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1 $arrow$ $\yen$ T (Yasutala Nagano) $arrow$ $\yen$ - 1 7?,,?,, (1),, (, ),, $\langle$2),, (3),, (4),,,, CFD ( ),,, CFD,,,,,,,,, (3),

2 $\overline{uv}$ 106 (a) (b) $=$ 1 - (5), 2,,,,, ( $=$ ) ( ), 1 $\overline{vuv}$, $\overline{vv\theta}$, $\langle$3) $D${aly-Harlow Wyngaard-Cot\ e,,

3 107 $=$ 2 - $\langle 7)$ (6) $\overline{u_{i }u_{j}u_{k}}=-c_{s}\frac{k}{\epsilon}\overline{u_{k}u_{\ell^{f_{\partial x_{\ell}}^{fi_{i}\mathcal{t}\overline{u_{j}}}}}}$ (1) $\overline{u_{i}u_{j}\theta}=-c_{\theta}\frac{k}{\epsilon}\overline{u_{j}u_{\ell}}\frac{\theta\overline{u_{l}\theta}}{\partial x_{\ell}}$ (2) $k=\overline{\tau 4u_{i}}/2$ $\epsilon=\nu\overline{(\theta u_{i}/\partial x_{j})^{2}}$, $k$,, $\langle$, 8x9),,,,,, (3) $(5)$,,,

4 $\overline{vv\theta}$ 108 (a) (b) 3 $\underline{=}$, 2, $\underline{=}$ $\overline{vuv},$,,,,, 1, $=$ $\overline{vv\theta}$ (3) $\overline{vuv}$, $\ovalbox{\tt\small REJECT}(\hat{v}\hat{u}\hat{v})=\int_{0}^{\infty}\int_{-\infty}^{\infty}\{P(\frac{x}{\hat{v}^{2}},\hat{v},\hat{\theta})+P(\frac{x}{\hat{v}^{2}},$ $- \hat{v},\hat{\theta})\}\frac{d\hat{\theta}d\hat{v}}{\hat{v}^{2}}$ $= \frac{1}{\pi}\sum_{\ulcorner-even}^{k\leq 4}C_{1\eta 0}\int_{0}^{\infty}H_{p}(\frac{x}{\hat{v}^{2}})H_{q}(\hat{v})\exp[-\frac{1}{2}\{(\frac{x}{\hat{v}^{2}})^{2}+\hat{v}^{2}\}]\frac{d\hat{v}}{\hat{v}^{2}}$ (3) $P_{III}( \hat{v}\hat{v}\hat{\theta})=\frac{1}{\pi}\sum_{q=even}^{k\leq 4}c_{w_{0}}\Gamma^{J}H_{q}(\hat{v})H_{f}(\frac{x}{\hat{v}^{2}})\exp[-\frac{1}{2}\{\hat{v}^{2}+(\frac{x}{\hat{v}^{2}})^{2}\}]\frac{d\hat{v}}{\hat{v}^{2}}$ (4) $(\wedge)$, $K=p+q+r$, Ims, $P$

