OPEN CHANNEL FLOW
where, V0.2 = velocity at a depth of 0.2 y0 from the free surface.
V0.8 = Velocity at a depth of 0.8 y0 from the free surface.
Terms related to open channel flow
(a) Depth of Flow (y) : Vertical distance between the lowest point of the channel section (bed of the channel) to the free liquid surface.
(b) Top width (T) : Width of channel section at free liquid surface.
(c) Wetted Area (A) : Cross sectional area of the channel normal to the direction of flow.
(d) Channel slope (S) : Inclination of channel bed is called channel slope and is given by,
S = tan a » sin a = (h/I)
where h = vertical fall in length l of channel.
(e) Wetted Perimeter and Hydraulic mean depth
R = (A/P)
(f) Hydraulic depth (D) : It is the ratio of wetted area A to the top width T.
D = (A/T)
(g) Hydraulic Grade Line (HGL)
(h) Energy Grade Line (EGL) or Total energy Line (TEL)
If the datum coincides with the channel bed at the section, the resulting expression is known as SPECIFIC ENERGY and is denoted by E. Thus.
Equation (i) & (ii) are basic equations for critical flow conditions in a channel.
where D = (A/T)
at critical flow F = 1 at y = yc
Velocity Formulae in Uniform Flow
(a) Chezy Equation
V = velocity of flow
R = hydraulic radius
S = bottom slope
Relationship between ‘C’ and friction factor ‘f’
hf = head loss due to friction
D = Diameter of pipe
f = friction factor
L = length of pipe
2. Manning’s Formula
where, n = a roughness coefficient known as Manning’s ‘n’.
Dimension of n is [L- 1/ 3T]
The manning’s formula is most widely used in uniform flow.
Relationship between Manning’s ‘n’ ‘c’ & ‘f’ :
(By comparing chezy and Manning’s formula)
(3) Ganguillet and Kutter Formula
where, n = manning’s coefficient.
HYDRAULICALLY-efficient channel sections
(a) Rectangular Section A = By = Constant
y = B/2 and R = y/2
(b) Trapezoidal Section
A = (B + ny) y = Constant
(i) R = (y/2)
(ii) The trapezoidal channel should be part of regular hexagon.
(iii) The side slopes should be 60° with the horizontal.
(iv) Half top width is equal to one of the sloping sides
(c) Triangular Section
(i) q = 45°
(ii) R = y/2√2
(d) Circular Section Case 1 : Condtion for maximum discharge :
y = depth of flow
(i) y » 0.95 D and
(ii) hydraulic radius, R » 0.29 D
Case 2 : Condition for maximum mean velocity
(i)y » 0.81D
(ii)R » 0.30D
Basic Assumptions of Gradually Varied Flow Analysis
(i) The pressure distribution at any section is assumed to be hydrostatic.
(ii) The resistance to flow at any depth is given by the corresponding uniform flow equation, such as Manning’s formula with the condition that the slope term to be used in the equation is energy slope and not the bed slope.
Dynamic Equation of GVF
Classification of Water Surface Profiles It is clear that the gradient of free surface dy/dx may be positive or negative depending on signs of numerator and denominator, which in turn depends on y, yc and yn. Whether the yn is below or above yc will depend on the classification of be slope. Surface profiles are classified by a letter and a number. The letter refers to the bed slope, whcih may be one of the fllowing : M Mild slope yn > yc C Critical slope yn = yc S Steep slope yn < yc H Horizontal So = O A Adverse So < O
RAPIDLY VARIED FLOW (RVF)
Hydraulic Jump Hydraulic Jump Is an example of steady RVF. A hydraulic jump occurs when a super critical stream meets a subcritical stream of sufficient depth. The super critical stream jumps up to meet its alternate depth. While doing so it generates considerable disturbances in the form of large scale eddies and a reverse flow roller with the result that the jump falls short of its alternate depth. A schematic sketch of jump is as shown below :
Section (1), the point of commencement of jump is called TOE of Jump. The distance between (1) and (2) is called LENGTH OF JUMP (L J). The two depth y1 and y2 at the ends of the jump are called SEQUENT DEPTHS. The specific force diagram resembles specific energy diagram.
From specific force curve we can find the sequent depths y1 and y2 for a given discharge in a given horizontal channel.
Condition for Critical Flow :
Hydraulic Jump in a rectangular channel :
(a) Sequent depth ratio
(i)y1y1 (y1 + y2) = 2yc3
Froude number at (i)
(b) Energy Loss (i)EL = E1 – E2
(c) Height of Jump = y2 – y1
(d) Length of jump = 5 to 7 times Height of Jump
(e) Ratio of energy loss to initial energy :
Classification of Jumps
|Undular Jump||1 – 1.7||EL/E1|
|Weak Jump||1.7 – 2.5|
|Oscillating Jump||2.5 – 4.5|
|Steady Jump||4.5 – 9|
|Stronger choppy||> 9.0|