Velocity transformation for compressible wall-bounded turbulent flows with and without heat transfer

In this work, a transformation, which maps the mean velocity profiles of compressible wall-bounded turbulent flows to the incompressible law of the wall, is proposed. Unlike existing approaches, the proposed transformation successfully collapses, without specific tuning, numerical simulation data from fully developed channel and pipe flows, and boundary layers with or without heat transfer. In all these cases, the transformation is successful across the entire inner layer of the boundary layer (including the viscous sublayer, buffer layer, and logarithmic layer), recovers the asymptotically exact near-wall behavior in the viscous sublayer, and is consistent with the near balance of turbulence production and dissipation in the logarithmic region of the boundary layer. The performance of the transformation is verified for compressible wall-bounded flows with edge Mach numbers ranging from 0 to 15 and friction Reynolds numbers ranging from 200 to 2,000. Based on physical arguments, we show that such a general transformation exists for compressible wall-bounded turbulence regardless of the wall thermal condition.

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.

Characteristics of large- and small-scale structures in the turbulent boundary layer over a drag-reducing riblet surface

Riblet is one of the most promising passive drag reduction techniques in turbulent flows. In this paper, hot-wire measurements on a turbulent boundary layer perturbed by a drag-reducing riblet surface are carried out to further understand the riblet effects on the turbulent flows and the drag reduction mechanism. Compared with the smooth case, different energy variations in the near-wall region and the logarithmic region are observed over riblets. Then, by using a spectral filter of a given wavelength, the time series of the hot-wire data are decomposed into large- and small-scale components. It is indicated that large-scale structures in the logarithmic region impose a footprint (amplitude modulating effect) on the near-wall small-scale structures. By quantifying this footprint, it is found that the interactions between large- and small-scale structures over riblets are weakened in the near-wall region. Furthermore, the bursting process of large and small scales is studied. For both large- and small-scale structures, a shorter bursting duration and a higher bursting frequency are observed over the riblet surface, which indicates that riblets impede the formation of large- and small-scale bursting events. The flow physics behind these phenomena are also discussed in detail.

The spatial structure of the logarithmic region in very-high-Reynolds-number rough wall turbulent boundary layers

Using super-large-scale particle image velocimetry (SLPIV), we investigate the spatial structure of the near-wall region in the fully rough atmospheric surface layer with Reynolds number$Re_{\unicode[STIX]{x1D70F}}\sim O(10^{6})$. The field site consists of relatively flat, snow-covered farmland, allowing for the development of a fully rough turbulent boundary layer under near-neutral thermal stability conditions. The imaging field of view extends from 3 m to 19 m above the ground and captures the top of the roughness sublayer and the bottom of an extensive logarithmic region. The SLPIV technique uses natural snowfall as seeding particles for the flow imaging. We demonstrate that SLPIV provides reliable measurements of first- and second-order velocity statistics in the streamwise and wall-normal directions. Our results in the logarithmic region show that the structural features identified in laboratory studies are similarly present in the atmosphere. Using instantaneous vector fields and two-point correlation analysis, we identify vortex structures sharing the signature of hairpin vortex packets. We also evaluate the zonal structure of the boundary layer by tracking uniform momentum zones (UMZs) and the shear interfaces between UMZs in space and time. Statistics of the UMZs and shear interfaces reveal the role of the zonal structure in determining the mean and variance profiles. The velocity difference across the shear interfaces scales with the friction velocity, in agreement with previous studies, and the size of the UMZs scales with wall-normal distance, in agreement with the attached eddy framework.

Wall-attached structures of velocity fluctuations in a turbulent boundary layer

Wall turbulence is a ubiquitous phenomenon in nature and engineering applications, yet predicting such turbulence is difficult due to its complexity. High-Reynolds-number turbulence arises in most practical flows, and is particularly complicated because of its wide range of scales. Although the attached-eddy hypothesis postulated by Townsend can be used to predict turbulence intensities and serves as a unified theory for the asymptotic behaviours of turbulence, the presence of coherent structures that contribute to the logarithmic behaviours has not been observed in instantaneous flow fields. Here, we demonstrate the logarithmic region of the turbulence intensity by identifying wall-attached structures of the velocity fluctuations ($u_{i}$) through the direct numerical simulation of a moderate-Reynolds-number boundary layer ($Re_{\unicode[STIX]{x1D70F}}\approx 1000$). The wall-attached structures are self-similar with respect to their heights ($l_{y}$), and in particular the population density of the streamwise component ($u$) scales inversely with $l_{y}$, reminiscent of the hierarchy of attached eddies. The turbulence intensities contained within the wall-parallel components ($u$ and $w$) exhibit the logarithmic behaviour. The tall attached structures ($l_{y}^{+}>100$) of $u$ are composed of multiple uniform momentum zones (UMZs) with long streamwise extents, whereas those of the cross-stream components ($v$ and $w$) are relatively short with a comparable width, suggesting the presence of tall vortical structures associated with multiple UMZs. The magnitude of the near-wall peak observed in the streamwise turbulent intensity increases with increasing $l_{y}$, reflecting the nested hierarchies of the attached $u$ structures. These findings suggest that the identified structures are prime candidates for Townsend’s attached-eddy hypothesis and that they can serve as cornerstones for understanding the multiscale phenomena of high-Reynolds-number boundary layers.

