CAE ( 6 ) 2 k ε LES (Large Eddy Simulation) DNS (Direct Numerical Simulation) Pr Pr 1 1 (Pr = 0.7) Pr 1 Pr 1
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1 CAE ( 6 ) 2 k ε LES (Large Eddy Simulation) DNS (Diret Numerial Simulation) Pr Pr 1 1 (Pr =.7) Pr 1 Pr 1
2 T u Pr < 1 Pr ~ 1 Pr > 1 Tw 1: Pr Tmi y x Tmo TFD HFD x 2: (Hydrodynamially Fully Developed = HFD) Thermally Fully Developed = TFD 3 3 (q w = λ( T/ y) y= = onst.) (T w = onst.) 3 2
3 u (y) Tmi Tw Tw T (x, y) 3: (1) (2) x y u, v u x + v y = ( p 2 ) x + ν u x + 2 u 2 y 2 ( y = 1 p 2 ) ρ y + ν v x + 2 v 2 y 2 u u x + v u y = 1 ρ u v x + v v (1) (2) ρ, ν [kg/m 3 ] [m 2 /s] (3) [4] u T x + v T y = α ( 2 T x + 2 T 2 y 2 ) α [m 2 /s] u(x, y), v(x, y) T (x, y) T (1), (2) (3) 3
4 y x u 2H 4: y = 2H y = ±H u x =, v = u x v (4) (4) (2) = 1 p ρ x + ν 2 u y 2 = 1 (5) p ρ y u(x, y) u(y), p(x, y) p(x) (4) ν d2 u dy = 1 dp 2 ρ dx (6) (6) y 2 u = 1 ( ) dp y 2 + C 1 y + C 2 (7) 2µ dx y = ±H u = (7) = 1 ( ) dp H 2 ± C 1 H + C 2 (8) 2µ dx C 1, C 2 C 1 = C 2 = 1 2µ ( ) dp H 2 dx (9) 4
5 (7) u(y) u = 1 ( ) dp (y 2 H 2 ) = 1 ( dp ) (H 2 y 2 ) (1) 2µ dx 2µ dx y = (dp/dx) > µ y = u max u max = u y= = 1 2µ ( dp ) H 2 (11) dx u max (1) ( ) u = u max 1 y2 H 2 (12) u m u m 1 H udy 2H H = u [ ] H max y y3 = 2 2H 3H 2 3 u max (13) H u max u m T m T (y) y T m = H H H u(y)t (y)dy H u(y)dy = H H u(y)t (y)dy 2u m H (14) u m (14) ρ p (14) u(y) T av = H H T (y)dy 2.4 Θ Θ(x, y) = T w(x) T (x, y) T w (x) T m (x) (15) 5
6 T w, T m Θ(x, y) Θ(y) (15) y Θ(x, y) x = (15) x [( Θ x = 1 dtw (T w T m ) 2 dx T ) ( dtw (T w T m ) (T w T ) x dx dt )] m dx = (16) (17) (T w T m ) T x = (T w T m ) dt w dx (T w T ) dt w dx + (T w T ) dt m dx = (T T m ) dt w dx + (T w T ) dt m dx x T/ x T x = T T ( ) m dtw + T ( ) w T dtm (19) T w T m dx T w T m dx (19) h q w h(t w T ) T T w T m h (18) q w h(t w T m ) (2) q w q w = λ T = λ T λ (21) y y y= H y=h T (x, y) 6
7 dq w dx = h (22) (2) dt w dx dt m dx = (22) (23) dt w dx = dt m dx (19) T/ x x T x = dt w dx = dt m dx (24) = F untion(x) (25) T/ x T w T m x (25) 5(a) T m W x x + dx x ṁ p T m (x + dx) = ṁ p T m (x) + 2q w W dx (26) ṁ, p [kg/s] [J/(kg K)] (26) ṁ = ρu m W 2H dt m dx = q w ρ p u m H (27) 5(a) q w = onst. dt m /dx = onst. dt w dx = (19) T x = T w T dt m T w T m dx (28) = F untion(x, y) (29) T/ x T/ x x y (29) 5(b) 7
