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

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1 SIMPLE Algorithm for Two-Dimensional Channel 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 ( ρu) ( ρu2 ) ( ρuv) P 2 2 e u u (2) Momentum: x-direction: + [ + ] = + µ ( + ) t x y x x2 y2 ( ρv) ( uv) ( v2 ) Pe y-direction: [ ] t + ρ x + ρ y = 2v 2v + µ ( + ) y x2 y2 T T T 2T 2T (3) Energy : + [ u + v ] = ( + ) t x y α x2 y2 For example: the geometry of the original problem (2-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 ( 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 - ) y (i+/2, j+) (i, j+/2) pt. a (i+, j+/2) (i-, j) (i-/2, j) (i, j) (i+/2, j) (i+, j) (i+3/2, j) (i, j-/2) pt. b (i+, j-/2) dy dx (i+/2, j-) Note: pressure grid ponts x x-dir. velocity (u) grid points y-dir. velocity (v) grid points

6 y (i-/2, j+) (i, j+) (i+/2, j+) (i-, j+/2) pt. c (i, j+/2) pt. d (i+, j+/2) (i-/2, j) (i, j-/2) (i, j) (i+/2, j) dx dy at point "a": vi+ /2, j+ /2 ( vi, j+ /2 + vi+, j+ /2) 2 at point "b": v i+ /2, j /2 ( vi, j /2 + vi+, j /2) 2 at point "c": u i /2, j+ /2 ( ui /2, j + ui /2, j+ ) 2 at point "d": ui+ /2, j+ /2 ( ui+ /2, j + u /2, 2 ) i+ j+ x

7 centered around point ( i+ / 2, j), the x-direction momentum equation is solved ( J.D. Anderson, JR., Computational Fluid Dynamics, McGraw Hill 995). ( ρu) ( ρu) n+ n i+ /2, j i+ /2, j ( ) ( ) ρuv ρuv p p = [ + ] 2 2 n n n n n n ui+ 3/2, j 2ui+ /2, j + ui /2, j ui+ /2, j+ 2ui+ / 2, j + ui+ /2, j + µ [ + ] 2 2 ( ) ( ) 2 n 2 n n n n n ρu i+ 3/2, j ρ u i-/2, j ( ) i+ /2, j+ ( ) i+ /2, j- i+, j i, j or ( ρu) = ( ρu) + n+ n i+ /2, j i+ /2, j A ( p p ) () n n i+, j i, j 2 n 2 n n n ( ρu ) i+ 3/2, j ( ρu ) i-/2, j ( ρuv ) i+ /2, j+ ( ρuv ) i+ /2, j- where A [ + ] 2 2 n n n n n n ui+ 3/2, j 2ui+ /2, j + ui /2, j ui+ /2, j+ 2ui+ /2, j + + µ [ ( ) ( ) u i + /2, j ]

8 similarly, centered around point ( i, j equation is solved. + / 2), the y-direction momentum ( ρv) ( ρv) n+ n i, j+ /2 i, j+ /2 ( ρv ) ( ρv ) ( ρvu ) ( ρvu ) p p = [ + ] 2 2 n n n n n n vi, j+ 3/2 2vi, j+ /2 + vi, j /2 vi+, j+ /2 2vi, j+ /2 + vi, j+ / 2 + µ [ + ] 2 2 ( ) ( ) 2 n 2 n n n n n i, j+ 3/2 i, j-/2 i+, j+ /2 i, j+ /2 i, j+ i, j n+ n n n or ( ρv) i, j+ /2 = ( ρv) i, j+ /2+ B ( pi, j+ pi, j) (2) where B ρv ( ρv ) ( ρvu) ( ρvu ) [ v 2v + v v 2v + v + µ + ( ) ( ) 2 n 2 n n n ( ) i, j+ 3/2 i, j-/2 i+, j+ /2 i, j+ /2 n n n n n n i, j+ 3/2 i, j+ /2 i, j /2 i+, j+ /2 i, j+ /2 i, j+ /2 [ ] 2 2 ]

