Shock-induced Separated Flows Research Articles

Bi-global stability of supersonic backward-facing step flow

Supersonic backward-facing step (BFS) flow is numerically studied using direct numerical simulation (DNS) and global stability analysis (GSA) with a free stream Mach number of 2.16 and a Reynolds number of 7.938 × 105 based on the flat-plate length L and free stream conditions. Two-dimensional BFS flow becomes unstable to three-dimensional perturbations as the step height h exceeds a certain value, while no two-dimensionally unstable mode is found. Global instability occurs with the fragmentation of the primary separation vortex downstream of the step. Two stationary modes and one oscillatory unstable mode are obtained at a supercritical ratio of L/h = 32.14, among which the two stationary modes originate from the coalescence of a pair of conjugate modes. The most unstable mode manifests itself as streamwise streaks in the reattached boundary layer, which is similar to that in shock-induced separated flow, although the flow separation mechanisms are different. Without introducing any external disturbances, the DNS captures the preferred perturbations and produces a growth rate in agreement with the GSA prediction in the linear growth stage. In the quasi-steady stage, the secondary separation vortex breaks up into several small bubbles, and the number of streamwise streaks is doubled. A low-frequency unsteadiness that may be associated with the oscillatory mode is also present.

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.

Anisotropic SST turbulence model for shock-boundary layer interaction

Menter SST k-ω is a Reynolds-averaged Navier–Stokes based two-equation turbulence model routinely used in industry for predicting aerodynamic flows. It shows excellent performance for low-speed flows, but gives inconsistent predictions for high-speed shock-induced separated flows. The model assumption of using a constant value of 0.31 for the structure parameter contradicts experimental observations. The model is also unable to predict Reynolds stress anisotropy generated by shock waves. In this work, we augment the SST model with quadratic eddy viscosity formulation of an explicit algebraic Reynolds stress model. A new relation for the structure parameter is proposed, making it a function of the local strain-rates and is no longer a constant in the regions of shock/turbulent boundary layer interaction (SBLI). Additional shock-physics is introduced using (Sinha et al., 2003) shock-unsteadiness model and an upper limit to the value of structure parameter is set in regions of shock waves. The new model, termed as SUQ-SST, is validated using a number of SBLI test cases ranging from supersonic to hypersonic speeds and near-incipient to fully-separated flows. Results show that the modifications do not alter the boundary layer prediction capability of the SST model. On the other hand, the new model gives significant improvement in predicting Reynolds stress anisotropy, flow separation, and surface properties in a wide range of SBLI flows.

Occurrence of global instability in hypersonic compression corner flow

Hypersonic flow over a two-dimensional compression corner with a Mach number of 7.7 and unit Reynolds number of 4.2 × 106 m−1 is numerically investigated. Special emphasis is given to the onset of global instability with respect to three-dimensional perturbations. Global stability analysis is performed for various ramp angles and wall temperatures. It is found that the shock-induced separated flow system becomes unstable when the ramp angle is beyond a certain value. The critical ramp angle increases slightly with the wall temperature, although the length of the separation region is significantly enlarged. The global instability is shown to be closely linked with the occurrence of secondary separation beneath the primary separation bubble. A criterion is established based on a scaled ramp angle defined in the triple-deck theory to predict the global stability boundary, which depends on the free-stream conditions and geometries only.

Control of Fin Shock Induced Flow Separation Using Vortex Generators

This work demonstrates the control of the shock-induced separated flow generated by a sharp fin placed on a cylindrical surface (fin-on-cylinder shock-wave/boundary-layer interaction) with an incident Mach 2.5 flow. Well-quantified distortions in the form of streamwise vortices are embedded into the boundary layer using single sub-boundary-layer vortex generators (VGs). At four device heights downstream from the VG trailing edge, the boundary layer exhibits a peak deficit in the streamwise velocity from the undistorted boundary layer of 60%. At 15 device heights, the boundary layer was energized to a peak surplus of 25% in the streamwise velocity. These two VG streamwise placements (four and 15 device heights) were employed to emphasize the impact of direct vortex interactions and boundary-layer energization on separation vortex control. It was found that having an energized boundary-layer incident on the fin-on-cylinder SBLI made very little change to the separation vortex size and the associated pressure distribution. However, placing the vortex generator much closer significantly mitigated the separation vortex size, and several azimuthal placement locations showed an elimination of the separation vortex. Cross-sectional velocity and vorticity fields showed a strong reduction in spanwise velocity and streamwise vorticity above the separation vortex in the successful configurations.

