CIVE 445 - ENGINEERING HYDROLOGY

CHAPTER 9B: STREAM CHANNEL ROUTING, KINEMATIC WAVES

9.2  KINEMATIC WAVES

  • Three types of unsteady open channel flow waves are common in engineering hydrology:

    1. kinematic,

    2. diffusion,

    3. dynamic

  • Kinematic waves are the simplest type of wave.

  • Dynamic waves are the most complex.

  • Diffusion waves lie somewhere in between kinematic and dynamic waves.
 

Kinematic wave equation

  • The derivation of the kinematic wave equation is based on the principle of mass conservation within a control volume.

  • The difference between outflow and inflow within one time interval is balanced by a corresponding change in volume.

  • In terms of finite intervals, it is:

    (Q2 - Q1) Δt + (A2 - A1) Δx = 0

  • The differential form can be written as:

    ∂Q/∂x + ∂A/∂t = 0

  • The equation of conservation of momentum (Eq. 4-22) contains local inertia, convective inertia, pressure gradient, friction, gravity, and a momentum source term.

  • The kinematic wave uses a statement of steady uniform flow in lieu of conservation of momentum.

  • This simplification limits the applicability of kinematic waves.

  • The Manning equation of steady uniform flow is:

    Q = (1/n) A R2/3Sf1/2

  • The Chezy equation of steady uniform flow is:

    Q = C A R1/2Sf1/2

  • The hydraulic radius R = A/P. Substituting this into the Manning equation leads to:

    Q = (1/n) (Sf1/2/P2/3) A5/3

  • Assume that n, Sf and P are constants, for simplicity.

  • This is the case of a wide channel with little bed movement.

  • This equation can be written as:

    Q = α Aβ

    in which α and β are parameters of the discharge-area rating.

    α = (1/n) Sf1/2/P2/3

    β = 5/3

  • Differentiating the discharge-area rating leads to the kinematic wave celerity:

    dQ/dA = αβAβ-1 = β(Q/A) = βV

    in which V = mean flow velocity.

  • Multiplying the continuity equation with the kinematic wave celerity equation (chain rule), leads to:

    ∂Q/∂t + (dQ/dA) ∂Q/∂x = 0

    or, alternatively:

    ∂Q/∂t + (βV) ∂Q/∂x = 0

  • These two equations describe flood wave movement (unsteady flow) under the kinematic wave assumption.

  • Kinematic waves travel with celerity βV.

  • The kinematic wave equation is a first-order partial differential equation.

  • Therefore, it cannot describe diffusion or attenuation.

  • Diffusion is a second-order process.

  • Since dQ/dA is the celerity, it can be replaced by dx/dt. Therefore:

    ∂Q/∂t + (dx/dt) ∂Q/∂x = 0

  • This is equal to the total derivative dQ/dt.

    dQ = (∂Q/∂t) dt + (∂Q/∂x) dx = 0

  • Since RHS = 0, it follows that Q remains constant in time for waves traveling with celerity dQ/dA.
 

Discretization of kinematic wave equation

  • The kinematic wave equation is nonlinear and of first-order.

  • It is nonlinear because the wave celerity varies with discharge.

  • The nonlinearity is usually mild, and for some applications, the equation can be considered to be linear.

  • The solution can be obtained by analytical or numerical means.

  • The simplest kinematic wave solution is a linear numerical solution.

  • It is necessary to select a numerical scheme and its properties.
 

Order of accuracy of numerical schemes

  • The order of accuracy of a numerical scheme measures the ability of the numerical scheme to reproduce the terms of the differential equation being solved.

  • In general, the higher the order, the better the reproduction of the differential equation.

  • Forward and backward finite-difference schemes have first-order accuracy.

  • Central schemes have second-order accuracy.

  • First-order schemes create numerical diffusion and dispersion.

  • Second-order schemes create only numerical dispersion.

  • A third-order scheme of the kinematic wave equation creates neither numerical diffusion or dispersion.

  • A third-order scheme reproduces exactly the terms of the differential equation.  

     


    Fig. 9-5

     

  • Second-order accurate numerical scheme, central differences in time and space: Example 9-3.

