Expansion Corner Research Articles

Correlations to Predict the Streamwise Influence Regions in Supersonic Turbulent Flows

Correlation functions that can accurately predict the extent of the streamwise regions of influence for twodimensional supersonic turbulent flows over compression ramps, expansion corners, and shock-impingement flowfields have been developed. These empirically developed correlations that involve known flow parameters can be used to determine the lengths of the streamwise (upstream and downstream) regions of influence. The correlations were obtained by analyzing numerically generated compression ramp, expansion corner, and shockimpingement flowfields. These flowfields were computed using an iterative parabolized Navier‐Stokes algorithm in conjunction with the relaxation eddy-viscosity turbulence model of Shang and Hankey. Regression analysis using the least-squares approach was applied to the numerically generated data to determine the correlation functions. The accuracy of the developed correlations is demonstrated by comparing them with experimental and numerical data, and they were found to be accurate in predicting the extent of both the upstream and downstream influence regions.

Refining Strategy of the Supersonic Turbulent Flow Over a Backward-Facing Step

ABSTRACTA modified error indicator is developed to study the supersonic turbulent flow over a backward-facing step. In the Cartesian coordinate system, the unsteady Navier-Stokes equations with a low-Reynolds-number k−ε turbulence model are solved. The modified error indicator, in which the unified magnitude of substantial derivative of pressure and unified magnitude of substantial derivative of weighted vorticity magnitude are incorporated, is applied to treat the mesh refining. To assess the present approach, the transonic turbulent flow around an NACA 0012 airfoil is performed. Based on the comparison with the experimental data, the accuracy of the present approach is confirmed. According to the high-resolutional result on the adaptive mesh, the structure of backstep corner vortex, expansion wave and oblique shock wave is distinctly captured.

Shock/Vortex Interactions Induced by Blast Waves

The problem of blast-wave propagation in an open-ended duct with a 90-deg sharp convex (expansion) corner is numerically investigated by the use of a high-resolution Euler solver. The complicated e owe elds induced by the blast-wave diffraction around the corner and by a secondary shock/vortex interaction are explored. Both a two-dimensional case and an axisymmetric case are considered. It is found that two vortices near the corner in the two-dimensionalcase(ortwo ring vorticesin theaxisymmetriccase) areinduced by theblast-wavediffraction.One vortexisstrongerandislocatedfartherdownstream thantheother.Thesetwovorticesrotatein oppositedirections. In the axisymmetric e ow, the major ring vortex is located farther downstream than that for the corresponding two-dimensional case, resulting in a lower pressure at the vortex centers. This is due to the relieving effect that is represented by an extra nonvanishing term in the vorticity transport equation in the axisymmetric case. The trajectories of the major planar vortex or ring vortex induced by different blast intensities are explored. Basically, the planar major vortex (or the major ring vortex in the axisymmetric case) moves downstream before a critical time and then turns toward the duct centerline after the critical time. The critical time roughly corresponds to the formation of a minor vortex.

Remeshing Strategy of the Supersonic Flow Over a Backward-Facing Step

AbstractA modified error indicator and an extended locally implicit scheme with anisotropic dissipation model on quadrilateral-triangular mesh are developed to study the supersonic flow over a backward-facing step. In the Cartesian coordinate system, the unsteady Euler equations are solved. The modified error indicator, in which the unified magnitude of density gradient and unified magnitude of gradient of vorticity magnitude are incorporated, is utilized to treat the new node spacing of mesh remeshing. To assess the accuracy of the extended locally implicit scheme with anisotropic dissipation model on quadrilateral-triangular mesh, two flow calculations which include the oblique-shock reflection at a wall and transonic flow around an NACA 0012 airfoil are performed. Based on the comparison with the related numerical and experimental data, the accuracy of the present approach is confirmed. According to the high-resolutional result on the adaptive mesh, the structure of backstep corner vortex, expansion wave and oblique shock wave is distinctly captured.

Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner

A computational technique which is based on a numerical-asymptotic expansion matching for computing the local singular behavior of a viscous flow around a sharp right-angle expansion corner is presented. Moffatt's (1964) asymptotic solution is extended and a matching with a time-marching finite-difference scheme of the Navier--Stokes equations is formulated. Local mesh refinement around the corner is required to meet the validity of the asymptotic solution. Flows in an expanding channel with expansion ratio D/d=3 at various Reynolds numbers 1≤Re≤700 are simulated. The results are compared with those from a standard finite-difference scheme that uses second-order forward/backward differences near the corner. It is found that the results of the standard scheme converge toward those of the present technique as the level of local refinement near the corner is increased. The time-dependent parameters of the first two terms of the asymptotic solution at the steady-state solution are also described for various cases of Re and D/d. It is demonstrated that the present method enhances the accuracy of the simulations and requires less refinements near the corners to achieve converged numerical results.

The effects of expansion strength on large-scale structures in compressible free shear layers

Planar visualizations of two compressible free shear layers were performed immediately downstream of centered expansions of differing strengths in order to assess the influence of expansion strength on the embedded large-scale structures. The free shear layers studied here were formed through the separation of an approach flow, either a Mach 2.0 stream or a Mach 2.5 stream, from a planar backstep. In addition to side-view and end-view visualizations, spatial correlations (computed from large image ensembles) and laser Doppler velocimetry surveys of the free shear layers were also examined to discern relationships between the structure dynamics and the underlying pre- and postexpansion velocity fields. The instantaneous images clearly illustrate that ellipsoidal, highly coherent structures were present in both shear layers downstream of the expansion corner. The dissimilar expansion strengths did not appear to produce qualitatively different structures in the shear layers; however, as compared to the weaker expansion, the stronger expansion did result in an increase in the growth rate of the large-scale structures, apparently from an augmentation of the 〈u′v′〉∂U/∂y production term in the TKE equation. Furthermore, quantitative measurements of the mean structure geometry, as determined from the spatial correlation fields, revealed that a stronger expansion strength resulted in a larger aspect ratio of the mean structures (i.e., the structures were stretched preferentially in the streamwise and transverse directions as compared to the spanwise direction during the expansion process). Quadrant decompositions of the instantaneous velocity fluctuations within the approach boundary layers and within the free shear layers indicated a definite increase in structure organization across the expansion region, which is in contrast with studies of expanded supersonic boundary layers without separation. The instantaneous image data, spatial correlations, and velocity decompositions uniformly suggest that the separation process itself, and not the expansion strength, is the primary influence on initial eddy structure in the postexpansion free shear layer.

The dynamics of a laminar flow in a symmetric channel with a sudden expansion

Bifurcation analysis, linear stability study, and direct numerical simulations of the dynamics of a two-dimensional, incompressible, and laminar flow in a symmetric long channel with a sudden expansion with right angles and with an expansion ratio D/d (d is the width of the channel inlet section and D is the width of the outlet section) are presented. The bifurcation analysis of the steady flow equations concentrates on the flow states around a critical Reynolds number Rec(D/d) where asymmetric states appear in addition to the basic symmetric states when Re [ges ] Rec(D/d). The bifurcation of asymmetric states at Rec has a pitchfork nature and the asymmetric perturbation grows like √Re − Rec(D/d). The stability analysis is based on the linearized equations of motion for the evolution of infinitesimal two-dimensional disturbances imposed on the steady symmetric as well as asymmetric states. A neutrally stable asymmetric mode of disturbance exists at Rec(D/d) for both the symmetric and the asymmetric equilibrium states. Using asymptotic methods, it is demonstrated that when Re < Rec(D/d) the symmetric states have an asymptotically stable mode of disturbance. However, when Re > Rec(D/d), the symmetric states are unstable to this mode of asymmetric disturbance. It is also shown that when Re > Rec(D/d) the asymmetric states have an asymptotically stable mode of disturbance. The direct numerical simulations are guided by the theoretical approach. In order to improve the numerical simulations, a matching with the asymptotic solution of Moffatt (1964) in the regions around the expansion corners is also included. The dynamics of both small- and large-amplitude disturbances in the flow is described and the transition from symmetric to asymmetric states is demonstrated. The simulations clarify the relationship between the linear stability results and the time-asymptotic behaviour of the flow. The current analyses provide a theoretical foundation for previous experimental and numerical results and shed more light on the transition from symmetric to asymmetric states of a viscous flow in an expanding channel. It is an evolution from a symmetric state, which loses its stability when the Reynolds number of the incoming flow is above Rec(D/d), to a stable asymmetric equilibrium state. The loss of stability is a result of the interaction between the effects of viscous dissipation, the downstream convection of perturbations by the base symmetric flow, and the upstream convection induced by two-dimensional asymmetric disturbances.

