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.     

Accurate estimate of disturbance amplitude variation from solution of minimal composite stability theory

Minimal composite theory (Proc. R. Soc. Lond., Ser. A, 453 2537–2549, 1979) shows that, at the lowest order in the reciprocal of the local flow Reynolds number R, the stability of a spatially developing similarity flow may be described by an ordinary differential equation in the similarity coordinate. It is, in principle, not possible to determine the dependence of the disturbance amplitude on the streamwise coordinate solely from such an ordinary differential equation. However, noting that, to O(R^−2/3), the dependence of the eigenfunction on the normal coordinate is identical in both the full non-parallel and minimal composite theories, and using a method due to Gaster, we show how the streamwise variation of disturbance amplitude can be determined to O(R⁻¹ without solving a partial differential equation, although knowledge of the partial differential operator is required. Comparison with the DNS results of Fasel and Konzelmann shows excellent agreement with the present results. Furthermore, especially in strong adverse pressure gradients, the present amplitude ratio estimates are within 3% of the full non-parallel theory, whereas the Orr–Sommerfeld results show an underestimate by 26%.This revised version was published online in May 2005. In the previous version, the published online date was missing. Moreover, the preliminary article pagination was deleted.     

On the receptivity and nonparallel stability of traveling disturbances in rotating-disk flow

The generation and evolution of small amplitude long wavelength traveling disturbances in rotating-disk flow is the subject of this paper. The steady rotational speed of the disk is perturbed so as to introduce high-frequency oscillations in the flow field. Secondly, we introduce surface imperfections on the disk such as roughness elements. The interaction of these two disturbances will generate the instability waves whose evolution is governed by parabolic partial differential equations which are solved numerically. It is found that, for the class of disturbances considered here (wavelength on the order of Reynolds number), eigensolutions exist which decay or grow algebraically in the radial direction. However, these solutions grow only for frequencies larger than 4.58 times the steady rotational speed of the disk. The computed receptivity coefficient shows that there is an optimum size of roughness for which these modes are preferentially excited. The width of these roughness elements in the radial direction is about 0.1r ₀ ^* where r ₀ ^* is the radial location of the roughness. It is also found that the receptivity coefficient is larger for a negative spanwise wave number than for a positive one. The cumulative wave pattern produced from the roughness site shows that the typical wave angles for these disturbances are about –26° with about seven waves around the circumference. This is in contrast with the wave angles of 10°–14° found for the 30 or so inviscid cross-flow vortices.This work was sponsored by NASA Langley Research Center under Contracts NAS1-18240 (P.B. and M.R.M.) and NAS1-18605 (P.H.).     

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.     

Turbulent wall pressure fluctuations over wavy surfaces

Measurements of the spectral characteristics of the wall pressure fluctuations produced by a turbulent boundary layer flow over solid sinusoidal surfaces of moderate wave amplitude to wave-length ratios have been obtained. The wave amplitudes were sufficiently small so that the flow remained attatched. The results show that the root mean square pressure level reaches a maximum on the adverse pressure gradient side of the wave at a position somewhat before the trough. Spectral analysis of the pressure fluctuations in narrow frequency bands reveals considerable differences in low and high frequency behavior. At low frequencies, the peak fluctuation amplitude was found at the trough whereas at high frequencies, the peak occurs just after the crest and a minimum is found at the trough. Pressure fluctuations having streamwise correlation lengths on the order of or larger than the wavelength of the surface do not return to their equilibrium (crest) amplitudes as they travel the length of a wave. Pressure fluctuations having streamwise correlation lengths about one order of magnitude less than a wavelength return exactly to their equilibrium amplitudes. Two-point correlation measurements show a decrease in longitudinal coherence on the adverse pressure gradient side of the wave at low frequencies and a considerable increase over a broad frequency range on the positive pressure gradient side. No change is found in the lateral coherence.List of symbols C _f skin friction coefficient - C _p pressure coefficient - C _n Fourier amplitudes of the pressure coefficient - C _dp pressure drag coefficient - d pinhole diameter - f frequency - h half the crest to trough distance - h ₊ nondimensional wave amplitude = - k _n wavenumber = - k fundamental wavenumber = - l _p pressure correlation length - p _s mean surface pressure - P ambient pressure - p fluctuating pressure - p ² mean square pressure - q dynamic head = 1/2 U ² - R space-time correlation - P Reynolds number based on wavelength = - R Reynolds number based on momentum thickness = - t time - R free stream velocity - U mean streamwise velocity - U _e streamwise velocity at the edge of the boundary layer - u ^* friction velocity = - x streamwise coordinate - y wall-normal coordinate - z spanwise coordinate - ⁺ non-dimensional wavelength = ^*) - phase of the cross-spectral density - ^* boundary layer displacement thickness - _long longitudinal coherency - _lat lateral coherency - wavelength of wavy surface - v kinematic viscosity - radian frequency = 2 f - spectral or cross-spectral density - _n phase of the Fourier series - density - time delay - _w wall shear stress - boundary layer momentum thickness     

