Module 3: BASICS OF CFD. Part A: Finite Difference Methods

Transcription

1 Module 3: BASICS OF CFD Part A: Finite Difference Methods

2 THE CFD APPROACH Assembling the governing equations Identifying flow domain and boundary conditions Geometrical discretization of flow domain Discretization of the governing equations Incorporation of boundary conditions Solution of resulting algebraic equations Post-solution analysis and reformulation, if needed

3 OUTLINE Basics of finite difference (FD) methods FD approximation of arbitrary accuracy FD formulas for higher derivatives Application to an elliptic problem FD for time-dependent problems FD on non-uniform meshes Closure

4 Basics of Finite Difference Methods FD methods are quite old and somewhat dated for CFD problems but serve as a point of departure for CFD studies The principal idea of CFD methods is to replace a governing partial differential equation by an equivalent and approximate set of algebraic equations Finite difference techniques are one of several options for this discretization of the governing equations In finite difference methods, each derivative of the pde is replaced by an equivalent finite difference approximation

5 Basics of Finite Difference Methods The basis of a finite difference method is the Taylor series expansion of a function. Consider a continuous function f(x). Its value at neighbouring points can be expressed in terms of a Taylor series as (1) f(x+ Δx) = f(x) + df/dx x (Δx) + d 2 f/dx 2 x (Δx 2 )/ d n f/dx n x (Δx n )/n +.. The above series converges if Δx is small and f(x) is differentiable For a converging series, successive terms are progressively smaller

6 FD Approximation for a First Derivative The terms in the Taylor series expansion can be rearranged to give Or df/dx x = [f(x+ Δx) - f(x)] / Δx - d 2 f/dx 2 x (Δx)/2 - -d n f/dx n x (Δx n-1 )/n -... (2) df/dx x [f(x+ Δx) - f(x)] / Δx + O(Δx) Here O(Δx) implies that the leading term in the neglected terms of the order of Δx, i.e., the error in the approximation reduces by a factor of 2 if Δx is halved. Equation (2) is therefore a first order-accurate approximation for the first derivative.

7 Other Approximations for a First Derivative Other approximations are also possible. Writing the Taylor series expansion for f(x- Δx), we have (3) f(x-δx) = f(x) - df/dx x (Δx) + d 2 f/dx 2 x (Δx 2 )/ d n f/dx n x (Δx n )/n + Equation (3) can be rearranged to give another first order approximation : (4) df/dx x [f(x) - f(x-δx)] / Δx + O(Δx) Subtracting (3) from (1) gives a second order approximation for df/dx x : (5) df/dx x [f(x+ Δx) - f(x-δx)] / (2Δx) + O(Δx 2 )

8 FD Approximations on a Uniform Mesh Consider a uniform mesh with a spacing of Δx over an interval [0, L] Denoting the mesh index by i, we can write Then f i = f(x i ) = f(i Δx) and f i+1 = f [(i+1) Δx] and so on (2) df/dx x [f(x+ Δx) - f(x)] / Δx = (f i+1 - f i )/Δx + O(Δx) (3) df/dx x [f(x) - f(x- Δx) ] / Δx = (f i - f i-1 )/Δx + O(Δx) (5) df/dx x [f(x+ Δx) - f(x-δx)] / (2Δx) = (f i+1 - f i-1 )/(2Δx) + O(Δx 2 ) are the forward one-sided backward one-sided central symmetric differencing formulas, respectively, for df/dx at x or node i One-sided formulas are necessary at ends of domains

9 Graphical representation of FD approximation of first derivative f-forward differencing b-backward differencing c-central differencing t-true slope

10 Higher Order Accuracy Higher order of accuracy of approximation can be obtained by including more number of adjacent points Let us seek a third-order, one-sided approximation for u(x). This requires four points and will be of the form (6) du/dx) i = [a u i + b u i+1 + c u i+2 + d u i+3 ]/ Δx + O(Δx 3 ) This is equivalent to writing (6) as (7) du/dx) i = [a u i + b u i+1 + c u i+2 + d u i+3 ]/ Δx + (0) d 2 u/dx 2 x (Δx) + (0)d 3 u/dx 3 x (Δx 2 ) + (e) d 4 u/dx 4 x (Δx 3 ) or (8) a u i + b u i+1 + c u i+2 + d u i+3 = + du/dx x (Δx) + (0) d 2 u/dx 2 x (Δx 2 ) + (0)d 3 u/dx 3 x (Δx 3 ) - (e) d 4 u/dx 4 x (Δx 4 ) How to find a, b, c and d?

