Free Interaction Theory Research Articles

Effects of wall temperature on hypersonic shock wave/turbulent boundary layer interactions

Wall temperature has a significant effect on shock wave/turbulent boundary layer interactions (STBLIs) and has become a non-negligible factor in the design process of hypersonic vehicles. In this paper, direct numerical simulations are conducted to investigate the wall temperature effects on STBLIs over a 34° compression ramp at Mach number 6. Three values of the wall-to-recovery-temperature ratio (0.50, 0.75 and 1.0) are considered in the simulations. The results show that the size of the separation bubble declines significantly as the wall temperature decreases. This is because the momentum profile of the boundary layer becomes fuller with wall cooling, which means the near-wall fluid has a greater momentum to suppress flow separation. An equation based on the free-interaction theory is proposed to predict the distributions of the wall pressure upstream of the corner at different wall temperatures. The prediction results are generally consistent with the simulation results (Reynolds number Reτ ranges from 160 to 675). In addition, the low-frequency unsteadiness is studied through the weighted power spectral density of the wall pressure and the correlation between the upstream and downstream. The results indicate that the low-frequency motion of the separation shock is mainly driven by the downstream mechanism and that wall cooling can significantly suppress the low-frequency unsteadiness, including the strength and streamwise range of the low-frequency motions.

On incident shock wave/boundary layer interactions under the influence of expansion corner

Cowl-induced incident shock wave/boundary layer interactions (ISWBLIs) under the influence of shoulder expansion represent one of the dominant phenomena in supersonic inlets. To provide a more comprehensive understanding of how an expansion corner affects the ISWBLI, a detailed experimental and analytical study is performed in a Mach 2.73 flow in this work. Pressure measurement, schlieren photography and surface oil-flow visualisation are used to record flow features, including the pressure distribution, separation extent and surface-flow topological structures. Our results reveal three types of ISWBLIs influenced by the expansion corner. When the shock intensity is weak, the separation is small scale with the expansion waves emanating from the expansion corner. This is the first type of expansion-corner-affected ISWBLI (EC-ISWBLI). When the incident shock wave is strong, large-scale separation occurs, accompanied by the disappearance of expansion waves, forming the second type of EC-SWBLI. The expansion corner induces a ‘lock-in’ effect in which the separation onset is consistently locked near the expansion corner regardless of the incident shock intensity and impingement position. The third type of EC-ISWBLI occurs when the shock is sufficiently strong and the impingement point is close to the expansion corner. In this interaction, the ‘lock-in’ effect ceases to manifest. Moreover, a shock polar-incorporating inviscid model is employed to elucidate the shock patterns. Two criteria are established by combining free interaction theory with this model. The first criterion provides valuable insights into the evolution of separations with a minimal overall pressure rise and the second criterion determines the threshold for the occurrence of the ‘lock-in’ effect.

On the double-sided shock diffractions in quiescent and supersonic crossflows

Shock diffraction is a widespread phenomenon in aerospace applications, such as shock tunnel nozzle and jet exciter exit, impacting their performance significantly. This paper focuses on the transient evolution of double-sided shock diffraction in both quiescent and supersonic crossflows by unsteady numerical simulations. The characteristics of the shock wave and the vortex are revealed. In the quiescent flow, the double-sided shock diffraction exhibits remarkable symmetry. The diffracted shock retains a self-similar nature, but its intensity distribution displays non-uniform characteristics, which gradually weakens from the center to both sides. The vortices on both sides also exhibit symmetrical behavior, with their trajectory behaving in linear tendency. When the supersonic crossflow interacts with the diffracted shock, an upward-moving separation shock and an asymmetric diffracted shock are generated. The vortices remain confined beneath the boundary layer and exhibit different shapes. Moreover, due to the rapid motion of the separation shock, the relative Mach number is introduced into the free-interaction theory (FIT) to predict the shock angle of the separation shock. The F(x¯) values corresponding to the separation point and pressure plateau are determined to be 3.04 and 4.68, respectively. The results evaluated by modified FIT show a good agreement with the values of simulation and experiment.

