Demo problem: Flow past a cylinder with a waving flag

In this example we consider the flow in a 2D channel that contains a cylinder with a waving "flag". This is a "warm-up problem" for the solution of Turek & Hron's FSI benchmark problem discussed in another tutorial.

The Problem

The figure below shows a sketch of the problem: A 2D channel of height $ H^*$ and length $ L^* $ contains a cylinder of diameter $ d^* $, centred at $ (X^*_c, Y^*_c) $ to which a "flag" of thickness $ H^*_{flag}$ and length $ L^*_{flag} $ is attached. We assume that the flag performs time-periodic oscillations with period $ T^*. $ Steady Poiseuille flow with average velocity $ U^* $ is imposed at the left end of the channel while we assume the outflow to be parallel and axially traction-free.

Sketch of the problem in dimensional variables.

We non-dimensionalise all length and coordinates on the diameter of the cylinder, $ d^* $, time on the natural timescale of the flow, $ d^*/U^* $, the velocities on the mean velocity, $ U^* $, and the pressure on the viscous scale. The problem is then governed by the non-dimensional Navier-Stokes equations

\[ Re \left( St \frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} \right) = - \frac{\partial p}{\partial x_i} + \frac{\partial }{\partial x_j} \left[ \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \right], \]

where $ Re = \rho U^* H_0^* / \mu $ and $ St = 1 $, and

\[ \frac{\partial u_i}{\partial x_i} = 0, \]

subject to parabolic inflow

\[ {\bf u} = 6 x_2 (1-x_2) {\bf e}_1 \]

at the inflow cross-section; parallel, axially-traction-free outflow at the outlet; and no-slip on the stationary channel walls and the surface of the cylinder, $ {\bf u} = {\bf 0} $. The no-slip condition on the moving flag is

\[ {\bf u} = \frac{\partial {\bf R}_w(\xi_{[top,tip,bottom]},t)}{\partial t} \]

where $ \xi_{[top,tip,bottom]} $ are Lagrangian coordinates parametrising the three faces of the flag. The flag performs oscillations with non-dimensional period $ T = T^* U^* / H^*_{tot} $. Here is a sketch of the non-dimensional version of the problem:

Sketch of the problem in dimensionless variables, showing the Lagrangian coordinates that parametrise the three faces of the flag.


The figure below shows a snapshot of the flow field (pressure contours and instantaneous streamlines) for a Reynolds number of $ Re=100 $ and an oscillation period of $ T= 10 $, as well as the corresponding fluid mesh. Note how oomph-lib's automatic mesh adaptation has refined the mesh in the high-shear regions near the front of the cylinder and at the trailing edge of the flag.

Mesh (top) and flow field (bottom).

The corresponding animation illustrates the algebraic node update strategy (implemented with an AlgebraicMesh, discussed in more detail in another tutorial) and the evolution of the flow field. Note that the instantaneous streamlines intersect the (impermeable) flag because the flag is not stationary.

The global parameters

As usual we use a namespace to define the (single) global parameter, the Reynolds number.

/// Global parameters
namespace Global_Parameters
/// Reynolds number
double Re=100.0;

Specification of the flag geometry

We specify the flag geometry and its time-dependent motion by representing its three faces by GeomObjects, each parametrised by its own Lagrangian coordinate, as shown in the sketch above. The geometry of the cylinder is represented by one of oomph-lib's generic GeomObjects, the Circle.

We enclose the relevant data in its own namespace and start by defining the period of the oscillation, and the geometric parameters controlling the flag's initial shape and its subsequent motion.

