Numerical calculation of transitional boundary layers

A computational procedure for compressible axisymmetric boundary layers, on bodies of revolution, in transition from laminar to turbulent flow, is introduced. The procedure is an extension of a former method, due to Patankar and Spalding. The flow field is computed by solution of four simultaneous equations for the momentum, the thermal energy, the turbulence energy amplitude and the turbulent scale. The results show good agreement with existing theoretical and experimental data.     

Computation of internal turbulent flow with a large separated flow region

An implicit two-equation turbulence solver, KEM. in generalized co-ordinates, is used in conjunction with the three-dimensional incompressible Navier–Stokes solver, INS3D, to calculate the internal flow in a channel and a channel with a sudden 2:3 expansion. A new and consistent boundary procedure for a low Reynolds number form of the κ-ε turbulence model is chosen to integrate the equations up to the wall. The high Reynolds number form of the equations is integrated using wall functions. The latter approach yields a faster convergence to the steady-state solution than the former. For the case of channel flow, both the wall-function and wall-boundary-condition approaches yield results in good agreement with the experimental data. The back-step (sudden expansion) flow is calculated using the wall-function approach. The predictions are in reasonable agreement with the experimental data.     

Self-preservation of rough-wall turbulent boundary layers

It is proposed that all fully rough-wall boundary layers should satisfy self-preservation more closely than a smooth-wall boundary layer. Previous work has shown that the self-preserving forms of the momentum and turbulent kinetic energy equations for a zero pressure gradient turbulent boundary layer, at sufficiently high Reynolds number, require that the wall shear stress is constant with x, and the layer thickness increases linearly with x. Measurements in two rough wall boundary layers suggest these conditions are met without assuming a form for the mean velocity distribution, and are more likely to exist in a fully rough wall layer than a smooth wall layer.     

Bursting frequency in turbulent boundary layers

Employing laser Doppler anemometry and VITA techniques, the bursting frequency in turbulent boundary layers has been measured over the Reynolds-number range 320 to 1470. The result indicates that the mean and non-dimensional bursting frequency scaled with the variables appropriate for the wall region was constant and independent of Reynoids number. When the same data are plotted using the outer variables of boundary layer to normalize the bursting frequency, the non-dimensional frequency increases as the Reynolds number increases. This is in agreement with the results of Blackwelder et al. (1983) who used hot wire anemometry and VITA technique. The project is supported by the National Natural Science Foundation of China     

Properties of the streamwise velocity fluctuations in the inertial layer of turbulent boundary layers and their connection to self-similar mean dynamics

Well-resolved streamwise velocity measurements are used to investigate three measures of self-similarity in the spatial inertial sublayer of turbulent boundary layers. The emergence of self-similarity in the inertial sublayer requires a high Reynolds number, and thus a relatively wide range of ${\delta }^{+}=\delta u_{\tau }/\nu$ $(1400\lesssim {\delta }^{+}\lesssim 20\text{,}000)$ is explored. The measures investigated include the Kullback–Leibler divergence (KLD) used in turbulent flow analysis by Tsuji et al. (2005), the logarithmic decrease of the even statistical moments studied by Meneveau and Marusic (2013), and the diagnostic plot of Alfredsson and Örlü (2010). These measures are compared with the analyses of Fife et al. (2005) that determine and exploit an invariant form of the mean momentum equation. A primary focus is on domain(s) where the self-similar behaviors are analytically predicted and empirically observed. The present findings indicate that the approximately constant KLD and approximately logarithmic moment profiles reside in a region that is interior to the bounds of the self-similar inertial domain associated with the mean momentum equation. Conversely, the bounds of the self-similar region on the diagnostic plot correspond closely to the theoretically estimated bounds. Results are briefly discussed relative to Townsend’s notion of outer layer similarity, and, on the inertial domain, the physical existence of uniform momentum zones segregated by narrow vortical fissures.     

Prediction of periodic boundary layers

A relatively simple, yet efficient and accurate finite difference method is developed for the solution of the unsteady boundary layer equations for both laminar and turbulent flows. The numerical procedure is subjected to rigorous validation tests in the laminar case, comparing its predictions with exact analytical solutions, asymptotic solutions, and/or experimental results. Calculations of periodic laminar boundary layers are performed from low to very high oscillation frequencies, for small and large amplitudes, for zero as well as adverse time-mean pressure gradients, and even in the presence of significant flow reversal. The numerical method is then applied to predict a relatively simple experimental periodic turbulent boundary layer, using two well-known quasi-steady closure models. The predictions are shown to be in good agreement with the measurements, thereby demonstrating the suitability of the present numerical scheme for handling periodic turbulent boundary layers. The method is thus a useful tool for the further development of turbulence models for more complex unsteady flows.     

