Adverse Pressure Gradient Research Articles

Dynamic mode decomposition and Koopman spectral analysis of boundary layer separation-induced transition

In the present work, dynamic mode decomposition (DMD) and Koopman spectral analysis are applied to flat plate particle image velocimetry experimental data. Experiments concerning separated-flow transition process were carried out in a test section allowing the variation of the Reynolds number (Re), the adverse pressure gradient (APG) and the free-stream turbulence intensity (Tu). The analysis accounts for two different Re numbers, two different Tu levels, and a fixed APG condition inducing flow separation, as it may occur in low pressure turbine-like conditions. For every flow condition, instantaneous velocity fields clearly show the formation of Kelvin–Helmholtz (KH) vortices induced by the KH instability. The most effective definition of the observable matrix for Koopman analysis able to characterize these vortices is addressed first for a reference Tu and Re number condition. Successively, the robustness of DMD and Koopman modal decomposition has been examined for different Tu levels and Re numbers. On a short time trace (10 KH periods), the Koopman analysis is shown to identify the proper KH vortex shedding frequency and wavelength for every condition tested, while DMD fails especially with low Tu and high Re. To validate the results on a longer time trace, a statistical analysis of the dominant unstable eigenvalues captured by the two procedures is successively performed considering several temporal blocks for different inflow conditions. Overall, the Koopman analysis always performs better than DMD since it finds a larger number of unstable eigenvalues at the KH instability frequency and wavelength.

Effect of an adverse pressure gradient on hypersonic wall pressure fluctuations

Measurements based on a plate-perpendicular fin model were conducted to investigate the effect of an adverse pressure gradient on hypersonic wall pressure fluctuations. The leading edge diameter of the perpendicular fin is designed to be 25 mm with a height of 200 mm. A linear sensor-array was used to measure the wall pressure fluctuation, and a numerical computation was carried out to validate the measurement. Wall pressure fluctuations were discussed in terms of two aspects: the time–frequency domain and spatial correlation. The flow types on the plate could be estimated by the sound pressure level distribution, and the dominant flow type that substantially contributes to the wall pressure fluctuations could be determined. The spatial correlation of wall pressure fluctuations was analyzed using the phase array technique, and two disturbance modes could be identified from the wavenumber map obtained using the beamforming algorithm. The investigation results revealed that the change in the unit Reynolds number caused by the variation in the inflow dynamic pressure leads to the change in the flow type and the distribution of wall pressure fluctuations. The small-scale vortices within the hypersonic laminar flow lead to the difference in the signs of the convective mode wavenumber and that generated by hypersonic turbulence.

Acoustics of rotor ingesting an axisymmetric turbulent boundary layer

The noise generation by a five-bladed rotor ingesting an axisymmetric turbulent boundary layer at the tail end of a body of revolution (BOR) is investigated numerically at Reynolds number of 1.9 × 106 based on the BOR length and free-stream Mach number of 0.059. The turbulent boundary layer on the nose and midsection of the BOR is computed using wall-modeled LES whereas that in the acoustically important tail-cone section under a strong adverse pressure gradient is wall resolved. The radiated acoustic field is calculated based on the Ffowcs Williams-Hawkings equation. The computed flow statistics and sound pressure spectra agree well with the experimental measurements at Virginia Tech. In addition to broadband turbulence-ingestion noise, haystacking spectral peaks near the blade passing frequency and its harmonics are well captured and related to correlated blade interactions with coherent turbulence structures. The accuracy of the classical Sears theory for rotor-noise prediction is evaluated using the upwash velocity field from the LES. The theory is shown to predict well the sound pressure levels at low to mid-frequencies including the haystacking peaks but less accurately at higher frequencies. The spatial and frequency characteristics of blade unsteady-loading dipoles and their correlations are investigated, and the appropriate Mach number scaling for the ingestion noise is identified.