/// Namespace for definition of flag boundaries
namespace Flag_definition
/// Period of prescribed flag oscillation
double Period=10.0;
/// Height of flag
double H=0.2;
/// Length of flag
double L=3.5;
/// x position of centre of cylinder
double Centre_x=2.0;
/// y position of centre of cylinder
double Centre_y=2.0;
/// Radius of cylinder
double Radius=0.5;
/// Amplitude of tip deflection
double Amplitude=0.33;
///Pointer to the global time object
Time* Time_pt=0;

We choose the motion of the flag such that it vaguely resembles that expected in the corresponding FSI problem: We subject the flag's upper and lower faces to purely vertical displacements that deform them into a fraction of a sine wave, while keeping the face at the tip of the flag straight and vertical. We implement this by prescribing the time-dependent motion of the two vertices at the tip of the flag with two functions:

/// Time-dependent vector to upper tip of the "flag"
Vector<double> upper_tip(const double& t)
double tmp_ampl=Amplitude;
if (t<=0.0) tmp_ampl=0.0;
Vector<double> uppertip(2);
uppertip[0]= Centre_x+Radius*sqrt(1.0-H*H/(4.0*Radius*Radius))+L;
uppertip[1]= Centre_y+0.5*H-
return uppertip;
/// Time-dependent vector to bottom tip of the "flag"
Vector<double> lower_tip(const double& t)
double tmp_ampl=Amplitude;
if (t<=0.0) tmp_ampl=0.0;
Vector<double> lowertip(2);
lowertip[0]= Centre_x+Radius*sqrt(1.0-H*H/(4.0*Radius*Radius))+L;
lowertip[1]= Centre_y-0.5*H-
return lowertip;
} // end of bottom tip

This information is then used in three custom-written GeomObject that define the time-dependent shape of the three faces. Here is the one describing the shape of the upper face:

/// GeomObject that defines the upper boundary of the flag
class TopOfFlag : public GeomObject
///Constructor: It's a 1D object in 2D space
TopOfFlag() : GeomObject(1,2) {}
///Destructor (empty)
/// \short Return the position along the top of the flag (xi[0] varies
/// between 0 and Lx)
void position(const unsigned& t,const Vector<double> &xi, Vector<double> &r)
// Compute position of fixed point on the cylinder
Vector<double> point_on_circle(2);
r[0] = point_on_circle[0]+xi[0]/L*(upper_tip(Time_pt->time(t))[0]-
r[1] = point_on_circle[1]+xi[0]/L*(upper_tip(Time_pt->time(t))[1]-
*sin(2.0* MathematicalConstants::Pi*Time_pt->time(t)/Period);
/// Current position
void position(const Vector<double> &xi, Vector<double> &r) const
return position(0,xi,r);
/// Number of geometric Data in GeomObject: None.
unsigned ngeom_data() const {return 0;}

[We omit the definition of the other two GeomObjects, BottomOfFlag and TipOfFlag in in the interest of brevity. Their definitions can be found in the source code.] We provide storage for the pointers to the four GeomObjects required for the representation of the flag,

/// Pointer to GeomObject that bounds the upper edge of the flag
TopOfFlag* Top_flag_pt=0;
/// Pointer to GeomObject that bounds the bottom edge of the flag
BottomOfFlag* Bottom_flag_pt=0;
/// Pointer to GeomObject that bounds the tip edge of the flag
TipOfFlag* Tip_flag_pt=0;
/// \short Pointer to GeomObject of type Circle that defines the
/// central cylinder.
Circle* Cylinder_pt=0;

and provide a setup function that generates the GeomObjects and stores the pointer to the global Time object that will be created by the Problem:

/// Create all GeomObjects needed to define the cylinder and the flag
void setup(Time* time_pt)
// Assign pointer to global time object
// Create GeomObject of type Circle that defines the
// central cylinder.
Cylinder_pt=new Circle(Centre_x,Centre_y,Radius);
/// Create GeomObject that bounds the upper edge of the flag
Top_flag_pt=new TopOfFlag;
/// Create GeomObject that bounds the bottom edge of the flag
Bottom_flag_pt=new BottomOfFlag;
/// Create GeomObject that bounds the tip edge of the flag
Tip_flag_pt=new TipOfFlag;
} // end of namespace that specifies the flag

The mesh

The figure below shows a sketch of the (unrefined) mesh and the enumeration of its boundaries. Various different implementations of the mesh exist, allowing the user to perform the node-update in response to the motion of the flag by a Domain/MacroElement - based procedure, or by using an algebraic node update. In either case, only the nodes in the elements that are shaded in yellow (or refined elements that are generated from these) participate in the node-update.

The (unrefined) mesh. Only nodes in the yellow regions participate in the node-update.

