2 (Preface) (potential flow) viscosty ( 0 (vorticity) (boundary layer) (shearing stress) (frictional stress) (frictional drag)) (laminar flow) (turbul

Transcription

1 Viscous Fluid Dynamics Yoshiaki NAKAMURA Professor of Chubu University Emeritus Professor of Nagoya University September 1, 2014

2 2 (Preface) (potential flow) viscosty ( 0 (vorticity) (boundary layer) (shearing stress) (frictional stress) (frictional drag)) (laminar flow) (turbulent flow) (separation) (Navier-Stokes) (CFD: Computational Fluid Dynamics) (Reynolds number) (eddy) (turbulence model) (eddy viscosity)

3

4 i

5

6 (viscosity) (potential flow) (shearing stress) 0 (governing equation) (potential equation; (Euler equation; Navier-Stokes equation; (Navier- Stokes equation) (continuum fluid) (Boltzmann equation) (Rarefied gas flow) ( ) 0 (specular reflection) (tangential direction to the wall surface) (diffuse reflection) ( )

7 (Newton-Stokes) (Newtonian fluid; ) (τ) τ = µ u (1.1) y u x u/ y (strain rate) (Non-Newtonian fluid) y v τ yx u x τ yx 1.1: (molecular viscosity) µ (oil) (gas dynamics theory) µ (1845; George Gabriel Stokes) [ ] /[ ] kg m/sec 2 m/sec [µ] = [τ]/[ u/ y] = m 2 = kg/(m sec) m (SI) : N sec/m 2 P a sec (P a:); 10 poise CGS : poise ( )(= g/cm sec) 2 (µ) kg/(m sec)

8 µ (Sutherland; 1893 ) ( ) 3/2 ( ) µ T T0 + S = µ 0 T 0 T + S µ (T ) µ 0 T 0 T 0 = 273[K], S = 111, µ 0 = [kg/(sec m)] Maxwell Rayleigh (dilute gas) power law µ µ 0 = (1.2) ( ) n T (1.3) n 0.5 < n < 1 µ 0 = [kg/(sec m)], T 0 = 273K, n = 2/3 T (kinematic viscosity) µ ρ ν ν µ ρ (1.4) [ν] [ν] = [µ] kg/(m sec) = [ρ] kg/m 3 = m 2 /sec ( ) kinematics ( ) 20 (ν) m 2 /s air , 760mmHg 1 (P a) P a = kP a = MP a(= hp a) 1cal = J (ratio of specific heat) (water) 1.2

9 : ρ kg/m 3 µ kg/s m ν m 2 /s k kcal/s m K = kj/(s m K) C p kcal/(kg K) = 1.004kJ/(kg K) () C v kcal/(kg K) = 0.716kJ/(kg K) γ = C p /C v : ρ kg/m 3 µ kg/s m ν m 2 /s k J/(s m K) = W/m K = kj/(s m K) C p kj/(kg K) () C v * kcal/(kg K) γ = C p /C v (stress)σ ij i, j i = 1, 2, 3, j = 1, 2, 3 x, y, z σ ij 2 (pressure)p (viscous stress)τ ij p (isotropy) p p p 1.2: p σ 11 σ 12 σ 13 p 0 0 τ 11 τ 12 τ 13 σ 21 σ 22 σ 23 = 0 p 0 + τ 21 τ 22 τ 23 (1.5) σ 31 σ 32 σ p τ 31 τ 32 τ 33

10 ( ) n (a tensor of order n) 3 n = 1 (scalar) = 3 (1.5) ( ) τ ij = λ div u δ ij + 2µ 1 ( ui + u ) j 2 x j x i λ δ ij δ ij = { 1 i = j 0 otherwise τ ij (u 1, u 2, u 3 ) (u 1, u 2, u 3 ) 3 (x 1, x 2, x 3 ) (x 1, x 2, x 3 ) x, y, z (x 1, x 2, x 3 ) = (x, y, z) τ 11 = τ xx, τ 12 = τ xy, (1.6) (1.7) u 2 x 2 x 3 o x 1 u 1 u 3 1.3: (x 1, x 2, x 3 ) (u 1, u 2, u 3 ) τ 11 τ 12 τ 13 τ 21 τ 22 τ 23 τ 31 τ 32 τ 33 = λ div u + 2µ u 1 2µ 1 ( u1 + u ) 2 2µ 1 ( u1 + u ) 3 x 1 2 x 2 x 1 2 x 3 x 1 2µ 1 ( u2 + u ) 1 λ div u + 2µ u 2 2µ 1 ( u2 + u ) 3 2 x 1 x 2 x 2 2 x 3 x 2 2µ 1 ( u3 + u ) 1 2µ 1 ( u3 + u ) 2 λ div u + 2µ u 3 2 x 1 x 3 2 x 2 x 3 x 3 (1.8) 1.6 (strain rate) S ij τ ij = λ div u δ ij + 2µ S ij (1.9)

11 6 1 S ij S ij = 1 2 ( ui + u ) j x j x i ( trace ) (1.10) Στ ii = 3λdiv u + 2µΣ u i x i = (3λ + 2µ) div u 0 λ 3 (x, y, z ) Στ ij = (λ + 23 ) 3 µ div u (λ + 23 ) µ bulk viscosity 0 Stokes (Stokes hypothesis) λ = 2 3 µ (1.11) 1.3 homogeneous (translation) 2 (rotation) 3 extensional strain 4 shear strain 5 (dilatation) S ij S xy = S yx S ij = S xx S xy S xz S yx S yy S yz S zx S zy S zz (1.12)

12 S ij S ij = 1 ( ui + u ) j 2 x j x i (1.13) S ij Ω ij Ω ij = 1 ( ui u ) j (1.14) 2 x j x i Ω ij = Ω ji (1.15) (antisymmetric tensor) (diagonal elements) 0 Ω 11 = Ω 22 = Ω 33 = 0 (1.16) S xx = u x }{{} x, S yy = v y }{{} y, S zz = w z }{{} z (1.17) div v = 0 u x + v y + w z = 0 (1.18) (1.17) 0 dilatation S xx + S yy + S zz = 0 (1.19) dilatation = u x + v y + w z (1.20) div v = (Shearing-strain rate) S xy = 1 ( v 2 x + u ), S yz = 1 y 2 ( w y + v z ), S zx = 1 ( u 2 z + w ) x (1.21)

13 8 1 3 (I 1, I 2. I 3 ) (invariant) (x, y, z) I 1 = trace S = S xx + S yy + S zz (1.22) I 2 = 1 2 [(trace S)2 trace S 2 ] = S xx S yy + S yy S zz + S zz S xx Sxy 2 Syz 2 Szx 2 (1.23) S xx S xy S xz I 3 = det S = S yx S yy S yz (1.24) S zx S zy S zz I 1 S 1.4 (Navier-Stokes equations) Navier(Claude Louis Marie Henri Navier) 1826 (1827?) Stokes(George Gabriel Stokes) Navier 1845 Navier molecular flow Stokes continuum flow x u(t, x, y, z) t x, y, z (x, y, z) t u u(t + δt, x + δx, y + δy, z + δz) = u + u t δt + u x δx + u y u δy + δz (1.25) z δx, δy, δz ( ) δx = u δt, δy = v δt, δz = w δt (1.26) Lagrange Lagrange 1.25 u(t + δt, x + δx, y + δy, z + δz) = u + u t δt + u x uδt + u y u vδt + wδt (1.27) z u u t t 0 Du Dt = lim u u t 0 t = u t }{{} + u u x + v u y + w u z }{{} (1.28) 2 D/Dt substantial derivative (Lagrange derivative) 1.28) (Euler) ( ) 2 Langrange Euler

14 (t = 0) u u = u( X, t) X t = 0 du/dt vortex method u=u( X, t 2 ) u=u( X, 0 ) t=t 2 x = X t=0 t=t 1 1.4: ( ) Lagrange (x, y, z) (x, y, z) u u = u ( x, t) ρ Dρ Dt = ρ + v grad ρ }{{} t }{{} (1.29) grad = v v = (u, v, w) 2 (1.28) 2 ma = F ( ) (1.30) x ρ Du ( Dt = X + σxx x + τ yx y + τ ) zx z (1.31)

15 10 1 ƒ y ƒ z ƒ x 1.5: y z ρ Dv ( Dt = Y + τxy x + σ yy y + τ ) zy z ρ Dw ( Dt = Z + τxz x + τ yz y + σ ) zz z (1.32) (1.33) y t yx s xx s xx x t yx 1.6: ρ (X, Y, Z) m ρ ρ D v Dt = X + σ (non-conservative form of the dynamic equation) (1.34) σ σ = σ xx τ xy τ xz τ yx σ yy τ yz τ zx τ zy σ zz (1.35) σ xx = p + λ div v {( }}{ u x + v y + w ) +2µ u z x (1.36)

16 ( u σ yy = p + λ σ zz = p + λ x + v y + w ) z ( u x + v y + w z + 2µ v y ) + 2µ w z (1.37) (1.38) λ Stokes λ = (2/3)µ 1.35 ( v τ xy = τ yx = µ x + u ) = 2µ 1 ( v y 2 x + u ) = 2µ S xy (1.39) y ( w τ yz = τ zy = µ y + v ) = 2µ 1 ( w z 2 y + v ) = 2µ S yz (1.40) z ( u τ zx = τ xz = µ z + w ) = 2µ 1 ( u x 2 z + w ) = 2µ S zx (1.41) x ρ Du = X p Dt x + [ ( µ 2 u x x [ ( u µ y y + v )] + x z ρ Dv = Y p Dt y + [ ( µ 2 v y y [ ( v µ z z + w )] + y x ρ Dw = Z p Dt z + [ ( µ 2 w z z x [ µ ( w x + u z )] + z )] div v [ ( w µ x + u )] z )] div v [ ( u µ y + v )] x )] div v [ ( v µ z + w )] y (1.42) (1.43) (1.44) x y z 1.42 ( u ρ t + u u x + v u y + w u ) ({}}{ ρ + u z t + ρu x + ρv y + ρw ) z = ρu t + ρuu x + ρuv y + ρuw z (1.45) t (ρ u) + div (ρ u u) = X + div σ (the momentum equation) (1.46) div σ (divergence)