High and Variable Drag in a Sinuous Estuary With Intermittent Stratification

AbstractIn field observations from a sinuous estuary, the drag coefficient based on the momentum balance was in the range of , much greater than expected from bottom friction alone. also varied at tidal and seasonal timescales. was greater during flood tides than ebbs, most notably during spring tides. The ebb tide was negatively correlated with river discharge, while the flood tide showed no dependence on discharge. The large values of are explained by form drag from flow separation at sharp channel bends. Greater water depths during flood tides corresponded with increased values of , consistent with the expected depth dependence for flow separation, as flow separation becomes stronger in deeper water. Additionally, the strength of the adverse pressure gradient downstream of the bend apex, which is indicative of flow separation, correlated with during flood tides. While generally increased with water depth, decreased for the highest water levels that corresponded with overbank flow. The decrease in may be due to the inhibition of flow separation with flow over the vegetated marsh. The dependence of during ebbs on discharge corresponds with the inhibition of flow separation by a favoring baroclinic pressure gradient that is locally generated at the bend apex due to curvature‐induced secondary circulation. This effect increases with stratification, which increases with discharge. Additional factors may contribute to the high drag, including secondary circulation, multiple scales of bedforms, and shallow shoals, but the observations suggest that flow separation is the primary source.

Extension of an algebraic intermittency model for better prediction of transition in separated layers under strong free-stream turbulence

A constitutive law describing the Reynolds stresses in boundary layers undergoing laminar-to-turbulent transition, constructed in previous work by elastic-net regression on an experimental data base, is used to improve an algebraic intermittency model for cases with transition in a separated layer influenced by a high level of free-stream turbulence. The intermittency model is combined with a k-ω turbulence model and the basic version, developed in previous work, functions well for bypass transition in attached boundary layers and for transition in separated boundary layers under a low free-stream turbulence level. The basic model version is extended by an additional production term in the transport equation for turbulent kinetic energy. A sensor detects the front part of a separated layer and activates the production term. The term expresses the effect of Klebanoff streaks generated upstream of separation on the Kelvin-Helmholtz instability rolls in the separated part of the layer. The Klebanoff streaks cause faster breakdown by the combined effects of a large adverse pressure gradient and an elevated free-stream turbulence level. The extended model does not alter the results of the basic model version for bypass transition in an attached boundary layer and for transition in a separated boundary layer under a low free-stream turbulence level. The extended model significantly improves the predictions of the previous model version for transition in a separated boundary layer under a high free-stream turbulence level.

Statistical characterization of free-stream turbulence induced transition under variable Reynolds number, free-stream turbulence, and pressure gradient

In this work, the free-stream turbulence (FST) induced transition of a flat plate boundary layer is studied using particle image velocimetry (PIV) under variable Reynolds number (Re), FST intensity, and adverse pressure gradient (APG). Overall, 10 different flow conditions were tested concerning the variation of these parameters. The streak spacing and the probability density function (PDF) of turbulent spot nucleation are computed for all cases. The streak spacing is shown to be constant in the transition region once scaled with the turbulent displacement and momentum thickness, with resulting values of around 3 and 5, respectively. Nucleation events are shown to occur near the position where the dimensionless streak spacing reaches such constant values. The streamwise position where most turbulent spots are formed is strongly influenced by the FST intensity level. Additionally, the PDF of spot nucleation becomes narrower with increase in the APG, while FST has the opposite effect. A common distribution of all the PDFs is provided as a function of a similarity variable accounting for the streak spacing, the shape factor of the boundary layer, and the FST intensity.

Instability mechanisms and intermittency distribution in adverse pressure gradient attached and separated boundary layers

Direct numerical simulation has been carried out on one attached and two separated boundary layer flows (involving small and large separation) under the influence of an adverse pressure gradient. A unified picture of the pre-transitional boundary layer for the three cases has been provided that reveals a “mixed-mode” instability, involving contribution from instability waves and streamwise streaks. A time-frequency analysis of the transitional velocity signals has been performed which shows that as the Reynolds number decreases, the character of the time traces evolves continuously from a “spotty” behavior (exhibiting distinct turbulent spots) for the attached case to a “non-spotty” behavior (involving more “uniform” distribution of turbulent fluctuations in time) for the large-separation case, encompassing the entire spectrum of transition scenarios. The variation of the intermittency factor within the transition zone is seen to compare well with the Narasimha universal intermittency distribution. We find that although the time variation of velocity for large separation is non-spotty (or more “uniform”), the spanwise variation of velocity is spotty, showing a clear clustering of high-wavenumber fluctuations separated by quasi-laminar regions. Thus, all three cases exhibit spottiness in the transition zone with different manifestations. We present a physical cartoon for the transition scenarios for the attached and separated cases, using the ideas of vortex-wall interaction and instability of spanwise vortical structures. We find that concentrated breakdown is exhibited by all the three cases near the transition onset, and the spot breakdown processes are broadly consistent with the postulates underlying the universal intermittency distribution.

