Hands on computer session: 1. Lid driven flow in a cavity [Time: 1 h 30 ] Objects choice of computational domain, problem adimensionalization and definition of boundary conditions; influence of the mesh refinement (uniform mesh vs. near wall refined mesh, increasingly refined meshes); evaluation of different discretization schemes: accurancy and dissipative effects; identification of reference data, calculation of quantities to verify simulation accuracy (streamfunction, velocity profiles along vertical and horizontal midlines); Richardson extrapolation as a discretization error extimator. Physical problem A square cavity filled with flow; the top wall moves with a uniform velocity U: isothermal laminar incompressible problem; primary central vortex and secondary vortices at the corners; complex flow patterns at increasing Reynolds number; stability bounds for Central Difference Scheme. Centro Interdipartimentale di Fluidodinamica e Idraulica 1
Non dimensional data top wall velocity U = 1 cavity width L = 2 cavity height H = 2 Reynolds number Re = 1000 and Re = 10000 fluid adimensional density rho = 1 fluid adimensional viscosity mu = 2/Re Worksheet Mesh generation Boundary conditions Solution method Results mesh.4a: uniform mesh 20x20x1 cells; mesh.4b: near wall refined mesh 20x20x1 cells; mesh.4c: near wall refined mesh 40x40x1 cells; mesh.4d: near wall refined mesh 80x80x1 cells; create a subdomain (a quarter domain) with block structure grid create near wall refinement (uniform space factor) use CGEN-REFLECT to replicate subdomain regions use symmetry B.C. to set up a bidimensional simulation use Wall B.C.(slip option) at the top wall use Wall B.C.(no-slip option) at the bottom and side walls steady state simulation at Re=1000 (mesh 4a,4b,4c), solve for V, P steady state simulation at Re=10000 (mesh 4c,4d), solve for V, P use of differencing scheme (UD,CD,MARS) considerations over velocity field (velocity vectors and stream function contour plots) extraction of u and v velocity component profiles along the vertical and the horizontal midlines comparison to the literature results Richardson extrapolation References Ghia, U., Ghia, K. N., and Shin C.T., (1982), High-Re solutions for incompressible flows using the Navier-Stokes equations and a multigrid method, J.Comput.Phys, 48, 387-411. Shon, J.L., (1988), Evaluation of FIDAP on some classical laminar and turbolent bench-marks, Int. J. Numer. Meth. Fluids, 8, 1469-1490. Nonino, C., and Croce, G., (1997), An equal-order velocity-pressure algorithm for incompressible thermal flows, part 2: validation, Numer. Heat Transfer, 32, 17-35. Centro Interdipartimentale di Fluidodinamica e Idraulica 2
Geometry and Results Figure 1: Refined grid 40x40 Figure 2: Streamfunction plot: right, Re=1000; left, Re=10000 [Ref.3] Figure 3: Streamfunction plot: right, Re=1000 (CDS); left, Re=10000 (MARS) Centro Interdipartimentale di Fluidodinamica e Idraulica 3
Re1000: u and v velocity components along vertical and horizontal midlines Figure 4: mesh with 20x20 cells: left,uniform mesh; right: near wall refined mesh.square[ref.3] Figure 5: Left: mesh with 20x20 cells; right: mesh comparison.square[ref.3]. Centro Interdipartimentale di Fluidodinamica e Idraulica 4
Re10000: u and v velocity components along vertical and horizontal midlines Figure 6: Meshes and discretization schemes comparison.square[ref.3]. Centro Interdipartimentale di Fluidodinamica e Idraulica 5