17 12 1 div σ = σ = ( i σ ij ) = 1 σ σ σ 31 1 σ σ σ 32 1 σ σ σ 33 (1, 2, 3) (x, y, z) i x i (1.47) ; viscous terms Euler Leonhard Euler 1757 div v = 0 µ = const ρ Du Dt ρ Dv Dt ρ Dw Dt 2 = (x, y, z) = X p x + µ 2 u (1.48) = Y p y + µ 2 v (1.49) = Z p z + µ 2 w (1.50) 2 u = 2 u x u y u z 2 (1.51) ( ) (1.48) ( ) PHILOSOPHIA NATURALIS PRINCIPIA MATHEMATICA Mathematical Principles of Natural Philosophy; (Isaac Newton, ) 1687 Part 1 Part 2 Part 3 ( ) (continuity equation) div = ρ + div (ρ v) = 0 (the continuity equation) (1.52) t ρ t + x (ρu) + y (ρv) + (ρw) = 0 (1.53) z

18 ρ t + u ρ x + v ρ y + w ρ + ρ div v = 0 (1.54) z ρ + v grad ρ + ρ div v = 0 (1.55) t Dρ + ρ div v = 0 (1.56) Dt 1.7: ( ) ( ρu + ρu ) ( y ρu y + ρv + ρv ) x ρv x = 0 x y ρu x + ρv y = 0 (1.57) t t t = 1 ( ρu + ρu ) ( y ρu y + ρv + ρv ) ( x ρv x + ρ + ρ ) x y ρ x y = 0 x y t ρ t + ρu x + ρv y = 0 (1.58) ( ) ρ (1.56) div v = div v = 0 v = 0 (1.59)

19 (the equation of state) (thermally perfect gas) p = RT, or pv = RT (the state equation) (1.60) ρ 1.3: v = 1 /ρ : (specific volume) R : R = R 0 /M (M: ) R 0 : gas constant R 0 = [J/(K mol)] M = 29 R = 287[m 2 /s 2 K] = 287[J/kg K] M = 29 g/mol kg M = kg/mol k b N b = mol R 0 ( ) k b = R o N b = J/K (1.61) pv = nr 0 T (1.62) n (the number of moles) R 0 p = 1 atm T = 0 C 1 (n = 1) V = 22.4 l (1.62) R 0 = 8.314J/(K mol) (1.63) V M ρ ρ = M V (1.64) (1.62) p M ρ = nr 0T p ρ = nr 0 M T (1.65) n M M(molecular weight) n = M M (1.66)

20 (1.65) p ρ = M M R 0 M T p ρ = R 0 M T (1.67) MKS M g/mol MKS (kg) R 0 M R 0 M 10 3 R M (1.63) R = R 0 J/(K mol) = 8314 M 10 3 kg/mol M J kg K (1.68) (1.69) (energy equation) ρ De i Dt ( + p div v = k T ) + ( k T ) + ( k T ) +µφ (1.70) x x y y z z }{{} e i k Φ { ( u ) 2 ( ) 2 ( ) } 2 ( v w v Φ = x y z x + u ) 2 ( w + y y + v ) 2 ( u + z z + w ) 2 x 2 ( u 3 x + v y + w ) 2 (1.71) z µφ (i, j) u µφ = τ xx x + τ u yx y + τ v xy x + τ v yy y 3 i = 1, 2, 3, j = 1, 2, 3 (1.72) µφ = τ ij u i x j (1.73) ( ) (dissipation) µφ = µ[2(s S S 2 33) + (2S 23 ) 2 + (2S 31 ) 2 + (2S 12 ) 2 ] + λ(s 11 + S 22 + S 33 ) 2 (1.74)

21 16 1 S ij 1.13 (1.70) (1.70) 2 x ( u ρ t + u u x + v u ) y y ( v ρ t + u v x + v v ) y = p x + x τ xx + y τ xy (1.75) = p y + x τ xy + y τ yy (1.76) (1.75) u (1.76) v ( u 2 + v 2 ρ + u u 2 + v 2 + v u 2 + v 2 ) t 2 x 2 y 2 = u p x v p y + u x τ xx + u y τ xy + v x τ xy + v y τ yy u2 + v 2 2 u 2 + v 2 2 ρ t + u2 + v 2 2 x ρu + u2 + v 2 2 (1.77) ρv = 0 (1.78) y (1.77) (1.78) ( u 2 + v 2 ) ρ + ( u 2 + v 2 ) ρu + ( u 2 + v 2 ) ρv t 2 x 2 y 2 = u p x v p y + u x τ xx + u y τ xy + v x τ xy + v y τ yy (1.79) 1 (the first law of thermodynamics) δq = δe i + pδv (1.80) ( δq δe i pδv v per unit volume δq { ( δq = k T ) + ( k T )} x x y y (1.81) 1.80 { = x ρ De i Dt }{{} ( k T x ) + ( k T )} y y ( u + ( p) x + v ) u + τ xx y x + τ u yx y + τ v xy x + τ v yy y (1.82)

22 p ( u ( p) x + v ) u + τ xx y x + τ u yx y + τ v xy x + τ v yy y = ( p + τ xx ) u x + ( p + τ yy) v y + τ u yx y + τ v xy x u = σ xx x + σ v yy y + τ u yx y + τ v xy x = σ v (1.83) σ (1.82) ρ De i Dt = q + σ v (1.84) q q = k T (1.85) 1.82 ρ De ( i Dt = ρ ei t + u e i x + v e ) i y (1.86) e i (1.86) (1.82) (1.82) ρ e i t + e i x ρu + e i ρv = 0 (1.87) y t ρe i + x (ρue i) + y (ρve i) (1.88) t ρe i + x ρue i + y ρve i ( u = div (k grad T ) p x + v ) u + τ xx y x + τ u yx y + τ v xy x + τ v yy y (1.89) (1.79) (1.89) { u 2 + v 2 } ρ + ρe i + { u 2 + v 2 t 2 x 2 } ρu + ρue i + { u 2 + v 2 } ρv + ρve i y 2 = div (k grad T ) x pu y pv + x uτ xx + y uτ yx + x vτ xy + y vτ yy (1.90) ( ) } 1 t 2 ρ u2 + ρe i + div {ρ u u2 2 + ρ ue i = div (k grad T ) div p u + div ( τ u) (1.91)

23 18 1 τ τ u ( τ u = τ xx τ xy τ yx τ yy ) ( ) ( ) u τ xx u + τ xy v = v τ yx u + τ yy v (1.92) ( ) B a ( ) b 11 a 1 + b 12 a 2 B a = (b ji a i ) = b 21 a 1 + b 22 a 2 (1.93) ( ) 1.91 ( ) 1 t 2 ρ u2 + ρe i + div { ( ) } u 2 ρ u 2 + h k grad T τ u = 0 (1.94) h (static enthalpy), h = e i + p ρ (1.95) total energy e t total enthalpy H t e t = 1 2 ρ u2 + ρe i, 1.94 H t = u2 2 + h (1.96) t e t + div (ρ uh t k grad T τ u) = 0 (the energy equation) (1.97) σ p H t τ (u, v, w, p, ρ, T, µ) 7 7 (x, y, z ) ( ) (1.91) (non-conservative form) ρ D ( ) u 2 Dt 2 + e i = div( q + σ u) (1.98) σ σ = pi + τ (1.99) I (unit matrix) τ

24 (1.34) u X = 0 v = u ρ D ( ) u 2 = u div σ (1.100) Dt 2 (1.98) (1.100) σ (1.99) ρ De i Dt = div( q + σ u) u div σ = σ u div q (1.101) ρ De i Dt + p div u = τ u div q (1.102) (1.84) σ p ρ Dρ + div ρ u = + ρ div u = 0 (1.103) t Dt div u = 1 Dρ ρ Dt = ρdv Dt (1.104) v specific volume v = 1/ρ (1.104) (1.102) ( ) Dei ρ Dt + pdv = τ u div q (1.105) Dt (1.105) T ds = de i + pdv (1.106) ρt DS Dt = τ u div q (entropy equation) (1.107) 1.5 control volume (divergence form) ρ + div (ρ u) = 0 (continuity equation) (1.108) t

25 20 1 t (ρ u) + div (ρ u u σ) = f (momentum equation) (1.109) div div = f σ ρ u u tensor product ρu 1 u 1 ρu 1 u 2 ρu 1 u 3 ρ u u = ρu 2 u 1 ρu 2 u 2 ρu 2 u 3 (1.110) ρu 3 u 1 ρu 3 u 2 ρu 3 u 3 [ ρ t ( 1 2 u u + e i )] [ ( ) ] 1 + div ρ u 2 u u + e i σ u + q = f u (energy euation) σ p (1.111) 1.8: V control volume V ( 1.8) (divergence theorem) div w dv = V div T dv = V A A n w da (1.112) n T da (1.113) A V n w T 2 { } ρ + div (ρ u) dv = t 2 V ρ t dv + div (ρ u) dv = 0 (1.114) ρ t dv + n (ρ u) da = 0 (1.115) A

26 { } V t (ρ u) + div (ρ u u σ) dv V fdv = 0 (1.116) V ρ u dv + n (ρ u u σ) da t A f dv = 0 (1.117) σ V { ( )} { ( ) } 1 1 ρ t 2 u u + e i dv + n ρ u A 2 u u + e i σ u + q da = f udv V (1.118) σ p Leibnitz Leibnitz f d f f ( r, t) dv = dt t dv + n u A f da (1.119) V A u A V A da u A = 0 f t dv = d f ( r, t) dv (1.120) dt V f/ t Leibnitz f V [ ] d ρ dv n ρ ( u A u) da = 0 (1.121) dt V A [ ] d ρ u dv n {( u A u) ρ u + σ} da = dt V A V f dv (1.122)

