One Dimensional Convection: Interpolation Models for CFD

Transcription

1 One Dimensional Convection: Interpolation Moels for CFD ME 448/548 Notes Geral Recktenwal Portlan State University Department of Mechanical Engineering ME 448/548: D Convection-Di usion Equation Overview Demonstrate the problem arising from central i erence approimation of the convection term.. Introuce upwin concept an provie a Matlab implementation. Introuce non-uniform mesh behavior, an show that the local Peclet number matters. Demonstrate numerical behavior with Matlab computations. ME 448/548: D Convection-Di usion Equation page

2 Moel Problem The one-imensional convection-i usion equation is (u ) S =0 0apple apple L () For S =0an the bounary conitions the eact solution to Equation () is (0) = 0 (L) = L, (2) 0 L 0 = ep(u/ ) ep(pe L ) (3) where Pe L = ul (4) ME 448/548: D Convection-Di usion Equation page 2 Eact Solution to the Moel Problem The parameter is Pe L = ul φ ME 448/548: D Convection-Di usion Equation page 3

3 Finite Volume Mesh Two-imensional mesh δ w δ Δy W P E i- i i+ y w Δ One-imensional mesh, with emphasis on cell sizes near the bounary. Δ 2 i = Δ m m 2 m m δ w,2 δ,2 δ w,m δ,m ME 448/548: D Convection-Di usion Equation page 4 Integrate the Moel Equation over a Control Volume Obtain a numerical moel of the convection-i usion equation (u ) S =0 by integrating with respect to over the control volume. δ w δ Z w (u ) Z w Z w S=0 (5) y Δy i- W w i P Δ E i+ ME 448/548: D Convection-Di usion Equation page 5

4 Integrate the Di usion Term Integrate i usion term once Z w (6) = e w y Δy i- W δ w w i P Δ δ E i+ = flu of across the east face, out of the C.V. e = flu of across the west face, into the C.V. w ME 448/548: D Convection-Di usion Equation page 6 Integrate the Di usion Term Replace the i usion flues with finite i erence approimations an e w D e = e E P = D e ( E P ) w P W w = D w ( P W ) e D w = w (7) w y Δy i- W δ w w i P Δ δ E i+ = E P w = P W. (8) W, P,an E are values at the noes, i.e.,thesearetheunknowns(theiscrete epenent variables) in the numerical moel. ME 448/548: D Convection-Di usion Equation page 7

5 Integrate the Di usion Term Assume that is uniform: e = w =. Z w = e w (9) D e ( E P ) D w ( P W ) (0) ME 448/548: D Convection-Di usion Equation page 8 Integrate the Source Term Integrate the source term, assuming that S is uniform over the control volume. Z w S S P p. () y Δy i- W δ w w i P Δ δ E i+ ME 448/548: D Convection-Di usion Equation page 9

6 Convection Term Overview. Integrate once: What is the value of at the interface? 2. Linear interpolation leas to central i erence 3. Central i erence causes unacceptable wiggles on a coarse mesh 4. Upwin interpolation suppresses the wiggles, but reuces the accuracy ME 448/548: D Convection-Di usion Equation page 0 Convection Term First Try The first step is easy Z w (u ) =(u ) e (u ) w (2) How o we efine (u ) e an (u ) w? (u ) e = prouct of u an at the east interface (u ) w = prouct of u an at the west interface ME 448/548: D Convection-Di usion Equation page

7 Use Linear Interpolation at the Interface Use linear interpolation to estimate e from P an E, antoestimate w from W an P where e = e E + ( e) P (3) w = w W + ( w) P (4) P e = an w = P w. E P P W ϕ W ϕ P W w P E ϕe ME 448/548: D Convection-Di usion Equation page 2 Reconstitute the Convection Terms Substituting Equation (3) an Equation (4) into Equation (2) an rearranging gives Z w (u ) = u e e( E P ) u w w ( W P )+u e P u w P (5) The last two terms in the preceing equation cancel because u is a uniform parameter, i.e. u e = u w. Z w (u ) = u e e( E P ) u w w ( W P ) (6) ME 448/548: D Convection-Di usion Equation page 3

