High Reynolds Number Boundary Layer Research Articles

Prandtl boundary layers for the Phan-Thien Tanner and Giesekus fluid

The Prandtl equations, arising naturally in the description of high Reynolds number boundary layers, have turned out to be quite difficult from the point of view of mathematical analysis. Recent work by the author has shown that the analogous problem for the upper-convected Maxwell fluid is actually better behaved, and the well-posedness of the boundary layer equations has been established. In this paper, boundary layers for the Phan-Thien–Tanner and Giesekus fluid are considered. It turns out that there are two fundamentally different types of boundary layers, which we shall call elastic and viscometric boundary layers. The elastic boundary layers will be analyzed in an analogous fashion as those for the upper-convected Maxwell fluid. On the other hand, for viscometric boundary layers, which occur only for the PTT fluid, the equations are equivalent to those for a power law fluid.

An improved dynamic non-equilibrium wall-model for large eddy simulation

A non-equilibrium wall-model based on unsteady 3D Reynolds-averaged Navier-Stokes (RANS) equations has been implemented in an unstructured mesh environment. The method is similar to that of the wall-model for structured mesh described by Wang and Moin [Phys. Fluids 14, 2043–2051 (2002)], but is supplemented by a new dynamic eddy viscosity/conductivity model that corrects the effect of the resolved Reynolds stress (resolved turbulent heat flux) on the skin friction (wall heat flux). This correction is crucial in predicting the correct level of the skin friction. Unlike earlier models, this eddy viscosity/conductivity model does not have a stress-matching procedure or a tunable free parameter, and it shows consistent performance over a wide range of Reynolds numbers. The wall-model is validated against canonical (attached) transitional and fully turbulent flows at moderate to very high Reynolds numbers: a turbulent channel flow at Reτ = 2000, an H-type transitional boundary layer up to Reθ = 3300, and a high Reynolds number boundary layer at Reθ = 31 000. Application to a separated flow over a NACA4412 airfoil operating close to maximum lift is also considered to test the performance of the wall-model in complex non-equilibrium flows.

Multiscale analysis of fluxes at the turbulent/non-turbulent interface in high Reynolds number boundary layers

Analysis of fluxes across the turbulent/non-turbulent interface (TNTI) of turbulent boundary layers is performed using data from two-dimensional particle image velocimetry (PIV) obtained at high Reynolds numbers. The interface is identified with an iso-surface of kinetic energy, and the rate of change of total kinetic energy (K) inside a control volume with the TNTI as a bounding surface is investigated. Features of the growth of the turbulent region into the non-turbulent region by molecular diffusion of K, viscous nibbling, are examined in detail, focussing on correlations between interface orientation, viscous stress tensor elements, and local fluid velocity. At the level of the ensemble (Reynolds) averaged Navier-Stokes equations (RANS), the total kinetic energy K is shown to evolve predominantly due to the turbulent advective fluxes occurring through an average surface which differs considerably from the local, corrugated, sharp interface. The analysis is generalized to a hierarchy of length-scales by spatial filtering of the data as used commonly in Large-Eddy-Simulation (LES) analysis. For the same overall entrainment rate of total kinetic energy, the theoretical analysis shows that the sum of resolved viscous and subgrid-scale advective flux must be independent of scale. Within the experimental limitations of the PIV data, the results agree with these trends, namely that as the filter scale increases, the viscous resolved fluxes decrease while the subgrid-scale advective fluxes increase and tend towards the RANS values at large filter sizes. However, a definitive conclusion can only be made with fully resolved three-dimensional data, over and beyond the large dynamic spatial range presented here. The qualitative trends from the measurement results provide evidence that large-scale transport due to the energy-containing eddies determines the overall rate of entrainment, while viscous effects at the smallest scales provide the physical mechanism ultimately responsible for entrainment. Data spanning over a decade in Reynolds number suggest that the fluxes (or the entrainment velocity) scale with the friction velocity (or equivalently the local turbulent fluctuating velocity), whereas Taylor microscale and boundary-layer thickness are the appropriate length scales at small and large filter sizes, respectively.