27 : [ ( ) ] { ( ) } d 1 1 ρ dt V 2 u u + e i dv n ρ ( u A u) A 2 u u + e i + σ u q da = f u dv (1.123) v T = a b a b T ij = a i b j ( ) ( n n σ ij σ x n 1 σ 11 σ 12 σ 13 n 1 σ 11 + n 2 σ 21 + n 3 σ 31 σ = σ y = n σ = n 2 σ 21 σ 22 σ 23 = n 1 σ 12 + n 2 σ 22 + n 3 σ 32 (1.124) σ z n 3 σ 31 σ 32 σ 33 n 1 σ 13 + n 2 σ 23 + n 3 σ 33 n = (n 1, n 2, n 3 ) σ σ n σ = σ n σ 11 σ 12 σ 13 σ = σ n = σ 21 σ 22 σ 23 n 1 n 2 = σ 11 n 1 + σ 12 n 2 + σ 13 n 3 σ 21 n 1 + σ 22 n 2 + σ 23 n 3 (1.125) σ 31 σ 32 σ 33 n 3 σ 31 n 1 + σ 32 n 2 + σ 33 n 3 σ = σ ij n j e i e i i ( ) u A n ( u A u) u A

28 Leibniz d dα = h(α) g(α) h(α) g(α) f (x, α) dx f (x, α) α dx + f [h(α), α] dh(α) dα dg(α) f [g(α), α] dα (1.126) ρ DV [ ] V Dt = ρ + (V ) V = p + µ 2 V (1.127) t D/Dt (substantial derivative) V (1.42) (1.44) (a pesudo-vector expression [ ( V V 2 ) ] ρ t + V ( V ) = p + µ 2 V (1.128) 2 (A B) = (A )B + (B )A + A rotb + B rota (1.129) A = B = V rot rot = ω ω = V [ ( V V 2 ) ] ρ t + V ω = p + µ 2 V (1.130) 2 2 V 2 V = ( V ) ( V ) = (divv ) ω (1.131)

29 (1.130) rot ω t (V ω) = ν 2 ω (1.132) = 0 ω (1.132) ) Helmholtz ω = V (1.133) (V ω) = (ω )V (V )ω (1.134) ω t (ω ) V + (V ) ω = ν 2 ω (1.135) D ω Dt = ω t + ( V ) ω (1.136) D ω Dt = ( ω ) V }{{} +ν 2 ω (1.137) 0 ω ( ) (cascade phenomenon) ( ) (Helicity) v ω (scalar product) H = v ω dv (1.138) H NS

30 : (cylindrical coodinates)(r, θ, z) (v r, v θ, v z ) a) (radial direction) [ vr ρ t + v v r r = p r + µ r + v θ r [ 2 v r r 2 ] v r θ + v v r z z v2 θ r + 1 r b) (circumferential direction) [ vθ ρ t + v v θ r = 1 r r + v θ r p θ + µ [ 2 v θ r 2 v r r v r r 2 θ v r z 2 v r r 2 2 v θ r 2 θ v θ θ + v v θ z z + v ] rv θ r + 1 r c) (axial direction) [ vz ρ t + v v z r = p z + µ r + v θ r [ 2 v z r 2 v θ r v θ r 2 θ v θ z v r r 2 θ v θ r 2 v z θ + v z + 1 r ] v z z v z r v z r 2 θ v z z 2 ] ] ] (1.139) (1.140) (1.141) ( ) (r, θ, z) (x, r, θ) (1.141) z x

31 26 1 v = 1 r r (rv r) + 1 v θ r θ + v z z = v r r + v r r + 1 v θ r θ + v z z ( ) = r, 1 r θ, z (1.142) (1.143) θ rδθ 2 2 = = 2 r r r r 2 θ z 2 = 1 ( r ) + 1r 2 r r r 2 θ z 2 (1.144) (vorticity) ω = rot v = v (1.145) ω r = 1 v z r θ v θ z ω θ = v r z v z r ω z = v θ r + v θ r 1 r v r θ = 1 r r (rv θ) 1 v r r θ () ( ) ( ) (1.146) r v t + (v ) v = 1 ρ p + ν 2 v (1.147) (cylindrical coodinates) (1.139) (1.141) a) v r t + (v ) v r v2 θ r = 1 ( p ρ r + ν 2 v r v r r 2 2 ) v θ r 2 θ (v ) = u r r + u θ r θ + u z z D/Dt Dv r Dt v2 θ r = 1 ( p ρ r + ν 2 v r v r r 2 2 ) v θ r 2 θ (1.148) (1.149) (1.150) D Dt = t + (v ) = t + v r r + v θ r θ + v z z (1.151)

32 b) θ v θ t + (v ) v θ + v rv θ r = Dv θ Dt + v rv θ r = 1 ρr ( p θ + ν 2 v θ + 2 v r r 2 θ v ) θ r 2 (1.152) c) z v z t + (v ) v z = Dv z Dt = 1 ρ p z + ν 2 v z (1.153) v = 1 r r (rv r) + 1 v θ r θ + v z z = 0 (1.154) ( 1) (advection trem) (v ) v v = v r e r + v θ e θ + v z e z (1.155) e r, e θ, e z r, θ, z (1.147) ( (v ) v = v r r + v ) θ r θ + v z (v r e r + v θ e θ + v z e z ) z = v r v r r e r + v 2 r / r e r + v θ v r r θ e r + v θv r r / θ e v r r +v z }{{} z e r + v z v r z e r + r, θ, z (e r, e θ, e z ) (x, y, z) e r r = 0, e r θ = e e θ θ, θ = e e z r, θ = 0, e r / θ ( 1.11 ) e θ e r z = 0 (1.156) r = r 2 r 1 r θ e θ r θ r e θ r θ r e θ θ e r = e θ (1.157) e θ θ e r (1.148) {(v ) v} r = (v ) v r v2 θ r (1.152) ( ) (1.158)

33 : e z O 1.12: ( 2) ( ) 1 (v ) v = ( v) v + 2 v2 (1.159) ( ) 2 v = ( v) ( v) (1.160) ( ) ω = v = e r r + e θ r θ + e z (v r e r + v θ e θ + v z e z ) (1.161) z 1.156

34 ( ) (1.161) ω = v = 1 r e r re θ e z r θ z v r rv θ v z (1.162) S 11 = S rr = v r r S 22 = S θθ = 1 v θ r θ + v r r S 33 = S zz = v z z S 12 = S rθ = 1 ( 1 v r v θ 2 r θ r v ) θ r S 13 = S rz = 1 ( vr 2 z + v ) z r S 23 = S θz = 1 ( 1 v z 2 r θ + v ) θ z (1.163) (1.164) (1.165) (1.166) (1.167) (1.168) τ ij = 2µS ij (1.169) θ (1.140) [ vθ ρ t + v v θ r r + v θ v θ r θ + v v θ z z + v rv θ r = 1 r p θ + µ [ 2 v θ r r ] v θ r v θ r 2 θ v θ z v r r 2 θ v θ r 2 ] (1.170) (θ ) v θ (swirling flow) θ (axisymmetric approximation) (z ) 0 θ = 0, (1.170) ρ v ( θ 2 t = µ v θ r v θ r r v ) θ r 2 r ρ rv θ t z = 0, v r = 0, v z = 0 (1.171) ( = µ r 2 v θ r 2 + v θ r v ) θ r (1.172) (1.173)

35 30 1 r ρ { t r (rv θ) = µ r 3 v θ r v θ r v θ r 2 1 v θ r r + v } θ r 2 r ρ 1 t r r (rv θ) }{{} ζ { 3 v θ = µ r v θ r r 2 1 v θ r 2 r + v } θ r 3 (1.174) (1.175) z ζ = ω z ζ 1 r r (rv θ) 1 v r r θ = 1 r r (rv θ) (1.176) (1.175) ζ 1.176) ζ r 2 ζ = ( vθ r r r + v ) θ r 2 ζ r 2 = 3 v θ r v θ r r 2 = 3 v θ r v θ r r 2 = 2 v θ r r v θ r v θ r 2 1 r 2 v θ r 1 r 2 v θ r + 2 r 3 v θ 2 r 2 v θ r + 2 r 3 v θ (1.177) 2 ζ r ζ r r = 3 v θ r v θ r r 2 1 v θ r 2 r + 1 r 3 v θ (1.178) ζ ρ ( 2 t ζ = µ ζ r ) ζ r r ( ζ ) (1.179) ( ) 1.137) 1.135) 1.7 ( ) (1.179) ζ t = ν 1 ( ( r ζ )) r r r ν ν = µ/ρ (1.180) ) ζ = C 0 t 1 exp ( r2 4νt v θ v θ = Γ ( )) 0 1 exp ( r2 2πr 4νt (1.180) (1.181) (1.182)

36 Γ 0 = C 0 π Γ 0 (1.182) Γ(= 2πrv θ ) )) Γ = 2πrv θ = Γ 0 (1 exp ( r2 4νt ( ) ( ) (1.183) 1.8 (r, θ, ϕ) (spherical coordinates) r θ (z ) ϕ ((x, y) ) 1.13: (r, θ, ϕ) v = 1 r 2 r (r2 v r ) + 1 r sin θ θ (v θ sin θ) + 1 v ϕ r sin θ ϕ ( ) (1.184) (v r, v θ, v ϕ ) (r, θ, ϕ) = r e r + ( r θ e θ + z e z (1.185) D Dt = t + v r r + v θ r θ + v ϕ r sin θ ϕ (1.186) µ

37 32 1 [ Dvr ρ Dt v2 θ + ] v2 r r = p r + µ θ [ Dv θ ρ Dt + v rvθ 2 v2 ϕ cot θ ] = 1 r r [ 2 v r 2v r r 2 2 v θ r 2 θ 2v θ cot θ r 2 [ p θ + µ 2 v θ + 2 v r r 2 θ 2 r 2 sin θ v θ r 2 sin 2 θ 2 cos θ r 2 sin 2 θ ] v ϕ ϕ ] v ϕ ϕ (1.187) (1.188) ϕ [ Dvϕ ρ Dt + v ] ϕv r + v θ v ϕ cot θ r 2 2 = 1 r 2 r = 1 [ p r sin θ ϕ + µ 2 v ϕ ( r 2 ) + r 1 r 2 sin θ v ϕ r 2 sin 2 ϕ + 2 v r r 2 sin 2 θ ( sin θ ) + θ θ 1 r 2 sin 2 θ ϕ + 2 cos θ ] v θ r 2 sin 2 θ ϕ (1.189) 2 ϕ 2 (1.190)

