Momentum Deficit Research Articles

A numerical study of self-similarity in a turbulent plane wake using large-eddy simulation

Turbulent wakes are known to develop self-similarly sufficiently far downstream from obstacles that generate them. It has long been assumed that the spreading rate of the wake in the self-similar regime is independent of the details of the body generating the wake, being dependent only on the total drag (or momentum deficit). This assumption seems to be in contradiction with some recent experiments. In this study we attempt to complement these experimental investigations through a numerical study of a time-developing wake. A numerical study has the advantage of eliminating many of the uncontrolled factors present in experiments and allowing precise control of initial conditions. Large-eddy simulations employing the recently developed dynamic localization model are used to extend previous results from direct numerical simulations. The large-eddy simulation results are compared to the direct numerical simulation database, wherever such comparisons are feasible, as a check of the method. Like the experiments, the large-eddy simulations suggest that non-unique self-similar states, characterized by different spreading rates and turbulent statistics, are possible and that they can be maintained for significant time periods. The study also demonstrates the predictive capability of the dynamic localization subgrid model.

MOMENTUM BUDGET AND SHELTER MECHANISM OF BOUNDARY-LAYER FLOW NEAR A SHELTERBELT

We use a nonhydrostatic shelterbelt boundary-layer turbulence model with Mellor–Yamada second-order closure to evaluate quantitatively the dynamic processes of surface boundary-layer flow perturbed by shelterbelts of different densities and to understand the shelter mechanism. We first analyze the drag exerted on air by shelterbelts of different densities, a root cause of any shelter function, and the resulting wind reduction. The results show that the effectiveness of a shelter is determined not only by its total drag but also by the distribution of the drag-generated momentum deficit in the sheltered area, and that medium-dense shelterbelts have the maximum shelter effect. We also analyze the horizontal momentum budget and find that the shelter mechanism is the product of several processes. The results reveal that strong vertical mean transport and the pressure gradient also play important roles in shelter efficiency. The pressure perturbation caused by the shelter extends far downstream of the shelter, and combines with advective transport to provide the larger shelter efficiency of medium-dense shelterbelts. We finally analyze the changes of perturbed pressure, turbulence, and vertical velocity with shelterbelt density to further clarify the shelter mechanism.

96/06769 Phase relationships in coal ash corrosion products

Four-element hot-wire probe measurements are used to examine the structure of the spanwise vorticity in the inner region of low Reynolds number zero pressure gradient turbulent boundary layers. Single-probe measurements were made over the range 1010 ≤ Rθ ≤ 4850(Rθ = θU∞/ν, where θ is the momentum deficit thickness, and U∞ is the free-stream velocity), while two probe measurements were made at Rθ ≈ 1010. The present results indicate that for y+ < 50 statistical moment profiles of ωZ scale on inner variables. Event duration analyses indicate that a nearly logarithmic increase in inner normalized time scales of the ωZ bearing motions occurs with increasing Rθ. Outside the buffer region, this Rθ dependence is effectively removed if the Taylor time scale is used to normalize the event durations. Two-point correlations with probe separations in the spanwise as well as wall-normal direction are presented. In addition, the structure of the associated two-dimensional (2-D) probability distributions are examined to reveal the statistically most significant contributions underlying these correlations. Wall-normal probe separation measurements indicate the increasing prevalence of adjacent regions of opposing sign ωZ as the wall is approached. Spanwise probe separation experiments indicate the predominance of single-sign contributions, as well as increasing spatial coherence nearer the wall. The present results are interpreted to indicate that the organized spanwise vorticity-bearing motions are distributed in planes parallel to the wall for y+ less than about 12, and decrease to a nearly fixed scale outside the buffer region.

Organized Turbulence Structures in the Near-Neutral Atmospheric Surface Layer

The Tiederman method, which was originally designed for unambiguous detection of “bursts” in the buffer layer of smooth surface laboratory boundary-layer flow, is shown to work equally well in the neutral atmospheric surface layer. It enables estimation of characteristic timescales of significant structures responsible for the momentum transport: mean duration of and mean interval between bursts (upward transport of momentum deficit) and mean duration and interval between sweeps (downward transport of momentum excess). Data representing near neutral conditions from three field experiments over flat and homogeneous areas but with very different roughness characteristics are used in the analysis. For two low vegetation sites (i.e., the measuring heights are very much larger than the roughness sublayer height), it is found that all above-mentioned timescales are constant for any height and identical at the two sites. Scaling the results with the friction velocity indicates the corresponding scaling height to be proportional to the scaling height of the neutral boundary layer. The third site is a forest site, and the measurements were taken in the roughness sublayer. Here canopy inflection scaling is found to apply. The structures identified with the Tiederman method are shown to be responsible for more than 90% of the momentum flux, most of which is accomplished by a mean flow component within the structures. Results are discussed in light of direct numerical simulation results for fully turbulent but low Reynolds number flow over a dynamically smooth surface and large-eddy simulation results for a high Reynolds number case, and these show encouraging consistencies for the logarithmic layer. Turbulence production at the surface is, however, fundamentally different in the rough atmospheric case compared to the smooth case and is presumably governed by canopy wind profile inflection instability.