Low-frequency dynamics in a shock-induced separated flow

The low-frequency unsteadiness in the direct numerical simulation of a Mach 2.9 shock wave/turbulent boundary layer interaction with mean flow separation is analysed using dynamic mode decomposition (DMD). The analysis is applied both to three-dimensional and spanwise-averaged snapshots of the flow. The observed low-frequency DMD modes all share a common structure, characterized by perturbations along the shock, together with streamwise-elongated regions of low and high momentum that originate at the shock foot and extend into the downstream flow. A linear superposition of these modes, with dynamics governed by their corresponding DMD eigenvalues, accurately captures the unsteadiness of the shock. In addition, DMD analysis shows that the downstream regions of low and high momentum are unsteady and that their unsteadiness is linked to the unsteadiness of the shock. The observed flow structures in the downstream flow are reminiscent of Görtler-like vortices that are present in this type of flow due to an underlying centrifugal instability, suggesting a possible physical mechanism for the low-frequency unsteadiness in shock wave/turbulent boundary layer interactions.

Unsteady Behavior of a Pressure-Induced Turbulent Separation Bubble

Wall static-pressure and longitudinal-velocity fluctuations are measured in a pressure-induced turbulent separation bubble generated on a flat test surface by a combination of adverse and favorable pressure gradients. The Reynolds number, based on momentum thickness upstream of separation, is at a free-stream velocity of . The results indicate that the flow is characterized by two separate time-dependent phenomena: a low-frequency mode, with a Strouhal number , which is related to a global “breathing” motion (i.e., contraction/expansion) of the separation bubble, and a higher-frequency mode, with a Strouhal number , which is linked to the roll-up of vortical structures in the shear layer above the recirculating region and their shedding downstream of the bubble. These two phenomena are reminiscent of the “flapping” and “shedding” modes observed in fixed-separation experiments, though their normalized frequencies are different. The breathing mode is also shown to be strikingly similar to the low-frequency unsteadiness observed in shock-induced separated flows at supersonic speeds.

Effect of Reynolds number on compressible convex-corner flows

An experimental study was conducted to investigate the effect of Reynolds number on compressible convex-corner flows, which correspond to an upper surface of a deflected flap of an aircraft wing. The flow is naturally developed along a flat plate with two different lengths, resulting in different incoming boundary layer thicknesses or Reynolds numbers. It is found that boundary layer Reynolds number, ranging from <TEX>$8.04{\times}10^4$</TEX> to <TEX>$1.63{\times}10^5$</TEX>, has a minor influence on flow expansion and compression near the corner apex in the transonic flow regime, but not for the subsonic expansion flow. For shock-induced separated flow, higher peak pressure fluctuations are observed at smaller Reynolds number, corresponding to the excursion phenomena and the shorter region of shock-induced boundary layer separation. An explicit correlation of separation length with deflection angle is also presented.

Unsteadiness in shock-induced separated flow with subsequent reattachment of supersonic annular jet

An axisymmetric truncated optimized contoured nozzle flow operating in overexpanded conditions is simulated in restricted shock separation regime. The numerical strategy adopted aims at computing essentially the largest dynamically active scales of the flow in order to analyze the unsteady mechanisms leading to the low-frequency peak of side loads observed in this case. It relies on an hybrid approach, based on a k–ω model, including realizability corrections and modified in the vein of delayed detached eddy simulation. The full model operability in LES mode is shown to be inhibited by a limited resolution, which admittedly limits the representativity of interactions between intermediate scales. However, the present approach is shown to lead to surprisingly good estimates of the wall-pressure signals and resulting rms of side-loads. A representative large-scale unsteadiness is reproduced and is found to be characterized by an irregular azimuthal asymmetry of the reattached flow, superimposed on large-amplitude axial oscillations. An essential ingredient driving the flow unsteadiness is related to the inhomogenous propagation of pressure disturbances through the subsonic three-dimensional trapped vortex downstream of the cap-shock system.

