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.     

Conservation relation of generalized growth rate in boundary layers

The elementary task is to calculate the growth rates of disturbances when the e^N method in transition prediction is performed. However, there is no unified knowledge to determine the growth rates of disturbances in three-dimensional (3D) flows. In this paper, we study the relation among the wave parameters of the disturbance in boundary layers in which the imaginary parts of wave parameters are far smaller than the real parts. The generalized growth rate (GGR) in the direction of group velocity is introduced, and the conservation relation of GGR is strictly deduced in theory. This conservation relation manifests that the GGR only depends on the real parts of wave parameters instead of the imaginary parts. Numerical validations for GGR conservation are also provided in the cases of first/second modes and crossflow modes. The application of GGR to the e^N method in 3D flows is discussed, and the puzzle of determining growth rates in 3D flows is clarified. A convenient method is also proposed to calculate growth rates of disturbances in 3D flows. Good agreement between this convenient method and existing methods is found except the condition that the angle between the group velocity direction and the x-direction is close to 90° which can be easily avoided in practical application.     

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.     

Infrared thermography investigations in transitional supersonic boundary layers

The study of boundary-layer transition in supersonic flows is conducted employing infrared thermography (IRT). Several models of swept wings are tested in a blow-down facility at Mach number 2.4. The effects of wing sweep and other parameters (angle of attack, leading-edge contour, presence/absence of surface roughness) are successfully observed. The transition front is clearly identified, demonstrating the utility of IRT for this type of study. The technique is particularly indicated for flows that are sensitive to surface alterations (roughness), such as transitional boundary layers, because it does not require interaction with the model or the flow under investigation. The additional advantage of no need for special apparatus, except for the infrared camera, makes IRT well suited for both wind-tunnel and in-flight testing. Practical problems and limitations encountered when dealing with IRT in high-speed flows are also discussed.     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

---

Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

---

a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

Low Reynolds number effects in open-channel turbulent boundary layers

The present study reports measurements of a turbulent boundary layer in an open-channel flow using fiber-optic laser Doppler anemometry. The Reynolds numbers based on momentum thickness and depth of flow are in the range 750≤Re _θ ≤2,400 and 15,300≤Re _h ≤54,200, respectively. It is shown that an accurate estimate of the wall shear stress can be made by fitting a fifth-order polynomial to the near-wall data. The effect of Reynolds number on the mean turbulence intensity and triple correlation is examined using both conventional scaling laws and the recent scaling laws proposed by George and Castillo. The present results show that different scaling laws lead to different conclusions on low Reynolds number effects.     

Smooth and rough turbulent boundary layers at high Reynolds number

A 2-D turbulent boundary layer experiment with zero pressure gradient (ZPG) has been carried out over a rough and a smooth surface using two cross hot-wire probes. Wind tunnel speeds of 10 m/s and 20 m/s were set up in order to investigate the effects of the upstream conditions and the Reynolds number on the downstream flow. For a given set of upstream conditions, such as the wind tunnel speed, trip wire size and location, the three components of the velocity field were measured from about 14 m from the inlet of the wind tunnel to 30 m downstream. This experiment is unique because it achieves Reynolds numbers as high as R120,000, for which measurements of the mean velocity are reported. It is shown that by fixing the upstream conditions, the mean deficit profiles collapse with the freestream velocity, , but to different curves depending on the upstream conditions and surface roughness. Moreover, the effects of the upstream conditions, the Reynolds number, and roughness are completely removed from the outer flow when the mean deficit profiles are normalized by the Zagarola/Smits scaling, . Consequently, the true asymptotic profile in the turbulent boundary layer is found in ZPG flow regardless of the range of Reynolds number, surface conditions and initial conditions.     

Thin shear boundary layers in flow of hot aluminium

The flow of hot aluminium in channels is investigated. The constitutive relation considered for the flow stress is the inverse hyperbolic sine function of the Zener–Hollomon parameter. Analytical solutions for the flow are derived. At high strain rates, an exponential velocity profile close to the channel walls is predicted indicating the existence of a thin shear boundary layer characterized by strong shear. The characteristic length scale for the exponential velocity profile is a function of material parameters in the constitutive relation and the inverse of the local pressure gradient. The analytical prediction of a thin shear boundary layer close to the channel walls for large strain rates is consistent with the observed microstructure in an extruded section.     