Spanwise vorticity structure in turbulent boundary layers

Four-element hot-wire probe measurements are used to examine the structure of the spanwise vorticity in the inner region of low Reynolds number zero pressure gradient turbulent boundary layers. Single-probe measurements were made over the range 1010 ≤ R θ ≤ 4850(R θ = θU ∞/ν, where θ is the momentum deficit thickness, and U ∞ is the free-stream velocity), while two probe measurements were made at R θ ≈ 1010. The present results indicate that for y + < 50 statistical moment profiles of ω Z scale on inner variables. Event duration analyses indicate that a nearly logarithmic increase in inner normalized time scales of the ω Z bearing motions occurs with increasing R θ. Outside the buffer region, this R θ dependence is effectively removed if the Taylor time scale is used to normalize the event durations. Two-point correlations with probe separations in the spanwise as well as wall-normal direction are presented. In addition, the structure of the associated two-dimensional (2-D) probability distributions are examined to reveal the statistically most significant contributions underlying these correlations. Wall-normal probe separation measurements indicate the increasing prevalence of adjacent regions of opposing sign ω Z as the wall is approached. Spanwise probe separation experiments indicate the predominance of single-sign contributions, as well as increasing spatial coherence nearer the wall. The present results are interpreted to indicate that the organized spanwise vorticity-bearing motions are distributed in planes parallel to the wall for y + less than about 12, and decrease to a nearly fixed scale outside the buffer region.

Field Measurement of Boulder Flow Drag

A field method for measuring the flow drag force on individual boulders on a riverbed is identified and tested. Drag is determined from the momentum deficit in the boulder wake, obtained by measuring the flow velocity in a traverse across the wake. A field test yielded a mean drag coefficient of 0.36, which is low compared with flume measurements of 1.1–1.5 for cubes. This may be the result of higher velocity water near the riverbed being sucked into and reenergizing the wake. Further tests are needed to define the conditions in which the method can be used and to indicate how well the assumptions behind the method can be accommodated in a natural flow environment.

Interaction of wake turbulence with a free surface

The turbulent wake of a flat plate aligned with a uniform water flow and extending through the free surface was investigated experimentally. Laser-Doppler velocimetry (LDV) measurements show good agreement with published data for a two-dimensional wake, except in a shallow layer near the free surface. In this surface layer, the wake width is observed to double while the wake centerline velocity remains essentially unchanged from its value at depth, resulting in a wake momentum deficit that is twice that at depth. Instantaneous, full-field measurements of the velocity were made using digital particle image velocimetry (DPIV) to elucidate the role of vortical structures in the development of the surface layer. DPIV measurements reveal that in the deep wake, vortex structures predominantly of opposite sign exist on opposite sides of the wake centerline and contribute to a velocity and vorticity field that is two-dimensional in the mean. However, near the surface, vortex structures tend to become either surface-normal or surface-parallel and contribute to a velocity and vorticity field that is highly three-dimensional in the mean. The resulting surface layer is characterized by surface–parallel structures interacting with their images above the surface to retard and widen the surface flow to a depth comparable to the size of the vortex structures. Histograms taken from many independent DPIV realizations of the flow characterize the distribution of vorticity in the wake and verify a mean flow consistent with the LDV measurements. The existence of these structures in the surface layer is further confirmed by flow visualization using laser-induced fluorescence.

