1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1

1 1.1......... 1............. 1.3... 1.4......... 1.5.............. 1.6................ Bownian Motion.1.......... Einstein.............. 3.3 Einstein........ 3.4..... 3.5 Langevin Eq.... 3.6................ 4.7.................... 4 3 5 3.1................... 5 3............ 5 3.3................. 5 3.4 Navie-Stokes............ 5 3.5................. 5 3.6 Benoulli.............. 5 3.7............ 5 3.8 Reynolds............. 6 3.9.......... 6 3.1.......... 6 3.11 Boltzmann....... 6 6.3........ 11 6.3.3 Kaman-Howath...... 11 6.3.4............... 11 6.4............ 11 6.4.1........ 11 6.4. Reynolds..... 11 6.4.3 Kolmogoov...... 1 6.4.4 Taylo............. 1 6.5................ 1 6.6 Kolmogoov -5/3 K41. 1 6.7 K6 K41............. 1 6.8... 13 7 13 8 14 8.1... 14 8.... 14 8.3............... 15 8.4.............. 15 8.5............... 15 8.6 K ɛ..... 15 8.7... 16 9 16 9.1................ 16 9........... 17 4 Reynolds 7 4.1 Reynolds Reynolds... 7 4. Reynolds........... 7 4.3 Pandtl............ 7 4.3.1........... 8 4.4 Hagen Poiseuille........ 8 5 9 5.1........ 9 5. Kolmogoov... 9 5.3 MHD.............. 9 6 1 6.1... 1 6......... 1 6.3 Kaman-Howath.......... 11 6.3.1 Navie-Stokes... 11 1

1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1 4 x =Dt 5 1.3 W l, n n l n 1 W l, n = n C n+l 6 n! = 7 n + l/! n l/! πn exp l n l 1 8 n Stiling log n! n + 1 log n n + 1 log π 1.4 x = la, t = nτ 9 W x, n x = W l, n x a 1 = exp x x 1 4πDt 4Dt D = a τ 1.5 = [L] [T ] 11 n +1 x n x a x + a W x, n +1= 1 W x a, n+ 1 W x + a, n 1 1.6 W t τ = a W x 13 n x n x n =x n x n 1 +x n 1 x n +...+x 1 x +x 14 ±a x n x n = na =Dt 15 /4/1 Bownian Motion.1 F x v = βfx β n t = Γ x = D n x x βfn 16

. Einstein Γ=D n βfn =, Ux = dxf x 17 x n = n exp Maxwell n = n exp Einstein βux D Ux kt 18 19 D = βkt Bown Renolds β = 1 1 6πη Einstein D = kt 6πη, x =Dt = kt 3πη t x.3 Einstein Bow P = nkt f = 1 n kt 3 n x f = f f v = f/6πη Γ= D n x + nf 6πη = 4 D = kt/6πη.4 pq W l, n = n C l p l q n l 1 l np exp πnpq pqn 1 x vt exp πdt πdt 5 D = pqa /τ v = pa/τ Stiling.5 Langevin Eq. ζ mftf t ft =, ftft =Aδt t 6 Langevin Foke-Plank 1: m =,Ft = W x =, ζ dx dt W t m =,Ft W ζ dx dt x = F ζ t, W t = ft 7 x = A t 8 ζ = D W x 9 /4/17 = F t+ft 3 x = A t 31 ζ = D W x x βfxw 3 m,ft = Langevin d x dx dt = ζ + ft 33 dt <x > < v>=< dx/dt > d <x > dt + ζ d <x > = v = A dt ζ = kt 34 3

