ME Computational Fluid Mechanics Lecture 5

Transcription

1 ME Computational Fluid Mechanics Lecture 5 Dr./ Ahmed Nagib Elmekawy Dec. 20, 2018

2 Elliptic PDEs: Finite Difference Formulation Using central difference formulation, the so called five-point formula for the Laplace's equation: A higher order formula using nine-point formula

3 Elliptic PDEs: Finite Difference Formulation Using five-point formula for Laplace's equation: where The boundary conditions:

4 Elliptic PDEs: Finite Difference Formulation The corresponding linear set of equations:

5 Elliptic PDEs: Finite Difference Formulation The set of equations in a matrix form:

6 Elliptic PDEs: Solution Algorithms Solving simultaneous linear algebraic equations are classified as Direct method: Gaussian elimination Cramer s rule Enormous amount of athematic operations (N x N) Large memory usage Some advanced techniques can be used for small problems Iterative method: It is based on guess and correct philosophy By an initial guessed solution, a new solution will be obtained. A newer solution will be obtained from the new solution. The procedure will be repeated until convergence is reached

7 Elliptic PDEs: Iterative Methods Iterative method: It is classified based on the formulation of the iterative scheme Point iterative method: similar to explicit methods of parabolic equations, where there is only one unknown at the next time/iteration step Line iterative method: similar to implicit schemes, where the formulation contains more than one unknowns

8 Elliptic PDEs: Point Iterative Methods 1- Jacobi Iteration Method Jacobi Iteration Method: Dependent variable is computed at each grid point as a function of initial guessed or previously computed values Therefore the discrete equation can be written as: This method is rarely used Analogy between Jacobi method and Explicit time dependent equation: Consider the parabolic equation: Using FTCS and using Δx = Δy, where D x =D y =D has to be 0.25 for stable solution of two-dimensional problems.

9 Elliptic PDEs: Point Iterative Methods 1- Jacobi Iteration Method Using D = 0.25, the discrete formulation becomes Applying the Jacobi method as Δx = Δy β=1, the iterative equation for the elliptic equation becomes: Although, the two equations are representing completely different phenomenon, but mathematically they are identical. This notice suggests that many techniques used in parabolic equations can be extended to obtain efficient algorithms for the solution of elliptic equations.

10 Elliptic PDEs: Point Iterative Methods 2- Point Gauss-Seidel Iteration Method In the Jacobi method, the new dependent variable at k+1 is computed from the dependent variable at k level. In the current method, the the new dependent variable at k+1 is computed from its neighbors once they are updated. This method is 100% faster in convergence than Jacobi method.

11 Elliptic PDEs: Line Iterative Methods 1- Line Gauss-Seidel Iteration Method In this formulation, the discrete equation is written with three unknowns at (i-1,j), (i,j), and (i+1,j): This method converges faster than Point Gauss-Seidel method by 50% in number of iterations. But it requires more time per iteration to be computed. Solution advances over iterations at the line direction much faster than the other direction as the boundary directly used with implicit scheme to get all line points.

12 Elliptic PDEs: Over-Relaxation Method 1- Point Successive Over-Relaxation The concept assumes that there is a trend where the initial state approaches steady state, the direction of the change can be used to extrapolate for the next iteration step Starting with Gauss-Seidel iteration method Adding to the right hand side, and collecting terms: As the solution advances, approaches, to accelerate the solution, the term in brackets is multiplied by ω, the relaxation parameter: For convergence, 0<ω<2, and if 0<ω<1 --> it is called under relaxation In general, optimum ω cannot be determined easily and needs numerical experimentations

13 Elliptic PDEs: Over-Relaxation Method 2- Line Successive Over-Relaxation Solution can also be accelerated using similar relaxation parameter used for PSOR Starting with Line Gauss-Seidel iteration method adding new terms and rearrange: In practice, trial and error is used to determine the optimum ω

14 Elliptic PDEs: Over-Relaxation Method 3- Alternating Direction Implicit (ADI) Method An iteration is considered when the tridiagonal system is solved for rows and then for columns or vice versa: For rows: For columns:

15 Programing Assignment 2 Consider an incompressible and inviscid fluid flow moving around an infinite cylinder.

16 Elliptic Equations in Polar Coordinates Consider an incompressible and inviscid fluid flow moving around an infinite cylinder. The inlet velocity is 1 m/s. The cylinder radius is 1 m. Consider a mesh starting from r = 1 m with r = 0.1 and a final radius of 10 m and θ=360/100=3.6. For polar coordinates

