Singuläre Störungsrechnung und ihre Anwendung in der Aerodynamik. Stefan Braun
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1 Singuläre Störungsrechnung und ihre Anwendung in der Aerodynamik Stefan Braun Institut für Strömungsmechanik und Wärmeübertragung Seminarvortrag TU Graz, 3. Nov. 211
2 method of matched asymptotic expansions ODE example: ε y +y +y =, y() =, y(1) = 1, < ε 1 exact solution: 3 y(x; ε) ε = x 1 boundary layer : y 1, δ 1 : ε δ δ + 1 δ ε
3 ε y + y + y =, y() =, y(1) = 1 outer region: x = O(1), ε (singular perturbation) regular expansion: y(x; ε) = y (x) + εy 1 (x) + O(ε 2 ), y, y 1 = O(1) y + y + ε(y 1 + y 1 + y ) + O(ε2 ) = O(1) : y + y =, y () =, y (1) = 1 y = e e x 3 e 2.5 y(x; ε) 2 inner region: x ε y outer region: x x 1
4 ε y + y + y =, y() =, y(1) = 1 inner region: X = x/ε = O(1), ε (stretching) local expansion: y(x; ε) = Y (X) + εy 1 (X) + O(ε 2 ), Y, Y 1 = O(1) 1 (Y ε + Y ) + Y 1 + Y 1 + Y + O(ε) = O(ε 1 ) : Y + Y =, Y () = Y = c (1 e X ) Van Dyke s matching rule: E { y (εx) + } = E {Y (x/ε) + } E n { }... expand its argument in powers of ε up to and including O(ε n ) keeping x, X fixed E {e e εx } = E {c (1 e x/ε )} e = c
5 ε y + y + y =, y() =, y(1) = 1 inner region: X = x/ε = O(1), ε (stretching) local expansion: y(x; ε) = Y (X) + εy 1 (X) + O(ε 2 ), Y, Y 1 = O(1) 1 (Y ε + Y ) + Y 1 + Y 1 + Y + O(ε) = O(ε 1 ) : Y + Y =, Y () = Y = c (1 e X ) Van Dyke s matching rule: E { y (εx) + } = E {Y (x/ε) + } E n { }... expand its argument in powers of ε up to and including O(ε n ) keeping x, X fixed E {e e εx } = E {c (1 e x/ε )} e = c
6 Sky and Water I, woodcut by M.C. Escher, 1938
7 uniformly valid approximation (composite expansion): y(x; ε) = y +Y e+o(ε) e e x +e (1 e X ) e e (e x e x/ε ) 3 e 2.5 y(x; ε) 2 y Y exact 1.5 composite solution 1.5 ε = x 1
8 Prandtl s hierarchical boundary layer concept LE L inviscid BL TE α ũ, ν D L u =, (u ) u = p + 1 Re u, ũ L Re = = L 1 ν lν far field: u 1, p, no slip condition: u = on P outer region: x, y = O(1), ε = Re 1/2 u = u (x, y) + ε u 1 (x, y) +, p = p (x, y) + ε p 1 (x, y) + inner region: x, Y = y/ε = O(1) u = U (x, Y) + ε U 1 (x, Y) +, p = P (x, Y) + ε P 1 (x, Y) +, v = ε V (x, Y) + ε 2 V 1 (x, Y) +
9 Prandtl s hierarchical boundary layer concept LE L inviscid BL TE α ũ, ν D L u =, (u ) u = p + 1 Re u, ũ L Re = = L 1 ν lν far field: u 1, p, no slip condition: u = on P outer region: x, y = O(1), ε = Re 1/2 u = u (x, y) + ε u 1 (x, y) +, p = p (x, y) + ε p 1 (x, y) + inner region: x, Y = y/ε = O(1) u = U (x, Y) + ε U 1 (x, Y) +, p = P (x, Y) + ε P 1 (x, Y) +, v = ε V (x, Y) + ε 2 V 1 (x, Y) +
10 Prandtl s hierarchical boundary layer concept LE L inviscid BL TE α ũ, ν D L u =, (u ) u = p + 1 Re u, ũ L Re = = L 1 ν lν far field: u 1, p, no slip condition: u = on P outer region: x, y = O(1), ε = Re 1/2 u = u (x, y) + ε u 1 (x, y) +, p = p (x, y) + ε p 1 (x, y) + inner region: x, Y = y/ε = O(1) u = U (x, Y) + ε U 1 (x, Y) +, p = P (x, Y) + ε P 1 (x, Y) +, v = ε V (x, Y) + ε 2 V 1 (x, Y) +
