Reynolds-number-dependence of the maximum in the streamwise velocity fluctuations in wall turbulence

A survey is made of the standard deviation of the streamwise velocity fluctuations in near-wall turbulence and in particular of the Reynolds-number-dependency of its peak value. The following canonical flow geometries are considered: an incompressible turbulent boundary layer under zero pressure gradient, a fully developed two-dimensional channel and a cylindrical pipe flow. Data were collected from 47 independent experimental and numerical studies, which cover a Reynolds number range of R _θ=U _∞ θ/v=300−20,920 for the boundary layer with θ the momentum thickness and R ⁺=u _*R/v=100-4,300 for the internal flows with R the pipe radius or the channel half-width. It is found that the peak value of the rms-value normalised by the friction velocity, u _*, is within statistical errors independent of the Reynolds number. The most probable value for this parameter was found to be 2.71±0.14 and 2.70±0.09 for the case of a boundary layer and an internal flow, respectively. The present survey also includes some data of the streamwise velocity fluctuations measured over a riblet surface. We find no significant difference in magnitude of the normalised peak value between the riblet and smooth surfaces and this property of the normalised peak value may for instance be exploited to estimate the wall shear stress from the streamwise velocity fluctuations. We also consider the skewness of the streamwise velocity fluctuations and find its value to be close to zero at the position where the variance has its peak value. This is explained with help of the equations of the third-order moment of velocity fluctuations. These results for the peak value of the rms of the streamwise velocity fluctuations and also the coincidence of this peak with the zero value of the third moment can be interpreted as confirmation of local equilibrium in the near-wall layer, which is the basis of inner-layer scaling. Furthermore, these results can be also used as a requirement which turbulence models for the second and triple velocity correlations should satisfy. The authors are indebted to Prof. P. Bradshaw for making available his list of references on this topic and for his remarks on “active” and “inactive” motions. We also gratefully acknowledge discussions with Prof. I. Castro regarding the value of σ _u ⁺ above rough walls.     

The response of a turbulent boundary layer to different shaped transverse grooves

The response of a turbulent boundary layer to three different shaped transverse grooves was investigated at two values of momentum thickness Reynolds numbers ( R =1000 and 3000). A 20-mm wide square, semicircular and triangular groove with depth to width ( d / w) ratio of unity was used. In general, the effects of the grooves are more significant at the higher R , with the most pronounced effects caused by the square groove. An increase in wall shear stress _w was observed just downstream of the groove for all three shapes. The increase in _w is followed by a small decrease in _w below the smooth-wall value before it relaxes back to the corresponding smooth-wall value at x / ₀3. At the higher R , the maximum increase in _w for the square groove is about 50% higher than for the semicircular groove and almost twice that for the triangular groove. The effect of the square groove on U / U ₀, u / U ₀ and v / U ₀ is much more significant than the effect of the semicircular and triangular grooves. There is an increase in the bursting frequency ( f _B⁺) on the grooved-wall compared to the smooth-wall case. The distribution of f _B⁺ downstream of the different shaped grooves is similar to the _w distribution.Symbols C _f skin friction coefficient, C _f2 _w/( ( U ₀)²) - C _f,0 skin friction coefficient on the smooth wall - d groove depth - D _h diameter of the idealized primary eddy inside the groove - D _h,s diameter of the idealized secondary eddies inside the groove - d _i internal layer thickness - E turbulent energy spectrum - f _B bursting frequency - f _B⁺ normalized bursting frequency, f _B⁺ f _B/( u )² - k wave number, k =2f/ U - q _i ⁺ contributing quadrant to the total Reynolds stress – uv , q _i ⁺ – uv _i /( u )², i =1, 2, 3, 4 - R Reynolds number based on , R U ₀ / - R Reynolds number based on , R U ₀ / - U mean velocity in the streamwise direction - U ₀ free stream velocity - U ⁺ normalized U by inner variable, U ⁺ U / u - u root-mean-square of velocity fluctuation in the streamwise direction - u ⁺ normalized u by inner variable, u ⁺ u / u - u friction velocity, u ( _w/ )^0.5 - – uv Reynolds stress - v root-mean-square of velocity fluctuation in the wall-normal direction - w groove width - x streamwise coordinate measured from the groove trailing edge - y wall-normal coordinate - y ⁺ normalized y by inner variables, y ⁺ yu / Greek symbols boundary layer thickness - ₀ boundary layer thickness just upstream of the groove, unless otherwise stated - fluid kinematic viscosity - momentum thickness - fluid density - _w wall shear stress     

