Mean Velocity Profiles Research Articles

Determination of Reynolds shear stress from the turbulent mean velocity profile and spectral characteristics of Orr-Sommerfeld -Squire equations

Theoretically, based on a waveguide model, the expression of the tangential stress is formulated for steady, two-dimensional incompressible fluid flow over a flat plate in turbulent boundary layer. It is dependent on some factors, one of them, the behaviour of the last damping mode eigenvalues, and eigenfunctions, that are deduced from solution Orr-Sommerfeld equation by spectral Chebyshev collocation Method. Verification of the latter method is investigated by comparison the deduced formula of turbulent tangential stress with experimental data. In addition to, weight factors in this expression are connected to define the condition of dynamical system solution for multiple 3-wave resonance. This system is solved numerically, and the dynamic invariant is normalized to obtain the time average of the square modulus harmonic, and sub harmonics amplitudes by theorem Birkhoff-Khinchin. Comparison is made between the time-averaged and the phase average for the square modulus of harmonic, and sub harmonic amplitudes that defined on the unit sphere, in the state of multiple 3-wave resonance.

Design and calibration of permanent magnet probes for the local measurement of velocity and temperature in a liquid metal backward facing step flow

The simultaneous and local measurement of velocity and the temperature of a non-isothermal liquid metal flow has been an ongoing research topic over decades. The motivation is to obtain a detailed panorama of a liquid metal flow for the validation of turbulent heat flux models. So-called permanent magnet probes were used in the past for the local measurement of velocity and temperature profiles in liquid sodium in rather canonical flow configurations. The next step is to measure velocity and temperature profiles in a more complex flow geometry, namely a vertical confined backward facing step. For this, the permanent magnet probe must be adapted regarding its design, calibration procedure and temperature correction method. Particularly, considering that for this experiment the eutectic alloy of gallium, indium and tin was used as a working fluid, instead of liquid sodium, as in the mentioned past experiments. The main design aspects for a permanent magnet probe found in the literature are summarized and applied to the present probe. A calibration strategy for the probe was developed and implemented for the measurement of mean velocity profiles. A wetting procedure for the probe is proposed. The measured probe sensitivity for all six used probes agrees well with the theoretical estimations. The highest uncertainty contribution to measured sensitivity is related to the typical wetting issues of gallium–indium–tin. Future implementation of permanent magnet probes in general gallium–indium–tin experiments can make use of the developed know-how shared in this work.Graphic abstract

Effects of liquid viscosity and bubble size distribution on bubble plume hydrodynamics

The present paper provides a complete databank of bubble plume hydrodynamics in various media characterized by different viscosities in the range 1–100mPa.s. Thus, the influence of the viscosity on bubble plume oscillations and on both liquid and gas velocity fields is studied. The role of the bubble size distribution at different viscosities was also analysed by using two different spargers: a membrane to create ellipsoidal millimetrice bubbles and a slugflow tube to create spherical cap bubbles. Mean velocity profiles, averaged over a large number of plume oscillation periods and corresponding root mean square profiles are discussed, in both phases. Furthermore, void fraction profiles, plume oscillation periods, and bubble size distributions are given for all experimental conditions. Finally, probability density functions and phase averaged profiles in the liquid phase provide additional information. The dependence of bubble plume hydrodynamics on liquid viscosity is huge. At large viscosity, the oscillating plume is damped, regardless of the sparger. Concerning the membrane sparger the viscosity increase modifies the plume oscillation period and its lateral plume expansion, whereas, in the case of the slugflow sparger, only the plume expansion is impacted. Liquid phase hyrdodynamics are weakly affected by the choice of sparger, especially at high viscosities.

Development of an Analytical Wall Function for Bypass Transition

The objective of the present work is to propose an extended analytical wall function that is capable of predicting the bypass transition from laminar to turbulent flow. The algebraic γ transition model, the k−ω turbulence model and the analytical wall function are integrated together in this work to detect the transition onset and start the transition process. The present analytical wall function is validated with the experimental data, the Blasius solution and the law of the wall. With this analytical wall function, the transition onset in the skin friction coefficient is detected and the growth rate of transition is properly generated. The predicted mean velocity profiles are found to be in good agreement with the Blasius solution in the laminar flow, the experimental data in the transition zone and the law of the wall in the fully turbulent flow.