11 outer region: potential flow, determine slip u (x, ) = u w (x) inner region: classical Prandtl s boundary layer U x + V Y =, U U x + V U Y = u wu w + 2 U Y 2 x = : initial cond., Y = : U = V =, U (x, ) = u w (x) boundary layer characteristics: wall shear τ w (x) = U Y displacement thickness δ (x) = b.c. for outer flow correction: v 1 (x, ) = ( Y= 1 U (x, Y) u w (x) ( ) δ (x)u w (x) ) dy
12 outer region: potential flow, determine slip u (x, ) = u w (x) inner region: classical Prandtl s boundary layer U x + V Y =, U U x + V U Y = u wu w + 2 U Y 2 x = : initial cond., Y = : U = V =, U (x, ) = u w (x) boundary layer characteristics: wall shear τ w (x) = U Y displacement thickness δ (x) = b.c. for outer flow correction: v 1 (x, ) = ( Y= 1 U (x, Y) u w (x) ( ) δ (x)u w (x) ) dy
13 hierarchical structure: u, v, p u 1, v 1, p 1 u 2, v 2, p 2 U, V U 1, V 1
14 hierarchical structure: u, v, p u 1, v 1, p 1 u 2, v 2, p 2 U, V U 1, V 1 breakdown I: abrupt change of boundary conditions example 1: trailing edge flow no slip symmetry condition v 1 p 1, u 1 x
15 example 2: shock-boundary layer interaction color streak photo: DLR
16 breakdown II: (regular) adverse pressure gradient p w = u w u w > boundary layer separation at x : τ w (x) a (x x) + b (x x) 2 +, a, b > massive sep. δ marginal sep.: a = τ w x x Goldstein 1948, Stewartson 197, Ruban & Stewartson et al. 1981
17 viscous-inviscid interaction I: Re = ũ L ν drag of finite flat plate triple deck inviscid BL ũ, ν L Ψ y Ψ yx Ψ x Ψ yy = P +Ψ yyy, P(x) = 1 π A (ξ) x ξ dξ, δ δ Re 1/8 A+ c d 1.1 Blasius exp. N.-S. calc. triple deck Re Stewartson & Messiter 1969/7
18 viscous-inviscid interaction I: Re = ũ L ν drag of finite flat plate triple deck inviscid BL ũ, ν L Ψ y Ψ yx Ψ x Ψ yy = P +Ψ yyy, P(x) = 1 π A (ξ) x ξ dξ, δ δ Re 1/8 A+ c d 1 exp. N.-S. calc..1 c d.1 Blasius triple deck Re c turbulent Re.1 1 1e+6 1e+7 1e+8 1e+9 1e+1 Re Stewartson & Messiter 1969/7
19 viscous-inviscid interaction II: massive separation outer (potential) region: Helmholtz-Kirchhoff free streamline flows k =, Brillouin Villat k < k mixing layer curvature: κ(x) = + κ w (x ) + O( x x ) x x p w (x) = k x x + 16 k 2 + O( x x ), 3, x x ±
20 viscous-inviscid interaction II: massive separation outer (potential) region: Helmholtz-Kirchhoff free streamline flows k =, Brillouin Villat k < k mixing layer curvature: κ(x) = + κ w (x ) + O( x x ) x x p w (x) = k x x + 16 k 2 + O( x x ), 3, x x ± interaction region: k = O(Re 1/16 ) Sychev 1972
21 circular cylinder, Re = ũ d ν triple deck laminar k = O(Re 1/16 ) S Sychev 1972 ũ, ν d turbulent k = O(1) S Scheichl, Kluwick & Smith 211 outer/inner interaction
22 viscous-inviscid interaction III: leading edge separation Re = ũ L ν, Re R = Re R L }{{} τ 2 1 c l = α ũ, ν, ρ LE L ρũ 2 L/2 O(α α ), c d = L D L inviscid BL D ρũ 2 L/2 TE O(Re 1/2 )... lam. O(ln 2 Re)... turb.
