Outer Scaling Research Articles

Direct numerical simulations of free convection beneath an air–water interface at low Rayleigh numbers

Direct numerical simulations of a cooling air–water interface were employed to determine the structure of the temperature, velocity, and vorticity fields in the thin thermal boundary layer formed at the free surface. The simulations were performed at low to moderate Rayleigh numbers. In this flow, the turbulence is initiated by the Rayleigh instability at the interface and is maintained by buoyant production. Visualizations of the flow reveal that the temperature field at the interface is composed of large warm patches surrounded by cooler dense fluid which accumulates in thin bands. The cool fluid associated with the bands initially falls in sheets, but rapidly forms descending tubes and plumes. The turbulence statistics were scaled both with outer and inner variables. The latter scaling is based on the so-called surface strain model which is essentially consistent with Townsend’s inner scaling. It is found that the temperature statistics collapse well using inner variables. On the other hand, the vertical velocity scales well with inner variables within the thermal boundary layer, but at greater depths it becomes more appropriate to use outer scaling. The anisotropic nature of the velocity statistics in the core of the flow is ascribed to the relatively low Rayleigh numbers used in the simulations. An explanation for this anisotropy is offered based on a detailed examination of the turbulence kinetic energy balances.

Direct numerical simulation of stagnation region flow and heat transfer with free-stream turbulence

A direct numerical simulation is performed for stagnation-region flow with free-stream turbulence. A fully implicit second-order time-advancement scheme with fourth-order finite differences and an optimized scheme are employed. The optimized scheme is developed to save computational cost. The free-stream turbulence is a precomputed field of isotropic turbulence. The present DNS results in the “damping” and “attached amplifying” regimes are found to be similar to those of the organized inflow disturbances. Emphasis is placed on the flow and temperature fields in the “detached amplifying” regime. The contours of instantaneous flow field illustrate that streamwise vortices are stretched in the streamwise direction by mean strain rate. The temperature field is also stretched in the streamwise direction near the wall. The surface contours reveal that the temperature field is influenced significantly by streamwise vorticity. Due to the dominance of the mean strain, the log-law region is not observed for ū and T̃, the inner scaling fails, but the outer scaling works. The single-point turbulence statistics and the turbulent statistics budgets are obtained. The flow statistics reflect the typical characteristics of stagnation-region flow which are generically different from those of other canonical shear flows. One of the typical features of the budgets is that the velocity pressure correlation and the turbulent transport play significant roles in the stagnation-region flow. Finally, the present simulation data are compared with experimental results. It is found that the effect of large-scale eddies on the enhancement of wall heat transfer is substantial in the turbulent stagnation-region heat transfer.

Time-dependent drag reduction and ageing in aqueous solutions of a cationic surfactant

The turbulent pipe flow of a highly dilute aqueous cationic surfactant solution is investigated by means of a pulsed ultrasound Doppler method with special emphasis on the wall boundary layer. The velocity profiles are recorded for several Reynolds numbers at varying ages of the solution. The wall shear stress ve- locities us used for the normalization of the velocity profiles are determined by fitting the measured profiles to the universal linear velocity profile in the viscous sub- layer. The theoretical pressure loss is then calculated from the numerical values of us and compared to the experimental values. Two different scaling methods are discussed for the velocity fluctuations concerning the correlation of the root-mean square values with the effect and the amount of drag reduction. It is shown that outer scaling with the mean velocity is appropriate for the detection of drag reduction in surfactant solutions, rather than inner scaling with the wall shear stress velocity, which is common practice in investigations of 'usual' turbulent flows.

