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This tutorial simulates incompressible flow around a cylinder in a 2D rectangular domain.
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* [Reynolds number](#reynolds-number): effect of different Re numbers
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* [Mesh refinement](#mesh-refinement): mesh subdivision
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* [Domain boundaries](#domain-boundaries): simulation and boundary conditions
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* [Drag and lift forces](#drag-and-lift-forces): post-processing of drag and lift forces
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* [Input files](#input-files): detailed description of case input files
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* [cylinder.dat ](#cylinderdat)
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* [cylinder.dom.dat](#cylinderdomdat)
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* [cylinder.geo.dat](#cylindergeodat)
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* [cylinder.set.dat](#cylindersetdat)
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* [cylinder.fix.dat](#cylinderfixdat)
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* [cylinder.ker.dat](#cylinderkerdat)
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* [cylinder.nsi.dat](#cylindernsidat)
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* [cylinder.post.alya.dat](#cylinderposalyadat)
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* * *
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* * *
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# <a class="anchor" id="Reynolds_number"></a>Reynolds number
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The case is run first at low Reynolds number and a steady state solution is found.
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![image](uploads/234671d394fe01ba5e58a703530adece/image.png)
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<center>cylinder case – steady state solution (Re=10)</center>
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By increasing velocity the flow becomes unstable and von Karman vortex phenomena are observed in the wake of the cylinder ([click here to see the animation](http://www.youtube.com/watch?feature=player_embedded&v=RnZXCHUSSuE) )
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![image](uploads/22328e8d21f65dd0edf91a3ef291720c/image.png)
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<center>cylinder case – von Karman patterns at Re=100</center>
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* * *
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* * *
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# <a class="anchor" id="Mesh_refinement"></a>Mesh refinement
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The mesh of the model is structured quadrilateral. DIVISION command is used to refine the mesh in order to obtain more accurate results.
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![image](uploads/9753039976c75a4ada43b41b46952bbe/image.png)
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<center>DIVISION=0</center>
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![image](uploads/2d51e6d0a505a0ad340b3091af4d912a/image.png)
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<center>DIVISION=2</center>
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* * *
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* * *
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# <a class="anchor" id="Domain_boundaries"></a>Domain boundaries
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![image](uploads/0d862624a54545f118af7fab2b2f4a73/image.png)
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<center>Boundary domains are shown in the next figure: (1) inflow, (2) outflow, (3) wall and (4) symmetry.</center>
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The simulation is transient, integrating from 0 to 200 time interval with a prescribed time step of 0.2\. Velocity is set to 1, the diameter of the cilynder is 1, density is 1 and dynamic viscosity 0.01, so that the Reynolds number is 100\. Units are SI.
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* * *
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* * *
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# <a class="anchor" id="Drag_lift_forces"></a>Drag and lift forces
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As a post process option, forces and moments acting on boundaries are specified to be output in order to calculate drag and lift forces acting on the cylinder surface. Next plots show drag and lift forces due to pressure and viscosity over time.
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![image](uploads/e980347394f09233de2bf36c0b47045b/image.png)
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<center>Drag & Lift Forces vs. Time (Re=100)</center>
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These plots have been obtained using the Open Source application gnuplot.
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* Type ‘gnuplot’ on a terminal and then:
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gnuplot> plot [0:200] [-0.6:0.3] './alya-sets.out' using 1:7 title 'pressure drag force' with lines, './alya-sets.out' using 1:8 title 'pressure lift force' with lines, './alya-sets.out' using 1:4 title 'viscosity drag force' with lines, './alya-sets.out' using 1:5 title 'viscosity lift force' with lines
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* This command produces the plot in the figure by reading the data from columns 1 (time), 4 (F_v_x), 5 (F_v_y), 7 (F_p_x) and 8 (F_p_y) in the file alya-sets.out.
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* Prior to this, alya-sets.out file must be generated with the following command:
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/path/Utils/user/alya-sets cylinder-boundary.nsi.set 59 60 61 62
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The application alya-sets reads the file cylinder-boundary.nsi.set and does the summation of boundary variables specified in casename.nsi.dat / BOUNDARY_SET, for all elements belonging to boundary sets nº 59, 60, 61 and 62 (cylinder boundary sets in this case). Results are written to file alya-sets.out.
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Simulation results show that lift oscillations stabilize at a frequency of 0.14Hz.
