Code - Fujin - Validation - Immersed Boundary Method

Uniform flow around a cylinder

 

Sketch of the computational domain and of the Cartesian coordinate system.

 

We consider a 2D rigid cylinder of radius R located in a computational domain of size \(15R \times 15R\), discretised with a resolution of \(68\) grid points per particle diameter. The incoming flow is uniform with velocity \(V_{in}\), providing a Reynolds number \(Re = D V_{in}/\nu\) of \(100\), \(150\) and \(200\).

config.h

#define _FLOW_COUETTE_
#define _FLOW_INOUT_
#define _FLOW_INOUT_UNBOUND_
#define _SOLVER_INOUT_
#define _MULTIPHASE_
#define _MULTIPHASE_IBM_
#define _MULTIPHASE_IBM_2D_

param.f90

! Total number of grid points in X, Y and Z
integer, parameter :: nxt = 4, nyt = 2048, nzt = 2048
! Size of the domain in X, Y and Z
real, parameter :: lx = 30.0/2048.*4., ly = 30.0, lz = 30.0
! Fluid viscosity and density
real, parameter :: vis = 1.0/200.0
real, parameter :: rho = 1.0
! Reference velocity (bulk velocity or wall velocity)
real, parameter :: refVel = 1.0
!Gravitational acceleration
real, dimension(3), parameter :: gr=(/0.0,0.0,-0.0/)
! Number of particles
integer, parameter :: ibmN = 1
! Particle radius and density
real, parameter :: ibmR = 0.5, ibmRho = 1.0, ibmRhoF = rho

input/input.in

# Simulation initial and final step
 1200000     9999999
 # Simulation time
 299.99999999463392
 # Simulation time-step
 2.5000000000000001E-004
 # Output frequency
 400     1000000       40000     1000000      400000
 # Simulation restart condition for the multiphase part
 F

initParticlePosition.dat

7.5    15.0
Vorte shedding behind the cylinder.

 

Strouhal number of a 2D cylinder in a unifrom flow. The line denotes the analytical result [1], the red symbols the numerical results from Ref. [2] and the blue symbols the numerical results. The empty blue symbols are numerical results obtained using the RKPM immersed boundary method.

 

Lateral migration of a circular cylinder in a shear flow

 

Sketch of the computational domain and of the Cartesian coordinate system.

 

We consider a 2D rigid cylinder of radius R located in a computational domain of size \(8R \times 50R\), discretised with a resolution of \(16\) grid points per particle diameter. The top and bottom walls move with opposite velocity \(\pm V_w\), providing a shear rate \(\dot{\gamma} = 2 V_w/8R\), while periodic boundary conditions are imposed in the remaining directions. The particle is neutrally buoyant, and the Reynolds number \(Re = \rho \dot{\gamma} R^2 / \mu\) is fixed equal to \(40\). Two different initial conditions are studied: \(z = 2R\) and \(6R\).

config.h

#define _FLOW_COUETTE_
#define _SOLVER_FFT2D_
#define _MULTIPHASE_
#define _MULTIPHASE_IBM_
#define _MULTIPHASE_IBM_2D_

param.f90

!Total number of grid points in X, Y and Z
integer, parameter :: nxt = 4, nyt = 8*50, nzt = 8*8
!Size of the domain in X, Y and Z
real, parameter :: lx = 4.0/8.0, ly = 50.0, lz = 8.0
!Fluid viscosity and density
real, parameter :: vis = 1.0*8.0/40.0, rho = 1.0
!Reference velocity (bulk velocity or wall velocity)
real, parameter :: refVel = 0.5
!Gravitational acceleration
real, dimension(3), parameter :: gr=(/0.0,0.0,-0.0/)
!Output frequency
integer,parameter :: iout0d = 20, iout1d = 1000000, iout2d = 4000, iout3d = 1000000, ioutfld = 1000000
integer, parameter :: ibmN=1
real, parameter :: ibmR=1.0, ibmRhoF=rho, ibmRho=ibmRhoF, ibmVol=pi*ibmR**2, ibmMas=ibmRho*ibmVol, ibmIne=0.5*ibmMas*ibmR**2

main.f90

!Simulation initial step
iStart = 0
!Simulation final step
iEnd   = 9999999
!Choose the timestep
dt = 0.000001

initParticlePosition.dat

0.5 2.0

 

Center of mass position of a 2D cylinder in a Couette flow. The line denotes the numerical results and the symbols those from Ref. [3].

 

Lateral migration of a circular cylinder in a pressure driven flow

 

Sketch of the computational domain and of the Cartesian coordinate system.