Measurements of turbulence quantities and bursting period in developing turbulent boundary layers on the concave and convex walls of a 90° square bend

The results of an experimental study of developing turbulent boundary layers on the concave and convex walls of a 90° bend of square cross-section using hot-wire anemometry are presented. The bend has a mean radius to height ratio of R/ H=1.17. The flow has a Reynolds number Re H =3.6×10 5, and at start of the bend the boundary layer thickness to the radius of curvature ratio is δ/ R cc=0.041 and δ/ R cx=0.054 on the concave and convex walls, respectively. Conditional sampling based on the VITA method applied to the streamwise velocity fluctuations is used to obtain the period of bursting events in the concave and convex boundary layers. Three scaling laws, namely, inner scaling, outer scaling, and the mixed scaling, are used to non-dimensionalize the bursting period. Results are presented for the conventional mean turbulence quantities and the period of bursting. Turbulence enhancement on the concave wall and the opposite effect on the convex wall are most pronounced in the outer layer, and the effect on the normal intensity is greater than on the streamwise intensity. On the convex wall, suppression of turbulence shear stress to insignificant levels occurs for normal distances from the wall greater than y/ δ≈0.4. The results of conditional sampling show the opposite effects of the concave and convex walls on the bursting frequency, and that none of the scaling laws results in the collapse of this parameter.

Reynolds number dependence of streamwise velocity spectra in turbulent pipe flow.

Spectra of the streamwise velocity component in fully developed turbulent pipe flow are presented for Reynolds numbers up to 5.7x10(6). Even at the highest Reynolds number, streamwise velocity spectra exhibit incomplete similarity only: while spectra collapse with both classical inner and outer scaling for limited ranges of wave number, these ranges do not overlap. Thus similarity may not be described as complete, and a region varying with the inverse of the streamwise wave number, k(1), is not expected, and any apparent k(-1)(1) range does not attract any special significance and does not involve a universal constant. Reasons for this are suggested.

Turbulent boundary layer pressure fluctuations at large scales

The pressure fluctuations underneath a turbulent boundary are an important excitation source for noise and vibration for both aircraft and ships. This presentation describes experimental results from a new study of flat-plate turbulent boundary layer pressure fluctuations in water at large scales and high Reynolds number. The experiments were performed at the U.S. Navy’s William B. Morgan Large Cavitation Channel in Memphis, TN on a polished flat plate 3.05 m wide, 12.8 m long, and 0.18 m thick. Flow velocity, skin friction, surface pressure, and plate acceleration measurements were made at multiple downstream locations at flow speeds ranging from 0.5 m/s to 19 m/s for a Reynolds number (based on downstream distance) range of several million to 200 million. Dynamic surface pressures were recorded with 16 flush mounted pressure transducers forming an L-array with streamwise dimension of 0.264 m and cross-stream dimension of 0.391 m. Measured 99% boundary-layer thicknesses were typically of order 0.10 m. Results for spatial and temporal correlation functions, as well as auto- and cross-spectra are presented and compared with prior lower-Reynolds-number results using either inner or outer variable scaling. [Work sponsored by the Defense Advanced Research Projects Agency and ONR Code 333.]

Scaling of velocity and temperature fluctuations in turbulent thermal convection

The scaling arguments presented by [J. Fluid Mech. 204 (1989) 1] to justify the Nu∼Ra 2/7 power law observed in high Rayleigh number thermal convection also have implications regarding the distribution of velocity and temperature fluctuations within the turbulent layer. Asymptotic matching of the inner scaling law and the outer scaling law implies that the root-mean-square (r.m.s.) vertical velocity fluctuation varies as σ w∼ln z, and the r.m.s. temperature fluctuation varies as σ θ ∼ln z or σ θ ∼ z −1/2 depending on the details of the modeling assumptions [Exp. Fluids 4 (1986) 121]. These results differ markedly from the Priestley similarity theory [Turbulent Transport in the Lower Atmosphere, 1959], in which σ w∼ z 1/3 and σ θ ∼ z −1/3. Consequently, the r.m.s. velocity and temperature profiles offer the possibility to distinguish between the two theories. We present measurements in wide-aspect-ratio convection over the range Ra=10 7–10 9 that support the logarithmic variation for both the temperature and velocity.