放電発光粒子を利用した極超音速粘性流れ場の可視化

Spark technique was applied to qualitative velocity profile visualization in hypersonic flows over compression and expansion corners. For comparison, Schlieren measurement and numerical simulation based on the two-dimensional full Navier-Stokes equations were performed. The computational results did not contradict with the Holden criterion. For shock wave visualization, quantitative agreement among the spark, the Schlieren, and the computational results was good. Further, we found that the spark column shapes observed photographically and the computational velocity profiles correlate to each other. The results demonstrate the capability of the technique.

Interaction Region of Turbulent Expansion-Corner Flow

Narasimha and Sreenivasan mentioned that the interaction region appears to be insensitive to the corner deflection angle for supersonic flow. Here, this statement is corroborated for a low supersonic Mach number in conjunction with the work of Lu and Chung, which allows conclusions to be drawn concerning Mach number effects on the extent of the interaction region for turbulent expansion corner flows. In addition, an interaction region based on surface pressure fluctuations is illustrated

Prandtl–Meyer flows with homogeneous condensation. Part 1. Subcritical flows

Prandtl–Meyer flows with heat addition from homogeneous condensation not exceeding a critical value (subcritical flows) are investigated by an asymptotic method in the double limit of a large nucleation time followed by a small droplet growth time. The physically distinct condensation zones, with detailed analytical structure, are displayed along streamlines and the flow field in each zone is determined utilizing the asymptotic solution of the rate equation along streamlines. In particular the nucleation wave front, which corresponds to states of maximum nucleation along streamlines, is accurately located independently of the particular condensation model employed. Results obtained using the classical nucleation equation together with the Hertz–Knudsen droplet growth law show, despite qualitative agreement, considerable differences between the nucleation wave fronts and measured onset conditions for the experiments of Smith (1971), because of intersecting characteristics in the heat addition zones. This shows the necessity of including an embedded oblique shock wave in the expansion fan of corner expansion flows for the cases investigated.

Visualization of Hypersonic Separated Flow by Afterglow Technique.

The afterglow technique was applied to visualize hypersonic flow past a compression corner and that past an expansion corner. The experiments were conducted at the Kobe University hypersonic shock tunnel facility. An electric discharge column was generated in a flow in which there was a region of reverse flow and in a flow which expanded, and afterglow was observed. Good quantitative agreement with a result of Schlieren visualization of a shock wave shape was obtained. The afterglow results were also compared with the results of calculations based on the two-dimensional full Navier Stokes equations. The technique can be used to obtain a qualitative flow velocity profile. Finally, several error sources were discussed.

Expansion corner effects on hypersonic shock wave/turbulent boundary-layer interactions

An experimental program has been conducted to investigate the behavior of hypersonic shock wave/ turbulent boundary-layer interactions in the vicinity of an expansion corner. Tests were run in the Calspan 96-in. shock tunnel at nominal Mach 11.5 freestream conditions for a unit Reynolds number of approximately 18 x 106/ft. Fully turbulent boundary layers were generated with a flat plate, and oblique shocks were generated with a planar shock generator canted at angles of 10, 12, and 15 deg relative to the freestream. Shock-impingement points were varied to create impingement points upstream, in the vicinity, and downstream of an expansion corner with an angle that was similar in magnitude to the shock generator. Wall static pressure and heat transfer were measured in the region of the shock interaction to determine the effects of the expansion corner on the character of the shock wave/boundary-layer interaction. Detailed data have been generated for comparison with computational codes and to enable a better understanding of hypersonic inlet design.

