Case 09. The hydraulic jump case
This test problem was suggested to us by Kate Hedstrom (U. Alaska), which tests numerical model’s advection scheme with supercritical flow condition. The then latest FVCOM FORTRAN 90 code (version 2.1.1) was used to determine if FVCOM could accurately capture the hydraulic jump. The numerical solutions presented here show excellent agreement with the analytic solution. The model setup for this test problem are available at the FVCOM website (needs a password) for users who is interested in repeating this experiment. A brief description of this test problem and comparison between FVCOMbased numerical and analytical solutions is given below. In addition, the numerical experiment results using ROMS are also cited here for comparison with FVCOM results.  
Problem Setting and Analytic Solution  
This problem involves supercritical fluid flow through a channel with a constriction bearing an angle of 8.95° to the horizontal, as shown in Fig. 1. Due to the constriction, the flow accelerates over the ramp, resulting in the formation of a straightline hydraulic jump emanating from the ramp corner. For certain prescribed initial conditions (Acrudo and GarciaNavarro, 1993; Choi, 2004), the jump angle and some flow metrics are known. It is a good test case for examining the performance of numerical model’s advection scheme in simulating a discontinuity (shock type) solution. We remark that a number of advection schemes have been reported to fail for this test problem, especially those using central difference scheme, which is used in most numerical ocean models.  




The shallow water equations with the initial and boundary conditions described above permit the formation of a steadystate hydraulic jump (Acrudo and GarciaNavarro, 1993; Choi, 2004). There will be two distinct regions of flow separated by an infinitesimally thin, straightline, hydraulic jump emanating from the ramp corner (Fig. 1). The region of interest for this study is the triangular, aftjump portion for which some properties are known: (1) the angle of the jump is 30° to the horizontal; (2) the along channel velocity component is 7.956 m/s; (3) the Froude number is 2.074; and (4) the freesurface height (η) is 0.500 m 

FVCOM Results and Comparison with Analytic solution  
In addition to the above analytic solution, the following additional "numerical" metrics are of interest: (5) the minimum η value anywhere in the domain (should be zero; i.e., no undershoots); (6) the maximum η value anywhere in the domain (should be 0.5; i.e., no overshoots); (7) the mean deviation of the simulated hydraulic jump line from the expected line; and (8) the mean thinkness of the jump. These metrics are numerically evaluated as follows: (i) for (5), (6), the minimum and maximum η values are simply searched for over the whole computational domain, (ii) for (1), (7), (8), 101 crosssections in the y direction are taken and the x,y locations corresponding to η=0.25 are found via linear interpolation. These sections are taken between 3 m after the ramp corner and 3 m ahead of the outflow boundary. A straightline leastsquares fit is performed on the x,y values and the angle of the jump, (1), is recovered from the line slope. As the exact angle of the jump (30.0°) is known, the exact y values corresponding to it for the set of x values of the crosssections can be evaluated, from which the mean absolute deviation in y values, (7), can be extracted. The jump thickness, (8) is evaluated by searching for the y coordinates corresponding to η=ηR, η=ηL where, ηR =0.375, ηL =0.125 also via linear interpolation and then taking their differences. Finally, (iii) the mean u, Fr and η quantities, (2)(4), are evaluated by first defining a triangular region, 1 m away from the shock, ramp and outer boundary and thereafter taking an areaweighted averaged of these quantities in this region. Only full grid cells are used in the computations of these averages after testing whether their centroids lie within this triangle. Three sets of experiments have been run, with successively refined grid resolution: Experiment 1: dx ~= 0.5, dt=0.002; Experiment 2: dx ~= 0.25, dt=0.001; Experiment 3: dx ~= 0.125, dt=0.0005. In each set of the experiment, the Smagorinsky horizontal viscosity parameter changes in the range of 0, 0.6, 1.2, 2.0, and 3.0. They are named using suffix a, b, c, d, and e respectively. Therefore, 15 experiments are run (Table 1). 

