On the inverse problem of blade design for centrifugal pumps and fans

The inverse problem of blade design for centrifugal pumps and fans has been studied. The solution to this problem provides the geometry of rotor blades that realize specified performance characteristics, together with the corresponding flow field. Here a three-dimensional solution method is described in which the so-called meridional geometry is fixed and the distribution of the azimuthal angle at the three-dimensional blade surface is determined for blades of infinitesimal thickness. The developed formulation is based on potential-flow theory. Besides the blade impermeability condition at the pressure and suction side of the blades, an additional boundary condition at the blade surface is required in order to fix the unknown blade geometry. For this purpose the mean-swirl distribution is employed. The iterative numerical method is based on a three-dimensional finite element method approach in which the flow equations are solved on the domain determined by the latest estimate of the blade geometry, with the mean-swirl distribution boundary condition at the blade surface being enforced. The blade impermeability boundary condition is then used to find an improved estimate of the blade geometry. The robustness of the method is increased by specific techniques, such as spanwise-coupled solution of the discretized impermeability condition and the use of underrelaxation in adjusting the estimates of the blade geometry. Various examples are shown that demonstrate the effectiveness and robustness of the method in finding a solution for the blade geometry of different types of centrifugal pumps and fans. The influence of the employed mean-swirl distribution on the performance characteristics is also investigated

Fabric response to stress probing in granular materials:Two-dimensional, anisotropic systems

The microstructure of granular materials has a significant influence on their macroscopic quasi-static strength and deformational behaviour. This microstructure is often quantified by a second-order fabric tensor that describes the primary orientational statistics of interparticle contacts. Here, it is investigated how the fabric tensor changes when samples are subjected to small (strain) loadings with different ‘directions’, i.e. probes. This is accomplished by the analysis of extensive sets of Discrete Element Method (DEM) simulations for various anisotropic, pre-peak two-dimensional samples, where both in-plane (i.e. coaxial with the current stress and fabric tensor) and out-of-plane, noncoaxial probes are considered. The results of DEM simulations show that the in-plane and out-of-plane fabric responses are effectively decoupled, i.e. they are only dependent on the in-plane and out-of-plane strain increment, respectively. The out-of-plane fabric increment is proportional to the out-of-plane strain increment whereas the in-plane fabric increment is linearly dependent on the in-plane strain increment. An accurate theoretical description (with a modest number of model parameters) has been developed that describes the fabric response to the imposed, in-plane as well out-of-plane, strain increments for the considered systems

Multiscale Analysis of the Stress State in a Granular Slope in Transition to Failure

By means of contact dynamics simulations, we analyze the stress state in a granular bed slowly tilted towards its angle of repose. An increasingly large number of grains are overloaded in the sense that they are found to carry a stress ratio above the Coulomb yield threshold of the whole packing. Using this property, we introduce a coarse-graining length scale at which all stress ratios are below the packing yield threshold. We show that this length increases with the slope angle and jumps to a length comparable to the depth of the granular bed at an angle below the angle of repose. This transition coincides with the onset of dilatation in the packing. We map this transition into a percolation transition of the overloaded grains, and we argue that in the presence of long-range correlations above the transition angle, the granular slope is metastable.Comment: 11 pages, 14 Fig, submitted to PR

Experimental and theoretical study of Rapid flow of cohesionless granular materials down inclined chutes

A theoretical and experimental study is performed of rapid, fully developed flows of cohesionless granular materials down inclined chutes with a rough base. Two flow types are studied in detail: (1) immature sliding flow, where a stagnant zone forms on the base of the chute, and (2) fully developed sliding flow, where no such zone is formed. A simple phenomenological theory is developed that predicts the flow type and the associated velocity profile. The theory models dynamic stresses induced by interparticle collisions as well as quasi-static stresses induced by friction acting on semi-permanent interparticle contacts. Hence it is developed for the so-called frictional-collisional regime. Employing a photographical method, the flow type and the velocity profile are determined experimentally for various chute angles in a test set-up in which the granular material is continuously circulated. Quantitative agreement between the theoretical and the measured velocity profile is reasonably good, although it appears that at the largest chute angle the side wall friction causes deviations between the theoretical and the measured velocity profile