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     

A family of high-order accuracy explicit difference schemes with branching stability for solving 3-D parabolic partial differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｏｆｔｅｎｍｅｅｔｔｈｅｐｒｏｂｌｅｍｏｆｓｏｌｖｉｎｇｅｑｕａｔｉｏｎｏｆｐａｒａｂｏｌｉｃｔｙｐｅｉｎｍａｎｙｆｉｅｌｄｓｓｕｃｈａｓｓｅｅｐａｇｅ ,ｄｉｆｆｕｓｉｏｎ ,ｈｅａｔｃｏｎｄｕｃｔｉｏｎａｎｄｓｏｏｎ .Ｉｎｔｈｅｃａｓｅｏｆ3＿ｄｉｍｅｎｓｉｏｎ ,ｔｈｅｍｏｄｅｌｉｓａｎｉｎｉｔｉａｌａｎｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍａｓｆｏｌｌｏｗｓ: ｕ ｔ = 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 　　　　 (0 <ｘ,ｙ,ｚ<1 ;ｔ>0 ) ,ｕ(ｘ ,ｙ,ｚ,0 ) =φ(ｘ ,ｙ ,ｚ)…     

On near-wall hot-wire measurements

A specially constructed hot-wire probe was used to obtain very near-wall velocity measurements in both a fully developed turbulent channel flow and flat plate boundary layer flow. The near-wall hot-wire probe, having been calibrated in a specially constructed laminar flow calibration rig, was used to measure the mean streamwise velocity profile, distributions of streamwise and spanwise intensities of turbulence and turbulence kinetic energy k in the viscous sublayer and beyond; these distributions compare very favorably with available DNS results obtained for channel flow. While low Reynolds number effects were clearly evident for the channel flow, these effects are much less distinct for the boundary layer flow. By assuming the dissipating range of eddy sizes to be statistically isotropic and the validity of Taylor's hypothesis, the dissipation rate ɛ _iso in the very near-wall viscous sublayer region and beyond was determined for both the channel and boundary layer flows. It was found that if the convective velocity U _c in Taylor's hypothesis was assumed to be equal to the mean velocity Uˉ at the point of measurement, the value of (ɛ⁺ _iso)₁ thus obtained agrees well with that of (ɛ ⁺)_DNS for y ⁺ ≥ 80 for channel flow; this suggests the validity of assuming U _c=Uˉ and local isotropy for large values of y ⁺. However, if U _c was assumed to be 10.6u _τ , the value of (ɛ⁺ _iso)₂ thus obtained was found to compare reasonably well with the distribution of (ɛ⁺ _iso)_DNS for y ⁺≤ 15. Received: 31 May 1999/Accepted: 20 December 1999     

The high accuracy explicit difference scheme for solving parabolic equations 3-dimension

ＩｎｔｒｏｄｕｃｔｉｏｎＦｒｏｍｐｒａｃｔｉｃａｌｐｒｏｂｌｅｍ,ｗｅｃｏｕｌｄｃｏｎｃｌｕｄｅｍａｎｙｐｒｏｂｌｅｍｓａｂｏｕｔｓｏｌｖｉｎｇｐａｒａｂｏｌｉｃｐａｒｔｉａｌｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ．Ｎｏｗｔｈｅｒｅａｒｅｍａｎｙｎｕ...     

A study of the three-dimensional spectral energy distribution in a zero pressure gradient turbulent boundary layer

Time-resolved particle image velocimetry (PIV) measurements performed in wall parallel planes at three wall normal locations, y ⁺ = 34, 108, and 278, in a zero pressure gradient turbulent boundary layer at Re _τ = 470 are used to illuminate the distribution of streamwise velocity fluctuations in a three-dimensional energy spectrum (2D in space and 1D in time) over streamwise, spanwise, and temporal wavelengths. Two high-speed cameras placed side by side in the streamwise direction give a 10δ × 5δ streamwise by spanwise field of view with a vector spacing of _boxclose = z^+ 37\Updelta x^+ = \Updelta z^+ \approx 37 and a time step of \Updelta t⁺=0.5\Updelta t^+=0.5. Although 3D wavenumber--frequency spectra have been calculated in acoustics studies, to the authors’ knowledge this is the first time they has been calculated and presented for a turbulent boundary layer. The calculation and normalization of this spectrum, its relation to 2D and 1D spectra, and the effects of the PIV algorithm on its shape are carefully analyzed and outlined.     