Large coherence of spanwise velocity in turbulent boundary layers

The spatial signature of spanwise velocity coherence in turbulent boundary layers has been studied using a series of unique large-field-of-view multicamera particle image velocimetry experiments, which were configured to capture streamwise/spanwise slices of the boundary layer in both the logarithmic and the wake regions. The friction Reynolds number of $Re_{\unicode[STIX]{x1D70F}}\approx 2600$ was chosen to nominally match the simulation of Sillero et al. (Phys. Fluids, vol. 26 (10), 2014, 105109), who had previously reported oblique features of the spanwise coherence at the top edge of the boundary layer based on the sign of the spanwise velocity, and here we find consistent observations from experiments. In this work, we show that these oblique features in the spanwise coherence relate to the intermittent turbulent bulges at the edge of the layer, and thus the geometry of the turbulent/non-turbulent interface, with the clear appearance of two counter-oriented oblique features. Further, these features are shown to be also present in the logarithmic region once the velocity fields are deconstructed based on the sign of both the spanwise and the streamwise velocity, suggesting that the often-reported meandering of the streamwise-velocity coherence in the logarithmic region is associated with a more obvious diagonal pattern in the spanwise velocity coherence. Moreover, even though a purely visual inspection of the obliqueness in the spanwise coherence may suggest that it extends over a very large spatial extent (beyond many boundary layer thicknesses), through a conditional analysis, we show that this coherence is limited to distances nominally less than two boundary layer thicknesses. Interpretation of these findings is aided by employing synthetic velocity fields of a boundary layer constructed using the attached eddy model, where the range of eddy sizes can be prescribed. Comparisons between the model, which employs an array of self-similar packet-like eddies that are randomly distributed over the plane of the wall, and the experimental velocity fields reveal a good degree of agreement, with both exhibiting oblique features in the spanwise coherence over comparable spatial extents. These findings suggest that the oblique features in the spanwise coherence are likely to be associated with similar structures to those used in the model, providing one possible underpinning structural composition that leads to this behaviour. Further, these features appear to be limited in spatial extent to only the order of the large-scale motions in the flow.

DNS of a turbulent Couette flow at constant wall transpiration up to

We present a new set of direct numerical simulation data of a turbulent plane Couette flow with constant wall-normal transpiration velocity $V_{0}$, i.e. permeable boundary conditions, such that there is blowing on the lower side and suction on the upper side. Hence, there is no net change in flux to preserve periodic boundary conditions in the streamwise direction. Simulations were performed at $Re_{\unicode[STIX]{x1D70F}}=250,500,1000$ with varying transpiration rates in the range $V_{0}^{+}\approx 0.03$ to 0.085. Additionally, a classical Couette flow case at $Re_{\unicode[STIX]{x1D70F}}=1000$ is presented for comparison. As a first key result we found a considerably extended logarithmic region of the mean velocity profile, with constant indicator function $\unicode[STIX]{x1D705}=0.77$ as transpiration increases. Further, turbulent intensities are observed to decrease with increasing transpiration rate. Mean velocities and intensities collapse only in the cases where the transpiration rate is kept constant, while they are largely insensitive to friction Reynolds number variations. The long and wide characteristic stationary rolls of classical turbulent Couette flow are still present for all present DNS runs. The rolls are affected by wall transpiration, but they are not destroyed even for the largest transpiration velocity case. Spectral information indicates the prevalence of the rolls and the existence of wide structures near the blowing wall. The statistics of all simulations can be downloaded from the webpage of the Chair of Fluid Dynamics.

Universality of the energy-containing structures in wall-bounded turbulence

The scaling behaviour of the longitudinal velocity structure functions $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$ (where $2p$ represents the order) is studied for various wall-bounded turbulent flows. It has been known that for very large Reynolds numbers within the logarithmic region, the structure functions can be described by $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}/U_{\unicode[STIX]{x1D70F}}^{2}\approx D_{p}\ln (r/z)+E_{p}$ (where $r$ is the longitudinal distance, $z$ the distance from the wall, $U_{\unicode[STIX]{x1D70F}}$ the friction velocity and $D_{p}$, $E_{p}$ are constants) in accordance with Townsend’s attached eddy hypothesis. Here we show that the ratios $D_{p}/D_{1}$ extracted from plots between structure functions – in the spirit of the extended self-similarity hypothesis – have further reaching universality for the energy containing range of scales. Specifically, we confirm that this description is universal across wall-bounded flows with different flow geometries, and also for both the longitudinal and transversal structure functions, where previously the scaling has been either difficult to discern or differences have been reported when examining the direct representation of $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$. In addition, we present evidence of this universality at much lower Reynolds numbers, which opens up avenues to examine structure functions that are not readily available from high Reynolds number databases.