Laminar Boundary-layer Profiles Research Articles

A characterization of unsteady effects for transonic turbine airfoil limit loading

To better understand the effect of vortex shedding on the nature of the trailing edge shock system during airfoil limit loading a computational fluid dynamic investigation was performed for a transonic turbine airfoil at sublimit, limit and supercritical conditions. Four modeling strategies were employed: steady state RANS, unsteady RANS, DDES, and turbulence model free. The influence of transient modeling approaches on the predicted mass-flow averaged total pressure loss coefficient, mass-flow averaged flow angles, and on the limit loading pressure ratio were found to be insignificant with the exception of the URANS model during critical loading. It was found that the URANS modeling approach failed to predict the transition from near trailing edge dominated vortex formation to base pressure vortex formation resulting in a drastic rise in predicted total pressure loss. Surface isentropic Mach distributions were predicted similarly for all modeling strategies, with the exception of the trailing edge base pressure region and points of shock impingement along the suction surface. A detailed review of the boundary layer states at the trailing edge was performed. It was found that all of the modeling approaches predicted laminar boundary layer profiles along the pressure surface trailing edge and turbulent profiles along the suction surface. The predicted base pressure distributions were also reviewed, showing the base pressure to decrease with increasing pressure ratio. The unsteady simulation approaches consistently predicted lower average surface pressures than the steady state RANS simulations. Qualitative images of the numerical Schlieren contours were presented and reviewed showing large differences in the prediction of vortex shape, size, and subsequent shock influence.

Generation of Excess Noise by Jets with Highly Disturbed Laminar Boundary-Layer Profiles

The generation of noise by jets with highly disturbed laminar boundary-layer profiles at the nozzle exit, also referred to as initially nominally laminar jets in the literature, is investigated using large-eddy simulation and linear stability analysis. Four jets at a Mach number of 0.9 and a Reynolds number of , one with a nonlaminar boundary-layer profile and three others with laminar profiles, are considered for exit peak turbulence intensities equal to 6% in the nonlaminar case and to 9% in the laminar ones. The jets with laminar boundary-layer profiles all radiate greater sound pressure levels than the jet with a nonlaminar profile but weaker initial disturbances. This particularly appears at high frequencies for the jets with a thinner boundary layer compared with the nonlaminar case. These results are shown to be related to the dependence on the shape of the boundary-layer profile of the most unstable frequencies downstream of the nozzle. For a laminar profile, these frequencies are similar to those obtained downstream in the mixing-layer profiles, whereas they are higher for a nonlaminar profile. Despite larger nozzle-exit flow disturbances, this leads to longer-term persistence of coherent large-scale structures in the shear layers, hence stronger velocity fluctuations and noise levels, for the present initially nominally laminar jets than for the other one.

Laminar boundary layer stability calculation for contoured Mach 6 nozzle

Stability of the laminar boundary layer on the surface of a hypersonic nozzle for the Mach number M = 6 of the Transit-M wind tunnel is calculated. The laminar boundary-layer profiles are obtained by solving the Navier-Stokes equations numerically within the framework of the Ansys Fluent software. N-factors of the Goertler vortices and of the first and second Mack modes are obtained in the approximation of the linear stability theory. It is demonstrated that the Goertler vortices are the most unstable disturbances for the nozzle under consideration. Empirical dependences of the local Reynolds number of the laminar-turbulent transition on the N-factor and unit Reynolds number are determined.

Effects of nozzle-exit boundary-layer profile on the initial shear-layer instability, flow field and noise of subsonic jets

