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ex_navierstokes1.m File Reference

Description

EX_NAVIERSTOKES1 2D Example for incompressible stationary flow in a channel.

[ FEA, OUT ] = EX_NAVIERSTOKES1( VARARGIN ) Sets up and solves stationary Poiseuille flow in a rectangular channel. The inflow profile is constant and the outflow should assume a parabolic profile ( u(y)=U_max*4/h^2*y*(h-y) ). 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
h           scalar {0.5}           Channel height
l           scalar {2.5}           Channel length
igrid       scalar 1/{0}           Cell type (0=quadrilaterals, 1=triangles)
hmax        scalar {0.04}          Max grid cell size
sf_u        string {sflag2}        Shape function for velocity
sf_p        string {sflag1}        Shape function for pressure
iphys       scalar 0/{1}           Use physics mode to define problem (=1)
iplot       scalar 0/{1}           Plot solution and error (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 cOptDef = { ...
   'rho',      1; ...
   'miu',      1e-3; ...
   'umax',     0.3; ...
   'h',        0.5; ...
   'l',        2.5; ...
   'igrid',    1; ...
   'hmax',     0.04; ...
   'sf_u',     'sflag2'; ...
   'sf_p',     'sflag1'; ...
   'iphys',    1; ...
   'iplot',    1; ...
   '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.
% Geometry and grid parameters.
 h         = opt.h;       % Height of rectangular domain.
 l         = opt.l;       % Length of rectangular domain.
% Discretization parameters.
 sf_u      = opt.sf_u;    % FEM shape function type for velocity.
 sf_p      = opt.sf_p;    % FEM shape function type for pressure.


% Geometry definition.
 gobj = gobj_rectangle( 0, l, 0, h );
 fea.geom.objects = { gobj };
 fea.sdim = { 'x' 'y' };   % Coordinate names.


% Grid generation.
 if ( opt.igrid==1 )
   fea.grid = gridgen(fea,'hmax',opt.hmax,'fid',fid);
 else
   fea.grid = rectgrid(round(l/opt.hmax),round(h/opt.hmax),[0 l;0 h]);
   if( opt.igrid<0 )
     fea.grid = quad2tri( fea.grid );
   end
 end
 n_bdr = max(fea.grid.b(3,:));           % Number of boundaries.


% Boundary conditions.
 dtol      = opt.hmax;
 i_inflow  = findbdr( fea, ['x<',num2str(dtol)] );     % Inflow boundary number.
 i_outflow = findbdr( fea, ['x>',num2str(l-dtol)] );   % Outflow boundary number.
 s_inflow  = ['2/3*',num2str(umax)];                                            % Definition of inflow profile.
 s_refsol  = ['4*',num2str(umax),'*(y*(',num2str(h),'-y))/',num2str(h),'^2'];   % Definition of velocity profile.


% 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.d{end,i_outflow}  = 0;   % Set pressure to zero on outflow boundary.

   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 );   % Call to stationary solver.


% Postprocessing.
 s_velm = 'sqrt(u^2+v^2)';
 s_err  = ['abs(sqrt((',s_refsol,')^2)-(',s_velm,'))'];
 s_len  = ['(x>',num2str(3/4*l),')'];
 if ( opt.iplot>0 )
   figure
   subplot(3,1,1)
   postplot(fea,'surfexpr',s_velm,'evaltype','exact')
   title('Velocity field')
   subplot(3,1,2)
   postplot(fea,'surfexpr','p','evaltype','exact')
   title('Pressure')
   subplot(3,1,3)
   postplot(fea,'surfexpr',[s_err,'*',s_len],'evaltype','exact')
   title('Error')
 end


% Error checking.
 if ( size(fea.grid.c,1)==4 )
   xi = [0;0];
 else
   xi = [1/3;1/3;1/3];
 end
 c_ind = find(evalexpr0(s_len,xi,1,1:size(fea.grid.c,2),[],fea))';
 err = evalexpr0(s_err,xi,1,c_ind,[],fea);
 ref = evalexpr0(['sqrt((',s_refsol,')^2)'],xi,1,c_ind,[],fea);
 err = sqrt(sum(err.^2)/sum(ref.^2));


 if( ~isempty(fid) )
   fprintf(fid,'\nL2 Error: %f\n',err)
   fprintf(fid,'\n\n')
 end


 out.err  = err;
 out.pass = err<0.05;
 if ( nargout==0 )
   clear fea out
 end