Response to “Comment on ‘Experimental study of skin friction drag reduction on superhydrophobic flat plates in high Reynolds number boundary layer flow’” [Phys. Fluid 25, 079101 (2013)

Response to “Comment on ‘Experimental study of skin friction drag reduction on superhydrophobic flat plates in high Reynolds number boundary layer flow’” [Phys. Fluid 25, 079101 (2013)

Comment on “Experimental study of skin friction drag reduction on superhydrophobic flat plates in high Reynolds number boundary layer flow” [Phys. Fluids 25, 025103 (2013)

First Page

Experimental study of skin friction drag reduction on superhydrophobic flat plates in high Reynolds number boundary layer flow

In this paper, we report the measurement of skin friction drag on superhydrophobic-coated flat plates in high Reynolds (Re) number boundary layer flows, using a high-speed towing tank system. Aluminum flat plates with a large area (4 feet × 2 feet, 3/8 in. thick) and sharpened leading/trailing edges (1 in. long) were prepared as a boundary layer flow model. Spray coating of hydrophobic nanoparticles was applied to make two different types of superhydrophobic coatings: one with low contact angle and high contact angle hysteresis, and the other with high contact angle and low contact angle hysteresis. Skin friction drag of the superhydrophobic plates was measured in the flow speed up to 30 ft/s to cover transition and turbulent flow regimes (105 < ReL < 107), and was compared to that of an uncoated bare aluminum plate. A significant drag reduction was observed on the superhydrophobic plate with high contact angle and low contact angle hysteresis up to ∼30% in transition regime (105 < ReL < 106), which is attributed to the shear-reducing air layer entrapped on the superhydrophobic surface. However, in fully turbulence regime (106 < ReL < 107), an increase of drag was observed, which is ascribed to the morphology of the surface air layer and its depletion by high shear flow. The texture of superhydrophobic coatings led to form a rugged morphology of the entrapped air layer, which would behave like microscale roughness to the liquid flow and offset the drag-reducing effects in the turbulent flow. Moreover, when the superhydrophobic coating became wet due to the removal of air by high shear at the boundary, it would amplify the surface roughness of solid wall and increase the drag in the turbulent flow. The results illustrate that drag reduction is not solely dependent on the superhydrophobicity of a surface (e.g., contact angle and air fraction), but the morphology and stability of the surface air layer are also critical for the effective drag reduction using superhydrophobic surfaces, especially in high Re number turbulent flow regimes.

Grid-point requirements for large eddy simulation: Chapman’s estimates revisited

Resolution requirements for large eddy simulation (LES), estimated by Chapman [AIAA J. 17, 1293 (1979)], are modified using accurate formulae for high Reynolds number boundary layer flow. The new estimates indicate that the number of grid points (N) required for wall-modeled LES is proportional to ReLx, but a wall-resolving LES requires ÑReLx13/7, where Lx is the flat-plate length in the streamwise direction. On the other hand, direct numerical simulation, resolving the Kolmogorov length scale, requires ÑReLx37/14.

Very high Reynolds number boundary layers over 3D sparse roughness and obstacles: the mean flow

Hot-wire and oil-film interferometry measurements are taken for 3D rough wall boundary layers at very high Reynolds numbers (61,000 < Re θ < 120,000) with low blockage ratios, 10 < δ/H < 135, and high roughness, 100 < H + < 4,900. The results cover flows over both rough walls and over obstacles and are compared with and provide extension to recent lower Reynolds number results. The validity of the Townsend ‘wall similarity hypothesis’ in relation to consistently increasing 3D roughness is interrogated. In agreement with recent work, Schultz and Flack (J Fluid Mech 580:381–405, 2007) and Castro (J Fluid Mech 585:469–485, 2007) found that, for relatively low roughness, Townsend’s hypothesis holds for the mean velocity field. With increasing roughness, the equilibrium layer diminishes and gradually vanishes. The viscous component of the wall shear stress decreases, while the turbulent component increases as the roughness effects extend across the boundary layer.