8 TFD Tw Tm = onst. Tw Tm Tmi (a) x Tw Tw =onst. Tm Tmi TFD (Thermally Fully Developed) (b) x 5: x 6 6 T w T T w (25) (3) 2 T x 2 T 2 y 2 (3) v = (3) u T x = T α 2 y 2 (31) (25) T/ x = dt m /dx (31) 2.6 T/ x 8
9 Tw Tw y qw x qw (a) (qw = onst.) Tw Tw y qw x qw (b) (Tw = onst.) 6: 2 T/ x 2 = (31) (31) (12), (13) 2 T = 3u ( ) ( ) m dtm 1 y2 (32) y 2 2α dx H 2 (32) A x A 3u ( ) m dtm 2α dx (33) y ( ) T = A y y3 + C y 3H 2 1 (34) ( ) y 2 T = A 2 y4 + C 12H 2 1 y + C 2 (35) C 1, C 2 y = T y = y = H T = T w (36) 1 y = 2 y = H T T w (36) (35) 9
10 C 1 = T w = A ( ) H 2 2 H4 12H 2 + C 2 = A 5H C 2 T (x, y) ( ) y 2 T (x, y) = A 2 y4 12H 5H2 + T 2 w (x) 12 [ ( = T w (x) AH2 )4 ( ) ] y y H H q w q w = λ T = 2λAH y y=h 3 (37) (38) (39) T m (x) (14) T m (x) = 1 H u T dy 2u m H H ( = 1 H 3u m 2u m H H 2 1 y2 H 2 ) { T w (x) AH2 12 [ ( )4 ( ) ]} y y dy (4) H H (4) η y/h η dy = Hdη (4) T m (x) = = 3T w(x) 4 ( 1 η 2 ) [ T w (x) AH ( 1 η 2 ) dη + AH2 16 ( η 4 6η )] dη 1 1 (η 2 1) ( η 4 6η ) dη = T w (x) 34AH2 15 (2), (39), (41) h h = q w T w T m = 2λAH/3 34AH 2 /15 = 35 λ 17 H (41) (42) Nu 2H Nu h (2H) λ = 35λ 17H 2H λ = 7 17 = (43) 1
11 1 2 Nu 1 (HFD, TFD) x / (D Re Pr) 7: (W ) D h D h = 4 = lim 2W H 4 W 2(W + 2H) = 4H (44) Nu = 8.24 Nu Nu = x D Gz (Re Pr D)/x Gz 7 Gz 1 >
12 Inlet y x Flow 2H 8: 3. Nu = u, v, p, T u = u, v = v, p = p, θ = (T T in) (45) U in U in ρuin 2 T in U in, T in x, y t x = x 2H, y = y 2H, t = t (2H/U in ) (46) u u u + u + v t x y = p x + 1 ( 2 u Re x + 2 u ) 2 y 2 v v v + u + v t x y = p y + 1 ( 2 v Re x + 2 v ) 2 y 2 θ θ θ + u + v t x y = 1 RePr ( 2 ) θ x + 2 θ 2 y 2 (47) (48) Re U in 2H/ν, Pr = ν/α (48) Re Pr Pe 12
13 3.2 q w q w = λ T ( ) Tin θ = λ y y= 2H y y = (49) λ [W/(m K)] (49) q w ( ) 2H qw = q w (5) λt in qw qw = θ y = θ i, j=1 θ i, j= (51) y y = (51) θ(i, ) θ(i, ) = θ(i, 1) + q w y (52) [ ] θ(i, N y + 1) q w h q w = h(t w T m ) (53) q w = Nu(θ w θ m ) (54) Nu = h 2H/λ x x Nu Nu m [ ] (54) (54) Nu = q w θ w θ m x θ w θ m (55) θ w = θ i, j= + θ i, j=1 2 (56) 13
