# The Courant Number and Unsteady Flow Computation

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

**Document Type:**Proceeding Paper

**Part of:**Hydraulic Engineering

**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:**Numerical methods | Hydraulic models | Curvature | Mathematical models | Unsteady flow | Water flow | Open channel flow | Wave velocity

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