Navier Stokes Equation

Lecture Notes 23 Navier-Stokes Equation – dimensional form − ∂ 2 vi Dv ∂p +η + fi = ρ i ∂xi ∂x j∂x j Dt Assume: Characteristic velocity v0 Characteristic length L Characteristic stress ηv0 / L Characteristic time v0 / L Choose non-dimensional variables v ' = v / v0 x ' = x / L p ' = v L p t' = 0 t ηv0 L or v = v ' v0 x = x ' L p = ∂ 1 ∂ = ∂x L ∂x ' f '= 2 L f ηv0 ηv0 L p' t ' = L t' v0 ∂ v0 ∂ = ∂t L ∂t ' Substitute into Navier-Stokes equation − ∂ 2 vi ' v Dvi ' 1 ηv0 ∂p ' 1 1 + 2 v0η + 2 v0η f i ' = ρ v0 0 L L ∂xi ' L L Dt ' ∂x j '∂x j ' L − ∂ 2 vi ' ρ v L Dvi ' ∂p ' + + fi ' = 0 η Dt ' ∂xi ' ∂x j '∂x j ' or where ρ v0 L = Re η ⇒ Re gives importance of inertial terms relative to viscous terms. Re 1 viscous forces ≈ balance acceleration negligible Re 1 inertia dominates Note: in dimensionless form, Re is the only parameter in the Navier-Stokes equation. ⇒ for given geometry (boundary conditions) ALL equivalent (non-dimensional) problems at same Re give same result! Examples: 1. Low Reynolds number flow past a cylinder. Re 1 Symmetry, like in the sphere problem. Figure 23.1 2. Re = 10 ∂vi = 0 (steady) ∂t vj ∂vi ≠0 ∂x j Asymmetry; eddies in wade. Figure 23.2 The figure below is experimental. 102 CYLINDER 10 CD 1 10-1 10-1 1 10 102 103 104 105 106 107 Re Figure 23.3. Variation of drag coefficient with Reynolds number for circular cylinder. Figure 23.3 Re < 1 CD ∼ 1 / Re D∼V 102 < Re < 3⋅105 CD ∼ const. D ∼ V2 Re ∼ 3 ⋅105 big drop in CD! Recall Earth’s mantle: Re ∼ 10-19 canoe: Re ∼ 2 ⋅105 Summary: For low Re, inertia not important Navier-Stokes equation linear “simple” results (analytic theory) For high Re, inertia important Navier-Stokes equation nonlinear time dependent complicated – experimental approach → empirical