UNIVERSITY OF CAMBRIDGE ENGINEERING TRIPOS PART IIA, MODULE 3D5 WATER ENGINEERING II. Open Channel Flow 1. Fundamentals 2. Steady Open Channel Flow 1) Uniform Flow and Flow Resistance 2) Rapidly Varied Flow 3) Gradually Varied Flow 3. Unsteady Open Channel Flow (to appear in IV) 1 Steady, uniform Steady, non-uniform Unsteady (to be studied after "pollutant transport") Oct 2018 2 Oct 2018
1. FUNDAMENTALS 1.1 OPEN CHANNEL FLOW VS. PIPE FLOW A · Pressurised flow Q A Cross section A-A Pressure at a cross-section can be of any value. Cross-sectional area is fixed. L A Q · Open channel flow Cross section A-A LA Also called free surface flow. Pressure at free surface equals to Pa , but the water level can rise or drop. Cross-sectional area varies with water depth. 1.2 BASIC ASSUMPTIONS IN THE STUDY B 2 V h ? A L ? x (chainage) 1 Zb Ph datum Side view Cross-sectional view · Small bed slope: cosp ~ 1, S+ = tan p ~ sin p, n~ z +h Head (source Concave-upward curve area) Mouth Steeper Gradient Longitudinal Profile Gentler Gradient Severn Thames Nile Source height 610 m Length 354 km Average Sb 0.0017 110 m 2700 m 346 km 0.00032 6650 km 0.00041 · One-dimensional U =- Ju .dA =U(x) A The vertical and lateral velocity components, v and w, are often small in comparison with the streamwise component. We only consider U as a representative velocity through a cross-section in analysing the flows. With such an area-averaged velocity, it is very easy to calculate the flow rate. E We assume that there is only one free surface elevation at one cross-section in our study, although it may not be true at river bends. Hence, the water surface position is only a function of x. · Hydrostatic pressure distribution 3 Oct 2018 4 Oct 2018
p = pgx (distance to free surface) The piezometric head is defined as z+ p pg whose unit is [m]. The water surface represents the piezometric head, and there is only one piezometric head at one cross-section. For example, the piezometers have the same reading at points 1 and 2. 1.3 BASIC TOOLS: control volume method, continuity, momentum and energy (Bernoulli) equations. 7 = 4x We choose the control volume to be a stream tube bounded by the free surface, river bed and banks. 1). Continuity - conservation of mass (volume) We often neglect the water volume change due to groundwater flow and evaporation because they occur relatively slowly. Steady flow: Q1 = Q2 Unsteady flow: difference between QAt and Q2At results in the volume change within time At . Q1 At - Q2 At = V(t + At) -V(t) 2). Energy head at a cross-section is defined as p pg 2g aU2 = n+ aU2 H = z+ -+ 2g U2 2g 5 where a is the kinetic energy correction factor and is sometimes referred to as the Coriolis coefficient. a takes into account the non-uniform velocity distribution in a