Analysis of shock train leading shock structure under oscillatory backpressure

The leading shock in a steady shock train takes one of two forms: oblique or normal. However, the phenomenon and mechanism of the normal-to-oblique transition of the leading shock in a forced shock train have not been widely reported. In this study, leading normal and oblique shocks are experimentally observed at the same position and similar velocities in a forced oscillation shock train with an incoming Mach number of 1.83. The normal-to-oblique transition is found to be closely related to the self-excited oscillation of the shock train. In addition, we find that the direct use of free-interaction theory cannot accurately predict the leading shock angle in a moving shock train. Thus, free-interaction theory requires appropriate correction for this scenario.

Effects of expansion waves on incident shock-wave/boundary-layer interactions in hypersonic flows

The effects of expansion waves on incident shock-wave/boundary-layer interactions (SWBLIs) at a Mach number of 4.96 are experimentally studied. The flow characteristics of the interaction zone under various intensities and positions of the incident shock wave and expansion waves are quantitatively analyzed. The expansion waves weaken the intensity of the shock waves encountered, thereby weakening the intensity of the SWBLIs. With an increasing distance between the expansion waves and the interaction zone, the total wall pressure jump and the interference length show a linear growth trend. However, the expansion waves do not affect the initial pressure jump of the separation, which is consistent with free-interaction theory. Finally, the scaling model proposed by Souverein et al. [“A scaling analysis for turbulent shock-wave/boundary-layer interactions,” J. Fluid Mech. 714, 505 (2013)] is simply modified using the measured value of the pressure jump. This correction provides a better approximate result for SWBLIs under the impact of expansion waves.

Switching between symmetric and asymmetric separation-induced shock reflections in an oscillatory duct flow

Separation-induced shock reflections in straight ducts are generally considered asymmetric for Mach numbers beyond 2.2, but our experiment shows that this is not always the case. A symmetric shock pattern, a proven outcome of following the free-interaction theory (FIT), is observed to appear in an oscillatory duct flow at a Mach number of 2.47. Interestingly, its existence is restricted to the period when the shocks move forward. Once a full retreat starts, it changes suddenly into an utterly asymmetric style that conforms to the past observation. This behavior indicates that the FIT fundamentally plays a limited role in supersonic duct flows.

On the scaling of three-dimensional shock-induced separated flow due to protuberances

Supersonic flow over three-dimensional bodies protruding out of the turbulent boundary layer was investigated by means of experiments and numerical computations. A parametric study was performed by varying the shape and dimensions of the protuberance, as well as the freestream Mach numbers (1.5, 2, 2.5, 2.89, and 3.5). Surface streak line visualization, surface pressure measurements, and time-resolved Schlieren visualization were employed along with Reynolds-averaged Navier–Stokes computations to elicit the complex flow features such as the separation line, shock pattern, and the horseshoe vortex, which greatly influence the flow dynamics in the separated region. The rise in surface pressure at mid-span due to separation (plateau pressure) was dependent only on the incoming flow parameters and independent of protuberance geometry. The two-dimensional free interaction theory, applied for normal shock-induced separation, closely predicts the mid-span plateau pressure. Although protuberances are of varying shapes and dimensions, the inviscid bow shock (obtained from Euler computations) provided generalized scales, whose effects on shock boundary layer interactions are analyzed. The radius of curvature of the inviscid shock on the wall plane at the nose, which is theoretically related to the local second derivative (along the shock) of pressure jump, was found to be a determining parameter of mid-span separation length (Lsep). Since the spanwise distance of the sonic point on the inviscid shock was found to be strongly correlated with its nose radius of curvature, it follows that the “strong” portion of the inviscid bow shock fixes the mid-span separation location. These observations concerning mid-span plateau pressure, and the role of strong shock portion in fixing mid-span separation, suggest that the Lsep shall be predicted from a modification of the scaling laws for the length of plateau pressure region in two-dimensional shock boundary layer interaction, with the inclusion of a spanwise relieving effect. A correlation is obtained relating the Lsep with various incoming flow parameters and inviscid shock nose radius. The mid-span vortex core position was found to be linearly related to the Lsep. The radius of curvature of the separation shock is, however, found to be influenced by the entire inviscid shock, including the “weak” portion.

