Prediction Of Wall Shear Stress Research Articles

Accurate Method for Estimating Wall-Friction Based on Analytical Wall-Law Model

A novel method is proposed for accurately determining the local wall friction through the near-wall measurement of time-average velocity profile in a Type-A turbulent boundary layer (TBL). The method is based on the newly established analytical wall-law in Type-A TBL. The direct numerical simulations (DNS) data of turbulence on a zero-pressure-gradient flat-plate (ZPGFP) is used to demonstrate the accuracy and the robustness of the approach. To verify the reliability and applicability of the method, a two-dimensional particle image velocimetry (PIV) measurement was performed in a ZPGFP TBL with a low-to-moderate Reynolds number (Re). Via utilizing the algorithm of single-pixel ensemble correlation (SPEC), the velocity profiles in the ZPGFP TBL were resolved at a significantly improved spatial resolution, which greatly enhanced the measurement accuracy and permitted us to accurately capture the near-wall velocity information. The accuracy of the approach is then quantitatively validated for the high Reynolds number turbulence using the ZPGFP TBL data. The research demonstrates that the current method can provide the precise estimation of wall friction with a mean error of less than 2%, which not only possesses the advantage of its insensitivity to the absolute wall-normal distance of the measuring point, but also its capability of providing an accurate prediction of wall shear stress based on fairly sparse experimental data on the velocity profile. The current study demonstrates that the wall shear stress can be accurately estimated by a velocity even at a single-point either measured or calculated in the near-wall region.

Enhancing Wind Turbine Blade Preventive Maintenance Procedure through Computational Fluid Dynamics-Based Prediction of Wall Shear Stress

Wind turbine blades are essential parts of wind energy systems and are frequently exposed to harsh environmental elements, such as strong winds, turbulence, and corrosive atmospheric elements. Over time, these circumstances may result in serious harm to blades, such as delamination and erosion, which may negatively affect the wind turbine’s functionality and durability. Accurate prediction of various types of damage is crucial to improve the toughness and lifespan of wind turbine blades and to maximize the overall effectiveness of wind energy systems. This article presents a novel computational fluid dynamics (CFDs)-based method for analyzing the distribution of wall shear stress on turbine blades, aimed at publicizing the yearly maintenance procedure. The investigation results from the CFDs, when compared with the current situation in a wind turbine farm in Thailand, confirmed that our wall shear stress modeling accurately predicted wind turbine damage. A maximum wall shear stress level higher than 5.00 Pa in the case of PA 90°, incoming air velocity 10.00 m/s, and 15 rpm was the main contribution to presenting the erosion and delamination from current drone inspection in wind turbine farms. In conclusion, these findings demonstrated the potential of using CFDs to predict wind turbine blade delamination and erosion, thereby significantly contributing to the development of specific and accurate yearly preventive maintenance. The proposed CFDs-based approach should serve as a sustainability tool for local human development, benefiting wind turbine engineers and operating technicians by providing them with a deeper understanding of the local flow conditions and wall shear stress distribution along wind turbine blades. This enables them to make informed decisions regarding blade design and maintenance.

Assessment of Machine Learning Wall Modeling Approaches for Large Eddy Simulation of Gas Turbine Film Cooling Flows: An a Priori Study

Abstract In this work, a priori analysis of machine learning (ML) strategies is carried out with the goal of data-driven wall modeling for large eddy simulation (LES) of gas turbine film cooling flows. High-fidelity flow datasets are extracted from wall-resolved LES (WRLES) of flow over a flat plate interacting with the coolant flow supplied by a single row of 7-7-7 shaped cooling holes inclined at 30 degrees with the flat plate at different blowing ratios (BR). The WRLES are performed using the high-order Nek5000 spectral element computational fluid dynamics (CFD) solver. Light gradient boosting machine (LightGBM) is employed as the ML algorithm for the data-driven wall model. Parametric tests are conducted to systematically assess the influence of a wide range of input flow features (velocity components, velocity gradients, pressure gradients, and fluid properties) on the accuracy of ML wall model with respect to prediction of wall shear stress. In addition, the use of spatial stencil and time delay is also explored within the ML wall modeling framework. It is shown that features associated with gradients of the streamwise and spanwise velocity components have a major impact on the prediction fidelity of wall model, while the effect of gradients of wall-normal velocity component is found to be negligible. Moreover, adding flow feature information from an x-y-z spatial stencil significantly improves the ML model accuracy and generalizability compared to just using local flow features from the matching location. Overall, highest prediction accuracy is achieved when both spatial stencil and time delay features are incorporated within the data-driven wall modeling paradigm.

