SIMPLE Algorithm for Two-Dimensional Flow - Fluid Flow and Heat Transfer

Transcription

1 SIMPLE Algorithm for Two-Dimensional Flow - Fluid Flow and Heat Transfer by Professor Jung-Yang San Mechanical Engineering Department National Chung Hsing University

2 Two-dimensional, transient, incompressible laminar flow neglecting body force: u v () Continuity: + = 0 x y ρ ( u) ( u ) ( uv) P e u u () Momentum: x-direction: + [ ρ ρ + ] = +µ ( + ) t x y x x y ρ ( v) ρ ( uv) ρ ( v ) Pe v v y-direction: + [ + ] = +µ ( + ) t x y y x y T T T T T (3)Energy : + [u + v ] = α ( + ) t x y x y For example: the geometry of the original problem (-D channel) is u y x v u L y L x

3 The SIMPLE (semi-implicit method for pressure-linked equations) algorithm can only be used to solve the velocity distribution of steady flow ( 穩態流 ). It can be arranged into a finite-difference scheme or a finite-volume scheme. In this class, a finite-difference scheme is developed for the SIMPLE algorithm. A large number of iterations involved in the calculation, thus the method is quite time-consuming. In the method, the Navier-Stokes equations for unsteady flow was used. Time progressing in calculation is treated as iteration steps.

4 As every grid point in the flow field is satisfied with the continuity equation and the momentum equations, the acquired velocity distribution is the true velocity distribution. The intermittent result during the iterations does not possess any physical meaning. The SIMPLE algorithm can also be used to solve the velocity distribution of three-dimensional flow whether the flow is laminar or turbulent.

5 Finite-Difference Approach (Staggered Grid - 交錯網格 ) y (i+/, j+) (i, j+/) pt. a (i+, j+/) (i-, j) (i-/, j) (i, j) (i+/, j) (i+, j) (i+3/, j) (i, j-/) pt. b (i+, j-/) dy dx (i+/, j-) Note: pressure grid points x x-dir. velocity (u) grid points y-dir. velocity (v) grid points

6 y (i-/, j+) (i, j+) (i+/, j+) (i-, j+/) pt. c (i, j+/) pt. d (i+, j+/) (i-/, j) (i, j-/) (i, j) (i+/, j) dx dy at point "a": v i+ /,j+ / (vi,j+ / + v i+,j+ /) at point "b": v i+ /,j / (vi,j / + v i+,j /) at point "c": u i /,j+ / (ui /,j + u i /,j+ ) at point "d": u i+ /,j+ / (ui+ /,j+ ui /,j ) + + x

7 centered around point (i + /, j), the x-direction momentum equation is solved ( J.D. Anderson, JR., Computational Fluid Dynamics, McGraw Hill 995). ( ρu) ( ρu) n+ n i+ /,j i+ /,j ( ρu ) ( ρu ) ( ρuv) ( ρuv) p p = [ + ] x y x n n n n n n ui+ 3/, j ui+ /,j + ui /,j ui+ /,j+ ui+ /,j + ui+ /,j +µ [ + ] ( x) ( y) n n n n n n i+ 3/,j i-/,j i+ /,j+ i+ /,j- i+,j i,j or ( ρu) = ( ρu) + A n+ n i+ /,j i+ /,j x (p p ) () n i+,j n i,j n n n n ( ρu ) i+ 3/,j ( ρu ) i-/,j ( ρuv) i+ /,j+ ( ρuv) i+ /,j- where A [ + ] x y n n n n n n ui+ 3/,j ui+ /,j + ui /,j ui+ /,j+ ui+ /,j + u + µ [ + ( x) ( y) i+ /,j ]

8 similarly, centered around point (i, j+ / ), the y-direction momentum equation is solved. ( ρv) ( ρv) n+ n i,j+ / i,j+ / ( ρv ) ( ρv ) ( ρvu) ( ρvu) p p = [ + ] y x y n n n n n n vi,j+ 3/ vi,j + / + vi,j / vi+,j+ / vi,j + / + vi,j+ / +µ [ + ] ( y) ( x) n n n n n n i,j+ 3/ i,j-/ i+,j+ / i,j+ / i,j+ i,j or n+ n n n ( ρv) i,j+ /= ( ρv) i,j+ /+ B (pi,j + p i,j) y () where n n n n ( ρv ) i,j+ 3/ ρ ( v ) i,j-/ ( ρvu) i+,j+ / ρ ( vu) i,j+ / B [ + ] y x n n n n n n vi,j + 3/ vi,j + / + vi,j / vi+,j+ / vi,j + / + vi,j+ / +µ [ + ] ( y) ( x)

