Open Channel Flow Varied Flow By DR Ezzat

Open Channel Flow “Varied Flow” By: DR. Ezzat El-Sayed G. SALEH Civil Eng. Dept. El Minia University

Classification of Water. Surface Profiles

¥A gradually-varied flow profile or gradually -varied water surface profile is a line indicating the position of the water surface. It is a plot of the flow depth as a function of distance along the flow direction.

We use the following notation to designate different water surface profiles: M A letter refers to the type of the channel bottom slope and a numeral to the relative position of the profile with respect to the critical-depth line , (CDL) and the normal-depth line (NDL), M The critical depth and the normal depth are ycr and yn, respectively.

Governing Equation The gradually varied flow equations in a prismatic channel having no lateral inflow or outflow are derived in this section by making the following simplifying assumptions: è The slope of the channel bottom is small. è The channel is prismatic channel and there is no lateral inflow or outflow from the channel. è The pressure distribution is hydrostatic at all channel sections. For clarity of presentation, an exaggerated vertical scale is used in the illustrations of this presentation. Thus, the slope of the channel bottom, even though small, may appear to be large in these illustrations.

Zones for Classification of Surface Profile Zone 1 Zone 2 Zone 3 ND L or CDL or NDL Bed Slo pe

Channel Bottom Slopes are Classified into the following Five(5) Categories ● Mild, Steep, Critical, Horizontal (zero slope) And Adverse (negative slope). ● The first letter of these names refers to the type i. e. , M for mild, S for steep, C for critical, H for horizontal and A for adverse slope.

Mild and steep slopes MA channel slope is usually ‘classified’ by comparing the uniform flow depth yn with the critical flow depth ycr. M When the uniform flow depth is larger than the critical flow depth, yn > ycr the uniform equilibrium flow is tranquil and sub-critical. The slope is called a mild slope. M For yn < ycr, the uniform flow is supercritical and the slope is steep.

For the Specified Discharge Q and Manning’s n Uniform flow: Fr <1 (sub-critical yn > Mild flow) ycr slope yn = Critical Uniform flow: Fr = 1 (critical slope ycr flow) Uniform flow: Fr > 1 (supercritical yn > Steep flow) ycr slope yn = horizontal normal depth is slope infinite yn Adverse slope normal depth is nonexist (negative slope) imaginary ent

●The region above both lines (the normal-depth and critical-depth) is designated as Zone 1; that between the upper and lower lines is designated as Zone 2, and the one between the lower line and the channel bottom is designated as Zone 3. ●For the mild and steep slopes, the NDL&CDL lines divide the space above the channel bottom into three regions, ●For the adverse, horizontal, and critical bottom slopes, there are only two regions since the normal depth does not exist, is infinite, or is the same as the critical depth, respectively.

Note that: ● The upper line is the normal-depth line if the channel bottom slope is mild, and the upper line is the critical-depth line if the bottom slope is steep. ● We have 12 different types of surface profiles: ~ three for the mild slope, ~ three for the steep slope, ~ two for the critical slope (zone 2 does not exist since yn = ycr and we do not consider the critical-depth line as a surface profile); ~ two for the horizontal slope (zone 1 does not exist since yn = ), and ~ two for the adverse slope (there is no zone 1, since yn does not exist).

Equation of Gradually Varied Flow numerator …………(1) denominator So Slope of the channel bottom, SE Slope of the energy-grade line, Fr The definition of the Froude number depends on the channel Geometry.

When y = yn : The energy-grade line, water surface, and channel bottom are parallel to each other in uniform flow; i. e. , SE = Sw = So It is clear from the Manning or Chezy equation that for specified discharge, Q, SE > So if y < yn. and SE < So if y > yn By using these two inequalities, we determine the sign of the numerator of Eq. (1) Whether the flow is sub-critical (Fr < 1) or super-critical (Fr > 1), we determine the sign of the denominator of Eq. (1)

How the Surface Profiles Approach the Normal and Critical Depths and the Channel Bottom? ¥ As y tends to yn (y → yn), SE → So. Therefore, it follows from Eq. (1) that dy/dx → 0 provided Fr 1 (i. e. , flow is not critical). In other words, the surface profile approaches the normal-depth line asymptotically. ¥ As y → ycr, Fr → 1 and the denominator of Eq. (1) tends to zero. Therefore, dy/dx tends to provided SE So. Thus, the water-surface profile approaches the criticaldepth line vertically.