Incorporating the complex rheological behavior of blood in microvascular network simulations: Two-phase modeling and a model reduction approach

Vessel flow resistance in microvascular network simulations is often computed using the single-phase approximation, where the no-slip boundary condition is imposed on the microvessel wall, and the "in vitro viscosity law" or "the relative hematocrit" relationships proposed by Pries et al. (1992) are utilized. However, these models have limitations due to their reliance on experimental data, which inherently contain measurement errors, thereby affecting their consistency and accuracy. To moderate these limitations, biphasic flow models can be employed, allowing for adaptable rheological descriptions of blood. Nevertheless, this approach presents challenges in terms of automation and computational cost. In order to address these challenges, we propose a monophasic steady-state model that consistently incorporates a wall-slip condition in place of the cell-free layer contribution. Model reduction is achieved by establishing an explicit relationship for the apparent slip length, which is calibrated through the regression of biphasic model results. Subsequently, simulations are conducted on microvessel network structures. The predictions of wall shear stress by the monophasic model exhibit deviations of less than 2 % from the expected values, within the range of vessel radii (10–75 μm) and discharge hematocrits (0.2–0.7). Furthermore, the computational time required is reduced by almost an order of magnitude compared to the biphasic model, rendering it suitable for multiscale biochemical simulations. The developed model closely approximates the flow resistance predictions of the biphasic model, while its modular nature allows for easy implementation of future modifications. Both models successfully reproduce experimental data by Lipowsky et al. (1978). Finally, we derive semi-analytical expressions for the apparent slip length and discuss why it is more appropriate to impose only the "relative hematocrit" relationship as a global constraint for calculating the hemodynamics in microvessels, rather than relying on the "in vitro viscosity law", which provides unreliable predictions at low values of the discharge hematocrit.

Evaluation of Computational Methodologies for Accurate Prediction of Wall Shear Stress and Turbulence Parameters in a Patient-Specific Aorta.

Background: Recent studies suggest that blood flow in main arteries is intrinsically disturbed, even under healthy conditions. Despite this, many computational fluid dynamics (CFD) analyses of aortic haemodynamics make the assumption of laminar flow, and best practices surrounding appropriate modelling choices are lacking. This study aims to address this gap by evaluating different modelling and post-processing approaches in simulations of a patient-specific aorta. Methods: Magnetic resonance imaging (MRI) and 4D flow MRI from a patient with aortic valve stenosis were used to reconstruct the aortic geometry and derive patient-specific inlet and outlet boundary conditions. Three different computational approaches were considered based on assumed laminar or assumed disturbed flow states including low-resolution laminar (LR-Laminar), high-resolution laminar (HR-Laminar) and large-eddy simulation (LES). Each simulation was ran for 30 cardiac cycles and post-processing was conducted on either the final cardiac cycle, or using a phase-averaged approach which utilised all 30 simulated cycles. Model capabilities were evaluated in terms of mean and turbulence-based parameters. Results: All simulation types, regardless of post-processing approach could correctly predict velocity values and flow patterns throughout the aorta. Lower resolution simulations could not accurately predict gradient-derived parameters including wall shear stress and viscous energy loss (largest differences up to 44.6% and 130.3%, respectively), although phase-averaging these parameters improved predictions. The HR-Laminar simulation produced more comparable results to LES with largest differences in wall shear stress and viscous energy loss parameters up to 5.1% and 11.6%, respectively. Laminar-based parameters were better estimated than turbulence-based parameters. Conclusion: Our findings suggest that well-resolved laminar simulations can accurately predict many laminar-based parameters in disturbed flows, but there is no clear benefit to running a HR-Laminar simulation over an LES simulation based on their comparable computational cost. Additionally, post-processing “typical” laminar simulation results with a phase-averaged approach is a simple and cost-effective way to improve accuracy of lower-resolution simulation results.

Challenges in Modeling Hemodynamics in Cerebral Aneurysms Related to Arteriovenous Malformations.