The influence of the nozzle-exit boundary-layer profile on high-subsonic jets is investigated by performing compressible large-eddy simulations (LES) for three isothermal jets at a Mach number of 0.9 and a diameter-based Reynolds number of $5\times 10^{4}$, and by conducting linear stability analyses from the mean-flow fields. At the exit section of a pipe nozzle, the jets exhibit boundary layers of momentum thickness of approximately 2.8 % of the nozzle radius and a peak value of turbulence intensity of 6 %. The boundary-layer shape factors, however, vary and are equal to 2.29, 1.96 and 1.71. The LES flow and sound fields differ significantly between the first jet with a laminar mean exit velocity profile and the two others with transitional profiles. They are close to each other in these two cases, suggesting that similar results would also be obtained for a jet with a turbulent profile. For the two jets with non-laminar profiles, the instability waves in the near-nozzle region emerge at higher frequencies, the mixing layers spread more slowly and contain weaker low-frequency velocity fluctuations and the noise levels in the acoustic field are lower by 2–3 dB compared to the laminar case. These trends can be explained by the linear stability analyses. For the laminar boundary-layer profile, the initial shear-layer instability waves are most strongly amplified at a momentum-thickness-based Strouhal number $St_{\unicode[STIX]{x1D703}}=0.018$, which is very similar to the value obtained downstream in the mixing-layer velocity profiles. For the transitional profiles, on the contrary, they predominantly grow at higher Strouhal numbers, around $St_{\unicode[STIX]{x1D703}}=0.026$ and 0.032, respectively. As a consequence, the instability waves rapidly vanish during the boundary-layer/shear-layer transition in the latter cases, but continue to grow over a large distance from the nozzle in the former case, leading to persistent large-scale coherent structures in the mixing layers for the jet with a laminar exit velocity profile.

Parabolized Stability Equations Code with Automatic Inflow for Swept Wing Transition Analysis

This paper aims at developing a parabolized stability equation transition analysis code coupled to the industrial in-house accurate Reynolds-averaged Navier–Stokes code developed at Israel Aerospace Industries for stability analysis of three-dimensional boundary layers over swept wings. In the first step, the parabolized stability equation derivation in a body-fitted coordinate system is exposed, as well as its classical initialization, using the solution procedure of the global Chebyshev eigenvalue problem for the linear stability equations. This initialization procedure demands large user interactions, and it prevents the parabolized stability equation method to be of practical use in an industrial environment. Therefore, one exposes in the second step a new way to automatically generate the inflow solution of the linear stability equations needed by the parabolized stability equations. This new robust approach paves the way for innovative automatic industrial use of the parabolized stability equation technology without the need to involve the user. This approach supplies the inflow conditions required for the parabolized stability equation marching procedure without the need to solve the standalone linear stability global eigenvalue problem. The solution of the linear stability equations at initialization is now obtained by iteratively converging the local linear stability solver starting from the new inflow empirical approximation obtained in two special coordinate systems for crossflow and Tollmien–Schlichting, respectively. These formulas represent some invariant functions of the stability characteristics of the exact Navier–Stokes laminar boundary-layer profiles instead of self-similar boundary-layer profiles like the Falkner–Skan–Cooke. Standard academic validation test cases and industrial results of the parabolized stability equation/Reynolds-average Navier-Stokes codes on a generic unmanned aerial vehicle swept wing are exposed.

Turbulent Rayleigh–Bénard convection in spherical shells

We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a Prandtl number of unity and for Rayleigh numbers up to $10^{9}$. In this geometry, curvature and radial variations of the gravitational acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from ${\it\eta}=r_{i}/r_{o}=0.2$ to ${\it\eta}=0.95$) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume separation allows us to accurately predict the boundary layer asymmetry for the various spherical shell configurations explored here. The mean temperature and horizontal velocity profiles are in good agreement with classical Prandtl–Blasius laminar boundary layer profiles, provided the boundary layers are analysed in a dynamical frame that fluctuates with the local and instantaneous boundary layer thicknesses. The scaling properties of the Nusselt and Reynolds numbers are investigated by separating the bulk and boundary layer contributions to the thermal and viscous dissipation rates using numerical models with ${\it\eta}=0.6$ and with gravity proportional to $1/r^{2}$. We show that our spherical models are consistent with the predictions of Grossmann & Lohse’s (J. Fluid Mech., vol. 407, 2000, pp. 27–56) theory and that $\mathit{Nu}(\mathit{Ra})$ and $\mathit{Re}(\mathit{Ra})$ scalings are in good agreement with plane layer results.

