# St Venant Equations

St Venant Equations Reading: Sections 9.1 9.2 Types of flow routing Lumped/hydrologic Flow is calculated as a function of time alone at a particular location Governed by continuity equation and flow/storage relationship Distributed/hydraulic Flow is calculated as a function of space and time throughout the system Governed by continuity and momentum equations 2 Distributed Flow routing in channels Distributed Routing St. Venant equations

Continuity equation Q A 0 x t Momentum Equation 1 Q 1 Q 2 y g g ( S o S f ) 0 A t A x A x What are all these terms, and where are they coming from? Assumptions for St. Venant Equations Flow is one-dimensional

Hydrostatic pressure prevails and vertical accelerations are negligible Streamline curvature is small. Bottom slope of the channel is small. Mannings equation is used to describe resistance effects The fluid is incompressible Continuity Equation Q = inflow to the control volume q = lateral inflow Q x Q Rate of change of flow with distance

Q dx x ( Adx) t Elevation View Change in mass Reynolds transport theorem 0 Plan View Outflow from the C.V.

d d V .dA dt c.v. c.s. Continuity Equation (2) Q A 0 x t (Vy ) y 0 x t

V y V y y 0 x x t Conservation form Non-conservation form (velocity is dependent variable) Momentum Equation From Newtons 2nd Law: Net force = time rate of change of momentum

d Vd VV .dA F dt c .v . c.s. Sum of forces on the C.V. Momentum stored within the C.V Momentum flow across the C. S. Forces acting on the C.V.

Elevation View Plan View Fg = Gravity force due to weight of water in the C.V. Ff = friction force due to shear stress along the bottom and sides of the C.V. Fe = contraction/expansion force due to abrupt changes in the channel cross-section

Fw = wind shear force due to frictional resistance of wind at the water surface Fp = unbalanced pressure forces due to hydrostatic forces on the left and right hand side of the C.V. and pressure force exerted by banks Momentum Equation d Vd VV .dA F dt c .v . c.s. Sum of forces

on the C.V. Momentum stored within the C.V Momentum flow across the C. S. 1 Q 1 Q 2 y g g ( S o S f ) 0 A t A x A x

Momentum Equation(2) 2 1 Q 1 Q y g g ( S o S f ) 0 A t A x A x Local acceleration term Convective acceleration term

Pressure force term Gravity force term Friction force term V V y V g g ( S o S f ) 0 t

x x Kinematic Wave Diffusion Wave Dynamic Wave Momentum Equation (3) 1 V V V y S o S f g t g x x Steady, uniform flow Steady, non-uniform flow Unsteady, non-uniform flow

## Recently Viewed Presentations

• 3 Differentiate between a) OMR and OCR b) Compiler and Interpreter c) MICR and Bar code Reader d) Flat Bed Plotter and Drum Plotter 4 a) What is Secondary Storage?How does it Different from Primary Storage? b) Explain the Memory...
• Let's take a look at some of the multiple high school pathway plans our SLSS student population can considerâ€¦ very popular university programs that our SLSS student population consider are: ENGINGEERING
• 67% of our master's degrees have accelerated tracks. Recruit more students into those paths. Develop accelerated tracks in remaining programs, as feasible. Completion rate is astonishing. CULTIVATE OUR OWN! MSU Undergrads Become MSU Grad Students. EVERYONE in this room plays...
• 1. Rubriblast (arrow) largest, with deep blue cytoplasm and round euchromatic nucleus 2. Prorubricyte (arrowhead) Similar, but smaller, no nucleoli 4. Metarubricyte: the smallest nucleated erythrocyte with picnotic nucleus 3. Basophilic rubricyte: nucleus with clumped chromatin 4. 3.
• How much core-business knowledge should leaders have?. How important is it that the Managing Director of McKinsey's was an outstanding consultant?. Do German automobile companies perform so well because engineers are on the shop floor and in the boardroom?
• Infants (0-2 years) Bowlby argued that infants have an inbuilt need to form an attachment with a carer. The quality of this attachment may affect emotional development for the rest of the child's life.
• Monsters in Greek mythology fell into two rough categories. Some stood as obstacles to the heroes and their deedsâ€”abominations that needed to be destroyed. The bull-headed Minotaur and the terrible Medusa are good examples of such monsters. Other Greek monsters...
• Para sahabat berkata, "Wahai Rasulullah, siapa al-ghuraba ini?" Rasulullah saw bersabda, "Mereka adalah orang-orang yang melakukan perbaikan ketika manusia sudah rusak (HR. Atthabrani) Rasulullah saw. bersabda: Pada hari kiamat nanti, akan datang sekelompok orang yang imannya begitu ...