Weathering of igneous rocks during shallow burial in an upland peat environment: observations from the Bronze Age Copney Stone Circle Complex, Northern Ireland

The stone circle complex at Copney, County Tyrone is a key Bronze Age site that forms part of the Mid-Ulster stone circle complex. This site was excavated in 1995 and since then, deterioration of the archaeological stonework has been a serious problem. Deterioration is visible as splitting/fracturing of stones, the development of a bleached outer margin, surface scaling and granular disintegration, and, in some cases, complete disintegration of individual stones to sandy regolithic material. Environmental conditions at this site exacerbate the deleterious action of weathering processes, with periodic waterlogging and sub-zero temperatures during winter months. The Copney stones comprise two igneous lithologies: quartz porphyry and porphyritic andesite, both Lower Ordovician (470 Ma) in age. Both rock types show extensive alteration by hydrothermal processes resulting in weakened stone fabrics exploited by weathering processes during burial and subsequent exposure. Mineralogy of buried and exposed material indicates that approximately 2500 years of burial in a peat bog has produced secondary porosity comprising extensive microfracture networks and dissolution voids, which permitted further ingress of moisture and acidic waters, thus, promoting chemical alteration of mineral constituents. The occurrence of completely grussified stones and arenisation of boulder surfaces at Copney indicates that the processes of grussification are not solely restricted to deep weathering environment but can be achieved by burial in a shallow, aggressive environment over periods measured in thousands of years.

Wavelet-based method for burst detection

A method to detect energetic structures based on wavelet analysis is outlined. Combined with a quadrant decomposition the method may provide information about the scale dependency of ejection and sweep events. One advantage of this method compared to previously published algorithms is that there are no thresholds or grouping times involved. The method was applied to a number of zero pressure gradient boundary layer flows covering a large range of Reynolds numbers. It is shown that the duration of the ejections, over the entire boundary layer, scales with inner variables for a wide range of turbulent scales. The time between the events scale with inner variables in the immediate vicinity of the wall, but for y+ greater than about 70, outer scaling proved to give the best collapse for the Re range investigated.

Surface Pressure Fluctuations Beneath Two- and Three-Dimensional Turbulent Boundary Layers

Surface pressure fluctuation measurements were made in two-dimensional turbulent boundary layers at two Reynolds numbers (Re θ = 7.3 × 10 3 and 2.34 × 104) and pressure-driven three-dimensional turbulent boundary layers at two Reynolds numbers (approach Re θ = 5.94 x 10 3 and 2.32 × 104). The collapse of spectral levels at middle and high frequencies and the effects of inner and outer boundary-layer scaling variables are shown for a wide range of Reynolds numbers (1.4 X 10 3 < 2.34 × 104) for the two-dimensional flows. Such scaling parameters do not collapse the pressure spectra beneath three-dimensional flows, which have a nearly constant, or flat, midfrequency range, and at some measurement stations and spectral levels within the flat- and high-frequency spectral ranges that significantly raise p'. Additionally, dimensional spectral levels within the flat-frequency range are independent of Reynolds number. Analysis based on the Poisson equation shows that the variation of the high-frequency spectral levels is related to the variation in near-wall mean velocity gradients and ν 2 structure due to the spanwise pressure gradient

Effects of Convection and Decay of Turbulence on the Wall Pressure Wavenumber-Frequency Spectrum

Cross-spectral and cross-correlation data from experiments and numerical simulations have shown the turbulent wall pressure convection velocity to vary with the streamwise sensor spatial separation. This variation is due to the spatial decay rates of turbulent structures in the inner and outer regions of the boundary layer. Its effect is shown to have a significant impact on the distribution of energy in the wavenumber-frequency spectrum Φ (k1, ω). The standard Corcos model is known to over predict the wavenumber-frequency spectrum at low wavenumbers. This is shown to result from its constant convection velocity assumption. The spectral levels at sub convective and lower wavenumbers are shown to be directly influenced by the spatial variation in convection velocity. Convection velocity measurements from past investigations that cover the range 285 ≤ Rθ ≤ 29,000 are compared, and an outer variable scaling is shown to effectively collapse the data.

