A Unified Fractional-Step, Artificial Compressibility and Pressure-Projection Formulation for Solving the Incompressible Navier-Stokes Equations

Abstract

This paper introduces a unified concept and algorithm for the fractional-step (FS), artificial compressibility (AC) and pressure-projection (PP) methods for solving the incompressible Navier-Stokes equations. The proposed FSAC-PP approach falls into the group of pseudo-time splitting high-resolution methods incorporating the characteristics-based (CB) Godunov-type treatment of convective terms with PP methods. Due to the fact that the CB Godunov-type methods are applicable directly to the hyperbolic AC formulation and not to the elliptical FS-PP (split) methods, thus the straightforward coupling of CB Godunov-type schemes with PP methods is not possible. Therefore, the proposed FSAC-PP approach unifies the fully-explicit AC and semi-implicit FS-PP methods of Chorin including a PP step in the dual-time stepping procedure to a) overcome the numerical stiffness of the classical AC approach at (very) low and moderate Reynolds numbers, b) incorporate the accuracy and convergence properties of CB Godunov-type schemes with PP methods, and c) further improve the stability and efficiency of the AC method for steady and unsteady flow problems. The FSAC-PPmethod has also been coupled with a non-linear, full-multigrid and fullapproximation storage (FMG-FAS) technique to further increase the efficiency of the solution. For validating the proposed FSAC-PP method, computational examples are presented for benchmark problems. The overall results show that the unified FSAC-PP approach is an efficient algorithm for solving incompressible flow problems.

Validation of a Magneto- And Ferro-Hydrodynamic Model for Non-Isothermal Flows in Conjunction With Newtonian and Non-Newtonian Fluids

“…This limitation can be explained by the fact that the MHD-FHD source terms are stiff source terms in the momentum and energy equations, therefore the numerical behaviour of higher-order methods [11] should be further investigated for MHD-FHD problems at high dimensionless magnetic numbers.…”

“…We assume that the magnetic state through the magnetization equation M = M (H) depends only on the magnetic field intensity H, however the magnetization depends on the temperature T in reality. Relying on this assumption which can be used for weak magnetic conductors such as blood [2], the effects of MHD and FHD is taken into account in the momentum equation (10), however, only the effect of MHD appears in the temperature equation (11). This is due to the fact that the temperature derivative of the magnetization vector M vanishes from the energy equation (3), because it is assumed that the magnetization is function of the magnetic field intensity, but not function of the temperature.…”

“…It can be seen in Figure 12 that the outlet velocity profile is not affected by the magnetic field strength, thus the profile is similar to the fully developed laminar velocity distribution for both Newtonian and non-Newtonian fluids. The temperature field behaviour is different compared to the non-isothermal Hartmann flow due to the fact that the effects of MHD and FHD terms in the momentum equation (10) has impact on the temperature distribution through convection even if the magnetocaloric effect of the FHD term does not appear in the temperature equation (11). The outlet temperature profile (see Figure 12) exhibits a non-linear behaviour compared to the outlet temperature distribution of the non-isothermal Hartmann flow whereas the profile becomes quasi-linear at the outlet section of the microfluidic channel (see Figure 6).…”

“…For this multiphysics problem, we assume again that the magnetic state through the magnetization equation M = M (H) depends only on the magnetic field intensity H and it is independent from the temperature field T . It means that the effects of MHD and FHD have been taken into account in the momentum equation (10) and the FHD effect have been neglected in the temperature equation (11). For an incompressible, laminar and non-isothermal non-Newtonian blood flow, the velocity and temperature field contours have been shown in Figures 14 and 15, respectively.…”

Investigation of Numerical Dissipation in Classical and Implicit Large Eddy Simulations

“…The aforementioned contributions made attempts to obtain very accurate results within the framework of LES and ILES methods to gain a deeper insight into the behaviour of the physics of turbulence. The reader can refer to the application of the ILES method in different contexts [14][15][16][17][18], where the quantification of numerical dissipation and dispersion could also be employed to improve the accuracy of the numerical solution.…”