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Unformatted text preview: ENU 4133 – Introduction to Turbulence March 7, 2011 Topics I Time Averaging I Reynolds Decomposition and RANS (Reynolds Averaged NavierStokes) I Law of the Wall and Velocity Profiles I Kolmogorov hypotheses and scales I Mixing length modeling of turbulence I Multiequation models of turbulence ( k , k ω ) I Computational techniques for turbulence (including large eddy simulation – LES & direct numerical simulation – DNS) Example Fluctuations Sufficient time for u and p to converge – couple/few seconds (White recommends 5). TimeAverage Operator f = 1 T Z T fdt (1) ~ V = 1 T Z T ~ Vdt (2) p = 1 T Z T pdt (3) Rules ( f , g scalar/vector/tensor functions, h scalar variable) f = f (4) f + g = f + g (5) f g = f g (6) fg 6 = f g (7) ∂ f ∂ h = ∂ f ∂ h (8) Reynolds Decomposition Fluctuations ~ V and p : ~ V ( ~ r , t ) = ~ V ( ~ r , t ) ~ V (9) p ( ~ r , t ) = p ( ~ r , t ) p (10) Mean fluctuation: ~ V = 1 T Z T ~ V ~ V dt (11) ~ V = ~ V ~ V (12) ~ V = ~ V ~ V = 0 (13) However, u v , u 2 , u p are not zero. Mean Flow Equation – RANS Homework problem: use Reynolds decomposition for p and u to obtain Equation 621 in the text, which is the scalar equation for u ρ D u Dt = ∂ p ∂ x + ρ g x + ∂ ∂ x μ ∂ u ∂ x ρ u 2 + ∂ ∂ y μ ∂ u ∂ y ρ u v + ∂ ∂ z μ ∂ u ∂ z ρ u w (14) Appearance of “turbulent stress” terms (turbulent convective acceleration) with no analogy in the regular NS equation. These stresses must be modeled to solve equations for turbulent flow. Duct & Boundary Layer Case ρ D u Dt ≈  ∂ p ∂ x + ρ g x + ∂ ∂ y μ ∂ u ∂ y ρ u v (15) τ = μ ∂ u ∂ y ρ u v (16) τ = τ lam + τ turb (17) τ = τ viscous + τ turb (18) τ turb vs. τ lam Dimensional Analysis for Wall Layer Velocity Profile u = u ( μ,τ w ,ρ, y ) (19) Note: ignores variation in τ with y for duct flow. Essentially, assumes that y << L, a characteristic length of the flow . Select μ , τ w , and ρ as repeating variables to obtain: u √ ρ √ τ w = F y √ τ w ρ μ (20) u √ ρ √ τ w = F y √ τ w ν √ ρ (21) u u ? = F yu ? ν (22) u + = F ( y + ) (23) u ? : friction/shear velocity. u + : universal velocity. Data Regions of Behavior Very near the wall y + < 5: u + = y + (24) Viscous (laminar) profile – viscous sublayer . Overlap layer y + > 30: u + = A ln y + + B (25) u + = 1 κ ln y + + B (26) Log layer – k : von K´ arm´ an constant. Value of the Theodore von K´ arm´ an Constant Must be obtained from experimental data. Results range from 0.38 to 0.43. The results of 0.40 and 0.41 are the most common. 0.41 is the most common value for singlephase, internal flows; 0.40 is more common for twophase flows and external flows. T&K go with 0.40, White with 0.41....
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 Spring '11
 Schubring
 Fluid Dynamics, Kinetic Energy, large eddies

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