Figure 1.
Numerical solver system organization.
Figure 1.
Numerical solver system organization.
Figure 2.
Numerical solver module diagram.
Figure 2.
Numerical solver module diagram.
Figure 3.
Poiseuille flow in a planar channel and the velocity component . We perform measurements on the line or at the grid points , , , and .
Figure 3.
Poiseuille flow in a planar channel and the velocity component . We perform measurements on the line or at the grid points , , , and .
Figure 4.
The snapshots: (a) of the velocity field for Oldroyd-B model with at ; (b) of the pressure field p for Oldroyd-B model with at ; and (c) of the stress tensor field for Oldroyd-B model with at .
Figure 4.
The snapshots: (a) of the velocity field for Oldroyd-B model with at ; (b) of the pressure field p for Oldroyd-B model with at ; and (c) of the stress tensor field for Oldroyd-B model with at .
Figure 5.
The snapshots: (a) of the velocity field for Hooke model with at ; (b) of the pressure field p for Hooke model with with at ; and (c) of the stress tensor field for Hooke model with with at .
Figure 5.
The snapshots: (a) of the velocity field for Hooke model with at ; (b) of the pressure field p for Hooke model with with at ; and (c) of the stress tensor field for Hooke model with with at .
Figure 6.
The snapshots: (a) of the velocity field for FENE model with , at ; (b) of the pressure field p for FENE model with , at ; and (c) of the stress tensor field for FENE model with , at .
Figure 6.
The snapshots: (a) of the velocity field for FENE model with , at ; (b) of the pressure field p for FENE model with , at ; and (c) of the stress tensor field for FENE model with , at .
Figure 7.
The evolution of horizontal velocity component for different models (Oldroyd-B model, Hooke model and FENE model) over time at point .
Figure 7.
The evolution of horizontal velocity component for different models (Oldroyd-B model, Hooke model and FENE model) over time at point .
Figure 8.
Stress component on the vertical channel wall during the steady state at : (a) for different of Hooke model; (b) for different number of Hooke model; and (c) for different mesh levels of Hooke model.
Figure 8.
Stress component on the vertical channel wall during the steady state at : (a) for different of Hooke model; (b) for different number of Hooke model; and (c) for different mesh levels of Hooke model.
Figure 9.
Velocity component on the vertical channel wall during the steady state at : (a) for different of Hooke model and Oldroyd-B model ; (b) for different number of Hooke model and Oldroyd-B model ; and (c) for different mesh levels of Hooke model and Oldroyd-B model .
Figure 9.
Velocity component on the vertical channel wall during the steady state at : (a) for different of Hooke model and Oldroyd-B model ; (b) for different number of Hooke model and Oldroyd-B model ; and (c) for different mesh levels of Hooke model and Oldroyd-B model .
Figure 10.
(a) Stress component ; and (b) velocity component on the y-axis during the steady state at for different number of FENE model.
Figure 10.
(a) Stress component ; and (b) velocity component on the y-axis during the steady state at for different number of FENE model.
Figure 11.
(a) Stress component ; and (b) velocity component on the y-axis during the steady state at for different mesh levels of FENE model.
Figure 11.
(a) Stress component ; and (b) velocity component on the y-axis during the steady state at for different mesh levels of FENE model.
Figure 12.
(a) Stress component ; and (b) velocity component on the y-axis during the steady state at for different of FENE model.
Figure 12.
(a) Stress component ; and (b) velocity component on the y-axis during the steady state at for different of FENE model.
Figure 13.
(a) Stress component ; and (b) velocity component on the y-axis during the steady state at for different dumbbell’s extension parameter b of FENE model.
Figure 13.
(a) Stress component ; and (b) velocity component on the y-axis during the steady state at for different dumbbell’s extension parameter b of FENE model.
Figure 14.
(a) Stress component ; and (b) velocity component on the y-axis during the steady state at for FENE model with the dumbbell’s extension parameter and Hooke model.
Figure 14.
(a) Stress component ; and (b) velocity component on the y-axis during the steady state at for FENE model with the dumbbell’s extension parameter and Hooke model.
Figure 15.
(a) Stress component on the y-axis; (b) stretch length D on the y-axis; (c) the distribution of molecules at point ; (d) corresponding probability density function (pdf) at point ; (e) the distribution of molecules at point ; (f) the distribution of molecules at point ; (g) the distribution of molecules at point ; (h) the distribution of molecules at point ; (i) corresponding probability density function (pdf) at point ; (j) corresponding probability density function (pdf) at point ; (k) corresponding probability density function (pdf) at point ; and (l) corresponding probability density function (pdf) at point .
Figure 15.
(a) Stress component on the y-axis; (b) stretch length D on the y-axis; (c) the distribution of molecules at point ; (d) corresponding probability density function (pdf) at point ; (e) the distribution of molecules at point ; (f) the distribution of molecules at point ; (g) the distribution of molecules at point ; (h) the distribution of molecules at point ; (i) corresponding probability density function (pdf) at point ; (j) corresponding probability density function (pdf) at point ; (k) corresponding probability density function (pdf) at point ; and (l) corresponding probability density function (pdf) at point .
Table 1.
The main partial differential equations discrete programming interface in OpenFOAM.
Table 1.
The main partial differential equations discrete programming interface in OpenFOAM.
Differential term | Explicit/implicit | Model expressions | Function name |
---|
laplace term | explicit/implicit | | laplacian(phi) |
| | | laplacian(Gamma,phi) |
time derivative term | explicit/implicit | | ddt(phi) |
| | | ddt(rho,phi) |
2-nd order time derivative term | explicit/implicit | | d2dt2(rho,phi) |
convection term | explicit/implicit | | div(psi,scheme) |
| | | div(psi,phi,word) |
| | | div(psi,phi) |
divergence term | explicit | | div(chi) |
gradient term | explicit | | grad(chi) |
| | | gGrad(phi) |
| | | lsGrad(phi) |
| | | snGrad(phi) |
| | | snGradCorrection(phi) |
source term | implicit | | Sp(rho,phi) |
Table 2.
The basic configuration of linear equations solver.
Table 2.
The basic configuration of linear equations solver.
Field variables | Solver | Preprocessor | Error limit |
---|
p | PCG | GAMG | |
| PBiCG | GAMG | |
| BICCG | DILU | |
| PBiCG | DILU | |
Table 3.
Numerical discrete format used in the solver.
Table 3.
Numerical discrete format used in the solver.
Equation terms | Discrete format | Accuracy |
---|
First-order time derivative | Euler | 1st-order |
Gradient | Gauss linear | 2nd-order |
Divergence | Gauss Minmod/linear | 2nd-order |
Laplace | Gauss linear corrected | 2nd-order |
Table 4.
Parameters used in simulations.
Table 4.
Parameters used in simulations.
Parameter | | | | |
---|
Value | 0.05 | 0.40 | 1.0 | 0.6 |
Table 5.
Mesh characteristics on different levels l used for the simulations.
Table 5.
Mesh characteristics on different levels l used for the simulations.
l | , | | Cells/direction | Total cell |
---|
1 | | 0.01 | | 4000 |
2 | | 0.01 | | 6250 |
3 | | 0.01 | | 16,000 |
4 | | 0.01 | | 25,000 |
5 | | 0.01 | | 64,000 |