The Expansion of the Bernoulli Equation to Finite Flow Cross Sections at One Level

The Bernoulli Equation is one of the most important equations in fluid mechanics. It states that for incompressible fluids at a static height, the sum of geodetic height, pressure head and velocity head remains constant along one stream line.

In engineering practice, however, statements along spatial coordinates are required for the pressure and velocity, depending on the geodetic height in finite flow cross sections. As an approximation of this, the Bernoulli Equation is also often used despite it only applying exactly along one streamline. In this context, it is possible to draw on the stream filament theory, in which a median streamline is regarded as representative of the entire cross section. The velocity used often corresponds to the median velocity, taken as a ratio of flow rate and cross section through which the flow passed. In order to minimise the emergence of errors, the velocity head is multiplied by a correction coefficient. In the literature, the energy flow coefficient and the momentum current coefficient appear as possible approaches. Which of these two coefficients is correct for this is a matter of controversy. The use of each approach leads to different results in practice.

In this paper, we present an approach that confirms the applicability of both approaches. Both coefficients must be based on the three-dimensional velocity vector rather than merely on the normal components of velocity. In addition, when observing energy flow, the median pressure has to be replaced by the median energy flow pressure. These two median pressures are generally different.

If the changes listed above are taken into account, this results in a Bernoulli Equation extended to finite cross sections and an energy flow equation associated with this. Both equations are precise and deliver the same results in relation to the description of flow processes in a finite flow cross section with asymmetric pressure and velocity distribution.

The considerations presented here are verified using potential flows. When observing linear momentum currents, the balance of forces is thus checked. The pressure and flow forces on the entrance and exit surfaces must be consistent with the pressure forces on the cover of the flow area under observation. In asymmetric flows, it is clear that the conservation of angular momentum is also achieved with this approach.

The balance equations specified must always be completed. The evidence required for this is indicated here.