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

Viscous Fluid Dynamics Yoshiaki NAKAMURA Professor of Chubu University Emeritus Professor of Nagoya University 2014 9 September 1, 2014

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) 1995 9

3 1 1 1.1............................................ 1 1.2.......................................... 4 1.3..................................... 6 1.4................................ 8 1.5............................... 19 1.6............................... 23 1.7................................. 25 1.8.................................. 31 2 33 2.1...................................... 33 2.2........................................ 39 2.3............................... 46 2.4...................................... 50 3 57 3.1........................................ 57 3.2........................................ 59 3.3........................................ 64 3.4........................................ 68 3.5........................................... 72 3.6....................................... 76 3.7.......................................... 81 4 91 4.1....................................... 91 4.2..................................... 92 4.3....................................... 95 4.4.......................................... 99 5 105 5.1................................. 107 5.2...................... 110 5.3............................. 112 5.4............................... 112 5.5................................. 117 5.6........................................ 119

i 5.7................................... 123 5.8........................................... 124

1 1 1.1 (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) ( )

2 1 1.1.1 (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: 1.1.2 (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) 0.8004 1.0 10 3 1.8 10 5

1.1. 3 1.1.3 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 = 1.716 10 5 [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 = 1.716 10 5 [kg/(sec m)], T 0 = 273K, n = 2/3 T 0 1.1.4 (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 0.9 10 3 1.0 10 6 1.5 10 5 air 1.1 15, 760mmHg 1 (P a) 1.01325 10 5 P a = 101.325kP a = 0.101325MP a(= 1013.25 hp a) 1cal = 4.18605J (ratio of specific heat) (water) 1.2

4 1 1.1: ρ 1.219 kg/m 3 µ 1.788 10 5 kg/s m ν 1.467 10 5 m 2 /s k 5.76 10 6 kcal/s m K = 2.42 10 5 kj/(s m K) C p 0.23990 kcal/(kg K) = 1.004kJ/(kg K) () C v 0.17100 kcal/(kg K) = 0.716kJ/(kg K) γ = C p /C v 1.4 1.2: ρ 999.6 kg/m 3 µ 1.0 10 3 kg/s m ν 1.01 10 6 m 2 /s k 0.610 J/(s m K) = 0.610 W/m K = 6.10 10 4 kj/(s m K) C p 4.182 kj/(kg K) () C v * kcal/(kg K) γ = C p /C v 1. 1.2 (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 33 0 0 p τ 31 τ 32 τ 33

1.2. 5 ( ) n (a tensor of order n) 3 n 0 3 0 = 1 (scalar) 1 3 1 = 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)

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 1.3.1 5 1 (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)

1.3. 7 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) 1.3.2 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 = 0 1.3.3 (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)

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

1.4. 9 (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)

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)

1.4. 11 ( 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 1.31 1.32 1.33 ρ Du = X p Dt x + [ ( µ 2 u x x 2 3 + [ ( u µ y y + v )] + x z ρ Dv = Y p Dt y + [ ( µ 2 v y y 2 3 + [ ( v µ z z + w )] + y x ρ Dw = Z p Dt z + [ ( µ 2 w z z 2 3 + 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)

12 1 div σ = σ = ( i σ ij ) = 1 σ 11 + 2 σ 21 + 3 σ 31 1 σ 12 + 2 σ 22 + 3 σ 32 1 σ 13 + 2 σ 23 + 3 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 2 + 2 u y 2 + 2 u z 2 (1.51) ( ) (1.48) ( ) PHILOSOPHIA NATURALIS PRINCIPIA MATHEMATICA Mathematical Principles of Natural Philosophy; (Isaac Newton, 1642-1727) 1687 Part 1 Part 2 Part 3 ( ) 1.4.1 (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

1.4. 13 2 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: ( ) 2 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 = 0 2 3 div v = 0 v = 0 (1.59)

14 1 1.4.2 (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 = 8.3145 [J/(K mol)] M = 29 R = 287[m 2 /s 2 K] = 287[J/kg K] M = 29 g/mol kg M = 29 10 3 kg/mol 10 3 1 k b N b = 6.02 10 23 mol R 0 ( ) k b = R o N b = 1.38066 10 23 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)