Quasi-Conical Free Interaction in Fin-Induced Shock Wave/Laminar Boundary-Layer Interactions

A numerical study was performed on the three-dimensional interaction between a laminar boundary layer and the shock wave induced by a sharp, unswept fin mounted perpendicularly on a flat plate at freestream Mach numbers of 2.64, 2.95, 3.44, and 3.93. The fin angles were 13, 16, and 19 deg. All the interactions were strong enough for the boundary layer to separate, and the separated flow features displayed a quasi-conical nature emanating from a virtual conical origin. The concept of quasi-conical fin-induced interactions, initially developed for turbulent interactions, was used to extend the free interaction theory to three-dimensional laminar interactions considered in the present study. The well-established scaling functions in free interaction theory were extended, and the results were found to agree reasonably with the theoretical trend. Surface-pressure distributions were plotted for the different cases studied and were found to have a plateau and dip that is typical of separated flows. The overlapping of the scaled pressure distributions reinforced the concept of quasi-conical free interactions for the range of Mach numbers and fin angles considered.

Theory-based prediction of separation angle and peak pressure for laminar separated hypersonic compression corner flows

Interactions between shock waves and the boundary layer will cause high pressure loads and severe heating locally on the material surface in hypersonic separation–reattachment flows. In this paper, theoretical modeling is employed to study the separation angle and peak pressure of hypersonic compression corner flows with separated regions at low-to-medium Reynolds number. According to the characteristics of wall pressure distribution and flow field structures, an equivalent double shocks inviscid flow model for peak pressure is introduced. Based on this model, combining the “free-interaction theory,” formulas for the separation angle and peak pressure with freestream parameters (Mach number and Reynolds number) and the geometry corner angle are derived, and the reasonability and effectiveness are also shown by the numerical data provided by the direct simulation Monte Carlo method. It is revealed that the separation angle is almost independent of the freestream Mach number, and the ratio of peak pressure to freestream pressure is approximately proportional to the square of the freestream Mach number under strong viscous interaction conditions. Moreover, it is indicated that present theoretical results can be extended to axisymmetric flows through the discussion of hypersonic separated flows generated by a hollow cylinder-flare geometry.

Improvement of the free-interaction theory for shock wave/turbulent boundary layer interactions

Free-interaction theory is widely used for the analysis and modeling of the flow structure for shock wave/turbulent boundary layer interactions (SWTBLIs). However, many studies have demonstrated that the value of the nondimensional pressure rise function at the plateau should not be treated as a universal constant, which is an assumption taken in the traditional free-interaction theory. Such an assumption brings huge uncertainty to the theoretical prediction of shock wave/boundary layer interaction flows. To improve the accuracy of free-interaction theory, numerical simulations on the incident shock wave/turbulent boundary layer interactions are carried out in this study over an extensive flow range (Ma0 = 2.0–5.0, Reδ = 7.4 × 104–7.29 × 105). Utilizing the simulated flow field structures and literature data, this paper analyzes the essential influencing factors for determining the plateau pressure. Two nondimensional parameters—the incompressible shape factor of the incoming boundary layer and the nondimensional separation-bubble height—are identified as the essential influencing factors for the nondimensional pressure rise function at the plateau. A new scaling rule is proposed by taking these two nondimensional parameters into consideration, and the experimental data of the SWTBLIs after scaling collapse well onto a single curve with an R2 value of 0.918. The experimental data used to validate the scaling rule include incident and ramp SWTBLIs and the leading SWTBLIs in shock trains. The proposed scaling rule can be used to establish more accurate theoretical predicting models for SWTBLIs.

Theoretical study on regular reflection of shock wave–boundary layer interactions

Abstract

Flow Similarity in Strong Swept-Shock/Turbulent-Boundary-Layer Interactions

The separation length employed to scale unsteady phenomena in strong (separated) nominally two-dimensional (2-D) shock/turbulent-boundary-layer interactions cannot be directly extended to three-dimensional (3-D) swept interactions, due to the quasi-conical symmetry of 3-D interactions. This paper examines the issue by considering large-eddy simulations of Mach 2 canonical flows arising from plate-mounted swept compression ramps and sharp fins at various interaction strengths, which have been the subject of concurrent experimental campaigns. Several key structural features are discussed that distinguish 3-D interactions from 2-D interactions in the context of free-interaction principles and quasi-conical symmetry, including their relationship to strong crossflow and open separation. The virtual-conical-origin concept is augmented by considering inceptive effects, which motivate the introduction of an inceptive origin. The dependence of the inceptive-origin/virtual-conical-origin relationship with interaction strength yields several insights and provides a more rigorous framework to quantify inception strength than previously available. Finally, examination of the spanwise dependence of the growth of the quasi-conical separated shear layer shows that the outer (main) layer exhibits spanwise-homogeneous symmetry consistent with 2-D free-interaction theory, whereas the inner layer exhibits conical symmetry, consistent with 3-D (conical) free-interaction theory. The ramifications for shear layer and shock unsteadiness, as well as acceleration of the inner layer and secondary separation, are discussed.

