2.29 Numerical Fluid Mechanics Fall 2011 Lecture 27

Transcription

1 REVIEW Lecture 26: Fall 2011 Lecture 27 Solution of the Navier-Stokes Equations Pressure Correction and Projection Methods (review) Fractional Step Methods (Example using Crank-Nicholson) Streamfunction-Vorticity Methods: scheme and boundary conditions Artificial Compressibility Methods: Scheme definitions and example Boundary Conditions: Wall/Symmetry and Open boundary conditions Finite Element Methods Introduction Method of Weighted Residuals: Galerkin, Subdomain and Collocation General Approach to Finite Elements: Steps in setting-up and solving the discrete FE system Galerkin Examples in 1D and 2D 1 p ui t x i 0 n1 n p p ( u ) i 0 t x i n1 * n1 n 1 * n1 ( ui ) p p ( ) p xi xi x i n ui ui n1 PFJL Lecture 27, 1

2 Project Presentations: Schedule 15 minutes each, including questions Notes: i) Project Office Hours As needed ii) Need Draft Titles by Monday Dec 12 iii) Reports latest at noon on Tue Dec 20 4 to 25 pages of text, single space, 12ft 1 to 10 pages of figures PFJL Lecture 27, 2

3 TODAY (Lecture 27): Finite Elements and Intro to Turbulent Flows Finite Element Methods Introduction, Method of Weighted Residuals: Galerkin, Subdomain and Collocation General Approach to Finite Elements: Steps in setting-up and solving the discrete FE system Galerkin Examples in 1D and 2D Computational Galerkin Methods for PDE: general case Variations of MWR: summary Isoparametric finite elements and basis functions on local coordinates (1D, 2D, triangular) Turbulent Flows and their Numerical Modeling Properties of Turbulent Flows Stirring and Mixing Energy Cascade and Scales Turbulent Wavenumber Spectrum and Scales Numerical Methods for Turbulent Flows: Classification Note: special recitation today on (Hybrid-) Discontinuous Galerkin Methods PFJL Lecture 27, 3

4 References and Reading Assignments Chapter 31 on Finite Elements of Chapra and Canale, Numerical Methods for Engineers, Lapidus and Pinder, 1982: Numerical solutions of PDEs in Science and Engineering. Chapter 5 on Weighted Residuals Methods of Fletcher, Computational Techniques for Fluid Dynamics. Springer, Chapter 9 on Turbulent Flows of J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics. Springer, NY, 3 rd edition, 2002 Chapter 3 on Turbulence and its Modelling of H. Versteeg, W. Malalasekra, An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Prentice Hall, Second Edition. Chapter 4 of I. M. Cohen and P. K. Kundu. Fluid Mechanics. Academic Press, Fourth Edition, 2008 Chapter 3 on Turbulence Models of T. Cebeci, J. P. Shao, F. Kafyeke and E. Laurendeau, Computational Fluid Dynamics for Engineers. Springer, PFJL Lecture 27, 4

5 General Approach to Finite Elements 1. Discretization: divide domain into finite elements Define nodes (vertex of elements) and nodal lines/planes 2. Set-up Element equations i. Choose appropriate basis functions i (x): ux u( ( ) a a i i ( x) i1 1D Example with Lagrange s polynomials: Interpolating functions N i (x) x2 x xx1 u a xun( 1 1 x) un ( x) where N ( x) and N x x x ( ) 0 a x x With this choice, we obtain for example the 2 nd order CDS and x2 Trapezoidal rule: du u u1 u2 1 a 2 and (x 2 x 1 ) dx x 2 x 1 u 1 u dx x 1 2 ii. Evaluate coefficients of these basis functions by approximating the solution in an optimal way This develops the equations governing the element s dynamics n 2 1 Node 1 Node 2 x 1 u 1 1 (i) u (ii) N 1 (iii) N 2 (iv) x 2 u 2 (i) A line element (ii) The shape function or linear approximation of the line element (iii) and (iv) Corresponding interpolation functions. Image by MIT OpenCourseWare. 1 Two main approaches: Method of Weighted Residuals (MWR) or Variational Approach Result: relationships between the unknown coefficients a i so as to satisfy the PDE in an optimal approximate way PFJL Lecture 27, 5