The significantly higher incidence of aneurysms in patients with arteriovenous malformations (AVMs) suggests a strong hemodynamic relationship between these lesions. The presence of an AVM alters hemodynamics in proximal vessels by drastically changing the distal resistance, thus affecting intra-aneurysmal flow. This study discusses the challenges associated with patient-specific modeling of aneurysms in the presence of AVMs. We explore how the presence of a generic distal AVM affects upstream aneurysms by examining the relationship between distal resistance and aneurysmal wall shear stress using physiologically realistic estimates for the influence of the AVM on hemodynamics. Using image-based computational models of aneurysms and surrounding vasculature, aneurysmal wall-shear stress is calculated for a range of distal resistances corresponding to the presence of AVMs of various sizes and compared with a control case representing the absence of an AVM. In the patient cases considered, the alteration in aneurysmal wall shear stress due to the presence of an AVM is considerable, as much as 19 times the base case wall shear stress. Furthermore, the relationship between aneurysmal wall shear stress and distal resistance is shown to be highly geometry-dependent and nonlinear. In most cases, the range of physiologically realistic possibilities for AVM-related distal resistance are so large that patient-specific flow measurements are necessary for meaningful predictions of wall shear stress. The presented work offers insight on the impact of distal AVMs on aneurysmal wall shear stress using physiologically realistic computational models. Patient-specific modeling of hemodynamics in aneurysms and associated AVMs has great potential for understanding lesion pathogenesis, surgical planning, and assessing the effect of treatment of one lesion relative to another. However, we show that modeling approaches cannot usually meaningfully quantify the impact of AVMs if based solely on imaging data from CT and X-ray angiography, currently used in clinical practice. Based on recent studies, it appears that 4D flow MRI is one promising approach to obtaining meaningful patient-specific flow boundary conditions that improve modeling fidelity.

Stochastic forcing for sub-grid scale models in wall-modeled large-eddy simulation

In the framework of wall-modeled large-eddy simulation (WMLES), the problem of combining sub-grid scale (SGS) models with the standard wall law is commonly acknowledged and expressed through multiple undesired near-wall behaviors. In this work, it is first observed that the static Smagorinsky model predicts efficiently the wall shear stress in a wall-modeled context, while more advanced static models like wall-adapting local eddy (WALE) viscosity or Sigma with proper cubic damping fail. It is, however, known that Smagorinsky is overall too dissipative in the bulk flow and in purely sheared flows, whereas the two other models are better suited for near-wall flows. The observed difficulty comes from the fact that the SGS model relies on the filtered velocity gradient tensor that necessarily comes with large errors in the near-wall region in the context of WMLES. Since the first off-wall node is usually located in the turbulent zone of the boundary layer, the turbulent structures within the first cell are neither resolved by the grid nor represented by the SGS model, which results in a lack of turbulent activity. In order to account for these subgrid turbulent structures, a stochastic forcing method derived from Reynolds-averaged Navier–Stokes (RANS) turbulence models is proposed and applied to the velocity gradients to better estimate the near-wall turbulent viscosity while providing the missing turbulent activity usually resulting from the WMLES approach. Based on such corrections, it is shown that the model significantly improves the wall shear stress prediction when used with the WALE and Sigma models.