1.4. 15 (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 0 10 3 M (1.63) R = R 0 J/(K mol) = 8314 M 10 3 kg/mol M J kg K (1.68) (1.69) 1.4.3 (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 = 2 + + + 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 1.82 2 (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 2 11 + S 2 22 + S 2 33) + (2S 23 ) 2 + (2S 31 ) 2 + (2S 12 ) 2 ] + λ(s 11 + S 22 + S 33 ) 2 (1.74)

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)

1.4. 17 1.70 1.82 1.80 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)

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

1.5. 19 (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

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

1.5. 21 { } 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)

22 1 1.9: [ ( ) ] { ( ) } 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

1.6. 23 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) 1.6 1.6.1 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 1.127 2 2 [ ( 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)

24 1 1.6.2 (1.130) rot ω t (V ω) = ν 2 ω (1.132) = 0 ω (1.132) 2 1.132) 1.135 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 2 1.137 3 ( ) (cascade phenomenon) ( ) (Helicity) v ω (scalar product) H = v ω dv (1.138) H NS

1.7. 25 1.7 1.10: (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 + 1 2 v r r 2 θ 2 + 2 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 + 1 2 v θ r 2 θ 2 + 2 v θ z 2 + 2 v r r 2 θ v θ r 2 v z θ + v z + 1 r ] v z z v z r + 1 2 v z r 2 θ 2 + 2 v z z 2 ] ] ] (1.139) (1.140) (1.141) ( ) (r, θ, z) (x, r, θ) (1.141) z x

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 2 + 1 r r + 1 2 r 2 θ 2 + 2 z 2 = 1 ( r ) + 1r 2 r r r 2 θ 2 + 2 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 1.127 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 1.148 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)

1.7. 27 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)

28 1 1.11: 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

1.7. 29 ( ) (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) 1.7.1 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 2 + 1 r ] v θ r + 1 2 v θ r 2 θ 2 + 2 v θ z 2 + 2 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 2 + 1 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)

30 1 r ρ { t r (rv θ) = µ r 3 v θ r 3 + 2 v θ r 2 + 2 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 3 + 2 2 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 3 + 1 2 v θ r r 2 = 3 v θ r 3 + 1 2 v θ r r 2 = 2 v θ r 2 + 1 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 2 + 1 ζ r r = 3 v θ r 3 + 2 2 v θ r r 2 1 v θ r 2 r + 1 r 3 v θ (1.178) 1.175 1.175 ζ ρ ( 2 t ζ = µ ζ r 2 + 1 ) 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 1.176 v θ v θ = Γ ( )) 0 1 exp ( r2 2πr 4νt (1.180) (1.181) (1.182)

1.8. 31 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

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)

33 2 2 (external flow) (internal flow) 2.1 2 (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 2 + 2 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)

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 2.1.1 (Couette Flow) Maurice Couette (1858-1943 ) (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)

2.1. 35 (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 [ ] 2.1.2 (Poiseuille Flow) y = ±b u = 0 (2.22)

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) 2.1.3 (Hagen-Poiseuille flow) 2 Hagen 1839 v θ = 0, θ = 0 (2.28) (1.141) ( d 2 u µ dr 2 + 1 ) du = dp r dr dx (2.29)

2.1. 37 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 r4 2 4 0 ( dp ) dx (2.35)

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)

2.2. 39 (2.31) r r = R µ f = 64 Re D (2.45) Re D Re D = V D ν (2.46) f = 0.316Re 1/4 3 10 3 Re 10 5 (2.47) f = 0.032 + 0.221Re 0.237 10 5 Re 3 10 6 (2.48) ( ) L L = 0.065 Re D (2.49) L = 20 40 D (2.50) 2.2 ( ) (r, θ, z) z t r (z ζ ( ζ 2 t = ν ζ r 2 + 1 ) 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)

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

2.2. 41 (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 0 2.58t 0 Γ 0 δ(r) (vortex filament) ( ) 2.5: (2.55) u θ 2.6 ν ( 2.5) t ( 2.6) η ν t

42 2 3 Circumeferential velocity 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 Radius r t=1000 t=2000 t=3000 t=4000 2.6: 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 2 + 1 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 2 + 1 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)