One-point statistics for turbulent pipe flow up to

We study turbulent flows in a smooth straight pipe of circular cross-section up to friction Reynolds number $({\textit {Re}}_{\tau }) \approx 6000$ using direct numerical simulation (DNS) of the Navier–Stokes equations. The DNS results highlight systematic deviations from Prandtl friction law, amounting to approximately $2\,\%$ , which would extrapolate to approximately $4\,\%$ at extreme Reynolds numbers. Data fitting of the DNS friction coefficient yields an estimated von Kármán constant $k \approx 0.387$ , which nicely fits the mean velocity profile, and which supports universality of canonical wall-bounded flows. The same constant also applies to the pipe centreline velocity, thus providing support for the claim that the asymptotic state of pipe flow at extreme Reynolds numbers should be plug flow. At the Reynolds numbers under scrutiny, no evidence for saturation of the logarithmic growth of the inner peak of the axial velocity variance is found. Although no outer peak of the velocity variance directly emerges in our DNS, we provide strong evidence that it should appear at ${\textit {Re}}_{\tau } \gtrsim 10^4$ , as a result of turbulence production exceeding dissipation over a large part of the outer wall layer, thus invalidating the classical equilibrium hypothesis.

Boundary-layer evolution over long wind farms

The structure of the internal boundary layer above long wind farms is investigated experimentally. The transfer of kinetic energy from the region above the farm is dominated by the turbulent flux of momentum together with the displacement of kinetic energy operated by the mean vertical velocity: these two have comparable magnitude along the farm opposite to the infinite-farm case. The integration of the energy equation in the vertical highlighted the key role of the energy flux, and how that is balanced by the growth of the internal boundary layer in terms of energy thickness with a small role of the dissipation. The mean velocity profiles seem to follow a universal structure in terms of velocity deficit, while the Reynolds stress does not follow the same scaling structure. Finally, a spectral analysis along the farm identified the leading dynamics determining the turbulent activity: while behind the first row the signature of the tip vortices is dominant, already after the second row their coherency is lost and a single broadband peak, associated with wake meandering, is present until the end of the farm. The streamwise velocity peak is associated with a nearly constant Strouhal number weakly dependent on the farm layout and free stream turbulence condition. A reasonable agreement of the velocity spectra is observed when the latter are normalised by the velocity variance and integral time scale: nevertheless the spectra show clear anisotropy at the large scales and even the small scales remain anisotropic in the inertial subrange.

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.

Complete and incomplete similarity for the mean velocity profile of turbulent pipe and channel flows at extreme Reynolds number

The recent availability of experimental data on turbulent pipe flows in the extreme Reynolds number range has shown strong evidence that when Re≫O(105), the flow exhibits a transition to new scaling laws for important physical quantities, such as the friction factor and the Reynolds stress. In this work, we discuss a transition concerning the scaling of the mean velocity profile (MVP) of turbulent pipe and channel flows. We show that for sufficiently large Re number, the MVP exhibits complete similarity asymptotics in bulk coordinates, as U/U¯→ΨE(1)(η). Then, using a recent friction power-law formulation for extreme-Re flows, we show that this is equivalent to an incomplete similarity asymptotics, as u+→Reτ12ΦE(1)(y+/Reτ). We then use the incomplete similarity asymptotics to relate the representation of the flow in bulk coordinates, (η,U/U¯), to the representation of the flow in inner coordinates, (y+,u+), so that u+ can be approximated by a Reτ-dependent functional as u+≈ΦE(y+,Reτ). We also propose a multiscale polynomial model of the MVP, which yields excellent approximations of available data in both bulk and inner coordinates.

Reconstruction of numerical inlet boundary conditions using machine learning: Application to the swirling flow inside a conical diffuser

A new approach to determine proper mean and fluctuating inlet boundary conditions is proposed. It is based on data driven techniques, i.e., machine learning approach, and its goal is to use any known information about the downstream flow to reconstruct the unknown or incomplete inlet boundary conditions for a numerical simulation. The European Research Community On Flow, Turbulence And Combustion (ERCOFTAC) test case of the swirling flow inside a conical diffuser is investigated. Despite its relatively simple geometry, it constitutes a very challenging test case for numerical simulations due to incomplete experimental data and to the delicate balance between core flow recirculation and boundary layer separation. Simulations are performed using both Reynolds averaged Navier–Stokes (RANS) and large-eddy simulations (LES) turbulence methods. The mean velocity and turbulence kinetic energy profiles obtained with the machine learning approach in RANS are found to be in very good agreement with the experimental measurements and the numerical predictions are greatly improved as compared to the previous results using basic inlet boundary conditions. They are indeed comparable to the best previous RANS using empirical ad hoc inlet conditions to accurately simulate the downstream flow. In LES, in addition to the mean velocity profiles, the machine learning approach also allows us to properly reconstruct the fluctuating part of the turbulent field. In particular, the methodology allows us to circumvent the lack of turbulent correlations associated with classical inlet synthetic turbulence.