9 n n n Assume a set of initial quessed (p ), ( ρu ) and ( ρv ) values, equations () n+ n+ and () can be calculated to yield a new set of ( ρu ) and ( ρv ) values, n+ n n n ( ρu ) i+,j / = ( ρu ) i+ /,j+ A [(p ) i+,j (p ) i,j] (3) x n+ n n n ( ρv ) i,j+ /= ( ρv ) i,j+ /+ B [(p ) i,j+ (p ) i, j] () 4 y let u u u ; v v ; p p p A A A ; B B B ' ' ' v ' ' Subtracting eq.(3) from eq.() and eq.(4) from eq.(), it yields ' n+ ' n ' ' n ' n ( ρu ) i+ /,j= ( ρu ) i+ /,j+ A [(p ) i+,j ( p ) i,j] (5) x ' n+ ' n ' ' n ' n ( ρv ) i,j+ /= ( ρv ) i,j+ /+ B [(p ) i,j+ (p ) i,j] (6) y ' ' ' where p, u, v are corrections ( 修正 ) of pressure and velocities respectively.

10 建議先將 與 設為 與 則會分別變為 : ' ' ' n ' n Patankar A, B, (pu ) (pv ) 0, eq. (5) eq. (6) ' n+ ' n ' n ( ρu ) i+ /,j= [(p ) i+,j (p ) i,j] (7) x ' n+ ' n ' n ( ρv ) i,j+ /= [(p ) i,j+ (p ) i,j ] (8) y 亦即 n+ n+ ' n ' n ( ρu) i+ /,j= ( ρu ) i+ /,j [(p ) i+,j (p ) i,j] (9) x n+ n+ ' n ' n ( ρv) i,j+ /= ( ρv ) i,j+ / [(p ) i,j+ (p ) i,j ] (0) y 同理, 對不同之點上列之關係亦存在, n+ n+ ' n ' n ( ρu) i /,j= ( ρu ) i /,j [(p ) i,j (p ) i,j] () x n+ n+ ' n ' n ( ρv) i,j /= ( ρv ) i,j / [(p ) i,j (p ) i,j ] () y

11 y 點 (i, j) 之質量守恆 (mass conservation) (i, j+) (i, j+/) (i-, j) (i-/, j) (i, j) (i+/, j) (i+, j) (i, j-/) dx dy (i, j-) ( ρu) ( ρv) Continuity equation: + = 0 x y x 差分, ( ρu) i+ /,j ( ρu) i /,j ( ρv) i,j+ / ( ρv) i,j / + = x y 0 (3)

12 將 eqs. (9)-() 代入 eq. (3), 可得 n+ ' n ' n n+ ' n ' n {( ρu ) i+ /,j [(p ) i+,j (p ) i,j]} {( ρu ) i /,j [(p ) i,j (p ) i,j]} x x + x n+ ' n ' n n+ ' n ' n {( ρv ) i,j+ / [(p ) i,j+ (p ) i,j]} {( ρv ) i,j / [(p ) i,j (p ) i,j ]} y y = 0 y 整理後, 亦即, a(p ) + b(p ) + b(p ) + c(p ) + c(p ) + d = 0 ' ' ' ' ' i,j i+,j i,j i,j+ i, j (4) 其中, a = [ + ] ; b = ; c = ( x) ( y) ( x) ( y) n+ n+ n+ n+ d = [( ρu ) i+ /,j ( ρ u ) i /,j] + [( ρv ) i,j+ / ( ρv ) i,j /] x y = mass source term

