DAY 19: Boundary Layer
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1 DAY 19: Boundary Layer flat plate : let us neglect the shape of the leading edge for now flat plate boundary layer: in blue we highlight the region of the flow where velocity is influenced by the presence of the solid surface
2 Boundary layer velocity profile Far from the surface, the fluid velocity is unaffected. In a thin region near the surface, the velocity is reduced Layer of fluid in the proximity of a boundary (condition) Zoom: in this layer of fluid we observe a velocity profile slow growth: d/dx << d/dy v << u
3 Boundary layer growth x The free stream velocity u 0 is undisturbed far from the plate but next to the plate, the flow is reduced by drag Farther in x along the plate, the effect of the drag is felt by a larger region of the stream (viscous effects), and because of this the boundary layer grows Fluid friction on the surface is associated with velocity reduction along the boundary layer o du dy y0
4 Local stress & total force, skin friction o du dy y0 This is different from the case of a Couette flow, where the gradient is defined by the two boundary conditions (thin film approx.) We need to find o y0 du/dy decreases with x & y (x) F s L x0 Bdx And there is more trouble
5 Boundary layer transition to turbulence At a certain distance along a plate, viscous forces become to small relative to inertial forces to damp fluctuations
6 Picture of boundary layer from text thickness of the boundary layer defined such to include 99% of the velocity variation ox du( x) dy y0 BLACKBOARD the laminar boundary layer 19A,B,C,D note that as du/dy decreases in x the shear stress decreases as well Goal : keep laminar regime on the airfoil, to reduce drag. 98%..so far, so what is the problem? CONTROL
7 B L thickness in laminar region & fluid properties Blasius solution viscosity Re x U O x UOx 5x x 5 Re x u O
8 Boundary layer transition How can we solve problems for such a complex system? δ(x) y0 We can think about key parameters and possible dimensionless numbers Important parameters: Viscosity μ, density ρ Distance, x Velocity u O Re u uox Reynolds number combines these into one number x note: increase in Cf implies a sudden growth in the BL O x
9 What is turbulence? turbulence is a state of fluid motion where the velocity field is : highly 3D, varying in space and time, hardly predictable, non Gaussian, anisotropic but somehow statistically organized coherent structures
10 The mean velocity profile in the smooth wall turbulent boundary layer : 1) viscous sublayer u = τ 0 y μ τ = μ du dy the velocity varies linearly, as a Couette flow (moving upper wall). Thus, the shear stress is constant: τ 0
11 scaling near wall turbulence We can define a velocity scale u* = τ ρ [m/s] characteristic of near wall turbulence u* = shear velocity or friction velocity we can rewrite the linear profile in the viscous sublayer as υ u u = yu υ where is a length scale (very small, remember υ u =O( ) m 2 /s, while u* is a fraction (~5-10%) of the undisturbed velocity U 0 δ boundary layer height we already have 2 velocity scales: 1) u* 2) U 0 How many length scale? 1) υ u 2) δ
12 viscous sublayer continued How thick is the viscous sublayer? it depends on the boundary layer... yes/no? as u* and υ define the viscous length scale, we can represent the extension of the viscous sublayer in terms of multiples (5-10) of the viscous scale (viscous wall units) δ υ = 5 υ u Note that as u* δ υ : the viscous sublayer becomes thinner Note: roughness protrusion (fixed physical scale) may emerge from the viscous sublayer and change the near wall structure of the flow δ υ
13 The mean velocity profile in the smooth wall turbulent boundary layer : 2) the logarithmic region here is another velocity scale standard deviation or r.m.s. velocity velocity scale of the energy containing eddies The mixing length theory: fluid particles with a certain momentum are displaced throughout the boundary layer by vertical velocity fluctuation. This generate the so called Reynolds stresses τ = ρu v think about the complication as compared to LAMINAR case τ = μ du dy
14 If we know the stress, we can obtain by integration the velocity profile τ = ρu v mixing length assumption (Prandtl: u = l du dy ) What does it mean? A displaced fluid parcel (towards a faster moving fluid) will induce a negative velocity u ~ v such that τ = ρu v = ρl 2 du/dy 2 l represent the scale of the eddy responsible for such fluctuation very important: we also assume that the size of the eddies l varies with the height l=ky : very reasonable, farther from the wall eddies are larger (attached eddy)
15 we thus have τ = ρk 2 y 2 du/dy 2 with u* = τ ρ integrating we obtain : u u = 1 k ln yu υ +C Logarithmic law of the wall!!! where u* depends on the flow and the surface k is the von Karman constant(?)= (k=0.41 is a good number) C is the smooth wall constant(?) of integration (C=5.5 is a good number) note that is a rough wall boundary layer = 1 ln u k where y 0 is the aerodynamic roughness length: it is a measure of aerodynamic roughness, not geometrical (surface) roughness u y y0 relating with y 0 is complicate
16 The mean velocity profile: where is it valid? from about 60 viscous wall units to about 15% of he boundary layer height it makes sense that the extension of the log layer has to be determined by both inner scaling and outer scaling
17 Laminar flow : only viscous friction Turbulent flow : small viscous friction as compared to momentum transfer by eddies τ = μ du dy τ = ρu v However at the small scales at any instant, viscosity still matters (cannot be neglected)
18 Laminar and Turbulent BL Analytical results BL growth Empirical results δ x = 5x 0.16x Re1/2 δ x = Re 1/7 shear stress coefficient assuming a 1/7 power law velocity distribution u/u 0 = (y/ δ) 1/7 cf = τ 0 = /2ρ(U 0 ) Re 1/2 cf = Re 1/7 and many others Re=xU 0 / υ as the distance x increases cf decreases Note that a different set of formula exist for the full plate (averaged over the length L)
19 figure_09_07
20 QUESTIONS?
21 Laminar, Turbulence, Induced Turbulence Laminar Turbulent Induced δ(x) 5x Re X 0.16x 1/ 7 Re X 0.16x 1/ 7 Re X c f 0 x O U 2 O 2 U 2 2 O c Re X f ln Re X / 7 Re X F S L x0 Bdx Area * U o 2 2 C f Re x C f F S BLU U O x O 2 2 U O x 1.33 Re L ln Re 0.06Re L L / 7 Re L Re L U O L
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