A high-resolution laser Doppler anemometer for three-dimensional turbulent boundary layers

 A new laser Doppler anemometer optimized for high spatial resolution near a wall is described. The instrument uses short focal length optics, a mirror probe in the flow, and side-scatter collection to produce a measuring volume 35 μm in diameter by 66 μm long. Data are presented for a two-dimensional boundary layer, demonstrating the instrument’s ability to measure Reynolds shear stresses as close to the wall as 0.1 mm, or y ⁺≈3. Received: 6 October 1995/Accepted: 9 April 1996     

Effect of local forcing on a turbulent boundary layer

An experimental study is performed to analyze flow structures behind local suction and blowing in a flat-plate turbulent boundary layer. The local forcing is given to the boundary layer flow by means of a sinusoidally oscillating jet issuing from a thin spanwise slot at the wall. The Reynolds number based on the momentum thickness is about Re _θ =1700. The effects of local forcing are scrutinized by altering the forcing frequency (0.011 ≤ f ⁺≤ 0.044). The forcing amplitude is fixed at A ₀=0.4. It is found that a small local forcing reduces the skin friction and the skin friction reduction increases with the forcing frequency. A phase-averaging technique is employed to capture the large-scale vortex evolution. An organized spanwise vortical structure is generated by the local forcing. The cross-sectional area of vortex and the time fraction of vortex are examined by changing the forcing frequency. An investigation of the random fluctuation components reveals that turbulent energy is concentrated near the center of vortical structures. Received: 17 March 2000/Accepted: 3 April 2001     

Moderately Strong Vorticity in a Bathtub-Type Flow

Theoretical and numerical analysis is performed for an inviscid axisymmetric vortical bathtub-type flow. The level of vorticity is kept high so that the image of the flow on the radial–axial plane (r–z plane)is not potential. The most significant findings are: (1) the region of validity of the strong vortex approximation is separated from the drain by a buffer region, (2) the power-law asymptote of the stream function, specified by Δψ∼r ^4/3Δz, appears near the axis when vorticity in the flow is sufficiently strong and (3) the local Rossby number in the region of the power-law is not very sensitive to the changes of the initial vorticity level in the flow and the global Rossby number. Received 3 April 2000 and accepted 29 September 2000     

A new high-order accuracy explict difference scheme for solving three-dimensional parabolic equations

In this paper, a new three-level explicit difference scheme with high-orderaccuracy is proposed for solving three-dimensional parabolic equations. The stabilitycondition is r＝△t/△x² ＝△t/△y²＝△t/△z²≤1/4, and the trumcation error is O(△t²+△x⁴).     

Three-dimensional wake structure measurement using a modified PIV technique

Three-dimensional vortical structures have been measured in a circular-cylinder wake using particle imaging velocimetry (PIV) for the Reynolds number range of 2×10³ to 1×10⁴. The PIV was modified, compared with the conventional one, in terms of its light sheet arrangement to capture reliably streamwise vortices. While in agreement with previous reports, the presently measured spanwise structures complement the data in the literature in the streamwise evolution of the near-wake spanwise vortex in size, strength, streamwise and lateral convection velocities, shedding new light upon vigorous interactions between oppositely signed spanwise structures. The longitudinal vortices display mushroom patterns in the (x, z)-plane in the immediate proximity to the cylinder. Their most likely inclination in the (x, y)-plane is inferred from the measurements in different (x, z)-planes. The longitudinal vortices in the (y, z)-plane show alternate change in sign, though not discernible at x/d > 15. They decay in the maximum vorticity and circulation rapidly from x/d = 5 to 10 and slowly for x/d > 10, and are further compared with the spanwise vortices in size, strength and rate of decay.     