11 Third-Order, One-sided Formula Expand u i+1, u i+2 and u i+3 in Taylor series about u i : (9a) u i+1 = u(x+ 1Δx) = u(x) + du/dx x (Δx) + d 2 u/dx 2 x (Δx) 2 / (9b) u i+2 = u(x+ 2Δx) = u(x) + du/dx x (2Δx) + d 2 u/dx 2 x (2Δx) 2 / (9c) u i+3 = u(x+ 3Δx) = u(x) + du/dx x (3Δx) + d 2 u/dx 2 x (3Δx) 2 / Find { a u i + b (9a) + c(9b) + d (9c) } and rearrange to get (10) a u i + b u i+1 + c u i+2 + d u i+3 = p u +q du/dx x (Δx) + r d 2 u/dx 2 x (Δx 2 ) + s d 3 u/dx 3 x (Δx 3 ) + t d 4 u/dx 4 x (Δx 4 ) (8) a u i + b u i+1 + c u i+2 + d u i+3 = + du/dx x (Δx) + (0) d 2 u/dx 2 x (Δx 2 ) + (0)d 3 u/dx 3 x (Δx 3 ) - (e) d 4 u/dx 4 x (Δx 4 ) Compare the coefficients of (8) and (10) to get a = -11/6 b = 3 c = -3/2 d = 1/3 or (11) du/dx) i = [-11 u i + 18 u i+1-9 u i u i+3 ]/ (6Δx) + O(Δx 3 )

12 Higher Derivatives Finite difference approximation for second derivative: or d 2 u/dx 2 ) i = [d/dx(du/dx)] i [ (du/dx) i+1/2 - (du/dx) i-1/2 ] / Δx [ (u i+1 - u i )/ Δx - (u i - u i-1 )/ Δx ] / Δx (12) d 2 u/dx 2 ) i [ (u i+1-2 u i + u i-1 ) ] / Δx 2 Taylor series evaluation of equation (11) shows that the approximation is second order accurate; thus, (12a) d 2 u/dx 2 ) i = [ (u i+1-2 u i + u i-1 ) ] / Δx 2 + O(Δx 2 ) Note that use of central differences for the second derivative requires three points, viz., (i-1), i, (i+1), for a second order accurate formula

13 Other Formulas for Higher Derivatives Using forward differencing throughout, one can get the following first order accurate formula involving three points for the second derivative: or d 2 u/dx 2 ) i = [d/dx( du/dx)] i [ (du/dx) i+1 - (du/dx) i ] / Δx [ (u i+2 - u i+1 )/ Δx - (u i+1 - u i )/ Δx ] / Δx (13) d 2 u/dx 2 ) i [ (u i+2-2 u i+1 + u i ) ] / Δx 2 + O(Δx) A central, second order scheme for the third derivative needs four points: (14) d 3 u/dx 3 ) i = [ (u i+2-2 u i u i-1 - u i-2 ) ] / (2Δx 3 ) + O(Δx 2 ) If p = order of derivative, q = order of accuracy and n = no of points, then n = p + q -1 for central schemes n = p + q for one-sided schemes

14 Mixed Derivatives Mixed derivatives can occur as a result of coordinate transformation to a non-orthogonal system (for example, to take account of non-regular shape of the flow domain). Straightforward application of the method for higher derivatives: or 2 u/ x y) i,j = [ / x ( u/ y)] i,j [( u/ y) i+1,j - ( u/ y) i-1,j ] / (2Δx) [ (u i+1,j+1 - u i+1,j-1 )/ 2Δy - (u i-1,j+1 - u i-1,j-1 )/ 2Δy ] / (2Δx) (15) 2 u/ x y)i,j [(u i+1,j+1 -u i+1,j-1 - u i-1,j+1 + u i-1,j-1 )] / (4 ΔxΔy) + O(Δx 2, Δy 2 ) A large variety of schemes possible