Confinement effects on regular–irregular transition in shock-wave–boundary-layer interactions

An oblique shock wave is generated in a Mach 2 flow at a flow deflection angle of$12^{\circ }$. The resulting shock-wave–boundary-layer interaction (SWBLI) at the tunnel wall is observed. A novel traversable shock generator allows the position of the SWBLI to be varied relative to a downstream expansion fan. The relationship between the SWBLI, the expansion fan and the wind tunnel arrangement is studied. Schlieren photography, surface oil flow visualisation, particle image velocimetry and high-spatial-resolution wall pressure measurements are used to investigate the flow. It is observed that stream-normal movement of the shock generator downwards (towards the floor and hence the point of shock reflection) is accompanied by (1) growth in the streamwise extent of the shock-induced boundary layer separation, (2) upstream movement of the shock-induced separation point while the reattachment point remains nearly fixed, (3) an increase in separation shock strength and (4) transition between regular and irregular (Mach) reflection without an increase in incident shock strength. The role of free interaction theory in defining the separation shock angle is considered and shown to be consistent with the present measurements over a short streamwise extent. An SWBLI representation is proposed and reasoned which explains the apparent increase in separation shock strength that occurs without an increase in incident shock strength.

A parametric study of laminar and transitional oblique shock wave reflections

High-resolution particle image velocimetry measurements were performed on laminar and transitional oblique shock wave reflections for a range of Mach numbers ($M=1.6{-}2.3$), Reynolds numbers ($Re_{x_{sh}}=1.4\times 10^{6}{-}3.5\times 10^{6}$) and flow deflection angles ($\unicode[STIX]{x1D703}=1^{\circ }{-}5^{\circ }$ or $p_{3}/p_{1}=1.11{-}1.64$). The laminar interactions revealed a long, flat and triangular shaped separation bubble. For relatively strong interactions ($p_{3}/p_{1}>1.2$), the bubble grows linearly in the upstream direction with increasing shock strength. Under these conditions, the boundary layer keeps an on average laminar velocity profile up to the shock impingement location, followed by a quick transition and subsequent reattachment of the boundary layer. For weaker interactions ($p_{3}/p_{1}<1.2$), the boundary layer is able to remain laminar further downstream of the bubble, which consequently results in a later reattachment of the boundary layer. The pressure distribution at the interaction onset for all laminar cases shows excellent agreement with the free-interaction theory, therefore supporting its validity even for incipiently separated laminar oblique shock wave reflections.

Application of free interaction theory in swept shock wave/turbulent boundary layer interactions

Fin-generated swept shock wave/turbulent boundary layer interactions were experimentally investigated via nanoparticle-based planar laser scattering method, surface oil flow visualizations, surface pressure measurements, and particle image velocimetry. Surface plateau pressure is also theoretically predicted. The plateau pressure of present quasi-conical swept shock interaction is dependent on the local skin friction coefficient and normal Mach number component, and can be predicted per a criterion deduced from free interaction theory. Visualizations and surface pressure measurements confirm that the surface plateau pressure data are equal to the pressure downstream of the separation shock; the upstream influence line is parallel to the impingement line of the separation shock on the flat plate. These observations support the free interaction theory as-applied to quasi-conical swept shock interactions.

Prediction dynamic model of shock train with complex background waves

Oblique shock waves are unavoidable in a rectangular hypersonic inlet, leading to a non-uniform flow field. While a significant body of the literature exists regarding the shock train modeling in a uniform incoming flow condition, few efforts have focused on the shock train behavior considering the influence of the shock wave boundary layer interactions. A low-order dynamic model of the shock train has been constructed with the help of the free interaction theory and a 1-D analysis approach. Experimental and numerical investigations have been carried out to evaluate the low-order model. The results show that the model has the capability of qualitatively analyzing the shock train behavior. In the cases with incident shocks, the rapid forward movement of the shock train has been observed by experiment. Besides this phenomenon was also modeled using the low-order model. Schlieren images show that when the shock train approaches the interaction zone, its behavior is characterized by oscillation and then follows a rapid forward movement with a linear increasing backpressure at 2.7 Ma. This phenomenon is analyzed theoretically based on the free interaction theory. Meanwhile with the help of the direct numerical simulation results from some existing studies, the flow structures in the interaction region and the following boundary layer also provide the evidence.