<v>=, v w < x >= A ζ t = kt ζ t 35 w t = w A x + ζvw 36 v w exp ζv /A t ζ 1 w W w exp v /kt, m,ft dv dt W exp x x /4Dt 37 = ζv + F x+ft 38 < x >= v t, < v >= ζv + F t, < x > < v x >=, < v >=A t 39 v w Foke-Plank w t.6 = v w x + v ζv F w + A w v 4 Langevin <vtvt >= A ζ e ζ t t,<vt >= kt 41 <vtv >= kt ζ β = 1 ζ = 1 kt ζ = 1 kt, < ftf >= ζktδt 4 dt < vtv > 43 dt < ftf > 44 m d dt <v>= mζ < v > +ee, J = ne < v > 45 < v > = ee/mζσ = J/E = ne /mζ jt = n i ev i d jt = ζjt+e dt m n ɛ i t 46 i=1 e4 m <ɛ i tɛ l t >= Aδ il δt t σ = 1 dt < jtj > 47 kt 1 σ = 1 dt < ɛtɛ > 48 kt.7 P = P t + t P t P t + t =bt P t+ft 49 ft bt Langevin d P t = ν P t+ft dt ν =1 bt/ t F t =ft/ t bt > 1 P 49 P t + t = bt P t + ft P t ft = 1 bt bt P 1/4/18 m d x dt + mω x = F t 5 Fouie mω x + mω 1 + iφ x = F 51 φ β = F ω x ω ω φ m ω ω + ω 4 5 φ x = v /ω kt ω φ mω ω ω + ω 4 53 φ φ ω ω ω ω 4

3 3.1 ρ + ρ v = 54 t ρ ρ = onst. 3. substantial mateial deivative 3.3 ρ v t v = 55 D Dt = + v 56 t + ρ v v = + 57 ρ v vdv = ρ v v nds 58 ρ gdv ρ v v v ρ t 59 σ xy :y x σ xy = µ dv x dy 6 σ nds = σ 61 3.4 Navie-Stokes σ = p I τ σ ij = pδ ij τ ij 63 τ i j vi τ ij = µ + v j 64 x j x i µ Navie-Stokes τ = µ v 65 D v Dt = 1 ρ p + ν v + g 66 ν = µ/ρ 3.5 1/4/5 v N-S Eq. t + v v = 1 ρ p + ν v φ g = φ ω v v + ω v = t + P + φ ν ω 67 p/ρ = P ω t + v ω =ω v + ν ω 68 3.6 Benoulli t =, g = φ, ν = v v ω = +P+φ 69 v +P+φ = onst. 7 ρ D v Dt = σ + ρ g 6 3.7 ν = ωds = onst. 5

3.8 Reynolds N-S Eq. U, L N-S Eq. v = U v,x= Lx,y= Ly, z = Lz, t =L/Ut, P = ρu P Dv Dt = P + ν UL v 71 R = UL/ν:Reynolds numbe 3.9 1/5/ v / ρ D v Dt = ρ v D v Dt = v p τ + ρ g 7 54 ρv t ρv v = p v+p v τ v+ τ v + ρ v g 73 1.. 3. 4. 5. 6. 5 µ vi τ v = + v j x j x i < 74 3.1 U dv ρu + ρv / v q q p v τ v ρ g v 73 ρ D Dt U = q p v τ v 75 3.11 Boltzmann δf δt f q, p, t Liouville df dt = f F δf + v f + t m v f = 76 δt Boltzmann n n fd v 77 g <g> fgd v <g> 78 fd v f, n, g g f t d v = t n<g> n t g f gv i d v = n<v i g> n v i g x i x i x i g F i f d v = n gf i 79 m v i m v i Boltzmann g t n<g> n + n <g v> t n< g v > n F m δf v g = g d v8 δt g =1 n t + n< v>= δf δt d v 81 6

