# The Courant Number and Unsteady Flow Computation

*by*Chintu Lai, U.S. Geological Survey, Reston, United States,

**Abstract:**The Courant number C, the key to unsteady flow computation, is a ratio of physical wave velocity, λ, to computational signal-transmission velocity, τ, i.e., C = λ/τ. In this way, it uniquely relates a physical quantity to a mathematical quantity. Because most unsteady open-channel flows are describable by a set of n characteristic equations along n characteristic paths, each represented by velocity λ

_{i}, i = 1,2,....,n, there exist as many as n components for the numerator of C. To develop a numerical model, a numerical integration must be made on each characteristic curve from an earlier point to a later point on the curve. Different numerical methods are available in unsteady flow computation due to the different paths along which the numerical integration is actually performed. For the denominator of C, the τ defined as τ = τ

_{0}= δx/δt has been customarily used; thus, the Courant number has the familiar form of C

_{τ}= λ/τ

_{0}. This form will be referred to as common Courant number in this paper. The commonly used numerical criteria C

_{τ}for stability, neutral stability and instability, are imprecise or not universal in the sense that r

_{0}does not always reflect the true maximum computational data-transmission speed of the scheme at hand, i.e., C

_{tau}is no indication for the Courant constraint. In view of this , a new Courant number, called the natural Courant number, C

_{n}, that truly reflects the Courant constraint, has been defined. However, considering the numerous advantages inherent in the traditional C

_{τ}, a useful and meaningful composite Courant number, denoted by C

_{τ}

^{*}has been formulated from C

_{τ}. It is hoped that the new aspects of the Courant number discussed herein afford the hydraulician a broader perspective, consistent criteria, and unified guidelines, with which to model various unsteady flows.

**Subject Headings:**Unsteady flow | Computing in civil engineering | Numerical methods | Wave velocity | Numerical models | Hydraulic models | Mathematics

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