Table 1. Some metrics of FVCOM numerical experiments  


where grid is horizontal grid number of the computation domain; dt is the time step in second; horcon is the Smagorinsky horizontal viscosity parameter; Zmax and Zmin are the maximum and minimum sea surface elevation (in m) in the domain; Zmean and Umean are mean sea surface elevation (in m) and mean velocity (in m/s) south of the jump; Frmean is the mean Froude number south of the hydraulic jump; Angle and thickness are angle (in °) and mean thickness (in m) of the hydraulic jump; dy is the mean deviation of the simulated hydraulic jump line from the theoretical line (in m).  
In all experiments, the volume averaged kinetic energy and available potential energy approach constant values in about 15 seconds. To make sure that a steady state is reached, the computation duration is 120 seconds for Experiment 1 and 2, and 50 seconds for Experiment 3. Without numerical viscosity or when numerical viscosity is small, there are small amplitude oscillations near the hydraulic jump (Fig. 2). This is the reason for the existence of overshootings (Zmax > 0.500) and understootings (Zmin < 0.000) (see Table 1). With increased numerical viscosity, the amplitudes of the oscillation are greatly reduced. Another thing to notice in the table is that these overshootings and undershootings are sporadically distributed in the calculation domain. They only occupy a small fraction of the domain. Therefore, the area mean sea surface elevation, area mean velocity magnitude, and area mean Froude number are not affected by these oscillations and are very close to the theoretical values regardless of grid refinement and the Smagorinsky horizontal viscosity parameter (horcon) used.  
The hydraulic jump angle shows no obvious improvement with grid refinement and it is also not very sensitive to the value of the Smagorinsky horizontal viscosity parameter. However, the jump thickness is greatly improved with finer grid and larger Smagorinsky horizontal viscosity parameter makes it thicker.  
Comparison between FVCOM and ROMS results  
ROMS results cited in this comparison were obtained from ROMS website. Their table 1 in Supercritical flow case of ROMS test base is used here. One distinction between the two models is that FVCOM can reach a steady state solution without inclusion of any viscosity, but base on the information from ROMS website, ROMS yields an oscillation solution if no viscosity was included. Therefore, the ROMS experiments all include the horizontal viscosity (Table 2).  
Table 2. Comparison of experiment metrics between FVCOM and ROMS.  


where the definition for all variables are the same as in FVCOM experiments except visc is horizontal viscosity coefficient used in ROMS. ROMS experiments using their secondorder and forthorder accurate horizontal dvection schemes are designated as ROMS (2nd order) and ROMS (4th order) respectively. For all experiments, FVCOM results show better matches to theoretical values than ROMS. For examples: a) For Zmean: theoretical value is 0.500. FVCOM shows a range from 0.499 to 0.500. ROMS is from 0.472 to 0.491. b) For Umean: theoretical value is 7.9556. FVCOM shows from 7.948 to 7.952. ROMS is from 7.945 to 7.985. c) For Frmean: theoretical value is 2.074. FVCOM shows from 2.072 to 2.073. ROMS is from 2.078 to 2.100. d) For jump angle: theoretical value is 30.000. FVCOM is from 29.952 to 30.030. ROMS is from 29.307 to 29.568. e) For dy: theoretical value is 0.000. For the finest grid case, FVCOM is less than 0.050 m, but ROMS was large than 0.1 m. ROMS results show that no improvement in dy as the horizontal resolution increases, but FVCOM shows that dy decreases as the horizontal resolution increases. f) For thickness: theoretical value is 0.000. FVCOM and ROMS results are on the same order. 

Summary  
Although not using a signpreserving advection scheme, FVCOM can resolve the hydraulic jump strength, width, and angle quite well FVCOM result is better than ROMS for this superctical flow case, which, we believe, can be attributed to the better performance of finite volume method to the discontinuity (shock type) solution. 