Field versus laboratory turbulent boundary layers

The universal logarithmic profile that describes the mean streamwise velocity in the inner layer of any wall-bounded flow is the best known example of the stated classical idea. In the present research, data from numerous numerical and physical experiments have been carefully analyzed, and it is shown that boundary layers, in fact, never achieve true self-preservation and that neither inner nor outer scaling is strictly valid. We present an informative summary of value.

On the applicability of various scaling laws to the turbulent wall jet

The spatial distribution of the mean velocity in a two-dimensional turbulent wall jet was measured for a variety of nozzle Reynolds numbers. It was determined that the bulk of the flow is self-similar and it depends on the momentum flux at the nozzle and on the viscosity and density of the fluid. The width of the nozzle which was commonly used to reduce these data has no part in the similarity considerations as has already been suggested by Narasimha et al. (1973). This type of self-similarity can be easily applied to determine the skin friction, which can otherwise only be determined with considerable difficulty. It was also shown that the ‘law of the wall’ applies only to the viscous sublayer. The Reynolds stress in the inviscid, inner portion of the flow is not constant thus the assumption of a ‘constant stress layer’ is not applicable. The applicability and universality of the ‘outer scaling law’ (i.e. Coles’ law of the wake) has been verified throughout the inviscid inner portion of the wall jet. The logarithmic velocity distribution cannot be derived by making the usual assumptions based on the constancy of the Reynolds stresses or on the thinness of the logarithmic region relative to the thickness of the inner layer.

Scaling of wall shear stress fluctuations in a turbulent duct flow

Instantaneous fluctuations of the wall shear stress have been measured in a fully developed turbulent duct flow over a factor of 10 variation in the Reynolds number. Distributions for the average frequency of detections obtained using the variable-interval time averaging technique suggest that scaling on outer variables or perhaps a mixture of outer and inner variables is more appropriate than scaling on inner variables when the Reynolds number is sufficiently large. The Reynolds number behavior of the average frequencies of zero crossings and local maxima of the shear stress fluctuation rules out the possibility of outer scaling and indicates a weak Reynolds number dependence when scaled on inner variables. These apparently conflicting trends are discussed in the context of published results on the scaling of wall shear stress fluctuations.

Developing turbulent boundary layers with system rotation

Measurements are presented for low-Reynolds-number turbulent boundary layers developing in a zero pressure gradient on the sidewall of a duct. The effect of rotation on these layers is examined. The mean-velocity profiles affected by rotation are described in terms of a common universal sublayer and modified logarithmic and wake regions.The turbulence quantities follow an inner and outer scaling independent of rotation. The effect appears to be similar to that, of increased or decreased layer development. Streamwise-energy spectra indicate that, for a given non-dimensional wall distance, it is the low-wavenumber spectral components alone that are affected by rotation.Large spatially periodic spanwise variations of skin friction are observed in the destabilized layers. Mean-velocity vectors in the cross-stream plane clearly show an array of vortex-like structures which correlate strongly with the skin-friction pattern. Interesting properties of these mean-flow structures are shown and their effect on Reynolds stresses is revealed. Near the duct centreline, where we have measured detailed profiles, the variations are small and there is a reasonable momentum balance.Large-scale secondary circulations are also observed but the strength of the pattern is weak and it appears to be confined to the top and bottom regions of the duct. The evidence suggests that it has minimally affected the flow near the duct centreline where detailed profiles were measured.

