LOW-ORDER MODEL FOR THE LOW-FREQUENCY UNSTEADINESS IN OBLIQUE-SHOCK/TURBULENT-BOUNDARY-LAYER INTERACTIONS Emile Touber Department of Aeronautics Imperial College London South Kensington Campus, London SW7 2AZ, U.K. e.touber@imperial.ac.uk Neil D. Sandham School of Engineering Sciences University of Southampton Southampton SO17 1BJ, U.K. n.sandham@soton.ac.uk ABSTRACT The interaction between a turbulent supersonic bound- ary layer and an impinging shock wave is investigated nu- merically and analytically. The reflected-shock low-frequency motions are well captured even when using a narrow simula- tion domain, supporting the argument that one underlying key mechanism for the low-frequency shock motions is two di- mensional. Based on a two-dimensional approach, a stochas- tic ordinary differential equation for the low-frequency cou- pling between the reflected shock and the boundary layer is obtained. The system is closed and applied to a wide range of input parameters. It is argued that the low-frequency shock motions are not necessarily a property of the forcing, either from upstream or downstream of the shock, but are simply an intrinsic property of the coupled dynamical system. INTRODUCTION The physical mechanisms at the origin of the observed low-frequency shock motions in shock wave/turbulent bound- ary layer interactions (SBLI) are not fully understood. A number of tentative explanations have been proposed, usu- ally falling into one of two categories: the first relates the low-frequency motions to specific events or flow structures from the upstream turbulent boundary layer, whereas the sec- ond looks for causal mechanisms within the interaction itself (i.e. downstream of the shock). In both cases, the difficulty re- sides in identifying a mechanism that can span timescales of the order of 10 1 δ 0 / ¯ u 1 to 10 2 δ 0 / ¯ u 1 , where δ 0 is the upstream boundary-layer 99% thickness and ¯ u 1 the upstream freestream velocity. The variety of the mechanisms proposed in the literature, together with the subsequent debate about the merits of one approach relative to another is symptomatic of the difficulty one has in identifying and then separating individual events from a (supposedly) non-linear (chaotic) system, where ac- tual causal events may well be impossible to detect. Instead of reasoning about the relevance of one assumed mechanism against numerical/experimental data, an attempt to charac- terise in a useful way the properties of the dynamical system arising from the coupling between the shock and the boundary layer is sought. The paper is organised as follows. The next section high- lights the main steps for the derivation of a low-order model for the shock-foot low-frequency motions. Next, some impli- cations of the model and its sensitivity to modelling errors are discussed. A STOCHASTIC LOW-ORDER MODEL FOR THE SHOCK-FOOT MOTIONS The momentum integral equation Starting from the Navier–Stokes equations, and upon in- tegrating the streamwise component of the momentum equa- tion in the wall-normal direction (denoted y), one can derive a general form of the Momentum Integral Equation (MIE) where none of the classical assumptions (e.g. constant pres- sure in the wall-normal direction, steady state . . . ) are used. The resulting MIE is then expressed in the following moving coordinate system: ξ ≡ x + l 0 − ε l 0 − ε + s (1) where all notations are described in figure 1. Hence, in what follows, ξ = 0 is the instantaneous shock-foot position, ε the shock-foot displacement with respect to its mean position and ξ = 1 the instantaneous location of the shock crossing. Note that due to the presence of the boundary layer, the shock does not reach the wall and the foot is defined as the linear exten- sion of the shock to the wall. The following assumptions are made to simplify the 1