Identification of the effects of the nozzle-exit boundary-layer thickness and its corresponding Reynolds number in initially highly disturbed subsonic jets

The influence of the nozzle-exit boundary-layer thickness in isothermal round jets at a Mach number of 0.9 and at diameter Reynolds numbers ReD ≃ 5 × 104 is investigated using large-eddy simulations. The originality of this work is that, contrary to previous studies on the topic, the jets are initially highly disturbed, and that the effects of the boundary-layer thickness are explored jointly on the exit turbulence, the shear-layer and jet flow characteristics, and the acoustic field. The jets originate from a pipe of radius r0, and exhibit, at the exit, peak disturbance levels of 9% of the jet velocity, and mean velocity profiles similar to laminar boundary-layer profiles of thickness δ0 = 0.09r0, 0.15r0, 0.25r0, or 0.42r0, yielding 99% velocity thicknesses between 0.07r0 and 0.34r0 and momentum thicknesses δθ(0) between 0.012r0 and 0.05r0. Two sets of computations are reported to distinguish, for the first time to the best of our knowledge, between the effects of the ratio δ0/r0 and of the Reynolds number Reθ based on δθ(0). First, four jets with a fixed diameter, hence at a constant Reynolds number ReD = 5 × 104 giving Reθ = 304, 486, 782, and 1288 depending on δ0, are considered. In this case, due to the increase in Reθ, thickening the initial shear layers mainly results in a weaker mixing-layer development with lower spreading rates and turbulence intensities, and reduced sound levels at all emission angles. Second, four jets at Reynolds numbers ReD between 1.8 × 104 and 8.3 × 104, varying so as to obtain Reθ ≃ 480 in all simulations, are examined. Here, increasing δ0/r0 has a limited impact on the mixing-layer key features, but clearly leads to a shorter potential core, a more rapid velocity decay, and higher fluctuations on the jet axis, and stronger noise in the downstream direction. Similar trends can be expected for high-Reynolds-number jets in which viscosity plays a negligible role.

Numerical study of the stabilisation of boundary-layer disturbances by finite amplitude streaks

Well-resolved large-eddy simulations of passive control of the laminar-turbulent transition process in flat-plate boundary-layer flows are presented. A specific passive control mechanism is studied, namely the modulation of the laminar boundary-layer profile by a periodic array of steady boundary-layer streaks. This has been shown experimentally to stabilise the exponential growth of Tollmien-Schlichting (TS) waves and delay transition to turbulence. Here we examine the effect of the steady modulations on the amplification of different types of disturbances such as TS-waves, stochastic noise and free-stream turbulence. In our numerical simulations, the streaks are forced at the inflow as optimal solutions to the linear parabolic stability equations (PSE), whereas the additional disturbances are excited via volume forcing active within the computational domain. The simulation results show, in agreement with experimental and theoretical studies, significant damping of unstable two-dimensional TS-waves of va...

Prandtl–Blasius temperature and velocity boundary-layer profiles in turbulent Rayleigh–Bénard convection

The shapes of the velocity and temperature profiles near the horizontal conducting plates' centre regions in turbulent Rayleigh–Bénard convection are studied numerically and experimentally over the Rayleigh number range 108 ≲ Ra ≲ 3 × 1011 and the Prandtl number range 0.7 ≲ Pr ≲ 5.4. The results show that both the temperature and velocity profiles agree well with the classical Prandtl–Blasius (PB) laminar boundary-layer profiles, if they are re-sampled in the respective dynamical reference frames that fluctuate with the instantaneous thermal and velocity boundary-layer thicknesses. The study further shows that the PB boundary layer in turbulent thermal convection not only holds in a time-averaged sense, but is most of the time also valid in an instantaneous sense.

Numerical simulation of aerodynamical noise generated by a cylindrical cavity

Cylindrical cavities are mounted at some positions on fuselage and wings of commercial transport aircraft for various service functions. Such cavities were identified as potential airframe noise sources generating high intensity whistles at frequencies situated in the audible range of 0.5‐0.8 kHz. The present paper focuses on the numerical simulation of the far field noise generated by the unsteady flow passing over a cylindrical cavity with an aspect ratio (diameter/height) equal to unity, using a numerical methodology combining a LES and a Kirchhoff integration method. The LES computation was done using ONERA's cell‐center/finite‐volume/CFD code named FLU3M. A laminar boundary layer profile was applied upstream the computational domain, without forcing any turbulence. The LES results were coupled with a 3D Kirchhoff method in the frequency domain. The computed far field noise was compared to dedicated experimental data, showing very good agreement, especially regarding the emission frequency. The final paper will present in detail the involved numerical methods, the computational procedure, and several comparisons to experimental data. This study was completed in the framework of the AEROCAV (AEROacoustique d'une CAVité cylindrique) project funded by the FRAE (Fondation Recherche Aéronautique et Espace).

