M ODEL ISSUES

With this equation F can be solved. A value of 1, 0 and >0 indicate an interior, empty and surface cell respec-tively.

Computational domain The VARANS equations are discretised with an orthogonal grid structure. The cell spacing can vary over the computational domain, allowing for a finer grid at certain locations. See Fig-ure 4.3.

Figure 4.3: Computational domain

All scalar quantities are defined in the centre of the cells. The scalar quantities are: pressure, turbulent kinetic energy, dissipation rate, VOF function and an openness function. The vector quantities of horizontal and vertical flow velocity and two additional openness functions are defined at the faces of the cells.

Because of the orthogonal grid choice, sawtooth-shaped surfaces arise when defining solid boundaries under a slope. Since the surfaces will be defined at the cell faces. This often leads to non-physical reflections at these boundaries. Therefore, the model uses a different method of partial cell treatment. The method consists of modelling solid objects as fluids with an infinite density. This introduces the openness functionsθ. This function is a ratio of space not occupied by an solid object over the total space. At the cell centreθcequals the ratio of ’open space’ over the cell area. At the cell top and right faces,θtandθrare defined as the ratio of length of ’open space’ to the total length of the cell face. With this function, the model can thus identify if a cell corresponds to a solid object, fluid domain or a solid-fluid domain. The openness function is is similar to the VOF function, with the only difference that the openness function does not change with time (solid boundaries do not move). In case a cell is identified as a fluid domain, the VARANS equations are solved with the density of water. In case a cell is identified as a solid domain, the VARANS equations are skipped. In case a cell is identified as a solid-fluid domain, the equations are solved with a special porosity value which is linked to the openness function.

4.2. M

During the first few tests with the numerical model several model settings were tested, such as the wave series generation, static and dynamic paddles, boundary absorption and turbulence model. Some problems were found, which are briefly discussed below.

40 4. EVALUATION OF THE NUMERICAL MODEL

Wave generation The Matlab GUI of the numerical model offers three options to generate the wave series for the boundary condition. The first option is to supply the type of wave to be generated (solitary, regular, irregular) and a number of relevant variables such as significant wave height, (peak) period and series length.

The second option is to load a previously generated wave series, which allows one to use exactly the same boundary condition in several different simulations. The third is to ’reconstruct’ a wave series from a user supplied file which contains a water level elevation time series. With this option an observed wave series in the field can be used as a boundary condition in the model, for example. The model reads the user input and calculates the corresponding variance density spectra. For some reason however, the tail of the spectra

’disappears’ somewhere in the calculation. This does not happen in all cases and the frequency where the tail

’disappears’ also seems to vary. This was discovered by calculating the variance density spectra of the input wave series and the wave series file which is written by the model. Figure 4.4 shows an example.

Figure 4.4: Wave series reconstruction error

Figure 4.4 shows the largest deviation between the two spectra tails of all cases where a wave series was re-constructed in this report. The effect on the final results is considered negligible since the variance density is small relative to the other frequencies.

Paddle The numerical model offers two options to impose the boundary condition. The first is a static paddle, which, simply said, raises the water level at the boundary to generate waves. The second is a dynamic paddle. The dynamic paddle simulates a paddle which would be used in a physical flume, by oscillating a paddle in horizontal direction. In the model this is achieved by changing the openness function of the boundary cells in time, which effectively creates a moving solid object at the boundary (see the previous section). With suggested initial and maximum positions from the manual (Cantabria, 2012), large deviations in the water level elevation time series halfway along the flume were observed. Several other settings were tested without better results. Since the static paddle did produce good results, the static paddle was chosen to be used in this report. The reason why the dynamic paddle did not work was not further investigated.

Turbulence closure model Turbulence is by default switched off in the model. This means that the Reynolds stresses in the (VA)RANS equation are not solved and are equal to zero for the whole simulation. Therefore turbulence is not modelled. If turbulence is switched on, the Reynolds stresses are solved with the k−² closure model. In flows where turbulence is important, this is necessary for a proper modelling of the flow. However, during the first tests some of the simulations encountered an error when the turbulence was switched on. It appeared that if a simulation was run long enough the turbulence closure model would eventually lead to an error. Therefore, turbulence had to be switched off for all models. It is assumed that turbulence in front of the structure does not play a major role, and that the effect of turbulence inside the structure is fully incorporated in the equations through the turbulent Forchheimer coefficient.

4.2. MODEL ISSUES 41

Turbulent Forchheimer coefficient The turbulent Forchheimer coefficientβ is calculated in the model as follows:

KC is the Keulegan-Carpenter number. β is in this way dynamically updated each time step based on the maximum velocity inside the porous media and the mean wave period. Changing the variable xxt in the input file allows the user to use another representative period value for this calculation. However, several model tests with varying values for T showed exactly the same results. Investigation of the source code showed that this is because the model first reads the xxt value from the input file and consequently overwrites this values with the mean period. The source code was adjusted to fix this issue and an additional output line for the command prompt was added to show the value of xxt for each iteration. See Figure 4.5.

Figure 4.5: Command prompt showing an adjusted source code

Inconsistencies between the manual and source code Some inconsistencies between the description of the volume averaging closure model (Forchheimer equation) in the manual and the actual equation used in the source code were found. The equation in the manual is as follows:

... =αν(1 − n)²

(the gravitational acceleration is incorporated as a separate vector in the equations)

With suggested values of: c_A= 0.34¹⁻ⁿ_n ,α = 200 and β = 1.1. From this equation the coefficient of the ex-tended Forchheimer equation can be read. They are listed along with the coefficients used in the source code and in Equation 2.10, see Table 4.1.

Fortunately, the equation in the source code corresponds with literature. Unfortunately, it was only later in this research discovered that there was a slight error in the manual. The manual suggest a value of c_A= 0.34¹⁻ⁿ_n , which is called the added mass coefficient in the manual. However, in the source codeγ is used as the added mass coefficient and should therefore not be multiplied by¹⁻ⁿ_n this coefficient was overestimated for all numerical simulations. However, the effect on the results is considered small because of the small influence of c in the first place, see Figure 2.9

42 4. EVALUATION OF THE NUMERICAL MODEL

Table 4.1: Forchheimer equation in manual vs. source code

Manual Source code Literature

In this section it is investigated if the imposed wave conditions in the numerical model resemble the wave conditions of the physical model tests. Van der Meer (1988) gave the three variance density spectra (PM, wide and narrow) for one specific set of significant wave height and average wave period, which were used as a boundary condition in the research. See Figure 4.6.

Figure 4.6: Three spectra used in the original physical model tests

The numerical program calculates an energy density spectrum from a significant wave height, peak period, and a peak enhancement factor. The peak enhancement factor actually determines the type of spectrum.

A value of 3.3 produces a JONSWAP spectrum and a value of 1.0 a Pierson-Moskowitz spectrum, see equa-tion 4.6. From this spectrum a wave series is calculated. In the numerical test matrix only a PM spectrum is used, so the peak enhancement factor should be set to 1.0.

E ( f ) = αg²(2π)⁻⁴f⁻⁵exp [−5