Computational flow analysis of a single peristaltic wave propagation in the ureter

Background and objectiveThe bladder receives the urine from the kidney and ureter. The series of peristaltic waves facilitate urine transport to the bladder. The peristaltic flow in the ureter is associated with fluid trapping and material reflux, which may cause an increase in bladder pressure. It is difficult to visualize the complex peristalsis phenomenon, in the ureter using image and radiography experiments. A numerical simulation will help in the understanding of urine bolus formation and its effect on the ureter wall. MethodsA three-dimensional computational fluid dynamic analysis is carried out to understand the flow physics associated with bolus formation and the effect of reflux on the ureter. ANSYS-CFX, a commercially available computational dynamics package is used to simulate the peristalsis. A single sinusoidal peristaltic wave traveling along a circular tube will yield the velocity, pressure, wall shear stress distributions inside the ureter. ResultsThe propagation of the peristaltic wave results in the backflow of urine near the inlet at the beginning of the flow. As the wave propagates towards the outlet, the flow rate decreases. It is observed that pressure distribution along the ureter axis will deteriorate towards the outlet. The contraction produces a very high-pressure gradient which causes the urine backflow. The trapping and the bolus formation cause a significant rise in bolus pressure, simultaneously developing negative pressure at the contraction neck. ConclusionsThe effect of peristalsis on the ureter biofluid dynamic behavior of the ureter is visualized in this study. It is established that the peristaltic contraction results in high-pressure formation at the bolus and negative pressure at the neck. It was found to be a maximum of 1.1 Pa at the bolus center and —1.13 Pa at the neck region. At the ureter pelvis junction, a higher wall shear of 0.095 Pa is observed as the wave starts to propagate. The velocity vectors show that the trapping of urine causes reflux and results in an adverse pressure gradient near the wall. A maximum pressure gradient of 485 Pa/meter was observed at the contraction of the ureter wall.

Micro-Ramp Control for Shock Train Structure and Oscillation

Micro-ramps for shock train control were investigated experimentally in a Mach 1.85 supersonic isolator flow using three device heights. A mechanical flap was mounted downstream and linearly choked, forcing the shock train to travel upstream of the micro-ramp from the exit. Wall static pressure measurements along the primary and corner regions obtained a more complete pressure distribution. Detailed structures of the shock train were recorded using schlieren visualization with circular, horizontal, vertical, and color knife edges. The results show that the wake and wave structures generated by the micro-ramp control the shock train. The taller the device, the stronger the wake and wave structures, and the greater the control. The wake and wave structures shorten or eliminate the Mach stem in the shock train leading edge, causing modification of shocks downstream from pseudonormal to oblique, inhibiting the thickening of the boundary layer that results from shock impingement, slowing down the contraction of the supersonic core flow, and allowing more high total pressure fluid to develop downstream. The wake structure was found to promote the oscillation and upstream motion of the shock train. The wave structure can restrain shock train oscillation and upstream propagation of the downstream adverse pressure gradient, enhancing the resistance backpressure of the flowfield downstream of the micro-ramp.

Turbulence Modeling in Hypersonic Turbulent Boundary Layers Subject to Convex Wall Curvature

Hypersonic boundary layers are crucial in aerospace applications such as hypersonic glide vehicles, rockets, and other advanced space vehicles. Hypersonic flows present unique transport phenomena, including nonnegligible flow compression/dilation, extra strain rates, and large momentum/thermal gradients. In this paper, the performance of three widely used turbulence models is compared, namely, the standard , the shear stress transport (SST) , and the Spalart–Allmaras (SA). Based on our turbulence modeling assessment plus the analysis of turbulent transport equation budgets over the experimental geometry from a previous study at a Mach number of 4.9, a moderate supremacy of SA over two equation models was found. To back our conclusions, previous experiments and direct numerical simulations at Mach numbers around 5 have been employed. Overall, the three considered models exhibited a consistent ability to predict first-order statistics both inside and outside the boundary layer. The SST variants were capable of describing the amplification of the constant shear layer induced by the presence of an adverse pressure gradient (APG). Furthermore, the SST model also replicated the second peak of turbulence production induced by the concave wall. There was a more aggressive distortion of the boundary layer by APG than by favorable pressure gradient (FPG) as compared with a zero pressure gradient (ZPG) boundary layer. A reasonable performance by Walz’s equation in the FPG region is also shown, whereas a notable lack of agreement is seen in the APG. Overall, one could argue for the SA model’s best compromise between accurate predictions, numerical stability, and mesh resolution insensitivity in the FPG and ZPG regions, particularly, in outer or integral boundary-layer parameters such as or . That being said, the two-equation models are far superior in terms of predicting near-wall parameters (such as or ) or their ability to accurately describe the physics of the hypersonic boundary layer for APG regions (for instance, outer-secondary peaks of turbulent kinetic energy production).