Hydrodynamic structure of accelerated turbulent boundary layers

The aim of the paper is to determine the velocity profile and friction law in turbulent boundary layers that develop under conditions of a negative longitudinal pressure gradient (dP/dx < 0). In contrast to the numerous studies devoted to this problem and based on semi-empirical closure of the hydrodynamic equations, general expressions (containing, of course, some empirical coefficients) will be obtained on the basis of dimensional and similarity arguments alone. In this sense, the results of the paper are a natural continuation of the analysis of decelerated turbulent wall flows by Kader and Yaglom [1, 2]. It is shown that the general dependences found in this manner agree well with numerous experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 29–37, May–June, 1983.I thank A. M. Yaglom for his interest in the work and valuable advice during it.     

Near-wall structure of three-dimensional turbulent boundary layers

 Modifications to near-wall turbulent boundary layer structure with increased three-dimensionality have been investigated through the use of hydrogen bubble wire flow visualization. Results indicate that three-dimensionality does not influence the strength or sign of near-wall streamwise vortices. Increased three-dimensionality does stabilize the near-wall structure resulting in less ejection type activity. The spanwise spacing between low-speed streaks also decreased slightly with increased cross-flow. Received: 15 October 1996/Accepted: 2 April 1997     

Regularization modeling for LES of separated boundary layer flow

Regularization models for the turbulent stress tensor are applied to mixing and separated boundary layers. The Leray and the NS-α models in large-eddy simulation (LES) are compared to direct numerical simulation (DNS) and (dynamic) eddy-viscosity models. These regularization models are at least as accurate as the dynamic eddy-viscosity model, and can be derived from an underlying dynamic principle. This allows one to maintain central transport properties of the Navier-Stokes equations in the model and to extend systematically toward complex applications. The NS-α model accurately represents the small-scale variability, albeit at considerable resolution. The Leray model was found to be much more robust, allowing simulations at high Reynolds number. Leray simulations of a separated boundary layer are shown for the first time. The strongly localized transition to turbulence that arises under a blowing and suction region over a flat plate was captured accurately, quite comparable to the dynamic model. In contrast, results obtained with the Smagorinsky model, either with or without Van Driest damping, yield considerable errors, due to its excessive dissipation.     

The control of turbulent end-wall boundary layers using surface suction

An experimental evaluation of the effects of spatially-limited (i.e. localized) surface suction on a turbulent junction flow was performed using Particle Image Velocimetry (PIV). The results indicate that surface suction can (1) weaken both the instantaneous turbulent vortex and its associated surface interactions in the symmetry plane, (2) effectively eliminate the presence of the average turbulent necklace vortex in the symmetry plane, and (3) weaken the average downstream extensions of the vortex. It was also established that suction effectively reduces the low frequency component of the Reynolds-stress in both the symmetry plane and trailing-edge cross-stream planes, and stabilizes the behavior of the trailing vortex legs. Received: 18 May 1998/Accepted: 26 March 1999     

Bubble migration in a turbulent boundary layer

The wall void peaking distribution observed in an upward turbulent bubbly boundary layer along a flat plate is generated by bubbles that move towards the plate, come into contact with the wall and then slide along it. This transverse ‘migration’ has been studied using flow visualization, high speed video and particle tracking techniques to measure the trajectories of mono-disperse air bubbles at very low void fractions. Investigations have been performed at four Reynolds numbers in the range 280 < Re_θ < 3000, covering both the laminar and turbulent regimes, with mono-disperse bubbles of mean equivalent diameter between 2 mm and 6 mm. Lagrangian statistics calculated from hundreds of trajectories show that the migration only occurs in the turbulent regime and for bubble diameters below some critical value: 3.5 mm < d_eqcrit < 4 mm. Above this size (We > 3), the interface deformation is such that bubbles do not remain at the wall, even when they are released at the surface. Also, bubble migration is shown to be non-systematic, to have a non-deterministic character in the sense that trajectories differ significantly, to increase with Reynolds number and to take place on a short time scale. A series of experiments with isolated bubbles demonstrates that these results are not influenced by bubble–bubble interactions and confirm that two-way coupling in the flow is limited. Flow visualizations show that the migration originates with the capture of bubbles inside the large turbulent structures of the boundary layer (‘bulges’). The bubbles begin to move towards the wall as they cross these structures, and the point at which they reach the wall is strongly correlated with the position of the deep ‘valleys’ which separate the turbulent ‘bulges’. The analysis of the mean Lagrangian trajectories of migrating bubbles confirms these observations. Firstly, the average time of migration calculated from these trajectories coincides with the mean transit time of the bubbles across the structures. Secondly, once the trajectories have been scaled by this transit time and the boundary layer thickness δ, they all have the same form in the region y/δ < 0.4, independent of the Reynolds number.     

