CP502 Advanced Fluid Mechanics Flow of Viscous Fluids and Boundary Layer Flow [ 10 Lectures + 3 Tutorials ] Computational Fluid dynamics (CFD) project Midsemester (open book) examination
What do we mean by ‘Fluid’? Physically: liquids or gases Mathematically: A vector field u (represents the fluid velocity) A scalar field p (represents the fluid pressure) fluid density (d) and fluid viscosity (v) R. Shanthini 18 Aug 2010
Recalling vector operations Del Operator: Laplacian Operator: Gradient: Vector Gradient: Divergence: Directional Derivative: R. Shanthini 18 Aug 2010
Continuity equation for incompressible (constant density) flow - derived from conservation of mass where u is the velocity vector u, v, w are velocities in x, y, and z directions R. Shanthini 18 Aug 2010
kinematic viscosity (constant) Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ kinematic viscosity (constant) density (constant) pressure external force (such as gravity) R. Shanthini 18 Aug 2010
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ ρ υ R. Shanthini 18 Aug 2010
Acceleration term: change of velocity with time Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ Acceleration term: change of velocity with time R. Shanthini 18 Aug 2010
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ Advection term: force exerted on a particle of fluid by the other particles of fluid surrounding it R. Shanthini 18 Aug 2010
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ viscosity (constant) controlled velocity diffusion term: (this term describes how fluid motion is damped) Highly viscous fluids stick together (honey) Low-viscosity fluids flow freely (air) R. Shanthini 18 Aug 2010
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ Pressure term: Fluid flows in the direction of largest change in pressure R. Shanthini 18 Aug 2010
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ Body force term: external forces that act on the fluid (such as gravity, electromagnetic, etc.) R. Shanthini 18 Aug 2010
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ change in velocity with time body force = advection + diffusion + pressure + R. Shanthini 18 Aug 2010
Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid ρ υ R. Shanthini 18 Aug 2010
Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid in Cartesian coordinates Continuity: Navier-Stokes: x - component: y - component: z - component: R. Shanthini 18 Aug 2010
Steady, incompressible flow of Newtonian fluid in an infinite channel with stationery plates - fully developed plane Poiseuille flow Fixed plate Fluid flow direction h x y Steady, incompressible flow of Newtonian fluid in an infinite channel with one plate moving at uniform velocity - fully developed plane Couette flow Fixed plate Moving plate h x y Fluid flow direction R. Shanthini 18 Aug 2010
Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid in cylindrical coordinates Continuity: Navier-Stokes: Radial component: Tangential component: Axial component: R. Shanthini 18 Aug 2010
Steady, incompressible flow of Newtonian fluid in a pipe - fully developed pipe Poisuille flow Fixed pipe 2a φ r z Fluid flow direction 2a R. Shanthini 18 Aug 2010
Steady, incompressible flow of Newtonian fluid between a stationary outer cylinder and a rotating inner cylinder - fully developed pipe Couette flow φ aΩ a b r R. Shanthini 18 Aug 2010