A library for wall-modelled large-eddy simulation based on OpenFOAM technology

This work presents a feature-rich open-source library for wall-modelled large-eddy simulation (WMLES), which is a turbulence modelling approach that reduces the computational cost of standard (wall-resolved) LES by introducing special treatment of the inner region of turbulent boundary layers (TBLs). The library is based on OpenFOAM and enhances the general-purpose LES solvers provided by this software with state-of-the-art wall modelling capability. The included wall models belong to the class of wall-stress models that account for the under-resolved turbulent structures by predicting and enforcing the correct local value of the wall shear stress. A review of this approach is given, followed by a detailed description of the library, discussing its functionality and extensible design. The included wall-stress models are presented, based on both algebraic and ordinary differential equations. To demonstrate the capabilities of the library, it was used for WMLES of turbulent channel flow and the flow over a backward-facing step (BFS). For each flow, a systematic simulation campaign was performed, in order to find a combination of numerical schemes, grid resolution and wall model type that would yield a good predictive accuracy for both the mean velocity field in the outer layer of the TBLs and the mean wall shear stress. The best result, ≈1% error in the above quantities, was achieved for channel flow using a mildly dissipative second-order accurate scheme for the convective fluxes applied on an isotropic grid with 27000 cells per δ3-cube, where δ is the channel half-height. In the case of flow over a BFS, this combination led to the best agreement with experimental data. An algebraic model based on Spalding’s law of the wall was found to perform well for both flows. On the other hand, the tested more complicated models, which incorporate the pressure gradient in the wall shear stress prediction, led to less accurate results. Program SummaryProgram Title: libWallModelledLESProgram Files doi:http://dx.doi.org/10.17632/m8dnsnp4nd.1Licensing provisions: GPLv3Programming language: C++Nature of problem: Large-eddy simulation (LES) is a scale-resolving turbulence modelling approach providing a high level of predictive accuracy. However, LES of high Reynolds number wall-bounded flows is prohibitively computationally expensive due to the need for resolving the inner region of turbulent boundary layers (TBLs) [1]. This inhibits the application of LES to many industrially relevant flows [2] and prompts for the development of novel modelling techniques that would modify the LES approach in a way that allows it to retain its accuracy (at least away from walls) yet significantly lower its computational cost. Solution method: Wall-modelled LES (WMLES) is an approach that is based on complementing LES with special near-wall modelling that allows to leave the inner layer of TBLs unresolved by the computational grid. Many types of wall models have been proposed [1,3], commonly tested within the framework of in-house research codes. Here, an open-source library implementing several wall models is presented. The library is based on OpenFOAM, which is currently the most widely used general-purpose open-source software for computational fluid dynamics. The developed library can be directly applied to both academic and industrial flow cases, leading to a wider adoption of wall modelling and better understanding of its strengths and limitations.[1] J. Larsson, S. Kawai, J. Bodart, and I. Bermejo-Moreno. Large eddy simulation with modeled wall-stress: recent progress and future directions. Mechanical Engineering Reviews, 3(1):1-23, 2016.[2] J. Slotnick, A. Khodadoust, J. Alonso, D. Darmofal, W. Gropp, E. Lurie, D. Mavriplis. CFD vision 2030 study: A path to revolutionary computational aerosciences, Tech. rep., NASA, 2014.[3] S. T. Bose and G. I. Park. Wall-modeled large-eddy simulation for complex turbulent flows. Annual Review of Fluid Mechanics, 50(1):535–561, 2018.

Uncertainty quantification of simulated biomechanical stimuli in coronary artery bypass grafts

Coronary artery bypass graft surgery (CABG) is performed on more than 400,000 patients annually in the U.S. However, saphenous vein grafts (SVGs) implanted during CABG exhibit poor patency compared to arterial grafts, with failure rates up to 40% within 10 years after surgery. Differences in mechanical stimuli are known to play a role in driving maladaptation and have been correlated with endothelial damage and thrombus formation. As these quantities are difficult to measure in vivo, multi-scale coronary models offer a way to quantify them, while accounting for complex coronary physiology. However, prior studies have primarily focused on deterministic evaluations, without reporting variability in the model parameters due to uncertainty. This study aims to assess confidence in multi-scale predictions of wall shear stress and wall strain while accounting for uncertainty in peripheral hemodynamics and material properties. Boundary condition distributions are computed by assimilating uncertain clinical data, while spatial variations of vessel wall stiffness are obtained through approximation by a random field. We developed a stochastic submodeling approach to mitigate the computational burden of repeated multi-scale model evaluations to focus exclusively on the bypass grafts. This produces a two-level decomposition of quantities of interest into submodel contributions and full model/submodel discrepancies. We leverage these two levels in the context of forward uncertainty propagation using a previously proposed multi-resolution approach. The time- and space-averaged wall shear stress is well estimated with a coefficient of variation of <35%, but ignorance about the spatial distribution on the wall elastic modulus and thickness leads to large variations in an objective measure of wall strain, with coefficients of variation up to 100%. Sensitivity analysis reveals how the interactions between the flow and material parameters contribute to output variability.

Single-point transition modeling using the laminar kinetic energy concept

This work investigates transition to turbulence using laminar kinetic energy modeling based in single-point RANS approach. Transport equations are discretized using the cell centered control-volume method and the system of algebraic equations is relaxed using SIMPLE algorithm. A modified version of the most known laminar kinetic energy model is proposed and compared with the original version and with an experimental correlation transition model. The numerical results show that the laminar kinetic energy approach has reasonable experimental correlation. The modified version presents improvements in prediction of skin friction coefficient in flows subjected to pressure gradients. However, the use of the laminar kinetic energy concept has shown weakness when dealing with detached induced transition. Probably the cause of that is the lack of physical modeling of this phenomenon. Laminar kinetic energy modeling is in continuous evolution and is a good alternative to deal with engineering simulation that needs a good prediction of wall shear stress in transitional flows.

