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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_MULTIPHASE2 Multiphase flow rising bubble example.
[ FEA, OUT ] = EX_MULTIPHASE2( VARARGIN ) A benchmark multiphase flow rising bubble example. Reference solution is given in the paper Quantitative benchmark computations of two-dimensional bubble dynamics. Int. J. Numer. Meth. Fluids, 60: 1259-1288,
Input Value/{Default} Description
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rho1 scalar {1e2} Density of the bubble
rho2 scalar {1e3} Density of the surrounding fluid
miu1 scalar {1} Viscosity of the bubble
miu2 scalar {1e1} Viscosity of the surrounding fluid
gy scalar {0.98} Gravitational constant
sigma scalar {24.5} Coefficient of surface tension
igrid scalar 0/{1} Cell type (0=quadrilaterals, 1=triangles)
hmax scalar {1/20} Max grid cell size
dt scalar {0.1} Time step size
sf_u string {sflag2} Shape function for velocity
sf_p string {sflag1} Shape function for pressure
sf_c string {sflag1} Shape function for the level set function
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 = { ...
'rho1', 1e2; ...
'rho2', 1e3; ...
'miu1', 1; ...
'miu2', 1e1; ...
'gy', 0.98; ...
'sigma', 24.5; ...
'igrid', 0; ...
'hmax', 1/20; ...
'dt', 0.1; ...
'sf_u', 'sflag2'; ...
'sf_p', 'sflag1'; ...
'sf_c', 'sflag1'; ...
'iplot', 1; ...
'itest', 0; ...
'fid', 1 };
[got,opt] = parseopt(cOptDef,varargin{:});
fid = opt.fid;
% Geometry and grid generation.
fea.sdim = { 'x' 'y' }; % Coordinate names.
fea.grid = rectgrid(round(1/opt.hmax),round(2/opt.hmax),[0 1;0 2]);
if ( opt.igrid==1 )
fea.grid = quad2tri( fea.grid );
end
n_bdr = max(fea.grid.b(3,:)) + 1; % Increment number of boundaries.
fea.grid.b(3,1) = n_bdr; % Set first boundary edge to boundary n_bdr.
% Problem definition.
dh = num2str(1.5*opt.hmax); % Smoothing region width.
dw = ['(c/',dh,'/(sqrt(cx^2+cy^2+eps)))'];
nx = ['(cx/(sqrt(cx^2+cy^2+eps)))'];
ny = ['(cy/(sqrt(cx^2+cy^2+eps)))'];
smhs = ['((0.5*(1+',dw,'+1/pi*sin(pi*',dw,')))*(',dw,'>-1)*(',dw,'<1)+(',dw,'>=1))']; % Smooth heaviside function.
smdel = ['0.5*(1+cos(pi*',dw,'))/',dh,'*(',dw,'>-1)*(',dw,'<1)']; % Smooth delta function.
rho = [num2str(opt.rho1),'+',num2str(opt.rho2-opt.rho1),'*',smhs];
miu = [num2str(opt.miu1),'+',num2str(opt.miu2-opt.miu1),'*',smhs];
sigma = num2str(opt.sigma);
% Add Navier-Stokes equations physics mode.
fea = addphys(fea,@navierstokes);
fea.phys.ns.eqn.coef{1,end} = { rho };
fea.phys.ns.eqn.coef{2,end} = { miu };
fea.phys.ns.bdr.sel(n_bdr) = 4; % Set pressure to zero on last boundary segment.
fea.phys.ns.sfun = { opt.sf_u opt.sf_u opt.sf_p };
% Source terms for gravity and surface tension effects.
fg = ['-(',rho,')*',num2str(opt.gy)];
fx1 = ['-',sigma,'*(1-(',nx,')^2)*',smdel];
fx2 = ['-',sigma,'*(-(',nx,')*(',ny,'))*',smdel];
fy1 = ['-',sigma,'*(-(',ny,')*(',nx,'))*',smdel];
fy2 = ['-',sigma,'*(1-(',ny,')^2)*',smdel];
% Add convection and diffusion physics mode for the level set equation.
fea = addphys(fea,@convectiondiffusion);
fea.phys.cd.sfun = { opt.sf_c };
fea.phys.cd.eqn.coef{2,4} = { 0.001 }; % Add stabilizing diffusion.
fea.phys.cd.eqn.coef{3,4} = { 'u' }; % Convection velocity in the x-direction.
fea.phys.cd.eqn.coef{4,4} = { 'v' }; % Convection velocity in the y-direction.
% Parse physics modes.
fea = parsephys(fea);
% Correct source terms.
fea.eqn.f.form{1} = [ 2 3];
fea.eqn.f.form{2} = [ 1 2 3];
fea.eqn.f.coef{1} = { fx1 fx2};
fea.eqn.f.coef{2} = {fg fy1 fy2};
% Implement slip boundary conditions on vertical walls.
fea.bdr.d{2}{2} = [];
fea.bdr.d{2}{4} = [];
fea.bdr.n{2}{2} = 0;
fea.bdr.n{2}{4} = 0;
% Parse problem.
fea = parseprob(fea);
% Call to time-dependent solver.
init = { '0', '0', '0', 'sqrt((x-0.5)^2+(y-0.5)^2)-0.25' };
fea.sol.u = solvetime( fea, ...
'fid', fid, ...
'tmax', 3.0*(~opt.itest), ...
'tstep', opt.dt, ...
'maxnit', 24*(~opt.itest)+1, ...
'icub', 3, ...
'init', init, ...
'ischeme', 3 );
% Postprocessing.
if ( opt.iplot>0 )
subplot(2,2,1)
postplot(fea,'surfexpr','sqrt(u^2+v^2)')
title('Velocity field')
subplot(2,2,2)
postplot(fea,'surfexpr','p','evalstyle','exaxct')
title('Pressure')
subplot(2,2,3)
postplot(fea,'surfexpr','c')
title('Level set field')
subplot(2,2,4)
postplot(fea,'isoexpr','c','isolev',[0 0],'arrowexpr',{'u','v'})
plot([0 1 1 0 0],[0 0 2 2 0],'k')
title('Interface')
end
out.err = [];
out.pass = [];
if ( nargout==0 )
clear fea out
end