6 General Approach to Finite Elements, Cont d 2. Set-up Element equations, Cont d Mathematically, combining i. and ii. gives the element equations: a set of (often linear) algebraic equations for a given element e: where K e is the element property matrix (stiffness matrix in solids), u e the vector of unknowns at the nodes and f e the vector of external forcing 3. Assembly: After the individual element equations are derived, they must be assembled: i.e. impose continuity constraints for contiguous elements This leas to: K u e e e where K is the assemblage property or coefficient matrix, u and f the vector of unknowns at the nodes and f e the vector of external forcing 4. Boundary Conditions: Modify K u = f to account for BCs 5. Solution: use LU, banded, iterative, gradient or other methods 6. Post-processing: compute secondary variables, errors, plot, etc f K u f PFJL Lecture 27, 6

7 Galerkin s Method: Simple Example Differential Equation 1. Discretization: Generic N (here 3) equidistant points along x Boundary Conditions Exact solution: y=exp(x) 2. Element equations: i. Basis (Shape) Functions: Power Series Boundary Condition Note: this is equivalent to imposing the BC on the full sum In this simple example, a single element is chosen to cover the whole domain the element/mass matrix is the full one (K=K e ) PFJL Lecture 27, 7

8 Galerkin s Method: Simple Example, Cont d ii. Optimal coefficients with MWR: set weighted residuals (remainder) to zero Remainder: R dy dx Galerkin set remainder orthogonal to each shape function: Denoting inner products as: leads to: which then leads to the Algebraic Equations: y ( f, g) f, g f gdx k1 (, ) 0, 1,..., Rx k N 1 0 N=3; d=zeros(n,1); m=zeros(n,n); exp_eq.m for k=1:n d(k)=1/k; for j=1:n m(k,j) = j/(j+k-1)-1/(j+k); end end a=inv(m)*d; y=ones(1,n); for k=1:n y=y+a(k)*x.^k end j PFJL Lecture 27, 8

9 Galerkin s Method Simple Example, Cont d 3-4. Assembly and boundary conditions: Already done (element fills whole domain) 5. Solution: For N=3; d=zeros(n,1); m=zeros(n,n); exp_eq.m for k=1:n d(k)=1/k; for j=1:n m(k,j) = j/(j+k-1)-1/(j+k); end end a=inv(m)*d; y=ones(1,n); for k=1:n y=y+a(k)*x.^k end L 2 Error: PFJL Lecture 27, 9

10 Comparisons with other Weighted Residual Methods Least Squares Subdomain Method Galerkin Collocation PFJL Lecture 27, 10

11 Comparisons with other Weighted Residual Methods Comparison of coefficients for approximate solution of dy/dx - y = 0 Scheme Coefficient a 1 a 2 a 3 Least squares Galerkin Subdomain Collocation Taylor series Optimal L 2,d Image by MIT OpenCourseWare. Comparison of approximate solutions of dy/dx - y = 0 Least x Galerkin Subdomain Collocation squares Taylor series Optimal L 2,d Exact y a - y 2,d Image by MIT OpenCourseWare. PFJL Lecture 27, 11

12 Galerkin s Method in 2 Dimensions Differential Equation Boundary Conditions y n (x,y 2 ) Test Function Solution (u o satisifies BC) Remainder j n (x 1,y) n (x 2,y) j Inner Product: Galerkin s Method n (x,y 1 ) x PFJL Lecture 27, 12

13 Galerkin s method: 2D Example Fully-developed Laminar Viscous Flow in Duct y Steady, Very Viscous Fluid Flow in Duct 1 Poisson s Equation, Non-dimensional: -1 1 x Shape/Test Functions -1 4 BCs: No-slip (zero flow) at the walls Shape/Test functions satisfy boundary conditions Again: element fills the whole domain in this example PFJL Lecture 27, 13