9 p u v and (2) can be calculated to yield a new set of ( ρu ) and ( ρv ) values, * n * n * n Assume a set of initial quessed ( ), ( ρ ) and ( ρ ) values, equations () * n+ * n+ * n+ * n * * n * n ( ρu ) i+ /2, j= ( ρu ) i+ /2, j+ A [( p ) i+, j ( p ) i, j] (3) * n+ * n * * n * n ( ρv ) i, j+ /2 = ( ρv ) i, j+ /2+ B [( p ) i, j+ ( p ) i, j] ( 4) y let ; v v ; p p p A A A ; B B B ' u * u u ' v * ' * ' * ' * Subtracting eq.(3) from eq.() and eq.(4) from eq.(2), it yields ' n+ ' n ' ' n ' n ( ρu ) i+ /2, j= ( ρ u ) i+ /2, j+ A [( p) i+, j ( p) i, j] (5) ' n+ ' n ' ' n ' n ( ρv) i, j+ /2 = ( ρv) i, j+ /2 + B [( p) i, j+ ( p) i, j] (6) ' ' ' where,, are corrections ( p u v ) of pressure and velocities respectively.

10 ' ' ' n ' n Patankar A, B, ( pu ) ( pv ) 0, eq. (5) eq. (6) : ' n+ ' n ' n ( ρu ) i+ /2, j= [( p) i+, j ( p) i, j] (7) ' n+ ' n ' n ( ρv ) i, j+ /2= [( p) i, j+ ( p) i, j] (8) n+ * n+ ' n ' n ( ρu) i+ /2, j= ( ρu ) i+ /2, j [( p) i+, j ( p) i, j] (9) n+ * n+ ' n ' n ( ρv) i, j+ /2 = ( ρv ) i, j+ /2 [( p) i, j+ ( p) i, j] (0) 理不列 n+ * n+ ' n ' n ( ρu) i /2, j= ( ρu ) i /2, j [( p) i, j ( p) i, j] () n+ * n+ ' n ' n ( ρv) i, j /2= ( ρv ) i, j /2 [( p) i, j ( p) i, j ] (2)

11 y (i, j) 量 (i, j+) (i, j+/2) (i-, j) (i-/2, j) (i, j) (i+/2, j) (i+, j) (i, j-/2) dx dy (i, j-) ( ρu) ( ρv) Continuity equation: + = 0 x y x ( ρu) i+ /2, j ( ρu) i /2, j ( ρv) i, j+ /2 ( ρv) i, j /2 + = 0 (3)

12 eqs. (9)-(2) eq. (3), * n+ ' n ' n * n+ ' n ' n {( ρu ) i+ / 2, j [( p) i+, j ( p) i, j]} {( ρu ) i / 2, j [( p) i, j ( p) i, j]} x + * n+ ' n ' n * n+ ' n ' n {( ρv ) i, j+ /2 [( p) i, j+ ( p) i, j]} {( ρv ) i, j /2 [( p) i, j ( p ) i, j ]} = 0 理, a( p ) + b( p ) + b( p ) + c( p ) + c( p ) + d = 0 ' ' ' ' ' i, j i+, j i, j i, j+ i, j (4), a= 2[ + ] ; b= ; c= ( d = [( ρu ) ( ρu ) ] + [( ρv ) ( ρv = mass source term ) ( ) ( ) ( ) ) ] * n+ * n+ * n+ * n+ i+ /2, j i /2, j i, j+ /2 i, j /2

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+ 2. 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 2-5 until convergence ('d' values at all grid points approach zero) is achieved. 7. When all the 'normalized' mass source terms (normalized 'd' values at all grid points) less than to be convergent. -5 0, the solution is considered 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 Dimensionless groups ( 參數 ): u* u/u ; v* v/u ; x* x/l ; y* y/l ; t* t/(l / u ) ; P* P / ( ρu 2) ; y y e e Re ρu ( D )/ = ρu ( 2L )/ ; T Dimensionless Governing Equations h y y * T T T w T Substituting back into the original equations, it yields u* v* () Continuity: + = 0 x* y* (2) Momentum: u* u* u* P* 2 2 * 2 * -direction: * * e u u x + u + v = + ( + ) t* x* y* x* Re x*2 y*2 v* v* v* y-direction: + u* + v* = t* x* y* P* 2 * 2 * e 2 v v + ( + ) y* Re x*2 y*2 w

16 the geometry of the scaled problem (2-D channel) is y* x* L x / L y

17 Energy equation (Neglecting axial conduction effect): ( u ) = where T T 2T k + v α α x y y2 ρcp T y x T w L y L x

18 Define: T T T * w T T w T* T* 2 2 T* ( u* + v* ) = ( ) x* y* Re Pr y*2 where Pr ν µ / ρ = = α α heat transfer Prandtl number y* x* T* = 0 L x / L y

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

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