38 (external flow) (internal flow) (x, y) 2 x y ( u ρ t + u u x + v u ) = p y x + µ 2 u (2.1) ( v ρ t + u v x + v v ) = p y y + µ 2 v (2.2) 2 2 = 2 x y 2 (2.3) = ( / x, / y) = ρ (u, v) (x, y) p µ x x = 0 (2.4) 2) t = 0 (2.5) (steady flow) 0 (unsteday flow) u x + v y = 0 (2.6)

39 34 2 y=h y=0 2.1: 1) 1 u x = 0 (2.7) 2.6 y v y = 0 (2.8) v = const. (2.9) v = 0 y y v (Couette Flow) Maurice Couette ( ) (x ) U (boundary conditions) h y = 0 u = 0, y = h u = U (2.10) x (2.1) 0 = p x + µ 2 u y 2 (2.11) v = 0 y (2.2) 0 = p y (2.12) p y y p = p(x) (2.13)

40 (2.11) p x = u µ 2 y 2 (2.14) x 2.7y (2.15) y C 1 y dp dx = u µd2 = const. (2.15) dy2 dp dx y = µdu dy + C 1 (2.16) dp y 2 dx 2 = µu + C 1y + C 2 (2.17) C 2 (2.10) y = 0 C 2 = 0 y = h C 1 = dp h dx 2 µu (2.18) h C 1, C 2 (2.18) (2.17) x u u = U y ( h + h2 dp ) y ( 1 y ) 2µ dx h h (2.19) 2 dp dx = 0 u = U h y (2.20) τ τ = µ u y = µu h (2.21) y x F x = 0 [ ] (Poiseuille Flow) y = ±b u = 0 (2.22)

41 36 2 y=b y=-b y=0 u=u(y) 2.2: (2.22) (2.17) C 1 C 2 C 1 = 0, C 2 = b2 2 dp dx (2.23) (2.17) x u u = 1 ( dp ) (b 2 y 2) ( ) (2.24) 2µ dx Q u y b ( Q = u dy = 2b3 dp ) 3µ dx Q u b u = 3Q 4b { ( y ) } 2 1 b ( y ) } 2 u = u max {1 b (2.25) (2.26) (2.27) (Hagen-Poiseuille flow) 2 Hagen 1839 v θ = 0, θ = 0 (2.28) (1.141) ( d 2 u µ dr ) du = dp r dr dx (2.29)

42 r R x u 2.3: z x 0 r = 0 du = 0 dr r = R u = 0 (2.30) (2.29) (2.30) u(r) = 1 ( dp ) (R 2 r 2) (2.31) 4µ dx dp = const. (2.32) dx dp dx = p 1 p 2 (2.33) l l p 1 p 2 r = 0 ( u max = R2 dp ) 4µ dx Q (2.34) Q = = = R 0 R 0 ( 1 = πr4 8µ u(r) 2πr dr 1 dp ( R 2 r 2) 2πr dr 4µ dx ) [ dp R 2 4µ dx 2π r 2 ] R r ( dp ) dx (2.35)

43 38 2 µ ū ū = 1 πr 2 Q = 1 ( dp ) R 2 (2.36) 8µ dx ( ) (2.36) p/ x ū ū p/ x Reynolds ū ( p/ x) 1/1.7 ū ( p/ x) 0.59 ( ) f(darcy friction factor) dp dx = f ρv 2 D 2 (2.37) V ( p 1 πr 2 p 2 πr 2 (2.38) τ w (2πR) L (2.39) L 2 p 1 πr 2 p 2 πr 2 = τ w (2πR) L (2.40) p L = τ 2 w R p = p 1 p 2 (2.41) 2.37) 2.42) dp dx = τ 2 w R f ρv 2 2 = τ w D 2 R (2.42) (2.43) f = 8τ w ρv 2 (2.44)

44 (2.31) r r = R µ f = 64 Re D (2.45) Re D Re D = V D ν (2.46) f = 0.316Re 1/ Re 10 5 (2.47) f = Re Re (2.48) ( ) L L = Re D (2.49) L = D (2.50) 2.2 ( ) (r, θ, z) z t r (z ζ ( ζ 2 t = ν ζ r ) ζ = ν 1 ( r ζ ) r r r r r (2.51) ν (1.137) 2 (1.137) ( ) 0 z ζ ζ = 1 r r (ru θ) 1 u r r θ u r ζ (2.52) ζ = u θ r + u θ r = 1 r r (ru θ) (2.53)

45 40 2 r ξ θ η ξ = 1 r u z θ u θ z, η = u r z u z r (2.54) 2 QŽ Q xƒä ªzŽ² É W 2.4: u θ [ 1 exp u θ = Γ 0 2πr ( r2 4νt )] (2.55) Oseen(1912) Γ 0 (r ) (circulation) ( ) (2.55) p(r) Γ Γ = v d r = 2πru θ (2.56) ( ) v θ t Γ ( Γ 2 t = ν Γ r 2 1 ) Γ r r (2.57) ( ) (2.57) (2.51) 2πr r

46 (2.57) Γ = Γ 0 {1 exp( η 2 )} (2.58) Γ η η η 2 = r2 4νt η = r 2 νt (2.59) η ( ) Γ v θ ζ ( ) (2.57) 2.3 (2.53) (2.58) ζ ζ = Γ ( ) 0 r 2 4πνt exp 4νt (2.60) ζ η ζ Stokes Γ r ( )] r 2 Γ = ζ 2πr dr = Γ 0 [1 exp (2.61) 4νt t 0 Γ 0 δ(r) (vortex filament) ( ) 2.5: (2.55) u θ 2.6 ν ( 2.5) t ( 2.6) η ν t

47 Circumeferential velocity Radius r t=1000 t=2000 t=3000 t= : 0 ν = 0 0 r > 0 : = 0 u θ = Γ 0 2πr r = 0 : = () (2.62) diffusion u θ ( u θ 2 u θ = ν t r u θ r r u ) θ r 2 Γ 2.57 u θ = u θ r = 2 u θ r 2 = Γ ( 2πr 1 Γ r Γ r 2 ) 1 2π r ( 1 2 Γ r r 2 1 Γ r 2 r 1 Γ r 2 r + 2 r 3 Γ ) 1 2π (2.63) 2 u θ r r u θ r u θ r 2 = 1 2 Γ r r 2 1 Γ r 2 r + 1 r 3 Γ Γ r 3 = 1 2 Γ r r 2 1 Γ r 2 r = 1 ( 2 Γ r r 2 1 ) Γ r r (2.64) 2.57 Γ(r, 0) = Γ 0 t = 0 Γ(0, t) = 0 t > 0 ) (2.65)

48 (similarity solution) η = r/ νt 2.55 r 4νt (solid rotation) ω u θ = ωr = Γ 0 8πνt r (2.66) r u θ Γ 0 2πr ( ) Taylor ( ) 1 r 2 v θ = ωr exp 2 ( ) 1 r ζ = ω(2 r 2 2 ) exp 2 (2.67) (2.68) (2.69) v θ ζ ω Taylor r ω (induced velocity) (Biot-Savart law) l l dl dl x Γ P (x, y, z) l u p (x, y, z) = Γ 4π r d l P r d l ( x ) r 3 (2.70) r = x x, r = r (2.71) B = µ 0I 4π r d s r 3 (2.72) B (W b/m 2 ) I s d s µ µ 0 I = Γ (2.73)

49 44 2 x' P (x,y,z) r v dl Q (x',y',z') ƒ 2.7: ( ) 2.70 vortex method r i r i (ξ, t) t = 1 4π L j=1 ( ri r j ) ( r j / ξ )s( r i r j, σ i, σ j )dξ r i r j 3 (2.74) r i, r j i, j s(y, σ i, σ j ) ( ) s(y, σ i, σ j ) = σ (core) 1 [ (σ 2 i + σ2 j )/2y2 ] 3/2 (2.75) (angular velocity vorticity) ω v θ ζ ζ = 1 r ζ ω 2 v θ = ωr (2.76) r (rv θ) = 1 ( ωr 2 ) = 1 2ωr = 2ω (2.77) r r r (ζ) = 2 (ω) (2.78)

50 ρ v2 θ r = p r (2.79) v ƒæ (a) (b) 2.8: ( a v θ = ωr r < a (2.80) v θ = ωa2 r (2.80) (2.79) r (2.81) (2.79) r r p 0 r > a (2.81) p r = ρω2 r p = ρω 2 r2 2 + p c (2.82) p r = ρω2 a 4 1 r 3 p = ρω2 a r 2 (2.83) 2 r = a( ) p p c p c = ρω 2 a 2 (2.84) p = ρω 2 r2 2 ρω2 a 2 (0 < r < a) (2.85) p = ρω2 a r 2 (a < r) (2.86)

51 : 2.3 U τ x ( / x = 0) x u t = ν 2 u y 2 (2.87) (y) v 0 u/ x = 0 u x + v y = 0 v y = 0 v = 0 (2.88) 0 v = 0 y v = u y u x ν t < 0 : u(t, y) = 0 t 0 : y = 0 u = U y u 0 (2.89) x η η = y 2 νt (2.90) η 2 νt y

52 (t, y) = (τ, η) (τ, η) τ = t η = y 2 νt (2.91) τ = τ(t, y) (2.92) η = η(t, y) (2.93) t = t(τ, η) (2.94) y = y(τ, η) (2.95) (t, x) (τ, η) chain rule t y = τ τ t + η η t = τ = τ τ y + η η 2 y 2 = y y = 1 2 ντ y η = τ η 2τ 4 νt 3 2 y = 1 2 νt η = 1 2 ντ η ( ) 1 η 2 = 1 ντ η 4ντ η (2.96) (2.97) 2 η 2 (2.98) (2.96) (2.98) (2.87) x u u = Uf(η) (2.99) U η f () η f + 2ηf = 0 (2.100) (2.89) (τ, η) 2.91 } η = 0 f = 1 (2.101) η f 0 (2.100) f f(η) = 2 η e η2 dη + 1 (2.102) π 0

53 π e η2 dη = 2 (2.103) u [ u = U 1 2 η ] [ e η2 dη = U 1 2 ( )] y erf π π 2 νt 0 (2.104) erf(x) error function erf(x) = x 0 e x2 dx (2.105) y v y ƒä o x (a) (b) 2.10: (vorticity) 2 z ω z ζ(= ω z ) ζ(y, t) = v x }{{} 0 u y = u η η y = 1 ( 2 2U ) exp { η 2} = νt π η 2.90 u (2.104) U } exp { y2 πνt 4νt (2.106) ζ y = 0 y 0 y (Cartesian Coodinates)