Effect of a blade end slot on supersonic compressor cascade hub-corner separation

The strong adverse pressure gradient and shock wave/boundary layer interaction that occurs in supersonic compressor cascade corner regions usually leads to large three-dimensional corner separations with extensive total pressure losses. This substantially restricts compressor load performance. In this paper, the effect of a blade end slot on corner separation caused by shock wave/boundary layer interaction was investigated. Because of the pressure difference between the pressure and suction sides, the slot enables one to induce jet flow into the endwall/blade suction-side corner region. Four different blade end slot locations are designed and their influences are investigated numerically. The mechanism of blade end slot influence on corner separation varies by position. The optimal location is slightly downstream of the initial separation point, which improves corner separation control and can reduce the cascade total pressure loss by 9.7% using the currently prescribed slot size.

Acoustic-roughness receptivity in subsonic boundary-layer flows over aerofoils

The generation of a viscous–inviscid instability through scattering of an acoustic wave by localised and distributed roughness on the upper surface of a NACA 0012 aerofoil is studied with a time-harmonic compressible adjoint linearised Navier–Stokes approach. This extends previous work by the authors dedicated to flat plate geometries. The key advancement lies in the modelling of the inviscid acoustic field external to the aerofoil boundary layer, requiring a numerical solution of the convected Helmholtz equation in a non-uniform inviscid field to determine the unsteady pressure field on the curved aerofoil surface. This externally imposed acoustic pressure field subsequently drives the acoustic boundary layer, which fundamentally determines the amplitudes of acoustic-roughness receptivity. A study of receptivity in the presence of Gaussian-shaped roughness and sinusoidally distributed roughness at Mach number $M_\infty =0.4$ and Strouhal numbers $\mathcal {S} \approx \{46,69,115\}$ shows the effects of various parameters, most notably angle of attack, angle of incidence of the externally imposed plane acoustic wave and geometry of surface roughness; the latter is varied from viewpoint of its placement on the aerofoil surface and its wavelength. The parametric study suggests that non-parallel effects are quite substantial and that considerable differences arise when using parallel flow theory to estimate the optimal width of Gaussian-shaped roughness elements to provoke the greatest response. Furthermore, receptivity amplitudes for distributed roughness are observed to be generally higher for lower angles of attack, i.e. for less adverse pressure gradients. It is also shown that the boundary layer is more receptive to upstream-travelling acoustic waves.

Spatial wake transition past a thin pitching plate.

The spatial transition of the wake behind a thin pitching plate in the thrust regime is studied. The drag-to-thrust transition is seen to occur at a threshold pitching frequency which becomes smaller for higher pitching angle and aspect ratio. The reverse von Kármán wake shows deflected asymmetric vortex pairing in two dimensions, while a bifurcated wake with a swirling ring is the signature in three dimensions, both resulting in nearly the same scaling for the power coefficient. The wake transition in the thrust regime occurs through three spatial regions comprising the reverse von Kármán vortex street, transitional zone, and twin jet region. Near the trailing edge, reverse von Kármán shedding is marked by linear decay of the spanwise wake width and growth of secondary instability. A large volume of fluid resulting from inward horizontal acceleration renders a weak form drag in the subsequent transition zone which is followed by vertical bifurcation into twin jet formation in the far wake. The spanwise pressure gradient is seen to guide the wake compression, with a streamwise adverse pressure gradient aiding it in the near wake.

Investigation of turbulent boundary layer flows with adverse pressure gradient by means of 3D Lagrangian particle tracking with Shake-The-Box