Reynolds and Mach Number Dependence of Hypersonic Blunt Body Laminar Near Wakes

Laminar separation on the aftbody of an adiabatic circular cylinder was numerically investigated over a range of freestream Mach and Reynolds numbers using the free open-source program OpenFOAM. Simulations were performed assuming laminar, continuum, thermally perfect flow of nitrogen gas. Simulation predictions have been compared to various empirical correlations in the literature. Empirical formulations were presented for the pressure minimum location, separation location, and separation length on the cylinder aftbody. The effects of both Mach and Reynolds numbers on the separation location were examined by correcting for the change in viscosity across the shock wave. This was accounted for explicitly in the presented empirical formulations by deriving an expression for the viscosity change across the shock. A free-interaction theory was used to examine the wall-pressure distribution due to lip-shock formation on the aftbody. Over a range of Mach and Reynolds numbers, it was found that the profile of a free-interaction parameter was predictable and nearly Reynolds number and Mach number independent. This free-interaction parameter was a correlation between the properties at the onset of the interaction and the ensuing pressure rise. This provided a means to predict the lip-shock formation, provided the boundary-layer properties at the pressure minimum were known.

On the transition between regular and irregular shock patterns of shock-wave/boundary-layer interactions

The reflection of strong oblique shock waves at turbulent boundary layers is studied numerically and analytically. A particular emphasis is put on the transition between regular shock-wave/boundary-layer interaction (SWBLI) and Mach reflection (irregular SWBLI). The classical two- and three-shock theory and a generalised form of the free interaction theory are used for the analysis of well-resolved large-eddy simulations (LES) and for the derivation of stability criteria. We found that at a critical deflection angle across the incident shock wave, the perturbations related to the turbulent boundary layer cause bi-directional transition processes between regular and irregular shock patterns for a free-stream Mach number of$\mathit{Ma}_{0}=2$. Computational results show that the mean deflection angle across the separation shock is decoupled from the incident shock wave and can be accurately modelled by the generalised free interaction theory. On the basis of these observations, and the von Neumann and detachment criteria for the asymmetric intersection of shock waves, we derive the critical incident shock deflection angles at which the shock pattern may/must become irregular. Numerical data for a free-stream Mach number of$\mathit{Ma}_{0}=3$confirm the existence of the dual-solution domain predicted by theory.

Aeroelastic stability analysis of flexible overexpanded rocket nozzle

The aim of this paper is to present a new aeroelastic stability model taking into account the viscous effects for a supersonic nozzle flow in overexpanded regimes. This model is inspired by the Pekkari model which was developed initially for perfect fluid flow. The new model called the “Modified Pekkari Model” (MPM) considers a more realistic wall pressure profile for the case of a free shock separation inside the supersonic nozzle using the free interaction theory of Chapman. To reach this objective, a code for structure computation coupled with aerodynamic excitation effects is developed that allows the analysis of aeroelastic stability for the overexpanded nozzles. The main results are presented in a comparative manner using existing models (Pekkari model and its extended version) and the modified Pekkari model developed in this work.

Viscous effects of shock reflection hysteresis in steady supersonic flows

AbstractIn this paper, the viscous effects of shock reflection hysteresis in steady supersonic flow is studied analytically, experimentally and numerically, taking the boundary layer developed over the reflecting surface into consideration. Based on a hypothesis that the interaction origin keep invariable during the shock–boundary layer interaction, the separation region is replaced by a virtual wedge with a fixed angle. Combined the free-interaction theory with the shock reflection hysteresis theory, a detailed analysis describing the viscous flow structures of shock reflection configurations is proposed. It is illustrated by mean of further analysis that a shock reflection hysteresis which is similar to the one that exists in the reflection of symmetric shock waves is found theoretically. Experimental results verify the analytical interaction model as well as the existence of two shock reflection patterns, although the hysteresis is not conformed by experiments. This paper also presents results of simulations for the hysteresis process which results from keeping the wedge angle constant and changing the free-stream Mach number, for confirming the hysteresis phenomenon.