Simultaneous velocity and concentration field measurements of passive-scalar mixing in a confined rectangular jet

Simultaneous velocity and concentration fields in a confined liquid-phase rectangular jet with a Reynolds number based on the hydraulic diameter of 50,000 (or 10,000 based on the velocity difference between streams and the jet exit dimension) and a Schmidt number of 1,250 were obtained by means of a combined particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) system. Data were collected at the jet exit and six further downstream locations. The velocity and concentration field data were analyzed for flow statistics such as turbulent fluxes, turbulent viscosity and diffusivity, and turbulent Schmidt number (Sc _T ). The streamwise turbulent flux was found to be larger than the transverse turbulent flux, and the mean concentration gradient was not aligned with the turbulent flux vector. The average Sc _T was found to vary both in streamwise and in cross stream directions and had a mean value around 0.8, a value consistent with the literature. Spatial correlation fields of turbulent fluxes and concentration were then determined. The R _u′ϕ′ correlation was elliptical in shape with a major axis tilted downward with respect to the streamwise axis, whereas the R _v′ϕ′ correlation was an ellipse with a major axis aligned with the cross-stream direction. Negative regions of R _u′ϕ′ were observed in the outer streams, and these negatively correlated regions decayed with downstream distance and finally disappeared altogether. The R _ϕ′ϕ′ correlation field was found to be an ellipse with the major axis inclined at about 45° with respect to the streamwise direction. Linear stochastic estimation was used to interpret spatial correlation data and to determine conditional flow structures. It is believed that a vortex street formed near the splitter plate is responsible for the negatively correlated region observed in the R _u′ϕ′ spatial correlations of turbulent fluxes. A positive concentration fluctuation event was observed to correspond to a finger of nearly uniform concentration fluid reaching out into the outer stream, whereas a negative event corresponds to a pocket of nearly uniform fluid being entrained from the outer stream into the center jet region. Large-scale vortical structures were observed in the conditional velocity fields with an elliptical shape and a streamwise major axis. The growth of the structure size increased linearly initially but then grew more slowly as the flow transitioned toward channel flow. Support of this work was provided by the National Science Foundation through grants CTS-9985678 and CTS-0336435 and by the Dow Chemical Company. The author greatly acknowledge Charles Lipp at Dow Chemical and Ken Junk at Emerson Fisher for their valuable assistance in the design and construction of the flow system.     

Turbulent boundary layer over a smooth wall with widely separated transverse square cavities

Laser-Doppler velocimetry (LDV) measurements and flow visualizations are used to study a turbulent boundary layer over a smooth wall with transverse square cavities at two values of the momentum thickness Reynolds number (R =400 and 1300). The cavities are spaced 20 element widths apart in the streamwise direction. Flow visualizations reveal a significant communication between the cavities and the overlying shear layer, with frequent inflows and ejections of fluid to and from cavities. There is evidence to suggest that quasi-streamwise near-wall vortices are responsible for the ejections of fluid out of the cavities. The wall shear stress, which is measured accurately, increases sharply immediately downstream of the cavity. This increase is followed by a sudden decrease and a slower return to the smooth wall value. Integration of the wall shear stress in the streamwise direction indicates that there is an increase in drag of 3.4% at bothR .Nomenclature C _f skin friction coefficient - C _fsw friction coefficient for a continuous smooth wall - k height of the cavity - k ⁺ ku / - R Reynolds number based on momentum thickness (U ₁ /v) - Rx Reynolds number based on streamwise distance (U ₁ x/) - s streamwise distance between two cavities - t time - t ⁺ tu ₂ / - U ₁ freestream velocity - mean velocity inx direction - u,v,w rms turbulent intensities inx,y andz directions - u local friction velocity - u _sw friction velocity for a continuous smooth wall - w width of the cavity - x streamwise co-ordinate measured from the downstream edge of the cavity - y co-ordinate normal to the wall - z spanwise co-ordinate - y ⁺ yu / - boundary layer thickness - ₀ boundary layer thickness near the upstream edge of the cavity - _i thickness of internal layer - kinematic viscosity of water - ⁺ zu / - momentum thickness     

