FEATool Multiphysics  v1.16.6 Finite Element Analysis Toolbox
ex_navierstokes4.m File Reference

## Description

EX_NAVIERSTOKES4 2D Example for incompressible flow over a backwards facing step.

[ FEA, OUT ] = EX_NAVIERSTOKES4( VARARGIN ) Stationary flow over a backwards

facing step. References

[1] P.M. Gresho and R.L. Sani, Incompressible Flow and the Finite Element Method, Volume 1 & 2, John Wiley & Sons, New York, 2000.

[2] A. Rose and B. Simpson: “Laminar, Constant-Temperature Flow Over a Backward Facing Step,” 1st NAFEMS Workbook of CFD Examples, Glasgow, UK, 2000.

Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
re          scalar {389}           Reynolds number
h           scalar {1}             Channel height
y           scalar {0.485}         Step height (fraction of channel height)
lc          scalar {7.92}          Channel length (fraction of channel height)
li          scalar {1.98}          Inlet length (fraction of channel height)
hmax        scalar {0.1}           Max grid cell size
sf_u        string {sflag1}        Shape function for velocity
sf_p        string {sflag1}        Shape function for pressure
iphys       scalar 0/{1}           Use physics mode to define problem (=1)
solver      string openfoam/su2/{} Use OpenFOAM, SU2 or default solver
iplot       scalar 0/{1}           Plot solution (=1)
.
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

See also
ex_navierstokes12

# Code listing

 cOptDef = { ...
're',       389;
'h',        1;
'y',        0.0049/0.0101;
'lc',       0.08/0.0101;
'li',       0.02/0.0101;
'hmax',     0.1;
'sf_u',     'sflag1';
'sf_p',     'sflag1';
'iphys',    1;
'solver',   '';
'iplot',    1;
'tol',      0.2;
'fid',      1 };
[got,opt] = parseopt(cOptDef,varargin{:});
fid       = opt.fid;

% Geometry and grid parameters.
h         = opt.h;       % Height of channel.
y         = opt.y;       % Height of step.
lc        = opt.lc;      % Length of channel.
li        = opt.li;      % Length of inlet.
% Model parameters.
umax      = 1;    % Maximum magnitude of inlet velocity.
rho       = 1;           % Density.
miu       = umax*2/3*h/opt.re;    % Molecular/dynamic viscosity.
% 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.
vert = [ -li*h      lc*h lc*h    0 0 -li*h;   ...
(1-y)*h (1-y)*h -y*h -y*h 0     0];
gobj = gobj_polygon( vert' );
fea.geom.objects = { gobj };
fea.sdim = { 'x' 'y' };    % Coordinate names.

% Grid generation.
fea.grid = gridgen(fea,'hmax',opt.hmax,'fid',fid);
n_bdr    = max(fea.grid.b(3,:));        % Number of boundaries.

% Boundary conditions.
dtol      = opt.hmax;
i_inflow  = findbdr( fea, ['x<',num2str(-li*h+dtol)] );   % Inflow boundary number.
i_outflow = findbdr( fea, ['x>',num2str( lc*h-dtol)] );   % Outflow boundary number.
s_inflow  = ['4*',num2str(umax),'*(y*(',num2str((1-y)*h),'-y))/',num2str((1-y)*h),'^2'];   % Definition of inflow profile.
u_init    = ['4*',num2str(umax),'*(y*(',num2str((1-y)*h),'-y))/',num2str((1-y)*h),'^2*(y>0)'];

% 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 };
if( ~strcmp(opt.solver,'openfoam') )
fea.phys.ns.eqn.coef{5,end} = { u_init };
end
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) = 4;
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.
if( opt.iphys==1 && strcmp(opt.solver,'openfoam') )
logfid = fid; if( ~got.fid ), fid = []; end
fea.sol.u = openfoam( fea, 'fid', fid, 'logfid', logfid );
fid = logfid;
elseif( opt.iphys==1 && strcmp(opt.solver,'su2') )
logfid = fid; if( ~got.fid ), fid = []; end
fea.sol.u = su2( fea, 'init', [s_inflow,0,0], 'fid', fid, 'logfid', logfid );
fid = logfid;
elseif( opt.iphys==1 && strcmp(opt.solver,'fenics') )
fea = fenics( fea );
else
fea.sol.u = solvestat(fea,'fid',fid,'maxnit',50,'nlrlx',1,'tolchg',1e-3);   % Call to stationary solver.
end

% Postprocessing.
s_velm = 'sqrt(u^2+v^2)';
if ( opt.iplot>0 )
figure
subplot(3,1,1)
postplot(fea,'surfexpr',s_velm,'evaltype','exact','isoexpr',s_velm)
title('Velocity field')
subplot(3,1,2)
postplot(fea,'surfexpr','p','evaltype','exact')
title('Pressure')
subplot(3,1,3)
h = postplot(fea,'surfexpr',['(u<-eps)*x/',num2str(y)]);
title('Separation length')
end

% Error checking.
s_expr = ['(u<-eps)*x/',num2str(y)];
[~,slen] = minmaxsubd( s_expr, fea );
if( ~isempty(fid) )
fprintf(fid,'\nRecirculation zone length: %3f (Ref: 7.93)\n\n',slen)
fprintf(fid,'\n\n')
end

out.slen = [slen, 7.93];
out.err  = abs(diff(out.slen))/out.slen(end);
out.pass = out.err<opt.tol;
if ( nargout==0 )
clear fea out
end