Two-dimensional Turbulent Boundary Layer Research Articles

New momentum integral equation applicable to boundary layer flows under arbitrary pressure gradients

By incorporating the traditionally overlooked advective term in the wall-normal momentum equation, a new momentum integral equation is developed for two-dimensional incompressible turbulent boundary layers under arbitrary pressure gradients. The classical Kármán's integral arises as a special instance of the new momentum integral equation when the pressure gradient is weak. The new momentum integral equation's validity is substantiated by direct numerical simulation data. Unlike the classical Kármán's integral, which is limited to predicting wall shear stress within mild pressure gradients, the new momentum integral equation accurately computes wall shear stress across a broad range of pressure gradients, even in the presence of strong adverse pressure gradients that lead to flow separation. Moreover, a new pressure parameter $\beta _\kappa$ is introduced through examining terms in the new momentum integral equation. This parameter naturally quantifies the pressure gradient's influence on turbulent boundary layers and offers guidance for applying the classical Kármán's integral. Additionally, to facilitate experimental determination of wall shear stress under strong pressure gradients, an approximate integral equation is proposed that relies solely on easily measurable variables. Validation against direct numerical simulation data demonstrates that this simplified equation provides reasonably accurate estimates of wall shear stress in turbulent boundary layers experiencing strong pressure gradients.

The effect of slip on the development of flow separation due to a bump in a channel based on a two-fluid turbulence model

In the paper, two-fluid turbulence model simulations were used to calculate the flow of a statistically two-dimensional turbulent boundary layer over a hump. In this work, the finite element method is used for the numerical implementation of the turbulence equations. Galerkin least squares (GLS) stabilization was used to stabilize the discretized equations. The results of the two fluid models were compared with experimental measurements by Matai, R., Durbin P. A. and the SST turbulence model which exists in the Comsol Multiphysics software package. The implementation of the Comsol Multiphysics software package showed the stability and good accuracy of the two-fluid turbulence model than the SST turbulence model.

Systematic study of the Reynolds number and streamwise spacing effects in two-dimensional square-bar rough-wall turbulent boundary layers

Two-dimensional square-bars roughness significantly influences the drag coefficient and coherent structures of turbulent boundary layer flow. A pitch ratio of 8 can induce the maximum drag effect for the turbulent boundary layer compared to other pitch ratios at a moderate friction Reynolds number. Hot-wire experiment results show that a wide range of pitch ratios has similar energy modulation by transferring energy from the largest scale structures to the smaller ones.

A new equivalent sand grain roughness relation for two-dimensional rough wall turbulent boundary layers

The effects of different geometries of two-dimensional (2-D) roughness elements in a zero pressure gradient (ZPG) turbulent boundary layer (TBL) on turbulence statistics and drag coefficient are assessed using single hot-wire anemometry. Three kinds of 2-D roughness are used: (i) circular rods with two different heights, $k= 1.6$ and 2.4 mm, and five different streamwise spacing of $s_{x}= 6k$ to $24k$ , (ii) three-dimensional (3-D) printed triangular ribs with heights of $k= 1.6$ mm and spacing of $s_{x}= 8k$ and (iii) computerized numerical control (CNC) machined sinewave surfaces with two different heights, $k= 1.6$ and 2.4 mm, and spacing of $s_{x}= 8k$ . These roughnesses cover a wide range of ratios of the boundary layer thickness to the roughness height ( $23 < \delta /k < 41$ ), where $\delta$ is the boundary layer thickness. All roughnesses cause a downward shift on the wall-unit normalised streamwise mean velocity profile when compared with the smooth wall profiles agreeing with the literature, with a maximum downward shift observed for $s_{x}= 8k$ . In the fully rough regime, the drag coefficient becomes independent of the Reynolds number. Changing the roughness height while maintaining the same spacing ratio $s_{x}/k$ exhibits little influence on the drag coefficient in the fully rough regime. On the other hand, the effective slope $(ES)$ and the height skewness $(k_{sk})$ appear to be major surface roughness parameters that affect the drag coefficient. These parameters are used in a new expression for $k_{s}$ , the equivalent sand grain roughness, developed for 2-D uniformly distributed roughness in the fully rough regime.