A large-scale 3D Lagrangian particle tracking (LPT) investigation of a turbulent boundary layer (TBL) flow developing across different pressure gradient regions is presented in this study. Three high-speed multi-camera imaging systems, LED illumination and helium-filled soap bubbles (HFSB) tracers have been adopted to produce time-resolved sequences of particle images over a large volume encompassing approximately 3 m in the streamwise direction, 0:8 m in the spanwise direction and 0:25 m in the wall-normal direction. Individual tracers have been reconstructed and tracked within the imaged volume by means of the Shake-The-Box algorithm (STB, Schanz et al. (2016)); the FlowFit data assimilation algorithm (Gesemann et al. (2016)) has been used to evaluate the spatial velocity gradients and to interpolate the scattered LPT results onto a regular grid. Thanks to the large size of the investigated volume and to the time-resolved nature of the recorded images, the entire spatial extent of the large-scale coherent motions within the logarithmic region of the TBL (i.e. superstructures) could be captured and their dynamics investigated during their development over several boundary layer thickness in the streamwise direction, from the zero pressure gradient region (ZPG) to the adverse pressure gradient region (APG). Two free-stream velocities were investigated, namely 7 and 14m=s, corresponding to Ret ~ 3,000 and 5,000 respectively. The results confirm the location and scale of the elongated high- and low-momentum structures in the logarithmic region, as well as their meandering in the spanwise direction. Two-point correlation statistics show that the width and spacing of the superstructures are not affected by the transition from the ZPG to the APG region. The analysis of the instantaneous flow realizations from both a Lagrangian and Eulerian perspective indicates the presence of significant fluid particle elements exchange across the interfaces of the large-scale structures.

Integral measures of the zero pressure gradient boundary layer over the Reynolds number range ≤Rτ<∞

A recently developed mixing length model of the turbulent shearing stress has been shown to generate a universal velocity profile that provides an accurate approximation to incompressible pipe flow velocity profiles over a wide Reynolds number range [B. J. Cantwell, “A universal velocity profile for smooth wall pipe flow,” J. Fluid Mech. 878, 834–874 (2019)]. More recently, the same profile was shown to accurately approximate velocity profiles in channel flow, the zero pressure gradient boundary layer, and the boundary layer in an adverse pressure gradient [M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows,” AIAA Paper No. 2021-0061, 2021 and M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers,” J. Fluid Mech. (unpublished) (2021)] The universal velocity profile is uniformly valid from the wall to the free stream at all Reynolds numbers from zero to infinity. At a low Reynolds number, the profile approaches the laminar channel/pipe flow limit. The primary measure of the Reynolds number used in this work is the friction Reynolds number Rτ=uτδ/ν. It is a little unusual to use Rτ for the boundary layer since it requires that the velocity profile be cutoff using an arbitrarily defined overall boundary layer thickness, δ. Because of the slow approach of the velocity to the free stream, different conventions used to define the thickness lead to different values of Rτ assigned to a given flow. It will be shown in this paper that, through its connection to channel/pipe flow, the universal velocity profile can be used to define a practically useful, unambiguous, measure of overall boundary layer thickness, called here the equivalent channel half height, δh. For Rτ>≈5000, the universal velocity profile defines a Reynolds number independent shape function that can be used to generate explicit expressions for the infinite Reynolds number behavior of all the usual integral boundary layer measures; displacement thickness, momentum thickness, energy thickness, overall boundary layer thickness, and skin friction. The friction coefficient Cf(Rδ2) generated by the universal velocity profile accurately approximates data over a wide range of momentum thickness Reynolds numbers collected by Nagib et al. [“Can we ever rely on results from wall-bounded turbulent flows without direct measurements of wall shear stress?,” AIAA Paper No. 2004-2392, 2004]. The universal velocity profile is used to integrate the von Kaŕmań boundary layer integral equation [T. von Kármán, “Uber laminaire und turbulente reibung,” Z. Angew. Math. Mech. 1, 233–252 (1921)] in order to generate the various thicknesses and friction velocity as functions of the spatial Reynolds number, Rx=uex/ν.

Investigation on the internal flow of ventilated partial cavity

This study presents investigations on the internal flow behaviors and mechanisms of the ventilated partial cavity created by air ventilation behind a backward-facing step. Both planar particle image velocimetry experiments and numerical simulations have been conducted over a range of freestream velocities (U∞) for varying ventilation rates (QAs). The experimental and numerical internal flows are reasonably concordance, revealing three distinct internal flow regions: the ventilation region, the entrained internal boundary layer region, and the reversed region sandwiched in-between. The three-dimensional internal flow structures and the internal pressure gradients are numerically revealed. The internal recirculation vortex in the ventilation region is found to be induced by the stream-wise adverse pressure gradients, while the transverse airflow near the closure of the cavity is attributed to the presentence of the span-wise pressure gradient. Based on the internal velocity and air flux profiles, and the corresponding internal boundary layer thickness, the entrained and reversed air fluxes are revealed to initially increase, attain a maximum, and then decrease along the cavity. For the increment of U∞, both entrained and reversed air fluxes increase. For the increment of QAs, the entrained air flux rises, while the reversed air flux declines. Such feature is further verified and explained by applying the Couette flow model with adverse pressure gradients, which provides a reasonable estimation of internal flow profiles and offers a theoretical explanation on the change of internal air flux due to the variation of U∞ and QAs.