Near-field flow structures and sideband instabilities of an initially laminar plane jet

The intrinsic characteristics of coherent structures in the near field of a plane jet are extensively studied by hot-wire measurements. The instability modes which are responsible for the dynamics of the coherent structures are found to exhibit distinct evolution characters at different transverse positions of the shear layer along downstream direction. The occurrence of multiple peaks in the energy spectra depicts the formation of the sideband instabilities in the early stage of the jet flow field. These sideband instabilities are investigated to be induced by the mechanisms of the nonlinear interactions between neighboring fundamental and subharmonic instabilities, and the feedback effects of the preferred mode near the end of the potential core. Also, from the spatial distributions of the instability modes over the jet flow field, Ho's subharmonic evolution model (1982) is further examined with more interpretations.List of symbols E (f) energy content of streamwise velocity fluctuation at spe cific frequency - f _e excitation frequency - f ₀ fundamental frequency - f _p preferred frequency - f _r response frequency in an excited jet - H height of the plane jet at the exit - U streamwise mean velocity - U ₀ mean velocity at the nozzle exit - U _c mean velocity at the jet center line - u streamwise RMS velocity fluctuation - u _p peak streamwise velocity fluctuation alongY axis - u (f) amplitude of streamwise velocity fluctuation at specific frequency - X, Y streamwise and transverse coordinates - Y _a transverse position whereU = aU _c ,a = 0.99, 0.9,..., etc. - Y _c transverse position at the jet center line - ₀ initial instability wave length (=U ₀/2f ₀) - 0 momentum thickness - ₀ initial boundary layer momentum thickness A version of this paper was presented at the 11th Symposium on Turbulence, University of Missouri-Rolla, Oct. 17–19, 1988     

Analysis of Streamwise Velocity Fluctuations in Turbulent Pipe Flow with the Use of an Ultrasonic Doppler Flowmeter

Data collected from several studies of experimental and numerical nature in wall-bounded turbulent flows and in particular in internal flows (channel and pipe flows, Mochizuki and Nieuwstadt [1]) at different Reynolds numbers R ⁺(Ru _*/ν), indicate that: (i) the peak of the rms-value (normalized by u _*) of the streamwise velocity fluctuations (σ _u ⁺|_peak) is essentially independent of the Reynolds number, (ii) the position of the rms peak value (y ⁺|_peak) is weakly dependent of the Reynolds number, (iii) the skewness of the streamwise velocity fluctuations (S _u ) is close to zero at the position in which the variance has its peak. A series of measurements of streamwise velocity fluctuations has been performed in turbulent pipe flow with the use of an Ultrasonic Doppler Velocimeter and our results support those reported in [1]. This revised version was published online in July 2006 with corrections to the Cover Date.     

Streamwise pseudo-vortical structures and associated vorticity in the near-wall region of a wall-bounded turbulent shear flow

Streamwise pseudo-vortical motions near the wall in a fully-developed two-dimensional turbulent channel flow are clearly visualized in the plane perpendicular to the flow direction by a sophisticated hydrogen-bubble technique. This technique utilizes partially insulated fine wires, which generate hydrogen-bubble clusters at several distances from the wall. These flow visualizations also supply quantitative data on two instantaneous velocity components, and w, as well as the streamwise vorticity, _x . The vorticity field thus obtained shows quasi-periodicity in the spanwise direction and also a double-layer structure near the wall, both of which are qualitatively in good agreement with a pseudo-vortical motion model of the viscous wall-region.List of symbols C _i ,c _i ,d _i constants in Eqs. (2), (3) and (4) - H channel width (m) - Re _H Reynolds number (= U _c H/) - Re Reynolds number (= U _c /) - T period (s) - t time (s) - U mean streamwise velocity (m/s) - U _c center-line velocity (m/s) - u friction velocity (m/s) - u, , w velocity fluctuations (m/s) - x, y, z coordinates (m) - ^* displacement thickness (m) - momentum thickness (m) - mean low-speed streak spacing (m) - kinematic viscosity (m²/s) - phase difference - _x streamwise vorticity fluctuation (1/s) - ( )⁺ normalized by u and - (^–) root mean square value - (^–) statistical average This paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