Temperature dissipation measurements in a fully developed turbulent channel flow

All three derivatives of the temperature fluctuation in a fully developed turbulent channel flow have been measured using a pair of parallel cold wires with fixed separations in either the x (streamwise), y (normal to the wall) or z (spanwise) direction. At several locations, measurements were also made, each for a range of y separations between the wires. These latter measurements were important for establishing a suitable method of correcting the data for the effect of wire separation, thus allowing the fixed separation data to be corrected. In the near-wall region, the corrected temperature dissipation is in good agreement with results from direct numerical simulations. The time scale ratio deduced from measurements and simulations agrees well with the theoretical wall value.     

Near-wall flow in a three-dimensional boundary layer on the endwall of a 30° bend

 Turbulence measurements are reported on the three-dimensional turbulent boundary layer along the centerline of the flat endwall in a 30° bend. Profiles of mean velocities and Reynolds stresses were obtained down to y ⁺≈2 for the mean flow and y ⁺≈8 for the turbulent stresses. Mean velocity data collapsed well on a simple law-of-the-wall based on the magnitude of the resultant velocity. The turbulence intensity and turbulent shear stress magnitude both increased with increased three-dimensionality. The ratio of these two quantities, the a ₁ structure parameter, decreased in the central regions of the boundary layer and showed profile similarity for y ⁺<50. The shear stress vector angle lagged behind the velocity gradient vector angle in the outer region of the boundary layer, however there was an indication that the shear stress vector tends to lead the velocity gradient vector close to the wall. Received: 16 July 1996/Accepted: 14 July 1997     

Influence of wavy surfaces on coherent structures in a turbulent flow

We describe how outer flow turbulence phenomena depend on the interaction with the wall. We investigate coherent structures in turbulent flows over different wavy surfaces and specify the influence of the different surface geometries on the coherent structures. The most important contribution to the turbulent momentum transport is attributed to these structures, therefore this flow configuration is of large engineering interest. In order to achieve a homogeneous and inhomogeneous reference flow situation two different types of surface geometries are considered: (1) three sinusoidal bottom wall profiles with different amplitude-to-wavelength ratios of α = 2a/Λ = 0.2 (Λ = 30 mm), α = 0.2 (Λ = 15 mm), and α = 0.1 (Λ = 30 mm); and (2) a profile consisting of two superimposed sinusoidal waves with α = 0.1 (Λ = 30 mm). Measurements are carried out in a wide water channel facility (aspect ratio 12:1). Digital particle image velocimetry (PIV) is performed to examine the spatial variation of the streamwise, spanwise and wall-normal velocity components in three measurement planes. Measurements are performed at a Reynolds number of 11,200, defined with the half channel height h and the bulk velocity U _B. We apply the method of snapshots and perform a proper orthogonal decomposition (POD) of the streamwise, spanwise, and wall-normal velocity components to extract the most dominant flow structures. The structure of the most dominant eigenmode is related to counter-rotating, streamwise-oriented vortices. A qualitative comparison of the eigenfunctions for different sinusoidal wall profiles shows similar structures and comparable characteristic spanwise scales Λ _z = 1.5 H in the spanwise direction for each mode. The scale is observed to be slightly smaller for α = 0.2 (Λ = 15 mm) and slightly larger for α = 0.2 (Λ = 30 mm). This scaling for the flow over the basic wave geometries indicates that the size of the largest structures is neither directly linked to the solid wave amplitude, nor to the wavelength. The characteristic spanwise scale of the dominant eigenmode for the developed flow over the surface consisting of two superimposed waves reduces to 0.85 H. However, a scale in the order of 1.3 H is identified for the second mode. The eigenvalue spectra for the superimposed waves is much broader, more modes contribute to the energy-containing range. The turbulent flow with increased complexity of the bottom surface is characterized by an increased number of dominant large-scale structures with different spanwise scales.     