The node update strategy is illustrated in the animation of the flow field and the mesh motion.

The driver code

The driver code has the usual structure for a time-dependent problem: We store the command line arguments, create a DocInfo object, and assign the parameters that specify the dimensions of the channel.

/// Driver code -- pass a command line argument if you want to run
/// the code in validation mode where it only performs a few steps
int main(int argc, char* argv[])
// Store command line arguments
// Set up doc info
DocInfo doc_info;
// Length and height of domain
double length=25.0;
double height=4.1;

We build the problem using a mesh in which the node update is performed by the AlgebraicMesh class. (The driver code also provides the option to use a Domain/MacroElement - based node update – this option is chosen via suitable #ifdefs; see the source code for details).

// Create Problem with AlgebraicMesh-based node update
<AlgebraicElement<RefineableQCrouzeixRaviartElement<2> > >
problem(length, height);

Next we set up the time-stepping (as usual, fewer timesteps are performed during a validation run which is identified by a non-zero number of command line arguments):

// Number of timesteps per period
unsigned nsteps_per_period=40;
// Number of periods
unsigned nperiod=3;
// Number of timesteps (reduced for validation)
unsigned nstep=nsteps_per_period*nperiod;
if (CommandLineArgs::Argc>1)
// Also reduce the Reynolds number to reduce the mesh refinement
double dt=Flag_definition::Period/double(nsteps_per_period);
/// Initialise timestep

We start the simulation with a steady solve, allowing up to three levels of adaptive refinement (fewer if we are performing a validation run):

// Solve adaptively with up to max_adapt rounds of refinement (fewer if
// run during self-test)
unsigned max_adapt=3;
if (CommandLineArgs::Argc>1)
/// Output intial guess for steady Newton solve

Finally, we enter the proper timestepping loop, allowing one spatial adaptation per timestep and suppressing the re-assignment of initial conditions following an adaptation by setting the parameter first to false. (See the discussion of timestepping with automatic mesh adaptation in another tutorial.)

// Do steady solve first -- this also sets the history values
// to those corresponding to an impulsive start from the
// steady solution
/// Output steady solution = initial condition for subsequent unsteady solve
/// Reduce the max number of adaptations for time-dependent simulation
// We don't want to re-assign the initial condition after the mesh
// adaptation
bool first=false;
// Timestepping loop
for (unsigned istep=0;istep<nstep;istep++)
// Solve the problem
// Output the solution
// Step number
}//end of main

The Problem class

The problem class provides an access function to the mesh (note the use of #ifdefs to choose the correct one), and defines the interfaces for the usual member functions, either empty or explained in more detail below.

//======= start_of_problem_class=====================================
/// Flow around a cylinder with flag
template<class ELEMENT>
class TurekNonFSIProblem : public Problem
/// \short Constructor: Pass length and height of domain.
TurekNonFSIProblem(const double &length,
const double &height);
/// Update the problem specs after solve (empty)
/// Update the problem specs before solve (empty)
/// After adaptation: Unpin pressures and pin redudant pressure dofs.
/// Update the velocity boundary condition on the flag
/// Access function for the specific mesh
RefineableCylinderWithFlagMesh<ELEMENT>* mesh_pt()
return dynamic_cast<RefineableCylinderWithFlagMesh<ELEMENT>*>
/// Access function for the specific mesh
RefineableAlgebraicCylinderWithFlagMesh<ELEMENT>* mesh_pt()
return dynamic_cast<RefineableAlgebraicCylinderWithFlagMesh<ELEMENT>*>
/// Doc the solution
void doc_solution(DocInfo& doc_info);
/// Height of channel
double Height;

The problem constructor

The initial assignment of zero velocity and pressure provides a very poor initial guess for the preliminary steady Newton solve. We therefore increase the maximum residuals and iteration counts allowed in the Newton iteration before creating the timestepper and setting up the GeomObjects required to parametrise the flag:

/// Constructor: Pass length and height of domain.
template<class ELEMENT>
const double &length, const double &height) : Height(height)
// Bump up Newton solver parameters to allow for crappy initial guesses
// Allocate the timestepper
add_time_stepper_pt(new BDF<2>);
// Setup flag/cylinder geometry

Next we build the mesh, passing the pointers to the various GeomObjects and the geometric parameters to its constructor.

