Surface Water Equations Continuity (NS) Kinematic Boundary Conditions

Surface Water Equations Integrate continuity equation over depth, term by term

Surface Water Equations Third term… (need KW boundary conditions) Regrouping, letting h = z2 - z1 with velocity constant with depth

Surface Water Equations Momentum (NS x-direction) Term-by-term integration First: Second:

Surface Water Equations Third: Fourth: need kinematic boundary conditions

Surface Water Equations Left side of momentum equation becomes:

Surface Water Equations In terms of shear stress, the right side is written Assume horizontal shear components are small

Surface Water Equations The first term is the unbalanced pressure force; when vertically averaged: (hydrostatic?)

Surface Water Equations The third term is the gravitational force:

Surface Water Equations The second term must be vertically integrated: Shear stress at the water surface is zero

Surface Water Equations Combining and multiplying by depth:

Surface Water Equations Combining all terms, the x-direction momentum equation for overland flow is Similarly, the y-direction equation is

Surface Water Equations With Some substitutions: p = hu , q = hv ql = r – f

Surface Water Equations The equations become:

Surface Water Equations Friction Slope terms: Darcy-Weisbach

Surface Water Equations Darcy-Weisbach continued… for laminar flow: So:

Surface Water Equations Mannings:

Surface Water Equations Vector (compact) notation:

Surface Water Equations Alternate Derivation: conservation of mass and momentum using Reynold’s Transport Theorem Continuity: Momentum:

Surface Water Equations 1-D St. Venant equations: conservation form local acceleration, convective acceleration, unbalanced pressure force, gravity force, and friction force

Surface Water Equations 1-D St. Venant Equations: non-conservation form

Surface Water Equations When can the kinematic wave approximation be used? In general: steep slopes uniform flow no backwater effects

Surface Water Equations For 1-D overland flow on a plane - kinematic wave number: and F0 < 2 Woolhiser and Liggett (1967)

Surface Water Equations For 1-D plane or channel flow Hager and Hager (1985)

Surface Water Equations Wave Celerity (speed) Kinematic waves occur when there is a unique relationshhip between flow depth and discharge: general form: from Manning’s:

Surface Water Equations differentiate and sub into continuity the total derivative of discharge is or

Surface Water Equations from this we see that: discharge increases with lateral inflow and kinematic wave celerity

Surface Water Equations Is the KW celerity equal to mean velocity? - no. in a wide rectangular channel, u = Q/h, and substituting Manning’s equation:

Surface Water Equations Dynamic wave celerity