Hypersonic turbulent expansion-corner flow with shock impingement

Mean and fluctuating surface pressure data were obtained in a Mach 8, turbulent, cold flow past an expansion corner subjected to shock impingement. The expansion corner of 2.5 or 4.25 deg was located at 0.77 m (30.25 in.) from the leading edge of a sharp-edged flat plate, while an external shock, generated by either a 2- or 4deg sharp wedge, impinged at the corner, or at one boundary-layer thickness ahead or behind the corner. The mean pressure distribution was strongly influenced by the mutual interaction between the shock and the expansion. For example, the upstream influence decreased when the shock impinged downstream of the corner. Also, the unsteadiness of the interactions was characterized by an intermittent region and a local rms pressure peak near the upstream influence. The peak rms pressure fluctuations increased with a larger overall interaction strength. Shock impingement downstream of the corner resulted in lower fluctuation peaks and also in a shorter region of reduced fluctuation levels. These features may be exploited in inlet design by impinging the cowl shock downstream of an expansion corner instead of at the corner. In addition, a limited pitot pressure survey showed a thinning of the boundary layer downstream of the corner.

Hypersonic, turbulent viscous interaction past an expansion corner

Turbulent, viscous interaction theory is used to understand the hypersonic flow past an expansion corner. By assuming a pressure law, the boundary-layer properties of the flow are obtained through simultaneous solution of a displacement thickness relationship and a coupling equation relating the effects of incidence and displacement thickness to the effective body shape. The pressure law is obtained through examination of available experimental data. The form of the pressure law is found to depend on the incidence. The pressure decay is dependent on both viscous and incidence effects. For weak incidence, viscous effects are important throughout the interaction, and they produce a gentle pressure decay. For strong incidence, viscous effects are important only near the front of the corner. In this latter case, the pressure decays rapidly followed by a long asymptotic downstream region; the overall pressure distribution then appears more similar to the in viscid case. Nomenclature A B C^ = constant of proportionality in linear viscosity-temperature relationship K - hypersonic similarity parameter, M^cc Kv = modified hypersonic similarity parameter, M^ dy e(x)/ dx M = Mach number P(x) = nondimensional pressure, p^/p^ p = pressure Re = unit Reynolds number T = temperature U = velocity (x, y) = physical coordinates along and normal to the incoming flow a = expansion corner angle y = ratio of specific heats 8 = boundary-layer thickness 6* = boundary-layer displacement thickness ^ = dummy variable of integration p = density % = weak turbulent, viscous interaction parameter, (M9 C / Re*)115

Observations of liquid jets injected into a highly accelerated supersonic boundary layer

Experiments were conducted to observe the cross-sectional structure and streamwise growth of round transverse liquid jets injected into a highly accelerated boundary layer in supersonic flow. The accompanying shock structure was also visualized. In one case, a round jet of acetone was injected into a fully turbulent Mach 2.5 boundary layer that was subsequently accelerated and partially laminarized through a sharp Prandtl-Meyer expansion corner. In the second case, a jet was injected into the laminarized Mach 3.2 boundary layer downstream of the expansion corner at the same jet-to-freestream momentum ratio. The jet and shock structure in both cases were visualized using schlieren optics. Wall-flow patterns were visualized using paints. It was found that the lateral spreading of jets injected downstream of the expansion fan was augmented close to the wall and had a cross-sectional structure significantly different from that of the jet injected upstream: the upstream jet spreads rapidly at the expansion corner in both the lateral and vertical directions.

