Prandtl mixing length (1) Presentation

The mixing length theory is a turbulence theory developed by Ludwig Prandtl (Prandtl 1925). Considering a shear flow, the fluid motion is characterised a velocity gradient in a direction normal to the mean flow direction. In the shear flow, some momentum (i.e. momentum per unit volume=ρ×V) is transferred from the region of high velocity to that of low-velocity. The fluid tends to resist the shear associated with the transfer of momentum, and the shear stress (within the fluid) is proportional to the rate of transfer of momentum. For large shear stresses, the fluid cannot sustain the viscous shear stress and turbulence develops across the entire shear flow. In turbulent flows, the momentum exchange coefficient νT and the shear rate ∂V/∂y may be related by introducing the concept of mixing length lm. The mixing length lm is the characteristic distance travelled by a particle of fluid before its momentum is changed by the new environment. At a solid boundary (e.g. y=0 for a boundary layer), the velocity is zero (no slip condition) and the fluid shear stress equals the boundary shear stress τo. The boundary shear stress τo represents the shear force per unit area transferred between the fluid and the solid boundary. The Prandtl mixing length theory is a fundamental concept for turbulent boundary layer and shear flows in applied fluid mechanics and fluid dynamics. Although the mixing length theory is more than 80 years old, it is an useful model based upon some simple physically-based concept. A basic understanding of the physical turbulent processes is critical in many real-word applications and discussed in a number of relevant Youtube video movies in the same Playlist at: {   / @hubert_chanson  } Fluid mechanics and hydraulics in Hubert Chanson Youtube channel {   / @hubert_chanson  } Applied hydrodynamics [Playlist] Advanced hydraulics of open channel flows [Playlist] Fundamentals of open channel hydraulics [Playlist] Environmental hydraulics of open channel flows [Playlist] Streamlines {   • Streamlines  } Reynolds number {   • Reynolds number  } Laminar and turbulent flows {   • Laminar and turbulent flows  } Boundary layer (1) Presentation Acknowledgements Professor Colin J. APELT References PRANDTL, L. (1925). "Über die ausgebildete Turbulenz." ('On Fully Developed Turbulence.').A.M.M., Vol. 5, pp. 136-139 (in German). PRANDTL, L. (1952). "Essentials of Fluid Dynamics with Applications to Hydraulics, Aeronautics, Meteorology and Other Subjects." Blackie & Son, London, UK, 452 pages. SCHLICHTING, H. (1979). "Boundary Layer Theory." McGraw-Hill, New York, USA, 7th edition. CHANSON, H. (2014). "Applied Hydrodynamics: An Introduction." CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 448 pages & 21 video movies (ISBN 978-1-138-00093-3). VOGEL-PRANDTL, J. (2014). "Ludwig Prandtl. A Personal Biography Drawn from Memories and Correspondence." Göttinger Klassiker der Strömungsmechanik, Universitätsverlag Göttingen, Germany, 263 pages (ISBN: 978-3-86395-160-3).