CIVE 445 - ENGINEERING HYDROLOGY

CHAPTER 9C: STREAM CHANNEL ROUTING, DIFFUSION WAVES

9.3  DIFFUSION WAVES

  • The Muskingum method and a numerical solution of the linear kinematic wave equation show striking similarities.

  • Both methods have the same type of routing equation.

  • The Muskingum method can calculate hydrograph diffusion.

  • The linear kinematic method can calculate hydrograph diffusion only by the introduction of numerical diffusion.

  • The latter is dependent on the grid size and type of scheme.

  • Kinematic wave theory can be enhanced by allowing a small amount of physical diffusion in its formulation.

  • This leads to an improved type of kinematic-with-diffusion wave, for short, a diffusion wave.

  • Some diffusion is usually present in natural stream channel situations.
 

Diffusion wave equation

  • The kinematic wave equation was derived using a statement of steady uniform flow in lieu of momentum conservation.

  • In the diffusion wave equation, we use a statement of steady nonuniform flow.

  • This leads to a slightly modified form of the Manning equation:

    Q = (1/n) A R2/3 [So - (dy/dx)]1/2

     


    Fig. 9-7

     

  • Derivation of the diffusion wave equation: Diffusion wave equation

  • Diffusion wave equation (continued)

  • The diffusion wave equation is:  

    ∂Q/∂t + (∂Q/∂A) (∂Q/∂x) = [Qo/(2TSo)] (∂2Q/∂2x)

     

  • The coefficient of the second-order term (RHS) is the hydraulic or channel diffusivity.

  • The hydraulic diffusivity is:  

    νh = Qo /(2TSo)] = qo /(2So)]

     

  • The hydraulic diffusivity νh is directly related to unit-width discharge and inversely related to channel slope.

  • The diffusion wave equation is a second-order parabolic differential equation.

  • It can be solved analytically, leading to Hayami's diffusion-analogy solution for flood waves.

  • It can be solved numerically with a Crank-Nicolson scheme.

  • Another approach: to match the hydraulic diffusivity with the numerical diffusion coefficient of the Muskingum scheme (kinematic wave equation).

  • This approach is the basis of the Muskingum-Cunge method (Section 9.4).
 

Applicability of diffusion waves

  • Most flood waves have a small amount of physical diffusion.

  • Therefore, they are better approximated by the diffusion wave than the kinematic wave.

  • Diffusion waves apply to a much wider range of problems than kinematic waves.

  • Where the diffusion wave fails, only the dynamic wave can properly describe the translation and diffusion of flood waves.

  • The dynamic wave is very strongly diffusive, especially for flows well in the subcritical regime.

  • To determine is a wave is a diffusion wave, it should satisfy the following inequality:

    tr So (g / do)1/2 > M

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

  • For 95 percent accuracy in one period of translation, a value of M = 15 is recommended.

  • Example 9-8.
 

Go to Chapter 9D.

 
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