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ex_navierstokes3.m File Reference

Description

EX_NAVIERSTOKES3 2D Example for incompressible stationary flow around a cylinder.

[ FEA, OUT ] = EX_NAVIERSTOKES3( VARARGIN ) Stationary flow around a cylinder. References

[1] John V, Matthies G. Higher-order finite element discretizations in a benchmark problem for incompressible flows. International Journal for Numerical Methods in Fluids 2001.

[2] Nabh G. On higher order methods for the stationary incompressible Navier-Stokes equations. PhD Thesis, Universitaet Heidelberg, 1998.

Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
rho         scalar {1}             Density
miu         scalar {0.001}         Molecular/dynamic viscosity
umax        scalar {0.3}           Maximum magnitude of inlet velocity
igrid       scalar {2}             Grid type: >0 regular (igrid refinements)
                                              <0 unstruc. grid (with hmax=|igrid|)
sf_u        string {sflag2}        Shape function for velocity
sf_p        string {sf_disc1}      Shape function for pressure
iphys       scalar 0/{1}           Use physics mode to define problem (=1)
iplot       scalar 0/{1}           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 cOptDef = { ...
             'rho',      1; ...
             'miu',      0.001; ...
             'umax',     0.3; ...
             'igrid',    2; ...
             'sf_u',     'sflag2'; ...
             'sf_p',     'sf_disc1'; ...
             'iphys',    1; ...
             'iplot',    1; ...
             'tol',      [0.01 0.2 0.01]; ...
             'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


% Model parameters.
 rho       = opt.rho;     % Density.
 miu       = opt.miu;     % Molecular/dynamic viscosity.
 umax      = opt.umax;    % Maximum magnitude of inlet velocity.
 umean     = 2/3*umax;    % Mean inlet velocity.
% Geometry and grid parameters.
 h         = 0.41;        % Height of rectangular domain.
 l         = 2.2;         % Length of rectangular domain.
 xc        = 0.2;         % x-coordinate of cylinder center.
 yc        = 0.2;         % y-coordinate of cylinder center.
 diam      = 0.1;         % Diameter of cylinder.
% Discretization parameters.
 sf_u      = opt.sf_u;    % FEM shape function type for velocity.
 sf_p      = opt.sf_p;    % FEM shape function type for pressure.


% Grid generation.
 fea.sdim = { 'x' 'y' };
 if( opt.igrid>=1 )
   fea.grid = cylbenchgrid( opt.igrid );
 else
   gobj1 = gobj_rectangle( 0, 2.2, 0, 0.41, 'R1' );
   gobj2 = gobj_circle( [0.2 0.2], 0.05, 'C1' );
   fea.geom.objects = { gobj1 gobj2 };
   fea = geom_apply_formula( fea, 'R1-C1' );
   fea.grid = gridgen( fea, 'hmax', abs(opt.igrid), 'fid', fid );
 end
 n_bdr = max(fea.grid.b(3,:));   % Number of boundaries.


% Boundary conditions.
 dtol      = sqrt(eps)*1e3;
 i_inflow  = findbdr( fea, ['x<=',num2str(dtol)] );     % Inflow boundary number.
 i_outflow = findbdr( fea, ['x>=',num2str(l-dtol)] );   % Outflow boundary number.
 s_inflow  = ['4*',num2str(umax),'*(y*(',num2str(h),'-y))/',num2str(h),'^2'];   % Definition of inflow profile.
 i_cyl     = findbdr( fea, ['sqrt((x-',num2str(xc),').^2+(y-',num2str(yc),').^2)<=(',num2str(diam/2+dtol),')'] );    % Cylinder boundary number.


% Problem definition.
 if ( opt.iphys==1 )

   fea = addphys(fea,@navierstokes);     % Add Navier-Stokes equations physics mode.
   fea.phys.ns.eqn.coef{1,end} = { rho };
   fea.phys.ns.eqn.coef{2,end} = { miu };
   fea.phys.ns.sfun            = { sf_u sf_u sf_p };     % Set shape functions.

   fea.phys.ns.bdr.sel(i_inflow)  = 2;
   fea.phys.ns.bdr.sel(i_outflow) = 3;
   fea.phys.ns.bdr.coef{2,end}{1,i_inflow} = s_inflow;   % Set inflow profile.
   fea = parsephys(fea);                 % Check and parse physics modes.

 else

   fea.dvar  = { 'u'  'v'  'p'  };       % Dependent variable name.
   fea.sfun  = { sf_u sf_u sf_p };       % Shape function.

