Types of Open Channel Flow | Specific Energy & Critical Depth of Flow
The flow of liquid with one of its surfaces open to the atmosphere is called the open channel flow. The liquid in an open channel flow is subjected to atmospheric pressure, therefore open channel flow can also be defined as the flow of liquid through a passage at atmospheric pressure. Types of open channel flow along with specific energy concepts are covered further.
Types of Open Channel Flow
The flow in an open channel is classified into the following types.
Steady and unsteady flow
Uniform and non-uniform flow
Laminar and turbulent flow
Sub-critical, critical, and super-critical flow
Steady and Unsteady Flow
In a steady flow, the flow characteristics such as flow depth, the velocity of flow, etc. at any point do not change with respect to time. Mathematically it is expressed as,
∂V/∂t = 0, ∂Q/∂t = 0
On the other hand, if the flow characteristics like the depth of the flow, the velocity of the flow, etc. at any point in an open channel change with respect to time then it is called an unsteady flow. Mathematically it is expressed as,
∂V/∂t ≠ 0, ∂Q/∂t ≠ 0
Uniform and Non-Uniform Flow
For a given length of the channel if the depth of the flow, the velocity of the flow, the slope of the channel, etc. remains constant then it is called a uniform flow. In other words, the flow characteristics remain constant with space. Mathematically it is expressed as,
∂y/∂S = 0, ∂V/∂S = 0
On the other hand, if the depth of flow, the velocity of the flow, etc. changes with respect to space i.e., along a section of the channel, then it is a non-uniform flow. Mathematically it is expressed as,
∂y/∂S ≠ 0, ∂V/∂S ≠ 0
In reality, combinations of the above-mentioned flows such as steady uniform flow, steady non-uniform flow, unsteady uniform flow, and unsteady non-uniform flow, exist.
Non-uniform flow is also called a varying flow and is further classified as,
Gradually varied flow
Rapidly varied flow
Gradually Varied Flow: If the depth of flow in a channel changes gradually over a long length of the channel then it is called a gradually varied flow.
Rapidly Varied Flow: If the depth of the channel changes abruptly over a very small length of the channel, then it is called a rapidly varied flow.
Laminar and Turbulent Flow
The flow in an open channel flow is said to be laminar if Reynold's number is less than 500 to 600. Reynold's number is given as,
Re = ρVR/μ,
V - mean velocity of flow
R - Hydraulic radius or hydraulic mean depth = cross-section of flow/wetted perimeter
ρ - density of the liquid
μ - viscosity of the liquid
If Reynold's number is greater than 2000, then it is called a turbulent flow. If Reynold's number remains between 500 to 2000 then the flow is said to be in a transition state.
Sub-Critical, Critical, and Super Critical Flow
This classification is based on the Froude number. The Froude number is expressed as,
Fe = V/(gD)^(1/2)
V - mean velocity of flow
D - hydraulic depth = wetted area/top width of the channel
If the Froude number is less than one then it is called a sub-critical flow. It is also called a tranquil flow.
The flow is critical if the Froude number is equal to one. If the Froude number is greater than one then the flow is super-critical.
Specific Energy and Specific Energy Curve
We know that the total energy of a flowing fluid per unit weight is given as,
Total energy = z + h + V^2/2g
If the channel bottom in which the fluid is flowing is taken as datum then the above equation will become,
Total energy = h + V^2/2g
The above equation is known as the specific energy of the flowing liquid. Therefore, specific energy is defined as the total energy per unit weight of the liquid with respect to the bottom of the channel.
Specific Energy Curve
The specific energy curve is a curve that shows the variation of specific energy with respect to depth. It is obtained as follows.
E = h + (V^2)/2g = Ep + Ek
Ep - the potential energy of the flow
Ek - kinetic energy of the flow
Let us consider a rectangular channel in which a steady non-uniform flow is taking place at a discharge of Q. Let b and h be the width and depth of the channel respectively. Discharge per unit width is given as,
q = Q/b = which is a constant
Velocity can be rewritten as, V = Q/A = Q/(b*h) = q/h
Substituting 'V' in the specific energy equation,
E = h + (q^2)/(2gh^2)
Ep = h
Ek = q^2/(2gh^2)
The above equation shows the variation of specific energy variation with depth. Therefore, with the help of this equation, the specific energy curve can be plotted by directly finding the specific energy values for a given discharge at different depths or by plotting the potential energy curve (which is a straight line) and kinetic energy curve (which is a parabola) separately and then combining them. The blue curve below represents the specific energy.
Critical Depth (hc)
Critical depth of flow is that of flow at which the specific energy is minimum. From the above graph, it is the depth of flow at point 'C'. Mathematically the critical depth can be obtained by differentiating the specific energy equation with respect to depth and equating it to zero.
dE/dh = o
d/dh[h + (q^2)/(2gh^2)] = 0 (q^2/2g = constant)
Differentiating and rearranging the terms, we get,
hc = (q^2/g)^(1/3)
Critical Velocity (Vc)
Critical velocity is the velocity of flow at critical depth. It is found from the critical depth formula as follows.
hc = (q^2/g)^(1/3)
Taking cube on both sides, we get,
hc^3 = q^2/g
we know that q = Q/b = b*h*V/b = hc*Vc
Substituting 'q' in the above equation and rearranging the terms, we get,
Vc = (g*hc)^(1/2)
Minimum Specific Energy (Emin)
It is obtained by substituting the critical depth value (hc) in the specific energy equation. It is expressed as,
Emin = 3hc/2
Except for the point at the minimum specific energy, there are two possible values of depth possible for any given value of specific energy. These two values are called alternate depths. One of the depths is greater than the critical depth while the other is less than the critical depth.
Question: The conjugate depths at a location in a horizontal rectangular channel, 4m wide, are 0.2m and 1m. The discharge in the channel is? (GATE 1991)
Conjugate depths are nothing but alternate depths. It is given that,
h1 = 0.2 m
h2 = 1 m
E = h + (q^2)/(2gh^2)
E1 based on h1 = 0.2 + (q^2/0.785)
E2 based on h2 = 1 + (q^2/19.62)
Equating the above two equations (as at conjugate depths the specific energy is same) and solving for q, we get,
q = 1.0849
Q = q * b = 4*1.0849 = 4.339 cum/s
Therefore, discharge = 4.339 cum/s
Hope you found all the required information on open channel flow. There is one more interesting sub-topi called "Hydraulic jump" in the open channel flow topic. Want us to cover hydraulic jumps? Comment "Hydraulic Jumps" or any other topic of your interest in the form below.