Turbulent wake and vortex shedding for a stack partially immersed in a turbulent boundary layer

The effect of the jet-to-cross-flow velocity ratio, R, on the turbulent wake and Kármán vortex shedding for a cylindrical stack of aspect ratio AR=9 was investigated in a low-speed wind tunnel using thermal anemometry. The cross-flow Reynolds number was Re_D=2.3×10⁴, the jet Reynolds number ranged from Re_d=7.6×10³ to 4.7×10⁴, and R was varied from 0 to 3. The stack was partially immersed in a flat-plate turbulent boundary layer, with a boundary layer thickness-to-stack-height ratio of δ/H=0.5 at the location of the stack. From the behaviour of the turbulent wake and the vortex shedding, the flow around the stack could be classified into three regimes depending on the value of R, which were the downwash (R<0.7), cross-wind-dominated (0.7R<1.5), and jet-dominated (R1.5) flow regimes. Each flow regime had a distinct structure to the mean velocity (streamwise and wall-normal directions), turbulence intensity (streamwise and wall-normal directions), and Reynolds shear stress fields, as well as the variation of the Strouhal number and the power spectrum along the stack height.     

Transition from Convective to Absolute Instability in a Porous Layer with Either Horizontal or Vertical Solutal and Inclined Thermal Gradients,and Horizontal Throughflow

Recently, in Diaz and Brevdo (J Fluid Mech 681: 567–596, 2011), further in the text referred to as D&B, we found an absolute/convective instability dichotomy at the onset of convection in a flow in a saturated porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. The control parameter in D&B triggering the destabilization is the vertical thermal Rayleigh number, R _v. In this article, we treat the parameter cases considered in D&B in which the onset of convection has the character of convective instability and occurs through longitudinal modes. By increasing the vertical thermal Rayleigh number starting from its critical value, R _vc, we determine the value R _vt of R _v at which the transition from convective to absolute instability takes place and compute the physical characteristics of the emerging absolutely unstable wave packet. In some cases, the value of the transitional vertical thermal Rayleigh number, R _vt, is only slightly greater than the critical value, R _vc, meaning that at the onset of convection the base convectively unstable state can be viewed as marginally absolutely unstable. However, in several cases considered, the value of R _vt is significantly greater than the critical value, R _vc, implying that the base state is not marginally but essentially absolutely stable at the point of destabilization.     

The effect of different X-wire calibration schemes on some turbulence statistics

The effective-angle and look-up-table calibrations are used to reduce X-wire datasets in two flows: decaying grid turbulence at R50 and the far field of a circular jet at R400 with a mean streamwise velocity of nearly 6 ms^–1 in each case. A static check, which involves mean values of the velocity components, indicates that the effective-angle calibration incorrectly reduces the range of the estimated velocity components and is therefore likely to be less reliable than the look-up-table approach. The two methods produce significantly different results for both small- and large-scale statistics of the streamwise and, more especially, transverse velocity fluctuations. For grid turbulence, the mean-squared temporal derivatives of the streamwise and transverse velocities are larger by 3% and 19%, respectively, when estimated with the look-up-table method compared to the effective-angle calibration. The increase in the mean-squared transverse velocity derivative yields results in closer agreement with isotropy. At velocities below 6 ms^–1, the effective-angle is a quantity that strongly depends on the velocity magnitude and direction.     