54 : ω = rot v = v = ( i / x + j / y + k / z) ( iu + jv + kw) ( w = i y v ) ( u + j z z w ) ( + x v k x u ) y (2.107) i, j, k x, y, z (unit vector) i i = 0, i j = k, ω x = w y v z, ω y = u z w x, ω z = v x u y (2.108) (diffusion) y δ δ ζ(0, t)δ = (2.106) y = 0 0 ζ(0, t)δ = ζ(y, t) dy (2.109) Uδ πνt (2.110) y 0 ζ(y, t) dy = 0 U πνt e y2 4νt dy = 0 ( ) 2U e y2 y 4νt d π 2 = 2U π νt π 2 = U (2.111) U Uδ πνt = U (2.112)

55 : 2.13: δ = πνt (2.113) δ t 2.90 η dδ dt = 1 πν 2 t (2.114) (shearing atress) ( ) u τ w = µ = ρu 2 ( ν ) 1 2 ν y y=0 π U 2 = ρu t πt 1 (2.115) t τ w t 1 2 x

56 y, v ŠO Í ñ S «u b _ O(ƒË/c) x, u 2.14: 2. x y u u x + v u y = 1 ( p 2 ) ρ x + ν u x u y 2 u v x + v v y = 1 ( p 2 ) ρ y + ν v x v y 2 u x + v y = 0 (2.116) x x = x(x 0) (2.117) (2.118) y = 0 : u = 0, v = 0 (2.119) y : u ũ, v ṽ (2.120) ũ, ṽ x y ũ, ṽ ũ = cx, ṽ = cy (2.121) c > 0 c < 0 W W = ϕ + iψ = Cz n (2.122) n = 2 2 θ θ = π/n n = 2 θ = π/2 z z = x + iy c = 2C 2 ϕ = 2 ϕ x ϕ y 2 = 0 (2.123)

57 52 2 ϕ ũ = ϕ x, ṽ = ϕ y (2.124) c ν c [ũ ] [c] = = m/s x m = 1 s [ν] = m2 s ( l y η l y = ν c (2.125) η = y l y = y c = y ν/c ν (2.126) x y u v u = cxf (η), v = νcf(η) (2.127) f η (2.116) ψ ψ = νcxf(η) (2.128) νc x f(η) ψ u, v (2.127) u = ψ y, v = ψ x (2.129) (2.127) u, v (x = 0) x p ρv2 = p 0 (2.130) p v p 0 p p x < 0 (2.131)

58 y ˆ³ ÍŒ ˆ³ ÍŒ ˆ³ Í Á œ b Ý _ x 2.15: 3 (favorable pressure gradient) (x, y) (ξ, η) ξ η (x, y) (ξ, η) (2.132) ξ = x (2.133) c η = ν y (2.134) (chain rule) x = ξ ξ x + η η 2 x 2 = ( ) = 2 ξ ξ y 2 y 2 = x = ξ = ξ ξ y + η η y = ( ) c c = c ν η ν η ν ξ x = 1, (2.117) x (2.135) ξ 2 (2.136) c (2.137) ν η 2 η 2 (2.138) ξ y = 0 (2.139) c 2 xf 2 c 2 xff = 1 p ρ x + c2 xf (2.140) (2.118) y νccff = 1 ρ p y c νcf (2.141)

59 u ũ = cx, v ṽ = cy (2.142) (2.117) x c 2 x = 1 p ρ x (2.143) p ( (2.141) x ξ 0 = 1 2 p ρ x y (2.144) y p = const (2.145) x c 2 x = 1 p ρ x (2.146) p p (2.146) (2.140) f + ff f = 0 (2.147) f η ( ) (2.147) Falkner-Skan Falkner-Skan β f + ff + β(1 f 2 ) = 0 (2.148) β = 2m m + 1 (2.149) m ( 2.8 u ( e x ) m = (2.150) u 0 L m = 1 (,u e x) β = 1 f + ff + 1 f 2 = 0 (2.151) m = 0(β = 0) u e = const Blasius ( ) β m

60 ƒà ~ƒî/2 m m+1 ~ƒî 2.16: (Falkner-Scan ) (2.16) ( ) (2.147) (2.119) (2.120) (2.126) (2.127) y = 0 η = 0 : u = 0 f = 0 (2.152) v = 0 f = 0 (2.153) y η : u cx f 1 (2.154) (2.147) f = m m = g g = fg + m (f, m, g) y F (2.155) d y dη = F (η, y) (2.156) y = (f, m, g) t (2.157) F = (F 1, F 2, F 3 ) t (2.158) F 1 = F 1 (f, m, g) = m (2.159) F 2 = F 2 (f, m, g) = g (2.160) F 3 = F 3 (f, m, g) = fg + m 2 1 (2.161) F 1, F 2, F 3 f, m, g f = 0, m = 0 for η = 0 (2.162) m 1 for η (2.163) η = 0 g η = 0 g η m 1 η = 0 g m = 1 (iteration) Newton

61 56 2 Runge-Kutta four-stage, fourth-order method Runge-Kutta dy = f(x, y) (2.164) dx y n+1 = y n + h ( f 0 + 2f 1 + 2f 2 + f 3) (2.165) 6 x x = x n y n x (x n+1 ) y n+1 f 0 = f(x n, y n ) f 1 = f(x n h, y n hf 0 ) f 2 = f(x n h, y n hf 1 ) f 3 = f(x n + h, y n + hf 2 ) h = x (1, 2, 2, 1) x < 0 ( x = (scalar) (y, f) ( y, f) y y n+1 y n x n h/2 x n+1 x h 2.17: ( )

62 (boundary layer) (laminar boundary layer) (turbulent boundary layer) (Reynolds number) Re Re = UL ν = U L ν ( µ ρ ) ( ) 1904 (L.Prandtl; ; ) 1 u/ y y u τ = µ u/ y ( ) 2 (wake)

63 58 3 ƒ ƒeƒ ƒv ƒ i Q ³ µ:rotv=0 j «ŠE w U º Q x ª l Ü Á Ä é Ìˆæ Œã iwake j ûœü É x Ï» ªŒƒ µ y U(x) ƒâ x U(x) Fƒ ƒeƒ ƒv ƒ ê Ì x 3.1: (total pressure) (control surface) δ ν (3.1) δ δ L L 1 (3.2) δ L ( ) δ L = O(ϵ), 1 = O(ϵ0 ) (3.3) O(ϵ) 1 ( ) O(ϵ) 0 0 ( ) d «ŠE wœú ³ «ŠE w L ƒœ 3.2:

64 ( (reference values) U chord length L ρu 2 (ρ ) L/U ( ( ) ρu 2, L/U 2 u x + v y = 0 U L + v δ = 0 v U δ L (3.4) y v U 1 x u t + u u x + v u y = 1 ρ p x + ν ( 2 ) u x u y 2 (3.5) U U + U L/U L + U L δ U = 1 ( p δ ρ x + ν U L }{{} 2 + U ) δ }{{ 2 } U 2 ν U L δ (3.6) U L 2 U δ 2 (3.7) U 2 L = ν U δ 2 ( ) 2 δ = ν = 1 L LU Re δ L = Re 1/2 (3.8) Re Re = LU ν (3.9)

65 y x y v t + u v x + v v y = 1 ρ p y + ν ( 2 ) v x v y 2 U 2 L U L δ L U + U U L δ L + U U L δ L δ δ δ L + U 2 L δ L + U 2 L δ L = 1 ρ = 1 p ρ y + ν p y + ν ( U ( U L δ L 2 + L 2 U L δ δ 2 ) δ L + U δ 2 δ ) L (3.10) (3.6) (3.6) δ/l (3.6) 1 δ/l x y y y U 2 L δ L 1 p ρ y (3.11) y δp δp ρ U 2 L δ L δ (3.12) δy δ δp ρu 2 ( ) 2 δ (3.13) L y 2 y p y = 0 (3.14) x (3.14) p(x, δ) = p(x, 0) (3.15)

66 y p= ˆê è 3.3: p : ƒ ƒeƒ ƒv ƒ ̈³ Í U : ƒ ƒeƒ ƒv ƒ Ì x p p(x) U(x) U 3.4: U t + U U x = 1 ρ U y = 0 (3.16) p x (3.17) P 3.14 y 3.17 x p x = P x U U x = 1 p ρ x ( U 2 x 2 + p ) = 0 ρ (3.18) (3.19) p ρu 2 = const. = p ρu 2 (3.20) constant

67 62 3 X=X0 ƒâ X1 X2 º ûœü X Ì 3.5: U(x) 3.20 p (3.19) p/ x (3.5) u x (3.5) u x + v y = 0 (3.21) u t + u u x + v u y = 1 ρ p x + ν 2 u y 2 (3.22) U u x + v y = 0 (3.23) u t + u u x + v u y = U U x + ν 2 u y 2 (3.24) y = 0 : u = v = 0 (3.25) y : u = U(x, t) U y = 0 (3.26) u u x + v u y = 1 ρ u x + v y = 0 (3.27) p x + ν 2 u y 2 (3.28) y = 0 : u = v = 0, y : u = U(x) (3.29)

68 : (x x = x 0 : u = u(x 0, y) (3.30) y v (x, y) 3.6 x y R x y u t + R R + y u u x + v u = R R + y v t + y + { 1 p ρ x + ν + 2R (R + y) 2 v x R R + y u v x + v v y { 2 v y 2 uv R + y R 2 (R + y) 2 2 u x u y R + y R v (R + y) 3 x u2 R + y R dr (R + y) 3 dx v + u y u (R + y) 2 } Ry dr u (R + y) 3 dx x = 1 p ρ y + ν 2R u (R + y) 2 x + 1 v R + y y } + R2 2 v (R + y) 2 x 2 v (R + y) 2 + R dr (R + y) 3 dx u + Ry dr v (R + y) 3 dx x (3.31) (3.32) div u = 0 R u R + y x + v y + v R + y = 0 (3.33)

