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Various concepts have been proposed or used in the development of Theological models for debris flow. The earliest model developed by Bagnold was based on the concept of the “dispersive” pressure generated by grain collisions. Bagnold's concept appears to be theoretically sound, but his empirical model has been found to be inconsistent with most theoretical models developed from nonNewtonian fluid mechanics. Although the generality of Bagnold's model is still at issue, debrisflow modelers in Japan have generally accepted Takahashi's formulas derived from Bagnold's model. Some efforts have recently been made by theoreticians in nonNewtonian fluid mechanics to modify or improve Bagnold's concept or model. A viable rheological model should consist both of a rateindependent part and a ratedependent part. A generalized viscoplastic fluid GVF model that has both parts as well as two major rheological properties i.e., the normal stress effect and soil yield criterion is shown to be sufficiently accurate, yet practical, for general use in debrisflow modeling. In fact, Bagnold's model is found to be only a particular case of the GVF model. Analytical solutions for steady uniform debris flows in wide channels are obtained from the GVF model based on Bagnold's simplified assumption of constant grain concentration.

Theoretical velocity profile and theoretical pressure and concentration distributions for steady uniform debris flow in wide channels are derived from a generalized viscoplastic fluid GVF model without imposing Bagnold's assumption of constant grain concentration. Good agreement between the theoretical velocity profile and the experimental data of Japanese scientists strongly supports the validity of both the GVF model and the proposed method of solution from the model. It is shown that both Bingham and Bagnold versions or submodels of the GVF model can be used to simulate debris flow at the dynamic state. Although Bagnold's dilatant submodel appears to fit the Japanese data better than the Bingham submodel for flow of noncohesive grains, the choice between them is by no means clearcut. No matter which submodel is used, a full understanding of the major factors that affect the behavior of debris flows is essential prior to its application. This paper thus lays theoretical ground for extensive further studies of such factors.

Two general power formulas, one for hydraulically smooth flows and the other for fully rough flows, are derived in a rational way from the widely accepted logarithmic formulas for the velocity profile and the DarcyWeisbach friction factor. A regression analysis based on the method of least squares is used to determine the valid range of the local velocity or normal distance from the wall in the power formula. Some older empirical formulas, such as Lacey's, Manning's, Blasius', and HazenWilliams', and their valid ranges, are actually explained analytically by the results. Incomplete selfsimilarity of the power law, in which the exponent and the associated coefficient vary with the similarity parameters, such as the Reynolds number and the relative roughness, is elucidated through the parametric representations of the power formulas and their counterparts based on the logarithmic law. This paper examines the concept and rationale behind the power formulation of uniform turbulent shear flows, thereby addressing some critical issues in the modeling of flow resistance based on the power law.

Two approaches are generally taken in deriving the stability limit for the Froude number (Fs)for laminar sheet flow: The first approach uses the OrrSommerfeld equation, while the second uses the crosssectionaveraged equations of continuity and motion. Because both approaches are based on shallowwater theory, the values of Fsobtained from both approaches should be identical, yet in the literature they are not. This suggests that a defect exists in at least one of the two approaches. After examining the governing equations used in both approaches, one finds that the existing crosssectionaveraged equation of motion is dependent on the frame of reference. To correct this defect, one can formulate a frameindependent equation of motion relative to a coordinate system moving with constant velocity, then derive a new expression for Fsthat is generally applicable to both laminar and turbulent flows in prismatic channels of arbitrary crosssectional geometry. For laminar sheet flow, the new expression for Fsobtained from the second approach yields Fs≃0.527, which agrees with that obtained from the first approach.

The theoretical momentum coefficient and energy coefficient for turbulent shear flow in circular pipes and wide channels are derived from the power law, then compared with their counterparts on the basis of the logarithmic law. For such unidirectional flows, the exponent m of the powerlaw velocity distribution is the sole parameter that determines the values of and . A comparison of the corresponding expressions for and derived using both laws shows that their values differ only slightly within a valid range of small m's. Use of a powerlawbased expression along with a powerlaw resistance formula in the crosssectionaveraged equation of momentum cannot only eliminate the theoretical incompatibility that results from mixing logarithmic and power laws in the equation but also remove the inconsistent assumption of β=1made in the equation. Improvement on the accuracy of the flow computation by incorporating a powerlawbased p expression in the equation of momentum is demonstrated through an example in which a new stability limit for the Froude number for turbulent shear flow in wide channels can be accurately derived without assuming β=1.

A new general rainfall intensitydurationfrequency formula is presented, utilizing a method similar to, but more accurate than one previously developed. The previously developed formula was based on the average depthduration ratio of about 40 percent and the mean depthfrequency ratio of 1.48. It is shown that this formula is only a particular form of the writer's more general formulation. The earlier formula, however, requires only the 10yr 1hr rainfall depth instead of the writer's three rainfall depths i.e., 10yr 1hr, 10yr 24hr, and 100yr 1hr. The ratios of 1hr to corresponding 24hr depth and 100yr to corresponding 10yr depth can be computed from the required three rainfall depths in the writer's method so that geographical variation of rainfall can be evaluated in terms of both ratios.

The SchwarzChristoffel transformation formula for flow past an eccentric, normal constriction of zero thickness in a straight channel can be proved to have a simple form of arctangent function from the properties of elliptic integrals. Use of this mapping formula with the aid of another mapping formula between the complex potential and the transformed variable enables one to plot the corresponding flow pattern. Using such transformation formulas, flow graphics for convergent flow are thus developed to plot the flow net and the velocity distributions along the streamlines and across the channel width. To minimize the number of flow patterns in the presentation, the transformation formulas are further normalized by using the channel width and the discharge per unit depth. The normalized flow pattern so plotted is shown to vary only with the two physical parameters, the opening ratio and the eccentricity. Typical convergent flow patterns are plotted to show the validity and applicability of the transformation formulas.