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Governing equations, kinematic wave, Courant number

Short answer

Fluid movement is governed by the Saint-Venant continuity and momentum equations. Kinematic wave approximation neglects local/convective acceleration and pressure terms, reducing momentum to S_f=S_0, giving unique Q=αA^β relation and wave celerity c=βV. Courant number C=cΔt/Δx; explicit schemes require C≤1 for stability.

Write down the governing equations used for analyzing the movement of fluid. What are the kinematic wave approximations of these governing equations? Also define Courant number.

Governing equations

For 1D unsteady open-channel flow (Saint-Venant equations):

Continuity (conservation of mass):

∂A∂t+∂Q∂x=q\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=q

Momentum (conservation of momentum):

∂Q∂t⏟local acc.+∂∂x ⁣(Q2A)⏟convective acc.+gA∂y∂x⏟pressure force−gAS0⏟gravity+gASf⏟friction=0\underbrace{\frac{\partial Q}{\partial t}}_{\text{local acc.}}+\underbrace{\frac{\partial}{\partial x}\!\left(\frac{Q^2}{A}\right)}_{\text{convective acc.}}+\underbrace{gA\frac{\partial y}{\partial x}}_{\text{pressure force}}-\underbrace{gAS_0}_{\text{gravity}}+\underbrace{gAS_f}_{\text{friction}}=0

where AA = flow area, QQ = discharge, yy = flow depth, qq = lateral inflow per unit length, S0S_0 = bed slope, Sf=n2Q∣Q∣A2R4/3S_f=\dfrac{n^2Q|Q|}{A^2R^{4/3}} = friction slope (Manning), gg = gravity. These are the Saint-Venant equations (1D unsteady open-channel flow).

In terms of velocity VV and depth yy for a wide rectangular channel:

∂y∂t+V∂y∂x+y∂V∂x=0,∂V∂t+V∂V∂x+g∂y∂x=g(S0−Sf)\frac{\partial y}{\partial t}+V\frac{\partial y}{\partial x}+y\frac{\partial V}{\partial x}=0,\qquad \frac{\partial V}{\partial t}+V\frac{\partial V}{\partial x}+g\frac{\partial y}{\partial x}=g(S_0-S_f)

Kinematic wave approximation

In many flood-routing problems on steep slopes the local acceleration, convective acceleration and pressure terms are small compared with the gravity and friction terms. Dropping them, the momentum equation becomes

S0−Sf=0⇒Sf=S0S_0-S_f=0\quad\Rightarrow\quad S_f=S_0

so the flow is always at its normal-depth value for the local area and a unique relation exists:

Q=αkAβ,β=53 (Manning)Q=\alpha_kA^{\beta},\qquad \beta=\frac53\ \text{(Manning)}

Together with continuity, ∂A∂t+∂Q∂x=q\dfrac{\partial A}{\partial t}+\dfrac{\partial Q}{\partial x}=q, this gives the kinematic wave equation

∂Q∂t+c ∂Q∂x=c q,c=dQdA=βV\frac{\partial Q}{\partial t}+c\,\frac{\partial Q}{\partial x}=c\,q,\qquad c=\frac{dQ}{dA}=\beta V

The wave moves downstream at speed cc without attenuation (no backwater effects).

Courant number

The Courant number is the ratio of the distance a wave travels in one time step to the grid spacing:

C=c ΔtΔxC=\frac{c\,\Delta t}{\Delta x}

For explicit schemes, stability requires C≤1C\le1. For the dynamic wave, c=V±gyc=V\pm\sqrt{gy}.

✓ Answer checked by Claude Opus 5.5 (high effort)

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