.. raw:: latex \newpage ##################### WES SHEDDING VORTEX ##################### ==== Mesh ==== .. math:: X = 38 .. math:: Y = -0.5 .. math:: Z = 12 center hole: .. math:: X = 5 .. math:: Z = 6 Diameter hole: .. math:: D = 1.5 ================= Model definitions ================= Code_saturne defines LamVisc (molecular viscocity) which is the dynamic viscosity = :math:`\eta` [#]_. Here :math:`\eta = 0.0075` .. [#] https://www.physicsforums.com/threads/moleculer-viscosity-eddy-viscosity.190265/ Density is :math:`\rho = 1` ================= Setup calculation ================= Files from https://www.code-saturne.org/forum/viewtopic.php?f=12&t=2275&sid=98048c925822c4d513c5c737cdad280 are downloaded to :: cd ~/Projects/2018-10-03-oil-damper-functionality/Engineering/2D-numerical/ as: - cylinder.med - cylinder-download.xml Then:: cd /home/marcel/WIENS/Projects/2018-10-03-oil-damper-functionality/Engineering/2D-numerical/ cd ~/Projects/2018-10-03-oil-damper-functionality/Engineering/2D-numerical/ mkdir cs-VIV-example-2 cd cs-VIV-example-2 code_saturne create -s Tuto_VIV -c VIV cd ../cs-VIV-example cp cylinder.med ../cs-VIV-example-2/Tuto_VIV/MESH/cylinder.med cp cylinder-download.xml ../cs-VIV-example-2/Tuto_VIV/VIV/DATA/Tuto_VIV.xml cd ../cs-VIV-example-2/Tuto_VIV/VIV/DATA ./SaturneGUI - load the Tuto_VIV.xml file via menu - When simply go to `prepare batch calculation` and press `start calculation` will make the calculation to fail. First, you have to go to the mesh page and read the mesh explicitly, then you can go to `prepare batch calculation` and press `start calculation` now, the calculation seems to be running - calculation finishes with message `finished with exit code 0` in terminal window where GUI was started. =============== Post-processing =============== - goto directory, something like (adapt to your needs):: cd /home/marcel/WIENS/Projects/2018-10-03-oil-damper-functionality/Engineering/2D-numerical/cs-VIV-example-2/Tuto_VIV/VIV/RESU/20181009-1442 cd monitoring - try the following commands one by one to view some plots:: xmgrace -block probes_CourantNb.dat -bxy 2:3 xmgrace -block probes_FourierNb.dat -bxy 2:3 xmgrace -block probes_LamVisc.dat -bxy 2:3 xmgrace -block probes_Mesh_displacement[X].dat -bxy 2:3 xmgrace -block probes_Mesh_Velocity[X].dat -bxy 2:3 xmgrace -block probes_Mesh_Velocity[Y].dat -bxy 2:3 xmgrace -block probes_Mesh_Velocity[Z].dat -bxy 2:3 xmgrace -block probes_mesh_vi1.dat -bxy 2:3 xmgrace -block probes_Pressure.dat -bxy 2:3 xmgrace -block probes_total_pressure.dat -bxy 2:3 xmgrace -block probes_Velocity[X].dat -bxy 2:3 xmgrace -block probes_Velocity[Y].dat -bxy 2:3 xmgrace -block probes_Velocity[Z].dat -bxy 2:3 - Open Salome (or Salome_Meca or Salome_CFD) - Load module Paravis (paraview) and select `new` - menu - file - open paraview file - goto directory /home/marcel/WIENS/Projects/2018-10-03-oil-damper-functionality/Engineering/2D-numerical/cs-VIV-example-2/Tuto_VIV/VIV/RESU/20181009-1442/postprocessing/ - select file `Results.case` OK - in the left pane, select `apply` - Select +Y view to see all, the pressure distribution should be given by default - press play in the animation pane to see the pressure variation over time - goto menu - view - toolbars - check `commmon` - select `warp by vector` from the common toolbar - click apply in the left pane - set scale factor to 1 in left pane ============ Flow analyis ============ So, to understand what