2.2. 43 (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 0 2.2.1 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 0 2.70 µ 0 I = Γ (2.73)

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 [1 + 0.413(σ 2 i + σ2 j )/2y2 ] 3/2 (2.75) 2.2.2 (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)

2.2. 45 2.2.3 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 4 1 2 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 4 1 2 r 2 (a < r) (2.86)

46 2 2.9: 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 = 0 2.87 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 2.90 2.87

2.3. 47 (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) 2.100 (2.89) (τ, η) 2.91 } η = 0 f = 1 (2.101) η f 0 (2.100) 2 2.101 f f(η) = 2 η e η2 dη + 1 (2.102) π 0

48 2 0 π e η2 dη = 2 (2.103) 2.102 2.99 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)

2.3. 49 2.11: 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 2.1102.111 Uδ πνt = U (2.112)

50 2 2.12: 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 2.4 2 2

2.4. 51 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 2 + 2 u y 2 u v x + v v y = 1 ( p 2 ) ρ y + ν v x 2 + 2 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 2 + 2 ϕ y 2 = 0 (2.123)

52 2 ϕ ũ = ϕ x, ṽ = ϕ y (2.124) 2.121 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 2.127 (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 2.121 (x = 0) x p + 1 2 ρv2 = p 0 (2.130) p v p 0 p p x < 0 (2.131)

2.4. 53 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)

54 2 2.140 2.141 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 2.143 c 2 x = 1 p ρ x (2.146) p p (2.146) (2.140) f + ff f 2 + 1 = 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

2.4. 55 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 2 1 1 (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

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 + 1 2 h, y n + 1 2 hf 0 ) f 2 = f(x n + 1 2 h, y n + 1 2 hf 1 ) f 3 = f(x n + h, y n + hf 2 ) h = x 2.165 (1, 2, 2, 1) 6 6 1 x < 0 ( x = 0 2.164 (scalar) (y, f) ( y, f) y y n+1 y n x n h/2 x n+1 x h 2.17: ( ) 2.147 4

57 3 3.1 0 (boundary layer) (laminar boundary layer) (turbulent boundary layer) (Reynolds number) Re Re = UL ν = U L ν ( µ ρ ) ( ) 1904 (L.Prandtl; ; 1875-1953) 1 u/ y y u τ = µ u/ y ( ) 2 (wake)

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:

3.2. 59 3.2 ( 3.2.1 (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 2 + 2 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 2 3.2 2 (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)

60 3 3.8 y x y v t + u v x + v v y = 1 ρ p y + ν ( 2 ) v x 2 + 2 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) 3.2.2 x (3.14) p(x, δ) = p(x, 0) (3.15)

3.2. 61 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 + 1 2 ρu 2 = const. = p + 1 2 ρu 2 (3.20) constant

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 3.2.3 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)

3.2. 63 3.6: (x x = x 0 : u = u(x 0, y) (3.30) y v 3.2.4 (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 2 + 2 u y 2 + 1 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)

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 3.31 3.34 R 3.3 x 3.3.1 (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)

3.3. 65 U ŠŽ U d1 x1 d2 x2 x 3.8: x u y x 3.3.2 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)

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) ( ) 3.52 3.29 (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) 3.3.3 Blasius Blasius (3.52) (3.1) η 0 u(y) (3.37) 2 u/ 2 y = 0, 2 u/ 2 η = 0

3.3. 67 3.1: U η = y f f = u f ηf f νx U 0.0 0 0 0.3321 0 1.0 0.1656 0.3298 0.3230 0.1642 2.0 0.6500 0.6298 0.2668 0.6095 3.0 1.397 0.8461 0.1614 1.141 4.0 2.306 0.9555 0.0642 1.516 5.0 3.283 0.9915 0.0159 1.674 6.0 4.280 0.9990 0.0024 1.714 7.0 5.279 0.9999 0.0002 1.720 8.0 6.279 1.0000 0.0000 1.721 u/u ( =f') u/u =f' 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2 3 4 5 h 6 7 8 9 3.9: 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 ( ) ( ) ( )

