Book contents
- Frontmatter
- Contents
- Preface
- Part I Introductory Material
- Part II Kinematics, Dynamics and Rheology
- Part III Waves in Non-Rotating Fluids
- Part IV Waves in Rotating Fluids
- Part V Non-Rotating Flows
- 19 Orientation to One-Dimensional Flow
- 20 Steady Channel Flow
- 21 Unsteady Channel Flow: Hydraulic Shock Waves
- 22 Gravitationally Forced Flows
- 23 A Simple Model of Turbulent Flow
- 24 Some Non-Rotating Turbulent Flows
- Part VI Flows in Rotating Fluids
- Part VII Silicate Flows
- Part VIII Fundaments
22 - Gravitationally Forced Flows
from Part V - Non-Rotating Flows
Published online by Cambridge University Press: 26 October 2017
- Frontmatter
- Contents
- Preface
- Part I Introductory Material
- Part II Kinematics, Dynamics and Rheology
- Part III Waves in Non-Rotating Fluids
- Part IV Waves in Rotating Fluids
- Part V Non-Rotating Flows
- 19 Orientation to One-Dimensional Flow
- 20 Steady Channel Flow
- 21 Unsteady Channel Flow: Hydraulic Shock Waves
- 22 Gravitationally Forced Flows
- 23 A Simple Model of Turbulent Flow
- 24 Some Non-Rotating Turbulent Flows
- Part VI Flows in Rotating Fluids
- Part VII Silicate Flows
- Part VIII Fundaments
Summary
In the previous two chapters, we investigated the nature of flow in a horizontal channel assuming (except for the study of flash floods in § 21.4) resistance to flow is negligibly small, and we were able to classify flows as either sub- or super-critical and to investigate the smooth transition from sub- to super-critical and the abrupt transition from super- to sub-critical. We focused on the flow of water (as opposed to other fluids) because it has a very low value of viscosity and in many cases may be treated as inviscid. However, all real flows involve resistance which tends to degrade the mechanical energy. Typically channeled flows are maintained by the release of gravitational potential energy as the material moves downslope.
In this chapter we consider the one-dimensional flow of a continuous material down a slope. The material might be water in a river, ice in a glacier, snow in an avalanche, debris in a rock slide, molten rock in a lava stream, mud in a landslide, muddy water in a turbidity current, etc. We will formulate the problem fairly generally in § 22.1, then specialize it to specific types of material in § 22.2.
Formulation
Most natural downslope flows, exemplified by rivers and glaciers, are much wider than they are deep and move along beds that are relatively flat. Typically a river is wider than deep by a factor of 100 (e.g., see Yalin, 1992). Also, rivers, glaciers and like flows typically are contained by banks that are raised and relatively steep. It follows that a first approximation of the flow bed is a shallow rectangle and that the velocity is independent of the cross-stream direction to dominant order.
Let's consider flow of material in a straight, flat-bottomed channel (at z=0) of constant width; as previously, the downstream direction is x, the (unimportant) cross-stream direction is y and the (nearly) vertical direction is z. The height of the surface is represented by z = h(x, t). In this chapter we will focus on flows that are steady and invariant in the x direction, having h constant and v = u(z)1x. Our goal is to determine the vertical structure of the flow, u(z), driven by a down-gradient component of gravity.
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- Geophysical Waves and FlowsTheory and Applications in the Atmosphere, Hydrosphere and Geosphere, pp. 218 - 226Publisher: Cambridge University PressPrint publication year: 2017