% Define equation system.
   cvelx = [num2str(rho),'*',fea.dvar{1}];   % Convection velocity in x-direction.
   cvely = [num2str(rho),'*',fea.dvar{2}];   % Convection velocity in y-direction.
   fea.eqn.a.form = { [2 3 2 3;2 3 1 1]       [2;3]                   [1;2]; ...
                      [3;2]                   [2 3 2 3;2 3 1 1]       [1;3]; ...
                      [2;1]                   [3;1]                   []   };
   fea.eqn.a.coef = { {2*miu miu cvelx cvely}  miu                    -1; ...
                      miu                    {miu 2*miu cvelx cvely} -1; ...
                      1                       1                      [] };
   fea.eqn.f.form = { 1 1 1 };
   fea.eqn.f.coef = { 0 0 0 };


% Define boundary conditions.
   fea.bdr.d = cell(3,n_bdr);
   [fea.bdr.d{1:2,:}]         = deal( 0);

   fea.bdr.d{1,i_inflow}      = s_inflow;

   [fea.bdr.d{:,i_outflow  }] = deal([]);

   fea.bdr.n = cell(3,n_bdr);
 end


% Parse and solve problem.
 fea       = parseprob(fea);             % Check and parse problem struct.
 jac.form  = {[1;1] [1;1] [];[1;1] [1;1] []; [] [] []};
 jac.coef  = {[num2str(rho),'*ux'] [num2str(rho),'*uy'] []; [num2str(rho),'*vx'] [num2str(rho),'*vy'] []; [] [] []};
 fea.sol.u = solvestat( fea, 'fid', fid, 'jac', jac, 'nlrlx', '(1+(it>2))/2' );   % Call to stationary solver.


% Postprocessing.
 s_velm = 'sqrt(u^2+v^2)';
 if ( opt.iplot>0 )
   figure
   subplot(2,1,1)
   postplot( fea, 'surfexpr', s_velm )
   title( 'Velocity field' )
   subplot(2,1,2)
   postplot( fea, 'surfexpr', 'p' )
   title( 'Pressure' )
 end


% Calculate benchmark quantities (line integration method).
 s_tfx = ['nx*p+',num2str(miu),'*(-2*nx*ux-ny*(uy+vx))'];
 s_tfy = ['ny*p+',num2str(miu),'*(-nx*(vx+uy)-2*ny*vy)'];
 s_cd  = ['2*(',s_tfx,')/(',num2str(rho),'*',num2str(umean),'^2*',num2str(diam),')'];
 s_cl  = ['2*(',s_tfy,')/(',num2str(rho),'*',num2str(umean),'^2*',num2str(diam),')'];
 i_cub = 10;
 c_d1  = intbdr(s_cd,fea,i_cyl,i_cub);
 c_l1  = intbdr(s_cl,fea,i_cyl,i_cub);
 dp    = evalexpr('p',[0.15 0.25;0.2 0.2],fea);


% Calculate benchmark quantities (volume integration method).
 bdrm   = fea.bdr.bdrm{1};
 ind_b  = [];
 ind_bm = [];
 for ii=i_cyl
   ind_b  = [ind_b  find(fea.grid.b(3,:)==ii)];
   ind_bm = [ind_bm find(bdrm(3,:)==ii)];
 end
 ind_c    = fea.grid.b(1,ind_b);
 ind_gdof = bdrm(4,ind_bm);

% Create field 'a' with values one on the cylinder and zero everywhere else.
 fea.dvar = [ fea.dvar {'a'}      ];
 fea.sfun = [ fea.sfun {'sflag2'} ];
 fea      = parseprob(fea);
 n_dof    = max(fea.eqn.dofm{1}(:));
 u_a      = zeros(n_dof,1);
 u_a(ind_gdof) = 1;
 fea.sol.u= [fea.sol.u;u_a];
 fea.eqn  = struct;
 fea.bdr  = struct;
 fea      = parseprob(fea);

 s_tfx    = ['ax*p+',num2str(miu),'*(-2*ax*ux-ay*(uy+vx))-(u*ux+v*uy)*a'];
 s_tfy    = ['ay*p+',num2str(miu),'*(-ax*(vx+uy)-2*ay*vy)-(u*vx+v*vy)*a'];
 s_cd     = ['2*(',s_tfx,')/(',num2str(rho),'*',num2str(umean),'^2*',num2str(diam),')'];
 s_cl     = ['2*(',s_tfy,')/(',num2str(rho),'*',num2str(umean),'^2*',num2str(diam),')'];
 c_d2     = intsubd(s_cd,fea,[],[],3);
 c_l2     = intsubd(s_cl,fea,[],[],3);


 if( ~isempty(fid) )
   fprintf(fid,'\n\nBenchmark quantities:\n\n')

   fprintf(fid,'Drag coefficient,    cd = %6f (l), %6f (v) (Ref: 5.579535)\n',c_d1,c_d2)
   fprintf(fid,'Lift coefficient,    cl = %6f (l), %6f (v) (Ref: 0.010619)\n',c_l1,c_l2)
   fprintf(fid,'Pressure,            dp = %6f (Ref: 0.117520)\n',dp(1)-dp(2))
 end


% Error checking.
 out.cd   = [c_d1 c_d2];
 out.cl   = [c_l1 c_l2];
 out.dp   = dp(1)-dp(2);
 out.err  = [abs(out.cd-5.579535)/5.579535; ...
             abs(out.cl-0.010619)/0.010619; ...
             abs(dp(1)-dp(2)-0.117520)/0.117520 0];
 out.pass = (out.err(1,2)<opt.tol(1))&&(out.err(2,2)<opt.tol(2))&&(out.err(3,1)<opt.tol(3));
 if ( nargout==0 )
   clear fea out
 end


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