Momentum Deficit Research Articles

An experimental investigation of breaking waves produced by a towed hydrofoil

This paper presents the results of experiments on breaking waves produced by towing a submerged, two-dimensional hydrofoil at constant depth and speed. The wave field consists of a breaker followed by a train of lower, non-breaking waves. The breaker has a small zone of turbulent water riding its forward slope; this zone is called the breaking region. Measurements were made of surface height profiles, the vertical distribution of mean horizontal velocity in the wake of the wave, and the vertical thickness of the wake. The results support the hypothesis that the breaking region imparts a shearing force along the forward slope equal to the component of its weight in that direction. The force produces a turbulent, momentum-deficient wake similar to the wake of a towed, two-dimensional body in an infinite fluid. The vertical thickness of the wake grows in proportion to the square root of distance behind the breaker. The momentum deficit is approximately equal to the maximum momentum flux of a Stokes wave with the same phase speed as the breaker. The surface profile measurements yield several results: the proper independent variables describing the wave are its speed and the slope of its forward face. The relation between breaking wavelength and speed follows the finite-amplitude Stokes wave equation. The amplitude and the vertical extent of the breaking region are both proportional to the phase speed squared; however, they are not functions of the slope of the forward face of the wave. The breaking region has a small oscillation in its length with a regular period of 4.4 the period of a wave with phase speed equal to the hydrofoil speed. The amplitude of the oscillation diminishes with time. It is believed that this oscillation is due to wave components produced when the foil is started from rest.

Surface-roughness effects on the mean flow past circular cylinders

Measurements of mean-pressure distributions and boundary-layer development on rough-walled circular cylinders in a uniform stream are described. Five sizes of distributed sandpaper roughness have been tested over the Reynolds-number range 7 × 104to 5·5 × 105. The results are examined together with those of previous investigators, and the observed roughness effects are discussed in the light of boundary-layer theory. It is found that there is a significant influence of surface roughness on the mean-pressure distribution even at very large Reynolds numbers. This observation is supported by an extension of the Stratford–Townsend theory of turbulent boundary-layer separation to the case of circular cylinders with distributed roughness. The pressure rise to separation is shown to be closely related, as expected, to the characteristics of the boundary layer, smaller pressure rises being associated with thicker boundary layers with greater momentum deficits. Larger roughness gives rise to a thicker and more retarded boundary layer which separates earlier and with a smaller pressure recovery.

A Modified Displacement-Body Method for Treating the Axisymmetric Strong-Interaction Problem

A modification of the displacement-body method is presented for calculating the high-Reynolds-number, strong-interaction flow field in the aft region of a body of revolution. The modification is based on a simple pressure mapping which effectively represents streamline curvature effects in the vicinity of the tail. The mapping is derived and the method applied to three representative axisymmetric bodies for which experimental data are available. The boundary-layer equations and potential flow equations are solved by a standard finite-difference and surface singularity method, respectively. Calculated results for the body pressure distribution, momentum deficit area, and body drag are found to agree well with experiment.

Energy Effectiveness of Arbitrary Arrays of Wind Turbines

Determination of power degradation due to interference between wind turbines in an array is of importance in the engineering and economic planning of wind farms. A computer model for an arbitrary array of turbines is described. The basic fluid mechanics are treated in a simple but rational way. The turbine wake expands downstream due to ambient turbulence and mechanically generated turbulence (caused by momentum gradients) and entrains momentum and mass. Drag or momentum deficit, though, is conserved. Ground effect is handled by imaging. The effect of ambient turbulence is shown to be much greater than of that due to the momentum deficit generated by the turbine. The basic equations use fundamental fluid mechanical expressions related to drag conservation and wake growth due to turbulent entrainment and a family of self-similar wake profiles derived from experiment. This approach fully defines the wake velocity field. The wake of each turbine is then determined, subject to all upstream interferences. Power outputs of selected arrays as functions of wind direction are presented. The model is very well supported by the limited data available, and has proven effective and easy to implement. Advanced models incorporating nonuniformities in wind and turbulence and tower shadow are also described.