Assessment of turbulence model performance for transonic flow over an axisymmetric bump

AbstractA computational study has been performed to evaluate the predictive capabilities of some existing eddy-viscosity (both linear, LEVM, and non-linear, NLEVM) and Reynolds stress transport turbulence models (RSTM) by reference to a transonic shock-induced separated flow over a 10% axisymmetric bump. The calculations have been carried out during the course of a collaborative research programme including both UK universities and industry. The findings of the project demonstrate that improved results can be obtained for such flows by using more advanced turbulence models. For linear eddy-viscosity models, only the SST approach gave good predictions of shock location, recirculation size and pressure recovery, although this was accompanied by deficiencies in the prediction of post-shock velocity profile shape. Non-linear eddy-viscosity models, particularly at the cubic level, provided a more consistent level of agreement with experiments over the range of shock location, wall pressure and velocity profile parameters. Some improvement was also seen in the prediction of turbulence quantities, although only a move to an RSTM closure model reproduced the measured peak stress levels accurately. It was notable that the use of low-Re variants of the models (instead of wall functions) produced no significant improvement in predictions. There are, however, some shortcomings in all models, particularly in the development of flow after reattachment, which was always predicted to be too slow.

High-Speed Turbulent Separated Flows: Consistency of Mathematical Models and Flow Physics

A large fraction of the experimental database used for validation of computational fluid dynamics (CFD) predictions of shock-induced turbulent separated flows consists of time-averaged measurements. If meaningful conclusions are to be drawn from comparisons of these time-averaged data with computations, it is important to understand how the data are generated physically. This can be difficult and time consuming but is an essential element in the selection of an appropriate mathematical model for the computation. For the separated compression ramp interaction, which is widely used as a CFD test case, global unsteadiness is a dominant phenomenon. Without modeling the unsteadiness, accurate predictions of the time-averaged wall pressure, outgoing velocity profiles, and other parameters will likely remain elusive, irrespective of the turbulence model.

Reynolds stress transport modelling of shock-induced separated flow

Predicting the interaction process in transonic flow between the inviscid free stream and the turbulent boundary layer is a challenging task for numerical simulation, which involves complex physical phenomena. In order to capture the physics, a turbulence model capable of accounting for physical phenomena such as streamline curvature, strong non-local effects and history effects, and large irrotational strains should be used. In the present work a second-moment Reynolds Stress Transport Model (RSTM) is used for computing transonic flow in a plane channel with a bump. An explicit time-marching Runge-Kutta code is used for the mean flow equations. The convecting terms are discretized using a third-order scheme (QUICK), and no explicit dissipation is added. For solving the transport equations for the Reynolds stresses u 2 , v 2 , and uv as well as k and ϵ an implicit solver is used which—unlike the Runge-Kutta solver—proved to be very stable and reliable for solving source dominated equations. Second-order discretization schemes are used for the convective terms. As the RSTM is valid only for fully turbulent flow, a one-equation model is used near the wall. The two models are matched along a pre-selected grid line in the fully turbulent region. The agreement between predictions and measurements is, in general, good.

Axisymmetric afterbody experiments for CFD validation

SummaryAn experiment is described which was undertaken specifically to provide CFD validation data for the case of transonic flow over nozzle afterbodies. The tests were undertaken with the AGARD standard 10° and 15° axisymmetric boat-tail geometries. Onset Mach numbers in the range 0·80-0·99 and subsonic and under-expanded jet plumes were employed in the tests. Test conditions were selected which provided a range of afterbody flow features from largely attached to shock-induced separated flows. A uniquely detailed set of surface pressure and flowfield data are presented. The flow data were acquired with a two-component laser Doppler anemometer (LDA) and define the mean and fluctuating flow components at about 500 spatial locations for each of these complex transonic flowfields. Additional information was recorded which fully defines the required computational boundary conditions.Also presented is a detailed study of the necessary attributes of windtunnel CFD validation data. It is demonstrated that relatively high blockage experiments using cost-effective windtunnels can be used to generate CFD validation data if proper account is taken of the model/tunnel interference.