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Strouhal number (nondimensional frequency) of viscosity and pressure lift oscillations is:
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```math
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\textnormal{St} = \frac{( \textnormal{f} \cdot \textnormal{d} )}{\textnormal{u}_{\infty}} = 0.14
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```
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Pressure lift is defined as:
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```math
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\textnormal{pressure lift} = \frac{\int_{cylinder} {-} pn\mathrm{d}\Gamma}{\rho u_{\infty}^2}
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```
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where _**n** is_ the outward normal to the cylinder. Its evolution is plotted in the green curve of the previous figure, showing a clear sinusoidal pattern which is a sign of a sustained vortex shedding process. The simulation gives an amplitude of the pressure lift of 0.26
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These results match fairly well with the values reported in the literature:
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* G. Houzeaux and J. Principe. A variational subgrid scale model for transient incompressible flows, International Journal of Computational Fluyd Dynamics, Vol. 22, No. 3, March 2008, 135-152.
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* C.H.K. Williamson and G.L. Brown. A Series is to represent the Strouhal-Reynolds number relationship of the cylinder wake, J. Fluids Struct. 12,1073 (1998)
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* * *
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* * *
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# <a class="anchor" id="Input_files"></a>Input files
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* * *
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## <a class="anchor" id="cylinder_dat"></a>cylinder.dat
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```
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$----------------------------------------------------------—
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RUN_DATA
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ALYA: cylinder $ case name
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RUN_TYPE: Preliminary, Frequency=1e6 \ $ initial solution, write restart file frequency
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END_RUN_DATA
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$----------------------------------------------------------—
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PROBLEM_DATA
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TIME_COUPLING: Global, Prescribed $ global time step defined here next
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TIME_INTERVAL= 0.0, 200.0 $ integration time domain from t=0 to t=200s
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TIME_STEP_SIZE= 0.2 $ prescribed integration time step
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NUMBER_OF_STEPS= 1e5 $ maximum number of time steps
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NASTIN_MODULE: On $ nastin module (Incompressible Navier-Stokes) is used
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END_NASTIN_MODULE
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PARALL_SERVICE: Off $ parallelization service is not used in this case
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END_PARALL_SERVICE
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</pre>
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END_PROBLEM_DATA
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$----------------------------------------------------------—
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```
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* * *
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## <a class="anchor" id="cylinder_dom_dat"></a>cylinder.dom.dat
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```
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$----------------------------------------------------------—
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DIMENSIONS
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NODAL_POINTS= 1280 $ number of nodes
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ELEMENTS= 1200 $ number of elements
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SPACE_DIMENSIONS= 2 $ 2D mesh
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TYPES_OF_ELEMENTS= 12 $ 4 node quadrilateral elements
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BOUNDARIES= 160 $ number of boundary edges
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END_DIMENSIONS
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$----------------------------------------------------------—
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STRATEGY
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INTEGRATION_RULE: Open $ open rule is the default
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DOMAIN_INTEGRATION_POINTS: 0 $ 0 = automatic, depending on each element type
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END_STRATEGY
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$----------------------------------------------------------—
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GEOMETRY
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GROUPS= 20 $ number of groups (for deflation based solvers)
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INCLUDE cylinder.geo.dat $ include geometry file (nodes, elements & boundaries)
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END_GEOMETRY
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$----------------------------------------------------------—
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SETS
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INCLUDE cylinder.set.dat $ include set file (groups for post-processing)
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END_SETS
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$----------------------------------------------------------—
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BOUNDARY_CONDITIONS, EXTRAPOLATE $ edge BC’s extrapolate to nodes
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INCLUDE cylinder.fix.dat $ include boundary conditions file
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END_BOUNDARY_CONDITIONS
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$----------------------------------------------------------—
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```
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* * *
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## <a class="anchor" id="cylinder_geo_dat"></a>cylinder.geo.dat
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```
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$----------------------------------------------------------—
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NODES_PER_ELEMENT
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1 4 $ element number, # nodes
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2 4
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3 4
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:
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:
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1199 4
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1200 4
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END_NODES_PER_ELEMENT
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$----------------------------------------------------------—
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ELEMENTS
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1 1169 1154 1159 1180 $ element number, node number 1, node number 2, ...