15 Example: 2-D Poisson Equation (16) 2 u/ x u/ y 2 = f 0 < x < L and 0 < y < W with Dirichlet boundary conditions: u (x,y) = g(x,y) on boundary Write 2 u/ x 2 ) i,j [(u i+1,j -2u i,j + u i-1,j )] / (Δx 2 ) + O(Δx 2 ) and 2 u/ y 2 ) i,j [(u i,j+1-2u i,j + u i,j-1 )] / (Δy 2 ) + O(Δy 2 ) and substitute in (16) to get (17) [(u i+1,j -2u i,j + u i-1,j )] / (Δx 2 ) + [(u i,j+1-2u i,j + u i,j-1 )] / (Δy 2 ) = f ij + O(Δx 2, Δy 2 ) With Dirichlet boundary conditions, equation (17) would be valid for 2 < i < N i 2 < j < N j Results in (N i -1) (N j -1) algebraic equations to be solved for u(i,j)

16 2-d grid

17 Boundary conditions Boundary conditions over AB: Dirichlet boundary conditions: u (x i,y 1 ) = g(x i,y 1 ) on AB Normal gradient specified: du/dy i, 1 = c 1 for all i at j = 1 Convective boundary condition: q i,1 = -k u/ y i,1 = h*(u inf u i,1 ); h and u inf given

18 Poisson Equation: Other Boundary Conditions Normal gradient specified, e.g, du/dy = c 1 for all i at j = N j Values of u(i, N j ) not known and have to be determined For these boundary points, du/dy (u i,nj-1 -u i,nj )/ Δy = c 1 or u i,nj - u i,nj-1 = c 1 * Δy first order accurate Equations for the interior points remain the same For second order accuracy, one can write du/dy (a u i,nj-2 + b u i,nj-1 + c u i,nj )/ Δy = c1 Convective boundary condition: q w = h*(u inf -u w ); h and u inf given This can be implemented by noting that q w = -k du/dy Thus, h*(u inf - u i,nj ) = -k*(a u i,nj-2 + b u i,nj-1 + c u i,nj )/ Δy which gives the necessary algebraic equation for the boundary point. (Ni-1) (Nj) algebraic equations to be solved for u(i,j)

19 Discretization of Time Domain Consider the unsteady heat conduction problem: T/ t = 2 T/ x 2 (18) Denote T(x,t ) = T(i Δx, n Δt) = T i n We seek discretization of eqn. (18) of the form (19) T/ t ) i n = 2 T/ x 2 ) i n Evaluate LHS of (19) using forward differencing as (20) T/ t ) n i = (T n+1 i - T in ) / Δt + O (Δt) But several options for RHS even if we choose, say, central differencing for 2 T/ x 2

20 Explicit and Implicit Schemes Put 2 T/ x 2 ) i,n = (T i+1,n - 2 T i,n + T i-1,n )/ Δx 2 + O(Δx 2 ) (21) and substitute (20) and (21) in (19) to get Explicit equation for T i,n+1 : (T i,n+1 - T i,n )/ Δt = (T i+1,n - 2 T i,n + T i-1,n )/ Δx 2 or T i,n+1 = T i,n + Δt/Δx 2 (T i+1,n - 2 T i,n + T i-1,n ) + O(Δt, Δx 2 ) (22) Put 2 T/ x 2 ) i,n = (T i+1,n+1-2 T i,n+1 + T i-1,n+1 )/ Δx 2 + O(Δx 2 ) (23) and substitute (20) and (21) in (19) to get Implicit equation for T i,n+1 : (T i,n+1 - T i,n )/ Δt = (T i+1,n+1-2 T i,n+1 + T i-1,n+1 )/ Δx 2 or (1+ 2 Δt/Δx 2 ) T i,n+1 = T i,n + Δt/Δx 2 (T i+1,n+1 + T i-1,n+1 ) + O(Δt, Δx 2 ) (24)

21 Put T i,n = (T i, n-1 + T i,n+1 )/2 to get the DuFort-Frankel scheme: Other Schemes (25) (T i,n+1 - T i,n-1 )/ (2Δt) = (T i+1,n - T i,n+1 +T i,n-1 + T i-1,n )/ Δx 2 + O(Δt 2, Δx 2 ) Explicit, second order accurate and unconditionally stable Evaluate RHS at (n+1/2) as (RHS,n + RHS,n+1)/2 and put in (19) to get the Crank-Nicolson scheme: (26) -r T i-1,n+1 +(2+2r) T i,n+1 - rt i+1,n+1 = rt i-1,n + (2-2r)T i,n + rt i+1,n where r = Δt/Δx 2 Implicit, second order accurate and unconditionally stable

