FEATool  v1.9
Finite Element Analysis Toolbox
ex_swirl_flow1.m File Reference

Description

EX_SWIRL_FLOW1 2D Axisymmetric laminar swirl flow.

[ FEA, OUT ] = EX_SWIRL_FLOW1( VARARGIN ) Axisymmetric swirl for in tubular region where the inner cylindrical wall is rotating. Comparison with analytical solution.

Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
rho         scalar {2}             Density
miu         scalar {3}             Molecular/dynamic viscosity
omega       scalar {5}             Angular rotational frequency (of inner wall)
ri          scalar {0.5}           Inner radius
ro          scalar {1.5}           Outer radius
h           scalar {3}             Height of cylinder
sf_u        string {sflag1}        Shape function for velocity
sf_p        string {sflag1}        Shape function for pressure
iplot       scalar 0/{1}           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 cOptDef = { 'rho',      2;
             'miu',      3;
             'omega',    5;
             'ri',       0.5
             'ro',       1.5;
             'h',        3;
             'sf_u',     'sflag1';
             'sf_p',     'sflag1';
             'iphys',    1;
             'iplot',    1;
             'tol',      [];
             'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


% Geometry and grid generation.
 fea.sdim = {'r' 'z'};

 ri = opt.ri;   % Inner radius.
 ro = opt.ro;   % Outer radius.
 h  = opt.h;    % Height of cylinder.
 fea.geom.objects = { gobj_rectangle(ri,ro,-h/2,h/2) };

 fea.grid = gridgen( fea, 'hmax', (ro-ri)/6, 'fid', fid );


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

   fea = addphys(fea,@swirlflow);
   fea.phys.sw.eqn.coef{1,end} = { opt.rho };
   fea.phys.sw.eqn.coef{2,end} = { opt.miu };
   fea.phys.sw.sfun            = [ repmat( {opt.sf_u}, 1, 3 ) {opt.sf_p} ];
   fea.phys.sw.bdr.sel = [5 1 5 2];
   fea.phys.sw.bdr.coef{2,end}{2,4} = opt.omega*ri;

   fea = parsephys(fea);

 else

   opt.sf_u = 'sflag2';
   opt.sf_p = 'sflag1';

   fea.dvar = { 'u', 'v', 'w', 'p' };
   fea.sfun = [ repmat( {opt.sf_u}, 1, 3 ) {opt.sf_p} ];
   c_eqn    = { 'r*rho*u'' - r*miu*(2*ur_r + uz_z  +   wr_z) + r*rho*(u*ur_t + w*uz_t) + r*p_r     = r*Fr - 2*miu/r*u_t + p_t + rho*v*v_t';
                'r*rho*v'' - r*miu*(  vr_r + vz_z) + miu*v_r + r*rho*(u*vr_t + w*vz_t) + rho*u*v_t = r*Fth + miu*(v_r - 1/r*v_t)';
                'r*rho*w'' - r*miu*(  wr_r + uz_r  + 2*wz_z) + r*rho*(u*wr_t + w*wz_t) + r*p_z     = r*Fz';
                'r*ur_t + r*wz_t + u_t = 0' };
   fea.eqn = parseeqn( c_eqn, fea.dvar, fea.sdim );

   fea.coef = { 'rho', opt.rho ;
                'miu', opt.miu ;
                'Fr',  0 ;
                'Fth', 0 ;
                'Fz',  0 };

% Boundary conditions.
   fea.bdr.d = { []  0 [] 0 ;
                 []  0 [] opt.omega*ri ;
                 0   0  0 0 ;
                 [] [] [] [] };
   fea.bdr.n = cell(size(fea.bdr.d));

% Fix pressure at p([r,z]=[ro,h/2]) = 0.
   [~,ix_p] = min( sqrt( (fea.grid.p(1,:)-ro).^2 + (fea.grid.p(2,:)-h/2).^2) );
   fea.pnt = struct( 'type',  'constr', ...
                     'index', ix_p, ...
                     'dvar',  'p', ...
                     'expr',  '0' );
 end


% Parse and solve problem.
 fea = parseprob( fea );
 fea.sol.u = solvestat( fea, 'fid', fid );


% Exact (analytical) solution.
 a = - opt.omega*ri^2 / (ro^2-ri^2);
 b =   opt.omega*ri^2*ro^2 / (ro^2-ri^2);
 v_th_ex = @(r,a,b) a.*r + b./r;


% Postprocessing.
 if( opt.iplot )
   subplot(1,2,1)
   postplot( fea, 'surfexpr', 'sqrt(u^2+v^2+w^2)', 'isoexpr', 'v' )

   subplot(1,2,2)
   hold on
   grid on
   r = linspace( ri, ro, 100 );
   v_th = evalexpr( 'v', [r;zeros(1,length(r))], fea );
   plot( r, v_th, 'b--' )
   r = linspace( ri, ro, 10 );
   plot( r, v_th_ex(r,a,b), 'r.' )
   legend( 'Computed solution', 'Exact solution')
   xlabel( 'Radius, r')
   ylabel( 'Angular velocity, v')
 end


% Error checking.
 if( ~got.tol )
   if( opt.sf_u(end) == '2' )
     opt.tol = 0.01;
   else
     opt.tol = 0.16;
   end
 end
 r = linspace( ri, ro, 100 );
 v_th = evalexpr( 'v', [r;zeros(1,length(r))], fea )';
 out.err  = norm( v_th - v_th_ex(r,a,b) );
 out.pass = out.err < opt.tol;


 if( nargout==0 )
   clear fea out
 end