Flow around a Cylinder Using the Sharp Interface Method#

Features#

  • Solvers: lethe-fluid-sharp (with Q1-Q1)

  • Steady-state problem

  • Explains how to set sharp interface immersed boundary to represent a particle

  • Displays the use of non-uniform mesh adaptation

Files Used in This Example#

  • Parameter file: examples/sharp-immersed-boundary/cylinder-with-sharp-interface/cylinder-with-sharp-interface.prm

Description of the Case#

Mesh#

In this example, we study the flow around a static cylinder in 2D using the sharp-interface method to represent the cylinder. The geometry of the flow takes the same basic case defined in Flow around a Cylinder. As such, we use the parameter file associated with Flow around a Cylinder as the base for this example. The exact dimensions for this example can be found in the following figure.

Simulation schematic

This case uses a cartesian structured rectangular mesh, and we define the position and radius of the immersed boundary.

The mesh is defined using the following subsection.

subsection mesh
    set type               = dealii
    set grid type          = subdivided_hyper_rectangle
    set grid arguments     = 2,1: 0,0 : 32 , 16  : true
    set initial refinement = 7
end

Boundary Conditions#

As for the Flow around a Cylinder, we define the boundary conditions to have an inlet on the left, two slip boundary conditions at the top and bottom, and an outlet on the right of the domain.

subsection boundary conditions
  set number = 3

  subsection bc 0
    set id   = 0
    set type = function
    subsection u
      set Function expression = 1
    end
    subsection v
      set Function expression = 0
    end
    subsection w
      set Function expression = 0
    end
  end
  subsection bc 1
    set id   = 2
    set type = slip
  end
  subsection bc 2
    set id   = 3
    set type = slip
  end
end

Initial Conditions#

The initial condition has been modified compared to the initial solution proposed in Flow around a Cylinder. We use the following initial condition to ensure that the particle’s boundary condition is satisfied.

subsection initial conditions
  set type = nodal
  subsection uvwp
    set Function expression = 0; 0; 0
  end
end

IB Particles#

The only thing that is left to define is the immersed boundary. In this case, we want to define a circular boundary of radius 0.5 center at (8,8) that has no velocity. We use the sphere to model the cylinder in 2D.

subsection particles
  set number of particles                     = 1
  set assemble Navier-Stokes inside particles = false
  subsection extrapolation function
    set stencil order = 2
  end
  subsection local mesh refinement
    set initial refinement                = 0
    set refine mesh inside radius factor  = 0.8
    set refine mesh outside radius factor = 1.2
  end
  subsection particle info 0
    set type             = sphere
    set shape arguments  = 0.5
    set integrate motion = false
    subsection position
      set Function expression = 8;8
    end
    subsection velocity
      set Function expression = 0;0
    end
  end
end

  • number of particles is set to one as we only want one particle.

  • stencil order is set to 2 as this is the highest order that is compatible with the FEM scheme and it does not lead to Runge instability. The highest order of stencil compatible with a FEM scheme is defined by the polynomial order of the scheme time the number of dimensions. In this case 2.

  • refine mesh inside radius factor is set to 0.8. This will create a mesh refinement around the particle that avoids having hanging nodes in the calculation and helps ensure an adequately fine mesh around the particle.

  • refine mesh outside radius factor is set to 1.2. This will create a mesh refinement around the particle that avoids having hanging nodes in the calculation and helps ensure an adequately fine mesh around the particle.

  • initial refinement is set to 0. In this case, the initial mesh is small enough to ensure that the mesh around the particle is sufficiently smaller than the particle. In this case, it is not necessary to pre-refine the mesh around the particle.

  • integrate motion is set to false because we are not interested in the dynamic of the particle as this is a steady case.

  • assemble Navier-Stokes inside particles is set to false because we are not interested in the flow inside of the particle.

  • position Function expression is set to 8;8 as the position of the particle is constant in time, and the center of the particle is at this position.

  • velocity Function expression is set to 0;0 as the velocity of the particle is 0 and the case is steady.

All the other parameters have been set to their default values since they do not play a role in this case.

Results#

The simulation of this case results in the following solution for the velocity and pressure field.

Velocity:

Simulation schematic

Pressure:

Simulation schematic

We get the following force applied on the particle for each of the mesh refinements, which is similar to the one obtained with a conformal mesh in Flow around a Cylinder. With the conformal mesh drag force applied to the particle is 7.123. The difference between the 2 can mostly be attributed to the discretization error.

particle_ID    T_z        f_x         f_y
        0   -0.006703   6.447400    0.004273
        0   -0.000389   6.775330    0.000379
        0   -0.000040   6.906123    0.000377
        0   -0.000053   6.962566    0.000310
        0   -0.000039   6.992112    0.000193

Note

The drag coefficient obtained in this case is higher than the drag coefficient for a cylinder at a Reynolds number of 1 as the size of the domain is not large enough relative to the diameter of the cylinder. The flow around the cylinder is then constrained by the lateral boundaries, and this increases the drag coefficient.