is happening, a theoretical assessment is made. The vibration time of the oscilation is: .. math:: Delta T = T_2 - T_1 = 76.0 - 67.2 = 8.8 [s] Hence, the frequency is: .. math:: f = \frac{1}{T} = \frac{1}{8.8} = 0.1136 [Hz] \equiv 0.714 \left[ \frac{rad}{s} \right] The Reynolds number is: .. math:: Re = \frac{V_{inf} D \rho }{\eta} = \frac{1 \cdot 1.5 \cdot 1 }{0.0075} = 200 The Strouhal number for a cylinder at :math:`Re = 200` is approximately :math:`Sr = 0.2` [#]_ So, the frequency of oscillation should be: .. [#] Bohl, Technische Stroemungslehre, Vogel, Wuerzburg, page 150 .. math:: f_{Sr} = \frac{Sr V_{inf}}{D} = \frac{0.2 1}{1.5} = 0.133 Which seems to be the frequency in Herz. It is slightly different from the frequency calculated above, but that was only for 1 cycle, so lets calculate the frequency over more cycles: .. math:: Delta T_{10} = \frac{T_N - T_1}{N} = \frac{152.0 - 67.2}{10} = 8.48 [s] .. math:: f_{10} = \frac{1}{T_{10}} = \frac{1}{8.48} = 0.118 [Hz] \equiv 0.741 \left[ \frac{rad}{s} \right] With this frequency, the Strouhal number could be: .. math:: Sr_{f_{10}} = \frac{0.118 \cdot 1.5}{1} = 0.177 This fits better with the Strouhal number graph shown on wikipedia [#]_ .. [#] https://en.wikipedia.org/wiki/Strouhal_number The undampted frequency of the structure is: .. math:: f_s = \frac{1}{2 \pi} \sqrt{\frac{k}{m}} = \frac{1}{2 \pi} \sqrt{\frac{2}{5}} = 0.101 [Hz] And the damped frequency is: .. math:: f_{sd} = f_s \sqrt{1 - \zeta^2} with: .. math:: \zeta = \frac{c}{2 \sqrt{k m}} = \frac{3}{2 \sqrt{2 \cdot 5}} = 0.474 So that: .. math:: f_{sd} = f_s \sqrt{1 - \zeta^2} = 0.101 \sqrt{1 - 0.474^2} = 0.101 \cdot 0.880 = 0.089 [Hz] The ratio between exciation frequency and natural frequency is: .. math:: \frac{f_{10}}{f_{sd}} = \frac{0.118}{0.089} = 1.332 Not that close for resonant behaviour. The ratio between dynamic amplitude and static amplitude for a damped system is (assuming sine force excitation where force is not influenced by amplitude, as is actually the case here): .. math:: A_f = \frac{1}{\sqrt{\left(1 - \left(\frac{f_{10}}{f_{sd}}\right)^2 \right)^2 + \left(2 \zeta \frac{f_{10}}{f_{sd}} \right)^2}} = 0.675 [-] ================= Test calculations ================= Several calculations are performed with different parameters, the table below gives the different parameters and the output results .. |tmax| replace:: :math:`t_{max} [s]` .. |fsd| replace:: :math:`f_{sd} [Hz]` .. |ff| replace:: :math:`\frac{f_{10}}{f_{sd}} [-]` .. |Af| replace:: :math:`A_f [-]` .. |Amax| replace:: :math:`A_{max} [-]` .. |zet| replace:: :math:`\zeta` .. |f20| replace:: :math:`f_{20}` == ==== = ===== ============ ======== ====== =================== ===== ===== ===== ====== # M C |zet| mesh postpro probes |tmax| termination |fsd| |ff| |Af| |Amax| == ==== = ===== ============ ======== ====== =================== ===== ===== ===== ====== 0 5 3 0.474 fixed_mesh default 400 OK 0.089 1.332 0.675 0.104 1 5 3 0.474 fixed_mesh cyl edge 400 OK 0.089 1.332 0.675 0.104 2 5 3 0.474 trans_coord cyl edge 400 OK 0.089 1.332 0.675 0.104 3 3.53 3 0.565 trans_coord cyl edge 400 OK 0.099 1.193 0.708 1.165 4 1.25 3 0.949 trans_coord cyl edge 14.8 X-dir vibration 0.064 1.854 0.234 N/A 5 1.25 0 0.000 trans_coord cyl edge 12.9 X-dir vibration 0.201 0.586 1.523 N/A 6 1.25 1 0.316 trans_coord cyl edge 13.4 X-dir vibration 0.191 0.618 1.367 N/A 7 2.0 1 0.250 trans_coord cyl edge 218.2 Z-dir vibr mesh def 0.154 