Error-Landscape Assessment of Large-Eddy Simulations: A Review of the Methodology

A review is presented of the error-landscape methodology. This approach evaluates the error-response surface of large-eddy simulations (LES) to essential model and numerical parameters by a systematic variation of these parameters. Using an error landscape constructed for LES of decaying homogeneous isotropic turbulence, it is shown that the determination of LES quality based on one error measure alone, can lead to misleading results, related to underlying error-balancing mechanisms. This problem can be avoided by considering a range of errors simultaneously, emphasizing different scales in the solution. Subsequently, the error-landscape method is further illustrated by comparing different numerical discretizations for Smagorinsky LES. Finally, a more complex case, i.e. a high (infinite) Reynolds number boundary layer, is considered.

Thin water films driven by air shear stress through roughness

A model of thin water film transport over small scale surface roughness is developed in the context of a high Reynolds number boundary layer theory. The surface water is found to be described by a lubrication equation. It is shown that small scale surface roughness can first effect the water flow at roughness heights which are much less than those of first nonlinear response in the air. A number of well known phenomenon are encountered when using this model, such as pooling of water between roughness elements and rivulet formation. A linearized subsonic heat transfer analysis is also presented, and water protuberances and roughness are found to enhance the ambient heat flux. Solitons are calculated for two-dimensional films, and a linear stability analysis shows that two-dimensional film fronts can become unstable and develop into rivulets.

Some predictions of the attached eddy model for a high Reynolds number boundary layer

Many flows of practical interest occur at high Reynolds number, at which the flow in most of the boundary layer is turbulent, showing apparently random fluctuations in velocity across a wide range of scales. The range of scales over which these fluctuations occur increases with the Reynolds number and hence high Reynolds number flows are difficult to compute or predict. In this paper, we discuss the structure of these flows and describe a physical model, based on the attached eddy hypothesis, which makes predictions for the statistical properties of these flows and their variation with Reynolds number. The predictions are shown to compare well with the results from recent experiments in a new purpose-built high Reynolds number facility. The model is also shown to provide a clear physical explanation for the trends in the data. The limits of applicability of the model are also discussed.

A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows

A scale-dependent dynamic subgrid model based on Lagrangian time averaging is proposed and tested in large eddy simulations (LES) of high-Reynolds number boundary layer flows over homogeneous and heterogeneous rough surfaces. The model is based on the Lagrangian dynamic Smagorinsky model in which required averages are accumulated in time, following fluid trajectories of the resolved velocity field. The model allows for scale dependence of the coefficient by including a second test-filtering operation to determine how the coefficient changes as a function of scale. The model also uses the empirical observation that when scale dependence occurs (such as when the filter scale approaches the limits of the inertial range), the classic dynamic model yields the coefficient value appropriate for the test-filter scale. Validation tests in LES of high Reynolds number, rough wall, boundary layer flow are performed at various resolutions. Results are compared with other eddy-viscosity subgrid-scale models. Unlike the Smagorinsky–Lilly model with wall-damping (which is overdissipative) or the scale-invariant dynamic model (which is underdissipative), the scale-dependent Lagrangian dynamic model is shown to have good dissipation characteristics. The model is also tested against detailed atmospheric boundary layer data that include measurements of the response of the flow to abrupt transitions in wall roughness. For such flows over variable surfaces, the plane-averaged version of the dynamic model is not appropriate and the Lagrangian averaging is desirable. The simulated wall stress overshoot and relaxation after a jump in surface roughness and the velocity profiles at several downstream distances from the jump are compared to the experimental data. Results show that the dynamic Smagorinsky coefficient close to the wall is very sensitive to the underlying local surface roughness, thus justifying the use of the Lagrangian formulation. In addition, the Lagrangian formulation reproduces experimental data more accurately than the planar-averaged formulation in simulations over heterogeneous rough walls.

