CFD Home Work
1. Normalize the 2D Navier-Stokes equations.
2. Prove the normalization of eq.(1.43) from the original formalution.
2b.Find the solution of the characteristics for 2D steady state Euler equations. Prove that the 2D Euler equations are hyperbolic at supersonic and elliptic at subsonic.
3. Find the eigenvalues of 2D UNSTEADY Euler equations. What is the type of the equations?
4. Use the unknown coefficient method to construct the following finite difference scheme for the 1st order derivative. Please also give the order of accuracy:
1.)3 point forward difference: Du/dx = aU(i) + bU(i+1)+cU(i+2)
2.)3 point backward difference: Du/dx = aU(i) + bU(i-1)+cU(i-2)
5. Discretize the 2D continuity equation in the conservative form in a Cartesian coordinate, using finite difference with the uniform mesh spacing. Use 1st order one side forward differencing for the temporal term, explicit central differencing for the spatial terms. What is the order of accuracy for the finite difference equation?
6. Discretize the 2D inviscid (Euler) momentum equation (in x-direction) in the conservative form in a Cartesian coordinate, using finite volume method with uniform mesh spacing. Use 1st order one side forward differencing for the temporal term, implicit central differencing for the spatial terms. What is the order of accuracy for the finite difference equation?
7. For ID wave equation Du/dt + c(Du/dx)=0, c>0, please analyze the stability using von Newmann analysis for the following two schemes. You will find that one of the schemes is stable and the other is not. For both the schemes, the temporal term is discretized with two point one side differencing.
1)Explicit scheme with the first order two point forward differencing for (dU/dx).
2)Explicit scheme with first order two point backward differencing for (dU/dx).
Fortran Introduction
8. Solve the 1 D wave equation numerically using:
1)Law-Wendroff One Step Scheme
2)MacCormack Two Step Scheme
3)Second-Order Upwind Scheme
4)4th Order Runge-Kutta Scheme
The wave equation is: du/dt + du/dx=0. The initial solution is u(x,0)=sin (2*n*pi*x/40), 0<=x<=40. The analytical solution will be u(x,t)=sin (2*n*pi*(x-t)/40). Periodic condition will be used at the two ends x=0 and x=40. Use 41 grid point mesh with Dx=1 and compute to t=18.
a)Solve the problem with n=1, 3 and CFL=1.0, 0.6, 0.3. Compare the solution and analytical solution and also compare the solutions between the results. Give your comments on the behavior of those results and the reason.
b)Compare the Maximum CFL number numerically allowed for each scheme that can be stable.
All the results should be plotted. Hand in your computer code with your results and your discussion.
You are encouraged to write your own code. However, you can also start with the attached FORTRAN code in case you are not familiar with programming.
9. Solve the 1 D heat equation as described in the course note. The results are attached here explicit scheme, implicit scheme The tri-diagonal matrix solver are attached here page 1, page 2 You need to write out all your formulations and plot the results for each time level.
10. Please give the iterative formulations for the Laplace equation discussed as the example in the class notes:
1)Vertical line Gauss-Seidel iteration sweep from i-imax to i=i0.
2)Horizontal line Gauss-Seidel iteration sweep from j=j0 to j=jmax.
11. Please use algebraic method to generate mesh for the nozzle give in AIAA Paper 99-3348. Use mesh size 51x25 in x and y direction.
1)Generate uniform mesh in both x and y-direction.
2)Generate the mesh clustered near the wall as shown in the class notes by adjusting the stretching parameter, a.
PROJECT 1:
Grid Generation Using Elliptic PDEs
Generate the grid for a cylinder with the outer boundary located at 5 times of the radius by solving the Laplace's equations.
Requirements:
Two cases: 1) uniform mesh; 2) clustered near the wall.
1)Use Gauss-Seidel Point Iteration
2)Use Gauss-Seidel Line Iteration
3)Use Gauss-Seidel Line Iteration with SOR (optional, 10% bonus)
In your report, you need to present:
1)Introduction (why to use different methods above)
2)Governing Equations
3)Discretization Schemes
4)Boundary Condition Treatment
5)Results and Discussion, including description of the detailed geometry, compare the efficiency of different methods (convergency history), compare the mesh quality, etc.
12. Solve the inviscid Burger's equation iin the conservative form for a solution with a discontinuity using:
1)Lax first order scheme
2)Lax-Wendroff One Step Scheme
3)MacCormack Two Step Scheme
4)4th Runge-Kutta Scheme
Use 51 mesh points in x-direction, repeat your solutions with CFL=1.0, 0.6, 0.3 and compare your solutions with the analytical solutions (given in your notes). The initial solution is u=1.0 for the first 11 mesh points, and u=0 for the rest of the mesh points.
All the results should be plotted. Hand in your computer code with your results and your discussion.
13. Please obtain the Jacobian matrix for 2D Euler equations. Prove that F=AQ, where F is the flux in x-direction, A is the Jacobian matrix, Q is the conservation variable vector.
PROJECT 2:
Following AIAA Paper 99-3348, solve 1D Euler equations for the 1D shock tube problem (Sod Problem) using Zha scheme and Roe scheme. Use control volume method, explicit Euler scheme, and 1st order accuracy in both spatial and temporal direction. Assume the pressure ration is 10:1. The initial pressure on the left side is 2.0E05 and on the right side is 2.0E04, the temperature is 303k on both sides, the shock tube length is 14m, the tube area iis 1m^2, initially, the diaphrame is located at x=7m. Compare your results with the analytical results attached at time level=0.01s. Use 201 uniform mesh points in space.
14. Please describe the boundary conditions for 3D subsonic inlet and exit based on the characteristic directions.
15. Please derive the expression of the following terms in generalized coordinates:
1)viscous shear stress terms tau_xx, tau_xz
2)inviscid flux F_bar and G_bar (Notes, p137)
16. Please give the flux vector splitting scheme of Zha-Bilgen (1993) on 3D generalized coordinates
17. Please give a numerical procedure to use wall function (Law of the Wall) as the wall boundary conditions for turbulent boundary layer.