Similarity and weak closing relations for symmetric free turbulence

In [1, 2], similarity behavior of free turbulent flows at large Reynolds numbers was analyzed. Here, the term similarity was applied to distributions of averaged quantities that depend on the coordinate in the direction of the flow through two scaling functions of length1 (x) and velocity U(x). In the study [3] of flow on a self-propelled body (momentumless wake), Naudascher used a similarity concept of more general type. In the present work, which takes the point of view of similarity with one scaling function of length and different amplitude functions for different quantities, flows are analyzed that have arbitrary (including zero) deficit of the total momentum with respect to the momentum of the external flow. The similarity distributions satisfy the equations for the mean velocity and the single-point moments of second order only under the condition that certain connections (“weak closing relations”) hold between the amplitude functions. The relations are different depending on the dominant physical processes.

Entrainment in turbulent wake flows

The stability of an axisymmetric turbulent wake produced behind a stationary sphere in a fluid moving with uniform velocity is investigated. The dispersion relation indicates that the interface is unstable at all wave numbers, if the wake is treated as a viscous medium. From the dispersion relation the surface-strain energy density spectrum of a known initial disturbance is obtained. The dynamic similarity of the distributions of the phase velocity and the surface-strain energy density at high Reynolds number is demonstrated. From the energy equation one obtains a critical wavenumber beyond which the energy transport to the wake through the interface is cut off. A critical time is obtained based on the assumption that the momentum deficit in the wake is due to the viscous drag. The critical time and the critical wavenumber are used to explain the existence of an entrainment cycle.

Comment on "Evolution of the Laminar Wake behind a Flat Plate and Its Upstream Influence"

D URING last few years considerable effort has been expended in an attempt to understand character of boundary-layer flow at a sharp trailing edge. The Introduction to Ref. 1 summarizes most of published investigations of steady incompressible laminar flow past trailing edge of a finite plate at zero incidence and high Reynolds number. In particular, asymptotic descriptions considered to be valid in limit as R -> oo (R = UL/v; U = external flow speed, L = plate length, v = kinematic viscosity) were given independently by Stewartson and by Messiter. For most part, Ref. 1 correctly points out deficiencies of earlier work, but several comments about theory derived by Stewartson and Messiter are either unjustified or incorrect. First of all, Ref. 1 states incorrectly that work of Ref. 4 assumed isobaric flow. The assumptions actually made were that: a) effect of trailing edge is confined to a distance O(R~L); and b) Navier-Stokes equations can be linearized about .the Blasius solution in this neighborhood. The serious error lay in neglecting discontinuity of streamline slopes in boundary layer at' x = 1; this was corrected in Refs. 2 and 3. Imai treated same problem as in Ref. 4 but with isobaric assumption. Incidentally, Ref. 4 is not without value for it still provides a useful guide to structure of region within a distance O(R ~ L) of trailing edge provided one adjusts constant 1 as explained in Ref. 2. Hakkinen and O'Neil, not mentioned in Ref. 1, have also discussed this region, in terms of expansions for Rr/L '-> oo. Second, it is stated in Ref. 1 that use of linearized airfoil theory is not justified because of large slopes of (effective) slender body. It is certainly true that slopes are large compared with streamline slopes predicted by ordinary boundary-layer theory, but slopes do remain small compared to 1. As explained in Refs. 2 and 3, u = 1 + 0(#~), v = 0(R~), du/dx = 0(R + ), etc. In fact, Eq. (7) of Ref. 1 essentially incorporates notion of linearization because term vfiu/dy is omitted. Also, as Fig. 6 of Ref. 1 makes clear, ~ 0.01 and therefore cp = -2(u1) with error ~ 0.0001. Third, main problem studied in Eqs. (4-7) of Ref. 1 contains formulation of Refs. 2 and 3 and reduces to it as R -> oo. Thus there is agreement in this limit. However, this does not provide justification for neglecting terms when R = 10. A number of terms are omitted in Ref. 1, but perhaps most important is pressure variation across boundary layer. It is established in Refs. 2 and 3 that this is 0(R~), just a fraction smaller than longitudinal pressure variation O(R~). Neglecting this term is justified in limit as R -* oo, but may well lead to numerical inaccuracy at R = 10. Fourth, pressure plotted in Fig. 6 of Ref. 1 shows a singularity at x = 1, and so dp/dx is seriously in error if x is very close to 1. The solutions to Eqs. (1-3) of Ref. 1 presumably require that pressure be known at the edge of Region I; however, pertinent details are not discussed. Furthermore, numerical work of Ref. 1 would have benefited from making use of an asymptotic analysis of outer region of influence of trailing edge. Conditions at infinity are notoriously difficult to handle numerically, and discrepancy between results of Ref. 1 and those of Goldstein, when x = 2, should be regarded with caution. Finally, Fig. 6 of Ref. 1 is in qualitative agreement with predictions of Refs. 2 and 3, in that a favorable pressure gradient is established for x < 1, an adverse pressure gradient at x = 1 + , and a favorable pressure gradient as (x— 1)#-+ oo. Fig. 6 shows that displacement effect leads to large dcp/dx for Ax « 0.2 (say) whereas Fig. 10 shows significant changes in skin friction for Ax « 0.0005. While dependence on R is not clearly established by these results, curves do show unequivocally existence of two length scales, again qualitatively in agreement with results of Refs. 2 and 3. The latter references do not contend that region O(jR~L) is irrelevant to structure of flow, but rather that dominant correction occurs on larger scale, and that even for R = 10 (say) largest effects are found in a qualitatively correct way by asymptotic analysis given. For example, correction to drag coefficient on larger scale is O(R ~ ), whereas effects on smaller scale lead to a change O(R ~). The correction may be calculated either by integrating skin friction along plate or by calculating a momentum deficit in wake. For consistency eigenfunction proposed in Eq. 5.10 of Ref. 2 is required.

