On the effect of the meridional contour shape on the power characteristics of a centrifugal compressor wheel

This work is concerned with the development of approaches to the optimal aerodynamic design of centrifugal compressor wheels, which is due to the use of centrifugal stages in compressors of modern aircraft gas turbine engines and power plants. The aim of this work is a computational study of the effect of the meridional contour shape of a centrifugal compressor wheel on its power characteristics. The basic method is a numerical simulation of 3D turbulent gas flows in centrifugal wheels on the basis of the complete averaged Navier¬–Stokes equations and a two-parameter turbulence model. The computational study features: varying the shape of the hub and tip part of the meridional contour over a wide range, formulating quality criteria as the mean integral values of the wheel power characteristics over the operating range of the air flow rate through the wheel, and a systematic scan of the independent variable range at points that form a uniformly distributed sequence. As a result of multiparameter calculations, it was shown that in the case of a flow without separation in the blade channels of a wheel with a given starting shape of the meridional contour, varying that shape has an insignificant effect on the wheel power characteristics. It is pointed out that in similar cases it seems to be advisable to aerodynamically improve centrifugal wheels by varying the shape of their blades in the circumferential direction rather than in the meridional plane. This conclusion was made using rather a “coarse” computational grid, which, however, retains the sensitivity of the computed results to a variation in the centrifugal wheel geometry. On the whole, this work clarifies ways of further aerodynamic improvement of centrifugal compressor impellers in cases where the starting centrifugal wheel is a well-designed wheel with a flow without separation in the blade channels. The results obtained may be used in the aerodynamic optimization of centrifugal stages of aircraft gas turbine engines.

A Note on the Continued Fraction of Minkowski

Denote by Θ1,Θ2, · · · the sequence of approximation coefficients of Minkowski’s diagonal continued fraction expansion of a real irrational number x. For almost all x this is a uniformly distributed sequence in the interval [0, 1/2 ]. The average distance between two consecutive terms of this sequence and their correlation coefficient are explicitly calculated and it is shown why these two values are close to 1/6 and 0, respectively, the corresponding values for a random sequence in [0, 1/2].

Bi-invariant sets and measures have integer Hausdorff dimension

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators

The work is concerned with the first-order linear autoregressive process which has a rectangular stationary marginal distribution. A derivation is given of the result that the time-reversed version is deterministic, with a first-order recursion function of the type used in multiplicative congruential random number generators, scaled to the unit interval. The uniformly distributed sequence generated is chaotic, giving an instance of a chaotic process which when reversed has a linear causal and non-chaotic structure. An mk-valued discrete process is then introduced which resembles a first-order linear autoregressive model and uses k-adic arithmetic. It is a particular form of moving-average process, and when reversed approximates in m a non-linear discrete-valued process which has the congruential generator function as its deterministic part, plus a discrete-valued noise component. The process is illustrated by scatter plots of adjacent values, time series plots and directed scatter plots (phase diagrams). The behaviour very much depends on the adic number, with k = 2 being very distinctly non-linear and k = 10 being virtually indistinguishable from independence.

Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators

The work is concerned with the first-order linear autoregressive process which has a rectangular stationary marginal distribution. A derivation is given of the result that the time-reversed version is deterministic, with a first-order recursion function of the type used in multiplicative congruential random number generators, scaled to the unit interval. The uniformly distributed sequence generated is chaotic, giving an instance of a chaotic process which when reversed has a linear causal and non-chaotic structure. An mk -valued discrete process is then introduced which resembles a first-order linear autoregressive model and uses k-adic arithmetic. It is a particular form of moving-average process, and when reversed approximates in m a non-linear discrete-valued process which has the congruential generator function as its deterministic part, plus a discrete-valued noise component. The process is illustrated by scatter plots of adjacent values, time series plots and directed scatter plots (phase diagrams). The behaviour very much depends on the adic number, with k = 2 being very distinctly non-linear and k = 10 being virtually indistinguishable from independence.