Experimental model-based closed-loop control of a separated boundary layer at high Reynolds number

Experimental robust closed-loop control of a two-dimensional turbulent boundary layer with substantial separation is investigated. An array of 22 round fluidic jets located upstream of the separation location is used as actuation to reattach the flow. Measurements of phase-averaged velocity with 2D2C Particle Image Velocimetry (PIV) and wall friction using hot-film anemometry are performed to characterize the flow dynamics under open-loop actuation to elaborate models of the flow transient. Different feedback model-based controllers (Proportional Integral, linear–quadratic, linear–quadratic-gaussian and H∞ regulators) are then designed and implemented experimentally. The performances in term of precision, reactivity and control cost of each of the controllers are presented and discussed. Only the H∞ controller is found to maintain high performances despite large upstream unsteady perturbations.

Verification and Validation of High-Fidelity Open-Source Simulation Tools for Supersonic Aircraft Aerodynamic Analysis

Abstract Small supersonic vehicle concepts used as research platforms to test new aerospace technologies, such as advanced propulsion systems or large sensor payloads, require major modifications to conventional, large-scale, manned, supersonic airframe design. High-fidelity numerical simulation of these concepts in academic settings often requires the use of in-house or available open-source tools instead of expensive commercial software or those with export-control restrictions. A verification and validation analysis of two widely-used open-source compressible-flow solvers, rhoCentralFoam (rCF) and su2, is performed for several flow problems relevant to the supersonic aerodynamics of small-scale, autonomous aircraft concepts. The one-dimensional shock tube problem, two-dimensional supersonic turbulent boundary layer, and three-dimensional delta wing are simulated with both solvers. The effects of flux scheme, flux limiters, and Courant–Friedrichs–Lewy (CFL) number on solution accuracy, stability, and solver speed are assessed. The solvers' limitations and their usefulness as supersonic aircraft design tools in a holistic sense are discussed.

Mean Parameters of an Incompressible Turbulent Boundary Layer on the Wind Tunnel Wall at Very High Reynolds Numbers

The mean parameters of a two-dimensional incompressible turbulent boundary layer with zero pressure gradient were measured on the smooth wall of the TsAGI T-128 transonic wind tunnel in the range Reθ = 5.3 × 104–2.9 × 105 velocity profiles, skin friction coefficient, and shape factor. Novel data were obtained in the range Reθ ≈ 2.35 × 105–2.9 × 105. At the maximum Reynolds number, the record value of the von Karman number Reτ ∼1 × 105 was reached and the logarithmic law was maintained up to y+ ≈ 1.3 × 104. The equilibrium state of the boundary layer was estimated using Clauser’s equilibrium parameter, which was G = 6.4–6.8 for high Reynolds numbers. The results of the investigation confirm the conclusions of other studies on the universality of dimensionless velocity profile in the boundary layer outer region. The results were compared with semi-empirical dependencies, direct numerical simulation (DNS) results and the results of other studies performed both on wind tunnel walls and on a flat plate. At high Reynolds numbers the agreement between the shape factor was within ΔH = ± 0.012 (± 1%), with a skin friction coefficient—Δcf= ± 0.00006 (± 3.7%).

Direct numerical simulation of a non-equilibrium three-dimensional turbulent boundary layer over a flat plate

Abstract

Turbulence in the rotating-disk boundary layer investigated through direct numerical simulations

Direct numerical simulations (DNS) are reported for the turbulent rotating-disk boundary layer for the first time. Two turbulent simulations are presented with overlapping small and large Reynolds numbers, where the largest corresponds to a momentum-loss Reynolds number of almost 2000. Simulation data are compared with experimental data from the same flow case reported by Imayama et al. (2014), and also a comparison is made with a numerical simulation of a two-dimensional turbulent boundary layer (2DTBL) over a flat plate reported by Schlatter and Örlü (2010). The agreement of the turbulent statistics between experiments and simulations is in general very good, as well as the findings of a missing wake region and a lower shape factor compared to the 2DTBL. The simulations also show rms-levels in the inner region similar to the 2DTBL. The simulations validate Imayama et al.’s results showing that the rotating-disk turbulent boundary layer in the near-wall region contains shorter streamwise (azimuthal) wavelengths than the 2DTBL, probably due to the outward inclination of the low-speed streaks. Moreover, all velocity components are available from the simulations, and hence the local flow angle, Reynolds stresses and all terms in the turbulent kinetic energy equation are also discussed. However there are in general no large differences compared to the 2DTBL, hence the three-dimensional effects seem to have only a small influence on the turbulence.