8 Final Form of the Discrete Convection Di usion Equation Substituting Equation (0), Equation () an Equation (6) into Equation (5) an simplifying gives a E E + a P P a W W = b (7) where a E = P (D e u e e ) (8) a W = P (D w + u w w ) (9) a P = a E + a W (20) b = S P (2) ME 448/548: D Convection-Di usion Equation page 4 Another Triiagonal System of Equations a P, a E, a W,2 a P,2... a E, a W,i a P,i a E,i i. 3 2 = b b 2. b i (22) a W,m a P,m m b m Coe cients in the first an last equations are ajuste to enforce the bounary conitions ( = 0) = 0 =) a P, =, a E, =0, b = 0 ( = L) = L =) a P,m =, a W,m =0, b m = L ME 448/548: D Convection-Di usion Equation page 5

9 Numerical Solution with the Central Di erence Scheme Central i erence solutions to Equation () for Pe L = 50, Pe =5. This is the steay solution. The oscillations are not an instability: the solution oes not grow as the result of iterations or time steps..5 φ 0.5 CDS solution eact PeL = 50.0, Pe = 5.0 CDS scheme, Marror = ME 448/548: D Convection-Di usion Equation page 6 Numerical Solution with the Central Di erence Scheme Central i erence solutions to Equation () for Pe L = 50, Pe =.7..5 CDS solution eact Increasing the number of noes has eliminate the wil oscillations in the numerical solution. φ 0.5 PeL = 50.0, Pe =.7 CDS scheme, Marror = ME 448/548: D Convection-Di usion Equation page 7

10 Consequences of Negative Coe cients Consier the possible values taken by a E where the interpolation coe cient e is Rearrange Equation (8) as a E = P (D e u e e ) (8) e = P /2 a E = P (D e u e e )= P = P e e P /2 u e Pe 2 ME 448/548: D Convection-Di usion Equation page 8 Consequences of Negative Coe cients We have a E = P e Pe 2 Since P e > 0 always, we observe the a E becomes negative when or Pe < 0 2 Pe > 2 ME 448/548: D Convection-Di usion Equation page 9

11 Consequences of Negative Coe cients When Pe > 2 the value of a E becomes negative an the eigenvalues of the coe an the numerical solution oscillates. cients become comple Note that we can keep Pe = u e e P < 2 by making P small enough. That is why reucing the mesh size (increasing the number of noes) eliminate the error in the sample calculation. ME 448/548: D Convection-Di usion Equation page 20 Upwin Di erencing: A cure with a cost. Negative a E can be avoie with juicious choice of e. Recallthat 0.9 an e = e E + ( e) P 0.8 φ Increasing u a E = P (D e u e e ) Physical reasoning suggests that e shoul be influence by u e. 3. Take inspiration from the analytical solution to Equation () As the velocity increases, the value of at the mipoint gets closer to the value of the upstream bounary. The Upwin Di erence Scheme gives a strong bias to the upstream value of. ME 448/548: D Convection-Di usion Equation page 2

12 Upwin Di erencing: A cure with a cost Strategy: Ignore the value of at the ownwin noe: ϕ W u w > 0 u e > 0 u w < 0 u e < 0 ϕ W ϕ P ϕ E ϕ P ϕ E W w P E W w P E u w > 0, u e > 0: u w < 0, u e < 0: w = e =0 w = W e = P w =0 e = w = P e = E ME 448/548: D Convection-Di usion Equation page 22 Upwin Di erencing: A cure with a cost ϕ W u w > 0 u e > 0 u w < 0 u e < 0 ϕ W ϕ P ϕ E ϕ P ϕ E W w P E W w P E Restate the rules e = ( 0 if ue 0 if u e < 0 w = ( if uw 0 0 if u w < 0 ME 448/548: D Convection-Di usion Equation page 23

13 Compare Upwin Solution with Central Di erence Solution Compare UDS an CDS for m = 2, Pe =5. No oscillations for the UDS scheme 2.5 Central Upwin Eact φ 0.5 Pe L = 50.0, Pe = e c =.583, e u = ME 448/548: D Convection-Di usion Equation page 24 Compare Upwin Solution with Central Di erence Solution Compare UDS an CDS for m = 32, Pe =.7. No oscillations for either scheme. 2.5 Central Upwin Eact φ 0.5 Pe L = 50.0, Pe =.7 0 e c = 0.268, e u = ME 448/548: D Convection-Di usion Equation page 25

14 Convergence stuy for Upwin an Central Di erence Schemes Solve the moel problem on a series of finer meshes Upwin Central Di erence m Pe e u e u ratio e c e c ratio ME 448/548: D Convection-Di usion Equation page 26 Convergence stuy for Upwin an Central Di erence Schemes m e u ratio e c ratio Marror Central Upwin ME 448/548: D Convection-Di usion Equation page 27