Laminarisation and re-transition of a turbulent boundary layer subjected to favourable pressure gradient

 Experimental results are reported for the response of an initially turbulent boundary layer (Re _θ≈1700) to a favourable pressure gradient with a peak value of K≡(−υ/ρU ³ _E ) dp/dx equal to 4.4×10^-6. In the near-wall region of the boundary layer (y/δ<0.1) the turbulence intensity u′ scales roughly with the free-stream velocity U _E until close to the location where K is a maximum whereas in the outer region u′ remains essentially frozen. Once the pressure gradient is relaxed, the turbulence level increases throughout the boundary layer until K falls to zero when the near wall u′ levels show a significant decrease. The intermittency γ is the clearest indicator of a fundamental change in the turbulence structure: once K exceeds 3×10^-6, the value of γ in the immediate vicinity of the wall γ _s falls rapidly from unity, reaches zero at the location where K again falls below 3×10^-6 and then rises back to unity. Although γ is practically zero throughout the boundary layer in the vicinity of γ _s =0, the turbulence level remains high. The explanation for what appears to be a contradiction is that the turbulent frequencies are too low to induce turbulent mixing. The mean velocity profile changes shape abruptly where K exceeds 3×10^-6. Values for the skin friction coefficient, based upon hot-film measurements, peak at the same location as K and fall to a minimum close to the location where K drops back to zero. Received: 28 January 1998/Accepted: 8 April 1998     

Double row vortical structures in the near field region of a plane wall jet

The qualitative and quantitative behaviour of double row vortical structures in the near field region of a plane wall jet are studied experimentally by flow visualization and hot-wire measurements. Ensemble averaging is employed to investigate the interaction of vortices with the wall. In the flow visualization study, a double row vortical structure, which includes a primary vortex formed in the outer layer region and a secondary vortex induced in the inner layer region, and the vortex lift-off phenomenon are clearly observed during the development of the wall jet. The phase averaged results of the velocity measurements show that the instability leading to induction of the secondary vortex is stimulated by the primary vortex. In the early stage of wall jet transition, the inflection point of the inner layer velocity profile moves transversely from the wall surface to the inner layer region due to passage of the well-organized primary vortex in the outer layer region. The inner layer instability is thus induced and the instability wave rolls up to form the secondary vortex. Furthermore, the secondary vortex will convect downstream faster than the primary vortex, and this difference in convective speed will lead to the subsequent phenomenon of vortex lift-off from the wall surface.List of symbols A1, A2, . . . primary vortex - B1,B2, . . . secondary vortex - fe forcing frequency - f fundamental frequency - H nozzle exit height - Re Reynolds number,U _j H/ - T period of the referred signal (=13.5 ms) - t, t time scale - U streamwise mean velocity - U _c convection speed - U _j jet exit velocity - U _m local maximum velocity - ut' streamwise turbulence intensity - uv turbulent shear stress - V transverse mean velocity - v transverse turbulence intensity - X streamwise coordinate - Y transverse coordinate - X _Ai streamwise location of vortexAi - X _Bi streamwise location of vortexBi - X _ave averaged streamwise location of the vortex - Y _m wall jet inner layer width, the distance from wall to whereU=U _m - Y _1/2 wall jet half-width, the distance from wall to whereU=1/2U _m in outer layer region - t time interval (=0.267 s) - phase averaged value     

