1 G. Ahmadi ME 639-Turbulence G. Ahmadi ME 639-Turbulence Outline Outline 4 4 Double Diffusive Convection Double Diffusive Convection 4 4 Thermal Convection Thermal Convection 4 4 Isotropic Turbulence Isotropic Turbulence 4 4 Bifurcation Bifurcation 4 4 Turbulence Turbulence G. Ahmadi ME 639-Turbulence S. A. Abu S. A. Abu - - Zaid Zaid and G. Ahmadi, and G. Ahmadi, Appl Appl . Math. Modeling, Vol. 13 (1989) . Math. Modeling, Vol. 13 (1989) 1 T 2 T 1 S 2 S Fluid Layer heat from below with a Fluid Layer heat from below with a salt concentration gradient salt concentration gradient G. Ahmadi ME 639-Turbulence ( ) ( ) ( ) u u u u 2 0 0 t 0 g p p ∇ ρν + ρ − ρ + − −∇ = ∇ ⋅ + ∂ ρ T T T 2 T t ∇ κ = ∇ ⋅ + ∂ u S S S 2 S t ∇ κ = ∇ ⋅ + ∂ u 0 = ⋅ ∇ u Governing Equations Governing Equations ( ) ( ) ( ) 0 0 0 S S T T 1 − β + − α − ρ = ρ
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G. AhmadiME 639-Turbulence G. AhmadiME 639-Turbulence
CountablyCountably Infinite Bifurcation Infinite Bifurcation of of NavierNavier--Stokes EquationStokes Equation
TurbulenceTurbulence
After Four Bifurcation After Four Bifurcation Solutions to Solutions to NavierNavier--Stokes Stokes Equation Become ChaoticEquation Become Chaotic
RuelleRuelle--TakensTakens
G. AhmadiME 639-Turbulence
G.I. Taylor & von G.I. Taylor & von KarmanKarman (1937)(1937)
““Turbulence is an irregular motion Turbulence is an irregular motion which in general makes its appearance which in general makes its appearance in fluids, gaseous or liquid, when they in fluids, gaseous or liquid, when they flow past solid surfaces or even when flow past solid surfaces or even when neighboring streams of the same fluid neighboring streams of the same fluid flow past or over one another.”flow past or over one another.”
G. AhmadiME 639-Turbulence
HinzeHinze (1959)(1959)
““Turbulent fluid motion is an irregular Turbulent fluid motion is an irregular condition of flow in which the various condition of flow in which the various quantities show a random variation quantities show a random variation with time and space coordinates, so with time and space coordinates, so that statistically distinct average values that statistically distinct average values can be discerned.”can be discerned.”
