Accuracy of Kinematic Wave and Diffusion Wave Approximations for Flood Routing. I: Steady Analysis

The applicability of the kinematic wave (KW) and diffusive wave (DW) approximations was investigated for steady flow in prismatic channels by using a second-order two-step Lax-Wendroff numerical scheme coupled with the characteristic method at the boundaries. Two types of downstream boundary conditions, critical-flow depth and zero-flow depth gradient, were considered together with the condition of discharge hydrograph reaching the steady state at the upstream end. The role of inertial, pressure, friction, and gravity forces was investigated for 16 test cases defined through the kinematic wave number $K$ and the Froude number ${\mathsf{F}}_0$ , whose range was assumed to be (3–30) and (0.1–1), respectively. Errors were computed by comparing the steady dimensionless profiles of the flow depth with those estimated by the dynamic wave solution. The accuracy of the two approximations was assessed through the mean value $\epsilon$ of the magnitudes of errors computed for the channel region where the solution was not significantly influenced by the boundary conditions. For critical flow at the downstream end and for $\epsilon$ less than 5%, the KW approximation was reasonably accurate for $K{\mathsf{F}}_0^2⩾1.4$ , whereas the DW solution for $K{\mathsf{F}}_0^2>0.6$ . However, close to the downstream end, the KW approximation gave large errors in flow depth, indicating that if an accurate solution there was needed, the KW solution should not be used. For the DW approximation, the maximum error in the flow depth also occurred close to the downstream end; for the lowest $K$ and ${\mathsf{F}}_0$ values it reached 13%. On the other hand, under the steady flow condition and zero-flow depth gradient at the downstream end, both the diffusive solution and the kinematic solution were found reasonably accurate for the $K$ and ${\mathsf{F}}_0$ values investigated. Therefore, the rule adopted for the critical depth during steady flow also holds for the zero-flow depth gradient.