Showing papers by "National Physical Laboratory published in 1971"

The application of integral equation methods to the numerical solution of some exterior boundary-value problems

Abstract: The application of integral equation methods to exterior boundary-value problems for Laplace’s equation and for the Helmholtz (or reduced wave) equation is discussed. In the latter case the straightforward formulation in terms of a single integral equation may give rise to difficulties of non-uniqueness; it is shown that uniqueness can be restored by deriving a second integral equation and suitably combining it with the first. Finally, an outline is given of methods for transforming the integral operators with strongly singular kernels which occur in the second equation.

Frequency of the hydrogen maser.

Abstract: THE accuracy of the unperturbed frequency of the hydrogen maser is limited by the uncertainty of the wall shift caused by the storage bulb coating1. Different samples of nominally the same material have been found to give different shifts2 and so it seems that a frequency determination should include an independent measurement of wall shift. The results of such determinations3–7, however, have a spread of 0.018 Hz which is large compared with the potential accuracy of the maser. The wall shift depends on the number of bounces of the atoms on the wall of the bulb and for a spherical bulb can be expressed as K/D where K is a constant for a particular coating material and D is the diameter of the bulb. If measurements are made with bulbs of different size and the frequency is plotted against I/D), the true frequency is that corresponding to infinite bulb size, that is 1/D = 0, and the slope of the line gives the constant K. The range of bulb sizes that can be used is limited by practical considerations and there is therefore an extrapolation error in addition to the error of measurement. This extrapolation error can be reduced by making measurements with two different coating materials and making the assumption that the values should coincide at 1/D = 0.

Calculation of boundary-layer development using the turbulent energy equation: compressible flow on adiabatic walls

Abstract: The basic method described by Bradshaw, Ferriss & Atwell (1967) is extended to compressible flow in two-dimensional boundary layers in arbitrary pressure gradient (excluding shock waves and expansion fans) by invoking Morkovin's hypothesis (Favre 1964) that the turbulence structure is unaffected by compressibility Using the same empirical functions as in incompressible flow, skin friction in zero pressure gradient is predicted to within 3% of Spalding & Chi's (1964) correlation for free-stream Mach numbers less than 5 Comparisons with experiments in pressure gradient are restricted by the lack of data, but, since Morkovin's hypothesis does not depend on pressure gradient, methods which use it (of which the present method seems to be the first) can be checked fairly adequately by comparisons with data in zero pressure gradientNo ‘compressibility transformations’ are needed, although the Crocco relation is used, provisionally, for the temperature: since the calculations take only about 20% longer than in incompressible flow, Morkovin's hypothesis does as much as any transformation could do It is pointed out that, in supersonic flow, surface curvature which is large enough to induce a significant longitudinal pressure gradient is also large enough to have a very significant effect on the turbulence structure

An investigation of the forces on flat plates normal to a turbulent flow

Abstract: Measurements on square and circular plates in turbulent flow show the mean base pressure to be considerably lower than that measured in smooth flow. Power spectral density measurements of the fluctuating component of the drag on square plates in both smooth and turbulent flow are presented. The measurements in turbulent flow show the importance of the ratio of turbulence scale to plate size. There is shown to be a strong correlation between the fluctuating drag force and the velocity fluctuations in the approaching flow. The distortion of the turbulence structure approaching a plate is also discussed.

Vortex shedding from circular cylinders at low Reynolds numbers

Abstract: Experiments on slightly tapered models of circular cross-section have shown that the vortex wake structure exists in a number of discrete cells having different shedding frequencies. Within each cell shedding is regular and periodic, the frequency being somewhat lower than that from a parallel cylinder of the same diameter. A similar type of wake behaviour has also been observed on a parallel model in a non-uniform mean flow. These results suggest that the discontinuities in the shedding law observed by Tritton could arise through non-uniformities in the flow.

Analytical expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions

Abstract: Hessian matrices play a key role in optimization. Knowledge of their behavior is useful both in giving insight into optimization problems and in designing algorithms to solve them. In this paper, analytical expressions are obtained for the eigenvalues and eigenvectors at the intermediate minima of barrier and penalty functions. This in turn leads to an analytical expression for the inverse of the Hessian matrix (it is singular) at the solution.

A study of grain-boundary grooving at the platinum/alumina interface

Abstract: Grain-boundary grooving at the interface between solid platinum and solid alumina has been studied and its relevance to spheroidisation and particle coarsening noted. Analysis of the equilibrium shape of the grooves in conjunction with zero creep studies on platinum shows that the platinum/alumina interfacial energy is 1050 ergs/cm2 at 1400° C. The rate of growth of the grooves shows that volume diffusion is the operative mass transport mechanism at all temperatures considered (1240 to 1540° C); however, there are indications that diffusion through the metal is rate-determining at lower temperatures (< 1410° C) while diffusion through the ceramic phase predominates at higher temperatures.

The QR and QL Algorithms for Symmetric Matrices

Abstract: The QR algorithm as developed by Francis [2] and Kublanovskaya [4] is conceptually related to the LR algorithm of Rutishauser [7]. It is based on the observation that if $$A = QR{\text{ and }}B{\text{ = }}RQ{\text{,}}$$ (1) where Q is unitary and R is upper-triangular then $$B = RQ = {Q^H}AQ,$$ (2) that is, B is unitarily similar to A. By repeated application of the above result a sequence of matrices which are unitarily similar to a given matrix A 1 may be derived from the relations $${A_s} = {Q_s}{R_s},{\rm{ }}{A_{s + 1}} = {R_s}{Q_s} = Q_s^H{A_s}{Q_s}$$ (3) and, in general, A s tends to upper-triangular form.

