The Navier–Stokes equations are nonlinear partial differential equations describing the motion of fluids. Due to their complicated mathematical form they are not part of secondary school education. A detailed discussion of fundamental physics—the conservation of mass and Newton’s second law—may, however, increase the understanding of the behaviour of fluids. Based on these principles the Navier–Stokes equations can be derived. This article attempts to make these equations available to a wider readership, especially teachers and undergraduate students. Therefore, in this article a derivation restricted to simple differential calculus is presented. Finally, we try to give answers to the questions ‘what is a fluid?’ and ‘what do the Navier–Stokes equations mean?’.
This article presents a detailed investigation of the three-dimensional flow field produced by a small centrifugal fan used for cooling of electronic components, for instance, in the automotive industry. A particle image velocimetry (PIV) system captured numerous cross sections of the fan jet for analysis of velocity distribution and jet development. General parameters, such as the spreading rate, and flow specifics, such as the full width at half maximum contours in planes parallel to the fan outlet and the jet's rotation rate, were thus evaluated. In addition, constant temperature anemometry enabled a thorough investigation of the flow field at the fan's outflow port. Hot-wire measurements complement the PIV results by providing the power spectral density of the turbulent kinetic energy at several locations. Our results demonstrate that the cross-flow profile at the fan outlet consists of two counter-rotating vortices. This leads to a jet with a nearly elliptical, rotating cross section, which does not propagate in a direction perpendicular to the outlet plane. Several aspects of these distinctive flow features are evaluated and presented to provide both fluid mechanics and heat transfer engineers with (i) data on the air jet produced by a small centrifugal fan and (ii) reference data for computational fluid dynamics simulations. Despite complex flow development in the near field of the jet and the nonaxisymmetric jet contour in the intermediate field, our results are in good agreement with published data on axisymmetric jets in terms of spreading rate and the development of turbulence spectra observed in the intermediate field.
Accurate flow measurement is a ubiquitous task in fields such as industry, medical technology, or chemistry; it remains however challenging due to small measurement ranges or erosive flows. Inspiration for possible measurement methods can come from nature, for example from the lateral line organ of fish, which is comprised of hair cells embedded in a gelatinous cupula. When the cupula is deflected by water movement, the hair cells generate neural signals from which the fish gains an accurate representation of its environment. We built a flow sensor mimicking a hair cell, but coupled it with an optical detection method. Light is coupled into a PDMS waveguide that consists of a core and a cladding with a low refractive index contrast to ensure high bending sensitivity. Fluid flow bends the waveguide; this leads to a measurable light loss. The design of our sensory system allows flow measurement in opaque and corrosive fluids while keeping production costs low. To prove the measurement concept, we evaluated the light loss while (a) reproducibly bending the fiber with masses, and (b) exposing the fiber to air flow. The results demonstrate the applicability of an optical fiber as a flow sensor.
The free jets of an axial and a centrifugal fan have been scanned by a specialized particle image velocimetry (PIV) set-up, which allows for volumetric scans of the time-averaged velocity field. Both of these fans have similar dimensions of approximately 70 mm x 70 mm x 25 mm. A classic PIV set-up was combined with a precise linear stage to move the fans through the laser fan beam in small steps, creating a dense array of measurement planes. Two components of the time-averaged velocity field are captured by the first 2.5D scan. Another scan, with the fan rotated by 90° about its outlet surface normal, captures the missing third velocity component. This article describes the details of the measurement set-up, and mentions measures concerning seeding, reflections, and calibration. In the signal processing stage, two independent sets of gathered image data have to be processed, producing two sets of velocity image frames. These are subsequently combined using gridded interpolation in order to obtain a 3D velocity field. Specifically devised software tools allow for a CFD-like analysis and visualization of the flow field. Typical parameters of the generated jets, like the spreading and rotation rates, are calculated from the measurement data and details of the outlet flow fields are investigated. The interpolated data are also used to analyze the influence of an assumed coarser measurement grid resolution on the results for the obtained outlet flow fields.
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