Some Reynolds number effects on two- and three-dimensional turbulent boundary layers

Experimental data for a two-dimensional (2-D) turbulent boundary layer (TBL) flow and a three-dimensional (3-D) pressure-driven TBL flow outside of a wing/body junction were obtained for an approach Reynolds number based on momentum thickness of Re _θ =23,200. The wing shape had a 3:2 elliptical nose, NACA 0020 profiled tail, and was mounted on a flat wall. Some Reynolds number effects are examined using fine spatial resolution (Δy ⁺=1.8) three-velocity-component laser-Doppler velocimeter measurements of mean velocities and Reynolds stresses at nine stations for Re _θ =23,200 and previously reported data for a much thinner boundary layer at Re _θ =5,940 for the same wing shape. In the 3-D boundary layers, while the stress profiles vary considerably along the flow due to deceleration, acceleration, and skewing, profiles of the parameter correlate well and over available Reynolds numbers. The measured static pressure variations on the flat wall are similar for the two Reynolds numbers, so the vorticity flux and the measured mean velocities scaled on wall variables agree closely near the wall. The stresses vary similarly for both cases, but with higher values in the outer region of the higher Re _θ case. The outer layer turbulence in the thicker high Reynolds number case behaves similarly to a rapid distortion of the flow, since stream-wise vortical effects from the wall have not diffused completely through the boundary layer at all measurement stations. Received: 9 June 2000/Accepted: 26 January 2001     

Influence of Reynolds number on characteristics of turbulent wall boundary layers

Hot-wire anemometer measurements, using two types of probes, are reported for wall boundary layer flows with particular attention being given to the near-wall region and to measurements at high Reynolds numbers up to R 15,000. To obtain accurate near-wall measurements, the influence of wall proximity on hot-wire readings was eliminated by using a highly insulating wall material. Measurements were carried out with a single hot-wire boundary layer probe to obtain the longitudinal velocity informatemperature-wake sensor for the cross flow tion and a hot-wire, information.The results provided in the paper include measurements of averaged properties like mean velocity, rms-quantities of velocity fluctuations, probability density distributions etc. Conditional averages are also provided in order to yield information related to coherent flow structures present in boundary layer flows. It is shown that these structure remain present up to the highest Reynolds number investigated in the present study. The conditionally averaged data provide quantitative information on the mechanisms that are involved in the production of turbulence in boundary-layer flows.     

A study in transitional flat plate boundary layers: measurement and visualization

 This study is concerned with transition in flat plate boundary layer flow. Sets of results are obtained as follows: (1) Very clear pictures of the formation and the development of the butterfly-like structures rather than ∧-structures in the K-regime of boundary layer transition are obtained. (2) A chain of ring like vortices, which generate the high-frequency spikes on the time traces of velocity and still present periodical behaviour, at the tip of each ∧-vortex, which is the part of the butterfly-like structure, are visualized for the first time. (3) A wave-like structure is observed to occupy the whole boundary layer, extending from the near-wall region to the outer edge of the boundary layer. Received: 24 September 1998/Accepted: 24 April 1999     

Numerical studies on evolution of secondary streamwise vortices in transitional boundary layers

Formation and evolution of secondary streamwise vortices in the compressible transitional boundary layers over a flat plate are studied using a direct numerical simulation method with high-order accuracy and highly effective non-reflecting characteristic boundary conditions. Generation and development processes of the secondary streamwise vortices in the complicated transitional boundary flow are clearly analyzed based on the of numerical results, and the effects on the formation of the ring-like vortex that is vital to the boundary layer transition are explored. A new mechanism forming the ring-like vortex through the mutual effect of the primary and secondary streamwise vortices is expressed.     

