Codes

AFiD:  A highly parallel finite difference code for Rayleigh-Bénard and Taylor-Couette flows (www.afid.eu). 

NAFiD:  A highly parallel finite difference code for wall-bouned turbulent flows of non-Newtonian fluids. 

The non-Newtonian solver is built on top of the open-source AFiD (www.afid.eu) code. The non-Newtonian fluids can be modelled by either viscoelastic Oldroyd-B or FENE-P, or Saramito elastoviscoplastic constitutive equations based on the conformation tensor. The code can simulate turbulent wall-bounded flow of viscoelastic, viscoplastic and elastoviscoplastic fluids. The algorithm is demonstrated to preserve the properties of symmetry, boundedness and positive definiteness of the conformation tensor up to large Weissenberg numbers (1e2) and high Rayleigh number (1e10).  A comparison with available direct numerical simulation results in the literature shows a very good agreement. Moreover, the results for the heat transport modfication for highly turbulent thermal convection with polymer additives agree quantitatively with previous experiments in a similar parameter range. 

References

• Song, J. *, Xu C., Shishkina, O.*, 2025 A finite difference method for turbulent thermal convection of complex fluids. J. Comput. Phys. 525, 113732.

Validation: Viscoelastic Rayleigh-Bénard convection

3D DNS at Ra=1e5, Pr=7, solvent viscosity ratio of 0.9. 

BC: No-slip top and bottom walls, periodic in horizontal directions. 

The computational domain is set to 16h×16h×h with a grid resolution of 256×256×128. 

Heat transport enhancement (HTE)=14.4% with L=25, Wi=10; 

Heat transport reduction (HTR)=-31.4% with L=100, Wi=40. 

Reference: Y. Dubief, V.E. Terrapon, Heat transfer enhancement and reduction in low-Rayleigh number natural convection flow with polymer additives, Phys. Fluids 32 (2020) 033103.

Validation: Elastoviscoplastic turbulent channel flow 

The Re_tau=180, Bingham number Bi=2.8, 5.6, with L=60, Wi=4, solvent viscosity ratio of 0.9.

BC: No-slip top and bottom walls, periodic in horizontal directions. 

The computational domain is set to 6h×2h×3h with a grid resolution of 1024×320×512. 

Reference: D. Izbassarov, M.E. Rosti, L. Brandt, O. Tammisola, Effect of finite Weissenberg number on turbulent channel flows of an elastoviscoplastic fluid, J. Fluid Mech. 927 (2021) A45. 

Test: Highly turbulent and elastic Rayleigh-Bénard convection 

3D DNS at Ra=1e10, Pr=4.3, L=50, with Wi=0, 30, 120, solvent viscosity ratio of 0.9.

BC: No-slip top and bottom walls, periodic in horizontal directions. 

The computational domain is set to h×h×h with a grid resolution of 768×768×512. 

The flow fields demonstrate a significant modification of the Newtonian flow structures by polymer additives both in the bulk and boundary layer regions, where the plumes become more coherent and the boundary layer becomes more stable. Time series of the convective heat transport shows that the contribution from the elastic part (Nu^e) to the total heat flux (Nu^t) is more than 2 times larger than that of the solvent viscous part (Nu^v). The time series of the polymer stretch reflected by trace(C) demonstrates that the positive definiteness and the upper boundedness of the polymer conformation tensor (0<trace(C)<L*L) is well satisfied during the calculation. For the present grid resolution the proportions of reduced-order points in the z-and x/y-directions are only about 3% and 1%, respectively. 

Test: Turbulent Rayleigh-Bénard convection of four different fluids

2D DNS at Ra=1e8, Pr=7, L=100, with solvent viscosity ratio of 0.97. 

BC: No-slip top and bottom walls, periodic in horizontal directions. 

The computational domain is set to 2h×h with a grid resolution of 768×384. 

Newtonian fluid: Wi=0, Bi=0; Viscoplastic fluid: Wi=0.01, Bi=50;

Viscoelastic fluid: Wi=10, Bi=0; Elastoviscoplastic fluid: Wi=10, Bi=50.

As compared to Newtonian fluid, all non-Newtonian fluids demonstrate significant reduction in both heat and momentum transport in turbulent thermal convection (HTR=-6.2%, -16.6% and -20.4%).

Computational performance: Scaling behaviour of the code on HoreKa  

3D DNS at Ra=1e10, Pr=7, Wi=10, L=50, with solvent viscosity ratio of 0.9. 

BC: No-slip top and bottom walls, periodic in horizontal directions. 

The computational domain is set to h×h×h with a grid resolution of 768×768x512. 

HPC systems of ``Hochleistungsrechner Karlsruhe''(HoreKa) at KIT. Specifically, on Horeka, each node has two Intel Xeon Platinum 8368 processors, at least 256 GB of local memory, local 960 GB NVMe SSD disks and two high-performance network adapters. For both Newtonian and viscoelastic simulations, the averaged wall time per step shows an almost linear reduction with increasing number of CPU cores up to 1e4. The time for viscoelastic computing is around 2.5∼5 times larger than the corresponding computing time for the Newtonian flows, which is a quite high efficiency when considering the six additional conformation tensor variables that need to be solved. Moreover, second figure displays a very good speedup of the code up to around 5000 cores for Newtonian and 2000 cores for viscoelastic calculations. To this end, the good scaling characteristics of the code will guide us to reduce the computation time by choosing proper CPU cores for tasks that have similar grid mesh sizes and thus the available resources can be used more efficiently.

NAFiD_cylinder:  A finite difference code for non-Newtonian thermal convection in the cylindrical cell 

The non-Newtonian solver is built on top of the original cylindrical AFiD code. The non-Newtonian fluids can be modelled by either viscoelastic Oldroyd-B or FENE-P, or Saramito elastoviscoplastic constitutive equations based on the conformation tensor. The code can simulate turbulent wall-bounded flow of viscoelastic, viscoplastic and elastoviscoplastic fluids. 

Test: Turbulent Rayleigh-Bénard convection with polymers

3D DNS at Ra=5e8, Pr=4.3, L=50, Wi=0, 10, 50, with solvent viscosity ratio of 0.9. 

BC: No-slip top and bottom, no-slip sidewalls. 

As compared to Newtonian fluid, all viscoelastic fluids demonstrate significant reduction in both heat and momentum transport in turbulent thermal convection (HTR=-20.7% and -23.5%).