14 (14) u θ (i, j) u θ y θ m [ ] θ m θ m = 1 1 u θdy u dy = 1 u θdy (57) 3.3 Nu = % y x SMAC (original version by Dr. T. Ushijima) Simplified Marker and Cell method Solving Heat Transfer in flow between parallel walls. impliit double preision(a-h,o-z) n parameter(n=2, nx=n*1, ny=n) p: u, v: phi: divup: up, vp: psi: the: (e.g. ) dimension p(:nx+1,:ny+1) +,u(:nx,:ny+1),v(:nx+1,:ny) +,phi(:nx+1,:ny+1),divup(1:nx,1:ny) +,up(:nx,:ny+1),vp(:nx+1,:ny) +,psi(:nx,:ny+1) +,the(:nx+1,:ny+1),the(:nx+1,:ny+1) loop loop=2 re re=5.d pr ( ) pr=.7d q_w=1.d 14
15 dx(=dy) dy=1.d/dble(n) dx=dy dx=dy*2.d dt dt=.2d dt=min(dt,.25*dx) ( ) dt=min(dt,.2*re*dx*dx) write(6,*) dt =,dt ddx=1.d/dx ddy=1.d/dy ddx2=ddx*ddx ddy2=ddy*ddy ddt=1.d/dt initial ondition iont=1 iont= if (iont.eq.1) then open(unit=9,file= fort.21 +,form= unformatted, status= unknown ) read(9) u,v,p lose(9) else do 131 j=,ny do 132 i=,nx u(i,j)=1.d v(i,j)=.d p(i,j)=.d the(i,j)=.d 132 ontinue 131 ontinue end if do 133 i=,nx p(i,ny+1)=.d u(i,ny+1)=.d the(i,ny+1)=.d 133 ontinue do 134 j=,ny p(nx+1,j)=.d v(nx+1,j)=.d the(nx+1,j)=.d 134 ontinue un=.d uw=1.d us=.d ue=.d vn=.d vw=.d vs=.d 15
16 ve=.d do 1 it=1,loop boundary ondition do 135 j=,ny right wall (east) or outlet v(nx+1,j)=v(nx,j) u(nx,j)=u(nx-1,j) p(nx+1,j)=. the(nx+1,j)=2.*the(nx,j)-the(nx-1,j) left wall (west) or inlet v(,j)=vw u(,j)=uw p(,j)=p(1,j) the(,j)=.d 135 ontinue u(,ny/2+1)=uw do 136 i=,nx lower wall (south) u(i,)=2.*us-u(i,1) v(i,)=vs p(i,)=p(i,1) the(i,)=the(i,1)+q_w*dy!heat flux onstant upper wall (north) u(i,ny+1)=2.*un-u(i,ny) v(i,ny)=vn p(i,ny+1)=p(i,ny) 136 ontinue the(i,ny+1)=the(i,ny)+q_w*dy!heat flux onstant do 81 j=1,ny do 82 i=1,nx nvt=(ddx*((the(i,j)+the(i-1,j))*u(i-1,j) + -(the(i,j)+the(i+1,j))*u(i,j)) + +ddy*((the(i,j)+the(i,j-1))*v(i,j-1) + -(the(i,j)+the(i,j+1))*v(i,j)))/2.d dift=(ddx2*(the(i+1,j)-2.*the(i,j)+the(i-1,j)) + +ddy2*(the(i,j+1)-2.*the(i,j)+the(i,j-1)))/(pr*re) the(i,j)=the(i,j)+dt*(nvt+dift) 82 ontinue 81 ontinue do 83 j=1,ny do 84 i=1,nx the(i,j)=the(i,j) 84 ontinue 83 ontinue 16