A model for the evolution of turbulent bulges

It is shown that vortices rising through a high-Reynolds number boundary layer can interact with the mean shear so as to yield a localized structure similar to the bulges seen in turbulent boundary layers. These bulges evolve so as to entrain irrotational flow into the boundary layer. It is also shown that vortices can initiate the evolution of detrainment, in which rotational fluid is ejected into the outer flow. The results of this work support the idea that horseshoe vortices can lead to the evolution of large-scale structure in the boundary layer.

Packet Structure of Surface Eddies in the Atmospheric Boundary Layer

A smoke visualization experiment has beenperformed in the first 3,m ofneutral and unstable atmospheric boundary layersat very large Reynolds number(ReΘ > 106). Under neutral atmosphericconditions mean wind profiles agreewell with those in the canonical flatplate zero-pressure-gradient turbulentboundary layer. The experiment was designedto minimize the temperaturedifference between the passive marker (smoke)and the air to ensure that anyobserved structures were due to vortical, ratherthan buoyant, motions. Imagesacquired in the streamwise–wall-normal planeusing a planar laser light-sheetare strikingly similar to those observed inlaboratory experiments at low to moderate Reynolds numbers. They reveal large-scaleramp-like structures withdownstream inclination of 3°–35°.This inclination isinterpreted as the hairpin packet growthangle following the hairpin vortexpacket model ofAdrian, Meinhart, and Tomkins.The distribution of this characteristicangle agrees with the results of experiments at far lower Reynolds numbers,suggesting a similarity in structures among low, moderate, and high Reynoldsnumber boundary layers at vastly different scales. These results indicate thatthe hairpin vortex packet model extends over a large range of scales. Theeffect of vertical heat transport in an unstable atmosphere on wall structuresis investigated in terms of the hairpin vortex packet model.

Noise radiated from a rotating submerged elastic cylindrical thin shell

Although the aeroacoustics of high Reynolds number boundary layers is reasonably well understood, less is known about the hydroacoustics of such flows, and the effect of fluid loading. The noise generated by the turbulent boundary layer around an elastic, thin-walled and cylindrical shell rotating in quiescent water was studied in the Georgia Tech. Underwater Acoustic Tank for Reynolds numbers up to 200&amp;lt;th&amp;gt;000. The steel shell, which is filled with air, has a diameter D of 0.625 m, a wall thickness of 0.004D, and an aspect ratio of unity; the tank dimensions are 19D by 12D by 11D. Extraneous noise sources (e.g., bearing and motor vibration) were isolated from the net signal to estimate flow noise. Radiated noise power was calculated from hydrophone data under a diffuse field assumption. To our knowledge, these results are unique in both their structural acoustics and fluid mechanics scaling.

A comparative study of near-wall turbulence in high and low Reynolds number boundary layers

The present study explores the effects of Reynolds number, over three orders of magnitude, in the viscous wall region of a turbulent boundary layer. Complementary experiments were conducted both in the boundary layer wind tunnel at the University of Utah and in the atmospheric surface layer which flows over the salt flats of the Great Salt Lake Desert in western Utah. The Reynolds numbers, based on momentum deficit thickness, of the two flows were Rθ=2×103 and Rθ≈5×106, respectively. High-resolution velocity measurements were obtained from a five-element vertical rake of hot-wires spanning the buffer region. In both the low and high Rθ flows, the length of the hot-wires measured less than 6 viscous units. To facilitate reliable comparisons, both the laboratory and field experiments employed the same instrumentation and procedures. Data indicate that, even in the immediate vicinity of the surface, strong influences from low-frequency motions at high Rθ produce noticeable Reynolds number differences in the streamwise velocity and velocity gradient statistics. In particular, the peak value in the root mean square streamwise velocity profile, when normalized by viscous scales, was found to exhibit a logarithmic dependence on Reynolds number. The mean streamwise velocity profile, on the other hand, appears to be essentially independent of Reynolds number. Spectra and spatial correlation data suggest that low-frequency motions at high Reynolds number engender intensified local convection velocities which affect the structure of both the velocity and velocity gradient fields. Implications for turbulent production mechanisms and coherent motions in the buffer layer are discussed.