The boundary layers of an unsteady incompressible stagnation-point flow with mass transfer

In the current work, the boundary layers of an unsteady incompressible stagnation-point flow with mass transfer were further investigated. Similarity transformation technique was used and the similarity equation group was solved using numerical methods. Interesting observation is that there are multiple solutions seen for negative unsteadiness parameters, β. The influences of mass transfer, unsteadiness parameter, and Prandtl numbers on velocity and temperature profiles, wall drag, and wall heat fluxes were investigated and analyzed. The asymptotic behaviors for the similarity equations in limiting situations were theoretically analyzed. It is found that solutions exist for all mass transfer parameters for β≥−1. For a certain mass transfer parameter, there are two solutions when β_c<β<0; there is one solution for (β=β_c)∪(β≥0); there is no solution for β<β_c, where β_c is a critical unsteadiness parameter dependent on mass transfer parameter.     

Periodicity of large-scale coherence in turbulent boundary layers

This study examines the pronounced periodicity of large-scale coherent structures in turbulent boundary layers, which are of the order of the boundary layer thickness (δ) and reside in the logarithmic and wake regions. To this end, a series of multi-camera planar particle image velocimetry (PIV) measurements are conducted in a streamwise/spanwise and streamwise/wall-normal planes at a friction Reynolds number of Re_τ ≈ 2500. The experiments are configured to capture a large field-of-view with velocity fields that cover a streamwise extent in excess of 15δ. The resulting vector fields reveal large-scale streamwise and spanwise organisation instantaneously, which is often lost when only examining mean statistics. By extracting the dominant streamwise and spanwise Fourier modes of the large-scale motions, a clearer picture of these structural organisations and coherence is presented. A targeted inspection of these dominant modes reveal that these features remain coherent over a significant fraction of the boundary layer thickness in the wall-normal direction, but only a fraction of them have coherence that extends to the wall (wall-coherent). Further, the spatial extents and the population density of these wall-coherent and wall-incoherent modes are characterised, with the former conforming to the attached eddy arguments of Townsend (1976) and the subsequent attached eddy models. Collectively, through the evidence gathered here, we provide a conceptual picture of the representative large-scale structures in turbulent boundary layers, which are likely to have implications on the type of representative structures to be used in structure-based models for these flows.     

Transitionally rough zero pressure gradient turbulent boundary layers

Near-wall measurements are performed to study the effects of surface roughness and viscous shear stresses on the transitionally rough regime (5 < k ⁺ < 70) of a zero pressure gradient turbulent boundary layer. The x-dependence is known from the eleven consecutive measurements in the streamwise direction, which allows for the computation of the streamwise gradients in the boundary layer equations. Thus, the skin friction is computed from the integrated boundary layer equation with errors of 3 and 5% for smooth and rough, respectively. It is found that roughness destroys the viscous layer near the wall, thus, reducing the contribution of the viscous stress in the wall region. As a result, the contribution in the wall shear stress due to form drag increases, while the viscous stress decreases. This yields Reynolds number invariance in the skin friction as k ⁺ increases into the fully rough regime. Furthermore, the roughness at the wall reduces the high peak of the streamwise component of the Reynolds stress in the near-wall region. However, for the Reynolds wall-normal and shear stress components, its contribution is not significantly altered for sand grain roughness.     

Near critical phenomena in laminar boundary layers

Asymptotic analysis of boundary layer separation in the limit of large Reynolds number Re→∞ has shown that in a number of cases which are of importance from a practical point of view solutions of the resulting interaction equations describing two-dimensional (2-D) steady flows exist up to a limiting value Γ_c of the relevant controlling parameter Γ only while two branches of solutions exist in a regime Γ<Γ_c. The present study aims at a better understanding of near critical flows |Γ-Γ_c|→0 and in particular the changes of the flow behaviour associated with the passage of Γ through Γ_c.     

Visualization of the structure of supersonic turbulent boundary layers

A series of flow visualizations has been performed on two flat-plate zero-pressure-gradient supersonic boundary layers. The two different boundary layers had moderate Mach numbers of 2.8 and 2.5 and Re 's of 82, 000 and 25, 000 respectively. A number of new visualization techniques were applied. One was a variation of conventional schlieren employing selective cut-off at the knife edge plane. Motion pictures of the flow were generated with this technique. Droplet seeding was also used to mark the flow, and high speed movies were made to show structure evolution. Still pictures were also taken to show details within the large-scale motions. Finally, Rayleigh scattering was used to construct planar images of the flow. Together, these techniques provide detailed information regarding the character and kinematics of the large-scale motions appearing in boundary layers in supersonic flow. Using these data, in concert with existing hot-wire data, some suggestions are made regarding the characteristics of the average large-scale motion.This work was supported by the Air Force Office of Scientific Research under Grant 89-0120, monitored by Dr. James M. McMichael. Also, the authors wish to thank Prof. R. B. Miles for his contributions to the Rayleigh scattering portion of this project.