Published online by Cambridge University Press: 05 June 2012
These are possible answers to the problems set at the end of each chapter in An Introduction to Ocean Turbulence. Alternative or simpler answers may be possible in some cases. Please advise the author if any errors are found.
Chapter 1
P1.1. Integrating over the area of the tube, the net flow is ∫0au ⋅ 2π r. dr = 4π U ∫ (r − r3/a2)dr = 4π U[r2/2 − r4/(4a2)]0a=π Ua2. By definition, this must be equal to the mean flow times the cross-sectional area, π a2, so U is equal to the mean flow. By conservation of the volume flux, the mean flow downstream of the transition from laminar to turbulent flow must also be equal to U.
The flux of kinetic energy upstream of the transition from laminar to turbulent flow is u(ρ u2/2) integrated over the cross-section of the tube, i.e., ∫0a (ρ u3/2)⋅ 2π r dr = 8π ρ U3∫0ar(1 − r2/a2)3 dr = − π ρ U3a2[(1 − r2/a2)4]0a = π ρ U3a2.
In the same way, the flux of the kinetic energy of the mean flow is (ρ U3/2)π a2 within the turbulent flow downstream of the transition, so the reduction in the flux is the difference between the flux upstream of the transition and that downstream, or (ρ U3/2) π a2. This represents a flux of kinetic energy to the turbulent motion, ignoring any work done by pressure forces.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.