The Green’s function of the mild-slope equation:: The case of a monotonic bed profile

In the present work the Green’s function of the mild-slope and the modified mild-slope equations is studied. An effective numerical Fourier inversion scheme has been developed and applied to the construction and study of the source-generated water-wave potential over an uneven bottom profile with different depths at infinity. In this sense, the present work is a prerequisite to the study of the diffraction of water waves by localized bed irregularities superimposed over an uneven bottom. In the case of a monotonic bed profile, the main characteristics of the far-field are: (i) the formation of a shadow zone with an ever expanding width, which is located along the bottom irregularity, and (ii) in each of the two sectors not including the bottom irregularity the asymptotic behavior of the wave field approaches the form of an outgoing cylindrical wave, propagating with an amplitude of order O(R^−1/2), where R is the distance from the source, and wavelength corresponding to the sector-depth at infinity. Moreover, the weak wave system propagating in the shadow zone is of order O(R^−3/2), and along the bottom irregularity consists of the superposition of two outgoing waves with wavelengths corresponding to the two depths at infinity.     

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     

Numerical simulation of receptivity of a hypersonic boundary layer to acoustic disturbances

Direct numerical simulations of the evolution of disturbances in a viscous shock layer on a flat plate are performed for a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵. Unsteady Navier-Stokes equations are solved by a high-order shock-capturing scheme. Processes of receptivity and instability development in a shock layer excited by external acoustic waves are considered. Direct numerical simulations are demonstrated to agree well with results obtained by the locally parallel linear stability theory (with allowance for the shock-wave effect) and with experimental measurements in a hypersonic wind tunnel. Mechanisms of conversion of external disturbances to instability waves in a hypersonic shock layer are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 84–91, May–June, 2007.     

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.     

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     

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     

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.     

Receptivity of hypersonic boundary layer due to fast-slow acoustics interaction

The objective of receptivity is to investigate the mechanisms by which external disturbances generate unstable waves. In hypersonic boundary layers, a new receptivity process is revealed, which is that fast and slow acoustics through nonlinear interaction can excite the second mode near the lower-branch of the second mode. They can generate a sum-frequency disturbance though nonlinear interaction,which can excite the second mode. This receptivity process is generated by the nonlinear interaction and the nonparallel nature of the boundary layer. The receptivity coefficient is sensitive to the wavenumber difference between the sumfrequency disturbance and the lower-branch second mode.When the wavenumber difference is zero, the receptivity coefficient is maximum. The receptivity coefficient decreases with the increase of the wavenumber difference. It is also found that the evolution of the sum-frequency disturbance grows linearly in the beginning. It indicates that the forced term generated by the sum-frequency disturbance resonates with the second mode.     

Equilibrium states in homogeneous turbulence with baroclinic instability

The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ?U _i /?x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise ${(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)}The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ∂U _i /∂x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise (?B/?x₂ = N_h² = -2WS){(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)} and vertical (?B/?x₃ = N_v²){(\partial B{/}\partial x_3\,{=}\,N_v^2)} directions. Computations based on the rapid distortion theory (RDT) are performed for several values of the rotation number R = 2Ω/S and the Richardson number R_i = N_v²/S² < 1{R_i\,{=}\,N_v^2/S^2 <1 }. It is shown that, during an initial phase, the energies and the buoyancy fluxes are sensitive to the effects of pressure and viscosity. At large time, the ratios of energies, as well as the normalized fluxes, evolve to an asymptotically constant value, while the pressure–strain correlation scaled with the product of the turbulent kinetic energy by the shear rate approaches zero. Accordingly, an analytical parametric study based on the “pressure-less” approach (PLA) is also presented. The analytical study indicates that, when R _i  < 1, there is an exponential instability and equilibrium states of turbulence, in agreement with RDT. The energies and the buoyancy fluxes grow exponentially for large times with the same rate (γ in St units). The asymptotic value of the ratios of energies yielded by RDT is well described by its PLA counterpart derived analytically. At R _i  = 0, the asymptotic value of γ increases with increasing R approaching 2 for high rotation rates. At low rotation rates, an important contribution to the kinetic energy comes from the streamwise kinetic energy, whereas, at high rotation rates, the contribution of the vertical kinetic energy is dominant. When 0 < R _i  < 1 and R 1 0{R\ne 0}, the asymptotic value of γ decreases as R _i increases so as it becomes zero at R _i  = 1.     