¥ As y→ , V → 0, and consequently both Fr and SE tend to zero. Hence, it follows from Eq. ( ) that dy/dx → So for very large values of y. Since we are assuming that So is small, we may say that the water surface profile almost becomes horizontal as y becomes large.

NDL M-1 y>yn and consequently SE<So CDL y>ycr and thus Fr<1. 0 Mild Slope Bed L evel y<yn and consequently SE>So y>ycr and thus Fr<1. 0 y<yn and consequently SE>So y<ycr and thus Fr>1. 0 M-3 M-2

yn>ycr and consequently So<Scr Fr <1. 0 NDL CDL Fr >1. 0 M 3 C D L= Critical Depth Line; N D L= Normal Depth Line; M 1 M 2 Bed Level Gradually Varied Flow Profiles on Mild Slope

y>yn and consequently SE<So CDL S-1 NDL y>ycr and thus Fr<1. 0 Steep Slope Bed L evel y>yn and consequently SE<So y<ycr and thus Fr>1. 0 y<yn and consequently SE>So y<ycr and thus Fr>1. 0 S-3 S-2

yn<ycr and consequently So>Scr S-1 Fr < 1. 0 C D L Fr > 1. 0 C D L=Critical Depth Line; Horizon tal S-2 NDL S-3 Bed Level N D L=Normal Depth Line; Gradually Varied Flow Profiles on Steep Slope

y>yn and consequently SE<So C-1 y>ycr and thus Fr<1. 0 &CDL Bed Critical Slope y<yn and consequently SE>So y<ycr and thus Fr>1. 0 NDL C-3 Leve l

yn= ycr and consequently So= Scr C 1 Fr < 1. 0 Fr > 1. 0 C D L=Critical Depth Line; Horizont al C 3 C D L& NDL Bed Level N D L=Normal Depth Line; Gradually Varied Flow Profiles on Critical Slope

yn CDL --- Bed Level Horizontal Slope y<yn and consequently SE>So y>ycr and thus Fr<1. 0 y<yn and consequently SE>So y<ycr and thus Fr>1. 0 H-3 H-2

yn and So= 0 Horizont al Fr < 1. 0 Fr > 1. 0 H 2 CDL H 3 Bed Level C D L=Critical Depth Line; N D L=Normal Depth Line; Gradually Varied Flow Profiles on Horizontal Slope

yn imaginary CDL vel e L d e B Adverse Slope y<yn and consequently SE>So y>ycr and thus Fr<1. 0 y<yn and consequently SE>So y<ycr and thus Fr>1. 0 A-3 A-2

Horizont al Fr < 1. 0 Fr > 1. 0 A-2 yn imaginary and So=-ve CDL A-3 l e v e L d e B C D L = Critical Depth Line; N D L = Normal Depth Line; Gradually Varied Flow Profiles on Adverse Slope

Class Bed Slope Depth Type Classification Mild So 0 y yn ycr 1 M-1 Mild So 0 yn y ycr 2 M-2 Mild So 0 yn ycr y 3 M-3 Critical So 0 y yn = ycr 1 C-1 Critical So 0 y yn = ycr 3 C-3 Steep So 0 y ycr yn 1 S-1 Steep So 0 y ycr yn 2 S-2 Steep So 0 y yn ycr 3 S-3 Horizontal So = 0 y ycr 2 H-2 Horizontal So = 0 y ycr 3 H-3 Adverse So 0 y ycr 2 A-2 Adverse So 0 y ycr 3 A-3

Water surface profiles

Cases of water-surface profiles

Cases of water-surface profiles (cont. )

Cases of water-surface profiles (cont. )

Cases of water-surface profiles (cont. )

Cases of water-surface profiles (cont. )

D. Ezzat el sayed