5. Secondary Current and Spiral Flow
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1 5. Secondary Current and Spiral Flow The curve of constant velocity for rectangular and triangular cross-section obtained by Nikuradse are shown in Figures and 2. In all cases the velocities at the corners are comparatively very large with stems from the fact that in all straight pipes of non-circular cross-section there exist secondary flows. These are such that the fluid flows towards the corner along the bisectrix of the angle and then outwards in both directions. The secondary flows continuously transport momentum from the centre to the corners and generate high velocities there. Schematic diagrams of secondary flows in triangular and rectangular pipes are shown in Fig. 3. It is seen that the secondary flow in the rectangular cross-section which proceeds from the wall inwards in the neighborhood of the ends of the larger sides and of the middle of the shorter sides creates zones of low velocity. They appear very clearly in the picture of curves of constant velocity in Fig. Such secondary flows come into play also in open channels, as evidenced by the pattern of curves of constant velocity in Fig. 4. The maximum velocity does not occur near the free surface but at about one fifth of the depth down of the free surface.
2 Fig.. Curves of constant velocity for pipe of rectangular cross-section, after Nikuradse Fig. 2. Curves of constant velocity for a pipe of equilateral triangular cross-section after Nikuradse a b Fig. 3. Schematic representation of Secondary flows in pipes of triangular and rectangular (open channel) cross-section
3 water level Fig. 4. Curves of constant velocity for a rectangular open channel after Nikuradse Secondary circulation is that flow wherein the velocity can be resolved into two components, one in the longitudinal direction of the channel and the other in transverse to the direction of the channel. The transverse component of the velocity gives rise to the secondary circulation. It can occur in both straight and curved channels and for different reasons. Secondary circulation is affected by temperature gradients, sediment, turbulence, non-uniformity of boundary shear, and the curvature of streamlines. Secondary circulation has been associated with turbulent flow in prismatic channels wherein the shear at the boundary is not constant. In straight circular pipes as shear at the boundary is constant for both laminar and turbulent flow the secondary circulation
4 has not been observed. When secondary circulation does occur, it seems to take place in an even number of cells as depicted in Figure 5. The non-uniformity of sediment across a channel has been associated with secondary circulation. Fig. 5. Secondary circulation in straight channel Secondary current is the flow taking place in transverse direction of the main flow. The secondary currents are of four types viz.. The 'weak' secondary currents in straight-non-circular channel sections and in pipes due to boundary resistance (figure 5). 2. Secondary flow developed due to non-uniform bed configuration as in case of alluvial channels. 3. The ' strong ' currents caused in bends due to centrifugal force.
5 SPIRAL FLOW O y INSIDE OUTSIDE WALL SECTION ON A-A ILLUSTRATION OF SECONDARY FLOW AND SPIRAL CURRENTS IN A 90 BEND 4. Secondary currents due to the unsteadiness of the oscillating boundary layer. The occurrence of the maximum velocity filament in a straight channel just below the free surface (see figure below) to the findings of secondary current.
6 Secondary currents Isovels (a) Open channel (b) Equivalent closed conduit Comparison of Open Channel Flow with Closed-Conduit Flow The lens shaped figure is drawn such that it is orthogonal to each isovel. It may be noted that the maximum velocity occurs slightly below the free surface. On the lens shaped line no velocity gradient exist. The shear on the free surface is negligible and their is no shear resistance to balance the component of the weight of the prism along the main flow direction. The equivalent closed conduit is symmetrical about the central line and the shear stress is distributed along the boundary line. Side Slope, m: =.5 : 4y y 0.750γySo 0.970γySo 0.750γyS o Tractive force distribution obtained using membrane analogy This distribution varies depending on the cross section and material Gibson, explained the origination of the secondary current. Darcy, Cunningham, Sterns, Moseley, Francis and Wood (Thandaveswara, 969) recognized the presence of this secondary current and superposition of the main flow leads to spiral flow. If there is any slight disturbance in approach flow conditions instead of double spiral, then single spiral exists. Kennedy and Fulton established that the secondary current has a definite effect on the frictional resistance of the channel. The second type of secondary currents were observed by Schlichting, Jacob, Schultz Grunov. The projection of spheres from the surface is just similar to the spherical sand particles fixed uniformly over the surface, then this type of secondary current can be expected when the sand roughness is used. The flow pattern which exists behind an obstacle placed in the boundary layer near a wall differs markedly from that behind an obstacle placed in the free stream. This
7 circumstance emerges clearly from an experiment performed by Schlichting and shown in figure. The experiment consisted in the measurement of the velocity field behind a row of spheres placed on a smooth flat surface. The pattern of curves of constant velocity clearly shows a kind of negative wake effect. The smallest velocities have been measured in the free gaps in which no spheres are present over the whole length of the plate; on the other hand, the largest velocities have been measured behind the rows of spheres where precisely the smaller velocities. 0d 0d 2 3 0d measuring station V [m/s) d d 5d Isovels behind a row of spheres as measured by Schlichting. Secondary flow in the boundary layer is marked behind (), as calculated by K. Schultz-Grunow. In the neighbourhood of the wall, the velocity behind the spheres is larger than that in the gaps. The spheres produce a "negative wake effect" which is explained by the existence of secondary flow. Diameter of spheres d= 4mm When the spacing of roughness is close, the wavy water surface will not exist as the formation of vortices will be confined to roughness elements and forms a pseudo-wall and does not affect the main flow.
