Bifurcation of Shock Induced Separation Structures
Authors: Czerwinska J. , Doerffer P. , Magagnato F. 
Published: November 2021

Abstract
The scope of this paper is to present bifurcation phenomenon which occur in the transonic flow of a curved channel. The channel geometry is nominally two-dimensional. Due to changes in the third dimension, which seems to be a bifurcation parameter, shock induced separation structures significantly change their behavior. The objective of this paper is to show the phenomenon as well as to present models and the explanation of the physical process causing bifurcation in shock induced separation structures.
The interest for consideration of the presented flow has its origin in turbomachinery. The channel flow is a simplification, which allows the study of basic phenomenon of boundary layer shock wave interactions. Shock induced separation structures occur on the walls of the channel.
For a specific geometry taken into account here, such structures appear only on the bottom and side walls. The changes in their character occur due to increase in the distance between side walls. Two channel width cases (50 mm and 150mm) have been particularly well studied. The cases will be later referred to as either narrow or wide passages respectively. For the first one the separation structures were wide and reach far down stream, while for a second case separation was much smaller. For both cases experiments and 3-D Navier-Stokes calculations with turbulence modeling were performed. The separation structures can be seen also as sets of focal and saddle points, which change their type (Takens-Bogdanov bifurcation) with the changes of the distance between side walls. The description of the flow structures and their bifurcation has established a base for one aspect. Another consideration is global bifurcation. The question is how to predict what structures should be expected for any given width between the two cases. Further, how does one predict if small changes of the side wall distance cause small changes in separation structures or if they have a more critical behavior. The experiment and also 3-D calculations are very expensive and time consuming and they are not able to answer this in an easy way. Hence, a theoretical model is presented, which can describe global bifurcation of flow structures.
In experiments as well as in the full 3-D Navier-Stokes calculations the third dimension of the separation region was significantly smaller and depends on the Reynolds number, as does the height of the λ-foot. This has given the idea to model only a separation region in a way similar to the Hele-Shaw flow. As a result a simpler two-dimensional equation was obtained. The solution can be achieved in an easier and more efficient way than the calculation of unsimplified cases. This gives possibility of an estimation of the flow for the other values of the channel width. The proposed paper presents a phenomenon observed in shock induced separation structures in transonic turbulent flow by means of experiments and numerical simulations. Further, it attempts to explain the physical processes involved in this phenomenon by deriving a theoretical model.