KENZO 8 8 31 8 1 3 4 5 6 Karman) 7 3 8 x 8 1 1.1.............................. 3 1............................................. 5 1.3................................... 5 1.4 /......................... 6 13 7 13.1.................................................. 7 13........................................... 8 13..1 x........................... 9 13............................................. 1 13..3 ( ).............................. 1 13.3.......................................... 14 ============================ 1
1 7 11.3.3 R e 3 R e x y u v ū, v u v 1 u = ū + u, v = v + v x u y v v u = ū + u, v = v (1.1) F ig.56 u ū y v = v ū ū u O t O x τ r τ l τ t τ r = τ l + τ t τ l Fig.56 x ρv u x ρv (ū + u ) 1 t t ū u v t ρv (ū + u )dt = ρū t = ρ t t t v dt + ρ t t u v dt u v dt ρ u v (1.) ū = 1 t t udt =, 1 t t u dt =, 1 t t v dt = (1.3) 1.τ r τ r = ρ u v (1.4) Reynolds stress u v τ τ = τ + τ r = µ dū dt ρ u v = ρν dū dt ρ u v (1.5) 1 7 N-S Q Q Fig.57 τ r
ρν(dū/dt) ρ u v τ F ig.57 H-P H-P u u max ū Boussinesq τ = ρ u v = ρɛ dū (1.6) 3 τ τ = ρν dū dt ρ u v = ρ(ν + ɛ) dū (1.7) ɛ ν 1.1 u v Prandtl 4 5 mixing length ρu v ( ) dū τ r = ρ u v = ρ l (1.8) l mixing length l u v = l dū (1.9) 6 3 1.6 Boussinesq 4 Ludwig Prandtl 1875-1953) 5 η η = (1/3)mρv l l 6 195 3
F ig.58 y l m(dū/) l ū(y + l m) ū(y) ū(y l m) l m(dū/) ū(y + l m). = ū(y) = ū(y l m). = x u(y) u(y) dū + l m u(y) dū l m τ r τ r x x y ū u(y) l m l m ū(y + l m ) =. u(y) dū + l m ū(y) = u(y) ū(y l m ) =. u(y) dū l m (1.1) y y + l m y y l m y dū ū(y + l m ) ū(y) = l m ū(y l m ) u(y) dū = l m (1.11) y ū(y) u ū ū = 1 { l m dū } + dū l m = l m dū (1.1) y y v ū v = ( ) u dū = ( ) l m (1.13) u v y u v y u v c u v = u v (1.14) u v = c ū v (1.15) 1.8 τ r 1.1 1.15 τ r = ρ u v = ρc ū v ρc lm dū ( ) dū = ρ l (1.16) 1.8l c l m l 7 1.6 τ = ρɛ(dū/) ɛ ɛ = l dū/ ɛ 7 l m Prandtl l 4
1. (P89) l.4 l =.4y l y l = κy (1.17) κ κ =.4 1.8 dū = 1 τr κy ρ (1.18) 1.3 τ r R e < 5 viscous sublayer 8 δ 7 τ w τ w = µ du = µu y (y δ ) (1.19) ρ τ w ρ = ν u y (1.) τ w /ρ [ML T L 3 /M 1 L 3 ] 1/ = [LT 1 ] τ w /ρ = u u y = δ u = u δ R δ u δ = u δ = R δ (1.1) u ν F ig.59 y δ u u δ x 8 5
τ r = τ w 9 τ r ( ) dū τ r = κ ρy dū = 1 τr κy ρ (1.) τ r u 1. dū = u 1 κ y ū = 1 u κ ln y + const 1 ū ln y + const =.33 κ u κ log 1 y + const (1.3) ū u y = 5.75 log u 1 + 5.5 (1.4) ν 1 log law 7 τ τ = µ dū = τ w = const τ w dū = τ w µ = ( ) τw 1 ρ ν = u ν u ū = u ν y + const y = ū = const = ū = u ν y ū = u y u ν wall cordinate y + = yu /ν [L T 1 /L T 1 ] u + = ū u = u y ν = y+ 1.4 / ū/u max u ( y ) 1/n ( = = 1 r 1/n (1.5) u max a a) 9 1 h u/u = 8.48 + 5.75 log 1 y/h by Nikuradse) 6
