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FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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EX_NAVIERSTOKES8 2D Example for axisymmetric incompressible stationary flow in a constricted circular pipe.
[ FEA, OUT ] = EX_NAVIERSTOKES8( VARARGIN ) Sets up and solves stationary axisymmetric Poiseuille flow in a constricted circular pipe. The inflow profile is constant and the outflow should assume a parabolic profile. Accepts the following property/value pairs.
Input Value/{Default} Description
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rho scalar {1} Density
miu scalar {1} Molecular/dynamic viscosity
uin scalar {1} Inflow velocity (constant/mean)
r scalar {1} Channel radius
l scalar {3} Channel length
hmax scalar {0.1} Max grid cell size
sf_u string {sflag1} Shape function for velocity
sf_p string {sflag1} Shape function for pressure
solver string 'openfoam'/{'} Use OpenFOAM or default solver
iplot scalar 0/{1} Plot solution and error (=1)
.
Output Value/(Size) Description
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fea struct Problem definition struct
out struct Output struct
cOptDef = { ...
'rho', 1;
'miu', 1;
'uin', 1;
'r', 1;
'l', 3;
'hmax', 0.05;
'sf_u', 'sflag1';
'sf_p', 'sflag1';
'solver', '';
'iplot', 1;
'fid', 1 };
[got,opt] = parseopt( cOptDef, varargin{:} );
fid = opt.fid;
% Geometry definition.
r = opt.r; % Pipe radius.
l = opt.l; % Pipe length.
gobj1 = gobj_rectangle( 0, r, 0, l*2/3, 'R1' );
gobj2 = gobj_rectangle( 0, r/2, l*2/3, l, 'R2' );
gobj3 = gobj_circle( [r l*2/3], r/2, 'C1' );
fea.geom.objects = { gobj1 gobj2 gobj3 };
fea = geom_apply_formula( fea, 'R1+R2-C1' );
fea.sdim = { 'r' 'z' }; % Space coordinate names.
% Grid generation.
fea.grid = gridgen( fea, 'hmax', opt.hmax, 'fid', fid );
% Boundary specifications.
i_inflow = 1; % Inflow boundary number.
i_outflow = 5; % Outflow boundary number.
i_symmetry = [3 6]; % Symmetry boundary numbers.
% Problem definition.
fea = addphys( fea, {@navierstokes 1} ); % Add Navier-Stokes equations physics mode.
fea.phys.ns.eqn.coef{1,end} = { opt.rho };
fea.phys.ns.eqn.coef{2,end} = { opt.miu };
fea.phys.ns.sfun = { opt.sf_u opt.sf_u opt.sf_p };
% Boundary conditions.
dtol = 1e-3;
i_in = findbdr( fea, ['z<=',num2str(dtol)] );
i_out = findbdr( fea, ['z>=',num2str(3-dtol)] );
i_sym = findbdr( fea, ['r<=',num2str(dtol)] );
fea.phys.ns.bdr.sel(i_in) = 2;
fea.phys.ns.bdr.sel(i_out) = 3;
fea.phys.ns.bdr.coef{2,end}{2,i_in} = opt.uin;
fea.phys.ns.bdr.sel(i_sym) = 5;
fea = parsephys(fea); % Parse physics mode.
% Parse and solve problem.
fea = parseprob(fea); % Check and parse problem struct.
if( strcmp(opt.solver,'openfoam') )
logfid = fid; if( ~got.fid ), fid = []; end
fea.sol.u = openfoam( fea, 'nproc', 1, 'fid', fid, 'logfid', logfid );
fid = logfid;
else
jac.form = {[1;1] [1;1] [];[1;1] [1;1] []; [] [] []};
jac.coef = {'r*rho_ns*ur' 'r*rho_ns*uz' []; 'r*rho_ns*wr' 'r*rho_ns*wz' []; [] [] []};
fea.sol.u = solvestat( fea, 'fid', fid, 'nsolve', 2, 'jac', jac );
end
% Error checking.
r = linspace( 0, opt.r/2, 20 );
z = 0.9*opt.l*ones( 1, 20 );
U = evalexpr( 'sqrt(u^2+w^2)', [r;z], fea )';
u_fac = 4; % Due to contraction to 1/2 radius.
U_ref = 2*opt.uin*u_fac*( 1 - ( r/(opt.r/2) ).^2 );
err = sqrt( sum((U-U_ref).^2)/sum(U_ref.^2) );
% Postprocessing.
if( opt.iplot>0 )
figure
subplot(1,3,1)
postplot( fea, 'surfexpr', 'sqrt(u^2+w^2)', 'arrowexpr', {'u' 'w'} )
hold on
plot( r, z, 'k--' )
title( 'Velocity field' )
subplot(1,3,2)
postplot( fea, 'surfexpr', 'p', 'evaltype', 'exact', 'isoexpr', 'p' )
title( 'Pressure' )
subplot(1,3,3)
plot( r, U, 'b-' )
hold on
plot( r, U_ref, 'r-' )
title('Velcity profile at z=0.9*l')
xlabel( 'Radius' )
legend( 'Computed', 'Reference', 'location', 'south' )
end
out.err = err;
out.pass = err<0.05;
if( nargout==0 )
clear fea out
end