One Dimensional Convection: Interpolation Models for CFD

Transcription

1 One Dimensional Convection: Interpolation Models for CFD ME 448/548 Notes Gerald Recktenwald Portland State University Department of Mechanical Engineering ME 448/548: 1D Convection-Diffusion Equation

2 Overview Demonstrate the problem arising from central difference approximation of the convection term.. Introduce upwind concept and provide a Matlab implementation. Introduce non-uniform mesh behavior, and show that the local Peclet number matters. Demonstrate numerical behavior with Matlab computations. ME 448/548: 1D Convection-Diffusion Equation page 1

3 Model Problem The one-dimensional convection-diffusion equation is d dx (uφ) d ( Γ dφ ) S = 0 dx dx 0 x L (1) For S = 0 and the boundary conditions the exact solution to Equation (1) is φ(0) = φ 0 φ(l) = φ L, (2) where φ φ 0 = exp(ux/γ) 1 φ L φ 0 exp(pe L ) 1 Pe L = ul Γ (3) (4) ME 448/548: 1D Convection-Diffusion Equation page 2

4 Exact Solution to the Model Problem The parameter is Pe L = ul Γ φ x ME 448/548: 1D Convection-Diffusion Equation page 3

5 Finite Volume Mesh Two-dimensional mesh δx w δx e y x i-1 W P E x i x i+1 y x x w x x e One-dimensional mesh, with emphasis on cell sizes near the boundary. x 2 i = x m 1 m 2 m 1 m δx w,2 δx e,2 δx w,m 1 δx e,m 1 ME 448/548: 1D Convection-Diffusion Equation page 4

6 Integrate the Model Equation over a Control Volume Obtain a numerical model of the convection-diffusion equation d dx (uφ) d dx ( Γ dφ ) dx S = 0 by integrating with respect to x over the control volume. δx w δx e x e xw d(uφ) dx dx x e xw d dx ( Γ dφ ) dx x e xw dx S dx = 0 (5) y y x x i-1 W x w x i P x x e E x i+1 ME 448/548: 1D Convection-Diffusion Equation page 5

7 Integrate the Diffusion Term Integrate diffusion term once x e xw d dx ( Γ dφ ) dx (6) dx = ( Γ dφ ) dx e ( Γ dφ ) dx w y y x x i-1 W δx w x w x i P x δx e x e E x i+1 ( Γ dφ ) dx e ( Γ dφ ) dx w = flux of φ across the east face, out of the C.V. = flux of φ across the west face, into the C.V. ME 448/548: 1D Convection-Diffusion Equation page 6

8 Integrate the Diffusion Term Replace the diffusion fluxes with finite difference approximations ( Γ dφ ) dx e Γ e φ E φ P δx e = D e (φ E φ P ) δx w δx e ( Γ dφ ) dx w Γ w φ P φ W δx w = D w (φ P φ W ) y x i-1 W x i P E x i+1 y and D e = Γ e δx e D w = Γ w δx w (7) x x w x x e δx e = x E x P δx w = x P x W. (8) φ W, φ P, and φ E are values at the nodes, i.e., these are the unknowns (the discrete dependent variables) in the numerical model. ME 448/548: 1D Convection-Diffusion Equation page 7

9 Integrate the Diffusion Term Assume that Γ is uniform: Γ e = Γ w = Γ. x e xw d dx ( Γ dφ ) dx = dx ( Γ dφ ) dx e ( Γ dφ ) dx w (9) D e (φ E φ P ) D w (φ P φ W ) (10) ME 448/548: 1D Convection-Diffusion Equation page 8

10 Integrate the Source Term Integrate the source term, assuming that S is uniform over the control volume. x e xw S dx S P x p. (11) y y x x i-1 W δx w x w x i P x δx e x e E x i+1 ME 448/548: 1D Convection-Diffusion Equation page 9