a y (r n R e = 3 1 3 1 5 n = 7 u = u max (y/a) 1/7 (1.6) 1/7 11 R e = 1 5 τ w ū/u max =.8.8 (1.7) ( ) 1/4 τ w =.ρu ν max (1.8) u max a 8 cm 6cm/sec 1 1 ν =.13cm /sec R e = ūd/ν6 /.13 = 938 3 1 3 < R e < 1 5 1/7 ( ) 1/4 u max = ū/.8 = 6/.8 = 75.cm/sec, τ w =.5ρu ν max =.83kg/m (1.9) u max a 13 13.1 Re = L/ν = / 1 du/ du/ = free stream main stream 1 Boundary layer 1 F ig.6 y x.99 δ(x 1) x 1 x 99% y δ 11 R e 1 7 n = 1/8, 1/9 1 194 7
laminar boundary layerturbulent boundary layer transition region R c = 5 1 5 13 13. t + u x + v y = 1 ( p ) ρ x + ν u x + u y v t + u v x + v v y = 1 ( p ) ρ y + ν v x + v y x + v y = (13.1) x y δ L x u, x L, v y x L v L δ, v x L δ, y δ, u x L, v y Lδ = L L δ u y δ (13.) (1) u () v (3) u (4) v [ ] [ u ν x + u y ν L + L ] L δ [ ] [ v ν x + v δ y ν L L + ] L L δ u x + v y ( L + δ ) L δ = L + L u v x + v v y δ ( L + δ ) L L = δ L L + δ L L (13.3) (13.4) (13.5) (13.6) δ L ( δ L 1) (a) (b) (a) (b), (c) (d) (c) (d) 13.1 t + u x + v y = 1 p ρ x + ν u y p y = x + v y = (13.7) 13 8
14 13.7 ( ρ ū ū ) ū + v = ρ p x y x + τ y τ = µ ū y ρu v p y = ū x + v y = (13.8) 9 L b ρ µ δ δ 13. u x L, ν u y ν δ (13.9) δ L ν δ δ νl (13.1) 15 3 / y y= k y = u = v = u x + v y = 1 ρ p x + ν u x = 1 p ρ x + ν u x u y = 1 p ρν x = 1 p µ x y / y = (1/µ)( p/ x)y + const const = k (... y = / y = k) y = u = u = 1 p µ x y + ky (13.11) 13..1 x y x δ x t = 14 ) 15 13.3 13.4 δ = 5.r νl 9
y u(y) y = δ u y =, y=δ y = y=δ y x p x = p x y=δ u x = 1 p y=δ ρ x y=δ.. p. = ρu x x = ρ y=δ x (13.1) x x (d/dx = ) x dp/dx = 13.1 ( 1 x + p ) = p + 1 ρ ρ = const = p + 1 ρ (13.13) const 3 4. 13.. δ u 99% y δ =.99 δ.99 3 16 17 (1) δ displacement thickness δ F ig.61 y δ u x = δ u Fig.61 16 17 δ.99 31 1
Q δ = Q δ = 1 ( u) (13.14) y > δ u δ () θ momentum thickness θ θ = 1 u( u) (13.15) ρ ρ θ = uρ( u) ρ( u) Fig.61 u ρθ ρθ (3) θ energy thickness θ θ = 1 3 u( u ) (13.16) H shape factor δ θ H=.59 H=1.4 H = δ θ (13.17) adverse pressure gradient H θ H H H=3.5 H=.4 31 1/7 δ θ 1/7 1.6 δ = 1 θ = 1 u = (y/δ) 1/7 { ( y ) } 1/7 ( u) = 1 = δ δ 8 ( y ) { 1/7 ( y ) } 1/7 u( u) = 1 = 7 δ δ 7 δ 11
13..3 ( ) 18 u x + v y = 1 ρ 13.3 y = y = δ p x + ν u y (13.18) x + v y = (13.19) x = 13.18 y = y = δ { u x + v } { } 1 p = y ρ x + ν u y 13.34 1.19 v ( δ y = = 13.34 ) x y = x + { u x + v } = y 13.34 1 13.1 1.19 ν u y = 13.) 1 p ρ x = ν y δ v y = v y=δ (13.) u x u y { u x } x d dx δ + = ν y ν y=δ y = 1 y= ρ τ w u x (13.1) ( u x x d ) = 1 dx ρ τ w (13.) u x x d dx = x (u ) (u) ud x dx + d dx = d [u(u )] ( u) x dx 13.) d dx u( u) + d dx δ θ 13.3 d dx ( θ) + δ d dx = τ w ρ... 18 dθ d + θ dx dx + δ d dx = τ w ρ ( + δ θ dθ dx + ( u) = 1 ρ τ w (13.3) dθ dx + ) ( + δ θ d θ dx = τ w ρ ) θ d dx = τ w ρ (13.4) 1