Acetone: a tracer for concentration measurements in gaseous flows by planar laser-induced fluorescence

The structure of fully-developed turbulence in a smooth pipe has been studied via wavenumber spectra for various friction velocities, namely, u ,=0.61 and 1.2 m/s (the corresponding Reynolds numbers based on centerline velocity and pipe radius being respectively 134,000 and 268,000) at various distances from the wall, namely y ⁺ = 70, 200,400 and 1,000. For each distance from the wall, correlations of the longitudinal component of turbulence were obtained simultaneously in seven narrow frequency bands by using an automated data acquisition system which jointly varied the longitudinal (x) and transverse (z) separations of two hot-wire probes. The centre frequencies of the bandpass filters used correspond to a range of nondimensional frequencies ⁺ from 0.005 to 0.21. By taking Fourier transforms of these correlations, three-dimensional power spectral density functions and hence wavenumber spectra have been obtained at each y ⁺ with nondimensional frequency ⁺ and nondimensional longitudinal and transverse wavenumbers k _x ⁺ and k _z ⁺ as the independent variables. The data presented in this form show the distribution of turbulence intensity among waves of different size and inclination. The data reported here cover a wave size range of over 100, spanning a range of wave angles from 2° to 84°. The effects of friction velocity and Reynolds number on the distribution of waves, their lifetimes and convection velocities are also discussed.List of symbols A wave strength function - C _x streamwise phase velocity - C _z circumferential phase velocity - f wave intensity function - k resultant wave number = [k _x ² + k _z ² ]^1/2 - k _x , k _z longitudinal (x) and transverse (z) wavenumber respectively - P(k _x ⁺ , k _z ⁺ , ⁺) power spectral density function in u - R radius of pipe - Re Reynolds number (based on centerline velocity and pipe radius) - R _uu (x ⁺, z⁺, ) normalized correlation function in u - R _unu (x ⁺, z⁺,¦⁺¦) normalized filtered correlation function in u, as defined in equation (1) - t time - U mean velocity in the x-direction - u, v, w turbulent velocities in the cartesian x, y and z directions respectively - û, v, turbulent velocities in the wave coordinate x, and directions respectively - u friction velocity - x, y, z cartesian coordinates in the longitudinal (along the pipe axis), normal (to the pipe wall) and transverse (along the circumference of the pipe) directions respectively, as defined in Fig. 1 - wave angle - difference between two quantities - v kinematic viscosity - time delay - circular frequency (radians/s) - + quantity nondimensionalized using u and v - overbar time average A version of this paper was presented at the 12th Symposium on Turbulence, University of Missouri-Rolla, 24–26 September, 1990     

Flow field analysis of a turbulent boundary layer over a riblet surface

The near-wall flow structures of a turbulent boundary layer over a riblet surface with semi-circular grooves were investigated experimentally for the cases of drag decreasing (s ⁺=25.2) and drag increasing (s ⁺=40.6). One thousand instantaneous velocity fields over riblets were measured using the velocity field measurement technique and compared with those above a smooth flat plate. The field of view was 6.75 × 6.75 mm² in physical dimension, containing two grooves. Those instantaneous velocity fields were ensemble averaged to get turbulent statistics including turbulent intensities and turbulent kinetic energy. To see the global flow structure qualitatively, flow visualization was also carried out using the synchronized smoke-wire technique under the same experimental conditions. For the case of drag decreasing (s ⁺=25.2), most of the streamwise vortices stay above the riblets, interacting with the riblet tips frequently. The riblet tips impede the spanwise movement of the streamwise vortices and induce secondary vortices. The normalized rms velocity fluctuations and turbulent kinetic energy are small near the riblet surface, compared with those over a smooth flat plate. Inside the riblet valleys, these are sufficiently small that the increased wetted surface area of the riblets can be compensated. In addition, in the outer region (y ⁺ > 30), these values are almost equal to or slightly smaller than those for the smooth plate. For the case of drag increasing (s ⁺=40.6), however, most of the streamwise vortices stay inside the riblet valleys and contact directly with the riblet surface. The high-speed down-wash flow penetrating into the riblet valley interacts actively with the wetted riblet surface and increases the skin friction. The rms velocity fluctuations and turbulent kinetic energy have larger values compared with those over a smooth flat plate. Received: 24 March 1999/Accepted: 10 March 2000     

Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems

We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R²{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as

A study of a 3-D double backward-facing step

An investigation of the flow over a three-dimensional (3-D) double backward-facing step is presented using a combination of both quantitative measurements from a particle image velocimetry (PIV) system and qualitative oil-flow visualizations. The arrangement of the PIV instrument allows for snap-shots of the (x, y) and (y, z) planes at various axial and spanwise positions. The measurements illustrate characteristics that are found in both two-dimensional (2-D) backward-facing steps and 3-D flows around wall mounted cubes. In particular, the development of a horseshoe vortex is found after each step alongside other vortical motions introduced by the geometry of the model. Large turbulence levels are found to be confined to a region in the center of the backstep; their mean square levels being much larger than what has been observed in 2-D backward-facing steps. The large turbulent fluctuations are attributed to a quasi-periodic shedding of the horseshoe vortex as it continuously draws energy from the spiral nodes of separation, which form to create the base of the horseshoe vortex. A combination of effects including the shedding of the first horseshoe vortex, the horizontal entrainment of air and the presence of two counter rotating vortices initiated at reattachment, are shown to cause the steering vector of the flow to jettison away from the surface in the first redeveloping region and along the center at z/h = 0. Oil-flow visualizations confirm these observations.

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The structure of Reynolds stress in the near-wall region of a fully developed turbulent pipe flow

The structure of the Reynolds stress in the near-wall region of a fully developed turbulent pipe flow, at a pipe Reynolds number of 8,923, was investigated. Because the closed circuit tunnel used glycerine as a working fluid, measurements could be readily made inside the viscous sublayer. Two laser Doppler velocimeter (LDV) systems were combined to measure the two point spatial correlation, R ₁₂, between the stream wise and radial velocities in a radial plane of the pipe. The correlation measurements extended over the region from y ⁺ of 2 to 64 in the direction normal to the pipe wall and covered more than 800 wall units in the streamwise direction. Two-dimensional maps of the correlation coefficient were established for six different distances of the streamwise velocity probe from the wall. The use of LDV systems allowed the measurements to be made for small spatial separations of the probes without fear of probe interference effects. A characteristic feature of the correlation contour maps, that maxium correlation arises for small non-zero separation of the probes, may not have been observed had invasive techniques been employed.     

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n < ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture.     

An experimental study of a two-dimensional plane turbulent wall jet

 Laser-Doppler measurements were conducted in a plane turbulent wall jet at a Reynolds number based on inlet velocity, Re ₀, of 9600. The initial development as well as the fully developed flow was studied. Special attention was given to the near-wall region, including the use of small measuring volumes and the application of specific near-wall data corrections, so that wall shear stresses were determined directly from the mean velocity gradient at the wall using only data below y ⁺=4. It was possible to resolve the inner peak in the streamwise turbulence intensity as well as the inner (negative) peak in the shear stress. Limiting values of (u′)⁺ and uv ⁺ were determined. Turbulence data from the outer region of the flow were compared to earlier hot wire measurements and large differences in the normal turbulence intensity and the shear stress were found. These differences can be attributed to high turbulence intensity effects on the hot-wires. Received: 17 October 1996 / Accepted: 8 December 1997