14 Galerkin s Method: Viscous Flow in Duct, Cont d Remainder: Inner product: Analytical Integration: x=[-1:h:1]'; y=[-1:h:1]; duct_galerkin.m n=length(x); m=length(y); w=zeros(n,m); Nt=5; for j=1:n xx(:,j)=x; yy(j,:)=y; end for i=1:2:nt for j=1:2:nt w=w+(8/pi^2)^2* (-1)^((i+j)/2-1)/(i*j*(i^2+j^2)) *cos(i*pi/2*xx).*cos(j*pi/2*yy); end end Galerkin Solution: Nt=5 3 terms in each direction Flow Rate: PFJL Lecture 27, 14

15 Computational Galerkin Methods: General Case Differential Equation: Residuals PDE: ICs: BCs: Boundary problem PDE satisfied exactly Boundary Element Method Panel Method Spectral Methods Global Test Function: Inner problem Boundary conditions satisfied exactly Finite Element Method Spectral Methods Time Marching: Mixed Problem Weighted Residuals PFJL Lecture 27, 15

16 Inner Product Different forms of the Methods of Weighted Residuals: Summary Discrete Form k 1,2,..., n Subdomain Method: R( x) dxdt 0 tv k Collocation Method: R( x ) 0 k Least Squares Method: R( ) R( ) d dt 0 a x x x i tv k In the least-square method, the coefficients are adjusted so as to minimize the integral of the residuals. It amounts to the continuous form of regression. Method of Moments: Galerkin: In Galerkin, weight functions are basis functions: they sum to one at any position in the element. In many cases, Galerkin s method yields the same result as variational methods PFJL Lecture 27, 16

17 How to obtain solution for Nodal Unknowns? Modal Basis vs. Interpolating (Nodal) Basis functions N u( ( xy), ) u j N( j xy), j1 N u( ( xy), ) a k k ( xy), k1 N u j a k k ( x, y k 1 j j ) u Φa 1 a Φ u N N u( ( xy), ) Φ 1 u j k ( xy), kj k1 j1 N N j kj k j1 k1 u Φ 1 ( xy), 1 Dimension 2 Dimensions N N j ( xy), Φ 1 k ( xy), k 1 kj PFJL Lecture 27, 17

18 Complex Boundaries Isoparametric Elements Y Isoparametric mapping at a boundary X 4 η = 1 η 3 D A 4 C 1 B 3 2 ξ = 1 ξ = 1 ξ B 1 η = 1 2 Image by MIT OpenCourseWare. PFJL Lecture 27, 18

19 Trial Function Solution Finite Elements 1-dimensional Elements Trial Function Solution Interpolation (Nodal) Functions Element A Element B Element C (a) u u u a a u u a2 a u u a3 u a u ~ N u = Σ N j (x)u j j = 1 Interpolation Functions N 2 = x - x 1 x 2 - x 1 (b) 1.0 x 1 x 2 x 3 x 4 x N 2 = x - x 3 x 2 - x 3 Element B N 3 = x - x 2 x 3 - x 2 N 2 = x - x 3 x 2 - x 3 N(x) N 2 = x - x 1 x 2 - x 1 N 3 = x - x 4 x 3 - x 4 N 3 = x - x 2 x 3 - x 2 x 1 x 2 x 3 x x 4 N 3 = x - x 4 x 3 - x 4 Image by MIT OpenCourseWare. 4 PFJL Lecture 27, 19

20 Finite Elements 1-dimensional Elements Quadratic Interpolation Functions One-dimensional quadratic shape functions Element A Element B 1.0 N 2 N 3 N4 N(x) 0 x 1 x 2 x 3 x 4 x x 5 Image by MIT OpenCourseWare. 4 PFJL Lecture 27, 20

21 Finite Elements in 1D: Nodal Basis Functions in the Local Coordinate System Please see pp in Lapidus, L., and G. Pinder. Numerical Solution of Partial Differential Equations in Science and Engineering. 1st ed. Wiley-Interscience, [See the selection using Google Books preview] PFJL Lecture 27, 21

