7. Steady Flows in Open Channels Flashcards
Definition of Variables
- consider a channel of large width which can be approximated as a two dimensional flow in the (x,z) plane
- the base of the channel can be described by the function z=Z(x)
- the water is at a depth h(x) above this base surface
- we define ξ(x) = Z(x) + h(x) as the height of the free surface
Conservation of Mass
In Words
- for a steady flow, (∂h/∂t=0), the net mass flux through a volume 𝛿V between x and x+𝛿x must be zero
- i.e. the mass that flowa in and out of 𝛿V must be zero
- so the volume flux through the channel is constant in space and time
Conservation of Mass
Equations
Q = ∫ u dz
-where the integral is taken between Z(x) and ξ(x) is constant in space and time
-further, assuming that u is independent of z:
Q = u(x) |ξ(x)-Z(x)| = u(x)h(x)
Bernoulli’s Theorem on the Free Surface
1/2 u² + gξ = 1/2 u² + g(h+Z) = constant
-OR equivalently:
gh (F²/2 + 1 + Z/h) = constant
Froude Number
F = u / √[gh]
-has no dimension, it determines how flows react to disturbances
Solving Channel Flow Problems
1) apply conservation of mass
2) apply Bernoulli’s equation
3) sub the result from the conservation of mass into Bernoulli’s equation
4) calculate the Froude Number to determine if flow is supercritical or subcritical
Flow Over a Hump
Supercritical Flow
- if F>1, then ~h1 = 1
- starting from ~h=1, the depth of the water first increases, then decreases with ~Z and returns to ~h=1
- this is called supercritical flow
Flow Over a Hump
Subcritical Flow
- if F<1, then ~h2=1
- as Z increases, the water surface goes down and when ~Z returns to zero, the depth of the water returns to h=1
- this is called subcritical flow
What causes the two different behaviours (supercritical and subcritical)?
- the equation u²/2 + g(h+Z) = constant expresses the conservation of energy (KE ad GPE)
- in order to flow over the hump, the fluid has two choices:
i) to increase its KE by reducing PE, i.e. decreasing h
ii) to increase its PE by reducing its KE, i.e. increasing h - the Froude number F=u/√[gh] is the ration of KE to PE
- if F>1 then KE>PE so it is easier to reduce KE as the fluid flows over the bump
- if F<1 then KE
Flow Over a Hump
Sub/Supercritical vs Critical Flows
Description
- in both supercritical and subcritical cases, the fluid flow returns to its original height and speed after passing over the bump
- however, if the maximal height of the bump Zmax=Zc, the flow can move across from one branch of the solution to the other
- stable transitions occur only in one direction, from subcritical to supercritical
Flow Over a Hump
Critical Flow - Subcritical to Supercritical
Description
-a subcritical flow upstream (F<1) is transformed smoothly into a supercritical flow downstream (F>1)
-the condition for this to happen, Zmax=Zc leads to:
h = hc = H*F^(2/3) at the top of the hump
Flow Over a Hump
Critical Flow
Froude Number at the Top of the Hump
-the local Froude number, f:
f=1
-at the top of the hump
Flow Over a Hump
Critical Flow
f
- the local Froude number, f, is a continuous function
- less than 1 upstream, greater than 1 downstream and equal to 1 at the top of the hump
Flow Through a Constriction
Description
-an alternative to varying the height of the base of the channel is to vary its breadth b(x)
Flow Through a Constriction
Supercritical
-if F>1, then the flow is supercritical
~h1=1 and ~h=h’H increases as ~b=b/B decreases (i.e. K increases)
-the height of the free surface rises through the constriction