Reference values u components along vertical midline Re=1000 Re=10000 y u u 1.0 1.0 1.0 0.9532 0.65928 0.47221 0.9376 0.57492 0.47783 0.9218 0.51117 0.4807 0.9062 0.46604 0.47804 0.7032 0.33304 0.34635 0.4688 0.18719 0.20673 0.2344 0.05702 0.08344 0.0-0.0608 0.03111-0.0938-0.10648-0.0754-0.4374-0.27805-0.23186-0.6562-0.38289-0.32709-0.7968-0.2973-0.38-0.8594-0.2222-0.41657-0.875-0.20196-0.42537-0.8906-0.18109-0.42735-1.0 0.0 0.0 v components along horizontal midline Re=1000 Re=10000 x v v 1.0 0.0 0.0 0.9376-0.21388-0.54302 0.9218-0.27669-0.52987 0.9062-0.33714-0.49099 0.8906-0.39188-0.45863 0.8126-0.5155-0.41496 0.7188-0.42665-0.36737 0.6094-0.31966-0.30719 0.0 0.02526 0.00831-0.5312 0.32235 0.27224-0.5468 0.33075 0.28003-0.6874 0.37095 0.3507-0.8124 0.32627 0.41487-0.8438 0.30353 0.43124-0.8594 0.29012 0.43733-0.875 0.27485 0.43983-1 0.0 0.0 Centro Interdipartimentale di Fluidodinamica e Idraulica 6
MACROS! Geometry.MAC *!!*** complete mesh generation by symmetry along x and y! get vmax mxve cset all cgen,2,vmax,cset,cset,1,vref,1,2 cset all cplot get vmaxnew mxve cgen,2,vmaxnew,cset,cset,1,vref,1,1 cset all cplot vmer all vcomp all ccomp all cset all cdirection,1,0,1,1,1 restructure,50000,,default, $y vset news vran 0,49000,1 vcomp all $y cset all cplot!umidplane-x0-gu.mac *!!*** y coordinate and u velocity component extraction on vertical midline to screen and to file.usr! vset subset gran -0.0001 0.0001,,, -0.01 0.01 greset y getv, None,SU gpost,vset,,,next frame,1,xreg,init,4 frame,1,xtitle,4.3,1.5 Y LOCATION frame,1,yreg,init,9 frame,1,ytitle,0.5,5.5 SU term,,x gdraw,1 rlabel,9,on SU Centro Interdipartimentale di Fluidodinamica e Idraulica 7
lint,9,9,3,1 symt,9,9,8,1 $rtab,9,y,y bart,9,9,90,1 gredraw oper,getv,x,1,1 oper,getv,y,1,2 oper,getv,z,1,3 oper,getv,su,4 savu,file.usr,both,user,vset I9,6X,4G16.9 close file.usr! Vmidplane-y0-GU.MAC *!!*** x coordinate and v velocity component extraction on horizontal midline to screen and to file.usr! vset subset gran,,, -0.0001 0.0001-0.01 0.01 greset y getv, None,SV gpost,vset,,,next frame,1,xreg,init,3 frame,1,xtitle,4.3,1.5 X LOCATION frame,1,yreg,init,9 frame,1,ytitle,0.5,5.5 SV term,,x gdraw,1 rlabel,9,on SV lint,9,9,3,1 symt,9,9,8,1 $rtab,9,y,y bart,9,9,90,1 gredraw oper,getv,x,1,1 oper,getv,y,1,2 oper,getv,z,1,3 oper,getv,sv,4 savu,file.usr,both,user,vset I9,6X,4G16.9 close file.usr Centro Interdipartimentale di Fluidodinamica e Idraulica 8
! U-V-midplane.MAC *!!*** comparison of u and v velocity profile from different file.pst! - load exe4-ud.pst greset y vset subset gran -0.0001 0.0001,,, -0.01 0.01 greset y numreg,100 getv, None,SU gpost,vset,,,1 sens dele all vset subset gran,,, -0.0001 0.0001-0.01 0.01 getv, None,SV gpost,vset,,,12 frame,1,xreg,init,14,9 frame,1,yreg,init,20,4 frame,1,xrange,-1,1 frame,1,xtitle,4.94,1.6 U frame,1,yrange,-1,1 frame,1,ytitle,0.71,5.51 V rlabel,20,on SV lint,20,11,3,1 symt,20,20,8,1 $rtab,20,y,n bart,20,20,90,1 rlabel,4,on Y LOCATION lint,4,4,3,1 symt,4,4,8,1 $rtab,4,y,n bart,4,4,90,1 term,,x gdraw,1 load exe4mars.pst vset subset gran -0.0001 0.0001,,, -0.01 0.01 getv, None,SU gpost,vset,,,24 sens dele all vset subset gran,,, -0.0001 0.0001-0.01 0.01 getv, None,SV Centro Interdipartimentale di Fluidodinamica e Idraulica 9
gpost,vset,,,36 frame,1,xreg,init,38,32,14,9 frame,1,yreg,init,44,27,20,4 frame,1,xrange,-1,1 frame,1,xtitle,4.94,1.6 U frame,1,yrange,-1,1 frame,1,ytitle,0.71,5.51 V rlabel,44,on SV lint,44,3,3,1 symt,44,44,8,1 $rtab,44,y,n bart,44,44,90,1 rlabel,27,on Y LOCATION lint,27,27,3,1 symt,27,27,8,1 $rtab,27,y,n bart,27,27,90,1 term,,x gdraw,1 Centro Interdipartimentale di Fluidodinamica e Idraulica 10