22 Finite Elements 2-dimensional Elements 4 η = 1 3 ξ = 1 ξ = 1 B 1 η = 1 2 Bilinear shape function on a rectangular grid Image by MIT OpenCourseWare. Image by MIT OpenCourseWare. Quadratic Interpolation (Nodal) Functions Linear Interpolation (Nodal) Functions η ξ - PFJL Lecture 27, 22

23 Two-Dimensional Finite Elements Example: Flow in Duct, Bilinear Basis functions Finite Element Solution y 2 Integration by Parts dx= dx Algebraic Equations for center nodes (for center nodes) PFJL Lecture 27, 23

24 Finite Elements in 2D: Nodal Basis Functions in the Local Coordinate System Wiley-Interscience. All rights reserved. This content is excluded from our Creative Commons license. For more information, see PFJL Lecture 27, 24

25 Finite Elements in 2D: Nodal Basis Functions in the Local Coordinate System Please see table 2.7a, Basis Functions Formulated Using Quadratic, Cubic, and Hermitian Cubic Polynomials, in Lapidus, L., and G. Pinder. Numerical Solution of Partial Differential Equations in Science and Engineering. 1st ed. Wiley-Interscience, PFJL Lecture 27, 25

26 Triangular Coordinates u(x, y) a 0 a 1,1 x a 1,2 y Finite Elements 2-dimensional Triangular Elements u u 1 Linear Polynomial Modal Basis Functions: y u 3 (i) u 2 x N 1 u 1 (x, y) a 0 a 1,1 x 1 a 1,2 y 1 1 x 1 y 1 a 0 u 1 u 2 (x, y) a 0 a 1,1 x 2 a y 1,2 2 1 x 2 y 2 a u 1,1 2 u 3 (x, y) a 0 a 1,1 x 3 a y 1 x 1,2 3 3 y3 a 1,2 u 3 y 0 (ii) 1 0 x Nodal Basis (Interpolating) Functions: N 2 u( x, y) u 1 N 1 ( x, y) u 2 N 2 ( x, y) u 3 N 3 ( x, y) y x N 1 (x, y) N 2 (x, y) N 3 (x, y) 1 2A T 1 2A T 1 2A T (x 2 y 3 x 3 y 2 ) ( y 2 y 3 ) x (x 3 x 2 ) y (x 3 y 1 x 1 y 3 ) ( y 3 y 1 ) x (x 1 x ) y 3 (x 1 y 2 x 2 y 1 ) ( y y ) x (x x ) y y N 3 1 (iii) (iv) A linear approximation function (i) and its interpolation functions (ii)-(iv). 0 Image by MIT OpenCourseWare. 0 x PFJL Lecture 27, 26

27 Turbulent Flows and their Numerical Modeling Most real flows are turbulent (at some time and space scales) Properties of turbulent flows Highly unsteady: velocity at a point appears random Three-dimensional in space: instantaneous field fluctuates rapidly, in all three dimensions (even if time-averaged or space-averaged field is 2D) Some Definitions Ensemble averages: average of a collection of experiments performed under identical conditions Stationary process: statistics independent of time For a stationary process, time and ensemble averages are equal Academic Press. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Three turbulent velocity realizations in an atmospheric BL in the morning (Kundu and Cohen, 2008) PFJL Lecture 27, 27

28 U8 Turbulent Flows and their Numerical Modeling Properties of turbulent flows, Cont d Highly nonlinear (e.g. high Re) High vorticity: vortex stretching is one of the main mechanisms to maintain or increase the intensity of turbulence High stirring: turbulence increases rate at which conserved quantities are stirred Stirring: advection process by which conserved quantities of different values are brought in contact (swirl, folding, etc) Mixing: irreversible molecular diffusion (dissipative process). Mixing increases if stirring is large (because stirring leads to large 2 nd and higher spatial derivatives). Turbulent diffusion: averaged effects of stirring modeled as diffusion Characterized by Coherent Structures CS are often spinning, i.e. eddies Turbulence: wide range of eddies size, in general, wide range of scales Instantaneous interface 1 ~ δ δ Turbulent flow in a BL: Large eddy has the size of the BL thickness Image by MIT OpenCourseWare. PFJL Lecture 27, 28