17 preditor step for u_ij (up-u)/dt=-dp/dx-duu/dx-duv/dy+(nabla)ˆ2 u write(6,*) up do 125 j=1,ny do 126 i=1,nx-1 vij=(v(i,j)+v(i,j-1)+v(i+1,j)+v(i+1,j+1))*.25 nvu=.5*ddx*(u(i,j)*(u(i+1,j)-u(i-1,j)) + -abs(u(i,j))*(u(i-1,j)-2.*u(i,j)+u(i+1,j))) + +.5*ddy*(vij*(u(i,j+1)-u(i,j-1)) + -abs(vij)*(u(i,j-1)-2.*u(i,j)+u(i,j+1))) nvu: nvu=ddx*((u(i+1,j)+u(i,j))**2 + -(u(i-1,j)+u(i,j))**2)/4.d + +ddy*((u(i,j+1)+u(i,j))*(v(i+1,j)+v(i,j)) + -(u(i,j)+u(i,j-1))*(v(i,j-1)+v(i+1,j-1)))/4.d fij=-ddx*(p(i+1,j)-p(i,j))-nvu + +ddx2*(u(i+1,j)-2.d*u(i,j)+u(i-1,j))/re + +ddy2*(u(i,j+1)-2.d*u(i,j)+u(i,j-1))/re up(i,j)=u(i,j)+dt*fij 126 ontinue write(6,63) (up(i,j),i=1,nx-1) 125 ontinue do 124 j=1,ny up(,j)=uw up(nx,j)=up(nx-1,j) 124 ontinue up(,ny/2+1)=uw*1.1 do 224 i=1,nx-1 up(i,)=2.*us-up(i,1) up(i,ny+1)=2.*un-up(i,ny) 224 ontinue for v_ij (vp-v)/dt=-dp/dy-duv/dx-dvv/dy+(nabla)ˆ2 v write(6,*) vp do 122 j=1,ny-1 do 123 i=1,nx uij=.25*(u(i,j)+u(i+1,j)+u(i,j+1)+u(i+1,j+1)) nvv=.5*ddx*(uij*(v(i+1,j)-v(i-1,j)) + -abs(uij)*(v(i-1,j)-2.*v(i,j)+v(i+1,j))) + +.5*ddy*(v(i,j)*(v(i,j+1)-v(i,j-1)) + -abs(v(i,j))*(v(i,j-1)-2.*v(i,j)+v(i,j+1))) nvv: nvv=ddx*((u(i,j+1)+u(i,j))*(v(i+1,j)+v(i,j)) + -(u(i-1,j+1)+u(i-1,j))*(v(i-1,j)+v(i,j)))/4.d + +ddy*((v(i,j+1)+v(i,j))**2 + -(v(i,j)+v(i,j-1))**2)/4.d gij=-ddy*(p(i,j+1)-p(i,j))-nvv + +ddx2*(v(i+1,j)-2.d*v(i,j)+v(i-1,j))/re + +ddy2*(v(i,j+1)-2.d*v(i,j)+v(i,j-1))/re vp(i,j)=v(i,j)+dt*gij 123 ontinue write(6,63) (vp(i,j),i=1,nx) 122 ontinue 17
18 do 121 i=1,nx vp(i,)=vs vp(i,ny)=vn 121 ontinue do 221 j=1,ny-1 vp(,j)=vw vp(nx+1,j)=vp(nx,j) 221 ontinue evaluate ontinuity write(6,*) evaluate ontinuity i= div=. do 112 j=1,ny do 111 i=1,nx divup(i,j)=ddx*(up(i,j)-up(i-1,j)) + +ddy*(vp(i,j)-vp(i,j-1)) div=div+divup(i,j)**2 i=i ontinue write(6,63) (divup(i,j),i=1,nx) 112 ontinue write(6,*) sqrt(div/dble(i)) solve the poisson equation (nabla)2 p=(nabla)up/dt by SOR write(6,*) solve the poisson equation for pressure initialisation do 17 i=,nx+1 do 18 j=,ny+1 phi(i,j)=.d 18 ontinue 17 ontinue eps=1.d-6 maxitr maxitr=nx*ny C C maxitr=nx*ny/1 alpha= alpha=1.7 do 1 iter=1,maxitr error=.d do 11 j=1,ny do 12 i=1,nx rhs=ddt*divup(i,j) resid=ddx2*(phi(i-1,j)-2.d*phi(i,j)+phi(i+1,j)) + +ddy2*(phi(i,j-1)-2.d*phi(i,j)+phi(i,j+1)) + -rhs den=2.d*(ddx2+ddy2) dphi=alpha*resid/den error=max(abs(dphi),error) phi(i,j)=phi(i,j)+dphi 12 ontinue 11 ontinue do 13 j=1,ny 18