69 64 3 y U d x 3.7: ρ t + R (uρ) R + y x + (vρ) + vρ y R + y = 0 (3.34) R R R R 3.3 x (H. Blasius; 1908 Prandtl ph.d ) 0 dp dx = 0 (3.35) U = U = const u x + v y u u x + v u y = 0 (3.36) = ν 2 u y 2 (3.37) y = 0 : u = v = 0, y : u = U (3.38)

70 U ŠŽ U d1 x1 d2 x2 x 3.8: x u y x x x y U δ(x) u u ( y ) = ϕ U δ ϕ x (3.39) 2 δ δ νt (3.40) 2 t x tu = x (3.41) (3.41) δ δ νx νt (3.42) U (3.39) δ (3.42) η η = y δ = y U νx (3.43) ψ ψ = νxu f(η) (3.44) f(η) u = ψ y, v = ψ x (3.45)

71 66 3 u, v (ξ, η) (x, y) (ξ, η) ξ = x, chain rule η = y U νx (3.46) u = ψ y = ψ ξ ξ y + ψ η η y = νxu f (η) v = ψ x = ψ ξ = 1 2 ξ x ψ η η x = 1 2 νu U νx = U f (η) (3.47) f(η) νxu f (η) η x x νu x ( f + ηf ) (3.48) x, y ξ, η x = ξ η 2ξ U = y νξ η 2 y 2 = y y = η U νξ U η νξ η = U νξ (3.49) (3.50) 2 η 2 (3.51) (3.47) (3.51) (3.37) f ff + 2f = 0 (3.52) (1908) Blasius s equation) ( ) (3.47),(3.48) y = 0 : u = 0, v = 0 η = 0 : f = 0, f = 0 (3.53) y : u = U η : f = 1 (3.54) 2 ( ) 3.52 (2.165) Blasius Blasius (3.52) (3.1) η 0 u(y) (3.37) 2 u/ 2 y = 0, 2 u/ 2 η = 0

72 : U η = y f f = u f ηf f νx U u/u ( =f') u/u =f' h : y v y = δ 0 (3.48) v = 1 νu 2 x ( f + ηf ) (3.55) 3.1 η f + ηf = 1.72 (3.56) v v ν = 0.86 (3.57) U xu 0 x y ( ) ( ) ( )

73 68 3 y v U 3.10: y v x y d 0 U u U=0.99U x 3.11: 99% 1/7 (one-seventh power law) ( ) u ( y ) 1/7 = (3.58) U δ 3.4 (Boundary layer thickness) % ( ) U 99% δ δ = δ 99 u = 0.99U (3.59) η 5.0 (3.60) (3.43) η = y U νx = 5 (3.61)

74 y 0 U u(y) ƒâ P 3.12: x y δ 99 νx δ 99 = 5 (3.62) U δ δ 3.62 δ Re δ = 5.0Re 1/2 x (3.63) Re δ δ Re x x Re δ = U δ ν, Re x = U x ν (3.64) Re δ = 0.14Re 6/7 x, Re δ = 0.37Re 4/5 x (3.65) Re x 1 1/2 ( ) 30m/s 5m (displacement thickness) δ 1 δ U u(< U) U δ 1 = y=0 (U u)dy (3.66)

75 70 3 δ 1 δ 1 = y=0 (1 u )dy (3.67) U 3.3 Blasius (3.43) (3.47) δ 1 = = = y=0 νx (1 f ) U dη νx [ ] η1 η f U 0 νx ( ) η 1 f(η 1 ) U (3.68) η 1 η 1 (3.1) η 1 = 5 νx δ 1 = 1.72 (3.69) U Re δ1 = 1.72Re 1/2 x (3.70) (3.62) (3.69) δ 1 δ 1/3 δ 1 δ = 0.34 ( ) (3.71) δ 1 = 1/8 = 0.13 ( ) (3.72) δ 1/7 ( (3.58)) (3.67) (roughness) momentum thickness δ 2 θ ρu δ 2 2 = ρ u(u u)dy (3.73) y=0 U u

76 y U 0 u ƒâ2 x 3.13: δ 2 θ = δ 2 θ = δ 2 = = u y=0 U η=0 ( 1 u U ) dy νx f (1 f ) dη (3.74) U η = 5 Blasius f θ νx θ = δ 2 = (3.75) U Re θ = 0.664Re 1/2 x (3.76) 1/3 ( ) δ x = 5 Rex ( ), δ x = Re 1/5 x ( ) (3.77) δ x = 1.72 Rex ( ), δ x = Re 1/5 x ( ) (3.78) θ x = Rex ( ), θ x = Re 1/5 x ( ) (3.79) Re x x Re x = U x/ν

77 72 3 n y t w s U f x 3.14: δ 1 δ 2 H shapefactor) H = δ 1 = δ 1 δ 2 θ = δ θ (3.80) H = 2.59 H = 1.4 H = 1.3) H adverse pressure gradient H = 3.5 H = (skin friction)

78 shearing stress frictional stress ( ) u τ w (x) = µ (3.81) y y frictional stress 3.3 Blasius τ w (x) = µ U νx y=0 ( ) η U f (η) η=0 (3.82) = αµu U νx (3.83) (3.1) α = f (0) = 0.33 (3.84) C f C f = τ w(x) ρu /2 2 = αν 2 U ν U νx = 0.66 U x x Re x ( U x/ν) (3.85) C f = 0.66 Rex (3.86) C f x C f = Re 1/5 x C f = Re 1/5 x Re x,cr (3.87) A Re x, A = Re x,cr (C f,turbulent C f,laminar ) (3.88)

79 74 3 s b x l x 3.15: 3 U p ˆ³ Í å p p p p s s p ˆ³ Í 3.16: D f viscous drag l D f = b τ w cos ϕ ds (3.89) s=0 b l ϕ x s D f cos ϕ ds = dx (3.90) lx lx D f = b τ w dx = bµ s=0 x=0 ( ) u dx (3.91) y y=0 l x x ( profile drag U 0 (d Alembelt s paradox) D f (3.91),(3.83) D = b = b l x=0 l x=0 τ w dx αµu U νx dx = 2bα ρµu 3 l (3.92)

80 C D f C = 1.328Re -1/2 D f Laminar (Blasius) C D f Turbulent (Plandtl) -1/5 = 0.074Re transition CDf= ( logrl) Re l : 2D = 4bα ρµu 3 l = 1.328b U 3 µρl (3.93) D f D f U 3/2 l 1/2 µ 1/2 ρ 1/2 l 1/ C Df D C Df = 2D f 1 2 ρu 2 S (3.94) (1/2)ρU 2 S wetted surface area S S = 2bl (3.95) D f (3.93) (3.94) C Df = U l ν = Rel (3.96)

81 76 3 s U Stagnation i b Ý _ j Separation i _ j s 3.18: y p 1 2 p p > p 1 2 _ Ç ß Å Í,kinetic energy ª ³ D s t irecirculated flow j x 3.19: Re l (3.96) Re l < Re l ( ) C Df = Re 1/5 (3.97) C Df = (log 10 Re) 2.58 ( M 2 ) 0.65 (3.98) M ( ) 3.6 separation) adverse pressure gradient (dp/dx > 0)

82 y u d : «ŠE wœú ³ d Ýu ( Ýy) >0 Ýu y=0 ( ) =0 Ýu ( ) Ýy y=0 _ Ýy <0 y=0 t 3.20: ( ) u = 0 (3.99) y y=0 u τ y x (3.100) (x ) y x (3.101) (3.19), dp dx > 0 (3.102) (3.28) y = 0 µ ( 2 ) u y 2 = dp y=0 dx (3.103)

83 78 3 y 0 u ŠÔ Í ƒxƒ [ƒy É Â È ª é u y 0 0 É È é Ýu Ýy u Ì ª z ª ã É Ê 2 Ýu 2 Ýy ƒ0 Œù z Í0 y 2 Ýu 2 Ýy 3.21: (3.28) y u u y x + u 2 u x y + v u y y + v 2 u y 2 = 1 ρ y 0 2 p x y + ν 3 u y 3 (3.104) u = 0, v = 0, u x = 0, v y = u x = 0, 2 p x y = 0 (3.105) ( 3 ) u y 3 y=0 u 2 2 u/ y 2 0 = 0 (3.106) (3.103) (adverse pressure gradient) dp dx > 0 ( 2 ) u y 2 > 0 (3.107) y=0 (favorable pressure gradient) dp dx < 0 ( 2 ) u y 2 < 0 (3.108) y=0 dp dx = 0 ( 2 ) u y 2 = 0 (3.109) y=0 (3.9) ( p/ x < 0) 3.21 ( p/ x > 0) 3.22

84 y 0 u K Ï È _ ª Å «é (inflexion point) u y 0 0 É È é Å å l ª Ž² Ì r Å N «é Ýu Ýy 3.22: 0 y Œù z Í0 2 Ýu 2 Ýy U x f q o 3.23: u dp y = 0 dx > [ ] dp/dx > 0 λ θ Cp = p p 1 2 ρu 2 = 1 4 sin 2 θ (3.110)

85 : subcritical : ê ª w supercritical : ê ª sub super 3.25: 3.25 subcritical supercritical pressure drag (critical) drag crisis 3.24 ( C p = 1 C p = 0 C p = +1

86 U U = 2 sin x = 2x 0.333x x 5 + (3.111) x ( R ϕ x = ϕ = x/r, x ) U ϕ = (3.111) Hiemenz(1911) U U = 1.814x 0.271x x 5 + (3.112) ϕ = U ϕ = 80 Stratford ( ) 2 (x x B ) 2 dcp C p (3.113) dx x B Thwaites x0 ( U x B x 0 0 U max ) 5 dx (3.114) x 0 ϕ sep (seperation) x B = , ϕ sep = 79.8 (3.115) 3.7 D Alembert 5 (aerodynamic force) (lift) (drag) (side force) (pitching moment) (rolling moment) (yawing moment) 3 3 (aerodynamic coefficient) (lift) (drag) U U

87 C L L C L = 1 2 ρ U S 2 C D D C D = 1 2 ρ U S 2 (3.116) (3.117) C M M C M = 1 2 ρ U Sl 2 (3.118) L D M S l (1/2)ρ U 2 C l( l) C L C D Reynolds (C L = C L (Re), C D = C D (Re)) (similarity) M( / ) C L = f(re, M) C D = g(re, M) C M = h(re, M) 3.26 C D C D Re = 10 3 C D = C D = 1.2 Re = drag crisis ϕ sep ( ) f (Strouhal number)st St = fd U (3.119)