  • Second-order accurate numerical scheme: Example 9-3 (continued)

  • The Courant number is defined as the ratio of wave celerity βV to the "grid" celerity Δx/Δt:

    C = βV / (Δx / Δt) = βV (Δt / Δx)

     

  • This example illustrates the properties of kinematic waves:

    • For C =1, there is no numerical diffusion or dispersion; the solution is a kinematic wave, translated by not diffused.

    • For C= 1.5, there is numerical diffusion and dispersion. The diffusion results in slight attenuation; the dispersion causes negative outflows in the tail of the calculated hydrograph.

  • The second-order accurate method is impractical because it may lead to negative outflows.  

     

  • First-order accurate numerical scheme, backward in time, backward in space: Example 9-4.

  • Example 9-4 (continued)

  • It is observed that offcentering the derivatives has caused a significant amount of numerical diffusion, with peak outflow 120.93 m3/sec.

  • Different schemes will lead to different answers, depending on the Courant number.
 

Convex method

  • The convex method of channel routing belongs to the family of linear kinematic wave methods.

  • Until 1982, it was part of the NRCS (ex SCS) TR-20 hydrologic model.

  • The routing equation for the convex method is obtained by discretizing the kinematic wave equation in a linear mode with forward differences in time and backward differences in space.

  • Convex method, forward in time, backward in space: Convex method.

  • Convex method (continued)

  • Example 9-5 (continued)

  • The convex method is relatively simple, but the answer is dependent on the routing parameter C.

  • This can be interpreted as a Courant number.

  • However, for values of C other than 1, the amount of numerical diffusion is unrelated to the physical diffusion.

  • The convex method is a crude approach to stream channel routing.
 

Kinematic wave celerity

  • The kinematic wave celerity is dQ/dA or βV.

  • A value β= 5/3 is applicable to a wide channel with Manning friction.

  • In 1900, Seddon concluded that the celerity of long disturbances (read kinematic and diffusion waves) was equal to dQ/dA.

  • Since dA= T dy, where T is the channel top width, the Seddon law is expressed in practice as:

    c = (1/T) (dQ/dy)

  • with c = kinematic wave celerity.

  • The kinematic wave celerity is a function of the slope of the discharge-stage rating and the channel top width.

  • Since both dQ/dy and T vary with stage, c varies with the stage.

  • If c = βV is a function of Q, then the kinematic wave equation is nonlinear.

  • Nonlinear solutions account for the variation of c with stage and flow level.

  • Linear solutions assume a constant value of c.

  • Note that there is a striking similarity between linear kinematic wave solutions and the Muskingum method.  

  • Theoretical β values other than 5/3 can be obtained for other friction formulations and cross-sectional shapes.

  • For laminar flow, β = 3.

  • For a wide channel with Chezy friction, β = 3/2.

  • Calculation of β as a function of frictional type and cross-sectional shape: Example 9-6

  • Example 9-6 (continued)
 

Kinematic waves with lateral inflow

  • Practical application of stream channel routing often require the specification of lateral inflows.

  • The lateral inflow could be concentrated (tributary flow), or distributed along the channel (groundwater exfiltration or infiltration).

  • A mass balance leads to:

    (∂Q/∂x) + (∂A/∂t) = qL

  • Multiplying this equation by dQ/dA (or βV) leads to:

    (∂Q/∂t) + (βV) (∂Q/∂x) = (βV) qL

 

Applicability of kinematic waves

  • The kinematic wave is a fundamental streamflow property.

  • Flood waves which approximate kinematic waves travel with celerity βV and are subject to negligible attenuation (diffusion).

  • In practice, flood waves are kinematic if they are of long duration or travel on a channel of steep slope.

  • Usually, slopes greater than 1% are kinematic.

  • Criteria for the applicability of kinematic waves to overland flow (Chapter 4, page 145) and stream channel flow has been developed.

  • The stream channel criterion is:

    (tr So Vo) / do > N

  • where tr= time-of-rise of the inflow hydrograph; So = bottom slope, Vo = average velocity, and do = average flow depth.

  • For 95 percent accuracy in one period of translation, a value of N = 85 is recommended.

  • Example 9-7
 

Go to Chapter 9C.

 
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