Non-dimensional Wavelength Research Articles

Waviness Deformation in Starved EHL Circular Contacts

By means of numerical simulations the deformation of transverse and isotropic harmonic waviness in EHL circular contacts under pure rolling has been studied in relation to the lubricant supply to the contact. In earlier work the deformation of waviness under pure rolling in a fully flooded contact was shown to depend on a single non-dimensional wavelength parameter. In terms of this parameter short wavelengths deform very little. In this paper the effect of starvation on this behavior is shown. First, the steady state smooth surface problem is discussed as an introduction and as a reference problem. It is illustrated in detail how the entire film thickness level decreases with decreasing lubricant supply. Subsequently, results are presented for the time dependent problem with waviness moving through the contact under pure rolling. The relative deformed amplitude of the waviness inside the contact is shown to depend on the same non-dimensional wavelength parameter as before, but also on the degree of starvation. A smaller lubricant supply leads to a larger reduction of the waviness amplitude inside the contact. Finally, it is shown that to an acceptable accuracy the relative deformed amplitude of the starved problem can be predicted by the formula for the fully flooded problem if the generalized wavelength parameter is modified using the reduction factor of the central film thickness for the starved steady state smooth contact. For this reduction factor an accurate formula is available and as a result also for starved contacts by means of a component wise approach a crude estimate of the deformed surface micro-geometry (roughness) inside a contact can be obtained quite easily now.

Surface roughness attenuation in line and point contacts

Surface roughness effects in line and point elastohydrodynamically lubricated contacts are compared and it is shown that the underlying behaviours in both types of contact are identical. Long wavelength components of roughness are attenuated; short wavelength components pass through the conjunction unaltered. It is also shown that the roughness attenuation, plotted against a non-dimensional wavelength, follows almost identical curves in both cases.

Flutter of an infinitely long panel in a duct

The aeroelastic stability is examined of an infinitely long panel of finite width enclosed in a duct such that both the upper and lower surfaces of the panel are exposed to an inviscid, and compressible flow. The panel behavior is accounted for by small deflection plate theory, whereas the aerodynamic forces acting on the panel are described by the classical linearized small disturbance potential theory. As such, a self-consistent theoretical model is constructed for the asymptotic behavior of the panel. Two panel boundary conditions are considered; the panel is assumed to be either simply supported, or clamped along the side edges. For the simply supported case, rather extensive numerical results have been obtained. The effects of Mach number, air/panel mass ratio, and duct dimension on the flutter velocity are determined. a a* b c CQ c* D E h h* hp L / /* M m P t U U* U*T W x y z y Nomenclature speed of sound a/c0 panel width wave speed reference wave speed, 2n(D/pmhb2)1/2 C/CQ plate stiffness modulus of elasticity height of duct nondimensional height of duct, h/b panel thickness panel length wavelength nondimensional wavelength, l/b Mach number, U*/a* number of modes pressure time flow velocity U/CQ critical flutter velocity panel deflection streamwise coordinate spanwise coordinate coordinate perpendicular to plane of panel separation constant nondimensional dynamic pressure, /x£/*2

The Theoretical, Discrete, and Actual Response of the Barnes Objective Analysis Scheme for One- and Two-Dimensional Fields

This paper examines the response of the Barnes objective analysis scheme as a function of wavenumber or wavelength and extends previous work in two primary areas. First, the first- and second-pass theoretical response functions for continuous two-dimensional (2-D) fields are derived using Fourier transforms and compared with Barnes' (1973) responses for one-dimensional (1-D) waves. All responses are nondimensionalized with respect to a smoothing scale length, such that the first-pass responses are a function only of nondimensional wavelength. The 2-D response is of the same functional form as the 1-D response, with the 2-D wavenumber substituted for the 1-D wavenumber. The 2-D response departs significantly from the 1-D value (for the same x-component of the wavelength) when the y-component of the wavelength is less than approximately ten scale lengths, a condition applying to most fields with closed centers as well as open waves with a significant latitudinal variation. Second, the continuous theoretical response for 1-D and 2-D waves is compared with the response for discrete applications of the scheme using uniformly spaced observations. This response is evaluated two different ways. The discrete theoretical response is found by discretizing the Barnes scheme with a 2-D “comb” function for cases in which interpolation points are either coincident with or midway between observation points. The response can then be evaluated with Fourier transforms in much the same way as for the continuous case. The discrete response evaluated in this manner attains a minimum at the Nyquist wavelength, with enhanced values for smaller unresolvable wavelengths resulting from aliasing. The discrete response is approximately equal to the sum of the responses for the original wavelength and for a primary aliased wavelength. At larger resolvable wavelengths, the discrete response approaches the continuous value as aliasing becomes negligible. The second means of examining the response for discrete applications is through a Fourier series analysis of fields interpolated by the Barnes scheme. The “observations” in this context are given by analytic functions on a uniform mesh which may or may not differ from the analysis grid. A given input wavelength leads to an analysis which contains waves at one or more aliased wavelengths in addition to the original wavelength. Components of the response corresponding to each of these wavelengths can be estimated as the Fourier amplitude of the interpolated field divided by the amplitude of the input wave. The actual response from a discrete application of the Barnes scheme confirms the results of the analysis of the discrete theoretical response; aliasing to longer wavelengths is seen for nonresolvable wavelengths in the “observations,” while the actual response is close to the theoretical value for well resolved wavelengths for both 1-D and 2-D fields. This analysis of the discrete and actual response supports the recommendations of Caracena et al. and Koch et al. for the relationship between the smoothing scale length and the observation spacing. Setting the smoothing scale length to approximately four-thirds of the observation spacing, the upper bound recommended by Caracena et al., yields a response close to the continuous value for resolvable wavelengths, with a reasonably small degree of aliasing. However, the case in which the smoothing scale length is equal to the observation spacing, the lower bound recommended by Caracena et al., retains an unacceptably high degree of aliasing in a typical two-pass application of the Barnes scheme.

