Numerical simulations of shock/boundary layer interactions using time-dependent modeling techniques: A survey of recent results

Abstract

Supersonic or hypersonic flows within and around flight vehicles inevitably involve interactions of strong shock waves with boundary layers. Flows within inlet/isolator configurations, and flows induced by control surface deflections are some examples. Such interactions are time dependent in nature and are often subject to low-frequency, large-scale motion that induces local pressure and heating loads. With recent increases in available computer power, it has now become possible to simulate such interactions at experimentally relevant Reynolds numbers using time-dependent techniques, such as direct numerical simulation (DNS), large-eddy simulation (LES), and hybrid large-eddy simulation/Reynolds-averaged Navier–Stokes (LES–RANS) methods. This paper will survey some recent work in this area and will describe insights in shock/boundary layer interaction physics gained by using these high-fidelity methods. Attention will be focused on studies that compare directly with experimental data at the same (or nearly the same) Reynolds number. Challenges in the application of these techniques to even more complicated high-speed flow fields are also outlined.

Introduction

The accurate prediction of high-speed flows can depend critically on the ability of the chosen model to account for shock wave/turbulent boundary layer interactions (SWTBLIs). For example, off-design aerodynamic performance of missiles can be affected by shock/boundary layer interactions induced by plume interference, and the performance of inlets and isolators for high-speed engine concepts is significantly affected by shock/boundary layer interactions. A clear understanding of the effects of the associated fluctuating pressure and heating loads in SWBTLIs is necessary to better refine control techniques, to predict failure scenarios, and to determine ranges of efficient operation. The canonical problem for shock/boundary layer interactions is supersonic flow over a 2-D compression–corner, sketched in Fig. 1. Years of study of such interactions have revealed several traits that, to this point, have defied accurate prediction using Reynolds-averaged Navier–Stokes models [1], [2], [3]. These include:

• unsteady motion of the separation region, which broadens the ‘plateau’ evident in time-averaged surface pressure distributions and induces high fluctuating pressure and heat transfer loads,

• a rapid acceleration of the velocity field toward the surface in the re-attachment/recovery regions;

• anisotropic Reynolds-stress amplification throughout the interaction,

• a ‘laminarization’ effect in the backflow region, and

• the sustained presence of three-dimensional vortical structures that appear downstream of the re-attachment line in stronger interactions.

Recently, there has been interest in applying high-fidelity models, such as large-eddy simulation (LES), direct numerical simulation (DNS), and hybrid large-eddy/Reynolds-averaged Navier–Stokes (LES–RANS) techniques to predict turbulent shock/boundary layer interactions. These have the potential for directly capturing the unsteadiness of the interactions at the important scales. To this author's knowledge, the first attempt to apply time-dependent modeling techniques to a SWTBLI was that of Hunt and Nixon [4], who conducted a ‘very large eddy simulation’ of the Mach 3 compression–corner interaction experimentally mapped by Dolling and Murphy [5]. The Hunt and Nixon study was performed on a very coarse mesh (by today's standards) but nevertheless, achieved some agreement with the experimental data. This study is noteworthy in that it introduced some techniques that still are used, in modified forms, in more recent simulations of SWTBLIs. These include recycling of turbulent fluctuations to sustain large-eddy motion, wall-function closures to enable calculations at high Reynolds number, and methods for extracting time-dependent surface pressure statistics in a manner consistent with experiment. It is of interest that Hunt and Nixon did not attempt to reduce the Reynolds number to better resolve the large-eddy dynamics. Later studies, such those conducted by Adams [6], [7], Knight and co-workers [8], [9], [10], Rizzetta and co-workers [11], [12], and Kannepalli et al. [13], did not follow this model, reducing the Reynolds number significantly to enable affordable, yet properly resolved calculations but removing the possibility of direct comparisons with experimental data as a consequence. These studies also introduced and refined techniques for generating turbulent inflow conditions [8], [14] and examined the effects of different subgrid closure models [8], [9], [10], [11], [12]. A selection of numerical schemes ranging from sixth-order compact schemes on structured meshes [6], [7], [11], [12] to third-order upwind-biased schemes on unstructured meshes [8] were also used. Indirect comparisons with experimental data revealed that in general, the lower Reynolds number DNS or LES calculations were not able to predict the characteristic ‘pressure plateau’ obtained in the vicinity of a region of axial separation. Furthermore, these calculations were unable to capture low-frequency, large-scale motions of the shock system that had been observed in experiments. In a more recent review [15], Knight et al. surveyed many of these initial applications of DNS and LES to SWTBLIs and concluded that further progress in the assessment of such techniques would require calculations that matched experimental conditions to a much higher degree of fidelity.

