ES 552w - Problems

Problem sets are preferably transmitted as e-mail attachments, alternatively placed in the instructors Perkins Hall 305 mailbox. The homework archive is available for reference by registered students.
PE assignments submitted after the archive is opened will not be considered.

Reference :		(1) - Courseware
		(2) - The Computational Fluid-Thermal Sciences (AJB, 2009)
		(3) - Fluid Dynamics Handbook (AJB, 1998, Ch. 28)
		(4) - The Computational Engineering Sciences (AJB, 2006)

Due
Class

1.1

Solve the steady viscous laminar-thermal incompressible non-D !! Navier-Stokes equations for 2-D fully developed non-isothermal flow between horizontal parallel plates with plate fixed temperatures T1 and T2 > T1. Assume the viscous dissipation term in DE is negligible.

2.1

(2, Ch. 2) 4.1, 4.2, 4.3, 4.4 (optional)

(2, Ch. 2) 5.1, 5.2

3.1

(2, Ch. 2) 6.1, 6.2, 6.3

(2, Ch. 2) 7.1, 7.2

(2, Ch. 2) 8.1, 8.2, 8.3

(2, Ch. 2) 10.1, 10.2

(2, Ch. 2) 11.1, 11.2, 11.3

(2, Ch. 3) 4.1, 5.1, 5.2, 5.3

4.1

(2, Ch. 4) 2.1

(2, Ch. 4) 3.1, 3.2

(2, Ch. 4) 4.1, 4.2, 4.3

(2, Ch. 4) 5.1

5.1

(2, Ch. 4) 9.1, 9.2

(2, Ch. 4) 9.3, 9.4, 9.5

(2, Ch. 4) 10.1, 10.2, 10.3, 10.4

(2, Ch. 4) 12.1, 12.3

6.1

(2, Ch. 5) 2.1, 2.2, 2.3, 2.4

(2, Ch. 5) 3.1

(1; 4, Ch. CMn) Proceed through the Taylor series process, page SVNS.3, to verify the vorticity Robin BC, egn. (CMn.49) Then confirm the lower order forms in (4, Ch. CMn) eqn. (C,m/50),(CMn.51).

(1) verify the details of the GWS^h template for {FQ}, page SVNS.5.

(1) Confirm the {N₁(z)} basis convection term hyper-matrix, page SVNS.5A.

(1) Detail the templates for GWS^h Newton jacobian, starting with the essence on page SVNS.6.

(1) For the Taylor weak statement omega-psi INS algorithm, verify the TS development leading to the Beta term, page SVNS.12.

(1) Examine the TWS aPSE quasi-Newton template for the coupled fluid-thermal problem statement, hence confirm the formulation correctness. Then copy and edit the quasi-Newton template to make it a Newton template for this problem statement.

7.0

For the pressure projection INS algorithm:

(1; 2, Ch. 5) 4.1, 4.2, 4.3

(1) Review, hence confirm the PPNS algorithm key formulation, iteration and IC strategies detailed on page PPNS.10.

(1) Verify the correctness of the PPNS template essence summarized on pages PPNS.12-13.

(1) Convince yourself of the correctness of the tensor product quasi-Newton jacobian, page PPNS.14, focusing in particular on the role of the metric data strings in formation of the convection term contributions.

(1) For the TWS^h generalization to PPNS for error identification, confirm the TS operations leading to L^m(q), page PPNS.16.

(1) Confirm the matrix expression for [A_jA_k], page PPNS.17, hence the divergence form at the bottom of the page.

(1) For the TKE closure model, confirm the law-of-the-wall BC expressions, page PPNS.28.

8.0

For free-surface flow INS algorithms:

(1) Confirm that DP_z ordering produces the form presented on page FSNS.1.

(1) Verify that DM is ill-posed as an ODE expression for vertical velocity, page FSNS.2.

(1) Confirm the non-D depth-averaged flow PDE system, page FSNS.5, hence the hyperbolic conservation law form, page FSNS.7.

(1) Verify the flux vector jacobian operations, page FSNS.9.

(1) Convince yourself of the accuarcy of the TWS^h + TS template for depth-averaged INS, pages FSNS.10-11.

(1) Review, hence confirm the aPSE template for unsteady, turbulent depth-averaged INS, c.f., page FSNS.15 for summary.

9.0

For the computational spectral theory:

(2. Ch. 5) 6.1, 6.2

(2. Ch. 5) 6.3, 6.4

(2. Ch. 5) 6.5, 6.6

(2. Ch. 5) 6.7, 6.8, 6.9

(2. Ch. 5) 9.2, 9.3, 9.4

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