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
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
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
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
Boundary layer transition to turbulence At a certain distance along a plate, viscous forces become to small relative to inertial forces to damp fluctuations
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
B L thickness in laminar region & fluid properties Blasius solution viscosity Re x U O x UOx 5x x 5 Re x u O
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
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
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
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(10-5 10-6 ) 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
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
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
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)
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(?)=0.395-0.415 (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
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
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)
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 = 0.664 1/2ρ(U 0 ) Re 1/2 cf = 0.027 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)
figure_09_07
QUESTIONS?
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 0.664 Re X f ln 2 0.455 0.06Re X 0.027 1/ 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 2 0.523 1520 Re 0.06Re L L 0.032 1/ 7 Re L Re L U O L