 

We consider a 2D rigid cylinder of radius R located in a computational domain of size \(8R \times 8R\), discretised with a resolution of \(32\) grid points per particle diameter. The flow is driven by an imposed pressure gradient, such that the resulting Reynolds number \(Re = \rho V_b 8R/\mu\) is equal to \(12.78\) and \(96.74\). The particle is neutrally buoyant and initially located at \(3.2R\) from the bottom wall.

config.h

#define _FLOW_CHANNEL_
#define _FLOW_CHANNEL_CPG_
#define _SOLVER_FFT2D_
#define _MULTIPHASE_
#define _MULTIPHASE_IBM_
#define _MULTIPHASE_IBM_2D_

param.f90

!Total number of grid points in X, Y and Z
integer, parameter :: nxt = 4, nyt = 256, nzt = 256
!Size of the domain in X, Y and Z
real, parameter :: lx = 1.0/128.0*4.0/2.0, ly = 1.0, lz = 1.0
!Fluid viscosity and density
real, parameter :: vis = 4.283E-04, rho = 1.0
!Reference velocity (bulk velocity or wall velocity)
real, parameter :: refVel = 0.0
!Imposed pressure gradient
real, parameter :: gradP = 2.337E-04
!Gravitational acceleration
real, dimension(3), parameter :: gr=(/0.0,0.0,-0.0/)
!Output frequency
integer,parameter :: iout0d = 20, iout1d = 1000000, iout2d = 4000, iout3d = 1000000, ioutfld = 1000000
integer, parameter :: ibmN=1
real, parameter :: ibmR=0.125, ibmRhoF=rho, ibmRho=ibmRhoF, ibmVol=pi*ibmR**2, ibmMas=ibmRho*ibmVol, ibmIne=0.5*ibmMas*ibmR**2

main.f90

!Simulation initial step
iStart = 0
!Simulation final step
iEnd   = 9999999
!Choose the timestep
dt = 0.001

initParticlePosition.dat

0.5 0.4

 

The particle trajectory in a pressure-driven flow at two different Reynolds numbers. The line denotes the numerical results and the symbols those from Ref. [4].

 

Sedimentation of a spherical particle

 

Sketch of the computational domain and of the Cartesian coordinate system.

 

We consider a rectangular domain filled with a liquid with density and viscosity equal to \(960 kg~m^{−3}\) and \(0.058 Pa~s\). The domain size is \(0.1 m \times 0.1m \times 0.16m\) in the \(x\), \(y\) and \(z\) directions, where \(x\) and \(y\) are the two horizontal directions, while \(z\) is the vertical direction parallel to the gravitational acceleration \(g = 9.81 m~s^{-2}\). A particle of diameter \(0.015m\) and density \(1120 kg~m^{−3}\) is initially positioned at \(\left(0.05 m, 0.05 m, 0.1275 m \right)\).

config.h

#define _FLOW_CHANNEL_
#define _FLOW_CHANNEL_CPG_
#define _SOLVER_FFT2D_
#define _MULTIPHASE_
#define _MULTIPHASE_IBM_
#define _MULTIPHASE_IBM_3D_

param.f90

!Total number of grid points in X, Y and Z
integer, parameter :: nxt = 192, nyt = 192, nzt = 304
!Size of the domain in X, Y and Z
real, parameter :: lx = 0.1, ly = 0.1, lz = 0.1/192.0*304.0
!Fluid viscosity and density
real, parameter :: vis = 0.058, rho = 960.0
!Reference velocity (bulk velocity or wall velocity)
real, parameter :: refVel = 0.0
!Imposed pressure gradient
real, parameter :: gradP = 0.0
!Gravitational acceleration
real, dimension(3), parameter :: gr=(/0.0,0.0,-9.8/)
!Output frequency
integer,parameter :: iout0d = 20, iout1d = 1000000, iout2d = 1000000, iout3d = 1000000, ioutfld = 1000000
integer, parameter :: ibmN=1
real, parameter :: ibmR = 0.015/2.0, ibmRhoF = rho, ibmRho = 1120.0

main.f90

!Simulation initial step
iStart = 0
!Simulation final step
iEnd   = 9999999
!Choose the timestep
dt = 0.0001

initParticlePosition.dat

0.05    0.05    0.1275

 

The sedimentation velocity of a spherical particle in a closed container. The line denotes the numerical results and the symbols those from Ref. [5].

 

References:

[1] C. H. K. Williamson. Defining a universal and continuous Strouhal-Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Physics of Fluids, 31(10):2742–2744, 1988.

[2] E. Guilmineau and P. Queutey. A numerical simulation of vortex shedding from an oscillating circular cylinder. Journal of Fluids and Structures, 16(6):773–794, 2002.

[3] J. Feng, H. H. Hu, and D. D. Joseph. Direct simulation of initial value problems for the motion of solid bodies in a newtonian fluid. part 2. couette and poiseuille flows. Journal of Fluid Mechanics, 277:271–301, 1994.

[4] T. W. Pan and R. Glowinski. Direct simulation of the motion of neutrally buoyant circular cylinders in plane poiseuille flow. Journal of Computational Physics, 181(1):260–279, 2002.

[5] A. Ten Cate, C. H. Nieuwstad, J. J. Derksen, and H. E. A. Van den Akker. Particle imaging velocimetry experiments and lattice-boltzmann simulations on a single sphere settling under gravity. Physics of Fluids, 14(11):4012–4025, 2002.