Experimental study of plane turbulent wakes in a shallow water layer

Shallow two-dimensional turbulent wake flows have been studied experimentally on a large water table. In the experiments, the ambient Reynolds number Re h = U a H/ ν, in which U a is the depth-averaged ambient velocity, H the water depth, and ν the kinematic viscosity, is large, well above a lower critical value of the order of 500 for open-channel flows so that the ambient base flow is fully turbulent. Different types of blunt bodies extending over the full depth are inserted in that base flow, including cylinders and flat solid and porous plates oriented transversely to the ambient flow. In all cases, the transverse body dimension D greatly exceeds the water depthy, D H ⪢ 1 . With that condition, the wake Reynolds number Re d = U a D/ ν is very large, greater than 10 4. The shallow near-wake characteristics of plane wakes from blunt bodies extending over the full water depth have been found to fall into one of three classes: (i) the vortex street (VS) type with an oscillating vortex shedding mechanism, (ii) the unsteady bubble (UB) wake type with flow instabilities growing downstream of a recirculating bubble attached to the body, and (iii) the steady bubble (SB) wake type with an attached bubble followed by a turbulent wake that contains no growing instabilities. When Re h > 1500, the flow classification is uniquely dependent on a shallow wake parameter, S = c f D H in which c f is a quadratic law friction coefficient. For circular cylindrical bodies the VS-UB transition is characterized by a critical value, S ca ≈ 0.2, and the UB-SB transition by S cc ≈ 0.5. Solid plates, oriented transversely, differ by a factor of 1.25. The shallow far-wake behavior has been investigated with a special variable porosity wake device that reduces the wake velocity deficit and completely suppresses the VS instabilities in the near-field. Thus, only UB and SB wake types are found in that case. Furthermore, the shallow plane wake is obsserved to “stabilize” for large downstream distances, x H , in the sense that the growth and maintenance of the large scale structures in the wake flow become suppressed and the wake collapses into a more ordered flow that, however, still contains small scale (of scale H) turbulence. This wake stabilization is controlled by two factors: first, the usual evolution in a turbulent wake that reduces the velocity deficit while increasing the wake parameter S, and secondly, the exponential loss of the momentum deficit flux in the wake due to bottom friction.

An experimental study of a turbulent wing-body junction and wake flow

Extensive measurements were conducted in an incompressible turbulent flow around the wing-body junction formed by a 3∶2 semi-elliptic nose/NACA 0020 tail section and a flat plate. Mean and fluctuating velocity measurements were performed adjacent to the wing and up to 11.56 chord lengths downstream. The appendage far wake region was subjected to an adverse pressure gradient. The authors' results show that the characteristic horseshoe vortex flow structure is elliptically shaped, with ∂ (W)/∂Y forming the primary component of the streamwise vorticity. The streamwise development of the flow distortions and vorticity distributions is highly dependent on the geometry-induced pressure gradients and resulting flow skewing directions. The primary goal of this research was to determine the effects of the approach boundary layer characteristics on the junction flow. To accomplish this goal, the authors' results were compared to several other junction flow data sets obtained using the same body shape. The trailing vortex leg flow structure was found to scale on T. A parameter known as the momentum deficit factor (MDF = (Re T)2 (θ/T)) was found to correlate the observed trends in mean flow distortion magnitudes and vorticity distribution. Changes in δ/T were seen to affect the distribution of u′, with lower ratios producing well defined local turbulence maxima. Increased thinning of the boundary layer near the appendage was also observed for small values of δ/T.

On accurately measuring statistics associated with small-scale structure in turbulent boundary layers using hot-wire probes

Spanwise vorticity measurements have been performed in zero-pressure-gradient boundary layers over the range 1010 < Rθ < 4850 (Rθ ≡ U∞ θ/ν, where U∞ is the free-stream velocity and θ is the momentum deficit thickness) using a four-wire probe. In addition, experiments quantifying the spatial and temporal resolution required to obtain an accurate statistical representation of the small-scale structure of wall-bounded turbulence were performed. Furthermore, a thorough investigation of statistical convergence for a variety of fluctuating quantities was performed. Comparisons with earlier high-resolution studies indicate that the maximum value of u′/uτ increases with increasing Reynolds number over the given Rθ range (u′ ≡ r.m.s. u, and uτ is the friction velocity). It is suggested that detecting this dependence provides a good measure of probe resolution. In general it was found that statistics of velocity gradients were distinctly more sensitive to finite probe size than velocity statistics. Wire spacing experiments suggest that Wyngaard's (1969) criterion is to a good approximation valid even under anisotropic conditions. Furthermore, it was found that instantaneously spatial averaging of ∂u/∂t caused significant attenuation in the resulting r.m.s., and that this averaging procedure is sensitive to the level of mean shear. A simple method of estimating how noise in the u-velocity signals enters into the ∂u/∂y signals is presented. The convergence study shows that statistical convergence criteria developed from free-shear flows severely underestimates the averaging times required in boundary layers. A table of general convergence criteria is provided.