The influence of compliant surfaces on bypass transition

The objective of the work was to investigate the effect of compliant surfaces on the receptivity and bypass transition of a boundary layer. Hot wire measurements in the pre-transitional and transitional boundary layers on nine different compliant and one rigid surface with identical geometries were made. The experiments were conducted in air and the compliant surfaces were manufactured from gelatine covered by a 10 μm protective PVC film. The laminar boundary layer profiles and growth rate results were the same for all the surfaces. However, the receptivity of the laminar boundary layer to freestream disturbances increased close to the leading edge of each compliant surface. Further downstream the majority of the compliant surfaces were successful in reducing the receptivity to a value below that for the rigid surface. The transition onset position on the compliant surfaces ranged from 3% downstream to 20% upstream of the rigid surface position. It was concluded that compliant surfaces with optimum properties can reduce receptivity and delay transition.

An Experimental Investigation of Nozzle-Exit Boundary Layers of Highly Heated Free Jets

An experimental investigation of the effects of nozzle operating conditions on the development of nozzle-exit boundary layers of highly heated air free jets is reported in this paper. The total pressure measurements in the nozzle-exit boundary layer were obtained at a range of jet Mach numbers from 0.1 to 0.97 and jet total temperatures up to 900 K. The analysis of results shows that the nozzle-exit laminar boundary-layer development depends only on the nozzle-exit Reynolds number. For the nozzle-exit turbulent boundary layer, however, it appears that the effects of the jet total temperature on the boundary-layer integral characteristics are independent from the effect of the nozzle-exit Reynolds number. This surprising finding has not yet been reported. Further, laminar boundary-layer profiles were compared with the Pohlhausen solution for a flat-wall converging channel and an acceptable agreement was found only for low Reynolds numbers. For turbulent boundary layers, the dependence of the shape factor on relative Mach numbers at a distance of one momentum thickness from the nozzle wall resembles Spence’s prediction. Finally, the calculated total pressure loss coefficient was found to depend on the nozzle-exit Reynolds number for the laminar nozzle-exit boundary layer, while for the turbulent exit boundary layer this coefficient appears to be constant.

Flux-difference split algorithm for unsteady thin-layer Navier-Stokes solutions

An efficient flux-difference split scheme for calculating steady and unsteady thin-layer Navier-Stokes solutions for airfoils and wings is presented. Steady solutions for a flat plate laminar boundary-layer profile are correctly modeled with only three grid cells internal to the boundary layer. The flux-difference split algorithm was used to evaluate the scheme for unsteady viscous calculations. The diffusive terms are time lagged in the solution process. The scheme is second-order accurate in space and first-order accurate in time due to the explicit treatment or the diffusive terms

The Measurement of Boundary Layers on a Compressor Blade in Cascade: Part 3—Pressure Surface Boundary Layers and the Near Wake

Using the facility described in Part 1 [23], 11 detailed velocity and turbulence intensity profiles are obtained on the pressure surface of a double circular arc compressor blade in cascade. Two profiles are obtained in the near wake. Laminar boundary layer profiles, which agree well with profiles calculated from Falkner–Skan theory at the local pressure gradient, persist through 57.2 percent chord. The measurements indicate that the onset of transition occurs near 60 percent chord—a value in good agreement with the sublimation flow visualization studies (see Part 1). The lack of a logarithmic region in the data measured at the last chord position (97.9 percent chord) indicates that transition is not complete. The thin laminar boundary layers near the leading edge lead to some measurement problems, which are characterized by large turbulence intensities, in using the laser-Doppler velocimeter (LDV). Close examination of this problem shows that a combination of velocity-gradient broadening and a vibration of the LDV measurement volume causes an elevation of the measured turbulence levels. Fortunately only small errors in mean velocity are introduced. Because of the detached boundary layer on the suction surface, both of the near-wake velocity profiles exhibit regions of backflow. As expected, these near-wake velocity profiles do not exhibit similarity when tested against criteria derived for the far wake.