0.766 1.774 0.306 8 1.6 1 0.280 trans_coord cyl edge 239.2 Z-dir vibr mesh def 0.171 0.691 1.538 0.283 9 3.53 1 0.188 trans_coord cyl edge 400 OK 0.118 1.003 2.649 0.293 == ==== = ===== ============ ======== ====== =================== ===== ===== ===== ====== With the following definitions: :|Amax|: The maximum amplitude of the vibration measured in post processing. This is not very accurate as only every :math:`10^{th}` step is saved. :trans_coord: short for: `transient_coordinates`, an option to set. :cyl edge: 6 probes are defined on the North (1) South (1) East (2) West (2) locations around the hole. East as West have 2 nodes, as they are not exactly on 0 or 180 degrees. The purpuse was to track the vibration of the cylinder more closely, but getting to output other coordinates than X are not successful so far. :X-dir vibration: from start heavy vibration in X-direction, seems that speed is so high that Mach number is very high, so that pressure `lines` appear in the solution, like in Vibration in X-direction (horizontal) showing pressure lines :Z-dir vibr mesh def: At some point in the simulation after a very long period of nice vibrations, the solutions becomes unstable. the amplitude in Z-direction becomes larger mesh starts to deform, in combination with low pressures in these deformed meshes. as in Vibration in Z-direction (vertical) showing mesh deformation and low pressure :N/A: Not applicable, because the solution does not go into vibration in Z-direction Observations ============ - Version 0..4 are heavily damped. Hardly any overshoot is expected, see :numref:`damper sdofdampedovershoot.png`. The cylinder movement is just forced by the flow. The flow goes to a certain maximum amplitude as shown for 3 in :numref:`damper Displacement in Z-direction of cylinder over time`. Clearly amplitude builds up until 50s, then increases linearly to 80s and then tops off to reach a maximum amplitude of 1.165. THIS IS IN LINE WITH LITERATURE... EXPAND FURTHER - In versions 5..9, the mass spring system is more dynamic :math:`\zeta < 0.32` Hence, some overshoot near resonance frequency can be expected. Dynamic over static amplitude overshoot for a single DOF dynamic damped system. When :math:`\zeta > 0.5` there is virtually no overshoot. .. _`damper Displacement in Z-direction of cylinder over time`: Displacement in Z-direction of cylinder over time for case 3. Second tests ============ A sweep is designed through frequency spectrum, to find fluid-structure response for these data: .. table:: Input parameters for the next set of tests. +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | # | m | c | k | fs [Hz] | zeta | dcorr | |fsd| | |ff| | |Af| | |Amax| | |f20| | +====+=======+=======+===+=========+=======+=======+==========+===========+========+========+========+ | 21 | 0.034 | 0.131 | 2 | 1.219 | 0.250 | 0.968 | 1.180 | 0.100 | 1.009 | N/A | N/A | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 22 | 0.136 | 0.261 | 2 | 0.609 | 0.250 | 0.968 | 0.590 | 0.200 | 1.036 | N/A | N/A | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 23 | 0.307 | 0.392 | 2 | 0.406 | 0.250 | 0.968 | 0.393 | 0.300 | 1.084 | N/A | N/A | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 24 | 0.546 | 0.522 | 2 | 0.305 | 0.250 | 0.968 | 0.295 | 0.400 | 1.158 | N/A | N/A | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 10 | 0.853 | 0.653 | 2 | 0.244 | 0.250 | 0.968 | 0.236 | 0.500 | 1.265 | N/A | N/A | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 11 | 1.228 | 0.784 | 2 | 0.203 | 0.250 | 0.968 | 0.197 | 0.600 | 1.415 | N/A | N/A | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 12 | 1.671 | 0.914 | 2 | 0.174 | 0.250 | 0.968 | 0.169 | 0.700 | 1.617 | 0.312 | 0.123 | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 13 | 2.183 | 1.045 | 2 | 0.152 | 0.250 | 0.968 | 0.147 | 0.800 | 1.858 | 0.318 | 0.122 | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 14 | 2.763 | 1.175 | 2 | 0.135 | 0.250 | 0.968 | 0.131 | 0.900 | 2.047 | 0.280 | 0.120 | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 15 | 3.411 | 1.306 | 2 | 0.122 | 0.250 | 0.968 | 0.118 | 1 | 2 | 0.241 | 0.119 | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 16 | 4.127 | 1.437 | 2 | 0.111 | 0.250 | 0.968 | 0.107 | 1.100 | 1.699 | 0.198 | 0.117 | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 17 | 4.912 | 1.567 | 2 | 0.102 | 0.250 | 0.968 | 0.098 | 1.200 | 1.344 | 0.162 | 0.116 | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 18 | 5.765 | 1.698 | 2 | 0.094 | 0.250 | 0.968 | 0.091 | 1.300 | 1.055 | 0.131 | 0.116 | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 19 | 6.685 | 1.828 | 2 | 0.087 | 0.250 | 0.968 | 0.084 | 1.400 | 0.842 | 0.108 | 0.116 | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ | 20 | 7.675 | 1.959 | 2 | 0.081 | 0.250 | 0.968 | 0.079 | 1.500 | 0.686 | 0.089 | 0.116 | +----+-------+-------+---+---------+-------+-------+----------+-----------+--------+--------+--------+ .. note:: case 10..13 the amplitude and frequency is determined using 5 cycles and 90% starting limit, where 14..20 use 20 cycles and 95% starting limit. These Cases are stored in files: Tuto_VIV_vXX.xml where XX is the version number (#). The cases are run via a script `runCodeSaturneCases.sh`:: #!/bin/bash cd ~/WIENS/Projects/2018-10-03-oil-damper-functionality/Engineering/2D-numerical/cs-VIV-example-2/Tuto_VIV/VIV/DATA code_saturne run -p Tuto_VIV_v10.xml code_saturne run -p Tuto_VIV_v11.xml code_saturne run -p Tuto_VIV_v12.xml code_saturne run -p Tuto_VIV_v13.xml code_saturne run -p Tuto_VIV_v14.xml code_saturne run -p Tuto_VIV_v15.xml code_saturne run -p Tuto_VIV_v16.xml code_saturne run -p Tuto_VIV_v17.xml code_saturne run -p Tuto_VIV_v18.xml code_saturne run -p Tuto_VIV_v19.xml code_saturne run -p Tuto_VIV_v20.xml All files are modifications of Tuto_VIV_v09.xml Observations in the second tests ================================ - peak resonance seems to happen around :math:`\frac{f_{10}}{f_{sd}} = 0.8 [-]`, so, there must be considerable damping from the fluid flow. That is also why there is a maximum amplitude. - :numref:`damper sdofversusSimulated.png` gives the response plotted together with the response of a single DOF damped system response. The real amplitude ratio is not known, as the static force of the fluid-structure system is unknown, so, the amplitude is scaled by a factor 7.0 for shape comparison. There seems to be more damping than :math:`\zeta=0.25`. But also the amplitude is much larger for the :math:`\zeta=0.25` line. Below a certain frequency range, a stable solution cannot be obtained. When using other values, like scaling with factor 4.0 and compare to :math:`\zeta=0.5`, the dots would cross the line. Scaled amplitude (green dots) plotted together with the amplitude response of a single DOF damped system, showing shape differences in the response. .. raw:: latex \newpage