An investigation of the scales in transitional boundary layers under high free-stream turbulence conditions

The scales in a transitional boundary layer subject to high (initially 8%) free-stream turbulence and strong acceleration (K as high as 9×10^–6) were investigated using wavelet spectral analysis and conditional sampling of experimental data. The boundary layer shows considerable evolution through transition, with a general shift from the lower frequencies induced by the free-stream unsteadiness to higher frequencies associated with near-wall-generated turbulence. Within the non-turbulent zone of the intermittent flow, there is considerable self-similarity in the spectra from the beginning of transition to the end, with the dominant frequencies in the boundary layer remaining constant at about the dominant frequency of the free-stream. The frequencies of the energy-containing scales in the turbulent zone change with streamwise location and are significantly higher than in the non-turbulent zone. When normalized on the local viscous length scale and velocity, however, the turbulent zone spectra also show good self-similarity throughout transition. Turbulence dissipation occurs almost exclusively in the turbulent zone. The velocity fluctuations associated with dissipation are isotropic, and their normalized spectra at upstream and downstream stations are nearly identical. The distinct differences between the turbulent and non-turbulent zones suggest the potential utility of intermittency based transition models in which these zones are treated separately. The self-similarity noted in both energy containing and dissipation scales in both zones suggests possibilities for simplifying the modeling for each zone.

---

Reynolds stress modelling of rectangular open‐channel flow

A Reynolds stress model for the numerical simulation of uniform 3D turbulent open‐channel flows is described. The finite volume method is used for the numerical solution of the flow equations and transport equations of the Reynolds stress components. The overall solution strategy is the SIMPLER algorithm, and the power‐law scheme is used to discretize the convection and diffusion terms in the governing equations. The developed model is applied to a flow at a Reynolds number of 77000 in a rectangular channel with a width to depth ratio of 2. The simulated mean flow and turbulence structures are compared with measured and computed data from the literature. The computed flow vectors in the plane normal to the streamwise direction show a small vortex, called inner secondary currents, located at the juncture of the sidewall and the free surface as well as the free surface and bottom vortices. This small vortex causes a significant increase in the wall shear stress in the vicinity of the free surface. A budget analysis of the streamwise vorticity is carried out. It is found that both production terms by anisotropy of Reynolds normal stress and by Reynolds shear stress contribute to the generation of secondary currents. Copyright © 2006 John Wiley & Sons, Ltd.     

Implementation of physical boundary conditions into computational domain in modelling of oscillatory bottom boundary layers

This paper discusses the importance of realistic implementation of the physical boundary conditions into computational domain for the simulation of the oscillatory turbulent boundary layer flow over smooth and rough flat beds. A mathematical model composed of the Reynolds averaged Navier–Stokes equation, turbulent kinetic energy (k) and dissipation rate of the turbulent kinetic energy (ε) has been developed. Control‐volume approach is used to discretize the governing equations to facilitate the numerical solution. Non‐slip condition is imposed on the bottom surface, and irrotational main flow properties are applied to the upper boundary. The turbulent kinetic energy is zero at the bottom, whereas the dissipation rate is approaching to a constant value, which is proportional to the kinematic viscosity times the second derivative of the turbulent kinetic energy. The output of the model is compared with the available experimental studies conducted in oscillatory tunnels and wave flume. It is observed that the irrotational flow assumption at the upper boundary is not realistic in case of water tunnels. Therefore, new upper boundary conditions are proposed for oscillatory tunnels. The data of wave flume show good agreement with the proposed numerical model. Additionally, several factors such as grid aspect ratio, staggered grid arrangement, time‐marching scheme and convergence criteria that are important to obtain a robust, realistic and stable code are discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Relation between the stress and couple-stress tensors in the microcontinuum theory of elasticity

The stress tensor is expressed in terms of an arbitrary symmetric tensor field of second rank and the couple-stress tensor. The stress and couple-stress tensors are represented by arbitrary tensor fields satisfying the homogeneous equilibrium equations. These tensors are also given in the form of the expressions satisfying the inhomogeneous equilibrium equations used in the microcontinuum theory of elasticity. The stress tensor functions are considered.