Momentum flux in breaking wavelets

A breaking wavelet is taken to consist of a roller and a trailing turbulent wake, both riding on an irrotational wave. The shear stress force on the separation streamline between the roller and the underlying flow is balanced mainly by the horizontal pressure force on the same streamline. The pressure force acts on the underlying flow and reduces wavelet momentum; the shear force generates the momentum deficit of the wake. In this manner, wavelet momentum is turned into shear flow momentum. In wind‐driven wavelets the shear force of the wind aids roller formation: rollers form at a relatively low approach momentum from boundary layer fluid generated by surface shear. Breaking wavelets have been modeled by superimposing the surface disturbance generated by a roller on a sinusoidal wave. The phase relationship of the two components determines how much momentum is extracted from the wave. The models show the characteristic asymmetric, forward leaning shape of breakers. The wave under the roller is shortened, so that the steepness of the breaker is even greater than it would be on account of the roller's presence alone. Ahead of the roller's toe, capillary waves are generated. On short waves these are of easily visible amplitude and serve to identify the presence of a roller.

On the sign of the instantaneous spanwise vorticity component in the near-wall region of turbulent boundary layers

A geometric feature of the vorticity field in the near-wall region is educed from an analysis of time-resolved spanwise vorticity measurements. The results of this analysis indicate the existence of a region near the wall in which the instantaneous z component of the vorticity vector is virtually always in the direction indicated by the mean shear. In accordance with the solenoidal condition of the vorticity field, this observed physical feature represents a limitation on the spatial structure of the vorticity field in the near-wall region. The present results also reinforce the conclusion of Klewicki (Ph.D. dissertation, Michigan State University, 1989) that, for the momentum deficit thickness Reynolds number range 1010≤Rθ≤4850, the statistical properties of the spanwise vorticity component are invariant under an inner variable normalization for y+ less than about 40.

An experimental investigation of the turbulent structure in a two-dimensional momentumless wake

Mean velocity profiles have been measured in the wake of a two-dimensional airfoil with and without mass injection through a slit along its rear end. In particular, four cases have been documented: (a) a pure wake (no injection), (b) a weak wake (some injection), (c) a momentumless wake (injection adjusted to provide a thrust which exactly cancels the model's drag), and (d) a weak jet (more injection than necessary to cancel the drag). These mean velocity profiles clearly show the difference in momentum deficit for the four cases. When non-dimensionalized, the velocity profiles are self-similar.Smoke-wire flow visualizations are also presented for both the near wake (0 < x/d < 30) and the far wake (45 < x/d < 75). The characteristic geometry of large-scale turbulent vortical structures is easily identifiable for the wake and jet cases, while the structures for the momentumless state are neither wake-like nor jet-like. Beyond about 45 diameters downstream, turbulent structures for the momentumless case become quite weak and can barely be observed.For the momentumless wake at Re = 5400, the axial, lateral, and transverse turbulence intensities as well as the Reynolds stress were measured. Similarity of the axial and the transverse turbulence intensities was observed; the overall shape of those profiles is Gaussian except for the very near-wake region. The mean centreline velocity difference decays much faster (x−0.92) than the axial turbulent intensity (x−0.81). Consequently, the mean shear practically disappears far downstream; the flow becomes nearly isotropic beyond about 45 body diameters from the model. This turbulence behaviour is quite different from that of plane wakes or jets but rather closer to the case of grid turbulence.

Experimental validation of the UPM computer code to calculate wind turbine wakes and comparison with other models

The UPM Computer Code to calculate wind turbine wakes is applied to test cases, both in wind tunnel and in field experiments with full scale machines, in order to validate the model used in the code. In the UPM model the wake of the wind turbine is supposed to be immersed in a non-uniform air stream corresponding to the atmospheric boundary layer. The turbulent transport coefficients are modelled by the k-ϵ method. Relevant parameters describing the process are wind velocity at turbine height, ground roughness, atmospheric stability, turbine dimensions and initial momentum deficit. The wind tunnel measurements correspond to experiments carried out at TNO. The full scale experiments chosen for comparison are those from Nibe wind turbines in Denmark. The results of the UPM Code are also compared with those obtained with other simpler models.

Density currents or density wedges: boundary-layer influence and control methods

Density currents and density wedges are two observed manifestations of interactions between an ambient flow and a horizontal buoyant intrusion. In a density current the buoyant pressure force is primarily balanced by the local form drag of the current head which has a blunt shape and abrupt depth change. In a density wedge a distributed interfacial drag is the primary balancing force, leading to a stretched-out shape and long-distance intrusions. A perturbation analysis of the approach flow to the inclined front of a density current shows that slight momentum changes caused by viscous effects in the ambient flow determine which of these two flow types is established. In a uniform ambient channel flow, any momentum deficit relative to the inviscid case will lead to a local flattening of the front and ultimate breakdown into a density wedge. On the other hand, a momentum surplus will support a steady-state density current. Several exploratory experiments on control of the ambient boundary layer through local non-uniformities were performed with the objective of achieving stable density-current forms with limited intrusion lengths. These methods include a small step, a barrier and suction and are applied for intrusions at either the bottom or surface of an ambient water flow. In all cases, good agreement is found with the force balances predicted by Benjamin's (1968) theory and its extension by Britter &amp; Simpson (1978) which accounts for entrainment in the wake zone of the head.