11 Convection Term Overview 1. Integrate once: What is the value of φ at the interface? 2. Linear interpolation leads to central difference 3. Central difference causes unacceptable wiggles on a coarse mesh 4. Upwind interpolation suppresses the wiggles, but reduces the accuracy ME 448/548: 1D Convection-Diffusion Equation page 10

12 Convection Term First Try The first step is easy x e xw d(uφ) dx dx = (uφ) e (uφ) w (12) How do we define (uφ) e and (uφ) w? (uφ) e = product of u and φ at the east interface (uφ) w = product of u and φ at the west interface ME 448/548: 1D Convection-Diffusion Equation page 11

13 Use Linear Interpolation at the Interface Use linear interpolation to estimate φ e from φ P and φ E, and to estimate φ w from φ W and φ P φ e = β e φ E + (1 β e )φ P (13) ϕ W φ w = β w φ W + (1 β w )φ P (14) ϕ P ϕ E where β e = x e x P x E x P and β w = x P x w x P x W. x W x w x P x e x E ME 448/548: 1D Convection-Diffusion Equation page 12

14 Reconstitute the Convection Terms Substituting Equation (13) and Equation (14) into Equation (12) and rearranging gives x e xw d(uφ) dx dx = u eβ e (φ E φ P ) u w β w (φ W φ P ) + u e φ P u w φ P (15) The last two terms in the preceding equation cancel because u is a uniform parameter, i.e. u e = u w. x e xw d(uφ) dx dx = u eβ e (φ E φ P ) u w β w (φ W φ P ) (16) ME 448/548: 1D Convection-Diffusion Equation page 13

15 Final Form of the Discrete Convection Diffusion Equation Substituting Equation (10), Equation (11) and Equation (16) into Equation (5) and simplifying gives a E φ E + a P φ P a W φ W = b (17) where a E = 1 x P (D e u e β e ) (18) a W = 1 x P (D w + u w β w ) (19) a P = a E + a W (20) b = S P (21) ME 448/548: 1D Convection-Diffusion Equation page 14

16 Another Tridiagonal System of Equations a P,1 a E,1 a W,2 a P,2 a E, a W,i a P,i a E,i φ 1 φ 2. φ i. = b 1 b 2. b i. (22) a W,m a P,m φ m b m Coefficients in the first and last equations are adjusted to enforce the boundary conditions φ(x = 0) = φ 0 = a P,1 = 1, a E,1 = 0, b 1 = φ 0 φ(x = L) = φ L = a P,m = 1, a W,m = 0, b m = φ L ME 448/548: 1D Convection-Diffusion Equation page 15

17 Numerical Solution with the Central Difference Scheme Central difference solutions to Equation (1) for P e L = 50, P e x = 5. This is the steady solution. The oscillations are not an instability: the solution does not grow as the result of iterations or time steps φ 0.5 CDS solution exact PeL = 50.0, Pex = 5.0 CDS scheme, Max error = x ME 448/548: 1D Convection-Diffusion Equation page 16

18 Numerical Solution with the Central Difference Scheme Central difference solutions to Equation (1) for P e L = 50, P e x = CDS solution exact Increasing the number of nodes has eliminated the wild oscillations in the numerical solution. φ PeL = 50.0, Pex = 1.7 CDS scheme, Max error = x ME 448/548: 1D Convection-Diffusion Equation page 17

19 Consequences of Negative Coefficients Consider the possible values taken by a E where the interpolation coefficient β e is Rearrange Equation (18) as a E = 1 x P (D e u e β e ) (18) β e = x P /2 δx e a E = 1 x P (D e u e β e ) = 1 x P = Γ e x P δx e ( ) Γe x P /2 u e δx e δx e ( 1 P e ) x 2 ME 448/548: 1D Convection-Diffusion Equation page 18

20 Consequences of Negative Coefficients We have a E = Γ e x P δx e ( 1 P e ) x 2 Since Γ e x P δx e > 0 always, we observe the a E becomes negative when ( 1 P e ) x 2 < 0 or P e x > 2 ME 448/548: 1D Convection-Diffusion Equation page 19