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) 3.4.1 99% ( ) U 99% δ δ = δ 99 u = 0.99U (3.59) 3.3.3 η 5.0 (3.60) (3.43) η = y U νx = 5 (3.61)

3.4. 69 y 0 U u(y) ƒâ P 3.12: x y δ 99 νx δ 99 = 5 (3.62) U δ 99 3.43 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 3.4.2 (displacement thickness) δ 1 δ U u(< U) U δ 1 = y=0 (U u)dy (3.66)

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) 3.4.3 momentum thickness δ 2 θ ρu δ 2 2 = ρ u(u u)dy (3.73) y=0 U u

3.4. 71 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 = 0.664 (3.75) U Re θ = 0.664Re 1/2 x (3.76) 1/3 ( ) δ x = 5 Rex ( ), δ x = 0.371 Re 1/5 x ( ) (3.77) δ x = 1.72 Rex ( ), δ x = 0.046 Re 1/5 x ( ) (3.78) θ x = 0.664 Rex ( ), θ x = 0.036 Re 1/5 x ( ) (3.79) Re x x Re x = U x/ν

72 3 n y t w s U f x 3.14: 3.4.4 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 = 2.4 3.5 (skin friction)

3.5. 73 3.5.1 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) 3.5.2 3.83 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 = 0.0576 Re 1/5 x C f = 0.0576 Re 1/5 x Re x,cr (3.87) A Re x, A = Re x,cr (C f,turbulent C f,laminar ) (3.88)

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: 3.5.3 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)

3.5. 75 C D f 10-2 -3 10 C = 1.328Re -1/2 D f Laminar (Blasius) C D f Turbulent (Plandtl) -1/5 = 0.074Re transition 0.455 CDf= ( logrl) 2.58 10-4 10 10 10 Re l 4 5 6 3.17: 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 1/2 3.5.4 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 = 1.328 U l ν = 1.328 Rel (3.96)

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 < 5 10 5 10 6 Re l ( ) C Df = 0.074 Re 1/5 (3.97) C Df = 0.455 (log 10 Re) 2.58 (1 + 0.144M 2 ) 0.65 (3.98) M ( ) 3.6 separation) adverse pressure gradient (dp/dx > 0)

3.6. 77 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: 3.6.1 ( ) u = 0 (3.99) y y=0 u τ y x (3.100) (x ) y x (3.101) 3.6.2 (3.19), dp dx > 0 (3.102) (3.28) y = 0 µ ( 2 ) u y 2 = dp y=0 dx (3.103)

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) 3.104 ( 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

3.6. 79 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: 3.21 3.22 u dp y = 0 dx > 0 3.22 [ ] dp/dx > 0 λ 3.6.3 2 3.24 θ Cp = p p 1 2 ρu 2 = 1 4 sin 2 θ (3.110)

80 3 3.24: 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

3.7. 81 U U = 2 sin x = 2x 0.333x 3 + 0.0167x 5 + (3.111) x ( R ϕ x = ϕ = x/r, x ) U ϕ = 104 110 (3.111) Hiemenz(1911) U U = 1.814x 0.271x 3 0.0471x 5 + (3.112) ϕ = 71.2 1.595U ϕ = 80 Stratford ( ) 2 (x x B ) 2 dcp C p 0.0104 (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 = 0.8606, 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) 3.7.1 (lift) (drag) U U

82 3 3.7.2 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) 3.7.3 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 = 1 10 3 10 5 C D = 1.2 Re = 3 10 5 drag crisis ϕ sep ( ) f (Strouhal number)st St = fd U (3.119)

3.7. 83 D U St = 0.2 ( ) Re 1 Oseen-Lamb C D = 8π Re[0.5 γ + ln(8/re)] (3.120) γ γ = 0.577216 C D 1 + 10.0Re 2/3 (3.121) 100 10 C D 1 0.1 10 0 10 2 10 4 10 6 R=VD/ 3.26: 100 C D 1 0.01 10 0 10 2 10 4 10 6 R=VD/ 3.27: 3.27 C D

84 3 C D 10 3 10 4 C D = 0.4 C D 0.4 Re = 3 10 5 (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 [ 1 + 3 Re 16 Re + 9 ] 160 Re2 ln Re (3.128) Stokes Oseen Re 1 0 Re 2 10 5 C D 24 Re + 6 1 + + 0.4 (3.129) Re

