1. CIS/ME 794Y A Case Study in Computational Science & Engineering 2-D Conservation of Mass uu dx dy vv (x,y)

2. CIS/ME 794Y A Case Study in Computational Science & Engineering 2-D Conservation of Momentum

3. CIS/ME 794Y A Case Study in Computational Science & Engineering 2-D Conservation of Momentum (contd.): Surface forces dx (x,y) dy  xy (dy)(1)  yx (dx)(1)

4. CIS/ME 794Y A Case Study in Computational Science & Engineering 2-D Conservation of Momentum (contd.) Similarly, The body forces are expressed as: where is the body force per unit volume. For example,

5. CIS/ME 794Y A Case Study in Computational Science & Engineering 2-D conservation of momentum (contd.) Or, in cartesian tensor notation, Where repeated subscripts imply Einstein’s summation convention, i.e.,

6. CIS/ME 794Y A Case Study in Computational Science & Engineering Conservation of momentum (contd.): The shear stress  ij is related to the rate of strain (i.e., spatial derivatives of velocity components) via the following constitutive equation (which holds for Newtonian fluids), where  is called the coefficient of dynamic viscosity (a measure of internal friction within a fluid): Deduction of this constitutive equation is beyond the scope of this class. Substituting for  ij in the momentum conservation equations yields:

7. CIS/ME 794Y A Case Study in Computational Science & Engineering Navier-Stokes equations for 2-D, compressible flow The conservation of mass and momentum equations for a Newtonian fluid are known as the Navier-Stokes equations. In 2-D, they are:

8. CIS/ME 794Y A Case Study in Computational Science & Engineering Navier-Stokes equations for 2-D, compressible flow in Conservative Form The Navier-Stokes equations can be re-written using the chain-rule for differentiation and the conservation of mass equation, as: (1) (2) (3)

9. CIS/ME 794Y A Case Study in Computational Science & Engineering Conservation of energy and species The additional governing equations for conservation of energy and species are: (4) (5)

10. CIS/ME 794Y A Case Study in Computational Science & Engineering Summary for 2-D compressible flow UNKNOWNS: , u, v, T, P, n i N+5, for N species EQUATIONS: Navier-Stokes equations (3 equations: conservation of mass and conservation of momentum in x and y directions) Conservation of Energy (1 equation) Conservation of Species ((N-1) equations for n species) Ideal gas equation of state (1 equation) Definition of density: (1 equation)

11. CIS/ME 794Y A Case Study in Computational Science & Engineering Recapitulation So far,  Formulation of case study:quasi 1-D compressible flow  Numerical solution techniques  Steady vs. Time-marching to steady state  Finite differences (FD). Time marching to steady state (a)Explicit schemes (McCormack, FTBS)  Easier to program  Restricted to small  t for stiff problems  May not yield a solution at all for really stiff systems.

12. CIS/ME 794Y A Case Study in Computational Science & Engineering Recapitulation (contd.) (b)Implicit schemes (LBI):  Harder to program  Allows use of larger  t even for stiff problems  May be the only way to find a solution for really stiff systems  Finite elements (FE), time marching to steady state (a)Linearization same as for LBI FD method (b)well-suited for complex geometries.

13. CIS/ME 794Y A Case Study in Computational Science & Engineering Recapitulation (contd.)  Both FD and FE techniques ultimately require solution of linear equations Mx = f  In the LBI method, M is a block tri-diagonal matrix  Solution of systems such as Mx = f using PETSc allows you to explore parallel solution vs. serial solution.  implications for performance  Iterative methods (ex. Conjugate gradient) are well- suited to parallelization.

14. CIS/ME 794Y A Case Study in Computational Science & Engineering Extension of LBI method to 2-D flows Non-dimensionalize the 2-D governing equations exactly as we did the quasi 1-D governing equations. Take geometry into account. For example, Center Body Outer Body

15. CIS/ME 794Y A Case Study in Computational Science & Engineering Let r i (x) represent the inner boundary, where x is measured along the flow direction. Let r o (x) represent the outer boundary, where x is along the flow direction. r i (x) r o (x)

16. CIS/ME 794Y A Case Study in Computational Science & Engineering The real domain is then transformed into a rectangular computational domain, using coordinate transformation: x y or r  

17. CIS/ME 794Y A Case Study in Computational Science & Engineering The coordinate transformation is given by: The governing equations are then transformed:

18. CIS/ME 794Y A Case Study in Computational Science & Engineering Or, and etc.

19. CIS/ME 794Y A Case Study in Computational Science & Engineering This will result in a PDE with  and  as the independent variables; for example, Recall that for quasi 1-D flow, we had equations of the form

20. CIS/ME 794Y A Case Study in Computational Science & Engineering Applying the LBI method yielded:  or,

21. CIS/ME 794Y A Case Study in Computational Science & Engineering Applying the same procedure to our transformed 2-D problem would yield: Recall that after linearization of the quasi 1-D problem, the resulting matrix system was:

22. CIS/ME 794Y A Case Study in Computational Science & Engineering Now, in 2-D, the linearization procedure will result in: Where each F i, G i, H i are themselves block tri-diagonal systems as in the quasi 1-D problem. In other words, etc.