8 g = m v mn < v >+ mn < v j v> n < F>= t x δf m v d v j δt 8 v =< v>+ v mn D Dt < v>= n< F> P τ + R 83 P = nm < v >/3 τ ij = nm < v i v j <v >δ ij /3 > R = m v δf δt d v 8 g = m v/ nm > t <v + nm <v v> n F v v mv δf = d v 84 δt v =< v>+ v nm t <v> + 3 p nm + <v> + 5 p < v>+τ < v>+ q = nf < v>+ R < v>+ Q + m δf <v> d v 85 δt q = nm <v v > andom motion Q = m δf v δt d v 81, 83 nm t <v> = m δf n <v> < v>+ d v δt mn < v > < v> < v>+n < F> < v> < v> p < v> τ + R < v> 86 73 85 86 3 p + 3 t p< v>+p < v>+ τ < v>+ q = Q 87 3 ndt Dt + p < v>= q τ < v>+ Q 88 1/5/16 4 Reynolds 4.1 Reynolds Reynolds Navie-Stokes v t + v v = 1 ρ p + 1 ρ τ v = V + v V = v p = P + p τ = T + τ Vi T ij = µ + V j x j x i v,τ ij = µ i + v j x j x i Reynolds Navie-Stokes V V + v v = 1 ρ P 1 ρ T 89 v = v v 89 V V = 1 ρ P T + ρ v v 9 ρ v v Reynolds Boltzmann P ij = nm < v i v j > 4. Reynolds Reynolds τ 1 µ V 1 x µ U L 91 ρv 1 v ρv 9 ρv 1 v τ 1 v R 93 U Reynolds Reynolds 4.3 Pandtl ξ x = ξ V 1 x 1 v th x = 7

<mv 1 ξ mv 1 > ρ m <mv 1 ξ mv 1 >v th ρ V 1 x ξv th = µ V 1 x 94 µ = ρξv th ρv l ρv = ρv 1 l ρv 1 + ρv 1 l ρv 1 ρ V 1 x l 95 l Pandtl ρ V1 x lv Reynolds ρv v V1 x v 1 v 1 v ρl V 1 V 1 96 x x ρl V 1 x ρlv 1, µ = ρv th ξ 97 ρlv µ = lv ν Reynolds 4.3.1 98 n v =, v =n, v n t + n v = n t + v n = 99 γ n e γt 1 γn + v n = v = n L n n L nγ 11 L n l= n = n l L n 1 Γ=<n v > D n = D n L n 13 11,1 D = l γ = γ k 14 k 11 4.4 Hagen Poiseuille N-S Eq. z ν v p d ρ = ν d + 1 d v z 1 dp d ρ dz = 15 v z = 1 dp 4ρν dz a 16 Hagen Poiseuille V a V = πv zd πa = a dp 17 8ρν dz λ dp dz = λ 1 ρv a 18 λ = 64 19 R R = V a/ν Reynolds Reynoolds R< 3 R > 3 λ =.3164R 1/4 11 8

1/5/3 Kolmogoov Kolmogoov 5 kolmogoov 5/3 5.1 Kolmogoov ɛ ɛ d v dt v v v3 111 L L k L 1 µ v τ v ρ ρ v ν l 11 l l ɛ l 1 ɛ L K ɛ V 3 K L K ν V K l 113 ɛ, ν L K = T K = ν 3 1/4 ɛ ν 1/ ɛ V K = νɛ 1/4 114 5. Kolmogoov Ek Ekdk = v 115 [Ek] = L3 T 116 [ɛ] = L T 3 117 ɛ, k Ek k α ɛ β L 3 =[Ek] = L α T 3=β α, =3β α = 5 3, β = 3 L T 3 β 118 Ek ɛ /3 k 5/3 Kolomogoov 5/3 ν v l νv k νek k νk 5/3 k k νk 1/3 k 119 k Kolomogoov 5/3 5.3 MHD MagnetohydodynamiMHD v b Ek, F k MHD v A = B µρ T [ɛ] = [v3 ] [L] = [v3 ] B T = L3 T 4 1 B 1 9

