Convergence Studies

!_!

! !ℎ_! (4.4)

where !_∗ denotes the friction velocity, the model constant is !_! = 0.09, the von Kármán constant is != 0.41 and ℎ_! denotes the radial length between the cylinders surface and the centre of the first cell adjacent to the cylinders surface.

4. At top and bottom boundaries, zero normal gradient is prescribed for !_!, !_!, pressure, ! and !.

4.2 Convergence Studies

The convergence studies are performed in two steps. Firstly, mesh and time step dependence studies are carried out for a single cylinder. In this step, a series of simulations of a stationary cylinder is conducted to investigate the necessary mesh density to accurately model the flow around the cylinder and in the wake region. New simulations are then performed for a vibrating single cylinder using the same set of grid resolution from the simulations of the stationary cylinder. In this case, due to the motion solver, the mesh is deformed during the simulations, and as a consequence the size of the cells can vary significantly in some regions of the domain. Hence, time step convergence study is also carried out for the case with the vibrating single cylinder to find the maximum Courant number setting that minimizes time step errors.

Secondly, grid convergence study is carried out for two coupled cylinders with

!= 90° and !/! = 0.1 configuration, and free to vibrate with 2-DoF. The selected mesh densities are based on the study carried out for the single cylinder cases. Based on the obtained results, similar grid resolutions are used for the other investigated configurations with two cylinders.

Figure 4.2 presents a complete view of a typical mesh used in the first stage of the convergence studies. The domain is composed of structured hexahedral elements. A high refinement is applied in the region around the cylinder and in the cylinder’s

wake. In the far field, the mesh is coarsened in order to reduce the computational cost.

In Figure 4.3, detailed views of the mesh around the cylinder are shown. The first cell height near the surface of the cylinder is set as 0.0005! which yields approximately averaged !^! (here !^! = ℎ_!!_∗ !) values in the range of 30 − 40 which is appropriate for the use of wall function.

Figure 4.2: Computational mesh for the single cylinder cases: mesh A3 (Table 4.1), 104536 cells

Figure 4.3: Mesh details around the cylinder for the single cylinder cases

For the simulations with the stationary cylinder, hydrodynamic force coefficients and the Strouhal number are used to evaluate the convergence. OpenFOAM has the

built-in function object forceCoeffs which extracts the hydrodynamic coefficients !_! and !_! of the cylinder at each time step. Expressions for the mean drag and mean lift coefficients are given as follows:

! is the total number of time steps averaged.

The corresponding root-mean-square values for the drag and lift coefficients are obtained by:

In the grid resolution test for the stationary single cylinder case, four sets of meshes with an increment of approximately 40% in the total number of elements are assessed.

A time step of ∆!= 0.002 is employed in the simulations keeping the maximum Courant number below 0.60. The total non-dimensional duration of the simulations is set to ! =200, where the dimensionless time is defined as:

!= !!_!

! (4.9)

Table 4.1 summarizes the mesh parameters along with the corresponding results for the analysed hydrodynamic quantities. The differences between the results obtained on the meshes A3 and A4 are smaller than 0.5% for !_!, !_!,!"# and !".

Table 4.1: Mesh convergence study for the stationary single cylinder Mesh No. of cells Time Step !_! !_!,!"# !" number is constrained to !"_!"# =0.5. Besides the hydrodynamic force coefficients, displacement amplitudes are also used to compare the results obtained from the simulations. The normalized maximum vertical vibration amplitude is given by:

!_!,!"# are given in Table 4.2. The differences in the calculated !_!, !_!,!"# and !" obtained on the meshes A3 and A4 are smaller than 1%. The !_!,!"# ! values differ 1.44%

between the simulations using the meshes A3 and A4.

In the time step sensitivity analysis, three simulations with the mesh A3 (see Table 4.2) are carried out with different maximum Courant numbers,

!"_!"# = 0.25, 0.5, 1.0 , and the obtained results are presented in Table 4.3. The

discrepancies in the observed !_! and !" values between the simulations using

!"_!"# = 0.5 and !"_!"# = 0.25 are within 1%. The differences in the calculated

!_!,!"# ! and !_!,!"# using !"_!"# = 0.25 compared with the values obtained using

!"_!"# = 0.5 are approximately 1.80% and 3.10%, respectively. Thus, based on the

convergence studies with the vibrating single cylinder, it is concluded that the mesh A3 with 104536 elements and !"_!"! = 0.5 provides sufficient grid and time step convergence. This setting of mesh and time step is selected for the present simulations of FIV of a single cylinder.

Table 4.2: Mesh convergence study for the vibrating single cylinder

Mesh No. of cells Time Step !_! !_!,!"# !" !_!,!"# ! A1 53595 !"_!"#=0.5 1.2848 0.3659 0.4992 1.1701 A2 74889 !"_!"#=0.5 1.3120 0.4728 0.4898 1.2907 A3 104536 !"_!"#=0.5 1.2503 0.5134 0.4844 1.2937 A4 146092 !"_!"#=0.5 1.2529 0.5185 0.4844 1.2751

Table 4.3: Time step convergence study for the vibrating single cylinder

Mesh No. of cells Time Step !_! !_!,!"# !" !_!,!"# ! A3 104536 !"_!"#=1.0 1.3188 0.5194 0.5033 1.3274 A3 104536 !"_!"#=0.5 1.2503 0.5182 0.4844 1.3055 A3 104536 !"_!"#=0.25 1.2612 0.4997 0.4844 1.3170

In the grid resolution analyses for the cases with two coupled cylinders, the computational domain is the same as that for the single cylinder cases. It consists of structured hexahedral elements. Figures 4.4 and 4.5 show a typical mesh for ! =90°

and ! ! =0.1 configuration used in the convergence studies. Three meshes with an increment of approximately 40% in the total number of elements are assessed.

Similarly to the cases with a single cylinder, the mesh is highly refined close the cylinders, and coarser cells are used in the far-field regions.

In the Table 4.4, the results of the grid sensitivity study for two vibrating coupled cylinders are presented. The maximum Courant number is constrained to 0.5 based on the results obtained from the time step convergence studies with the vibrating single

Figure 4.4: Computational mesh for the two coupled cylinders cases with != 90° and

! ! = 0.1 configuration: mesh B3 (Table 4.4), 114321 cells

Figure 4.5: Mesh details around the cylinders for the two coupled cylinder cases with

!= 90° and ! ! =0.1 configuration

cylinder (Table 4.3). The reduced velocity !_!= 6 is used in the simulations. At this

!_!, high amplitudes of the cylinder’s vibration are expected, facilitating the assessment of the dynamic simulations convergence. Discrepancies in the results between the meshes B2 and B3 are within 1.45% for !_!^, !_!,!"#and!". The deviation in the observed !_!,!"# ! using the mesh B3 compared to the obtained result using the mesh B2 is 2.38%. Considering that further mesh refinement results in negligible

Table 4.4: Mesh convergence study for the two vibrating rigidly coupled cylinders simulations of FIV of two coupled cylinders with different ! and ! !.