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2 1170 1153 1154 1169
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3 1171 1152 1153 1170
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:
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:
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1199 791 739 740 792
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1200 843 785 739 791
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END_ELEMENTS
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$----------------------------------------------------------—
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COORDINATES
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1 1.200000e+01 1.200000e+01 $ node number, coordinate X, coordinate Y
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:
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:
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1279 -5.600000e+00 -1.200000e+01
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1280 -6.000000e+00 -1.200000e+01
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END_COORDINATES
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$----------------------------------------------------------—
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BOUNDARIES, ELEMENT
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1 790 727 301 $ boundary element #, node #s, element # boundary belongs to
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2 727 681 302
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3 681 631 303
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:
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:
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159 730 731 1099
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160 784 730 1100
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END_BOUNDARIES
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MATERIALS, NUMBER=0, DEFAULT=1 $ all elements have material 1 END_MATERIALS CHARACTERISTICS END_CHARACTERISTICS
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$----------------------------------------------------------—
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```
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* * *
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## <a class="anchor" id="cylinder_set_dat"></a>cylinder.set.dat
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```
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$----------------------------------------------------------—
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ELEMENTS $ element sets definition
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1 6 $ element number, set number
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2 6
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3 6
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:
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:
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:
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1199 17
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1200 17
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END_ELEMENTS
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$----------------------------------------------------------—
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BOUNDARIES $ boundary sets definition
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<pre class="fragment">1 21 $ boundary element number, boundary set number
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2 21
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3 21
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:
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:
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:
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159 62
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160 62 $ 1 set for each domain boundary
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END_BOUNDARIES
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$----------------------------------------------------------—
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NODES $ node sets definition (for postprocesing)
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389, 270, 113, 611, 220
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END_NODES
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$----------------------------------------------------------—
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```
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* * *
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## <a class="anchor" id="cylinder_fix_dat"></a>cylinder.fix.dat
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```
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$----------------------------------------------------------—
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ON_BOUNDARIES
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<pre class="fragment">1 4 $ boundary element number, boundary number
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2 4
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3 4
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:
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:
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:
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159 3
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160 3
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END_ON_BOUNDARIES
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$----------------------------------------------------------—
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```
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* * *
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## <a class="anchor" id="cylinder_ker_dat"></a>cylinder.ker.dat
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```
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$---------------------------------------------------------—
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PHYSICAL_PROBLEM
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PROPERTIES $ fluid physical properties
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DENSITY: CONSTANT, PARAMETERS = 1.0 $ density
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VISCOSITY: CONSTANT, PARAMETERS = 0.01 $ dynamic viscosity (μ)
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END_PROPERTIES
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END_PHYSICAL_PROBLEM
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$---------------------------------------------------------—
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NUMERICAL_TREATMENT
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MESH
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DIVISION=0 $ mesh subdivisions (0 = no subdivisions)
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END_MESH
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ELSEST $ element search strategy
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STRATEGY: BIN $ BIN divide mesh into boxes to find elements that host
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witness points (suited for structured meshes)
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NUMBER: 100,10 $ 2D domain divided in 100 x 10 boxes
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END_ELSEST
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END_NUMERICAL_TREATMENT
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$---------------------------------------------------------—
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OUTPUT_&_POST_PROCESS
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ON_LAST_MESH
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STEPS=10 $ write post-process file every ‘STEPS’ time steps
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END_OUTPUT_&_POST_PROCESS
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$---------------------------------------------------------—
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```
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* * *
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## <a class="anchor" id="cylinder_nsi_dat"></a>cylinder.nsi.dat
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```
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$---------------------------------------------------------—
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PHYSICAL_PROBLEM