Ek k α ɛ β L 3 L 3 β 1 =[Ek] = L α T T 4 B 3=3β α, =4β α = 3, β = 1 11 Ek ɛb 1/ k 3/ Kaihnann 6. Kaman-Howath A, B < u A p B >: < u A u B >: < u A u A u B >: 17 6 Kaman-Howath : : : B i =A i B ij =A 1 i j + B 1 δ ij B ijk =A i j k + B k δ ij +C j δ ik + D i δ jk 18 6.1 Reynolds Kamann-Howath N-S Eq. t u i ν u i = u j 1 p 1 x j ρ x j OpeatoL, M, L 1 L u = M u u + L 1 p 13 L < u>= M < u u >+L 1 <p> 14 u Reynolds Pandtl < u u > < u> 14 L < u u >= M < u u u >+L 1 <p u> 15 L < u u u >= M < u u u u >+L 1 <p u u > 16 Kaman-Howath 15 Q ij Q ij = u ia u jb AB Q ij =Q ji A 1,B 1 1/5/3 =,, Q ij Q ll = u la u lb Q nn = u na u nb Q ij = i j Q ll Q nn +Q nn δ ij 19 S ij,k S ij,k = u ia u ja u kb =,, S ij,k = S ji,k C = D S S 11,1 = Q lll S,1 = S 33,1 = Q nnl S 1, = S 13,3 = Q lnn S ij,k = i j k 3 Q lll Q nnl Q lnn + kδ ij Q nnl + iδ jk + j δ ik Q lnn 13 1

S ij,k = S k.ij 131 6.3 Kaman-Howath K-H Eq. 6.3.1 Navie-Stokes N-S Eq. A B N-S Eq. u i t + u i u k = 1 p + ν u i x k ρ x i u j t + x u j u k = 1 p + ν u j 13 k ρ x i u j, u i t ν Q ij = = S ik,j S i,jk k S ik,j + S jk,i T ij 133 k pu j = p u i = 6.3. i Q ij = Q nn = Q ll + Q ll = 1 Q ll Q ii = i S ik,j = Q nnl = 1 Q lll Q lnn = 1 4 Q lll S ik.i = k 6.3.3 Kaman-Howath +3 Q ll 134 + 4 Q lll 135 134 i = j 133 +3 t Q ll ν + +3 Q ll 135 +3 + 4 Q lll [ t ν + 4 ] Q ll = + 4 Q lll 136 Kaman-Howath N-S Eq. t u i ν u i = u i u k 1 x k ρ p 137 x i u Al u Bl = Q ll u Al u Al u Bl = Q lll 6.3.4 k Q ij =u ia u jb = + Φ ij ke i k d 3 k 138 S ij,k Φ ij,k 133 t Φ ij = ik α Φ iα,j +Φ jα,i νk Φ ij 139 t Ek, t =4πk6 Γk, t νk Ek, t 14 Ek, tdk = u i / Γk, t Φ iα,j 6.4 6.4.1 1/6/6 l e v e mve/ ρlev 3 e/ µ v y l e µl ev e = µl e vet t t ρl3 e v e µl e v e ρl e µ l e ν 6.4. Reynolds 141 Reynolds ρu i u j l E v E Reynolds ρve l E ρve 3 l E 11

ve 3 /l E ɛ ɛ K-H Eq.136 Q ll t + 4 Q lll 14 t E v E t E v3 E l E t E l E v E 6.4.3 Kolmogoov K-H Eq.136 Q ll ν t + 4 Q ll 143 t K v K t K ν v K l K ɛ t K l K /v K v K ν l K lk R K v Kl K ν v K ν l K ɛ ν3 l 4 K 1 Kolomogoov 114 GOY i k u i k N-S Eq. d dt u i k+νk u i k= i k j δ il k ik l k u j pu l q p+ q= k 144 p + q = k GOYGledze-Ohkitani-Yamada k i = i k i =,..N 1 u i t i ii iii d dt + νk u i =+i 1 i u i+1 u i+ + i u i 1u i+1 + 3 i u i 1u i 145 1 i = k i, i = 1 k i 1, 3 i = 1 k i 1 N =1 N 1 =, = N 1 =3 = 3 1 = GOY Kolmogoov 6.4.4 Taylo Q ll = MioSale : λ l Q ll Q ll Q ll MaoSale : Λ l Q ll d l K <λ l < Λ l 6.5 N-S Eq. 6.6 Kolmogoov -5/3 K41 ɛ K41 ɛ ɛ v p /ɛ/k p/3 k 6.7 K6 K41 1/6/13 ɛ ɛ ɛ x, t = 3 4π 3 ɛ x + y, tdy 146 y < 1

