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Ebook Fluid mechanics (2nd edition): Part 2

(BQ) Part 2 book "Fluid mechanics" has contents: Stream filament theory, potential flows, supersonic flow, boundary layer theory, creeping flows, solutions of the boundary layer equations, oblique shock wave, plane potential flow, steady compressible flow,. and other contents. | 9 Stream Filament Theory 9.1 Incompressible Flow We shall now follow on with our earlier statement that in many technically interesting problems the entire flow region can by represented as a single streamtube, and the behavior of the flow is then characterized by its behavior at a median streamline. Within the framework of this assumption, the flow quantities are only functions of the arc length s along the streamline, and possibly of the time t. Thus the flow quantities are assumed constant over the cross-section of the streamtube. Now this assumption does not have to be satisfied for the entire streamtube (at least not in steady flow), but only in those sections of the streamtube where we wish to calculate the flow in this quasi-one-dimensional approximation. Therefore the flow must be at least piecewise uniform, i.e. essentially constant over the cross-section, and also may not change too strongly in the flow direction: this assumes that the cross-section is a slowly varying function of the arc length s. In between these uniform regions the flow can exhibit a three-dimensional character, but cannot be computed there using stream filament methods. The assumption of constant flow variables over the cross-section requires that the friction effect is negligible, because we know from Chap. 6 that the flow quantities vary considerably over the cross-section of streamtubes bounded by walls if the flow is dominated by frictional effects, as is the case in fully developed pipe flow. Even in these flows, the concept of stream filament theory can be applied if the distribution of the flow quantities over the cross-section is known, or else it must be possible to make reasonable assumptions about these distributions. In particular attention must be paid in the calculation of quantities averaged over the cross-section: the averaged velocity calculated from the continuity equation, which we used as the typical velocity in the resistance laws cannot be used in the balances of energy 2 and .