22 Explicit vs Implicit Methods Explicit methods are simple to program and allow marching forward in time point by point from given initial condition Implicit methods are more difficult to program and require simultaneous solution of algebraic equations at each time step to get the solution Explicit methods are generally less stable than implicit methods and may give unphysical solutions if the marching time step is too large Too-large a time step, which is possible with implicit methods, may result in less accuracy and instability in non-linear problems

23 Non-uniform Meshes The methods discussed above can be extended to non-uniform meshes but may have to be done with care so as not to lose an order of accuracy : du/dx)i = {ui+1 (Δxi-1) 2 - ui-1(δxi) 2 + ui[(δxi-1) 2 - (Δxi) 2 ]} / {(Δxi-1) (Δxi) (Δxi-1 + Δxi)} + O{(Δxi) 2 } where Δxi = xi+1 - xi etc Highly non-uniform and distorted meshes should be avoided where possible

24 Closure Finite difference techniques can be used to systematically develop approximate formula for derivatives of a function on a structured mesh Formulas of desired accuracy level can be obtained for a derivative provided sufficient number of points are included in the formula Discretized equations of partial differential equations can be obtained by replacing the derivatives with finite difference formulas The resulting equations are coupled of algebraic equations which may be non-linear for non-linear equations and require simultaneous solution The issue of stability is brought into play for time-dependent problems

25 Need for Analysis Discretization of governing equations using FD or FE or FV techniques gives a set of discretized equations involving variable values at neighbouring points These discretized equations are the ones that are actually solved to determine the variable values at grid nodes Discretization is not unique and for the same PDE, different discretized equations can be obtained, e.g. discretization can be done to different orders of accuracy Are all discretizations equivalent? Or are some superior? How can we ensure that the numerical solution is correct?

26 A Simple Case Study Consider the linear convection equation u t u + a x = 0 n+ 1 n n n i i ( i i 1 FTBS : u = u σ u u ) n+ 1 n n n i i i+ 1 i FTCS: u = u σ ( u u 1) / 2 σ = aδt/δx Courant no n+ 1 n n n i = i ( i+ 1 i FTFS : u u σ u u )

27 The True Solution Take a = 1 m/s; initial conditions : u(x,0) = f(x) where f(x) = 1 for 0.1 x 0.2 and f(x) = 0 otherwise True solution: initial pulse gets convected in x-direction at 1 m/s u(x,t) = u(x-at) u x

28 FTBS Solution for σ 1 at t = 0.1 s f(x, 0.1) x (m) Peak location predicted correctly, but the square pulse has been rounded Decreasing amplitude with decreasing σ

29 FTBS Solution for σ 1 at t = 0.1 s 4 Fig I.1b. FTBS scheme with various Courant numbers at t=0.1 s f(x, 0.1) x (m) Both shape and amplitude predicted poorly for σ > 1

30 FTCS Solution for σ 1 at t = 0.1 s Discretization second-order accurate in space but... 8 Fig I.2a FTCS scheme with various Courant numbers at t=0.1s 6 4 f(x, t0) x (m)

31 FTFS Solution for σ 1 at t = 0.05 s Solution wildly fluctuating for all Courant numbers f(x, t0) Fig I.2b FTFS scheme with various Courant numbers at t=0.05 s x (m)

32 Need for Analysis Identified Different solutions ranging from exact to thoroughly unsatisfactory obtained for this simple, linear equation Behaviour of the solution appears to depend on choice of Δt and Δx Need a formal analysis to determine the choice of parameters to yield an accurate solution Three conditions for a good solution: Consistency : discretized equation pde Stability : exact solution of disc eqn computed solution Convergence: computed solution exact solution of pde Equivalence theorem of Lax (1954): For a well-posed linear initial value problem with a consistent discretization, stability is necessary and sufficient for convergence Convergence assures us that grid independent solution = exact solution of the governing equation