// Build mesh (with AlgebraicMesh-based node update)
Problem::mesh_pt()=new RefineableAlgebraicCylinderWithFlagMesh<ELEMENT>
length, height,

We perform two rounds of uniform refinement (the Newton solver does not converge on coarser meshes) before creating an error estimator for the subsequent automatic mesh adaptation.

// Refine uniformly
// Set error estimator
Z2ErrorEstimator* error_estimator_pt=new Z2ErrorEstimator;

Both velocity components are imposed by Dirichlet conditions (either via a no-slip condition or via the imposed inflow profile) on all boundaries, apart from the outflow cross-section (mesh boundary 1), where the axial velocity is unknown.

// Set the boundary conditions for this problem: All nodes are
// free by default -- just pin the ones that have Dirichlet conditions
// here.
//Pin both velocities at all boundaries apart from outflow
unsigned num_bound = mesh_pt()->nboundary();
for(unsigned ibound=0;ibound<num_bound;ibound++)
unsigned num_nod= mesh_pt()->nboundary_node(ibound);
for (unsigned inod=0;inod<num_nod;inod++)
// Parallel, axially traction free outflow at downstream end
if (ibound != 1)
} // done bc

Next we complete the build of the elements, passing the relevant pointers to physical parameters,

// Pass pointer to Reynolds number to elements
unsigned nelem=mesh_pt()->nelement();
for (unsigned e=0;e<nelem;e++)
// Upcast from GeneralisedElement to the present element
ELEMENT* el_pt = dynamic_cast<ELEMENT*>(mesh_pt()->element_pt(e));
//Set the Reynolds number
el_pt->re_pt() = &Global_Parameters::Re;
//Set the Womersley number (assuming St=1)
el_pt->re_st_pt() = &Global_Parameters::Re;

and impose the steady Poiseuille profile at the inlet (mesh boundary 3).

// Setup Poiseuille flow along boundary 3
unsigned ibound=3;
unsigned num_nod= mesh_pt()->nboundary_node(ibound);
for (unsigned inod=0;inod<num_nod;inod++)
double ycoord = mesh_pt()->boundary_node_pt(ibound,inod)->x(1);
// Set Poiseuille velocity
double uy = 6.0*ycoord/Height*(1.0-ycoord/Height);

Finally, we pin the redundant pressure degrees of freedom (see another tutorial for details), and assign the equations numbers.

// Pin redudant pressure dofs
// Assign equations numbers
cout <<"Number of equations: " << assign_eqn_numbers() << endl;
} // end of constructor

Actions before adapt

After each adaptation, we unpin and re-pin all redundant pressures degrees of freedom (see another tutorial for details). Since the inflow profile is parabolic, it is interpolated correctly from "father" to "son" elements during mesh refinement so no further action is required.

/// Actions after adapt
template <class ELEMENT>
// Unpin all pressure dofs
// Pin redundant pressure dofs
} // end_of_actions_after_adapt

Actions before the timestep

Before each timestep we update the nodal positions in the mesh and re-apply the no-slip condition at the nodes on mesh boundaries 5 - 7.

//==== start_of_actions_before_implicit_timestep==========================
/// Actions before implicit timestep: Update velocity boundary conditions
template <class ELEMENT>
// Update the domain shape
// Moving leaflet: No slip; this implies that the velocity needs
// to be updated in response to flag motion
for( unsigned ibound=5;ibound<8;ibound++)
unsigned num_nod=mesh_pt()->nboundary_node(ibound);
for (unsigned inod=0;inod<num_nod;inod++)
// Which node are we dealing with?
Node* node_pt=mesh_pt()->boundary_node_pt(ibound,inod);
// Apply no slip
} //end_of_actions_before_implicit_timestep

Post Processing

The function doc_solution(...) documents the results.

/// Doc the solution
template<class ELEMENT>
ofstream some_file;
char filename[100];
// Number of plot points
unsigned npts;
// Output solution
} // end_of_doc_solution

Comments and Exercises


Source files for this tutorial

PDF file

A pdf version of this document is available.