Euler equations, derived in 1755, are among the most important equations for general fluid analysis. Although this fundamental governing set of equations looks simple at first glance for incompressible, inviscid fluid flow, it does lead to a crucial numerical discretization dilemma related to existence, uniqueness, stability and oscillation problems. Take gradual pipe expansion flow, for instance: in the process of deriving Euler equations, the seemingly innocent artificial equalization of crosswise and forward-biased streamwise pressure will give serious discretization error compared with use of the Bernoulli equation. Here, the wall pressures, P W = P + d P, disappear from the Euler equations but later bring about troublesome problems, and even violate the fundamental energy principle. Various numerical measures have been devised to overcome such problems, such as artificial viscosity and streamline upwinding, but these are awkward and have high risk of flow field contamination in an attempt to somehow manipulate some energy dissipation when dealing with incompressible fluid flow. In this research, for the sake of finding the real culprit, we re-scrutinize the original derivation of Euler equations and compare the discretized Euler equation with a factorized Bernoulli equation, in order to find the actual pathology that leads to these serious numerical discretization dilemmas with the Euler equations.
Fish have appeared since Precambrian more than 500 million years ago. Yet, there are still much untamed areas for fish propulsion research. The swordfish has evolved a light thin/high crescent tail fin for pushing a large amount of water backward with a small velocity difference. Together with a streamlined forward-enlarged thin/high body and forwardbiased dorsal fin enclosing sizable muscles as the power source, the swordfish can thus achieve unimaginably high propulsion efficiency and an awesome maximum speed of 130 km/h as the speed champion at sea. This paper presents the innovative concepts of ''kidnapped airfoils'' and ''circulating horsepower'' using a vivid neat-digit model to illustrate the swordfish's superior swimming strategy. The body and tail work like two nimble deformable airfoils tightly linked to use their lift forces in a mutually beneficial manner. Moreover, they use sensitive rostrum/lateral-line sensors to detect upcoming/ambient water pressure and attain the best attack angle to capture the body lift power aided by the forwardbiased dorsal fin to compensate for most of the water resistance power. This strategy can thus enhance the propulsion efficiency greatly to easily exceed an astonishing 500%. Meanwhile, this amazing synergy of force/beauty also solves the perplexity of dolphin's Gray paradox lasting for more than 70 years and gives revelations for panoramic fascinating future studies.
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