Some flow characteristics of lobed forced mixers at higher velocity ratios

The flow characteristics of two types of lobed forced mixers, the unscalloped and the scalloped mixers, have been examined at velocity ratios higher than unity, in relation to the variation of mass flux uniformity, the decay of the streamwise vorticity, the variation of turbulent kinetic energy and the growth of the shear layer with distance from the trailing edge. Three trailing edge configurations have also been considered for each type of mixer, namely a square wave, a semi-circular wave and a triangular wave. The analysis showed that the strength of the streamwise vorticity shed at the trailing edge and the subsequent decaying rate with downstream distance are found to be very important in studying the mixing effectiveness of the lobed mixers.List of Symbols C _I normalized streamwise circulation, _s/U _r h tan - _s streamwise circulation - k turbulent kinetic energy = 1/2(u²+v²+w²) - Re Reynolds number, U _r /=2.27×10⁴ - h (=) Lobe height, 33 mm - U ₁, U ₂ mean velocity of the slow and fast streams - U _r reference mean velocity, (U ₁ + U ₂)/2 = 10 m/s - U, u streamwise mean and the corresponding rms velocities - V, v horizontal mean and the corresponding rms velocities - W, w vertical mean and the corresponding rms velocities - x,y, z streamwise, horizontal and vertical directions - A _wake cross-sectional area bounded by the wake region. The wake region boundary is defined at the region bounded by one half of a lobe along the y/ direction and at the locations along the z/ direction where U ₂/U _r2 and U ₁/U _r1<0.95. - nominal lobe wavelength, 33 mm - half of the included divergent angle of the penetration region, 22° - U uniformity factor - momentum thickness Financial supports from the Applied Research Grant is gratefully acknowledged. The contribution of Mr. J. K. L. Teh, Dr. J. H. Yeo and Mr. T. H. Yip to the work presented here are sincerely appreciated.     

Feedback control of vortex shedding from a circular cylinder by cross-flow cylinder oscillations

Feedback control of vortex shedding from a circular cylinder in a uniform flow at moderate Reynolds numbers is studied experimentally with the cylinder subjected to feedback cylinder oscillations in cross-flow direction. The cylinder oscillation is digitally phase shifted with respect to the shedding vortex and is controlled by velocity feedback from the shear layer of the cylinder wake. Possible attenuation of vortex shedding is demonstrated by hot-wire measurements of the flow field and its mechanisms are studied by simultaneous data sampling and flow visualization with the smoke wire method and a laser-sheet illumination technique. Measurement results reveal substantial reduction in the fluctuating reference velocity at the optimum phase control. Flow visualization study indicates that the shear layer roll-up and the eventual vortex formation are dynamically attenuated under the control which results in a modification of the near wake.List of symbols A amplitude of cylinder oscillation - D cylinder diameter - E _u power spectrum function for fluctuating velocity u - frequency - R radius of circular cylinder - t time - u streamwise mean velocity - u streamwise fluctuating velocity - U streamwise mean velocity of main flow - u _r mean reference velocity - u _r fluctuating reference velocity - u _rf fluctuating reference velocity after filtering - y _c cylinder displacement - x, y, z coordinates from the cylinder center (Fig. 1) - feedback coefficient - phase lag The authors would like to express thanks to Professor K. Nagaya for his advice for designing electromagnetic actuators in the present experiments.     

Effect of velocity ratio on plane mixing layer development: Influence of the splitter plate wake

An experimental study has been conducted to investigate the effect of velocity ratio on the approach of a plane mixing layer to self-similarity. Plane mixing layers with five different velocity ratios (0.5, 0.6, 0.7, 0.8 and 0.9) were generated in a newly designed mixing layer wind tunnel with both initial boundary layers tripped. For each velocity ratio, mean flow and turbulence measurements were obtained at eight streamwise locations with a single cross-wire probe. The results indicate that the splitter plate wake plays a very dominant and, in some cases, a lasting role in the development of the mixing layer. For velocity ratios between 0.5 and 0.7, self-similarity of the mixing layer was observed with the asymptotic states comparable. Mixing layers with the higher velocity ratios failed to achieve a self-similar state within the measurement domain, although a slow approach to it was apparent. The development distance decreased with increasing velocity ratio up to 0.7, after which it appeared to increase. Almost all of the observed effects may be attributed to the presence of the splitter plate wake and its complex interaction with the mixing layer.List of symbols C _f boundary layer skin friction coefficient - H boundary layer shape factor - r velocity ratio of the two streams, (=U ₂/U ₁) - Re _L Reynolds number, (=UL/v) - R correlation coefficient in least squares fit - U, V, W mean velocity in the X, Y, Z directions, respectively - U ^* velocity parameter, [=(U–U ₂)/(U ₁–U ₂)] - U ₀ velocity difference, (=U ₁–U ₂) - U _e free-stream velocity in the wind tunnel - u, , w fluctuating velocity components in the X, Y, Z directions, respectively - u, , w instantaneous velocity in the X, Y, Z directions, respectively, e.g. u=U+u - X ₀ virtual origin of the mixing layer - X, Y, Z cartesian coordinates for streamwise, normal, and spanwise directions, respectively - Y ₀ centerline of mixing layer from error function fit - mixing layer width from error function fit - ₉₉ initial boundary layer thickness - similarity coordinate [=(Y–Y ₀)/] - initial boundary layer momentum thickness - modified velocity ratio [=(1–r)/(1+r)] - _n initial instability wavelength in the mixing layer - spreading parameter [=1/(d/dX)] - ₀ spreading parameter for single-stream mixing layer - - (overbar) Time-averaged quantity - ( )_max maximum value at given X-station - ( )_min minimum value at given X-station - ( )₁ value for high-speed side - ( )₂ value for low-speed side     