88 D U St = 0.2 ( ) Re 1 Oseen-Lamb C D = 8π Re[0.5 γ + ln(8/re)] (3.120) γ γ = C D Re 2/3 (3.121) C D R=VD/ 3.26: 100 C D R=VD/ 3.27: 3.27 C D

89 84 3 C D C D = 0.4 C D 0.4 Re = (drag crisis) C D = 0.09 Re 1 (Stokes) D D = 6πµRV (3.122) R µ C D C D = Re = ρ2rv /µ D 1 2 ρv S = 24 (3.123) 2 Re Stokes Stokes v t = K 1 p + ν v (3.124) ρ v = 0 (3.125), ω ( ) ω t p = 0 (3.126) = ν ω (3.127) Oseen C D = 24 [ Re 16 Re + 9 ] 160 Re2 ln Re (3.128) Stokes Oseen Re 1 0 Re C D 24 Re (3.129) Re

90 Re 1 C Df C Dp (3.130) Re 1 C Df C Dp C Df C Dp (induced drag) (trailing vortex) (parasite drag) (skin friction drag) (wetted surface) (pressure drag) 9 (interference drag) 2 ( ) (trim drag) (profile drag) 2 (cooling drag) (base drag) (wave drag) CFD Re 10 4 ( C D = 2.0) ( (2.1) ((1.6) (1.2) (2.3) (1.2)(1.7) (1.6)(2.0) Re = C D = Re = 10 6 C D

91 86 3 U CD 2.0 CD : ( Re 10 4 (C D = 1.07) ( 60 (0.5) (1.17) (1.4) C D = 1.2 (0.4) l/d = 0.5 C D = (streamlined form) 2 C D = 2.0 C D = 1.1 C D = C D = : 1 C D = : 1 C D = : 1 C D = 0.25 C D = : 1 C D = : 1 C D = : 1 C D = 0.1

92 cube disk cup cone 3.29: L D L/D 1 induced drag 2 parasite drag profile dragfrictional drag pressure drag 1.0 Cl max Ž (stall) Cl ƒ 12 (deg) 3.30: (NACA0012 Re = ) (wing section; airfoil) 12% NACA0012 ( 30% )

93 88 3 Re = α = 12 C L = 1.0 C D C L = 0 C D = 0.01 C L = 0.8 ( 8 ) C D = C L C D C L C D C D (L/D) L/D 7 25 : L/D = 7.5 B747: L/D = 17 B52 L/D = 22 L/D = range down range R (Breguet range equation) R = V C L D ln W 0 W 1 (3.131) V W 0 ( ) W 1 (W 1 = W 0 W f ) T C dw dt = CT (3.132) R R = 550η C C l C D ln W 0 W 1 (3.133) η W 0 W 1 W f W 1 = W 0 W f L/D L/D (cross range) L/D L/D = 1L/D = 4.5 (Lift drag ratio) FX FX Re = 10 6 α = 10 C L = , C D = , C M = C L C D C D 10 4

94 (stall) 3 NACA ( 18%) NACA ( 12%) NACA ( 6%) ( ) (viscous drag) 50% (induced drag) 40% ( 42% 8% ) (winglet) 40% 37% (wave drag) 18% (turbulent boundary layer) (laminar boundary layer) B C D = B787 C D = C D0 (zero-lift drag coefficient) (biplane) C D0 = B52 C D0 = ( ) V max 3 P C D0 S (3.134) P power S

95 (ballistic coefficient) C BC = M SC D (3.135) M S C D (bluff body) (bullet) C BC =

96 x «ŠE w x ª ̈æ T U 4.1: internal enery e i ( conduction of heat) κ convection of heat u T/ x + v T/ y heat through friction : τ ij u i / x j radiation ( 800 C hypersonic flow heat transfer (condensation) evapolation ( ) (1/2)kT (k )(3/2)kT (1/2)RT (R ) e i = (3/2)RT ( )

97 t heat Q V = x y z (the First Law of Thermodynamics Q = E + W dq dt = de dt + dw dt (4.1) 1 E (total energy = internal energy + kinetic energy) 2 Fourier 1 dq T = q = k A dt n (4.2) Ê ÏA n q 4.2: A: area k: thermal conductivity 15 C k = J/m sec deg n: Q: q: (heat flux

98 y dz ü Á Ä é M Ê z dx dy control volume x o Ä M Ê 4.3: x ( k T ) [ y z + k T x x + ( k T ) ] x y z = ( k T ) x y z (4.3) x x x x, y z t [ ( Q = t }{{} V k T ) + ( k T ) + ( k T )] x x y y z z x y z (4.4) de i de t { de t dei = ρ V dt dt + 1 d ( u 2 + v 2 + w 2)} (4.5) 2 dt { ( dw σx = dydz t uσ xx + u + u ) ( x δx σ xx + σ )} xx x δx = V t x (uσ xx) (4.6) 4.4 { dw = V t x (uσ xx + vτ xy + wτ xz ) + y (uτ yx + vσ yy + wτ yz ) + } z (uτ zx + vτ zy + wσ zz ) (normal stress) σ p (u, v, w) (4.7) σ xx = p 2 µ div v + 2 µ u 3 x σ yy = p 2 µ div v + 2 µ v 3 y σ zz = p 2 µ div v + 2 µ w 3 z (4.8) (4.9) (4.10)

99 94 4 dz σ xx σ xx + δ σ xx dy dx δ x δ x 4.4: shearing stress ( v τ xy = τ yx = µ x + u ) y ( w τ yz = τ zy = µ y + v ) z ( u τ zx = τ xz = µ z + w ) x (4.11) (4.12) (4.13) ρ De ( i + p div v = k T ) + ( k T ) + ( k T ) + µφ (4.14) Dt x x y y z z D/Dt Φ dissipation function µφ = τ ij u ij x ij = u x τ xx + v x τ xy + w x τ xz + u y τ yx + v y τ yy + w y τ yz + u z τ zx + v z τ zy + w { ( u ) 2 ( ) 2 v = 2µ + + x y ( v + µ x + u ) 2 + µ y ( w z ) 2 } ( w y + v z 2 ( u 3 µ x + v y + w z ) 2 ( u + µ z + w x ) 2 z τ zz ) 2 (4.15) Φ 2 2 u 2 /l 2 τ xx, τ yy, τ zz σ xx, σ yy, σ zz ( ) µ ( ) (4.14) S (1.102) (1.107) ρt DS Dt = ( k T ) + ( k T x x y y ) + z ( k T ) + µφ (4.16) z

100 Thermodynamic Properties e i h de i = C v dt (4.17) ( ) p dh = C p dt = C v dt + d = C v dt + d (pv) (4.18) ρ h = e i + p ρ = e i + pv (4.19) v (specific volume) v = 1 ρ (4.20) calorically perfect gas C v = const C p = const (4.21) C p : Specific heat at constant pressure (4.22) C v : Specific heat at constant volume (4.23) C p 1.0 kj/kg K C v 0.72 kj/kg K R = C p C v γ = C p /C v R γ C p = C v = C (4.24) C p C v h de i = C dt (4.25) dh = C dt + dp = C dt + vdp ρ (4.26) h e i + p ρ = e i + pv (4.27) e i h ρ De i Dt = ρ D ( h p ) = ρ Dh Dt ρ Dt ρ D ( ) p DT = ρc p Dt ρ Dt ρ D ( ) p Dt ρ (4.28)

101 ρ De i Dt + p div v = ρc DT p Dt ρ D ( ) p + p div v Dt ρ ( DT 1 = ρc p Dt ρ Dp ρ Dt p ) Dρ ρ 2 + p div v Dt DT = ρc p Dt Dp ( ) 1 Dt + p Dρ ρ Dt + div v }{{} 0 DT = ρc p Dt Dp Dt 4.14 ρc p DT Dt Dp Dt = x k k ( ) (4.29) ( k T ) + ( k T ) + ( k T ) + µφ (4.30) x y y z z ρc p DT Dt = Dp Dt + k 2 T + µφ (4.31) (4.30) (4.31) Dp/Dt p = ρrt (4.35) (4.36) (4.39) ( ) Dp Dt Dp Dt = RT Dρ = RT Dt + RρDT Dt (4.32) ( ρβ DT ) + Rρ DT Dt Dt = 0 (4.33) div v = 0 div v = 0 v = 1 Dρ ρ Dt ( ) Dρ ρ Dt DT T Dt p (4.34) (4.35) β (the coefficient of volume expansion) β = ( ρ/ T ) p ρ β = ( v/ T ) p v (4.36)

102 v (v = 1/ρ) v = β DT Dt (4.37) β p = RT (4.38) ρ β = 1 T v = 1 T DT Dt p p v = ρr DT Dt (4.39) (4.40) (4.41) C p C v = R (4.42) p v = ρ(c p C v ) DT Dt (4.43) 4.14 ρc p DT Dt = k 2 T + µφ (4.44) k C p C v C p C v thermal diffusivity a a k ρc p (4.45) DT ρc p Dt = k 2 T (4.46) DT Dt = a 2 T (4.47) (4.47) (forced convection)

103 u T x + v T y = a 2 T y 2 (4.48) x x y y = 0 T = T w (4.49) y = δ T T = T T y = 0 (4.50) T w δ T u ν T 4.50 u u x + v u y = ν 2 u y 2 (4.51) θ = T T w T T w (4.52) y = 0 θ = 0 (4.53) y = δ T θ = 1 (4.54) Pohlhausen Blasius η(3.43) ψ(3.44) η = y U 4.48 νx, ψ = νxu f(η) (4.55) d 2 θ dη 2 + P r 2 f dθ dη = 0 (4.56) ( η 0 exp θ(η) = P r 2 0 exp ( P r 2 ) β 0 f(α)dα dβ ) β 0 f(α)dα dβ (4.57)

