Surface Water Equations

Surface Water Equations
Continuity (NS) Kinematic Boundary Conditions

Integrate continuity equation over depth, term by term

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

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

Third: Fourth: need kinematic boundary conditions

Left side of momentum equation becomes:

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

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

The third term is the gravitational force:

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

Combining and multiplying by depth:

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

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

The equations become:

Friction Slope terms: Darcy-Weisbach

Darcy-Weisbach continued… for laminar flow: So:

Mannings:

Vector (compact) notation:

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

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

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

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

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

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

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

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

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

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

Dynamic wave celerity

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