Experimental validation of heat/mass transfer analogy for two-dimensional laminar and turbulent boundary layers

Heat and mass transfer from a solid surface into a fluid stream are both diffusion driven processes. Their constant property mathematical model equations are identical for identical flow and equivalent boundary conditions, when Lewis number is equal to unity. Problems with engineering applications can be transformed from one domain to the other for practical and economic advantages. Mass transfer measurement using naphthalene sublimation can be faster, of higher resolution, economical, and more accurate than a direct heat transfer measurement. The diffusion rates by heat in air and naphthalene in air are quantitatively different. Hence, the advantages of naphthalene sublimation measurement are justified when the heat/mass transfer analogy is experimentally and/or analytically verified and an analogy factor (F=Nu/Sh) is determined.Heat/mass transfer analogy is experimentally verified in this study for two dimensional laminar and turbulent boundary layer flows over a flat plate. A thermal boundary layer technique is used to measure local heat transfer coefficient (Nu) and a naphthalene sublimation measurement is used to evaluate local mass transfer coefficient (Sh) for two similar plates subjected to the same boundary layer flow. The heat/mass transfer analogy factor is calculated using the measured Nu and Sh. It is found to agree with the analytical prediction of 0.677 within 2% for the laminar case and is found to be constant at 0.667 for the turbulent flow.

ABOUT EQUILIBRIUM FLOWS IN A TURBULENT BOUNDARY LAYER

Equilibrium and quasi-equilibrium flows in a two-dimensional turbulent boundary layer of an incompressible fluid are considered. For these regimes with the help of the experimental data for the Clauser velocity profile parameter G, a two-parametric function G(β,H) was obtained, where β (the Clauser parameter) is the non-dimensional velocity gradient and H is the ratio of the displacement thickness to the momentum thickness. This relationship allows the problem of computing the aforementioned flows to be closed. The region of maximally allowable values of H(G), where the obtained results are valid, was determined.

Empirical Modeling of Pressure Spectra in Adverse Pressure Gradient Turbulent Boundary Layers

This study investigated the unsteady aerodynamic forcing that occurs near the trailing edge of airfoils with chordwise Reynolds numbers between 2 and 6 million. Static pressure, unsteady surface pressure, and velocity measurements were made throughout the trailing-edge region of an airfoil model with three interchangeable, symmetric trailing-edge sections. The data of the present study were used to develop a new model for estimation of fluctuating surface pressure autospectra beneath two-dimensional turbulent boundary layer flow experiencing an adverse pressure gradient of variable strength. This new model has been used to compare with data from six separate experimental investigations from literature, and has been compared against a separately developed empirical model of the fluctuating surface pressure. Additional empirical modeling of the coherent extents of the fluctuating pressure field in the longitudinal and lateral directions, as well as the convection velocity, is presented as a function of nondimensional variables dependent on the adverse pressure gradient flow condition.

Numerical Investigation on Resistance Reducing Effect by Mass Injection through Porous Wall

A numerical study of flow in two-dimensional turbulent boundary layer flow with mass injection through a porous wall is conducted by Reynolds averaged numerical simulation (RANS) in this paper. Numerical results with available experimental data show that Wilcox (2006) k-w turbulence model can give good prediction in this complex flow. The comparison between mass injection and no mass injection for the local frictional resistance coefficient are performed. Numerical investigation indicates that mass injection may reduce the resistance significantly, which is confirmed by comparison of velocity profiles qualitatively.

Gross separation approaching a blunt trailing edge as the turbulence intensity increases.