ɛ ln ɛ L l n L l 1 = L/Γ l = l 1 /Γ l n = L/Γ n ɛ = ɛ ɛ 1 ɛ ɛ n ɛ = ɛ n = ɛ ɛ 1 ɛ ɛ ɛ 1 ɛ n 1 ln ɛ = lnɛ +lne 1 +lne + +lne n 147 e i ɛ i /ɛ i 1 ln ɛ 1 P ɛ = exp ln ɛ m 148 πσ ɛ σ m =lnɛ σ, σ = A + µ ln L/ µ ln L/ ɛ /3 = ɛ /3 ɛ /3 P ɛ dɛ ɛ /3 ɛ n µ/9 L 149 v ɛ/k 1/3 Ek v k ɛ/3 k 5/3 k µ/9 15 K41-5/3 µ/9 v p /ɛ/k p/3 k µpp 1/3 6.8 1/k N N = k D D k D /k 3 ɛ ɛ ɛ = k3 k D ɛ = ɛ k 3 D ɛ /3 = ɛ /3 k D /k 3 = ɛ /3 k D 3/3 Ek = ɛ /3 k 5 /3 = ɛ /3 k D 3/3 k 5/3 151 1/6/ 7 Reynolds Reynolds x y U s x lx x Lx Ux, y L l L U y/lx U s x x, y U, V, u, v U s, L, l U x + V y = V = OU s l/lreynolds N-S Eq. y U V V +V x y + x uv+ y v = 1 P ρ y +ν U x + V y 15 y v = 1 P ρ y 153 L l, Reynolds Rynolds U s l L u l 156 x v = 1 P 154 ρ x N-S Eq. x U U x + V U y + x u v + y uv = ν U x + V y 155 154 U U x + V U y + uv = 156 y O U s u = O L l ξ = y/l, lx, U s x U = U s fξ 157 uv = Us gξ 158 13

f = O1, g = Ol/L V y V = U dy = l x ξ 156 dus dx f U s dl l dx ξf dξ 159 ξ ξ g = l du s U s dx f dl dx ξff l du s U s dx f fdξ+ dl dx f ξf dξ 16 x l = onst., du s U s dx dl dx = onst. l x x y x Us l = onst. l x U s 1/ x ν T Reynolds R T U s l/ν T Reynolds U uv = ν T y Us g = ν T l f g = f 16 f = 1 ξ R T f + f fdξ f = seh ξ/ = 8 R T 161 e ξ/ +e ξ/ 16 163 8.1 Navie-Stokes Diet Numeial Simulation DNS DNS Reynolds UL/ν Kolmogoov l K ν 3 /ɛ 1/4, t K ν/ɛ 1/ ɛ v 3 /l l k ν 3/4 L = R 3/4 UL T L/U t K ν 1/ T = R 1/ UL R 9/4 R 1/ = R.75 U = 1m/s 36km/h ν = 1 5 m /s L =m R 1 6 1 13.5 1 3 8. 1/6/7 Kaman-Howath Reynolds Reynolds Rynolds 14

8.3 Reynolds x U y U y l u v U y l Reynolds uv = l U y U y 164 Businesque Reynolds uv = ν T U y ν T 165 Reynolds Reynolds 8.4 N-S Eq. U i t + U j U i = P x j x i ρ + ν x j x j U i u i t + U u i U i j + u j = p x j x j x i ρ + ν x i Reynolds x j u j u i 166 ui + u j x j x i x i u i u j u i u j 167 K 1 u i 168 u j ɛ τ ij = ν uj u i uj u i x i x i x j x i x j K t + U K U j i = R ij x i x i x j [ pu j ρ + U i u j uj νu i + u i x i x j 169 ] 17 ɛ ɛ ɛ t + U i = W H j + ν ɛ 171 x i x j ui u k W ν + u j u j Ui +ν u i u i u j x j x j x k x i x k x k x j x k ν U i x j x k H j ν ρ 8.5 +νu j U j x k U i x j x k 17 u j p u i u i + νu j 173 x k x k x k x k K = u i u i / Reynolds K ν T R ij = u i u j = 3 Kδ Ui ij ν T + U j 174 x j x i Pandtl u i u j l U U x x ν T lu l K ν T U x ν T = Kl 175 l K 17 K t + U U i U j j + R ij x j x i = νt K K3/ x j σ k x j l 176 σ K, σ k ɛ = K3/ l K N-S Eq.166 U i x i, σ K l 8.6 K ɛ ɛ K ɛ ɛ µ ν T = C µ K ɛ 171 ɛ t + U ɛ ɛ i = C ɛ1 x i K R U i ɛ ij C ɛ x j K + x j ɛ νt σ ɛ 177 ɛ x j 178 15