21 Consequences of Negative Coefficients When P e x > 2 the value of a E becomes negative and the eigenvalues of the coefficients become complex and the numerical solution oscillates. Note that we can keep P e x = u e x P Γ e < 2 by making x P small enough. That is why reducing the mesh size (increasing the number of nodes) eliminated the error in the sample calculation. ME 448/548: 1D Convection-Diffusion Equation page 20

22 Upwind Differencing: A cure with a cost 1. Negative a E can be avoided with judicious choice of β e. Recall that and φ e = β e φ E + (1 β e )φ P φ Increasing u a E = 1 x P (D e u e β e ) Physical reasoning suggests that φ e should be influenced by u e. 3. Take inspiration from the analytical solution to Equation (1) x As the velocity increases, the value of φ at the midpoint gets closer to the value of the upstream boundary. The Upwind Difference Scheme gives a strong bias to the upstream value of φ. ME 448/548: 1D Convection-Diffusion Equation page 21

23 Upwind Differencing: A cure with a cost Strategy: Ignore the value of φ at the downwind node: u w > 0 u e > 0 u w < 0 u e < 0 ϕ W ϕ W ϕ P ϕ E ϕ P ϕ E x W x w x P x e x E x W x w x P x e x E u w > 0, u e > 0: u w < 0, u e < 0: β w = 1 β e = 0 β w = 0 β e = 1 φ w = φ W φ e = φ P φ w = φ P φ e = φ E ME 448/548: 1D Convection-Diffusion Equation page 22

24 Upwind Differencing: A cure with a cost u w > 0 u e > 0 u w < 0 u e < 0 ϕ W ϕ W ϕ P ϕ E ϕ P ϕ E x W x w x P x e x E x W x w x P x e x E Restate the rules β e = { 0 if ue 0 1 if u e < 0 β w = { 1 if uw 0 0 if u w < 0 ME 448/548: 1D Convection-Diffusion Equation page 23

25 Compare Upwind Solution with Central Difference Solution Compare UDS and CDS for m = 12, Pe x = 5. No oscillations for the UDS scheme Central Upwind Exact φ 0.5 Pe L = 50.0, Pe x = e c = 1.583, e u = x ME 448/548: 1D Convection-Diffusion Equation page 24

26 Compare Upwind Solution with Central Difference Solution Compare UDS and CDS for m = 32, Pe x = 1.7. No oscillations for either scheme Central Upwind Exact φ 0.5 Pe L = 50.0, Pe x = e c = 0.268, e u = x ME 448/548: 1D Convection-Diffusion Equation page 25

27 Convergence study for Upwind and Central Difference Schemes Solve the model problem on a series of finer meshes Upwind Central Difference m x P e x e u e u ratio e c e c ratio ME 448/548: 1D Convection-Diffusion Equation page 26

28 Convergence study for Upwind and Central Difference Schemes m e u ratio e c ratio Max error Central Upwind x ME 448/548: 1D Convection-Diffusion Equation page 27

29 Improving Accuracy with a Non-uniform Mesh A non-uniform mesh can improve accuracy of a numerical solution. Nodes should be close together in regions where the gradient is higher. Demonstrate the idea with a stretched mesh L x 1 x 2 x 3 x n ME 448/548: 1D Convection-Diffusion Equation page 28

30 Stretched Mesh (1) A growing or shrinking cell size, and hence node spacing, can be achieved by specifying where r is a fixed constant. x i+1 x i = r (23) r > 1 causes node spacing to increase as i increases. r < 1 causes node spacing to decrease as i increases. L x 1 x 2 x 3 x n ME 448/548: 1D Convection-Diffusion Equation page 29