Effects of the bifurcation angle on the steady flow structure in model saccular aneurysms

The effects of the bifurcation angle on the steady flow structure in a straight terminal aneurysm model with asymmetric outflow through the branches have been characterized quantitatively in terms of laser-Doppler velocimetry (LDV)-measured mean velocity and fluctuating intensity distributions. The bifurcation angles investigated were 60°, 90°, and 140° and the Reynolds number based on the bulk average velocity and diameter of the afferent vessel was 500. It is found that the size of the recirculating zones in the afferent vessel, the flow activity (both mean and fluctuating motions) inside the aneurysm, and the shear stresses acting on the aneurysmal wall increase with increasing bifurcation angle. More importantly, both LDV-measured and flow-visualized results of the present study suggest the presence of a critical bifurcation angle below which the aneurysm is susceptible to thrombosis, whereas above this the aneurysm is prone to progression or rupture.List of symbols a aneurysm height - b distance from orifice to fundus - c orifice diameter - D afferent conduit diameter - d fundus diameter - Hz frequency unit = cycle/second - L length of bifurcation zone - Re Reynolds number = U · D/v - U streamwise mean velocity - U _m streamwise bulk mean velocity - u streamwise fluctuating component - X ^* normalized streamwise coordinate: X ^* 0: X ^* = X/a; X ^* <0: X ^* = X/L - Y ^* normalized transverse coordinate: Y ^* = Y/D - Z ^* normalized spanwise coordinate: Z ^* = Z/D - kinematic viscosity - _b angle of bifurcation - _c critical bifurcation angle     

A Navier–Stokes Approximation of the 3D Euler Equation with the Zero Flux on the Boundary

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T ₀ > 0, ν ₀ > 0 and a unique continuous family of strong solutions u ^ν (0 ≤ ν < ν ₀) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T ₀). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ^ν and curl ² u ^ν .      

Streamwise evolution of an inclined cylinder wake

The streamwise evolution of an inclined circular cylinder wake was investigated by measuring all three velocity and vorticity components using an eight-hotwire vorticity probe in a wind tunnel at a Reynolds number Re_d of 7,200 based on free stream velocity (U _∞) and cylinder diameter (d). The measurements were conducted at four different inclination angles (α), namely 0°, 15°, 30°, and 45° and at three downstream locations, i.e., x/d = 10, 20, and 40 from the cylinder. At x/d = 10, the effects of α on the three coherent vorticity components are negligibly small for α ≤ 15°. When α increases further to 45°, the maximum of coherent spanwise vorticity reduces by about 50%, while that of the streamwise vorticity increases by about 70%. Similar results are found at x/d = 20, indicating the impaired spanwise vortices and the enhancement of the three-dimensionality of the wake with increasing α. The streamwise decay rate of the coherent spanwise vorticity is smaller for a larger α. This is because the streamwise spacing between the spanwise vortices is bigger for a larger α, resulting in a weak interaction between the vortices and hence slower decaying rate in the streamwise direction. For all tested α, the coherent contribution to [`(v<sup>2</sup>)] \overline{{v^{2}}} is remarkable at x/d = 10 and 20 and significantly larger than that to [`(u²)] \overline{{u^{2}}} and [`(w<sup>2</sup>)]. \overline{{w^{2}}}. This contribution to all three Reynolds normal stresses becomes negligibly small at x/d = 40. The coherent contribution to [`(u²)] \overline{{u^{2}}} and [`(v²)] \overline{{v^{2}}} decays slower as moving downstream for a larger α, consistent with the slow decay of the coherent spanwise vorticity for a larger α.     