Since the publication of [15], there have been several attempts to conduct DNS, LES, and hybrid LES–RANS simulations of SWTBLIs at conditions that enable direct comparisons with experimental data. The present paper is concerned with reviewing a selection of these recent studies and in highlighting some of the insights into the physics of SWTBLIs that have resulted from this work. The canonical configuration remains the compression–corner, though some attempts at simulating other cases (impinging shocks, crossing shocks, sonic injection into a crossflow) have been or are being made. For validation of DNS strategies for SWTBLI prediction, the challenge is also in obtaining detailed experimental data at sufficiently low Reynolds numbers. Recently, Bookey et al. [16], [17] have conducted a series of experiments involving compression–corner and impinging oblique–shock interactions at Mach 3, Re_δ=3.74×10⁴ and have collected mean velocity data and mean and fluctuating surface pressure measurements. In a series of papers [18], [19], [20], [21], [22], Martin and co-workers have conducted DNS simulations of this experiment and have conducted detailed analyses of the flow structure. The Martin et al. database [23] and some the results of their analyses will be discussed in subsequent sections. Pirozzoli and Grasso [24] have conducted a DNS study of an impinging oblique–shock interaction at a Reynolds number close to, but lower than that considered earlier by Garnier et al. [25], [26] and have presented a few quantitative comparisons with experimental surface measurements.

The large-eddy simulations of Garnier et al. [25], [26] represent one of the first successful attempts to conduct large-eddy simulations of a SWTBLI at conditions that match experiment (Mach 2.3, Re_δ=6.0×10⁴). The configuration is an impinging shock interaction, generated through an 8° wedge, which results in small region of axially separated flow. The results showed generally good agreement for mean and rms velocity profiles upstream and downstream of the interaction, as well as good predictions of surface skin friction and the evolution of the compressible displacement thickness through the interaction. Comparisons between results obtained using a mixed-scale model [27] and the monotonically integrated large-eddy simulation (MILES) approach revealed only slight differences. A more comprehensive study of Mach 2.95 flow over a 25° compression–corner at Re_δ=6.36×10⁴ has been reported by Loginov et al. [28]. These calculations are conducted using a wall-resolved large-eddy simulation model, with the approximate deconvolution model (ADM) of Stolz and Adams [29] used as a subgrid closure. Selected results from this investigation are highlighted in this review.

Hybrid LES–RANS models have also been applied to shock/boundary layer interactions, primarily by Edwards and co-workers [30], [31], [32], [33], [34], [35]. The advantage of a hybrid LES–RANS model is that the near-wall flow is modeled using RANS concepts and as such, the resolution requirements reduce to those sufficient to resolve the eddies in the outer part of the boundary layer. The Reynolds number restrictions encountered in DNS and wall-resolved LES calculations are primarily associated with the need to resolve near-wall turbulent motion, and if other approaches can be used effectively in this area, the computational savings can be immense. The disadvantage is the same—the use of RANS concepts introduces a new level of uncertainty associated with the RANS closure and how it interacts with the resolved turbulence. It should be mentioned that the hybrid LES–RANS strategies developed in [30], [31], [32], [33], [34], [35] are distinct from the detached-eddy simulation (DES) idea of Spalart et al. [36]. By design, attached boundary layers in DES are to be modeled completely as RANS—the newly proposed ‘delayed DES model’ [37] maintains this focus by introducing a flow-dependent blending function that serves to render the ‘DES switch’ to a large-eddy simulation mode of operation less dependent on the mesh spacing. In contrast, the hybrid LES–RANS models of [30], [31], [32], [33], [34], [35] (and specifically the later versions) attempt to engineer a flow-dependent transition from RANS to LES as the boundary layer shifts from logarithmic behavior to wake behavior. The models of [30], [31], [32], [33], [34], [35] can therefore be viewed as a type of wall-layer closure embedded within a large-eddy simulation for the bulk of the flow, and as such, all of the algorithmic and modeling issues pertinent to large-eddy simulation and DNS are applicable. Earlier applications of hybrid LES–RANS models to SWTBLIs generated from compression–corners (Princeton 20° compression–corner [38], [39]; Zheltovodov et al. 25° compression–expansion corner [40], [41]) yielded reasonable agreement with surface quantities and mean velocity profiles, but no comparisons with second-moment quantities were made. A more recent study [35] has attempted to utilize hybrid LES–RANS methods to examine the dynamics of a SWTBLI interaction generated by Mach 5, Re_δ=87.7×10⁴ flow over a 28° compression–corner [42], [43], [44]. Some results from this investigation are reviewed herein, as well as results from simulations of supersonic turbulent boundary layers that serve to test the Reynolds-number independence of the LES–RANS models.

In summary, this paper will review recent results from three different types of time-dependent modeling strategies—DNS (Martin group) wall-resolved LES (Adams group) and hybrid LES–RANS (Edwards group)—as applied primarily to compression–corner experiments at different Reynolds numbers. Rather than discussing each case independently, the review will highlight the use of each model in predicting particular aspects of SWTBLIs: the structure of the incoming boundary layer, mean-flow properties, instantaneous flow structure, the evolution of the Reynolds-stress tensor, and shock-system dynamics. Implications and challenges for future work will conclude the review.