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Unformatted text preview: 1 Lecture 4 – Classification of Flows Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org © André Bakker (20022006) © Fluent Inc. (2002) 2 Classification: fluid flow vs. granular flow • Fluid and solid particles: fluid flow vs. granular flow. • A fluid consists of a large number of individual molecules. These could in principle be modeled as interacting solid particles. • The interaction between adjacent salt grains and adjacent fluid parcels is quite different, however. 3 Reynolds number • The Reynolds number Re is defined as: Re = ρ V L / μ. • Here L is a characteristic length, and V is the velocity. • It is a measure of the ratio between inertial forces and viscous forces. • If Re >> 1 the flow is dominated by inertia. • If Re << 1 the flow is dominated by viscous effects. 4 Effect of Reynolds number Re = 0.05 Re = 10 Re = 200 Re = 3000 5 Newton’s second law • For a solid mass: F = m. a • For a continuum: • Expressed in terms of velocity field u(x,y,z,t). In this form the momentum equation is also called Cauchy’s law of motion. • For an incompressible Newtonian fluid, this becomes: • Here p is the pressure and μ is the dynamic viscosity. In this form, the momentum balance is also called the NavierStokes equation. f t + ∇ = ∇ + ∂ ∂ σ ρ . . u u u acceleration Mass per volume (density) Force per area (stress tensor) Force per volume (body force) u u u u 2 . ∇ + ∇ = ∇ + ∂ ∂ μ ρ p t 6 Scaling the NavierStokes equation • For unsteady, low viscosity flows it is customary to make the pressure dimensionless with ρ V 2 . This results in: 2 * * * * * 2 * * * * * 2 * * . ( , , ) ( / , / , / ) / ; / ; /( / ) 1 . Re p t x y z x L y L z L V p p V t t V L p t ρ μ ρ ∂ + ∇ = ∇ + ∇ ∂ = = = = ∂ + ∇ = ∇ + ∇ ∂ u u u u u u u u u u with dimensionless variables : this becomes : 7 Euler equation • In the limit of Re A• the stress term vanishes: • In dimensional form, with μ = 0, we get the Euler equations: • The flow is then inviscid. p t ∇ = ∇ + ∂ ∂ u u u . ρ * * * * * . p t ∂ + ∇ = ∇ ∂ u u u 8 Scaling the NavierStokes equation  viscous • For steady state, viscous flows it is customary to make the pressure dimensionless with μ V/L. This results in: . Re / / ) / , / , / ( ) , , . * 2 * * 2 * * * * * * * 2 = ∇ + ∇ → ∇ + ∇ = ∇ = = = ∇ + ∇ = ∇ u u u u u u u u u p p L V p p V L z L y L x z y x p : vanishes term convective the Re of limit the In : becomes this ( : variables ess dimensionl with * μ μ ρ 9 NavierStokes and Bernoulli • When: – The flow is steady: – The flow is irrotational: the vorticity – The flow is inviscid: μ = 0 • And using: • We can rewrite the NavierStokes equation: as the Bernoulli equation: u u u u 2 . ∇ + ∇ = ∇ + ∂ ∂ μ ρ p t / =...
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This note was uploaded on 12/04/2010 for the course M MM4CFD taught by Professor N/a during the Fall '10 term at Uni. Nottingham.
 Fall '10
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