Aortic dissection simulation models for clinical support: fluid-structure interaction vs. rigid wall models.

BackgroundThe management and prognosis of aortic dissection (AD) is often challenging and the use of personalised computational models is being explored as a tool to improve clinical outcome. Including vessel wall motion in such simulations can provide more realistic and potentially accurate results, but requires significant additional computational resources, as well as expertise. With clinical translation as the final aim, trade-offs between complexity, speed and accuracy are inevitable. The present study explores whether modelling wall motion is worth the additional expense in the case of AD, by carrying out fluid-structure interaction (FSI) simulations based on a sample patient case.MethodsPatient-specific anatomical details were extracted from computed tomography images to provide the fluid domain, from which the vessel wall was extrapolated. Two-way fluid-structure interaction simulations were performed, with coupled Windkessel boundary conditions and hyperelastic wall properties. The blood was modelled using the Carreau-Yasuda viscosity model and turbulence was accounted for via a shear stress transport model. A simulation without wall motion (rigid wall) was carried out for comparison purposes.ResultsThe displacement of the vessel wall was comparable to reports from imaging studies in terms of intimal flap motion and contraction of the true lumen. Analysis of the haemodynamics around the proximal and distal false lumen in the FSI model showed complex flow structures caused by the expansion and contraction of the vessel wall. These flow patterns led to significantly different predictions of wall shear stress, particularly its oscillatory component, which were not captured by the rigid wall model.ConclusionsThrough comparison with imaging data, the results of the present study indicate that the fluid-structure interaction methodology employed herein is appropriate for simulations of aortic dissection. Regions of high wall shear stress were not significantly altered by the wall motion, however, certain collocated regions of low and oscillatory wall shear stress which may be critical for disease progression were only identified in the FSI simulation. We conclude that, if patient-tailored simulations of aortic dissection are to be used as an interventional planning tool, then the additional complexity, expertise and computational expense required to model wall motion is indeed justified.

Wall Shear Stress Prediction Using Computational Simulation on Patient Specific Artery with Aneurysm

An aneurysm is formed when a blood vessel becomes dilated or distorted. It will cause the vessel to expand to a size greater than its original diameter. In this study, Wall Shear Stress (WSS) of cerebral artery with aneurysm was predicted using Computational Fluid Dynamics (CFD). WSS in the artery is one of the indicators for brain artery disease progression. Based on the results, the maximum value of blood velocity and WSS on patient specific artery with aneurysm are 3.23 m/s and 60.1 Pa, respectively. The location of high WSS is before and after the aneurysm bulge. The WSS is above the normal physiological value where the artery wall is exposed to high stress. Hence, the vessel at this location is anticipated to become weaker and could be further dilated.

A comparative study of turbulence models in a transient channel flow

The performance of a number of low-Reynolds number turbulence models is evaluated against direct numerical simulations (DNS). All models are applied to an unsteady flow comprising a ramp-type excursion of flow rate inside a closed channel. The flow rate is increased linearly with time from an initial Reynolds number of 9308 (based on hydraulic diameter and bulk velocity) to a final Reynolds number of 29,650. The acceleration rate is varied to cover low, intermediate and high accelerations. It is shown that among the models investigated, the k–ε models of Launder and Sharma (1974) and Chang et al. (1995) [28] and the γ–Reθ transition model of Langtry and Menter (2009) [38] capture well the key flow features of these unsteady turbulent flows. For the cases of low and intermediate acceleration rates, these three models yield predictions of wall shear stress that agree well with the corresponding DNS data. For the case of high acceleration, the γ–Reθ model of Langtry and Menter (2009) [38] and the k–ε model of Launder and Sharma (1974) yield reasonable predictions of wall shear stress.

