Numerical Solution Techniques in Mechanical and Aerospace Engineering

Transcription

1 Numerical Solution Techniques in Mechanical and Aerospace Engineering Chunlei Liang LECTURE 9 Finite Volume method II 9.1. Outline of Lecture Conservation property of Finite Volume method Apply FVM to general conservation Law Generation and handling of unstructured grids Can write a 1D FVM code to solve hyperbolic equations 9.. Conservation of FVM Let us consider the D steady continuity equation (9.1) (ρu) = 0 Let us consider two adjacent CVs P and E shown in Fig. (1). Integrate the above equation over CV of P, we get (9.) enws { ρu x + ρv } dxdy = {(ρu) y e (ρu) w } y + {(ρv) n (ρv) s } x where overbar means the averaged value through linear interpolation at the edge center. For instance, to compute the first term on the right hand side of the above expression, we use the following approximation 1

2 CHUNLEI LIANG, FINITE VOLUME METHOD II Figure 1. A typical stencil for the D FVM. (9.3) (ρu) e y = ρ e u e y 1 (ρ P + ρ E ) 1 (u P + u E ) y Now, if we integrate the continuity equation over the CV of E, we get (9.4) ee ne e ss { ρu x + ρv } dxdy = {(ρu) y ee (ρu) e } y+{(ρv) ne (ρv) se } x Adding Eqs. 9. and 9.4 together, we get, (9.5) w s se ee ne n { ρu x + ρv } dxdy = {(ρu) y ee (ρu) w } y + {(ρv) ne (ρv) se + (ρv) n (ρv) s } x Thus the discretized equation for the control volume that surronds both P and E, involves only mass flow rates through the boundary faces. The flow rates through the e face (common face between CVs P and E) simply cancel each other and do not appear in the final expression. The same practice can be applied for the non-conservative form (9.6) ρ u x + u ρ x + ρ v y + v ρ y = 0

3 LECTURE 9. FINITE VOLUME METHOD II 3 Integrate Eq. 9.6 over control volume P, we can evaluate the flow rate through face e, (9.7) (ρ P u e + u P ρ e ) y = [ ] 1 ρ P (u 1 E + u P ) + u P (ρ E + ρ P ) y Integrate Eq. 9.6 over control volume E, we can evaluate the flow rate through face e, [ ] 1 (9.8) (ρ E u e + u E ρ e ) y = ρ E (u 1 E + u P ) + u E (ρ E + ρ P ) y Adding the above two expressions, the net flow is different from zero at face e. Thus the non-conservative formulation introduces artificial source/sink of mass without physical meaning. Similar analysis can be applied to the momentum equations Apply FVM for general conservation law Consider the general conservation law problem, represented by the following PDE, u (9.9) + f (u) = 0. t Here, u represents a vector of states and f represents the corresponding flux tensor. Again we can sub-divide the spatial domain into CVs or cells. For a particular cell, i, we take the volume integral over the total volume of the cell, v i, which gives, u (9.10) v i t dv + f (u) dv = 0. v i On integrating the first term to get the volume average and applying the divergence theorem to the second, this yields dū i (9.11) v i dt + f (u) n ds = 0, S i where S i represents the total surface area of the cell and n is a unit vector normal to the surface and pointing outward. Rearrange the volume term v i : (9.1) dū i dt + 1 f (u) n ds = 0. v i S i