On the modelling of sand bedforms using the semivariogram

AbstractThis study shows the usefulness of the semivariogram for modelling sand ripples created by water flows of varied flow intensity. A combination of two mathematical functions is fitted to each sample semivariogram, that is an exponential (or stochastic) component and a periodic component. The parameters of each of these components have direct physical meaning. A non‐dimensional ratio combining the two parameters of the exponential model is interpreted as a regularity index (which increases with the degree of regularity of bedform arrangement). This regularity index is inversely related to the Froude number of the flow. The non‐dimensional wavelength, estimated from the dominant periodic function, is also inversely and closely related to the Froude number. The wave height, accurately estimated from properties of the two fitted components, is a direct function of flow velocity and is also proportional to the standard deviation of bed elevations. The bedform shape introduces a considerable discrepancy between the generally assumed normal frequency distribution and the empirical distributions of bed height. The series of bed elevations are generally characterized by a mixture of normal distributions having the same variance but different means. The calculation of a covariance assuming a constant and single mean (as in spectral analysis) can therefore be misleading and the problem may be avoided by using the semivariogram.

On the Non-linear Evolution of Görtler Vortices in Non-parallel Boundary Layers

The non-linear development of finite amplitude Görtler vortices in a non-parallel boundary layer on a curved wall is investigated using perturbation methods based on the smallness of e, the non-dimensional wavelength of the vortices. The crucial stage in the growth or decay of the vortices takes place in an interior viscous layer of thickness O(ε2) and length O(ε). In this region the downstream velocity component of the perturbation contains a mean flow correction of the same order of magnitude as the fundamental which is driving it. Moreover, these functions satisfy a pair of coupled non-linear partial differential equations which must be solved subject to some initial conditions imposed at a given downstream location. It is found that, depending on whether the boundary layer is more or less unstable downstream of this location, the initial disturbance either grows into a finite amplitude Görtler vortex or decays to zero. For the Blasius boundary layer on a concave wall it is found that Görtler vortices can only develop if the rate of increase of curvature of the wall is sufficiently large. In this case the finite amplitude solution which develops initially in an ε-neighbourhood of the position where the disturbance is introduced changes its structure further downstream. This structure is investigated at a distance O(εδ) (with 0< δ<1) downstream of the above ε-neighbourhood. In this régime the downstream fundamental velocity component has an elliptical profile over most of the flow field. However, in two thin boundary layers located symmetrically either side of the centre of the viscous layer the fundamental velocity component decays exponentially to zero. The locations of these layers are determined by an eigenvalue problem associated with the one-dimensional diffusion equation. The mean flow correction persists both sides of the boundary layer and ultimately decays exponentially to zero. This large amplitude motion is not sensitive to the imposed initial conditions and appears to be the ultimate state of any initial disturbance. However, in the initial stages of the growth of the vortex, some surprising flows are possible. For example, it is possible to set up a vortex flow similar to that observed by Wortmann (1969) which consists of a sequence of cells inclined at an angle to the vertical.

On the radiation and scattering of short surface waves. Part 3

The radiation properties of partially immersed three-dimensional bodies, in time-periodic motion, are examined in the short-wave asymptotic limit ε → 0, where ε is a non-dimensional wavelength. The method of matched expansions is used to specify an outer approximation, away from the surface wave region, and an inner approximation where the potential, in the vicinity of the obstacle and free surface, depends only on the local geometry. Finally, the radially spreading surface wave field is estimated by ray-theory arguments. Explicit details are given for the heaving and rolling of a circular dock and for the heaving motion of a hemisphere. Some speculations are made regarding the scattering properties of such obstacles.

On the radiation and scattering of short surface waves. Part 1

The radiation and scattering of time-periodic surface waves by partially immersed objects is investigated in the short-wave asymptotic limit ε → 0, where ε is a non-dimensional wavelength. Details are given for the prototype radiation and scattering problems in which the fluid has infinite depth and the body is a two-dimensional dock of finite width and zero thickness. The solutions are then generalized to deal with other two-dimensional geometries, with the restriction that the ends of the obstacle are horizontal for a distance of many wavelengths. The method of matched expansions is used. A first approximation ϕ0, presumed to be a good estimate for the potential throughout most of the fluid region, is obtained by replacing the free-surface condition by its formal limit ϕ0 = 0. In the vicinity of the ends of the obstacle, the correct surface condition is used but the geometry of the problem is simplified. The remaining surface layers are dealt with by superimposing on the function ϕ0 regular wave trains of the appropriate amplitude.