A novel rational description of incompressible two-dimensional time-mean turbulent boundary layer (BL) flow separating from a bluff body at an arbitrarily large globally formed Reynolds number, Re, is devised. Partly in contrast to and partly complementing previous approaches, it predicts a pronounced delay of massive separation as the turbulence intensity level increases. This is bounded from above by a weakly decaying Re-dependent gauge function (hence, the BL approximation stays intact locally), and thus the finite intensity level characterizing fully developed turbulence. However, it by far exceeds the moderate level found in a preceding study which copes with the associated moderate delay of separation. Thus, the present analysis bridges this self-consistent and another forerunner theory, proposing extremely retarded separation by anticipating a fully attached external potential flow. Specifically, it is shown upon formulation of a respective distinguished limit at which rate the separation point and the attached-flow trailing edge collapse as [Formula: see text] and how on a short streamwise scale the typical small velocity deficit in the core region of the incident BL evolves to a large one. Hence, at its base, the separating velocity profile varies generically with the one-third power of the wall distance, and the classical triple-deck problem describing local viscous-inviscid interaction crucial for moderately retarded separation is superseded by a Rayleigh problem, governing separation of that core layer. Its targeted solution proves vital for understanding the separation process more close to the wall. Most importantly, the analysis does not resort to any specific turbulence closure. A first comparison with the available experimentally found positions of separation for the canonical flow past a circular cylinder is encouraging.

Experiments and Computations of Localized Pressure Gradients with Different History Effects

The study of high-Reynolds-number wall-bounded turbulent flows has become a very active area of research in the past decade, where several recent results have challenged current understanding. In this study, four different localized pressure gradient configurations are characterized by computing them using four Reynolds-averaged Navier–Stokes turbulence models (Spalart–Allmaras, , shear stress transport, and the Reynolds stress model) and comparing their predictions with experimental measurements of mean flow quantities and wall shear stress. The pressure gradients were imposed on high-Reynolds-number, two-dimensional turbulent boundary layers developing on a flat plate by changing the ceiling geometry of the test section. The computations showed that the shear stress transport model produced the best agreement with the experiments. It was found that what is called “numerical transition” (a procedure by which the laminar boundary conditions are transformed into inflow conditions to characterize the initial turbulent profile) causes the major differences between the various models, thereby highlighting the need for models representative of true transition in computational codes. Also, both experiments and computations confirm the nonuniversality of the von Kármán coefficient . Finally, a procedure is demonstrated for simpler two-dimensional computations that can be representative of flows with some mild three-dimensional geometries.

Measurements of roughness noise

Experiments have been performed on the roughness noise produced by a two-dimensional turbulent wall jet boundary layer flowing over short fetches of sandpaper roughness. A range of rough surface sizes were studied from hydrodynamically smooth through fully rough. Velocity measurements were made to document the form of the wall jet boundary layer and the influence of the roughness upon it. Acoustic measurements showed background noise levels to be very low so that the sound produced by the rough surfaces could be clearly detected with signal to noise ratios as large as 20 dB. Even hydrodynamically smooth roughness was found to produce noise, conclusively indicating the presence of scattering as a source mechanism. Variations of the roughness noise spectra with flow speed and roughness size are found to be inconsistent with any simple parameter scaling. Boundary layer wall pressure fluctuation measurements made within the roughness fetches reveal a spectral form quite different than the roughness noise, and fluctuation levels some 50–70 dB higher. Despite these differences the wall pressure and roughness noise are found to be very simply related, at least at lower frequencies (<6 kHz). The roughness noise spectrum varies closely as the product of the wall pressure spectrum, the frequency squared, and the mean-square roughness height. This is the scaling predicted by scattering theory and implies a major simplification to the problem of roughness noise prediction for stochastic surfaces.

Effect of Upstream Shear on Flow and Heat (Mass) Transfer Over a Flat Plate—Part I: Velocity Measurements

A parametric study investigates the effects of wall shear on a two-dimensional turbulent boundary layer. A belt translating along the direction of the flow imparts the shear. Velocity measurements are performed at 12 streamwise locations with four surface-to-freestream velocity ratios (0, 0.38, 0.52, and 0.65) and a momentum-based Reynolds number between 770 and 1776. The velocity data indicate that the location of the “virtual origin” of the turbulent boundary layer “moves” downstream toward the trailing edge of the belt with increasing surface velocity. The highest belt velocity ratio (0.65) results in the removal of the “inner” region of the boundary layer. Measurements of the streamwise turbulent kinetic energy show an inner scaling at locations upstream and downstream of the belt, and the formation of a new self-similar structure on the moving surface itself. Good agreement is observed for the variation in the shape factor (H) and the skin friction coefficient (cf) with the previous studies. The distribution of the energy spectrum downstream of the belt indicates peak values concentrated around 1 kHz for the stationary belt case in the near wall region (30&lt;y+&lt;50). However, with increasing belt velocity, this central peak plateaus over a wide frequency range (0.9–4 kHz).