5 $\check$ 109 $\langle$3) $P( \hat{u},\hat{v},\hat{\theta})=\frac{1}{(2\pi)^{3/2}}\sum_{p,q,r=0}^{\infty}c_{pqr}h_{p}(\hat{u})h_{q}(\hat{v})h_{r}(\hat{\theta})\exp\{-\frac{1}{2}($ $+\hat{v}^{2}+\hat{\theta}^{2})\}$ (5), C, $H_{n}(\chi)=(-1)^{n}\exp(\chi^{2}/2)$ $d^{n}\{\exp(-\chi^{2}/2)\}/d\chi^{n}$ 2 (3)(4),, $P$,, $\underline{=}$, $(u\hat, \hat{v})$, $(u$, W W $(\hat{u}$,, 3 (5) (, ), $(u, v)$, $=$, $\overline{vv\theta}$ $\overline{vuv}$ (3) 3, ( 2 : $u<0,$ $v>0$ ) (, $u$ $i$,, $Qi$ $\overline{(\chi)_{2}}$ $\overline{\chi}$ ), (3) $(5)$ ( $i$ $\overline{(\hat{u}^{\ell}\hat{v}^{m}\hat{\theta}^{n})}_{i}=\sigma_{u,1}^{\ell}\sigma_{v,i}^{m}\int_{0}^{\infty}[\int_{0}^{\infty}\{\int_{-\infty}^{\infty}\hat{u}^{\ell}\hat{v}^{m}\hat{\theta}^{n}p(\sigma_{u,i}\hat{u},\sigma_{v,i}\hat{v},\hat{\theta})d\hat{\theta}\}d\hat{v}]d\hat{u}$ $= \frac{2}{(2\pi)^{3/2}}$ $\sum_{p,qr=d,n+r=even}^{k\leq 4}\sigma_{u,i}^{\ell+p}\sigma_{v,i}^{m+q}C_{pqr}B_{\ell,p}B_{m_{2}q}B_{n,r}$ (6) $\grave$ C $B_{j,k}= \int_{0}^{\infty}\chi^{j}h_{k}(\chi)\exp(-\not\in/2)d\chi,$ $\sigma_{x}$ 1 $(+1,$ $x\geqq 0;-1,$ $x<$ $0)$ $\overline{(vuv)_{i}}$ 4, $((\overline{vuv})_{i}$ $\overline{(vv\theta)_{i}}$ (6) : $l=$

6 110 (a) (b) 4 1, $m=2,$ : $n=0;\overline{(vv\theta)_{i}}$ $\overline{uv}$ $P=0,$ $m=2,$ $n=1)$ 4, $\overline{vv\theta}$ $\overline{vuv}$, $Q2$ ( ) $Q4$ ( ), $\overline{vuv}$ $\overline{vv\theta}$ ( ), ( $Q1$ $Q3$ ),,,,,,, (6) $3$ $\equiv$, : $i)=-$, ; h $)$,,,, 4

7 111,, $=$, $\overline{vuv}$ 4 $\overline{vuv}\simeq\overline{(vuv)_{2}}+\overline{(vuv)_{4}}$ (7), (5), (6) $\overline{vv\theta}$ $\overline{vuv}$,, $\overline{\hat{v}\hat{u}\hat{v}}=c[s(u)+\sigma_{\overline{uv}}(\pi/2)s(v)]$ (8) $\overline{\hat{v}\hat{v}\hat{\theta}}=c[\sigma_{\overline{\tau\theta}}(\pi/2)s(v)+s(\theta)]$ (9) $C=1/[3\{(\pi/2)^{2}-1\}]$ (10) $(=\overline{\hat{\chi}^{3}})$, $S(\chi)$ $\chi$ $\sigma_{\overline{uv}}$, $\sigma_{\overline{v\theta}}$ $=$ $=$ :i) $=$ ;ii) ; i\"u) $-=$, ( ) 5 6, ( ) 6, 5, 6,,

8 112 (a) (b) (c) 5 ( ) 6 $=$ $=$, (10) ( 1 $\overline{vuv}$ ) 7, (1),

9 $\partial$ $\epsilon$ ,,,, $X^{12)}$ (11), 2, (, ), 2 (13) 2 $k-\epsilon$ $\nu_{t}$,, $\alpha_{t}$ ( ui $\theta=\alpha$ t $\overline{\theta^{2}}$ $\epsilon$ $\Theta$/ $\partial$, $\epsilon_{\theta}=\alpha\overline{(\partial\theta/\partial x_{j})^{2}}$, Prt $=$ $\alpha$t,, 2 (14),