Zero-crossings in turbulent signals

A primary motivation for this work arises from the contradictory results obtained in some recent measurements of the zero-crossing frequency of turbulent fluctuations in shear flows. A systematic study of the various factors involved in zero-crossing measurements shows that the dynamic range of the signal, the discriminator characteristics, filter frequency and noise contamination have a strong bearing on the results obtained. These effects are analysed, and explicit corrections for noise contamination have been worked out. New measurements of the zero-crossing frequency N0 have been made for the longitudinal velocity fluctuation in boundary layers and a wake, for wall shear stress in a channel, and for temperature derivatives in a heated boundary layer. All these measurements show that a zero-crossing microscale, defined as Λ = (2πN0)−1, is always nearly equal to the well-known Taylor microscale λ (in time). These measurements, as well as a brief analysis, show that even strong departures from Gaussianity do not necessarily yield values appreciably different from unity for the ratio Λ/λ. Further, the variation of N0/N0 max across the boundary layer is found to correlate with the familiar wall and outer coordinates; the outer scaling for N0 max is totally inappropriate, and the inner scaling shows only a weak Reynolds-number dependence. It is also found that the distribution of the interval between successive zero-crossings can be approximated by a combination of a lognormal and an exponential, or (if the shortest intervals are ignored) even of two exponentials, one of which characterizes crossings whose duration is of the order of the wall-variable timescale ν/U2*, while the other characterizes crossings whose duration is of the order of the large-eddy timescale δ/U∞. The significance of these results is discussed, and it is particularly argued that the pulse frequency of Rao, Narasimha &amp; Badri Narayanan (1971) is appreciably less than the zero-crossing rate.

Laterally converging flow. Part 2. Temporal wall shear stress

Instantaneous measurements of the wall shear stress were made in the laterally converging duct also used for mean measurements in part 1 and were analysed by conditional sampling and by conditional averaging. The sidewalls of the duct were adjusted to provide (i) a straight duct of constant rectangular cross-section and (ii) laterally (spanwise) converging ducts resulting in streamwise acceleration of the flow. The Reynolds number varied from 7600 to 47 200 and the dimensionless acceleration parameter Kv = (ν/V2)dV/dx ranged from 0 to 3·4 × 10−6, yielding a variation of the flow regime from fully turbulent to nearly laminar. The typical burst pattern, or conditionally averaged time history of the wall shear stress, resembled the time history of the streamwise velocity component deduced at y+ = 15 by Blackwelder and Kaplan using the same general technique. For fully developed flows, inner or wall scaling of the bursting frequency was found to be less dependent upon Reynolds number than outer scaling; other characteristics examined varied with both inner and outer scaling. For converging flows measurements of bursting characteristics essentially confirmed the indicated flow regimes deduced in part 1 and showed that the measured characteristic that was most affected by acceleration was the bursting frequency. All characteristics varied with acceleration, but the variation was generally less when normalized by wall variables rather than when normalized by outer variables.

Effects of Scale Porosity, Second‐Phase Oxides, and Doping in the High‐Temperature Oxidation of Cobalt and Dilute Cobalt‐Chromium Alloys

The oxidation mechanisms of high‐purity cobalt and dilute Co‐Cr alloys have been investigated over a 950°–1350°C temperature range in oxygen atmospheres of 10–760 Torr. The 99.999% cobalt exhibited parabolic oxidation behavior for weight gains of up to 30 mg oxygen/cm2 rectangular specimen surface. Parabolic kinetics were interrupted when fissures developed in the oxide scale as a result of mechanical stresses at the metal/scale interface or through anisotropic decomposition along grain boundaries. Molecular oxygen then short‐circuited to the porous inner scaling layer in accelerated attack of fresh Co surfaces. Diffusion of Co2+ through the scale is considered to be the primary rate determining process during periods of steady‐ state oxidation. The effect of porosity at the metal/scale interface prior to perforation of the outer scaling layer is slight, but is believed to result in a small reduction of the oxidation rate due to a reduction in the contact surface through which cations enter the oxide. Co alloys containing 0.5, 3, 7, and 10 w/o (weight per cent) Cr also obey parabolic kinetics during formation of thinner scaling layers. Porosity and localized loss of scale adherence are aggravated by the presence of and particles in the inner scaling layer to the extent that scale rupture mechanisms quickly obscure parabolic kinetics for chromium contents greater than 3 w/o at temperatures of 1150°C and above in 100 Torr oxygen. Under conditions where initial periods of steady‐state parabolic oxidation were discernible on thermobalance curves, maximum reaction kinetics are associated with 1–2 w/o Cr additions (which approach the Cr3+ solubility limit in ) as rationalized by Wagner's semiconductor valence theories.

Outer Iteration Scaling in Neutron Transport Codes

Outer Iteration Scaling in Neutron Transport Codes