3.7. 85 3.7.4 3.122 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 2 3.28 Re 10 4 ( C D = 2.0) ( (2.1) ((1.6) (1.2) (2.3) (1.2)(1.7) (1.6)(2.0) Re = 5 10 4 C D = 6 10 3 Re = 10 6 C D 4.4 10 3

86 3 U CD 2.0 CD 1.2 2.1 1.7 1.6 1.6 1.2 2.0 2.3 3.28: 2 3 3 3.29 ( 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 = 1.15 1.0 0.90 2 0.85 4 0.87 2 3 3 3.7.5 (streamlined form) 2 C D = 2.0 C D = 1.1 C D = 0.15 2 C D = 1.2 2 : 1 C D = 0.6 4 : 1 C D = 0.35 8 : 1 C D = 0.25 C D = 0.3 2 : 1 C D = 0.2 4 : 1 C D = 0.15 8 : 1 C D = 0.1

3.7. 87 1.07 1.17 cube disk 0.81 1.4 cup 0.5 0.4 60 cone 3.29: 3 3.7.6 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 = 6 10 6 ) (wing section; airfoil) 12% NACA0012 ( 30% )

88 3 Re = 6 10 6 α = 12 C L = 1.0 C D C L = 0 C D = 0.01 C L = 0.8 ( 8 ) C D = 0.018 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 = 30 50 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) 10 6 10 7 FX FX63-167 Re = 10 6 α = 10 C L = 1.6566, C D = 0.0238, C M = 0.1449 C L C D C D 10 4

3.7. 89 (stall) 3 NACA63 3 018 ( 18%) NACA63 1 012 ( 12%) NACA64 006 ( 6%) ( ) 3.7.7 (viscous drag) 50% (induced drag) 40% ( 42% 8% ) (winglet) 40% 37% (wave drag) 18% 2 5 3 2 (turbulent boundary layer) (laminar boundary layer) B747 0.8 6 10 7 C D = 0.031 B787 C D = 0.024 C D0 (zero-lift drag coefficient) (biplane) C D0 = 0.0378 B52 C D0 = 0.012 ( ) V max 3 P C D0 S (3.134) P power S

90 3 3.7.8 (ballistic coefficient) C BC = M SC D (3.135) M S C D (bluff body) (bullet) C BC = 0.12 1.0

91 4 4.1 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 ( )

92 4 4.2 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 = 0.0242J/m sec deg n: Q: q: (heat flux

4.2. 93 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)

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) 4.4 4.5 4.74.1 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

4.3. 95 4.3 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)

96 4 4.28 4.14 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)

4.3. 97 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)

98 4 4.3.1 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)

4.4. 99 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) 4.58 4.60 ( 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

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 = 0.18 10 3 (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)

4.4. 101 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 2 + 2 T y 2 + 2 T z 2 + µφ (4.80) 4.76 4.78 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)

102 4 Gr Gr = Re gβ T L3 ν 2 (4.89) Re = U L ν (4.90) 4.87 1/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 + 4.94 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)

4.4. 103 Ra = Gr P r < 10 9 ( ), Ra = Gr P r 10 9 ( ) (4.99) (heat transfer) ( Nu Nu = 0.59Ra 1/4 10 4 < Ra < 10 9 ( ) (4.100) Nu = 0.13Ra 1/3 10 9 < Ra < 10 12 ( ) (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 0.714 20 C 7.03 20 C 10100 20 C P r P r > 1 > P r = 1 = P r < 1 <

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:

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

106 5 u u- t 0 t t0 +ƒ t 5.2: 5.1 5.4) 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)

5.1. 107 5.1 (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) 5.1.1 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)

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 5.1.2 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

5.1. 109 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)

110 5 5.2 ( 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) 5.255.2, = 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) 5.8 5.8 x u + y v + z w = 0 (5.30) x u = x u = 0 (5.31) (5.30) (divergence) 0 5.29 5.30 x u + y v + z w = 0 (5.32) 0 5.26 5.28 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)

5.2. 111 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

112 5 5.3 5.30 5.33 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) 5.50 5.51 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)