Abstract

A numerical study was conducted using the finite difference technique to examine the mechanism of energy transfer as well as turbulence in the case of turbulent flow fully developed in the channel with substantially constant wall temperature and constant heat flow conditions. The methodology of solving the thermal problem is based on the equation of energy for a fluid of constant properties by placing itself in the equilibrium hypothesis of an axisymmetric. The global equations and the initial and boundary conditions acting on the problem are configured in dimensionless form in order to predict the characteristics of the turbulent fluid flow inside the tube. Thus, using Thomas' algorithm, a program in FORTRAN version 95 was developed to numerically solve the discretized form of the system of equations describing the problem. Finally, using this elaborate program, we were able to simulate the flow characteristics, changing some parameters involved such as Reynolds number, the Prandtl number and the Peclet number, along the longitudinal coordinate to obtain important results of the thermal model. These are discussed in detail in this work. Comparisons between literatures correlation data and the calculated simulation indicate that the results are a good match to previously published quantities.

Abstract

In this paper, cavity flow is simulated numerically. Forced convection in different Reynolds numbers between 100 and 5000 is simulated. Different and complex thermal boundary conditions are applied and various parameters are calculated numerically. Up wall and down walls are in constant temperature and left and right walls are thermal insulation in the first thermal boundary condition. The Left and the down walls are in constant temperature and the temperature of the up and the right walls changes linearly in the second thermal boundary condition. For the third thermal boundary condition, the left and the down walls are in constant temperature and the temperature of the up and the right walls changes sinusoidally. For this purpose, a code is written in the FORTRAN software. Streamlines, isotherms, velocity and temperature in centerlines of the cavity, Nusselt number and mean Nusselt number are obtained and shown in different figures and one table. Grid independence is surveyed and some obtained results are validated with other researchers' work. In these simulations, the Prandtl number is considered to be 0.71 because of the air’s Prandtl number. For time discretization, a fifth-order Runge-Kutta is used and for convective fluxes, averaging scheme with fourth-order damping term is applied.

Abstract

In this work, a variational multiscale finite element formulation is used to approximate numerically the lid-driven cavity flow problem for high Reynolds numbers. For Newtonian fluids, this benchmark case has been extensively studied by many authors for low and moderate Reynolds numbers (up to Re=10,000), giving place to steady flows, using stationary and time-dependent approaches. For more convective flows, the solution becomes unstable, describing an oscillatory behavior. The critical Reynolds number which gives place to this time-dependent fluid dynamics has been defined over a wide range 7, 300 ≲ Re ≲ 35, 000, using different numerical approaches. In the non-Newtonian case, the cavity problem has not been studied deeply for high Reynolds number (Re > 10, 000), specifically, in the oscillatory time-dependent case. A VMS formulation is presented to be validated using existing results, to determine flow conditions at which the instability appears, and lastly, to establish new benchmark solutions for high-Reynolds numbers fluid flows using the power-law model. Obtained results show a good agreement with those reported in the references, and new data related with the oscillatory behavior of the flow has been found for the non-Newtonian case. In this regard, time-dependent flows show dependence on both Reynolds number and power-law index, and the unsteady starting point has been determined for all studied cases. It is determined that the critical Reynolds number (Rec) that defines the first Hopf bifurcation for Newtonian fluid flow is ranged between 8,100 ≲ Rec ≲ 8, 250, whereas for power-law indexes n=0.5 and n=1.5, it is 7,100 ≲ Rec ≲ 7,200 and 18,250 ≲ Rec ≲ 18,500, respectively.

Abstract

Separate flow friction formulations for laminar and turbulent regimes of flow through pipes are in common use in engineering practice. However, variation of different parameters in a system of conduits during conveying of fluids can cause changes in flow pattern from laminar to fully turbulent and vice versa. Because of that, it is useful to unify formulations for laminar and turbulent hydraulic regimes in one single coherent equation. In addition to a physical interpretation of hydraulic friction, this communication gives a short overview of already available Darcy’s flow friction formulations for both laminar and turbulent flow and additionally includes two simple completely new approximations based on symbolic regression.