23 laminar separation bubble Eppler 387 airfoil, α = 2, Re = 1 5 : inviscid u w(x) y U δ BL laminar separation incipient bubble bursting coherent Λ-vortex structure vortex disintegration x smoke flow visualization: Cole, Mueller
24 classical boundary layer theory outer (potential) region: τ = d/ L =.1, α = 5.1 y x 1 c.5 p x c p = p p ρũ 2 O(1) /2
25 inner (viscous) region: 1 τ w α = α c = x.6 τ w = U Y x.6 δ δ = Y= α c 4 α = 3 5 LE ( 1 U ) dy u w (x) v 1 (x, ) = (u w δ ) x
26 marginal separation theory Ruban & Stewartson et al X Re 1/5 (x x ), Γ Re 2/5 (α α c ), x x, α α c A 2 X 2 +Γ = λ X 2 (A h)/ ξ 2 X dξ γ (ξ X) 1/2 (A h)/ T X dξ γ (X ξ) 1/4 v w dξ (X ξ) 1/4 δ δ (x )+Re 1/5 [c 1 A(X, T)+c 2 X]+, τ w Re 1/5 c 3 A(X, T)+ Hackmüller, Kluwick 199: flow control devices h(x, T), v w(x, T) h v w
27 marginal separation theory Ruban & Stewartson et al X Re 1/5 (x x ), Γ Re 2/5 (α α c ), x x, α α c A 2 X 2 +Γ = λ X 2 (A h)/ ξ 2 X dξ γ (ξ X) 1/2 (A h)/ T X dξ γ (X ξ) 1/4 v w dξ (X ξ) 1/4 δ δ (x )+Re 1/5 [c 1 A(X, T)+c 2 X]+, τ w Re 1/5 c 3 A(X, T)+ Hackmüller, Kluwick 199: flow control devices h(x, T), v w(x, T) 6 A(X) Γ = X
28 marginal separation theory Ruban & Stewartson et al X Re 1/5 (x x ), Γ Re 2/5 (α α c ), x x, α α c A 2 X 2 +Γ = λ X 2 (A h)/ ξ 2 X dξ γ (ξ X) 1/2 (A h)/ T X dξ γ (X ξ) 1/4 v w dξ (X ξ) 1/4 δ δ (x )+Re 1/5 [c 1 A(X, T)+c 2 X]+, τ w Re 1/5 c 3 A(X, T)+ Hackmüller, Kluwick 199: flow control devices h(x, T), v w(x, T) 6 A(X) Γ = 3 2 A() Γ c X Γ
29 near critical flows: Γ Γ c perturb. amp. Strouhal-number h c = v wc = A() Γ c Re 1/1 Γ A c b saddle-node bifurcation X A A c (X) + Γ Γ c b(x)c s [2u(t) 1] + [h, v w ] [h c, v wc ](X) + Γ Γ c [h 1, v w1 ](X, t) Γ Γ c
30 flow control: suction slot optimization suction rate: V = LV = const v wc V L V X c X 6 Γ5.5 c V = 1 V = L V = 2 L = X c 4
31 near critical perturbed 2D flow Γ < Γ c : du dt = u u 2 + a sin(ωt) 2 u(t) 1 a = (1.44, 1.5) 1.6 a c(ω) parametric resonance -1-2 sin(2t) t.2 a c(ω ) t ω
32 finite time blow up 2 v w 1 X v w T s T 4 A(X, T) A(X, ) = A(X; Γ = 2) T = 7-8 X s X blow-up asymptotics: Smith 1982 A(X, T) ε 2/3Â(ˆX) +, X X s = ε 4/9 ˆX, ε = T s T unique blow up profile Â(ˆX)! Scheichl, Braun, Kluwick 28
33 finite time blow up 2 v w 1 X v w T s T 4 A(X, T) A(X, ) = A(X; Γ = 2) T = 7-1 X s -4 ε 2/3 A(X, T) T = 7 Â(ˆX) X ε 4/9 (X X s) blow-up asymptotics: Smith 1982 A(X, T) ε 2/3Â(ˆX) +, X X s = ε 4/9 ˆX, ε = T s T unique blow up profile Â(ˆX)! Scheichl, Braun, Kluwick 28
34 bypass transition - asymptotic view marginal separation inviscid triple deck stage Euler stage turbulent b.l. boundary layer
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