29 Welander s scrapbook. Stirring and Mixing Welander P. Studies on the general development of motion in a two-dimensional ideal fluid. Tellus, 7: , His numerical solution illustrates differential advection by a simple velocity field. A checkerboard pattern is deformed by a numerical quasigeostrophic barotropic flow which models atmospheric flow at the 500mb level. The initial streamline pattern is shown at the top. Shown below are deformed check board patterns at 6, 12, 24 and 36 hours, respectively. Notice that each square of the checkerboard maintains constant area as it deforms (conservation of volume). Wiley. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Source: Fig. 2 from Welander, P. "Studies on the General Development of Motion in a Two-Dimensional, Ideal Fluid." Tellus 7, no. 2 (1955). PFJL Lecture 27, 29

30 Energy Cascade and Scales British meteorologist Richarson s famous quote: Big whorls have little whorls, Which feed on their velocity, And little whorls have lesser whorls, And so on to viscosity. Dimensional Analyses and Scales (Tennekes and Lumley, 1972, 1976) Largest eddy scales: L, T, U = Distance/Time over which fluctuations are correlated and U = large eddy velocity Viscous scales:, τ, u = viscous length (Kolmogorov scale), time and velocity scales Hypothesis: rate of turbulent energy production rate of viscous dissipation Length-scale ratio: Time-scale ratio: Velocity-scale ratio: L O u L T 3/4 / ~ (Re L ) Re L ' 1/2 / ~ O(Re L ) U u 1/4 / ' ~ O(Re L ) η (Kolmogorov microscale) Image by MIT OpenCourseWare. PFJL Lecture 27, 30

31 Turbulent Wavenumber Spectrum and Scales Turbulent Kinetic Energy Spectrum S(K): 2 ' ( ) 0 In the inertial sub-range, Kolmogorov argued by dimensional analysis that S SK (, ) A K K 2/3 5/ u S K dk A 1.5 found to be universal for turbulent flows Turbulent energy dissipation Turb. energy 2 u u u Turb. time scale L L Komolgorov microscale: Size of eddies depend on turb. dissipation ε and viscosity 3 Dimensional Analysis: 3 1/4 Academic Press. All rights reserved. This content is excluded from our Creative Commons license. For more information, see PFJL Lecture 27, 31

32 Numerical Methods for Turbulent Flows Primary approach (used to be) is experimental Numerical Methods classified into methods based on: 1) Correlations: useful mostly for 1D problems, e.g.: Moody chart or friction factor relations for turbulent pipe flows, Nusselt number for heat transfer as a function of Re and Pr, etc. 2) Integral equations: Integrate PDEs (NS eqns.) in one or more spatial coordinates Solve using ODE schemes (time-marching) 3) Averaged equations Averaged over time or over an (hypothetical) ensemble of realizations Often decompositions into mean and fluctuations: Leads to a set of PDEs, the Reynolds-averaged Navier-Stokes (RANS) equations ( One-point closure methods) f f(re, ) Nu (Re,Pr, Ra) u u u'; PFJL Lecture 27, 32

33 Numerical Methods for Turbulent Flows Numerical Methods classification, Cont d: 4) Large-Eddy Simulations (LES) Solves for the largest scales of motions of the flow Only Approximates or parameterizes the small scale motions Compromise between RANS and DNS 5) Direct Numerical Simulations (DNS) Solves for all scales of motions of the turbulent flow (full Navier- Stokes) The methods 1-to-5 make less and less approximations, but computational time increases Conservation PDEs are solved as for laminar flows: major challenge is the much wider range of scales (of motions, heat transfer, etc) PFJL Lecture 27, 33

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