8 y k s s Isolated - roughness flow (k/s) - Form drag dominates The wake and the vortex are dissipated before the next element is reached. The ratio of (k/s) is a significant parameter for this type of flow
9 y k s Wake interference flow (y/s) s s When the roughness elements are placed closer, the wake and the vortex at each element will interfere with those developed by the following element and results in complex vorticity and turbulent mixing. The height of the roughness is not important, but the spacing becomes an important parameter. The depth 'y' controls the vertical extent of the surface region of high level turbulence. (y/s) is an important correlating parameter. y k j j j j s s s k is surface roughness height s is the spacing of the elements j is the groove width y is the depth of flow Quasi smooth flow - k/s or j/s becomes significant acts as Pseudo wall Quasi smooth flow is also known as skimming flow. The roughness elements are so closed placed. The fluid that fills in the groove acts as a pseudo wall and hence flow essentially skims the surface of roughness elements. In such a flow (k/s) or (j/s) play a significant role. Concept of three basic types of rough surface flow In the following paragraphs 3rd type of secondary current has been discussed briefly. The third type of secondary currents will come into picture while the fluid flows in a curved channel. The fluid in a curved channel will be subjected to centrifugal force. Due to this centrifugal force, a pressure gradient normal to the direction of the main flow is created. Then the particles near the inside wall are thrown outside and they reach the outside boundary moving in transverse direction. Thus a sort of centripetal lift will be created causing the heaving up of the fluid. If the flow is irrotational and the fluid enters with uniform velocity into bend, then it is analogous to the potential vortex.
10 v Vr =CONSTANT r O r i r c B r 0 VELOCITY DISTRIBUTION IN POTENTIAL FLOW IN A CURVED CHANNEL But in actual case due to the presence of shear stress at the boundary, the velocity of main flow decreases abruptly at the boundary setting a velocity gradient in the boundary layer. It may be observed that the energy in the boundary regions is less than in the potential zone. It follows that at the outside of the bend the pressure intensity falls away abruptly towards the wall, unless a secondary flow takes place in the direction of outer wall. Continuity equation requires an inward flow along the side walls to compensate since the pressure gradient normal to the wall is exactly opposite to that of potential motion. The spiral flow motion induced by the centrifugal force is very pronounced and irregular in the bend. The complicated pattern of flow is caused by the superposition of secondary current in the bend over the spiral flow of the approach channel. The spiral flow of bend begins as a lateral boundary current near the point where the stream line curvature begins and at the bottom inside corner of the bend. This type of spiral motion also called helicoidal flow and was recognized by Thomson in 876 and was demonstrated by him in the laboratory in an 80 circular bend with rectangular channel section in 879. This was supported further by Engles, Beyerhams and others. During 883 to 990 several researchers while investigating the flow
11 characteristics in meanderings observed the action of scouring and deposition in the river bends. Several investigators (refer to Thandaveswara's Thesis, 969) mostly conducted the experiments in channel whose aspect ratios were of the same order of magnitude. Thus the mean flow occurring was essentially three dimensional in character. But Betz, Wilcken, Maccol and Wattendrof conducted experiments in two dimensional channel (rectangular conduit). Watterdrof showed the potential character of the spiral flow and drew the following conclusions. (i). There is only slight increase in channel resistance due to the presence of bends as indicated in pipe bends. (ii). The velocity distribution follows free vortex law. (iii). Rayleigh's stability criterion based on the calculation of mixing length and exchange factor showed the instability and increased mixing at the outer walls of the curved channels and decreasing mixing and stability at the inner wall. (iv). If the depth to breadth ratio is large enough so that the lateral currents occupy only a relatively small part of the area of the cross-section near the bottom and if form losses are ignored near the bend, then the bend loss scarcely exists. 5. Strength of spiral The term "Strength of Spiral" is defined as the percentage ratio of the mean kinetic energy of the lateral motion to the kinetic energy of flow and is denoted by S xy. 2 V xy 2 2g ( V xy ) S m m xy = * 00 = * V V 2g The strength of secondary current can be qualitatively estimated to be proportional to the extent of distortion of isovels. The concentration of velocity near boundary means the secondary flow concentration near boundary. This bears the hypothesis that the mechanism of secondary motion arises out of the boundary shear turbulence. It may be noted that the approach flow plays an important role and has a direct effect on the number of spirals, strength of spiral and other characteristics of spiral flow.
12 Following equations relate the deflection angle α along the centre line of bed, geometry of the channel and the hydraulic properties of flow, in channel bends. (i) (ii) (iii) For a smooth rectangular bend P r tan α =7.4 for 2000 R c 0.25 e Re For a smooth triangular channel P r c tanα =3.4 for 2000 R 0.25 e 5000 Re In general, P r c Re tanα =K If the channel is wide then y r 05. c 4 R 0.25 e tanα =K But Russian authors found that for a rectangular wide channel In general for a wide rectangular channel, y tanα = r c tanα =K ( ) b Φ R 0 e r c tanα =K 0 b y Φ rc Ks for smooth flow for rough flow tanα =K f 0 a b rc
13 where f = friction coefficient and "a" is an exponent >. The last equation can be 8g expressed in Chezy terms of coefficient C= f in the form a 8g b tanα K 2 0 C r c = The value of tanα can be assumed to indicate the strength spiral to some scale. Reference: Thandaveswara B.S., "Characteristics of flow around a 90 open channel bend", M.Sc (Engineering), Department of Civil and Hydraulic Engineering, Indian Institute of Science, Bangalore, 969.
Prof. B.S. Thandaveswara. Superelevation is defined as the difference in elevation of water surface between inside (1)
36.4 Superelevation Superelevation is defined as the difference in elevation of water surface between inside and outside wall of the bend at the same section. y=y y (1) 1 This is similar to the road banking
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