Second-order nonlinear differential equations with infinite set of periodic solutions

For the differential equation u″ = f(t, u, u′), where the function f: R × R ² → R is periodic in the first variable and f (t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found. Published in Neliniini Kolyvannya, Vol. 11, No. 4, pp. 495–500, October–December, 2008.     

Near-wall hot-wire measurements

This work continues the studies of Khoo et al. (Exp. Fluids 29: 448–460, 2001), where experiments were performed in turbulent-channel and flat-plate boundary-layer flows using near-wall hot-wire probes. The probability density function (pdf) of the wall-shear stress and streamwise velocity fluctuations in the viscous sublayer, buffer region and beyond were compared and analyzed. The convective velocity U _c of the streamwise velocity fluctuations in the very near-wall region was obtained using a two-point correlation technique. It was found that in the viscous sublayer, U _c is approximately constant at 13u _τ and 15u _τ , respectively, for the channel and boundary-layer flows. Spectra data for the viscous sublayer are presented for the first time, and the normalized spectral plots for different flow conditions collapse at high frequencies or wavenumbers, thus indicating the possible presence of small-scale universality at different Reynolds numbers. The integral time scale corresponding to the streamwise velocity fluctuations in the viscous sublayer is also presented. Received: 18 October 2000/Accepted: 2 April 2001     

Streamwise evolution of a high-Schmidt-number passive scalar in a turbulent plane wake

The concentration fluctuation c of diluted fluorescein dye, a high-Schmidt-number passive scalar (Sc=ν/D ≈ 2000, ν and D are the fluid momentum and dye diffusivities, respectively), is measured in the wake of a circular cylinder using a single-point laser-induced fluorescence (SPLIF) technique. The streamwise decay rate of the mean and rms values of c is slow in comparison to that of θ, the temperature fluctuation for which the molecular Prandtl number Pr=ν/κ is about 0.7 (κ is the thermal diffusivity). The comparison between mean and rms distributions of c and θ highlights the combined role the Reynolds and Schmidt numbers play in terms of dispersing the scalar. The streamwise evolution of the probability density functions (pdfs) of c and θ suggest that while p(θ) is approximately Gaussian in the intermediate wake (x/d ≈ 80), p(c) is strongly non-Gaussian, and depends on both x/d and Re. The skewness of c is larger than that of θ along the wake centreline. Arguably, the asymmetry of p(c) reflects the relatively strong organisation of the large-scale motion in the far-wake. Received: 27 July 2000/Accepted: 22 December 2000     

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

In the intermixing region behind circular cylinders with stepwise change of the diameter

The flow within the intermixing region behind circular cylinders with stepwise change of the diameter of diameter ratio d/D of 0.5 has been examined. Based on the statistical analysis and conditional sampling of the velocity fluctuations and of flow visualization, the vortex wakes associated with the big and small cylinders have been established. Both wakes are found under the dominant primary mode, which corresponds to the vortex shedding Strouhal number of two dimensional cylinder, and the less dominant secondary mode. The Strouhal number of the secondary mode of the big vortex wake is higher than that of the primary mode and the opposite is found for the small vortex wake. Both vortex wakes and their modes are found convecting downstream and into region behind the other cylinder. Both wakes are observed to be different from that of two dimensional cylinder.List of symbols D, d diameter of big and small cylinder - f frequency - R ₁₂ (f) cross-power spectral function - R ₁₁, R ₂₂ auto-power functions - Re _D, Re_d Reynolds numbers U ₀ D/v, U ₀ d/v - t time relative to triggering instant - U ₀ freestream mean velocity - U, V, W streamwise, lateral and spanwise mean velocity, respectively - u, v, w streamwise, lateral and spanwise velocity fluctuations, respectively - U _f phase velocity - U _T convection velocity - u _R, v_r recovered u and v velocity fluctuations - uv Reynolds stress - x, y, z streamwise, lateral, and spanwise coordinates, respectively - separation - ₁₂ ² (f) coherence function - _R recovered coherent vorticity fluctuation - phase - ₁₂ (f) phase spectral function