An Integral Solution for Heat Transfer in Accelerating Turbulent Boundary Layers

An equilibrium thermal wake strength parameter is developed for a two-dimensional turbulent boundary layer flow and is then used in the combined thermal law of the wall and the wake to give an approximate temperature profile to insert into the integral form of the thermal energy equation. After the solution of the integral x momentum equation, the integral thermal energy equation is solved for the local Stanton number as a function of position x for accelerating turbulent boundary layers. A simple temperature distribution in the thermal “superlayer” is part of the present modeling. The analysis includes a dependence of the hydrodynamic and thermal wake strengths on the momentum thickness and enthalpy thickness Reynolds numbers, respectively. An approximate dependence of the turbulent Prandtl number, in the “log” region, on the strength of the favorable pressure gradient is proposed and incorporated into the solution. The resultant solution for the Stanton number distribution in accelerated turbulent flows is compared with experimental data in the literature. A comparison of the present predictions is also made to a finite difference solution, which uses the turbulent kinetic energy—turbulent dissipation model of turbulence, for a few cases of accelerating flows.

Turbulence in a three‐dimensional wall‐bounded shear flow

AbstractA new turbulent flow with distinct three‐dimensional characteristics has been designed in order to study the impact of mean‐flow skewing on the turbulent coherent vortices and Reynolds‐averaged statistics. The skewing of a unidirectional plane Couette flow was achieved by means of a spanwise pressure gradient. Direct numerical simulations of the statistically steady Couette–Poiseuille flow enabled in‐depth explorations of the turbulence field in the skewed flow. The imposition of a modest spanwise gradient turned the mean flow about 8° away from the original Couette flow direction and this turning angle remained nearly the same over the entire cross section. Nevertheless, a substantial non‐alignment between the turbulent shear stress angle and the mean velocity gradient angle was observed. The structure parameter turned out to slightly exceed that in the pure Couette flow, contrary to the observations made in some other three‐dimensional shear flows.Coherent flow structures, which are known to be associated with the Reynolds shear stress in near‐wall regions, were identified by the λ2‐criterion. Instantaneous and ensemble‐averaged vortices resembled those found in the unidirectional Couette flow. In the skewed flow, however, the vortex structures were turned to align with the local mean‐flow direction. The conventional symmetry between Case 1 and Case 2 vortices was broken due to the mean‐flow three‐dimensionality. The turning of the coherent vortices and the accompanying symmetry‐breaking gave rise to secondary and tertiary turbulent shear stress components. By averaging the already ensemble‐averaged shear stresses associated with Case 1 and Case 2 vortices in the homogeneous directions, a direct link between the educed near‐wall structures and the Reynolds‐averaged turbulent stresses was established. These observations provide evidence in support of the hypothesis that the structural model proposed for two‐dimensional turbulent boundary layers remains valid also in flows with moderate mean three‐dimensionality. Copyright © 2009 John Wiley &amp; Sons, Ltd.

0203 突起列による平板境界層の乱流遷移 : 乱流くさび内の乱流量の発達(OS2-1 乱流の計測・解析・制御,オーガナイズドセッション)

Experiments were performed to investigate the effect of a line of roughness elements on flat-plate boundary layer transition. Each roughness element was a cylinder 2 mm in both diameter and height. Eleven elements formed a row in the spanwise direction. Wedge-shaped turbulent regions ("turbulence wedges") developed downstream from the respective roughness elements. Further downstream, two adjacent wedges merged together and then a two-dimensional turbulent boundary layer was formed. Mean and fluctuating velocities were measured by hot-wire anemometers. From the velocities an intermittency factor and turbulent dissipation, etc. were obtained. The manner how the three-dimensional turbulence wedges change into two-dimensional boundary layer were investigated. At first, the fluctuating velocity on the generator of roughness element and center between roughness elements overshot Spalart distribution. And further downstream, both distribution of fluctuating velocity undershot it.