Damping of surface pressure fluctuations in hypersonic turbulent flow past expansion corners

Surface pressure fluctuations of Mach 8 turbulent flow past a 2.5- and a 4.25-deg expansion corner maintained a Gaussian distribution but were severely attenuated by the expansion process. The pressure fluctuations did not recover to those of an equilibrium turbulent flow even though the mean pressures reached downstream inviscid values in four to six boundary-layer thicknesses. The fluctuations were convected with a velocity comparable to that on a flat plate, and they maintained their identities longer for the stronger expansion. The damping of pressure fluctuations at hypersonic Mach numbers, even by small corner angles, may be exploited in fatigue design.

Multiple distortion of a supersonic turbulent boundary layer

Two experiments were performed to study the response of a supersonic turbulent boundary layer to successive distortions. In the first experiment (Case 1), the flow passed over a forward-facing ramp formed by 20 0 compression corner followed by a 20 0 expansion corner located about 40 0 downstream, where 00 is the incoming boundary layer thickness. In the second experiment (Case 2), the forward-facing ramp was constructed of curved compression and expansion surfaces with the same turning angles and total step height as in Case 1. The radii of curvature for the compression and expansion surfaces were equal to 1200. In both experiments, the flow relaxation was observed over a distance equal to 120 0 • In this relaxation region, the mean and turbulent flow behavior of the boundary layer was measured. The mean velocity profile was found to be altered by the distortion. Recovery of the profile began near the wall and occurred rapidly, but in the outer part of the boundary layer, recovery proceeded slowly. Turbulence measurements revealed a dramatic reduction in the turbulence shear stress and a progressively decaying streamwise Reynolds stress profile.

Condensation phenomena in supersonic nozzles

The heat addition in a Laval-nozzle flow and in a Prandtl-Meyer corner expansion caused by condensation of water vapor in moist air is investigated. In both flow fields at the onset of condensation density, pressure and temperatur increase due to the release of the latent heat of vaporization. Experiments show that the energy addition is related to a relaxation process, and that the changes of the flow parameters can be expected in three different forms. In steady flow the thermodynamic state may change smoothly or it is discontinuous. At high rates of heat addition the flow becomes unsteady and it follows a repetitive cycle. The comparison of steady and unsteady flow fields allows the extraction of simple similarity rules by a dimensional analysis in conjunction with the laws of gasdynamic.

Local Heat Transfer Downstream of an Abrupt Expansion in a Circular Channel With Constant Wall Heat Flux

Measurements have been made of the local heat transfer coefficients to an air flow downstream of an axisymmetric abrupt expansion in a circular pipe with a constant wall heat flux. The experimental technique uses an electrically heated thin gold film on a plastic substrate. The flow upstream of the expansion was unheated and fully developed. Runs were made with small diameter to large diameter ratios of 0.267 to 0.800 and over the Reynolds numbers range of 5,300 to 87,000 (based on downstream diameter). The results include measurements near the expansion corner region where no previous measurements have been reported. These provide clear evidence of a secondary recirculation. Comparisons are also made with previous experimental results in the region of reattachment.

A Theoretical and Experimental Investigation of a Transonic Projectile Flowfield

Abstract : The transonic flow field about a secant-ogive-cylinder-boattail with a turbulent boundary layer has been studied. A joint theoretical and experimental effort is presented which compares the results of a generalized axisymmetric Navier-Stokes code, a composite inviscid boundary-layer/shock interaction solution method, and experiment. The experimental longitudinal pressure distribution at M= 0.94, and 0.97 for alpha = 0.0 degree are generally well predicted by both theoretical techniques although the Navier-Stokes solutions are shown to be superior in describing the detail, such as, upstream effects of expansion corners and the position and magnitude of minimum pressure regions. Both theoretical solutions predict the boundary layer velocity profiles very well in all cases with the largest differences occurring just downstream of the boattail corner. Comparisons of displacement thickness and skin friction distributions are also presented.