Cross-channel transfer of linear momentum in smooth rectangular channels

The object of this investigation was to study the lateral transfer of momentum from the central region to the vfertical-wall regions in smooth rectangular channels. The first step in the method was to determine the shear stress along the channel periphery. This shear stress was then incorporated in the momentum equation to solve for the lateral cross-channel transfer of linear momentum. The shear stress along the channel periphery was computed by two methods: one made use of the Preston-tube technique, and the other, the Karmn special study was made of the cross-channel transfer of linear momentum. The writer's analysis was based on a study of the shearstress distribution along the channel bottom and walls. LINEAR MOMENTUM TRANSFER All shear flows are characterized by a net transfer of longitudinal linear momentum from regions of higher linear momentum to regions of lower linear momentum. The simplest example is that of laminar flow in a pipe. In such flow, linear momentum is transported toward the wall region by a molecular transfer. Where flow in a pipe is turbulent, the linear momentum is transferred to the wall region by the movement of microscopic masses of fluid. Another example is that of overbank flow. In this type of flow, linear momentum is transported from the main channel section to the overbank section both by turbulence and by vertically oriented giant eddies (at least with shallow depths in the overbank section) which form at the junction plane of the main channel and overbank sections. The existence of these vertically oriented giant eddies was observed by Wilroy (1953) and Rodriguez-Diaz (1954) in their studies of the hydraulic jump in a nonrectangular open channel. The channel studied by Wilroy and Rodriguez-Diaz had vertical walls and a crosschannel slope of the bottom; the depth across the channel was therefore variable. For a hydraulic jump to occur in such a channel, a massive transfer of linear momentum from the deeper to the shallower part is required. Rodriguez-Diaz observed that, under certain conditions, turbulence plus steady cross currents were sufficient to effect the required transfer. On the other hand, if the required rate of transfer was too great, vertically oriented giant eddies formed which were capable of massive transport of linear momentum toward the shallower side. Another mechanism which can transport linear momentum consists of line vortices alined with axes along the channel. The superposition of these line vortices on the flow is called secondary circulation. Prandtl (1952) presented a hypothesis about the location of these TRANSFER OF MOMENTUM, RECTANGULAR CHANNELS B3 vortex pairs in various noncircular conduits. This secondary circulation is an effective supplement to turbulence in transferring linear momentum toward the wall regions in flow in noncircular conduits. In a rectangular open channel, linear momentum must be transferred both downward toward the floor and laterally toward the walls. The lateral transport is necessitated by the existence of boundary shear on the walls. Where flow is two dimensional the net transport is in only one direction toward the floor. Conceptually the flow is divided into two three-dimensional flow regions near the walls and into a two-dimensional flow region in the central part of the channel. The mean and the fluctuating velocity patterns in a two-dimensional region have been studied by Laufer (1951), who conducted experiments using a rectangular smooth conduit. The existence of a threedimensional flow region near the wall is reflected in the following flow characteristics: Mean velocity, turbulence, apparent shear stresses, and boundary shear stress. Boundary shear stress is a flow characteristic of major interest to engineers. For example, most engineering analyses are based on a gross-streamtube analysis, for which the total boundary shear force is generally required. As a further example, movement of particles on the bed and on the banks of a stream is directly related to the boundary shear stress. Determination of the boundary shear-stress distribution and the cross-channel transfer of linear momentum in rectangular open channels was the object of this investigation. [Summation of the"! [~ Rate of transport ~j [~ Rai external forces 1= of linear momentum I I of lit on a free body J |_out of the free bodyj [_ mt( LINEAR MOMENTUM EQUATION The linear momentum equation for steady flow is simply: te of transport near momentum (1) in o the free body _. For purpose of analysis a rectangular parallelepiped (fig. 1) was chosen as a fluid free body. Face 1 is at the junction of the fluid and the bed. Face 2 is the free surface. Face 3 is a vertical plane along the centerline. Face 4 is a vertical plane parallel to the centerline at a distance of b z from the centerline. Faces 5 and 6 are the end faces of the free body and are separated by the axial distance dx. The linear momentum equation in the x-direction, along the axis of the channel, is formulated for steady uniform flow. The external forces in the a?-direction on the free body are: (a) The pressure forces on faces 5 and 6; (b) the boundary shear force on face 1, dx I TO&( dz) and (c) the weight force, y(b z) (yQ sin 0)dx. Because J& the flow is uniform the pressure force on face 5 exactly equals that on face 6. 772-646 O 65 2 B4 LABORATORY STUDIES OF OPEN-CHANNEL FLOW