1. 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.,

2. 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:

3. 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:

4. 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)

5. 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)

6. 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)

7. 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

8. 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)

9. 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  

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

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

12. 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

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

14. 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:

15. 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.

16. CIS/ME 794Y A Case Study in Computational Science & Engineering At this point, we have a choice: –We can solve the full system as is, i.e. an (M+N)x(M+N) linear sparse system. Or –We can split the operator and apply the Alternating Direction Implicit (ADI) method to reduce the 2-D operator to a product of 1- D operators in each of the coordinate directions and solve each alternately, at each time step.

17. CIS/ME 794Y A Case Study in Computational Science & Engineering Douglas-Gunn ADI scheme Recall that after Crank-Nicolson differencing in time, linearization, and discretization of the spatial derivatives, we have: Split the operator by implicit factorization, approximate to either order (  t) or (  t) 2 as in the original discretization errors. 

18. CIS/ME 794Y A Case Study in Computational Science & Engineering Note that: Thus, operator splitting yields: Defining, we have:

19. CIS/ME 794Y A Case Study in Computational Science & Engineering Note that now, the solution of and is identical to solving two equivalent quasi 1-D problems in each of the coordinate directions  and . The Douglas-Gunn ADI scheme after implicit factorization can be shown to be unconditionally stable in 3-D as well as long as the convective term is absent, but is conditionally stable with the convective term present.

20. CIS/ME 794Y A Case Study in Computational Science & Engineering The conditional stability of this scheme worsens and may vanish if there are periodic boundary conditions. A virtue of the Douglas-Gunn ADI approach is that the same boundary conditions used for can be be used for . This is a result of consistent splitting of the operator. Other operator splitting schemes exist that are inconsistent - the same BCs used for cannot be used for .

21. CIS/ME 794Y A Case Study in Computational Science & Engineering In the present case study problem, our governing equations are of the form: Applying the LBI method to this equation yields: The Douglas-Gunn operator splitting then yields:

22. CIS/ME 794Y A Case Study in Computational Science & Engineering Still, no matrix inversion is required. The solution procedure can be implemented as follows: –Set –   solve for  in “xinv” –Next, solve for  n+1 -  n in “rinv”

23. CIS/ME 794Y A Case Study in Computational Science & Engineering 2-D LBI Code with Douglas- Gunn operator splitting

24. CIS/ME 794Y A Case Study in Computational Science & Engineering Sample solutions for 2-D LBI (x-y geometry)

25. CIS/ME 794Y A Case Study in Computational Science & Engineering Key features 10,000 time steps at  t = 10 -4 (non-dimensional) then for 100,000 time steps at  t = 2x10 -3 (non- dimensional). –L ref = 1 cm.; P ref = 1.013 x 10 5 Pa; T ref = 300 K 260 x 50 grid (x * r) Artificial Dissipation:or