Boundary Layers Flashcards
What is a boundary layer?
Layer around ball with flow flowing around it where flow cannot be, of width 𝛿
What is the equation for the boundary layer 𝛿?
𝛿/d = sqrt(μ/ρv0d) = 1/(Re^1/2), where d is the diameter of the ball and v0 is the velocity of the flow
What is the pressure on the surface of the ball like at different points?
On left side at 270, P> at 315 > at 360
What is dP/dx greater/less tan upstream/downstream?
dP/dx < 0 upstream, >0 downstream
What does the upstream pressure gradient do?
Acts to accelerate the flow along surface in direction of streamlines. Called favourable pressure gradient
What does the downstream pressure gradient do?
Opposes the flow: adverse pressure gradient.
What is the N-S equation near the boundary of a slab?
(v.∇)v(x) = -1/ρ *dP/dx + μ/ρ *d^2v(x)/dy^2, at all y=0, v= 0, so d^2v(x)/dy^2 = 1/μ *dP/dx
What do we assume in this N-S equation and what do we do then?
Assume that dP/dx = A, so can then integrate twice to get an equation, then use boundary conditions to find the constants
How does v(x) vary with the constant A?
If A<0, v(x) is +ve, if A>0, v(x) is negative if A >= 2v0/𝛿^2
What does the graph of v(x) against x look like for favourable pressure and for adverse pressure?
Favourable pressure: starts at 0 and curves upwards. Adverse pressure: starts at 0, goes negative then curls back round to positive and then curves upwards
What is the separation point?
The point at which the flow may be halted or reversed due to adverse pressure. Can happen when A = 2v0/𝛿
What is it called when there is a separation point?
Boundary separation.
What does the flow diagram with boundary separation look like?
Normal flow but with 1 stagnation point at left side of ball, and then 2 lung shaped flows on right side of all, with separation points x(s) at top and bottom of ball
When does turbulence develop?
If energy at wavenumber k = 2π/λ is E(k), then turbulence exhibits cascade (cascade to higher k. In high Re flow boundary separation leads to turbulence downstream of the separation.
For potential flow, what is the pressure at θ = 0 and θ = π on the balls surface?
P(θ = 0) = P0 +1/2 ρv^2, P(θ = π) = P0 +1/2 ρv^2
For real flow, what is the pressure at θ = 0 and θ = π on the balls surface?
P(θ = 0) = P0, P(θ = π) = P0 +1/2 ρv^2
What does the pressure at θ = 0 and θ = π for real flow suggest?
That there is a net force the right: in cylinder frame is a net drag.
Does laminar flow give drag?
No
What does a region of adverse pressure downstream lead to?
A stagnation point and boundary separation
What does the diagram of irrotational flow look like and what does this mean?
Simple flow above and below with 2 stagnation points. Surface pressure fully recovers downstream, so no drag.
With boundary separation and turbulence, what does the diagram of flow look like?
Flow above and below, one stagnation point and then turbulence lines coming from top and bottom of ball and following flow.
With boundary separation and turbulence, what is the pressure like at different points on the ball?
Pa > Pb, Pc -> drag, where a is on left of ball, b is on top and c is on right
What is the equation for force on surface area S of the ball?
F = b*ρv^2/2 *S, where b is the drag coefficient
What is the first way to reduce drag?
Narrower region of turbulence reduces drag. Do this by delaying the separation by streamlining the flow.
What is the second way to reduce drag?
Make the boundary layer turbulent. Turbulent eddies have a net effect of recirculating flow into boundary layer. This delays boundary separation so x(s) moves further downstream (e.g. gold ball with dimples).
What is the drag crisis?
Where the Reynolds number Re = ρvD/μ increases and there is a sudden drop in drag coefficient - as v increases suddenly the drag drops.
What does the graph of drag coefficient against Re look like for a smooth sphere?
Straight then sudden drop in drag at certain Re
What is the explanation for the drag crisis?
As Re increases the boundary layer thickness reduces. Shear stresses increases and boundary layer becomes turbulent. Separation points move further downstream and reduce drag.