Calculation of transonic flows with separation past arbitrary inletsat incidence

A method of calculating transonic separated flows past arbitrary inlets at incidence and yaw is presented. The inviscid flow solver is based on the transonic potential equation, which is approximated and solved numerically on a boundary-conforming coordinate system. The multrigrid finite-volume method is used to solve the flow equation. The inviscid flow technique is coupled with an interactive boundary-layer method that uses an inverse formulation to calculate shock-induced separated flows

Advances in the computation of transonic separated flows over finite wings

Transonic flow fields around a low-aspect-ratio wing in a wind tunnel have been simulated using an Euler/Navier-Stokes zonal approach. By using a fast parabolic-type grid generator, a global grid with an H-H topolgy is generated around the wing in which the far-field boundaries match the tests section of the wind tunnel. This global grid, which is a coarse grid, was then subdivided into separate zones. The grid zones near the wing were suitably clustered for viscous resolution, and the Reynolds-averaged Navier-Stokes equations were solved. In the rest of the flow field the Euler equations were solved. Particular emphasis was placed on simulating shock-induced separated flows. Problem areas in the state-of-art computational techniques in wind-tunnel simulations were identified, and each problem was studied in an isolated manner in order to avoid combined effects of multiple sources. The lessons that were learned from those studies were then combined to yield more accurate wind-tunner simulations and to further the understanding of separated flows. As a technology demonstration, one solution with extremely fine grid resolution (1.1 × 10 6 total grid points) was achieved using the Cray-2-superdomputer. Of particular importance for flow physics, a transonic mushroom-type separated flow with counterrotating vortices that closely resembles the experimental pattern, was demonstrated for the first time.

Space-time correlations of wall pressure fluctuations in shock-induced separated turbulent flows

An experimental study of the structure of wall pressure fluctuations is performed in shock-induced separated supersonic turbulent flows. Two test conditions are studied for external Mach numbers of 2.25 at low Reynolds number and Mach 3.3 at high Reynolds number. Piezoelectric pressure transducers, isolated from the main facility to minimize vibration effects, are used to sense the pressure fluctuations. Pressure fluctuations are given for several locations and analyzed in the frequency domain. The results exhibit the usual behavior. Space-time correlations are measured and subsequently analyzed using coherence analysis. The results are discussed in terms of structural modifications induced by the shock. Typical results are given for several positions in the incoming boundary layer, at the mean shock location, and in the recirculation region. Some new features concerning the behavior of characteristic scales at the shock traverse are identified, and the results are compared and scaled for both test conditions.

Shock-induced separated flows on the lee surface of delta wings

Experiments were conducted to study shock-induced separated flows on the lee surface of delta wings with sharp leading edge at supersonic speeds. Two sets of delta wings of different thickness (10° and 25° normal angle), each with leading edge sweep angles varying from 45° to 70°, were tested. The measurements, carried out in a Mach number range from 1.4 to 3.0, included oil flow visualisations (on both sets of wings) and static pressure distributions (on the thicker wings only). Using the test results, some features of shock-induced separated flows, including in particular the boundary between this type of flow and fully attached flow, have been determined. The experimental results indicate that this boundary does not seem to show any significant dependence on wing thickness within the limit of thicknesses tested. It is shown that this boundary can be predicted for thin delta wings using a well known criterion for incipient separation in a glancing shock wave boundary layer interaction, namely that a pressure rise of 1.5 is required across the shock. Comparison of the predicted boundary with experimental results (from oil flow visualisations) shows good agreement.

An Experimental Verification of Laser-Velocimeter Sampling Bias and Its Correction

The existence of “sampling bias” in individual-realization laser velocimeter measurements is experimentally verified and shown to be independent of sample rate. The experiments were performed in a simple two-stream mixing shear flow with the standard for comparison being laser-velocimeter results obtained under continuous-wave conditions. It is also demonstrated that the errors resulting from sampling bias can be removed by a proper interpretation of the sampling statistics. In addition, data obtained in a shock-induced separated flow and in the near-wake of airfoils are presented, both bias-corrected and uncorrected, to illustrate the effects of sampling bias in the extreme.

Steady and Unsteady Transonic Flow

An investigation of the transonic flow over a circular arc airfoil was conducted to obtain basic information for turbulence modeling of shock-induced separated flows and to verify numerical computer codes which are being developed to simulate such flows. The investigation included the employment of a laser velocimeter to obtain data concerning the mean velocity, the shear stress, and the turbulent kinetic energy profiles in the flowfield downstream of the airfoil midchord where the flow features are more complex. Depending on the free-stream Mach number, the flowfield developed was either steady with shock-wave-induced separation extending from the foot of the shock wave to beyond the trailing edge or unsteady with a periodic motion also undergoing shock-induced separation. The experimental data were compared with the results of numerical simulations in which a computer code was employed that solved the time-dependent Reynolds' averaged compressible Navier-Stokes equations.