Author(s): Ulf Teschke; Fabian Ruhr

Linked Author(s): Ulf Teschke

Keywords: Bernoulli equation; Correction coefficient; Energy flux; Potential flow

Abstract: The Bernoulli equation is one of the most important equations in fluid mechanics. It states for incompressible fluids that the sum of geodetic, pressure and velocity head is constant along a streamline. In engineering practice, however, statements are required along a spatial coordinate for the pressure and the speed depending on the geodetic head for finite flow cross-sections. The Bernoulli equation is often used as an approximation for this, although its application is exact only along a one specific streamline. In this context, one speaks of the stream filament theory, whereby a mean streamline is regarded as representative of the entire cross-section. The speed used often corresponds to the mean speed as the quotient of the volume flow and the cross-section through which the flow passes. In order to minimize the error that occurs, the speed level is multiplied by a correction coefficient. It can be shown that the Bernoulli equation expanded in this way and the energy flow equation give the same result. The respective coefficients have to be related to the three-dimensional velocity vector instead of just to the normal component of the velocity. In addition, when considering the energy flow, the mean pressure must be replaced by the mean energy flow pressure. Both mean pressures are generally different. Same applies to the geodetic head which is replaced by the averaged energy geodetic head when considering the energy flow . If the changes specified above are taken into account, a Bernoulli equation extended to finite cross-sections and an associated energy flow equation result. Both equations are exact and deliver the same results based on the description of a flow process in a finite flow cross section with non-uniform pressure and velocity distribution! Only the summands within the equation will have different values. The considerations presented have already been checked by Teschke for the potential flow of a symmetrical flow expansion. The considerations presented are transferred to an arbitrary flow section within an asymmetrical flow tube of the potential vortex flow. In contrast to the symmetrical flow expansion, the correction coefficients introduced are now location-dependent. It is shown that even under these circumstances, the extended Bernoulli equation and the averaged energy flow equation give the same result. Furthermore, the behavior of the summands in the expanded equation can now be examined more individually.

DOI: https://doi.org/10.3850/IAHR-39WC252171192022595

Year: 2022