31 Stretched Mesh (2) 1. The total length L of the region to be subdivided is given. 2. Choose the stretching ratio r, and the number of control volumes n. Constraint: The widths of the control volumes must add up to L. L = n x i = x 1 + r x 1 + r 2 x r n 1 x 1 i=1 = x 1 ( 1 + r + r r n 1) The sum of terms can be rewritten. S = 1 + r + r r n 1 = 1 rn 1 r (24) Thus, L = x 1 S or x 1 = L S. (25) ME 448/548: 1D Convection-Diffusion Equation page 30

32 Stretched Mesh (3) To create a mesh with a geometric progression of sizes 1. Specify L, r, and n. 2. Compute x 1 from Equation (25). 3. Compute the remaining cell widths with a loop. Note that the values of r and n interact in determining the actual spacing. It is possible to get distorted meshes by changing r with n held constant, or changing n with r held constant. ME 448/548: 1D Convection-Diffusion Equation page 31

33 Stretched Mesh (3) Examples: Use L = 1, n = 10 and vary r r = 1.05 n = 10 r = 0.95 n = 10 r = 1.10 n = 10 r = 0.90 n = 10 r = 1.20 n = 10 r = 0.80 n = 10 r = 1.30 n = 10 r = 0.70 n = 10 r = 1.40 n = 10 r = 0.60 n = 10 ME 448/548: 1D Convection-Diffusion Equation page 32

34 Stretched Mesh (4) Examples: Use L = 1, n = 20 and vary r r = 1.05 n = 20 r = 0.95 n = 20 r = 1.10 n = 20 r = 0.90 n = 20 r = 1.20 n = 20 r = 0.80 n = 20 r = 1.30 n = 20 r = 0.70 n = 20 r = 1.40 n = 20 r = 0.60 n = 20 For large n, avoid extreme distortion by using r 1. ME 448/548: 1D Convection-Diffusion Equation page 33

35 Stretched Mesh (4) Examples: Use L = 1, r = 0.90, r = 0.95 and vary n r = 0.90 n = 10 r = 0.95 n = 10 r = 0.90 n = 20 r = 0.95 n = 20 r = 0.90 n = 30 r = 0.95 n = 30 r = 0.90 n = 40 r = 0.95 n = 40 For large n, avoid extreme distortion by using r 1. ME 448/548: 1D Convection-Diffusion Equation page 34

36 Stretched Solution Central Upwind Exact n = 24, r = 0.90 n = 24, r = Central Upwind Exact 1 1 φ φ 0.5 Pe L = 50.0, Pe x = Pe L = 50.0, Pe x = e c = 0.042, e u = e c = 0.007, e u = x x Observations: Careful choice of stretching parameters can yield good accuracy on a modest mesh. No obvious way of predicting the accuracy when the exact solution is not available. ME 448/548: 1D Convection-Diffusion Equation page 35

37 Summary (1) We showed that Conventional finite volume schemes store dependent variables at cell centers. Advection terms require a secondary scheme for interpolating dependent variables at cell faces. The central difference scheme can cause oscillations The upwind difference scheme suppresses oscillations, but introduces additional truncation error. ME 448/548: 1D Convection-Diffusion Equation page 36

38 Summary (2) We did not include the following details No proofs were given for truncation error estimates. Many other schemes exist QUICK (quadratic upstream interpolation for convective kinematics), ENO (essentially non-oscillatory), TVD (total variational diminishing), Second-order upwinding, Streamline upwinding, and others. ME 448/548: 1D Convection-Diffusion Equation page 37

39 Summary (3) Users need to be aware of trade-offs Oscillatory higher order schemes cause solutions to fail, i.e., not converge Vendors may use a default scheme that is stable, but inaccurate, instead of a more accurate scheme Users may choose a stable and inaccurate scheme just to get the code to converge. That s not a good idea, especially if the accuracy of the solution is important. Don t suppress the wiggles They re trying to tell you something! P. Gresho and R. Lee, 1981 ME 448/548: 1D Convection-Diffusion Equation page 38