Kinetic energy budget of secondary motions in sinusoidally-tempered vertical turbulent pipe flow

Solar power towers represent one of the most promising concentrated solar power technologies for future large-scale renewable energy generation. In such plants, a heliostat field focuses solar radiation onto one side of the central vertical receiver, imposing a highly non-uniform heat flux on the fluid as it ascends and descends within the receiver tubes. To get a better understanding of the secondary motions occurring in such flows, the present study investigates the kinetic energy budget of secondary motions in a sinusoidally tempered vertical turbulent pipe flow using direct numerical simulations. Thus, the incompressible Navier-Stokes equations with the Boussinesq approximation and the energy equation in their dimensionless form are discretized using a finite-volume method with fourth-order spatial accuracy, and are integrated in time employing a semi-implicit second-order Euler-Leapfrog scheme. In the present case the characteristic parameters are set to Reb=5300, Pr=0.71, Gr=9500000. The flow geometry is a smooth-walled pipe with no-slip and impermeability boundary conditions at the wall and a sinusoidally-varied wall temperature As expected, the fluid moves upward from the warmest to the coldest flow region, traveling through the pipe center where the the maximum temperature gradient occurs, and returns downward along the wall. Figure 1(b,c,d) show some examples of different terms of the kinetic energy budget of these secondary motions. Figure 1(b) depicts the production term, which is responsible for the energy transfer between secondary motions and the turbulent fluctuations. The production mechanism causes the strongest energy transfer into the turbulent velocity field in the region of maximum velocity gradients, i.e. near the lower part of the pipe wall, whereas in the remaining near-wall regions of the pipe, a reverse transfer occurs, feeding energy back from the fluctuating velocity field into the mean secondary flow. The dissipation from the secondary motions presented in Figure 1(c) is strictly negative and restricted to regions close to the pipe wall, attaining its minimum where the secondary motions are most intense.