13 Step by Step Procedure for the SIMPLE Algorithm n. Guessed values of (p ) at all the "pressure" grid points. Also, n n arbitrarily set values of ( ρu ) and ( ρv ) at the "velocity" grid points. n+ n+. Using equations (3) and (4), solve for ( ρ u ) a at all corresponding internal grid points. nd ( ρv ) n+ n+ 3. Substitute these values of ( ρu ) and ( ρv ) into equation (4), and solve for p' values at all appropriate internal grid points. 4. Calculate new "p n+ " values at all internal grid points. n+ n [p = (p ) + p']

14 n+ n+ n+ 5. Designate the p, ( ρu ) and ( ρv ) values obtained above n n n as the new values of (p ), ( ρu ) and ( ρv ) and solve the momentum equations [eqs. (3) and (4)] again. 6. Repeat steps -5 until convergence ('d' values at all grid points approach zero) is achieved. 7. When all the 'normalized' mass source terms (normalized 'd' -4-6 values at all grid points) less than 0 0, the solution is considered to be convergent. 8. The larger the selected value, the faster the convergence of the solution. But as the selected value exceeds a certain level, the solution of the iteration might be divergent.

15 u u/u ; v v/u ; x x/l y ; y y/l ; t t/(l / u ) ; P P /(ρu ) ; y y e e Re ρu (D )/ μ= ρu (L )/ μ ; T Dimensionless Governing Equations Dimensionless groups ( 無因次化參數 ): h y T T T w T Substituting back into the original equations, it yields u v () Continuity: + = 0 x y () Momentum: u u u P e u u x-direction: + u + v = + ( + ) t x y x Re x y v v v y-direction: + u + v = t x y w P e v v ( + + ) y Re x y

16 Example : Flow in a Two-Dimensional Channel the geometry of the scaled problem (-D channel) is u = v = 0 y x L x / L y

17 Energy equation: ρ T T T T T k + [u + v ] = α ( + ) where α t x y x y cp T y x T w L y L x

18 Define: T Tw x y t T, x =, y =, Lx = L, Ly = S, t = T Tw S S (S/U ) u v ρu (S) ( / ) = μρ u =, v =, Re =, Pr, U U µ α The dimensionless and normalized energy equation can be expressed as follows: T T T T T = (u + v ) + ( + ) t x y RePr x y (u T ) (v T ) T T = [ + ] + ( + ) x y Re Pr x y T = y x T = 0 L x / L y

19 Numerical Method (temperature and pressure are located at the same grid point ) t T, n, n = ( )[T (i, j) T (i, j)] = ( ) { x x [u(i, j) + u (i, j)] T (i, j) } (ut) [u(i,j) u(i,j)] T(i,j) (v T ) [v(i, j) v (i, j )] T (i, j ) = ( ) { y y [v (i, j ) + v (i, j )] T (i, j ) } + + T T (i, j) T (i, j) T (i, j) = x ( x ) T T (i, j ) T (i, j) T (i, j ) = y ( y ) + +

20 Substituting back, T, n+ (i,j), n, n t [u(i,j) + u(i+,j)] T (i+,j) T (i,j) ( ){ = x, n [u (i,j) + u (i,j)] T (i,j) } ( ){ y, n t [v (i, j) v (i, j )] T (i, j ), n [v (i, j ) + v (i, j )] T (i, j ) }, n, n, n t T (i+,j) T (i,j) + T (i,j) + ( ) { RePr ( x ), n, n, n T (i, j ) T (i, j) T (i, j ) } ( y ) (5)

21 j = Ny y half channel width x j = Equal-size finite-difference grids

22 Nytotal Main layer y i =Nys+ i = Nys Sub-layer ys x i = Different-size finite-difference grids in main layer and sub-layer

23 Example : Flow in Front of a Block i = i = Nx j = Ny j = Nmid y block j i x x = L/S j =

24 Step by Step Procedure for the SIMPLE Algorithm n. Starting from n =, guess (T ) values at all the "pressure" grid points. n+. Using equation (5), solve for (T ) at all corresponding internal grid points. 3. Calculate the "d" values at all internal grid points. n+ n [d = (T ) (T ) ] n+ n 4. Designate the (T ) values obtained above as the new (T ) values and solve the energy equation [eq. (5)] again. 5. Repeat steps -4 until convergence ('d' values at all grid points approach zero) is achieved When all the 'd' values at all grid points) less than 0, the solution is considered to be convergent.