Accurate Prediction of Wall Shear Stress in a Stented Artery: Newtonian Versus Non-Newtonian Models

A significant amount of evidence linking wall shear stress to neointimal hyperplasia has been reported in the literature. As a result, numerical and experimental models have been created to study the influence of stent design on wall shear stress. Traditionally, blood has been assumed to behave as a Newtonian fluid, but recently that assumption has been challenged. The use of a linear model; however, can reduce computational cost, and allow the use of Newtonian fluids (e.g., glycerine and water) instead of a blood analog fluid in an experimental setup. Therefore, it is of interest whether a linear model can be used to accurately predict the wall shear stress caused by a non-Newtonian fluid such as blood within a stented arterial segment. The present work compares the resulting wall shear stress obtained using two linear and one nonlinear model under the same flow waveform. All numerical models are fully three-dimensional, transient, and incorporate a realistic stent geometry. It is shown that traditional linear models (based on blood's lowest viscosity limit, 3.5 Pa s) underestimate the wall shear stress within a stented arterial segment, which can lead to an overestimation of the risk of restenosis. The second linear model, which uses a characteristic viscosity (based on an average strain rate, 4.7 Pa s), results in higher wall shear stress levels, but which are still substantially below those of the nonlinear model. It is therefore shown that nonlinear models result in more accurate predictions of wall shear stress within a stented arterial segment.

The influence of the endothelial surface layer on the prediction of wall shear stress in small arteries

The influence of the endothelial surface layer on the prediction of wall shear stress in small arteries

A turbulence model study for simulating flow inside tight lattice rod bundles

Performances of various turbulence models are evaluated for calculation of detailed coolant velocity distribution in a tight lattice fuel bundle. The individual models are briefly outlined and compared with respect to the prediction of wall shear stress and velocity field, for a fully developed flow inside a triangular lattice bundle. Comparisons clearly show the importance of proper modeling of the turbulence-driven secondary flows in subchannels. A quadratic k– ɛ model, which showed promising capability in this respect, is adjusted in its coefficients, and the adjusted model is applied to fully developed flow in an infinite triangular array, with various Reynolds numbers. The results show that the inclusion of adequate anisotropy modeling enables to accurately reproduce the wall shear stress distribution and velocity field in tight lattice fuel bundles.

Remarks on numerical prediction of wall shear stress in entry flow problems

AbstractToday, commercial CFD codes are widely used to simulate many different entry flow problems. The flow in the developing zone undergoes a transition from a specified velocity profile at the inlet section to a fully developed profile in the region far from the inlet. Previous investigations have shown that the hydrodynamic variables, such as velocity and pressure magnitudes, along the centreline converge to a mesh independent solution even when coarse grid distributions are utilised. However, the present work shows that the local velocity profile is highly dependent on grid resolution in the vicinity of solid boundaries. It is shown that failure to account for the grid resolution can result in significant errors in the predicted solid–fluid interaction, e.g. wall shear stress calculation. A remedy to overcome this problem is subsequently given. Copyright © 2004 John Wiley &amp; Sons, Ltd.

An evolutionary approach to Wall Shear Stress prediction in a grafted artery

Abstract Restoring the blood supply to a diseased artery is achieved by using a vascular bypass graft. The surgical procedure is a well documented and successful technique. The most commonly cited hemodynamic factor implicated in the disease initiation and proliferation processes at graft/artery junctions is Wall Shear Stress (WSS). WSS distributions are predicted using numerical simulations as they can provide quick and precise results to assess the effects that alternative graft/artery junction geometries have on the WSS distributions in bypass grafts. Validation of the numerical model is required and in vitro studies, using laser Doppler anemometry (LDA), have been employed to achieve this. Numerically, the Wall Shear Stress is predicted using velocity values stored in the computational cell near the wall and assuming zero velocity at the wall. Experimentally obtained velocities require a mathematical model to describe their behavior. This study employs a grammar based evolutionary algorithm termed Chorus for this purpose and demonstrates that Chorus successfully attains this objective. It is shown that even with the lack of domain knowledge, the results produced by this automated system are comparable to the results in the literature.

Wall shear stress prediction in three-dimensional turbulent boundary layers

A method is developed to infer the wall shear stress for three-dimensional turbulent boundary layers based on the assumption that the resultant surface shear stress and the effective velocity based on Prahlad's model correlates the velocity profile into its two-dimensional form. Existence of the near wall region similarity has been demonstrated for three-dimensional turbulent boundary layers.

Shear-dependent deformation of erythrocytes in rheology of human blood.

Shear-dependent deformation of erythrocytes in rheology of human blood.