Approximation of the Couette-Poiseuille flow on an annular domain in compressible regime for a hyperloop pod

Fanno flow is one of the most studied problems in fluid mechanics, as it models the pressure losses on domains in which the characteristic length is considerably lower than the height. These types of flows are present in several applications in the industry, such as pipe flows in pneumatic and hydraulic circuits, oil pipelines or air conditioning and water distribution systems. One potential application can be found in hyperloop design, where flow at high speeds moves between the small gap between the capsule and the tube wall. In this case, the domain is annular-shaped instead of a single cylinder, and the interior wall moves at the capsule speed. Although this quasi-1D flow is strongly affected by the wall friction, it cannot be modelled using the standard Fanno expression. Thus, to model the pressure drop without extensive CFD techniques, a simplified 1D analytical expression is developed to estimate the friction losses. The model assumes a constant ratio between the shear stress on the interior and the exterior walls throughout the channel. Then, instead of the hydraulic diameter as in the standard Fanno expression, a new definition of an equivalent diameter is performed to consider the annular-shaped domain and the movement of the interior wall. The new equivalent diameter is always higher than the original used for Fanno flows (hydraulic one), even twice when the wall speed equals the mean speed. This effect leads to smaller pressure losses due to the movement of one of the walls and indicates that the wall movement is always beneficial. To validate the results obtained from the model, they are compared with an equivalent CFD simulation, leading to a maximum deviation of 5% on the Mach number and 7% on the total friction coefficient.