10 114, 1/3,, $k-\epsilon 2$ (15),, $\partial\overline{u}/\partial y$ $\overline{uv}\simeq 0$ $\langle$ (16) 17) $\epsilon$, (18), CFD, $X$ ffi (1) Kline, S J et al, The stmcture of turbulent boundary layers J Huid Mech, 30 (1967),pp (2) Lumley, J L, Ttirbulence modeling Trans ASME$\cdot$ pp J Applied Mech, 50 (1983), (3) Nagano, Y and Tagawa, M, Statistical characteristics of wall turbulence with a passive scalar J Fluid Mech, 196 (1988), pp (4) Kasagi, N, Hirata, M and Nishmo, K, Streamwise pseudovortical stmctures and associated vorticity in the near-wffi region of a wall-bounded turbulent shear flow $Exp$ $\prime 4$ Fluids (1986), pp (5) Nagano, Y and Tagawa, M, A structural turbulence model for triple products of velocity and scalar J Fluid Mech, 215 (1990), pp

11 115 (6) Launder, B E, On the computation of convective heat transfer in complex turbulent flows, Trans ASME$\cdot$ J Heat Transfer, 110(1988), pp (7) $La\iota mder$, B E, Phenomenological modelling: Present $\ldots\ldots$ and Fhture? Iaecture Notes $\acute{\iota}n$ (J L $si_{c3}$ $L$ 7 $y$ ed), $357(1989),$ $pp$ $ ,$ Sprin$ger-Verlag$ (8) Hishida, M and Nagano, Y, Turbulence measurements with symmetrically bent V-sharped hot-wires Part 1: Principles of operation, Trans ASME$\cdot$ Engng, 110(1988), pp J Fls$\iota$i& (9) Hishida, M and Nagano, Y, $T\iota irbulent$ measurements with symmetrically bent V- sharped hot-wires $Pa$ rt 2: Measuring velocity components and turbulent shear stresses, $\mathcal{i}\succ ans$ ASME$\cdot$ J Fluids Engng, 110(1988), pp (10) Nagano, Y and Tagawa, M, Turbulence model for triple velocity and scalar $corr\triangleright$ lations r $ent$ Shear 1 $ows7$ ed F $($ $D$ $rst$ et $gj)_{f}(1991),$ $pp47-62$, Springer- Verlag (11) Hishida, M and Nagano, Y, Stmcture of turbulent temperature and velocity fluctuations in the thermal entrance region of pipe Heat Transfer 1978, 2 (1978), pp $-$ , Hemisphere (12) Nagano, Y and Hishida, M, Improved form of the $k-\epsilon$ model for wall turbulent shear flows Trans ASME: J $Flui$ $gng,$ $109(1987),$ $pp$ $ $ (13) Nagano, Y and Kim, C, A $t_{w(>4}quation$ model for heat transport in wall turbulent shear flows Trans ASME$\cdot$ J Heat $Trar\iota sfer$ $110$ (1988), pp

12 116 (14) Youssef, M S, Nagano, Y and Tagawa, M, A two-equation heat transfer model for predicting turbulent thermal fields under arbitrary wall thermal conditions, Int J Heat Mass Transfer, 35 (1992), pp (15) Nagano, Y and Tagawa, M, An improved $k-\epsilon$ model for boundary layer flows Trans ASME$\cdot$ J Fluids Engng, 112 (1990), pp (16) Tsuji, T and Nagano, Y, Characteristics of a turbulent natural convection boundary layer along a vertical flat plate Int J Heat Mass Transfer, 31 (1988), pp (17) Tsuji, T and Nagano, Y, Turbulence measurements in a natural convection boundary layer along a vertical flat plate Int J Heat $Mass$ Transfer, 31 (1988), pp (18) Nagano, Y, Yin, Y and Tsuji, T Numerical prediction of turbulent buoyant flows Proc $7ffl$ Symp on $T$ $rb$ lent Shear Flows, (1989), $pp$ , Stanford Univ

NUMERICAL CALCULATION OF TURBULENT OPEN-CHANNEL FLOWS BY USING A MODIFIED /g-e TURBULENCE MODEL By Iehisa NEZU and Hiroji NAKAGA WA Numerical calculat

NUMERICAL CALCULATION OF TURBULENT OPEN-CHANNEL FLOWS BY USING A MODIFIED /g-e TURBULENCE MODEL By Iehisa NEZU and Hiroji NAKAGA WA Numerical calculation techniques of turbulent shear flows are classified