Simulation of laboratory experiments for vortex dynamics at shallow tidal inlets using the fine resolution environmental hydrodynamics (Frehd) model

Abstract

In this paper, we present the results of numerical simulations matched to laboratory experiments of tidal, starting-jet vortices forming at idealized, barotropic inlets. The laboratory experiments are for sinusoidally varying inflow/outflow along a wide, flat basin. Inlet configurations include simple inlets with negligible channel length as well as a longer inlet channels through a long barrier island and jettied inlet. Laboratory observations are made using particle image velocimetry, and we analyze the laboratory and numerical model velocity fields to compare starting-jet trajectory, diameter, and total circulation. The numerical simulations use the Fine Resolution Environmental Hydrodynamic model, solving the three-dimensional, hydrostatic, barotropic equations of motion. Our goal is to identify the minimum model attributes necessary to match the vortex properties. This is accomplished using a third-order upwind scheme for advection, a constant bottom friction drag coefficient, and a one-equation turbulence model for transport of turbulent kinetic energy. The optimal value of the bottom drag coefficient matches the time-average value at the basin inflow/outflow boundaries, and the optimal dissipation coefficient, $C_{me}$, was an order of magnitude smaller than literature values for a steady open-channel flow. We show that the drag coefficient mainly affects the advection speeds of the outer coastal waters, which control the penetration of the tidal vortices away from the inlet, and the lower turbulence dissipation rate results from the low turbulent dissipation in the large, shallow starting-jet vortices.

Appendix: A model results using first-order upwind scheme

During initial set-up of the Frehd model, we performed a comprehensive model calibration using a first-order upwind scheme for the advection. We simulated a wide range of bottom friction values, turbulence closure models and model coefficients, grid resolutions, and we tested different model physics, including the non-hydrostatic solution. None of these tests were able to capture all of the starting-jet vortex properties simultaneously. We present here a summary of the main attributes of our best-fit calibration using first-order upwind for the 3 cm water depth idealized inlet case (i.e., the same case as used in our calibration in Sect. 2.4, above). We use the same coarse- and fine-grid meshes here as in the main paper.

We adjust the advection speed of the starting-jet vortices using the drag coefficient $C_D$ for bottom friction. In Fig. 19 we see that bottom friction controls the advection speed of the vortices. Smaller values of $C_D$ allow the vortices to propagate farther downstream. The ideal value for first-order upwind lies between $C_D = 0.005$ and $C_D = 0.01$, which is lower than the best-fit value for the model using third-order upwind. This lower optimal value of $C_D$ indicates that there is more model dissipation present in the first-order upwind model than for third-order upwind. Figure 20 shows the dependence of the vortex trajectory on the grid size for $C_D = 0.005$. Decreasing the grid resolution allows the vortex to propagate farther with the same bottom friction value. This is expected as a finer-grid mesh will have somewhat lower numerical dissipation and allow for identification of smaller features of the vortices. Based on evaluation of the vortex advection alone, these figures show that the right combination of grid size and bottom friction can match the trajectory of the laboratory vortices. This agreement, however, breaks down for vortex size and circulation.

Fig. 19

Centroid location in space for $C_D$ values of 0.001 (orange triangles), 0.005 (red circles), and 0.01 (dark orange squares). The black plus sign show the PIV laboratory data; the vertical dashed line is the edge of the PIV field of view

Fig. 20

Centroid location in space for coarse grid (red circles), fine grid (orange squares), and PIV laboratory data (black plus signs); the vertical dashed line is the edge of the PIV field of view

In Fig. 21 we plot the results for the coarse-grid model for vortex diameter and total circulation for three values of $C_D$, and in Fig. 22 we show the same results for two values of the turbulence closure parameter $C_{me}$. Three important conclusions emerge from these figures. First, the model generally over-predicts the size of the vortices and under-predicts the total circulation, with no combination of parameters able to match both properties simultaneously. This means that the model also severely under-predicts the vorticity and rotation rate of the starting-jet vortices. Second, unlike the model using third-order advection, the results here show that vortex size and circulation are sensitive to the bottom friction. Third, the model shows very weak sensitivity to the turbulence closure parameter $C_{me}$. Together, these latter two observations stem from the fact that more energy is dissipated by the numerical advection scheme than in the turbulence model.

Fig. 21

a Primary vortex diameter and b total circulation within the primary vortex for $C_D$ values of 0.001 (orange triangles), 0.005 (red circles), and 0.01 (dark orange squares). The black plus sign show the PIV laboratory data

Fig. 22

Centroid location in space for $C_{me}$ values of 0.08 (red circles) and 0.008 (orange squares). The black plus sign show the PIV laboratory data

The dependence of the dissipation on the numerical scheme is more evident for simulations at different resolution, as shown for vortex size and total circulation in Fig. 23. The higher-resolution grid allows the model to track the vortex size and circulation well through the first half of the tidal period, but the vortices do not decay away properly in the second-half of the tide. Instead, they persist as large, energetic structures. Because most of the dissipation is numerical, this behavior cannot be fixed by adjusting the turbulence closure parameters. Our conclusions is that the high numerical dissipation of the first-order advection scheme renders the turbulence model ineffective, which makes it impossible to capture the vortex properties correctly. This result further underscores our main conclusion from the model that advection of these starting-jet vortices is controlled by bottom friction affecting the surrounding coastal current and turbulent dissipation within the vortex, which occurs at a lower rate than for dissipation of turbulence kinetic energy for a steady, open channel flow.

Fig. 23

a Primary vortex diameter and b total circulation in the primary vortex for coarse grid (red circles), fine grid (orange squares), and PIV laboratory data (black plus signs)