CFD Techniques and Thermo-Mechanics Applications by Zied Driss Brahim Necib & Hao-Chun Zhang

Author:Zied Driss, Brahim Necib & Hao-Chun Zhang
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

1.2 Free Convective Flows

The equations of continuity and momentum variation, which describe flows of viscous incompressible fluids, form a mixed elliptic–parabolic system of equations for velocity and pressure. In this case, the continuity equation includes only the velocity components. Therefore, there is no direct connection with pressure, which is performed via density in the case of compressible fluids (Wesseling 2000). The methods for solving the Navier–Stokes equations for viscous incompressible fluids are classified in pressure-based and pressure–velocity coupling methods.

To simulate flows of viscous compressible fluids in a wide range of Mach numbers in the presence of low-velocity flow subregions (stagnation and flow attachment regions, and recirculation zones), the full Navier–Stokes equations are used. The numerical solution of the Navier–Stokes equations at low Mach numbers runs into computational difficulties that result in slow convergence and reduced accuracy of the numerical solution.

To overcome the computational difficulties arising during the simulation of low-speed flows, preconditioning methods are widely used. These methods equalize the orders of the Jacobian eigenvalues in the original equations for all (Choi and Merkle 1993; Volpe 1993; Weiss and Smith 1995; Turkel et al. 1997; Turkel 1987, 1993). Preconditioning modifies the terms with the time derivative in the momentum equations. When the flow reaches the steady state, the solution to the modified (preconditioned) system coincides with the solution to the original gas dynamics equations. A method that eliminates stiffness of the preconditioned Navier–Stokes equations in the computations on elongated cells in the boundary layer was proposed in Buelow et al. (1994). A review and comparative analysis of various approaches to the preconditioning of the gas dynamics equations can be found in Turkel et al. (1997), Volkov (2009), and Turkel (1993).

For the simulation of unsteady flows, the dual time-stepping method is used (Rogers and Kwak 1990; Jameson 1991). At each integration step with respect to the physical time, additional iterations on the pseudo-time are used. The integration with respect to the physical time (outer iterations) and pseudo-time (inner iterations) is performed using different schemes (e.g., implicit and explicit ones). To improve the convergence of the inner iterations, various approaches may be used (e.g., the multigrid method or residual smoothing) (Venkateswaran and Merkle 1995; Rumsey et al. 1996; Zhang et al. 2004). In Bijl et al. (2002), the dual time-stepping method is modified to enable the use of multistep difference schemes with respect to the physical time. The solution of various complex problems is described in Tang et al. (2003), and a comparison of various approaches can be found in Jameson (2009).

The analysis of stability shows that the integration step with respect to the pseudo-time is 2/3 of the physical time step (Zhao 2004), which results in the slow convergence in the case of constraints on the physical time step (e.g., when explicit difference schemes are used). To remove the constraints on the time step, implicit schemes both with respect to physical time and pseudo-time are used (fully implicit dual time-stepping schemes) (Luo et al. 2001; Zhao et al. 2002).

The preconditioning method

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