Characteristics of subgrid-resolved-scale dynamics in anisotropic turbulence, with application to rough-wall boundary layers

Large-eddy simulation (LES) of the atmospheric boundary layer (ABL) using eddy viscosity subgrid-scale (SGS) models is known to poorly predict mean shear at the first few grid cells near the ground, a rough surface with no viscous sublayer. It has recently been shown that convective motions carry this localized error vertically to infect the entire ABL, and that the error is more a consequence of the SGS model than grid resolution in the near-surface inertial layer. Our goal was to determine what first-order errors in the predicted SGS terms lead to spurious expectation values, and what basic dynamics in the filtered equation for resolved scale (RS) velocity must be captured by SGS models to correct the deficiencies. Our analysis is of general relevance to LES of rough-wall high Reynolds number boundary layers, where the essential difficulty in the closure is the importance of the SGS acceleration terms, a consequence of necessary under-resolution of relevant energy-containing motions at the first few grid levels, leading to potentially strong couplings between the anisotropies in resolved velocity and predicted SGS dynamics. We analyze these two issues (under-resolution and anisotropy) in the absence of a wall using two direct numerical simulation datasets of homogeneous turbulence with very different anisotropic structure characteristic of the near-surface ABL: shear- and buoyancy-generated turbulence. We uncover three important issues which should be addressed in the design of SGS closures near rough walls and we provide a priori tests for the SGS model. First, we identify a strong spurious coupling between the anisotropic structure of the resolved velocity field and predicted SGS dynamics which can create a feedback loop to incorrectly enhance certain components of the predicted velocity field. Second, we find that eddy viscosity and “similarity” SGS models do not contain enough degrees of freedom to capture, at a sufficient level of accuracy, both RS-SGS energy flux and SGS-RS dynamics. Third, to correctly capture pressure transport near a wall, closures must be made more flexible to accommodate proper partitioning between SGS stress divergence and SGS pressure gradient.

High Reynolds number boundary layer chaos.

Using tools for the analysis of nonlinear chaotic observations, we have determined the embedding dimension required for data from measurements of wall pressure fluctuations in high Reynolds number boundary layer flow around an axisymmetric body. The Reynolds number based on the boundary layer size is as large as 10[sup 5], and the number of degrees of freedom excited in the fluid is certainly very large. Yet the active degrees of freedom seen by the pressure sensors in the transitional and turbulent parts of the flow are as low as 8 to 10. We discuss the source of these results in terms of sensor size and the known coherent structures in these shear flows.

Topology and transport properties of large-scale organized motion in a slightly heated rough wall boundary layer

Large-scale temperature discontinuities are detected in a high Reynolds number boundary layer over a slightly heated rough wall. In the streamwise and normal plane of the measurements, the discontinuities form (as for smooth wall boundary layers) the boundaries between successive spanwise vortexlike structures, flattened against the wall. The regions immediately downstream from the discontinuities are particularly important for the transport of heated, low-momentum fluid away from the wall. The proportions contributed by the detected organized motion to the velocity and temperature variances are similar to those in smooth wall boundary layers. The proportions contributed to mean products, especially to the Reynolds shear stress, are significantly higher than over smooth walls at similar Reynolds numbers. The detected organized motion in the rough wall boundary layer resembles that in a smooth wall boundary layer of similar skin friction coefficient (and hence much lower Reynolds number).