104 y ƒ V x d Í Y = -ƒïƒ V g 4.5: 4.4 (natural convection) 10 6 x, y, z ( ρ u u x + v u y + w u ) = p z x + X + µ 2 u (4.58) ( ρ u v x + v v y + w v ) = p z y + Y + µ 2 v (4.59) ( ρ u w x + v w y + w w ) = p z z + Z + µ 2 w (4.60) (X, Y, Z) (body force) (X, Y, Z) (X, Y, Z) = (ρg x, ρg y, ρg z ) (4.61) g = (g x, g y, g z ) (4.62) g (g x, g y, g z ) (x, y, z) ( ρ u u x + v u y + w u ) z ( ρ u v x + v v y + w v ) z ( ρ u w x + v w y + w w ) z = p x + ρg x + µ 2 u (4.63) = p y + ρg y + µ 2 v (4.64) = p z + ρg z + µ 2 w (4.65) ρ p T ρ, p, T

105 100 4 Boussinesq ρ = ρ ( p, T ) ( ) ( ) ρ ρ ρ = ρ + (T T ) + (p p ) T p γ = ρ ρ β (T T ) + c 2 (p p ) }{{} (4.66) coefficient of thermal expansion β β = 1 ( ) ρ (4.67) ρ T p = ρrt (4.68) β β = 1 T (4.69) β = (1/K) ( ρ/ρ ( ) ρ = 1 p T RT = γ γrt = γ c 2 (4.70) p/ρ U 2 ) ρ ρ U 2 = γ c 2 (4.71) ( ) ρ = U 2 p c 2 γ = M γ 2 (4.72) M (M = 0.01) p, ρ ( ρ/ρ T/T ( ) ρ 1 = ρ β = ρ (4.73) T T ) ρ T = ρ T ( ) ρ T = 1 (4.74) u x + v y + w z = 0 (4.75)

106 x, y, z ( ρ u u x + v u y + w u ) z ( ρ u v x + v v y + w v ) z ( ρ u w x + v w y + w w ) z = p x ρ g x βθ + µ 2 u (4.76) = p y ρ g y βθ + µ 2 v (4.77) = p z ρ g z βθ + µ 2 w (4.78) θ ( ρc p u T x + v T y + w T z θ = T T (4.79) ) ( 2 ) T = k x T y T z 2 + µφ (4.80) p p x p x + ρ g x = x (p ρ g x x) = p x (4.81) p p ρ g x = p (4.82) y 2 z ρu 2 ( u u L x + v u y + w u ) z θ p = p ρ (g x x + g y y) (4.83) p = p ρ (g x x + g y y + g z z) (4.84) = ρu 2 L p x ρgβ T g xθ + µu L 2 2 u (4.85) θ = T T T (4.86) u u x + v u y + w u z = p x gβ T L U 2 g x θ + 2 gβ T L U 2 = ν LU 2 u (4.87) ( ) 2 gβ T L3 ν ν 2 = Gr 1 U L Re 2 (4.88)

107 102 4 Gr Gr = Re gβ T L3 ν 2 (4.89) Re = U L ν (4.90) /Re u u x + v u y + w u z u v x + v v y + w v z u w x + v w y + w w z = p x Gr Re 2 g xθ + 1 Re 2 u (4.91) = p y Gr Re 2 g yθ + 1 Re 2 v (4.92) = p z Gr Re 2 g zθ + 1 Re 2 w (4.93) µφ 4.80 () ρ C p U T L ( u T x ρ C p U T L ( u T ) x + = Φ klt ρ C p U T L 2 2 T + ) = k T L 2 2 T + U 2 µφ (4.94) L2 U µ ρ C p T L Φ (4.95) U µ ρ C p T L = µ U 2 = 1 (γ 1)U 2 = (γ 1) M 2 (4.96) ρ LU C p T R e γrt R e M 0 2 T klt ρ C p U T L 2 = k ρ C p U L = k µc p = 1 ( ) 1 1 = P r R e P e µ ρ U L (4.97) P e P e = P r Re (4.98) R e = UL/ν Reynolds number G r = gβ T L 3 /ν 2 Grashof number P r = µc p /k Prandtl number R a = G r P r Rayleigh number) P e = P r R e (Peclet number)

108 Ra = Gr P r < 10 9 ( ), Ra = Gr P r 10 9 ( ) (4.99) (heat transfer) ( Nu Nu = 0.59Ra 1/ < Ra < 10 9 ( ) (4.100) Nu = 0.13Ra 1/ < Ra < ( ) (4.101) ( ) (Nu) α x Nu x = αx k (4.102) α = q w q w T w T (4.103) (St: Stanton number) St = α ρc p U (4.104) St = Nu Re P r (4.105) ( ) P r C C C P r P r > 1 > P r = 1 = P r < 1 <

109 104 4 u ĉ T ĉ T Pr>1 u ĉ T ĉ T Pr=1 u ĉ T ĉ T Pr<1 4.6:

110 105 5 (laminar flow) (turbulent flow) Re(= UL/ν) (turbulent flow) M ü Œv v( x) t(žžšô) 5.1: ( ) x u u = u + u (5.1) u u v = v + v w = w + w (5.2) p = p + p T = T + T compressible flow T ρ } ρ = ρ + ρ (5.3) T = T + T u = 1 t t0+ t t 0 u dt (5.4) t t

111 106 5 u u- t 0 t t0 +ƒ t 5.2: ) u = u + u (5.5) u = 0 (5.6) v = 0, p = 0, ρ = 0, T = 0 (5.7) u, v, w, p, ρ, T (5.4) f g f g f = f f + g = f + g f g = f g = f g f/ s = f/ s fds = fds (5.8) s 5.8 [ ] (mass-weighted average density-weighted average ) Favre à = ρā ρ (5.9) à A A A = à + A (5.10) A ρa = 0 (5.11)

112 (x y S x x Ì ò da : Ê Ï z 5.3: S x ) ( ( 5.3 ) x, y, z x : dm x = ρu u da y : dm y = ρu v da z : dm z = ρu w da (5.12) 5.8 x dm x = da ρu 2 = da ρ (u + u ) 2 = da ρ (u 2 + u 2 ) (5.13) 2uu = 0 y z 2ūu = 2ū u = 0 (5.14) dm y = da ρ(u v + u v ) (5.15) dm z = da ρ(u w + u w ) (5.16) law of action and reaction momentum) da da x σ xx = ρ (u u + u u ) y τ xy = ρ (u v + u v ) z τ xz = ρ (u w + u w ) (5.17)

113 108 5 ƒñ f ƒñ f š ƒð f xx 5.4: σ xx = ρu u τ xy = ρu v τ xz = ρu w (5.18) (apparent stress), (Reynolds stress) Osborne Reynolds 1885 (1895?) 2 S y, S z u v u v 2 (correlation) u v u v > 0 y u(y) u 2 y 2 y 1 v u 1 x 5.5: y dū/dy > 0 v > 0 x u 1 (eddy) ( 5.5 ) u 1 < u 2 y = y 2 x u 2

114 u 1 u 1 u 2 < 0 u < 0 u v u v < 0 (5.19) ensemble average: u v < 0 or u v > 0 (5.20) y u(y) u 2 y 2 v y 1 u 1 x 5.6: v < 0 x u 2 eddy ( 5.6 ) (y = y 1 ) u 2 u 1 > 0 u > 0 u v < 0 (5.21) u v < 0 or u v > 0 (5.22) ρ τ xy = ρu v > 0 (5.23) τ yx = µ u y > 0 (5.24)

115 ( u ρ t + x uu + y uv + ) z uw ( v ρ t + x vu + y vv + ) z vw ( w ρ t + x wu + y wv + ) z ww u x + v y + w z = 0 (5.25) , = p x + µ 2 u (x ) (5.26) = p y + µ 2 v (y ) (5.27) = p z + µ 2 w (z ) (5.28) x (u + u ) + y (v + v ) + z (w + w ) = 0 (5.29) x u + y v + z w = 0 (5.30) x u = x u = 0 (5.31) (5.30) (divergence) x u + y v + z w = 0 (5.32) x, y, z ( u ρ t + u u x + v u y + w u ) = p [ ] z x + ρu u µ 2 u + + ρu v + ρu w (5.33) x y z ( v ρ t + u v x + v v y + w v ) = p [ ] z y + ρu v µ 2 v + + ρv v + ρv w (5.34) x y z ( w ρ t + u w x + v w y + w w ) = p [ ] z z + ρu w µ 2 w + + ρv w + ρw w (5.35) x y z ρ Du Dt = p x + µ 2 u + ( x τ xx + y τ yx + ) z τ zx (5.36)

116 ρ Dv Dt ρ Dw Dt = p ( y + µ 2 v + = p z + µ 2 w + ) x τ xy + y τ yy + z τ zy ( x τ xz + y τ yz + ) z τ zz (5.37) (5.38) ρ D Dt v = grad p + µ 2 v + T R (5.39) v = (u, v, w) t t T R τ xx τ xy τ xz T R = τ yx τ yy τ yz τ zx τ zy τ zz = ρu u ρu v ρu w ρv u ρv v ρv w ρw u ρw v ρw w (5.40) σ xx = p + 2µ u x ρu u (5.41) ( u τ xy = µ y + v ) ρu x v (5.42) ( u τ xz = µ z + w ) ρu x w (5.43) σ yy = p + 2µ v y ρv v (5.44) ( v τ yz = µ z + w ) ρv y w (5.45) σ zz = p + 2µ w z ρw w (5.46) τ xy = τ yx τ xz = τ zx τ yz = τ zy (eddy viscosity) (molecular viscosity) u v ρ + ( ρṽ) = 0 (5.47) t ρṽ t + ( ρṽ ṽ + pi T T ) = 0 (5.48) T T = ρv v (5.49) ṽ, v

117 x u x + v y u u x + v u y = 0 (5.50) = 1 p ρ x + { ν u } y y u v (5.51) y y?? 3.22) u, v, p u, v, p (5.52) ( ν u ) y y y ( ν u ) y u v (5.53) τ l τ t ( τl y ρ + τ ) t (5.54) ρ τ l = µ u y, τ t = ρu v (5.55) closure problem ( ) Tollmien-Schlichting instability 2 (TS ) Re = U e δ /ν < 420 (U e δ ) U e δ δ TS TS 5.4 (J.Boussinesq, 1897) τ t = ρu v = µ t du dy (5.56)