Stereoscopic Micro PIV Investigation of Velocity Boundary Layer Near Piston Top of a Tumble Enhanced SI IC Engine

Stereoscopic Micro PIV Investigation of Velocity Boundary Layer Near Piston Top of a Tumble Enhanced SI IC Engine

Experimental Study of the Sidewall Effect on Three-Dimensional Turbulent Wall Jet

Abstract An experimental study on the effect of sidewalls on the flow characteristics of a three-dimensional turbulent square wall jet is carried out at a Reynolds number of 25,000. The sidewalls are defined as the two parallel plates along the vertical jet centerline. Four different sizes of sidewall enclosure (here after referred to as ‘SWE’) are placed at the lateral positions (z) of ±3.5h, ±4h,±4.5h, and ±5h from the vertical jet centerline plane, where h is the height of square jet. The mean characteristics of fluid flow in wall normal (y) and lateral (z) directions at different downstream locations (x/h=0.2−45) are measured using a hotwire anemometer. The velocity measurements are also performed in the z – y lateral plane at four downstream locations (x/h=30, 35, 40, and 45). Results indicate that the mean velocity profile in lateral and wall normal directions behaves differently depending on the size of SWEs. The decay rate of the maximum mean velocity increases with decrease in size of SWEs after the downstream location (x/h≥20). The decay rate of the maximum mean velocity increases about 5.6% in 140 mm SWE as compared to 200 mm SWE. It is noted that spread of the jet in wall normal and lateral directions increases with decrease in size of SWEs after the attachment of the flow stream on the sidewalls. In the far-field location of the present case, the smaller size of SWE (140 mm SWE) has 10.7% and 23.1% higher spread rate as compared to larger size of SWE (200 mm SWE) in wall-normal and lateral directions, respectively. It is also seen that the self-similar profile gets delayed in wall-normal direction as compared to lateral direction for all the cases. The wall-normal self-similar profile is obtained early with increase in the size of SWEs and it is obtained at x/h=30, 27, 24, and 20 for 140 mm, 160 mm, 180 mm, and 200 mm SWEs, respectively. The flow stream seems to climb the sidewall and this tendency increases with increase in size of SWEs.

Urban Boundary Layers Over Dense and Tall Canopies

Wind-tunnel experiments were carried out on four urban morphologies: two tall canopies with uniform height and two super-tall canopies with a large variation in element heights (where the maximum element height is more than double the average canopy height, h_{max}=2.5h_{avg}). The average canopy height and packing density are fixed across the surfaces to h_{avg} = 80~hbox {mm}, and lambda _{p} = 0.44, respectively. A combination of laser Doppler anemometry and direct-drag measurements are used to calculate and scale the mean velocity profiles with the boundary-layer depth delta . In the uniform-height experiment, the high packing density results in a ‘skimming flow’ regime with very little flow penetration into the canopy. This leads to a surprisingly shallow roughness sublayer (depth approx 1.15h_{avg}), and a well-defined inertial sublayer above it. In the heterogeneous-height canopies, despite the same packing density and average height, the flow features are significantly different. The height heterogeneity enhances mixing, thus encouraging deep flow penetration into the canopy. A deeper roughness sublayer is found to exist extending up to just above the tallest element height (corresponding to z/h_{avg} = 2.85), which is found to be the dominant length scale controlling the flow behaviour. Results point toward the existence of a constant-stress layer for all surfaces considered herein despite the severity of the surface roughness (delta /h_{avg} = 3 - 6.25). This contrasts with the previous literature.