The Measurement of Boundary Layers on a Compressor Blade in Cascade: Part 1—A Unique Experimental Facility

A unique cascade facility is described which permits the use of laser-Doppler velocimetry (LDV) to measure blade boundary layer profiles. Because of the need for a laser access window, the facility cannot reply on continuous blade pack suction to achieve two-dimensional, periodic flow. Instead, a strong suction upstream of the blade pack is used in combination with tailboards to control the flow field. The distribution of the upstream suction is controlled through a complex baffling system. A periodic, two–dimensional flow field is achieved at a chord Reynolds number of 500,000 and an incidence angle of 5 deg on a highly loaded, double circular arc, compressor blade. Inlet and outlet flow profiles, taken using five-hole probes, and the blade static-pressure distribution are used to document the flow field for use with the LDV measurements (see Parts 2 and 3). Inlet turbulence intensity is measured, using a hot wire, to be 0.18 percent. The static-pressure distribution suggests both separated flow near the trailing edge of the suction surface and an initially laminar boundary layer profile near the leading edge of the pressure surface. Probe measurements are supplemented by sublimation surface visualization studies. The sublimation studies place boundary layer transition at 64.2 ± 3.9 percent chord on the pressure surface, and indicate separation on the suction surface at 65.6 percent ± 3.5 percent chord.

Laminar Separation, Reattachment, and Transition of the Flow Over a Downstream-Facing Step

Results of an experimental investigation of the laminar flow of air over a downstream-facing step are presented. The experiments include visual observations of smoke filaments (in the viscous layer), qualitative velocity fluctuation measurements, and mean velocity profiles. Results are reported over a range of 0.36 – 1.02 cm in step height, 0.61 – 2.44 m/sec in free stream velocity at the step, and 0.16 – 0.51 cm in boundary layer displacement thickness at the step. Laminar flow to reattachment of a free shear layer is observed for subsonic flow and two criteria for which transition to turbulence at reattachment exists are presented. The laminar reattachment length is not a constant number of step heights as for turbulent flow, but varies with Reynolds number and boundary layer thickness at the step. The shape of the velocity profile at reattachment is found to be similar to the shape of a laminar boundary layer profile at separation and the boundary layer profiles downstream of reattachment are similar to those in a laminar boundary layer developing toward separation except that they are traversed in the reverse sense.

Laminar-compressible boundary layer on a blunt body.

T equations for the compressible two-dimensional laminar boundary layer, with heat transfer and arbitrary pressure gradient, originally set forth by Stewartson, have been solved by Cohen and Reshotkofor a number of pressure gradients and wall-to-f reestream stagnation temperature ratios. The purpose of this note is to compare some experimental data with the velocity profiles and displacement thicknesses computed from these solutions for an actual body, with nonsimilar pressure distribution and adiabatic wall. This experiment was conducted during a study of the effects of three-dimensional roughness on transition of the laminar boundary layer on a cooled blunt body. It was necessary during the course of the study to determine accurately the laminar boundary-layer profile and thickness over the body surface; however, a search of the literature failed to provide an experimental verification of the accuracy of using the Cohen and Reshotko solutions to calculate velocity profiles for the conditions of interest. Kemp et al. found that the laminar heat-transfer rates to blunt, highly cooled bodies could be predicted by assuming local similarity, but data for velocity profiles, and for conditions near adiabatic wall temperatures, are not available. The test body used was a hemisphere-cylinder with an isothermal wall temperature equal to the freestream stagnation temperature Tw = T0, in a flow of air having a freestream Mach number of 2.01. This case satisfies approximately the conditions set for the solutions given in Ref. 1: Prandtl number equal to one, linear viscosity-temperature relation across the boundary layer, and an isothermal surface. The solutions were applied in a point-by-point fashion along the surface of the body using the measured static pressure distribution. This pressure distribution, for the hemisphere-cylinder model used in the experiment, can be seen in Fig. 1. The measured Joressure distribution is compared with the Newtonian prediction,