N-S Eq. Reynolds 174 K K t + U U i U j j + R ij x j x i = νt K ɛ 179 x j σ k x j C µ, C ɛ1 C ɛ, σ ɛ 8.7 Lage Eddy Simulation, 1, G G U = Gx x ux dx = G u 18 u i = U i + u Rynolds G u i u j = G U i U j +G u i U j + U i u j + u i u j = U i U j + L ij + M ij 181 N-S Eq. U i t + U i U j = 1 P x j ρ x i x j L ij x j M ij + U i L ij = G U i U j U i U j M ij = Ui 3 K δ ij + ν SG + U j x j x i 18 183 L ij Lenad ν SG = C S 4/3 ɛ 1/3 u K = G i u j =C K /3 ɛ 1/3 184 C S, C K ɛ ɛ = M ij U i x j 185 Lage Eddy Simulation 9 1/7/4 Kolmogoov 9.1 Se.4.4 λ P. dx = λρv 186 d λ = 64 187 R λ =.316 R 1/4 188 λ N-S Eq. v t + v v = 1 ρ P + ν v x λ 1 x, v λ v, t λ 3 t, P/ρ λ 4 P/ρ, ν λ 5 ν, λ 1,λ, N-S Eq. λ λ 1 3 = λ λ 1 1 = λ 1 1 λ 4 = λ 5 λ 1 λ λ 3 = λ 1 λ 1, λ 4 = λ, λ 5 = λ 1 λ x λ 1 x, v λ v, t λ 1 λ 1 t, P/ρ λ P/ρ, ν λ 1λ ν, 189 dp/ρ dx = Cd α V β ν γ C λ λ 1 1 = λ α 1 λβ λ 1λ γ 19 α = 1 γ, β = α 191 dp dx = Cd 1 γ V γ ν γ = C V d ν γ dv γ dp dx = C V d F R 16

9. τ τ MHD τ v mn + v v = p + j B t n + n v = t v = E + v e B = η j p e en B t = E p 3/ η T 3/ 19 n x λ 1 x, v λ v, t λ 3 t, n λ 4 n, B λ 5 B, E λ 6 E, P λ 7 P, j λ 8 j, η λ 9 η 193 λ 4 λ λ 1 3 = λ 4 λ λ 1 1 λ 7λ 1 1 = λ 8 λ 5 λ 4 λ 1 3 = λ 1 1 λ 4λ λ 6 = λ λ 5 = λ 9 λ 8 = λ 7 λ 1 1 λ 1 4 λ 8 = λ 1 1 λ 5 λ 5 λ 1 3 = λ 1 1 λ 6 λ 9 = λ 3/ 7 λ 3/ 4 194 λ = λ P x λ 4 x, v λv, t λ 5 t, n λ 8 n, B λ 5 B, E λ 6 E, P λ 1, j λ 9 j, η λ 3 η 195 τ n T B a τ = Cn p T q B a s 196 C λ 5 = λ 8p λ q λ 5 λ 4s p + q + 5 4 s = 5 4 197 τ n p T q B a p+ q + 5 4 + 5 4 na p T a q +Ba 5/4 +1 B 1 198 τ = 1 B F na,t a, Ba 5/4 17