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PROBLEM_DEFINITION
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TEMPORAL_DERIVATIVES: On $ transient problem
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CONVECTIVE_TERM: On $ off for Stokes flow (negligible for high viscosity fluids)
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VISCOUS_TERM: LAPLACIAN $ suitable for constant viscosity fluid assumption
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END_PROBLEM_DEFINITION
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END_PHYSICAL_PROBLEM
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$---------------------------------------------------------—
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NUMERICAL_TREATMENT
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ELEMENT_LENGTH: Minimum $ element length for Alya to calculate critical time step and stabilization
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STABILIZATION: ASGS $ numerical stabilization method
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TIME_INTEGRATION: Trapezoidal, ORDER: 2, EULER=20 $ time integration scheme
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SAFETY_FACTOR: 100.0 $ multiply global time step: Alya calculates critical time
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step as required by explicit solvers, which is suited for transient analysis but makes stationary solutions converge very slowly
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STEADY_STATE_TOLER: 1e-12 $ onvergence tolerance for steady state
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NORM_OF_CONVERGENCE: LAGGED_ALGEBRAIC_RESIDUAL
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MAXIMUM_NUMBER_OF_IT: 1e5 $ max number of inner NS iterations
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CONVERGENCE_TOLERANCE: 1e-3 $ convergence tolerance for inner NS loop
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ALGORITHM: SCHUR $ NS solution algorithm (uncouples p & V)
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END_ALGORITHM
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MOMENTUM $ velocity solver
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ALGEBRAIC_SOLVER
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SOLVER: GMRES, KRYLOV=10 $ solver suited for asymmetric matrix
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CONVERGENCE: ITERA=1000, TOLER=1.0e-10, ADAPTIVE, RATIO=0.01 $ max iter #, convergence criteria
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ADAPTIVE, RATIO=0.001 means that the loop will end also if the difference of the norm of convergence value changes less than 0.1% in two consecutive iterations
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OUTPUT: CONVERGENCE $ solver convergence file (.cso) is generated
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PRECONDITIONER: DIAGONAL $ matrix preconditioner type
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END_ALGEBRAIC_SOLVER
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END_MOMENTUM
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CONTINUITY $ pressure solver
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ALGEBRAIC_SOLVER
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SOLVER: DEFLATED_CG, COARSE: SPARSE $ CG solvers are suited for symmetric matrix
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CONVERGENCE: ITERA=1000, TOLER=1.0e-10, ADAPTIVE, RATIO=0.01 $ max iter #, convergence criteria
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ADAPTIVE, RATIO=0.001 means that the loop will end also if the difference of the norm of convergence value changes less than 0.1% in two consecutive iterations
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OUTPUT: CONVERGENCE $ solver convergence file (.cso) is generated
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PRECONDITIONER: DIAGONAL $ matrix preconditioner type
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END_ALGEBRAIC_SOLVER
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END_CONTINUITY
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END_NUMERICAL_TREATMENT
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$---------------------------------------------------------—
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OUTPUT_&_POST_PROCESS
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START_POSTPROCES_AT STEP = 0 $ initial step to post process
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POSTPROCESS VELOCITY STEPS = 1 $ post process velocity every 1 step (ker.dat prevails)
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POSTPROCESS PRESSURE STEPS = 1 $ post process pressure every 1 step (ker.dat prevails)
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POSTPROCESS SCHUR STEPS = 1 $ post process schur every 1 step (ker.dat prevails)
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BOUNDARY_SET
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FORCE $ obtain pressure & velocity forces acting on boundaries
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TORQUE, CENTER=0,0,0 $ obtain torque of forces acting on boundaries, related to CENTER
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MASS $ obtain mass flow through boundaries
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END_BOUNDARY_SET
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END_OUTPUT_&_POST_PROCESS
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$---------------------------------------------------------—
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BOUNDARY_CONDITIONS
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PARAMETERS
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INITIAL: COARSE $ uses groups defined in dom.dat to generate an approximate steady initial solution
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END_PARAMETERS
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CODES, NODES
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$ flow around the cylinder $ boundary conditions
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1 11 1.000000 0.000000 $ Ux = 1 at inlet (left domain boundary)
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2 00 0.000000 0.000000 $ free condition for Ux and Uy (p~0) at the left outlet
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3 11 0.000000 0.000000 $ wall condition: Ux = Uy = 0 around the cylinder
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4 01 0.000000 0.000000 $ symmetry condition: Ux = free , Uy = 0
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1 & 4 11 1.000000 0.000000 $ inlet corners
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2 & 4 01 0.000000 0.000000 $ outlet corners
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END_CODES $ (note that BC’s apply to boundaries, not to boundary sets)
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END_BOUNDARY_CONDITIONS
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$---------------------------------------------------------—
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```
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* * *
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## <a class="anchor" id="cylinder_post_alya_dat"></a>cylinder.post.alya.dat
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```
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$----------------------------------------------------------------—
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DATA
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FORMAT: visit $ also valid for ParaView
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MARK_ELEMENTS: type $ to create automatic layers in post process according to some criterion
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ELIMINATE_BOUNDARY_NODES: yes $ for parallel runs, to avoid node duplicity between subdomain
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BOUNDARY: ON $ to post process boundary mesh
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SUBDOMAINS, ALL $ subdomains (parallel partitions) to post process
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END_SUBDOMAINS
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END_DATA
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$----------------------------------------------------------------—
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```
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* * *
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