Rotational states of 158Dy, 164Er, 168Yb and 174Hf excited by evaporation reactions

A series of rotational bands in 158Dy, 164Er, 168Yb and 174Hf were studied by excitation with neutron evaporation reactions initiated by protons, deuterons, 3He and α-particles with energies between 16 and 28 MeV. The resulting prompt γ-ray spectra were analyzed with a curved crystal diffraction spectrometer and with lithium-drifted germanium detectors. Transitions from levels in the ground state bands were seen up to 10 + in 168Yb, 12 + in 158Dy and 174Hf and 14 + in 164Er. The γ-vibrational bands in three of the nuclei were strongly excited and several β-bands were weakly excited. The energy levels deduced from the measurements were successfully fit by the empirical formulas of Holmberg and Lipas and of Sood for the ground state band as well as for the higher bands. The cross sections for transitions in the ground state band agree with the statistical model calculations made. Within the validity of the calculations, no large angular momentum deficit was observed.

Mean Flow Calculations Behind Arbitrarily Spaced Cylinders

A finite array of arbitrarily spaced parallel rods is considered. It is assumed that momentum deficit may be superposed locally. The momentum deficit is computed behind each cylinder—considering it as a single cylinder—using one of the accepted theories. Then the momentum deficits at each point due to all the cylinders are superposed. From this an overall velocity distribution is calculated. Experiments with various rod diameters and spacings were conducted. The results were within 3 percent of the theoretically predicted local velocity, and under 10 percent of the predicted local velocity defect (at locations of significant velocity deficits).

Toward an Acoustically Motivated Length Scale for Noise Environments beneath Disturbed Boundary Layers

This paper deals with moderate noise disturbances associated with shock. It is hoped that this work constitutes a bridge between the well-predictable noise due to undisturbed boundary layers and the yet unpredictable noise due to intense shock-boundary layer interactions. Moderately disturbed data is extracted from work published by Speaker and Ailman [NASA CR-486 (1966)], that data being variously scaled to show that the high-frequency behavior is well handled by nondimensionalization according to momentum deficit thickness θ. The prominance of θ substantiates that wall noise originates close to the wall, which makes possible a modification of the displacement thickness δ*, creating what appears to be another satisfactory acoustic scale. The indication is that boundary-layer acoustic tests should be associated with aerodynamic measurements that emphasize the inner, stress-dominated portions of the boundary layer if information is to be obtained pertinent to the relations between flow disturbances and wall noise.