Experimental Control of Turbulent Boundary Layers with In-plane Travelling Waves

The experimental control of turbulent boundary layers using streamwise travelling waves of spanwise wall velocity, produced using a novel active surface, is outlined in this paper. The innovative surface comprises a pneumatically actuated compliant structure based on the kagome lattice geometry, supporting a pre-tensioned membrane skin. Careful design of the structure enables waves of variable length and speed to be produced in the flat surface in a robust and repeatable way, at frequencies and amplitudes known to have a favourable influence on the boundary layer. Two surfaces were developed, a preliminary module extending 152 mm in the streamwise direction, and a longer one with a fetch of 2.9 m so that the boundary layer can adjust to the new surface condition imposed by the forcing. With a shorter, 1.5 m portion of the surface actuated, generating an upstream-travelling wave, a drag reduction of 21.5% was recorded in the boundary layer with Re _τ =?1125. At the same flow conditions, a downstream-travelling produced a much smaller drag reduction of 2.6%, agreeing with the observed trends in current simulations. The drag reduction was determined with constant temperature hot-wire measurements of the mean velocity gradient in the viscous sublayer, while simultaneous laser Doppler vibrometer measurements of the surface recorded the wall motion. Despite the mechanics of the dynamic surface resulting in some out-of-plane motion (which is small in comparison to the in-plane streamwise movement), the positive drag reduction results are encouraging for future investigations at higher Reynolds numbers.     

Scattering of elastic waves by a periodic array of cracks in 3-D space

The problem of scattering of normal incident time harmonic plane elastic waves by a co-planar periodic array of cracks in 3-D space is investigated. The scattered waves consist of a superposition of an infinite number of wave modes [M, N]^T and [M, N]^L,M. N=0, 1, 2, , but only a finite number of them are propagating wave modes. The numerical calculation has been made for rectangular cracks and P wave incidence. The reflection coefficient of [O, O] order,R ₀ ³ , has been studied in detail for various wave numbers and parameters of the geometry for the problem. The reliability of the numerical calculation has been checked by an application of the balance of rates of energies. For an elongated rectangular crack,R ₀ ³ in the corresponding 2-D problem in [2] is recovered. The dynamic stress intensity factors around the crack edge have been obtained. The results as the wave number goes to zero have been compared with those in the correspoding static case. Good agreement is observed.     

On the Convection in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow. Part I. Normal Modes

In this second part of our analysis of the destabilization of transverse modes in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow, we apply the mathematical formalism of absolute and convective instabilities to studying the nature of the transition to instability of such modes by assuming on physical grounds that the transition is triggered by growing localized wavepackets. It is revealed that in most of the parameter cases treated in the first part of the analysis (Brevdo and Ruderman 2009), at the transition point the evolving instability is convective. Only in the cases of zero horizontal thermal gradient, and in the cases of zero vertical throughflow and the horizontal Rayleigh number R _h < 49, the instability is absolute implying that, as the vertical Rayleigh number, R _v, increases passing through its critical value, R _vc, the destabilization tends to affect the base state throughout and eventually destroys it at every point in space. For the parameter values considered, for which the destabilization has the nature of convective instability, we found that, as R _v, increases beyond the critical value, while the horizontal Rayleigh number, R _h, and the Péclet number, Q _v, are kept fixed, the flow experiences a transition from convective to absolute instability. The values of the vertical Rayleigh number, R _v, at the transition from convective to absolute instability are computed. For convectively unstable, but absolutely stable cases, the spatially amplifying responses to localized oscillatory perturbations, i.e., signaling, are treated and it is found that the amplification is always in the direction of the applied horizontal thermal gradient.