Multi-resolution, time-resolved PIV measurements of a Decelerating Turbulent Boundary Layer near Separation

We report on measurements of the time-evolving velocity profile of a turbulent boundary layer subjected to a strong adverse pressure gradient (APG) at Reynolds numbers up to Reθ ≈ 55 000 with an upstream friction Reynolds number exceeding Reτ ≈ 10 000. Near the point of flow separation high-resolution imaging at high camera frame rates captured the time evolving velocity profile using the so-called “profile-PIV” technique in a nested imaging configuration of two cameras operating at different image magnifications. One camera used an image magnification better than unity to resolve the viscous scales directly at the wall while the remainder of the roughly 200 mm thick boundary layer is simultaneous captured by the second camera. In the APG the variance of the stream-wise velocity exhibits no “inner peak” commonly found in turbulent boundary layers without pressure gradient influence. Spectral analysis further shows that the peak energy within the boundary layer shifts away from the wall toward lower frequencies. The overlap between the simultaneously imaged areas allows to assess and, to first order, correct for the effect of spatial smoothing on statistical quantities, spectra and related quantities. A multi-frame cross-correlation algorithm was used to process the extensive data base. In addition, a newly developed 2D-2C “Shake-The-Box” algorithm (STB) provided highly resolved particle tracking data beyond the reach of conventional PIV processing.

Investigation of Large Scale Motions in Zero and Adverse Pressure Gradient Turbulent Boundary Layers Using High-Spatial-Resolution PIV

Particle image velocimetry (PIV) has been used to capture the high-spatial-resolution (HSR) two-component, two-dimensional (2C-2D) velocity fields of a zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) and of an adverse-pressure-gradient (APG) TBL. Proper Orthogonal Decomposition (POD) is performed on the measured velocity fields to characterize the velocity fields as large or small scale motions (LSMs or SSMs), with further characterisation of the LSMs into high and low momentum events. This paper reports the findings of the PIV experiment and the subsequent analysis of the high Reynolds number ZPG and APG TBLs

Fluorescent PIV using Atomized Liquid Particles

It is shown that aerosolized fluorescent particles generated using a Venturi-type atomizer, from a solution of fluorescent Kiton Red 620 dye in a water/glycol fluid, provide effective flow seeding for fluorescent PIV. The atomized liquid particles were found to be of acceptable size for PIV purposes, with 92% of detected particles by number concentration measuring < 1 μm in diameter. A PIV application was conducted in a wind tunnel (freestream velocity U∞ = 27 m/s), using the particles for measurement of the boundary layer flow approaching a forward-facing step (approach boundary layer momentum thickness Reynolds number of Reθ = 5930), to identify potential benefits in near-wall regions normally affected by unwanted laser reflections from tunnel surfaces. Particles were generated from solutions with dye molar concentrations of 2.5 × 10−3 and 1.0 × 10−2 mol/L, and PIV images were obtained for both elastic Mie scattering and filtered, Stokes-shifted fluorescent light. Raw images indicate that the fluorescence yield of the 1.0 × 10−2 mol/L solution provides PIV images with high contrast, even in the near-surface regions where Mie scattering images are highly affected by surface reflections. Boundary layer profiles are processed in the adverse pressure gradient region leading up to the forward-facing step, where the fluorescent PIV performed comparably to the most optimized Mie scattering PIV; both obtained data as near to the wall as 30 μm, or 2 viscous wall units in our flow of interest. These results indicate that the new seeding method holds excellent promise for near-surface measurement applications with more complicated three-dimensional geometries, where it is impossible to arrange PIV cameras to reject surface-scattered light.

Boundary layer separation and local behavior for the Steady Prandtl equation

In this paper, we prove the boundary layer separation of the steady Prandtl equation for Oleinik's data ensuring the local existence of the solution under suitable adverse pressure gradient, and confirm Goldstein's hypothesis on the separation rate. We also study the local behavior of the solution near the separation point, which is related to Open problem 5 proposed by Oleinik and Samokhin (P.501, [25]).