Detached-Eddy Simulation of Shock Wave/Boundary-Layer Interactions in a Planar Transonic Nozzle

Boundary conditions have to be imposed at the edge of the computational domain, that are made of the walls of the transonic nozzle, the sides in spanwise direction, the inlet station and the exit station. These prescriptions are needed in the Finite Volume approach only to evaluate the fluxes near the boundary and several ways to enforce them are possible. One of the most important problems is that the domain is bounded, whereas free-stream conditions are known at infinity and the constraint imposed can lead to spurious wave reflections, that may cause instability.

A standard approach to try to counter this problem is recurring to non-reflective boundary conditions [72], which linearize the problem in normal direction on each boundary face. As a result it is possible to obtain a diagonal system of linear convective equations in characteristics variables, with the possibility to identify waves that enter or leave the computational domain. Once categorized, correct characterics-based boundary conditions can be imposed to minimize numerical reflection, obviously not completely avoiding it owing to the fact that the domain is bounded. This method is capable of guarateeing good accuracy and stability.

Entering the details of the boundary conditions imposed to solve the numerical problem of this work, the following three conditions have been used:

Subsonic Inlet Boundary In the inlet station, a subsonic flow has been prescribed by the imposition of the Mach number M = 0:46 and on the basis of the relations of steady quasione- dimensional isentropic flows.

Wall Boundary Reflecting boundary conditions [39] have been used, just imposing the velocity at the wall to reflect its value, in such a way that the sum of the velocity in the first inner cell and its ghost cell (artificial cells outiside the computational domain) is zero.

Outlet Boundary At the exit station, a characteristics-based boundary condition prescribing the backpressure is assigned, imposing the ratio of chamber total pressure on ambient static pressure NPR = p0=pa. Besides, in order to avoid any acoustic coupling, a numerical sponge is imposed from the exit station to the end of the domain in longitudinal direction.

Periodic Trasversal Boundary Even if in the experimental analyisis of the case taken into consideration, a secondary flow in span direction is present, in order to simplify the computational setup by neglecting sidewalls and imposing periodic boundary conditions in the spanwise direction.

The evidence of three-dimensional effects is clear from the oil traces in the numerous experiments on the transonic nozzle, where an influx of low-speed fluid from the sides towards the center of the channel causes a considerable thickening of the centerline boundary layer. In any case, the choice of periodic boundary conditions can be justified by the fact that the effect of these 3D phenomena on the low frequency shock unsteadiness, whose study is one of the main purposes of this work, has not been clarified and hence its consideration can be delayed to future works.