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
24 - Some Non-Rotating Turbulent 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 this chapter we consider some non-rotating turbulent flows that are a bit more complicated than the flow of water down a slope considered in § 23.6. These include turbulent katabatic winds driven by thermal buoyancy (§ 24.1), avalanches driven by snow suspended in air (§ 24.2) and cumulonimbus clouds driven by the release of latent heat as water vapor condenses (§ 24.3).
Turbulent Katabatic Winds
In § 22.2.5, we investigate the katabatic wind down a slope in the case that the flow is laminar and, using typical parameter values, found that the flow is very likely turbulent rather than laminar. When flow is turbulent, the diffusivity coefficients for momentum and heat are not constant, but instead vary linearly with elevation. In this section, we will revisit the problem formulated in § 22.2.5, but with variable diffusivities, using Reynolds analogy to set the turbulent thermal diffusivity equal to the turbulent diffusivity of momentum.1
The governing equations now are
where z is elevation above the ground, u is the downslope speed, is the dimensionless perturbation temperature, is the reduced gravity, s the down-slope thermal gradient, is the temperature contrast, is the turbulent diffusivity, is a small dimensionless parameter is the velocity scale and is the roughness scale. As before, these equations are to be solved on the domain subject to the conditions u And as before, we can combine the two equations into a single complex equation, although the scalings are somewhat different. In the present case
while the complex equation for W= T ∗ −iu/U is
where is the dimensionless vertical distance and is the scaled boundary roughness, subject to conditions W(0) = 1 and W(∞) = 0.
The problem for the turbulent katabatic winds is a bit more challenging than for the laminar winds because our complex ordinary differential equation now has a variable coefficient. We can get this equation in “standard” form by introducing a new independent variable; let
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- Geophysical Waves and FlowsTheory and Applications in the Atmosphere, Hydrosphere and Geosphere, pp. 242 - 250Publisher: Cambridge University PressPrint publication year: 2017