Analysis of the details of total mechanical energy balance and turbulent flow characteristics in irregular open-channels

Abstract

The mechanical energy balance and energy loss have attracted significant attention in the literature. The general form of the energy equation has been previously presented; however, in most applications the simplified 1-D form without considering the details of turbulent flows has been applied. In this study, a form of the total mechanical energy balance equation for 3-D turbulent flows in open-channels is investigated in which all turbulence parameters involving the effects of the secondary currents can be evaluated by using equalizations that have not previously been applied to the total mechanical energy equation and RANS turbulence models. Based on the equalizations, the existing correlations in the total energy equation are replaced with a combination of mean flow variables and known turbulence parameters from turbulent flow numerical simulations. Therefore, the details of the total mechanical energy balance for 3-D turbulent flows in open-channels with different geometries can be analyzed using the proposed methodology. Then, an explicit relationship for the energy loss, usually determined empirically, is derived from the developed energy equation. In order to investigate the utilization of the analytical relationships, 3-D numerical simulations are performed using OpenFOAM to solve the Navier–Stokes equations with the RSM turbulence model. By applying the analytical relationship and the simulation results, the mechanical energy losses are obtained and compared with the energy losses from the previous energy equations, the effect of the various parameters on the total mechanical energy balance is studied, the role of wall resistance in the mechanical energy is investigated and efforts are made to identify the parameters affecting the local and friction losses of turbulent flows in the irregular open-channels. Finally, a deeper understanding of the details of the energy transformation process and the mechanical energy loss of the complex flows is achieved.

Appendix 1

The PIMPLE algorithm is a combination of SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) and PISO (Pressure Implicit with Splitting of Operator). All these algorithms are iterative solvers (PISO and PIMPLE are both used for transient cases; whereas, SIMPLE is used for steady-state cases). The PIMPLE algorithm is as SIMPLE algorithm for every time step, in which outer correctors are the iterations, and once converged will move on to the next time step until the solution is complete. Better stability is obtained from PIMPLE in comparison to PISO for this reason, especially when dealing with large time steps where the maximum Courant number may be more than 1, or when the nature of the solution is inherently unstable.

In other words, in PIMPLE algorithm, for each time step, a steady state solution that converges is sought after, using a specified number of correction loops. After that, all other transport equations are solved. It would be equal to the PISO algorithm with the specified number of correction loops. However following this, the algorithm loops back over the entire time step and solves the PISO algorithm again with a new initial guess [28, 29]. A simplified flowchart of the algorithm is shown in Fig. 

Fig. 14

Simplified flowchart of the PIMPLE algorithm

14. As mentioned earlier, the advantage of the use of the PIMPLE algorithm is that the Courant number can be much larger than 1 and no under-relaxation is required for a stable solution. This allows for larger time steps, but the computational effort for each time step is more than PISO algorithm.