19 phi(,j)=phi(1,j) phi(nx+1,j)=. 13 ontinue do 14 i=1,nx phi(i,)=phi(i,1) phi(i,ny+1)=phi(i,ny) 14 ontinue if (error.lt.eps) goto ontinue 998 ontinue write(6,*) iter =, iter, it, error pause if (iter.ge.maxitr) write(6,*) maximum iteration exeeded! orretor step do 15 j=1,ny do 151 i=1,nx-1 u(i,j)=up(i,j)-dt*ddx*(phi(i+1,j)-phi(i,j)) 151 ontinue 15 ontinue do 152 j=1,ny-1 do 153 i=1,nx v(i,j)=vp(i,j)-dt*ddy*(phi(i,j+1)-phi(i,j)) 153 ontinue 152 ontinue do 16 j=1,ny do 161 i=1,nx p(i,j)=p(i,j)+phi(i,j) 161 ontinue 16 ontinue hek the ontinuity for n+1 th step write(6,*) hek the ontinuty for n+1 th step i= div=. do 155 j=1,ny do 156 i=1,nx divup(i,j)=ddx*(u(i,j)-u(i-1,j))+ddy*(v(i,j)-v(i,j-1)) i=i+1 div=div+divup(i,j)**2 156 ontinue write(6,63)(divup(i,j),i=1,nx) 63 format(2(1x,e1.2)) 155 ontinue write(6,*) sqrt(div/dble(i)) ********************************************* i=nx ********************************************* in=nx tm=.d um=.d do 17 j=1,ny tm=tm+.5d*(u(in,j)+u(in-1,j))*the(in,j)*dy 19
20 um=um+.5d*(u(in,j)+u(in-1,j))*dy 17 ontinue tm=tm/um tw=.5d*(the(in,)+the(in,1)) s_nu=q_w/(tw-tm) write(6,*) s_nu 1 ontinue output do 491 i=,nx psi(i,)=.d do 492 j=1,ny+1 psi(i,j)=psi(i,j-1)+.5*dy*(u(i,j-1)+u(i,j)) 492 ontinue 491 ontinue do 51 j=1,ny do 52 i=1,nx x=dx*(dble(i)-.5) y=dy*(dble(j)-.5) u=.5*(u(i,j)+u(i-1,j)) v=.5*(v(i,j)+v(i,j-1)) p=p(i,j) divu=divup(i,j) psi=.5*(psi(i,j)+psi(i-1,j)) u,v P write(1,699) x,y,u,v,p,divu,psi,the(i,j) 699 format (8(1X,E12.5)) 52 ontinue write(1,*) 51 ontinue do 53 j=1,ny-1 do 54 i=1,nx-1 x=dx*dble(i) y=dy*dble(j) u=.5*(u(i,j)+u(i,j+1)) v=.5*(v(i,j)+v(i+1,j)) omega=dx*(v(i+1,j)-v(i,j))-dy*(u(i,j+1)-u(i,j)) psi=.5*(psi(i,j)+psi(i,j+1)) C x y u,v write(11,699) x,y,u,v,omega,psi 54 ontinue write(11,699) 53 ontinue write(21) u,v,p end 2
21 y/(2h)=.5 u / Uin 1..5 y/(2h)=.1 y/(2h)= x / (2H) : (T - Tin)/Tin.6.4 y/(2h)=.5 y/(2h)= x / (2H) : 5(a) 21
22 y / (2H) x / (2H) 1 11: u y / (2H) x / (2H) : θ 1., (1995), 2. JSME, (25), 3. White, F. M. Heat and Mass Transfer, (1988), Addison-Wesley. 4., , (24.1), pp ( 22
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