The Wake of A Jackdaw (Corvus Monedula) in Slow Flight

ABSTRACT The wake of a jackdaw in slow forward flight is described. The three-dimensional velocity field was investigated qualitatively and quantitatively by analysis of multiple-flash stereophotographs of the motion of neutrally buoyant helium bubbles. The best description of the wake structure appears to be a chain of planar, nearcircular, discrete, small-cored, vortex loops, each produced by vorticity shed during a single downstroke. However, the momentum measured in such a wake is approximately 35 % of that required for weight support under these flight conditions. Some evidence is presented that this apparent wake momentum deficit may arise because the description of the real wake vorticity distribution is too simplistic. The implications of these results for theoretical models of bird flight are briefly discussed.

Perturbation analysis and measurements of building wakes in a stably stratified turbulent boundary layer

Velocity and temperature wakes behind obstacles deeply submerged in a stably stratified turbulent boundary layer with one face perpendicular to the flow direction were investigated using wind-tunnel tests and mathematical analysis. For stably stratified approach flow, the effects of a building wake characterized by momentum deficit are to decrease mean velocity and mean temperature but to increase turbulence intensity and temperature fluctuation intensity. On the other hand, along the centerline of a building wake, mean velocity and mean temperature increase and turbulent intensity and temperature-fluctuation intensity decrease in the presence of longitudinal horseshoe vortices. A perturbation technique for determining the temperature distribution in the wake of an isolated building was developed using momentum and vortex wake arguments. Hunt's theory for velocity distribution in a “momentum” and “vortex” wake and a modified method developed by the writers for temperature distributions agreed with measurements of mean velocity and temperature distributions.

An observational estimate of gravity wave drag from the momentum balance in the middle atmosphere

The zonal average momentum budget in the middle atmosphere (up to 0.1 mbar) is computed for 7 months of satellite observations in order to determine the forcing needed to obtain a balance. This momentum residual includes forcing by waves with small zonal scales, such as gravity waves. The results indicate that the forcing needed in the lower mesosphere reaches peak values of about 20 m s−1 d−1, which is large compared to the Rayleigh friction used in that part of the atmosphere in numerical models, such as that of Holton and Wehrbein (1980a). The seasonal change follows that of Rayleigh friction; the northern hemisphere momentum deficit is large easterly in winter, decreases in the spring, and becomes small westerly in late spring. However, the largest momentum deficits are in middle and high latitudes (50°–80°N) in winter, whereas the largest Rayleigh frictional damping is in the vicinity of the jet (30°–50°N). The observations were also compared with the parameterized acceleration of the mean flow by gravity waves (Lindzen, 1981; Holton, 1983). The observations of the wintertime momentum residual are similar to the values used in the circulation model of Holton (1983). Derivation of the parameterization constant from the satellite observations results in an equivalent zonal wave number for gravity waves that is substantially smaller than that used by Holton (1983) in his model. Possible reasons for this difference are that many of the gravity waves break at levels above the region where data are available and that gravity waves with nonzero phase speeds may be present.

Drag minimization on a body of revolution through evolution

This study deals with the determination of the minimum drag profile of a body of revolution. With the restriction that the boundary layer is not allowed to separate, the profile is described by a continuous, linear line-singularity distribution defined at a number of points placed parallel in a uniform stream with zero angle of attack. The drag calculation is based on the momentum deficit in the boundary layer at the trailing edge of the body. By gradually changing the body profile with the Evolution Strategy the minimum drag profile is found for a constant fineness ratio. For a particular case, the drag coefficient is reduced by 30 percent by means of this procedure.

Light-particle emission in the reaction 32S + 27A1 AT 135 and 190 MeV

The properties oflight particles emitted by the 32S + 27Al reaction at 135and 190 MeV bombarding energies were studied by means of a coincidence spectrometer. The spectrometer consisted of two large-area ionization chambers which measured the energy, momentum, mass and nuclear charge of the heavy reaction products. By requiring the conservation of those quantities the energy, momentum, mass and charge deficits were determined which are representative of the unobserved light particles. The analysis of the momentum deficit in the event plane did not yield an indication for a fast, direct process of light-particle emission. The alternative analysis in the fragments rest system confirmed the statistical nature of the emission process. The out-of-plane angular correlations were used to determine the spins of the particle-emitting fragments.