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