ES 645w - Problems

Problem sets are preferably transmitted as an email attachment. Alternatively, they may be placed in the instructors Perkins 305 campus mailbox. The problem set archive is available to registered students. The pertinent assignment will be opened the class day following the due date.
Problem sets are not accepted after the archive is opened!

Texts:		(1) Turbulence Modeling for CFD, 3rd edition (Wilcox, 2006)
		(2) Turbulent Flows (Pope, 2000)
		(3) LES of Turb. Incomp. Flows (John, 2004)
References:		(4) Analysis of Turbulent BLs (Cebeci & Smith, 1974)
		(5) Fluid Dynamics Handbook (Baker, 1998 , Ch. 28)
		(6) The Computational Fluid-Thermal Sciences (Baker, 2009)
		(7) Computational Fluid Mechanics and Heat Transfer (Anderson, Tannehill and Pletcher, 1999)
		(8) Analytical SFS tensor/vector closure for rational LES thermal Navier-Stokes (Baker, Grubert and Orzechowski, 2009)

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 fixed temperatures T1 and T2 > T1. Assume the viscous dissipation term in DE is negligible.

1.2

(1) 1.3, 1.4, 1.6, 1.12, 1.15

2.0

For the GWS^h FE algorithm for 2D steady BL and PNS flow,

2.1

(6, Ch. 4) 9.1, 9.2, 9.3, 9.4, 9.5)

2.2

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

2.3

(6, Ch. 4) 12.1, 12.3

2.4

verify the aPSE k-eps-tau12, BL template for {FTKE}, {FEPZ} and {TXY} solved coupled

2.5

verify the k-eps-tau12 algorithm jacobian in the aPSE template. (Hint: use the chain rule when differentiating Ret with respect to k, eps and tau12, then re-express the results in terms of Ret * differential.)

3.0

For the continuity constraint INS algorithm

3.1

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

3.2

verify the aPSE k-eps-tau11, tau 12, tau 22 template for {FK}, {FEPS} and {FTAUIJ}.

3.3

verify the k-eps-tauij algorithm jacobian in the aPSE k-eps template. (Hint: use the chain rule when differentiating Ret with respect to k, eps and tauij, then re-express the results in terms of Ret * differential.)

(1, Ch. 2, 3) 2.1, 2.7, 2.9, 3.9, 3.10, 3.11

(1, Ch. 4) 4.3, 4.4, 4.20

(1, Ch. 5) 5.1, 5.3, 5.7, 5.8

7.1

(1, Ch. 6) 6.2, 6.3

7.2

(1, Ch. 6) Equation (6.25) expresses a non-linear algebraic Reynolds stress model (ASM), wherein the normal stresses are admitted to be non-isotropic dependent upon the flow field. Expand out equation (6.25) in three-dimensions.

7.3

(6, Ch. 4) The general tensor field ASM model is eq. (4.101), which is then PNS-ordered for 3-D flow to the lead two orders of signifiance, eq. (4.103) - (4.104). Determine the terms in ASM (1, eq.(6.25)) that produce the key terms detailed in the PNS ASM (6, eq. (4.103) - (4.104)).

(1, Ch. 8) 8.5

(2, Ch. 2 - 4) 2.8, 2.9, 2.10, 3.1, 3.2, 3.9, 3.36, 3.37, 4.3

10.

(2, Ch. 13) 13.1, 13.2, 13.19, 13.28, 13.30

11.1

(3) For the second order TS approximation to the Gaussian filter Fourier transform, (4.16) - (4.17), detail the steps leading to the Taylor approximation for the first 3 rational LES tensors, CLES.16-17.

11.2

(3) Repeat the 11.1 process for the second-order rational approximation leading to the second order Pade approximation, CLES.17.

11.3

(3) Proceed through the transfer details to confirm that the TS, Pade2 and Pade4 filter approximations lead universally to the forms given on CLES.19.

11.4

(8) Read this reference to establish completion of the rational LES theory implementation via a continuous Galerkin weak form algorithm. in preparation for the Powerpoint presentation on this subject to be given in the Class 28 meeting in CFD Lab.

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