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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).
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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