15 Improving Accuracy with a Non-uniform Mesh A non-uniform mesh can improve accuracy of a numerical solution. Noes shoul be close together in regions where the graient is higher. Demonstrate the iea with a stretche mesh L Δ Δ 2 Δ 3 Δ n ME 448/548: D Convection-Di usion Equation page 28 Stretche Mesh () A growing or shrinking cell size, an hence noe spacing, can be achieve by specifying where r is a fie constant. i+ i = r (23) r> causes noe spacing to increase as i increases. r< causes noe spacing to ecrease as i increases. L Δ Δ 2 Δ 3 Δ n ME 448/548: D Convection-Di usion Equation page 29

16 Stretche Mesh (2). The total length L of the region to be subivie is given. 2. Choose the stretching ratio r, an the number of control volumes n. Constraint: The withs of the control volumes must a up to L. L = nx i= i = + r + r r n = +r + r r n The sum of terms can be rewritten. S =+r + r r n = rn r (24) Thus, L = S or = L S. (25) ME 448/548: D Convection-Di usion Equation page 30 Stretche Mesh (3) To create a mesh with a geometric progression of sizes. Specify L, r, ann. 2. Compute from Equation (25). 3. Compute the remaining cell withs with a loop. Note that the values of r an n interact in etermining the actual spacing. It is possible to get istorte meshes by changing r with n hel constant, or changing n with r hel constant. ME 448/548: D Convection-Di usion Equation page 3

17 Stretche Mesh (3) Eamples: Use L =, n = 0 an vary r r =.05 n = 0 r = 0.95 n = 0 r =.0 n = 0 r = 0.90 n = 0 r =.20 n = 0 r = 0.80 n = 0 r =.30 n = 0 r = 0.70 n = 0 r =.40 n = 0 r = 0.60 n = 0 ME 448/548: D Convection-Di usion Equation page 32 Stretche Mesh (4) Eamples: Use L =, an vary r r =.05 r = 0.95 r =.0 r = 0.90 r =.20 r = 0.80 r =.30 r = 0.70 r =.40 r = 0.60 For large n, avoi etreme istortion by using r. ME 448/548: D Convection-Di usion Equation page 33

18 Stretche Mesh (4) Eamples: Use L =, r =0.90, r =0.95 an vary n r = 0.90 n = 0 r = 0.95 n = 0 r = 0.90 r = 0.95 r = 0.90 n = 30 r = 0.95 n = 30 r = 0.90 n = 40 r = 0.95 n = 40 For large n, avoi etreme istortion by using r. ME 448/548: D Convection-Di usion Equation page 34 Stretche Solution 2.5 Central Upwin Eact n = 24, r =0.90 n = 24, r = Central Upwin Eact φ φ 0.5 Pe L = 50.0, Pe = Pe L = 50.0, Pe = e c = 0.042, e u = e c = 0.007, e u = Observations: Careful choice of stretching parameters can yiel goo accuracy on a moest mesh. No obvious way of preicting the accuracy when the eact solution is not available. ME 448/548: D Convection-Di usion Equation page 35

19 Summary () We showe that Conventional finite volume schemes store epenent variables at cell centers. Avection terms require a seconary scheme for interpolating epenent variables at cell faces. The central i erence scheme can cause oscillations The upwin i erence scheme suppresses oscillations, but introuces aitional truncation error. ME 448/548: D Convection-Di usion Equation page 36 Summary (2) We i not inclue the following etails No proofs were given for truncation error estimates. Many other schemes eist. QUICK (quaratic upstream interpolation for convective kinematics),. ENO (essentially non-oscillatory),. TVD (total variational iminishing),. Secon-orer upwining,. Streamline upwining,. an others. ME 448/548: D Convection-Di usion Equation page 37

20 Summary (3) Users nee to be aware of trae-o s Oscillatory higher orer schemes cause solutions to fail, i.e., not converge Venors may use a efault scheme that is stable, but inaccurate, instea of a more accurate scheme Users may choose a stable an inaccurate scheme just to get the coe to converge. That s not a goo iea, especially if the accuracy of the solution is important. Don t suppress the wiggles They re trying to tell you something! P. Gresho an R. Lee, 98 ME 448/548: D Convection-Di usion Equation page 38