Topic02_PDE. 8/29/2006 topic02_pde 1. Computational Fluid Dynamics (AE/ME 339) MAE Dept., UMR

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1 MEAE 9 Computational Fluid Dnamics Topic0_ topic0_ Partial Dierential Equations () (CLW: 7., 7., 7.4) s can be linear or nonlinear Order : Determined b the order o the highest derivative. Linear, nd order s are classiied as the elliptic, hperbolic Parabolic tpe. Example: u u u u A A A B x z x u u + B + B + C u + D = z topic0_

2 Coeicients A, A, A Ma be +, - or zero. u is the dependent variable and x,, z are the independent variables. Note that we do not have an cross-derivative terms. Classiication: Elliptic: A, A, A are non-zero and have the same sign, then is o the elliptic tpe topic0_ Hperbolic: A, A, A are non-zero and o mixed sign, the is hperbolic Parabolic: I one o A, A, A, sa A is zero and the rest are o the same sign, and i B is non-zero, the is parabolic. Since A i, B i, C and D ma be unctions o x, and z, the classiication ma depend on position in space. In man CFD problems, one o the independent variables will be time and the rest will be space coordinates such as x,, z or or transormed variables such as ξ, η, ζ topic0_ 4

3 Two-Dimensional Examples Elliptic u x u + = 0 Hperbolic Parabolic u u u = + t x = + t x u u u topic0_ 5 Numerical solution o requires a inite number o points to Descretize the equations. Examples: See Figure in the next slide topic0_ 6

4 89006 topic0_ 7 Solution requires initial and boundar conditions depending on the Problem. Indices i, j, n can be used to label the nodes in x,, t directions(see ig.) I the origin has i=0, j=0 and n = 0, then the node i, j, n has coordinates i x, j, n t, where x,, t are the uniorm intervals between nodes along x,, t coordinate directions. Let uxt (,,) u be the exact solution o the and ijn,, be the approximations to be determined at each grid point. v i, j, n topic0_ 8 4

5 The derivatives o the original are approximated using the smbol v i, j, n and the discretization intervals x,, t. The procedure leads to a set o algebraic equations o which are then solved. Fine grids can be used to obtain solutions u i, j, n. Examples o s common in engineering v i, j, n v i, j, n that are close to.unstead heat conduction equation D orm: T T k = c p ρ x x t topic0_ 9 T - temperature k - thermal conductivit ρ - densit c p - speciic heat I k is a constant, the equation becomes Where α k c ρ p T α = x T t is the thermal diusivit topic0_ 0 5

6 In CFD, normalization o variables are oten used to improve the solution (or proper scaling o the variables). Let ξ = x αt, τ = L L or heat conduction in a rod o length L. Figure. The then becomes T = ξ T τ It is also possible to non-dimensionalize the dependent variable T topic0_ Talor s Expansion. d Let ( x, ). = h ( ) ( ) ( x + h) = ( x ) + h x, x + x, x! h x x ( 0 ( 0) ) +, +...! Where d (, ) = (, ) x x x x Talor series is as ollows. () () d x x x x (, ) = (, ) topic0_ 6

7 The higher order derivatives in Eq.() can be determined b Dierentiating Eq.() b chain rule. i. e., d d = +. x Example : (x,) is a unction o x alone. d = x topic0_ ( x ) ( x ) x, = x, =, = n ( x ), = 0 ( x0 0) ( x ) x0, 0 = x0, =, = n ( x ), = topic0_ 4 7

8 The unction at the neighboring point (x = x0+h) becomes x + h = x + hx + h x Example : (x,) is a unction o alone. d ( x, ) = = h topic0_ 5 x d x, = + = 0 + = 4 Similarl x, = 8 x, = ( + ) n x, = n topic0_ 6 8

9 h h x h x h x x x!! ( + ) = ( h) ( h) = ( x0 ) + h !! topic0_ 7 Talor series can also be used or non-linear higher order equations. Example: d d d x + = 0 d = ( x, ) = + x = x = x topic0_ 8 9

10 iv = ( + x + x ) iv = 4 x x Consider the initial conditions At x = = = 0, (0), (0) topic0_ 9 (0) = (0) 0 (0) = (0) = (0) (0) (0) = = iv (0) (0) 4 = + 4 = = topic0_ 0 0

11 Since Talor series gives h h h k k k k k = ! +! +! + For k = 0, becomes 4 h h h = h Letting h=0., we get = 0. + = topic0_ Calculation o Need Calculate w.r.t. h. b dierentiating the expression or h = h h = =.097 iv,, etc. can now be calculated as beore and topic0_ can be obtained.