4 4 CHUNLEI LIANG, FINITE VOLUME METHOD II This equation is now an ODE. One can apply the Runge-Kutta explicit type schemes we learned in Lecture 1 to solve this ODE. The values for the edge fluxes can be reconstructed by interpolation or extrapolation of the cell averages. The actual numerical scheme will depend upon problem geometry and mesh construction. Again, FVMs are conservative as cell averages change through the edge fluxes. In other words, one cell s loss is another cell s gain! an exercise of MUSCL type scheme Consider 1D advection problem, following Eq. (9.1), (9.13) du i dt + 1 [ ] F F = 0. i+ x 1 i 1 i Use piecewise constant approximation, we can use Rusanov scheme to approximate the above flux terms: (9.14) F i 1 (9.15) F i+ 1 = 1 = 1 {[ ( ) ( )] [ ]} F u R + F u L a i 1 i 1 i 1 u R u L. i 1 i 1 {[ ( ) ( )] [ ]} F u R + F u L a i+ 1 i+ 1 i+ 1 u R u L. i+ 1 i+ 1 where the local propagation speed, a i± 1, is the maximum absolute value of the eigenvalue of the Jacobian of F (u (x, t)) over cells i, i ± 1. MUSCL stands for Monotone Upstream-centered Schemes for Conservation Laws, and the term was introduced in a seminal paper by Bram van Leer (van Leer, 1979). van Leer, B. (1979), Towards the Ultimate Conservative Difference Scheme, V. A Second Order Sequel to Godunov s Method, J. Com. Phys.., 3, In MUSCL, we reconstruct face value states through (9.16) u L = u i+ 1 i +0.5φ (r i ) (u i+1 u i ), u R = u i+ 1 i+1 0.5φ (r i+1 ) (u i+ u i+1 ), (9.17) u L = u i 1 i φ (r i 1 ) (u i u i 1 ), u R i 1 = u i 0.5φ (r i ) (u i+1 u i ), (9.18) r i = u i u i 1 u i+1 u i.

5 LECTURE 9. FINITE VOLUME METHOD II 5 The function φ (r i ) is a limiter function. An example is a symmetric limiter proposed by Prof. Van Leer in 1974: (9.19) φ vl (r) = r + r 1 + r ; lim r φ vl(r) = Generation and handling of unstructured grids If you know how to use GAMBIT, you can certainly use it for grid generation. Being the first-time user of GAMBIT, you can add the following line into your.bashrc file in your accounts at either legendre or confucious export PATH=$PATH:/usr/local/Fluent.Inc/bin Then type source.bashrc Now you are ready to launch a graphical interface by invoking gambit & For learning purpose, we can start from a simple and free mesh generator (AUTOMESHD). A trial version of this package based on Windows and simple manual can be downloaded from In this section, we will demonstrate a generation mechanism of unstructured grids under the help of this simple package. We will then visualize this unstructured grid using TECPLOT. More advanced users can use the same strategy to generate grids from GAMBIT and handle them using the same way in TECPLOT. Consider the Stommel ocean circulation problem with a square basin shown in Fig. (). Now we add an island shown in Fig. 3 into the ocean basin. Specifying approximately 1000 points, we define a nearly uniform unstructured grid for the ocean basin with an island shown in Fig 4. A sample grid generated from AUTOMESHD can be downloaded from A sample C++ code to handle this grid and visualize it in TEC- PLOT can be downloaded from mae86/unstructured_read_write.cpp A grid visualized by TEC360 is shown in Fig. 5.

6 6 CHUNLEI LIANG, FINITE VOLUME METHOD II Figure. Use five points to define an ocean basin. Figure 3. Use six points to define a circular island Exercise of Lecture 9 No home work submission of this lecture. You can do the following exercise.

7 LECTURE 9. FINITE VOLUME METHOD II 7 Figure 4. Unstructured grid for the stommel ocean circulation problem. Figure 5. Unstructured grid plotted in TECPLOT for the stommel ocean circulation problem.

8 8 CHUNLEI LIANG, FINITE VOLUME METHOD II Solve 1D Burgers equation using MUSCL scheme Revise the sample code of Lecture 5 to use i) first-order FVM (firstorder piecewise constant, i.e. Ui+1/ R = U Avg i+1 and U i+1/ L = U Avg i 1 ) and Rusanov flux and ii) Rusanov flux and MUSCL-type reconstruction for the 1D inviscid Burgers equation. Boundary conditions are u = 0.5 at x = 1 and u = 0.5 at x = 1. Animate the results and check the CFL condition. Compare the results obtained by the upwind scheme using FDM in Lecture 5 to the ones you obtained by Rusanov schemes i) and ii) at a common final time instant. The boundary conditions now are given at two faces on two end nodes respectively. You shall initialize values on CV centers.