Manning’s equation and two-dimensional flow analogs

Abstract

summary Two-dimensional (2D) flow models based on the well-known governing 2D flow equations are applied to floodplain analysis purposes. These 2D models numerically solve the governing flow equations simultaneously or explicitly on a discretization of the floodplain using grid tiles or similar tile cell geometry, called ‘‘elements”. By use of automated information systems such as digital terrain modeling, digital elevation models, and GIS, large-scale topographic floodplain maps can be readily discretized into thousands of elements that densely cover the floodplain in an edge-to-edge form. However, the assumed principal flow directions of the flow model analog, as applied across an array of elements, typically do not align with the floodplain flow streamlines. This paper examines the mathematical underpinnings of a four-direction flow analog using an array of square elements with respect to floodplain flow streamlines that are not in alignment with the analog’s principal flow directions. It is determined that application of Manning’s equation to estimate the friction slope terms of the governing flow equations, in directions that are not coincident with the flow streamlines, may introduce a bias in modeling results, in the form of slight underestimation of flow depths. It is also determined that the maximum theoretical bias, occurs when a single square element is rotated by about 13, and not 45 as would be intuitively thought. The bias as a function of rotation angle for an array of square elements follows approximately the bias for a single square element. For both the theoretical single square element and an array of square elements, the bias as a function of alignment angle follows a relatively constant value from about 5 to about 85, centered at about 45. This bias was first noted about a decade prior to the present paper, and the magnitude of this bias was estimated then to be about 20% at about 10 misalignment. An adjustment of Manning’s n is investigated based on a considered steady state uniform flow problem, but the magnitude of the adjustment (about 20%) is on the order of the magnitude of the accepted ranges of friction factors. For usual cases where random streamline trajectory variability within the floodplain flow is greater than a few degrees from perfect alignment, the apparent bias appears to be implicitly included in the Manning’s n values. It can be concluded that the array of square elements may be applied over the digital terrain model without respect to topographic flow directions.