Numerical Methods for PDEs Homework 2

Transcription

1 Numerical Methods for PDEs Homework 2 Instructor : Prof. D. Willis david willis@uml.edu In this homework problem you will investigate finite difference methods on rectangular and nonrectangular 2D domains using MATLAB (or any other programming language of your choosing). You will approximate the porous media flow inside of a coffee pod, similar to those used by singlecup coffee brewers: Figure 1: (left) An example coffee pod and (right) A depiction of how it works. ( ( add.jpg) The diagram below illustrates the coffee pod. porous coffee media: Higher pressure water is driven through the Figure 2: (left) The flow inside a single brew coffee pod will be modeled as a porous-media flow. 1

2 The 2D physical domain will approximate one of these pods (in reality this should be modeled as a three-dimensional problem here, we will assume that the 2D model is adequate to model the basic physical principles). Since a pod is symmetric about the vertical centerline (x = 0), we shall only consider half of the domain in our analysis. The domain will be defined by the pod height, H, the bottom width W B, the top width, W T, and the hole size W H, as shown in the image below: Figure 3: (left) The physical domain. The physics of the flow through the coffee in the domain will be considered as pressure driven porous-media flow. To derive the equation, we start with the differential conservation of mass equation: q = q x x + q y y = 0, (1) where q is the velocity at a given point, q x is the x component of the velocity and q y is the y component of the velocity. Introducing Darcy s law to relate the flow velocity field to the pressure gradient field through the porous media: q = κ p (2) µ where, p is the pressure at a given location in the domain, µ is the viscosity of the fluid and κ is the permeability of the porous media. Substituting equation 2 into equation 1, the result is: q = ( κµ ) p = κ µ 2 p = 2 p = 0. (3) The equation governing the pressure in the porous media is the Laplace equation, which should be familiar to you at this point: 2 p = 0 (4) 2

3 The solution which you are finding is the pressure as a function of position in the coffee pod, p(x, y). In order to solve this equation numerically, boundary conditions are required. The pressure, p, at the inlet and exit holes are known. For the sake of simplicity, we will perform the analysis in gauge pressure, so: p = P on Γ T op p = 0 on Γ Hole The boundary conditions on the symmetry plane must reflect the symmetry assumption, hence: p n = 0 The boundary conditions on all solid boundaries must prevent flow from passing through the surface, hence: q ˆn = 0 ( κµ ) n p = 0 p n = 0 The following image illustrates the domain and boundary: Figure 4: (left) The physical domain with governing partial differential equation and boundary conditions. 3

4 Please answer the following questions. Submit written answers on paper. Plots and calculated results should be printed on paper. Computer code should be delivered in your dropbox.com folder. 1. Write the bilinear mapping functions, x(ξ, η) and y(ξ, η) that you will use. Present the linearsystem (matrix and vectors) uses H, W T, and W B to define the mapping coefficients for the bilinear mapping functions. 2. Write comprehensive pseudo code to illustrate how you will determine numerical values of the bilinear mapping coefficients. 3. Implement your mapping (only the bilinear mapping) using Matlab or other coding language. Demonstrate that your mapping works by mapping a collection of points in the ξ η space to the physical space for a collection of different H, W T and W B values (present 2-3 plots to confirm your mapping process works). 4. Show using a drawing and a pseudo code how you will go between a point with indices i, j in your (computational or physical) domain to a node number (ie. how you will take a point from i, j index definition to a node number so that you can access the appropriate row & column of a matrix). 5. List ALL of the second order finite difference equations that will be used to approximate the partial derivatives in the equation governing the computational domain solution. 6. Write the finite difference equation represents the governing equation for each internal node in the domain. Draw the associated 9-point finite difference stencil. Create a 3 3 table containing the coefficients (AA, BB, CC, etc) for each point used in the finite difference stencil for node i, j. 7. Write a detailed pseudo-code that describes the Matlab/computer code process for setting up and solving this finite difference problem. 8. Membrane deflection FD-code to start: Write your own finite difference solver (you may closely follow the example from class, but for your sake, try to write a code that differs from that in class) to solve the deflection of a membrane on a non-rectangular domain (set H = 2, W B = 0.5 and W T = 1.0). You will solve the following equation: 2 u = 1 (6) with boundary conditions, u = 0 on all domain boundaries. represents the deflection of the membrane. For this question, u 9. Illustrate that the deflection calculated using this finite difference solver has second order error convergence as a function of grid size (convergence is O( x 2 )). Please plot this result using an appropriate plot (eg. log, semilog, plot, etc.) 4

5 10. Simple(r) coffee pod problem: You will next try to solve the coffee pod problem for a rectangular domain. The coffee pod problem is different than the membrane problem due to boundary conditions. First describe on paper how the boundary conditions will be applied in the A-matrix for the rectangular domain finite difference problem (you should ensure a second order accurate solution is maintained so, second order FD expressions should be used for all derivatives). 11. Still using the rectangular domain (H = 2, W T = W B = 1 and W H = 0.25). Introduce in your code the appropriate boundary conditions for the porous-media flow in the relevant locations of the A-matrix. Solve the problem using Matlab (or other programming language of choice) twice. Once for a grid with 65 nodes and second for 129 nodes respectively in the x-direction (I only prescribe x-direction nodes to allow you flexibility to use whatever you want for NNodes in y). Assume P = 1 at the top of the domain. Plot the pressure distribution solution. Does the solution look reasonable & converged? 12. Now, describe on paper how you would introduce the boundary conditions when a nonrectangular domain is used (ie. the pod domain). You don t need to implement the nonrectangular domain boundary conditions to receive full credit for this assignment. 13. Bonus Points 1: Indicate how you would calculate the flow rate of coffee through the pod using your finite difference code. 14. Bonus Points 2: Implement the boundary conditions for a non-rectangular domain. Show that the flow rate calculation converges with refinement for a non-rectangular domain defined by: Set the pressure at the top to P = 1. Set the width of the bottom to W B = 1 Set the width of the hole to W H = 0.25 Set the width of the top to W T = 1.5 Set the height to H = Bonus Points 3: Implement the boundary conditions for a non-rectangular domain. Perform the pod analysis for a range of geometries defined by: Set the pressure at the top to P = 1. Set the width of the bottom to W B = 0.5 Set the width of the hole to W H = 0.25 Vary the length of the top boundary and height to maintain a constant pod area A pod = 2. This ensures a constant amount of coffee in the pod. Calculate and plot the coffee flow rate as a function of the pod height. What do you notice? 5