The strength of carbon fibres

Abstract: The available information on the structure and properties of high strength carbon fibres is reviewed, and some new data are presented, showing the effects of boron doping and neutron irradiation on the properties of PAN-based carbon fibres. Theories relating the Young's modulus of the fibre to its microstructure are examined, and it is concluded that their relationship is qualitatively understood. Variations in electrical resistivity with different treatments may also be explained satisfactorily, again, in qualitative terms. The strength of carbon fibres is less well understood, however. It has been suggested that the fibre strength is governed by the presence or absence of stress-raising flaws, but while it is clear that such flaws can markedly reduce the strength, there is no clear estimate of the strength of an unflawed fibre. In this paper, we examine an alternative failure mechanism, initiated by shearing of the graphite crystallites in the fibre, and we conclude that such a mechanism may control the strength of the more graphitic fibres. Increases in the strength of carbon fibres may thus be achieved, not only by reducing the number and severity of the flaws, but also by applying the principles of solid-solution or dispersion hardening, and by reducing the graphite crystallite size (“grain refining”).

Investigation of Drag Reduction by Certain Algae

Abstract: INVESTIGATIONS were carried out between March and September 1969 to determine the drag reduction effects produced by certain species of algae in No. 3 towing tank at the Ship Division of the National Physical Laboratory. The tank is 48 feet wide, 25 feet deep and 1,300 feet long and contains the water added in August-September 1957. Small additions are regularly made to compensate for evaporation losses. The tank is constantly used for hydrodynamical experiments and is daily skimmed by towing a plastic tube along the tank to remove surface scum. The tube extends across the width of the tank and below it is suspended a muslin curtain. Samples from this scum were studied and species were tested, for drag reduction effects.

Grain boundary dislocations: Their effect on vacancies and sliding

Abstract: When a grain interior dislocation meets a grain boundary it can move along the latter by a mixture of climb and glide. Vacancies are then cmitted or absorbed and sliding is caused in amounts depending on the orientation of the dislocation's Burgers vector, which in turn depends on the grain orientation. The consequence is that emission and absorption of vacancies by grain boundaries, and also sliding, depend on the orientation with respect to the applied stress of the grains as well as of the grain boundaries.

The implicit QL algorithm

Abstract: In [1] an algorithm was described for carrying out the QL algorithm for a real symmetric matrix using shifts of origin. This algorithm is described by the relations $$\matrix{ {{Q_s}({A_s} - {k_s}I) = {L_s},} & {{A_{s + 1}} = {L_s}Q_s^T + {k_s}I,} & {{\rm{giving}}} & {{A_{s + 1}} = {Q_s}{A_s}Q_s^T,} \cr } $$ (1) where Q s is orthogonal, L s is lower triangular and k s is the shift of origin determined from the leading 2×2 matrix of A s .

Some Comments from a Numerical Analyst

Abstract: A description is given of life with A. M. Turing at the National Physical Laboratory in the early days of the development of electronic computers (1946--1948). The present mood of pessimism among numerical analysts resulting from difficult relationships with computer scientists and mathematicians is discussed. It is suggested that in light of past and present performance this pessimism is unjustified and is the main enemy of progress in numerical mathematics. Some achievements in the fields of matrix computations and error analysis are discussed and likely changes in the direction of research in numerical analysis are sketched.

Measurements of submillimetre wavelength radiation emitted by the stratosphere

Abstract: Observations have been made of the submillimetre wavelength radiation emitted by the lower stratosphere, using techniques of Fourier transform spectroscopy, while flying in a Comet 2E aircraft. Using the method of phase-modulation high quality spectra have been obtained at several altitudes over the spectral range 20 to 66 cm−1, and these have enabled identification of many emission lines due to O2 and O3, in addition to those of H2O. Also several unidentified emission lines have been observed. The meteorological applications of this spectroscopic technique are considered. In particular, a direct method of comparing the emission of ‘strong’ water-vapour and oxygen lines enables a calculation of mixing ratio of water vapour in the stratosphere to be made, and a value of (2.1 ± 0.3) × 10−6 g/g is derived above an altitude of 12.2 km. A similar comparison for ozone has given an integrated concentration of 0.35 ± 0.10 cm atm in the vertical path above the observer. The merits of these observations in measuring stratospheric mixing ratios are discussed with respect to alternative techniques, in particular with reference to balloon-borne studies.

Steady and unsteady wind loading of buildings and structures

Abstract: Wind loads are conveniently divided into steady (time-average) loads and unsteady (time-dependent) loads The latter arise from fluctuating forces due to turbulence as well as from self-excited aerodynamic instability The relevant characteristics of atmospheric winds—speed profile and turbulence—and their dependence on the local terrain are briefly discussed For design based on steady wind loadings the design wind speed is dependent on the acceptable degree of risk Force and pressure coefficients may be influenced by Reynolds number, surface roughness, wind characteristics and proximity to other structures Unsteady loadings due directly to turbulence are assessed through the concept of aerodynamic admittance Only the vortex-shedding and galloping mechanisms leading to aerodynamic instability are reviewed here, together with some design features for avoiding such excitations To predict the response of even conventional structures while in the design stages further information is needed on the speeds and turbulence characteristics of natural winds, especially of those over cities Further investigations concerning the aerodynamic admittance and the aerodynamic excitation of practical structures are also needed Research on these properties involves wind-tunnel studies in which the separate effects of intensity and scale of turbulence must be determined, and improved techniques for simulating the characteristics of natural winds in wind tunnels are required A major difficulty in the application of these aerodynamic data to the prediction of structural response is the uncertainty in the assessment of values of the structural damping