Nce403 mod unit3
- 1. Equation of gradually varied flow and its limitations, Flow classification and surface profiles, Integration of varied flow equation by analytical, graphical and numerical methods, Flow in channels of non-linear alignment specifically for the case of a bend. 1/7/2017 1MODASSAR ANSARI
- 2. BY MODASSAR ANSARI 2nd Year Department of civil Engineering SUBJECT- HYDRAULICS & HYDRAULIC MACHINES SUBJECT CODE-NCE 403 1/7/2017 2MODASSAR ANSARI
- 3. The flow in an open-channel is termed as gradually varied flow (GVF) when the depth of flow varies gradually with longitudinal distance. Such flows are encountered both on upstream and downstream sides of control sections. Analysis and computation of gradually varied flow profiles in open- channels are important from the point of view of safe and optimal design and operation of any hydraulic structure. 1/7/2017 3MODASSAR ANSARI
- 4. Find Change in Depth wrt x 1/7/2017 2 2 1 2 1 2 2 2 o f V V y S x y S x g g energy equation for non- uniform, steady flow 12 yydy 2 2 f o V dy d S dx S dx g P A T dy y dy dx S dy dx S g V dy d dy dy of 2 2 2 2 2 1 2 1 2 2 o f V V S dx y y S dx g g Shrink control volume 4MODASSAR ANSARI
- 5. Governing equation So and Sf are positive when sloping down in direction of flow y is measured from channel bottom dy/dx =0 means water depth is _______ 1/7/2017 2 1 Fr SS dx dy fo yn is when o fS S= constant 5MODASSAR ANSARI
- 6. Mild slope (yn>yc) ◦ in a long channel subcritical flow will occur Steep slope (yn<yc) ◦ in a long channel supercritical flow will occur Critical slope (yn=yc) ◦ in a long channel unstable flow will occur Horizontal slope (So=0) ◦ yn undefined Adverse slope (So<0) ◦ yn undefined 1/7/2017 6MODASSAR ANSARI
- 7. Normal depth Steep slope (S2) Hydraulic Jump Sluice gate Steep slope Obstruction 1/7/2017 2 1 Fr SS dx dy fo S0 - Sf 1 - Fr2 dy/dx + + + - + - - - + 0 1 2 3 4 0 1 2 3 4 E y yn yc 7MODASSAR ANSARI
- 8. 1/7/2017 S0 - Sf 1 - Fr2 dy/dx 1 + + + 2 + - - 3 - - + 2 1 Fr SS dx dy fo 8MODASSAR ANSARI
- 9. 1/7/2017 xS g V yxS g V y fo 22 2 2 2 2 1 1 of SS g V g V yy x 22 2 2 2 1 21 energy equation solve for x 1 1 y q V 2 2 y q V 2 2 A Q V 1 1 A Q V rectangular channel prismatic channel 9MODASSAR ANSARI
- 10. 1/7/2017 2 2 4/3f h n V S R = 2 2 4/3 2.22 f h n V S R = 2 f 8 f h V S gR = Manning Darcy-Weisbach Si units english units 10MODASSAR ANSARI
- 11. Limitation: channel must be PRISMATIC (channel geometry is independent of x so that velocity is a function of depth only and not a function of x) Method identify type of profile (determines whether Dy is + or -) choose Dy and thus yi+1 calculate hydraulic radius and velocity at yi and yi+1 calculate friction slope given yi and yi+1 calculate average friction slope calculate Dx 1/7/2017 11MODASSAR ANSARI
- 12. Given a depth at one location, determine the depth at a second given location Step size (x) must be small enough so that changes in water depth aren’t very large. Otherwise estimates of the friction slope and the velocity head are inaccurate Can solve in upstream or downstream direction Usually solved upstream for subcritical Usually solved downstream for supercritical Find a depth that satisfies the energy equation 1/7/2017 xS g V yxS g V y fo 22 2 2 2 2 1 1 12MODASSAR ANSARI
- 13. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 05101520 elevation(m) distanceupstream(m) bottom surface yc yn 1/7/2017 S1 S3 is there a curve between yc and yn that increases in depth in the downstream direction? 13MODASSAR ANSARI
- 14. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0510152025303540 elevation(m) distanceupstream(m) bottom surface yc yn If the slope is mild, the depth is less than the critical depth, and a hydraulic jump occurs, what happens next? 1/7/2017 Rapidly varied flow! When dy/dx is large then V isn’t normal to cs Hydraulic jump! Check conjugate depths 14MODASSAR ANSARI
- 15. All the complications of pipe flow plus additional parameter... _________________ Various descriptions of energy loss ◦ Chezy, Manning, Darcy-Weisbach importance of Froude Number ◦ Fr>1 decrease in e gives increase in y ◦ Fr<1 decrease in e gives decrease in y ◦ Fr=1 standing waves (also min e given Q) 1/7/2017 free surface location 0 1 2 3 4 0 1 2 3 4 E y 15MODASSAR ANSARI
- 16. Methods of calculating location of free surface (Gradually varying) ◦ Direct step (prismatic channel) ◦ Standard step (iterative) ◦ Differential equation Rapidly varying ◦ Hydraulic jump 1/7/2017 2 1 Fr SS dx dy fo 16MODASSAR ANSARI
- 17. Specific energy Two depths with same energy ◦ How do we know which depth is the right one? ◦ Is the path to the new depth possible? 1/7/2017 2 2 1 2 1 2 2 2o f V V y S x y S x g g + + D = + + D 2 2 2 q y gy = + g V yE 2 2 2 2 2 Q y gA = + 17MODASSAR ANSARI
- 18. 1/7/2017 0 1 2 3 4 0 1 2 3 4 E y 18MODASSAR ANSARI