x = const d/dx = 13.3 τ w = d dx ρu( u) (13.5) 3 (δ y)y u = δ (13.6) δ x dθ dx + ) ( + δ θ d θ dx = τ w ρ x θ θ = 1 (δ y)y = δ u( u) = 1 { 1 τ w ( ) τ w = µ = µ y y= y 13.37 dθ dx = τ w ρ 15 dθ dx = τ w ρ (13.7) (δ y)y δ { (δ y)y δ { } = 3 δ 8 15 δ = 15 δ } (δ y)y δ } (δ y)y δ = µ δ (δ y) y= = µ δ dδ dx = 1 ( ) µ ρ 1 dδ δ 15 dx =... δdδ = 15ν dx δ = 3 νx + const ν, ν = µ/ρ δ x = δ = const = x 3νx δ(x) δ(x) = 33 δ u = a + by + cy τ a =, b = /δ, c = /δ 13.5 τ = ρ d dx y = u = = a y = δ u = = a + bδ + cδ y = δ y = = b + cδ u = [(y/δ) (y/δ) ] u( u) = ρ d ( ) dx 15 δ = dδ ρ 15 dx ( ) du τ = µ = µ y= d [ { ( y ) ( y ) }] = µ δ δ y= δ 13
dδ ρ 15 dx = µ δ... δdδ = 15 ν dx δdδ = 15 ν x dx νx δ = 5.48 τ τ = µ δ =.73ρ ν x 13.3 34 1/7 τ ( 1/4 τ =.5ρ ν δ(x)) 31 13.4 { ) dθ τ = ( dx + + δ θ d θ dx = 7 { dδ 7 dx + 3.86 δ d dx } ρ = } ρ { 7 dδ 7 dx + ( + 7 ) 7 8 7 7 δ } d ρ dx τ ( ) 1/4 ν τ =.5ρ (13.8) δ(x) d/dx = 7 dδ 7 dx =.5ρ ( ) 1/4 ν 7 δ(x) 7 δ 1/4 dδ =.5 x ( ν ) 1/4.. ( ν ) 1/5. δ(x) =.371 x 1/5 (13.9) x 4/5 x 13.3 p/ x = / t = u x + v y = ν u y, 9 δ νx δ x, y ξ, η 19 ξ = x, η = (y/) = y δ νx 19 η = y/δ y/ x + v y = (13.3) 14
ψ = νxf(η) (13.31) f(η) x, y ξ, η ϕ(ξ, η) ϕ(ξ, η) x ϕ(ξ, η) y = ϕ ξ ϕ(ξ, η) y = ξ x + ϕ η η x = ϕ ξ + η x ϕ νx η = ϕ ξ ξ y + ϕ η η y = 1 ( ) ( ) ϕ = 1 y y 4 νx u, v ϕ η x = ξ η x η y = 1 νx η ϕ y y = 1 4 νx η (13.3) u = ψ y = 1 ψ νx η = 1 f (13.33) v = ψ ( x = ξ η ) ψ = 1 ν x η x (f η f) ( x = ξ η x u y = 8νx f η ) f (η) = η 4x f, f y = 4 νx f (13.34) ff + f = (13.35) η y = u = v =, y = u = y = u =, v = η = f =, f = y = u = η = f = f(η) η f(η) = A + A 1 1! η + A! η + A 3 3! η3 (13.36)... f (η) = A 1 + A η + A 3! η + f() = A =, f () = A 1 = f(η) = A! η + A 3 3! η3 + A n n! ηn +, f (η) = A η + A 3! η + + A n f (η) = A + A 3 η + + 13.35 ( ) A A 3 + A 4 η + + A 5 η +! η (n 1)! ηn 1 + A n (n )! ηn +, f (η) = A 3 + A 4 η + A 5! η + + A n (n 3)! ηn 3 ( 4A A 3 + A 6 3! ) η 3 + = A 3 =, A 4 =, A + A 5 =, 4A A 3 + A 6 = ( A 6 = ), A 7 = 15
f(η) f(η) = A! η A = A 1/3 5! η5 + 11A3 8! { (A 1/3 η) 3! η 8 375A4 η 11 + 11! (A1/3 η) 5 5! + 11(A1/3 η) 8 8! [ = A 1/3 Y Y 5 + 11Y 8 ] 11 375Y +! 5! 8! 11! } 375(A1/3 η) 11 + 11! ( Y = A 1/3 η) = A 1/3 F (Y ) (13.37) A ( ) f lim f Y = lim = A /3 η lim Y Y η F (Y ) Y η = f = A /3 lim Y F (Y ) = A = lim Y F (Y ) 3/ 13.37 A = 1.384 (13.38) f (η) = f () = A = 1.384 τ w 13.34 τ w = µ ( ) = y y= 4 νx f () = 4 ( ν ) 1/ ρ νx f () 1.384 =.6641 x (13.39) δ xν δ(x) = 5. (13.4) xν xν δ (x) = 1.7, θ =.664 (13.41) ( ) 8 OK (!?) 9 16