17 For polar coordinates Elliptic Equations in Polar Coordinates v r = 1 ψ r θ, v θ = ψ (1) r The continuity equation in cartesian coordinates for an incompressible flow is u x + v y + w z = 0 The continuity equation in polar coordinates is 1 r rv r r + 1 r v θ θ = 0 (2)

18 By substitution of eq(1) in eq(2), we get 1 r 1 ψ r θ + 1 ψ r r r r θ 2 ψ ψ θ r = 1 r 2 ψ θ r = 0 = 1 r r θ 1 r So, the continuity is satisfied. Assume the following boundary conditions: ψ = r = 1 sin θ ψ = r sin θ Γ r 2π 1 r ψ r θ ln farfield, r = 10 m

19 Consider the rotation vector in cartesian coordinates: e x e y e z ω = x y z u v w the rotation vector in polar coordinates: ω = 1 r v z θ v θ z ω = e r + e r e θ e z 1 1 r r r θ z v r v θ v z 1 v z r r v r e z θ + 1 r v θ r 1 r v r θ e z

20 For 2d irrotational flow v θ r 1 r v r θ = 0 ω z = 1 r Substitute 1 r ψ r 1 1 ψ r θ = 0 r r r θ 1 ψ r r r r ψ r 2 θ 2 = 0 The Laplace equation for polar coordinates is: 1 ψ r r r r ψ r 2 θ 2 = 0

21

22

23

24

25

26

27

28 Sequence of Programs For all following methods we will use the following

29 Apply the following schemes: Point Jacobbi Method Point Gauss Seidal Point Successive Over Relaxation (PSOR) Line Successive Over Relaxation (LSOR) Plot the streamlines by using Matlab by using the following: Γ = 0, 2π, 4π and 8π.

30 Hyperbolic PDEs: Introduction Several governing equations in fluid mechanics are ruled by hyperbolic equations: First order wave equation Second order wave equation

31 Hyperbolic PDEs: Explicit Formulations Euler s FTFS Unconditionally unstable Euler s FTCS Unconditionally unstable The Upwind differencing scheme (FTBC) Stable when C 1, where C=a Δt/Δx

32 Hyperbolic PDEs: Explicit Formulations The Lax method Stable when C 1 Midpoint Leapfrog method Stable when C 1 The Lax-Wendroff method Stable when C 1

33 Hyperbolic PDEs: Implicit Formulations Euler s BTCS O(Δt, Δx 2 ) Unconditionally stable Implicit Upwind differencing method O(Δt, Δx) Unconditionally stable Crank-Nicolson method O(Δt 2, Δx 2 )

34 Hyperbolic PDEs: Mutli-Step Formulations The method uses finite difference method at split time level. The method is being referred to as predictor corrector method, where in the first step, a temporary value for the dependent variable is predicted and in the second step, a corrected value is computed providing the final value of the dependent variable. Richtmyer/Lax-Wendroff Multi-Step Method O(Δt 2, Δx 2 ) Stable when C 2

35 Hyperbolic PDEs: Mutli-Step Formulations The MacCormack Method O(Δt 2, Δx 2 ) It uses forward differencing in the predictor step While the corrector step is backward differencing Stable when C 1 Note: multi-step is called splitting method when applies to multidimensional problems and reduces finite difference equation to two sets of tri-diagonal systems. Multi-step is called predictor-corrector when it solves the problem in a sequence of time.

36 Hyperbolic PDEs: Application to Linear Problems Consider the model problem: where a, the wave speed and equal to 250 m/s The boundary conditions:

37 Hyperbolic PDEs: Application to Linear Problems Using FTBS (upwind difference): For stable solution Courant number c 1 If c=1 solution: Investigating solution for various time steps

38 Hyperbolic PDEs: Application to Linear Problems Solution using c=1 Best results at c=1. Why?

39 Hyperbolic PDEs: Application to Linear Problems Using Lax-Wendroff: For stable solution Courant number c 1 If c=1 solution: Investigating solution for various time steps

40 Hyperbolic PDEs: Application to Linear Problems Solution using c=1 Best results at c=1. Why?

41 Programing Assignment 3 A shock wave moving within 300 m starting with 2 m/s for 50 m and the rest space is shocked with 1 m/s speed. And the wave is moving with 300 m/s speed. We are going to study the behavior of that wave within different times and different stability conditions. Consider the following wave equation:-

42 42

43 43

44 44

45 45

46 46

47 47

48 48

49 49