STATISTICAL CHARACTERISTICS OF PRESSURE FLUCTUATION IN SHOCK WAVE AND TURBULENT BOUNDARY LAYER INTERACTION

Shock wave and turbulent boundary layer interaction widely exists in the internal and external flow of high-speed aircraft. The aerodynamic performance and flight safety of aircraft are seriously affected by the strong pressure fluctuation in the interaction region. To investigate statistical characteristics of fluctuating pressure, the interaction between an incident shock of 33.2° and a spatially developed Mach 2.25 turbulent boundary layer is analyzed by means of direct numerical simulation (DNS). The numerical results have been carefully validated against with previous experiment and DNS at similar flow conditions in terms of mean velocity profile, turbulence intensity and wall pressure distribution. Statistics at the wall and in the outer layer, including fluctuation intensity, power spectral density, two-point correlation and space-time correlation, are quantitatively compared. The differences between them are analyzed in detail. It is found that the effect of the shock interaction on the wall-pressure fluctuation and the fluctuating pressure in the outer layer are utterly different. Based on the analysis of the power spectra density, the fluctuations in the separated region are both characterized by the low-frequency content, but in the reattachment region, the peak frequency of outer pressure fluctuations quickly shifts to higher frequency, with the low-frequency energy of wall-pressure fluctuation still being predominant. It is identified that the two-point correlations of pressure fluctuation at the wall and in the outer layer are both more elongated in the spanwise direction than that in the streamwise direction. The integral scale at the wall is generally increased, while the one in the outer layer increases sharply after passing the shock and then gradually decreases. The analysis of space-time correlation indicates that the iso-correlation contours are similar to the elliptical distribution and the convection velocity deduced by the correlation is dramatically decreased. Downstream of the interaction, the convection velocity in the outer layer is higher than that of wall-pressure fluctuation.

Numerical Investigation of an Unsteady and Anisotropic Turbulent Flow Downstream a 90° Bend Pipe with and without Ribs

In this work, a numerical study of the dynamical behavior of unsteady and anisotropic turbulent flow downstream a 90° bended pipe was presented. For this purpose, comparative computations are carried out employing two flow configurations, bend pipe with ribs and bend pipe without ribs with a curvature radius ratio Rc/D=2.0. In the bend pipe with ribs, the pitch ratios Pt/e=40 and the rib height to pipe diameter e/D is 0.1. This model has been utilized to assess the effect of ribs on flow where the presence of the ribs leads to a complex velocity field with regions of flow separation upstream and downstream of the ribs. The Reynolds-Averaged Navier–Stokes (RANS) approach is employed and the computational model is validated by comparisons with the existing experimental data. The simulations are conducted with the commercials CFD software FLUENT for Dean number varying from 5000 to 40000. The result analysis shows that the higher resistance generated by the ribs produced relatively larger velocity gradient (∂U/∂y) compared to the case of bend pipe without ribs where a more uniform mean velocity profile is observed. The turbulence intensities are higher in the ribbed bend pipe compared to those in the non-ribbed case and depend faintly on the Dean number. The levels of the Reynolds shear stresses are significantly enhanced by the ribs compared to the case without ribs. This increasing is explained by significantly higher levels of turbulence production over those ribs produced by large values of ∂U/∂y.

Urban exposure upstream fetch and its influence on the formulation of wind load provisions

With the rapid expansion of city area, concern has increased about the atmospheric boundary layer in urban area, which is influenced by the complex city morphology. This paper aims at providing a fundamental understanding of the influence of city upstream fetch length and exposure roughness on the mean velocity and turbulence intensity profiles. Wind tunnel tests of two real city models have been carried out. The results indicate that in urban areas mean velocity profiles are influenced by an upstream fetch length up to 750 m; and not influenced by the exposure morphology when the fetch length goes farther than 1250 m. Turbulence intensity profiles are more sensitive to upstream exposure and require longer upstream fetches to become steady. The evolution of mean velocity profiles of different fetch lengths and the influence of upstream roughness elements in the variation of turbulence intensity have been analyzed. The roughness length z0 of Kowloon, Hong Kong is determined, and issues related to the specification of minimum upstream fetch in current wind load provisions are tackled.

Variable-Order Fractional Models for Wall-Bounded Turbulent Flows.

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.

