally refined meshes was different from that of the previous low-resolution meshes, affecting the river's discharge capacity, re-calibration of the model is required. Therefore, we advise first carrying out the local mesh refinement before executing the calibration procedure. During the calibration, the roughness coefficient will be altered to compensate for, among others, errors in the discretized bathymetry. Therefore, the calibrated roughness coefficients for the low-resolution mesh should not be applied to a locally refined mesh.
This especially applies to triangular meshes and river sections with narrow meander bends since these were found to be most sensitive to a change in mesh resolution compared to triangular meshes and wide meander bends. It can be time consuming to calibrate various meshes after each local mesh refinement if the effect of multiple river interventions must be studied. However, strictly speaking, this is the correct way to obtain a fair indication of the effect of a river intervention on local water levels which is highly important in the light of designing appropriate flood protection measures.

| DISCUSSION
In this study, we simulated the hydrodynamics in the schematised river meanders and the Grensmaas with different mesh shapes and mesh resolutions. In the schematised river meanders, water depth differences varied in the order of millimeters to centimetres between the various meshes. For the Grensmaas on the other hand, we found substantially larger water level differences which varied in the order of centimetres to decimetres. The latter findings correspond well with the case studies of Mohamad, Lee, and Raksmey (2016)  Therefore, it is presumed that the findings of this study also hold for other natural river meanders and other alternative hydrodynamic simulation software programs such as MIKE 21 FM and SRH-2D.
Regarding the evaluation of the effectiveness of river interventions (e.g., flood mitigation strategies), calibrating the model based on this future scenario was not possible since historical data was required to perform the calibration procedure, which in the case of a river intervention is only available after the realisation of such an intervention (Berends, Straatsma, Warmink, & Hulscher, 2019).
Therefore, we advise to first execute a local mesh refinement at the location of the future river intervention. Ideally, this mesh refinement should have a sufficiently high resolution such that the river's cross section is accurately captured. This mesh can be calibrated based on the current bathymetry by altering the bed roughness until the model output is close to measurements to compensate for the numerical effects and the errors caused by the relatively lowresolution mesh in the remaining parts of the model domain. These calibrated roughness values can then be used in the model including the updated bathymetry with intervention to limit the introduction of additional mesh effects.
F I G U R E 8 Cross-sectional view of the simulated water levels and depth-averaged flow velocities for the highest discharge range at CS 1 and CS 2. Regarding the names: Grensmaas stands for the Grensmaas stretch; Cur, Tri and Hybr for respectively curvilinear (as much as possible), triangular and hybrid; and HR and LR for high and low resolution, respectively. [Color figure can be viewed at wileyonlinelibrary.com]

| CONCLUSION
In this study, our objective was to assess the influence of mesh characteristics on the hydraulic outcomes of river meander models. We assessed the numerical effects by constructing various meshes for schematised river meanders with a flat bed in the transverse flow direction. The results showed that the numerical effects were proportional to mesh cell sizes, depth-averaged flow velocity, rapid flow changes and the orientation of the mesh lines with respect to the flow direction. The latter two factors were more pronounced in sharper bends. In a more realistic case study, we examined the combined effect of the numerical diffusion, false diffusion and the bathymetry discretization. The results of the case study demonstrated that the differences in simulated hydrodynamics become significantly greater between meshes when variations in bed levels were considered.
Therefore, we conclude that the bathymetry discretization effects were more important than the numerical effects.
In contrast to narrow river sections, simulated water level differences became substantially smaller between meshes if large floodplains were included in both the schematised river meander as well as in the Grensmaas. The results showed that due to the presence of wide floodplains, relatively smaller depth-averaged flow velocities were obtained through the river bends in comparison to narrower river sections. As a result of a large floodplain effect, mesh-generated effects in the main channel were less pronounced and hence dampened throughout the river cross-section. Therefore, we conclude that mesh-generated effects are proportional to the discharge per unit width.
To assess the impact of a local mesh refinement on the outcomes of the 2D depth-averaged model, we refined two river bends in the Grensmaas. The results demonstrated that a local increase in mesh resolution contributes to converging water levels and depth-averaged flow velocities towards those of the higher resolution variants locally.
It is recommended to first refine a certain part of the mesh before calibrating the model. Refining a mesh after calibration results in having calibrated roughness coefficients that correspond with those of the non-refined mesh. This recommendation especially applies to triangular meshes, and to rivers with narrow meander bends, since mesh resolution has a significant effect on model output in these situations.
T A B L E 2 Predicted water levels in the main channel at the bend apexes (CS 1 & CS2) for the two discharges ranges by the three locally refined meshes Note: Regarding the names: Grensmaas stands for the Grensmaas stretch; Cur, Tri and Hybr for respectively curvilinear (as much as possible), triangular and hybrid; HR and LR for high and low resolution, respectively; and Loc and Ref for locally and refined, respectively.