Direct numerical simulation of a turbulent plane Couette-Poiseuille flow with zero-mean shear

Direct numerical simulations of a turbulent Couette-Poiseuille flow with zero-mean-shear at the moving wall (SL-flow) is performed to examine flow features compared to those for a turbulent pure Poiseuille flow (P-flow). Profiles of the streamwise mean velocity, indicator function and ratio of production to dissipation show that the logarithmic region is significantly elongated for the SL-flow compared to that for the P-flow at a similar Reynolds number. In addition, the magnitudes of the Reynolds stresses are found to be larger in both inner and outer layers for the SL-flow than those for the P-flow. The spanwise spectra of the production term in the turbulent kinetic energy equation are examined to provide a structural basis for explaining the statistical behaviors. In addition, because the growth of the energy-containing motions extends to the outer layer further for the SL-flow due to the presence of a positive mean shear throughout the entire wall layer, the self-similar behavior of the energy balance between the production and transport terms with respect to the self-similar wavenumber is found far from the wall. We also find the increase in the number of uniform momentum zones in the SL-flow, revealing the hierarchical distribution of the energy-containing eddies which are composed of multiple uniform momentum zones. These coherent motions lead to the elongation of the logarithmic region for the SL-flow. Finally, investigation of the turbulent energy transfer process in a spectral domain for the SL-flow demonstrates importance of outer layer very-long structures, and these structures attribute to the energy transport process in an entire flow field.

Computational efficiency of CFD modeling for building engineering: An empty domain study

In computational wind engineering, Large Eddy Simulation (LES) operates with superior accuracy. However, this comes with a higher computational cost compared to Reynolds Averaged Navier-Stokes (RANS) models. We propose multiple strategies to reduce the computational cost without compromising accuracy. The turbulence synthesis approach is adopted for efficient inflow generation. To alleviate the issues with tedious and resource-consuming computations, the concept of ‘interpolation’ is endorsed. This approach allows the usage of identical inflow information for different arrangements of meshing. The effect of employing larger time-step and coarser mesh, on the performance of LES with wall functions, is also investigated. Several LES cases are studied, initially with special near-wall treatment, and later, with three different Subgrid Scale (SGS) models. The profiles of mean velocities, turbulence intensities, and integral length scales are examined, in addition to the velocity spectra. The horizontal homogeneity is assessed for all cases. The results show that the wall-adapting eddy viscosity (WALE) SGS model with LES can reduce the computational time at a lower error range. Besides, among three investigated hybrid models, the Detached Eddy Simulation (DES) and the Delayed Detached Eddy Simulation (DDES) provide a significant reduction in the computational cost, compared to LES with the dynamic one-equation eddy-viscosity SGS model, when the objective is to achieve a similar level of accuracy.

Investigation of the Turbulent Boundary Layer Structure over a Sparsely Spaced Biomimetic Spine-Covered Protrusion Surface.

Multiperspective particle image velocimetry was used to investigate the turbulent boundary layer structure over biomimetic spine-covered protrusion (BSCP) samples inspired by dorsal skin of pufferfish. The comparison of BSCP samples of two sparse “k-type” arrangements (aligned and staggered) with roughness height k+ = 5–7 (nearly hydraulically smooth) and smooth case were manufactured in bulk Reynolds number Reb = 37,091, 44,510. The negative value of the roughness function ΔU+ shows a downward shift of the mean velocity profile of BSCP samples, which shows a drag reduction effect. The results of turbulent statistics present strong fluctuation over the aligned case in the streamwise direction, while little influence is observed in the wall-normal and spanwise direction, which promotes turbulence stability. The same phenomenon was found based on the probability density function of fluctuation velocity that the suppression of turbulent flow is better over the staggered case. It is obvious that the shear stress induced is governed by the streamwise fluctuations. Furthermore, the Q-criterion and the λci-criterion improved with vorticity ω were introduced for vortex identification, which indicates less prograde vortex population and weaker swirling strength over BSCP samples than over the smooth one. Finally, the spatial coherent structure appeared similar and more orderly over the staggered case in the streamwise and wall-normal direction based on the analysis of two-point correlations Ruu. These results provide further guidance to reveal the mechanism of drag reduction on the BSCP surface.

Evaluation of the mixing quality of high-viscosity yield stress fluids in a tubular reactor

Abstract In this study, the mixing quality of high-viscosity yield stress fluid (Carbopol aqueous solution) under laminar and turbulent flow regimes was evaluated through a numerical experimental study. A three-dimensional computational fluid dynamics large-eddy simulation (CFD-LES) model was employed to capture large-scale vortex structures. The proposed CFD model was validated by the experimental data in terms of mean velocity profiles and velocity-time history. Thereafter, the CFD model was applied to simulate the residence time distribution using the tracking technique: tracer pulse method and step method. In addition, the non-ideal flow phenomena caused by molecular diffusion and eddy diffusion were evaluated. The effects of the rheological properties on the mixing performance were also investigated. The presented results can provide useful guidance to enhance mass transfer in reactors with high-viscosity fluids.