Math 270E: Assignment 9

Assigned Wednesday, Nov. 19, 2008, due Wednesday, Nov. 26, 2008


[1] Modify the programs in Assignment 8 to solve the Navier-Stokes equations

\[ \begin{array}{c}\dfrac{\partial \vec u}{\partial t} + (\vec u \cdot \nabla ) \vec u = -\nabla P + \nu \Delta \, \vec u\\ \, \\ {\rm div}(\vec u) = 0 \end{array}\]

in the rectangular region $[0,1] \times [0,1]$ with velocity boundary conditions as specified in the following figure

$\qquad \qquad \qquad \qquad$


and homogeneous initial data $\vec u(x,y,0)=0$. For the approximation of the second derivatives near the boundary, you may use either the formulas obtain with (6.2.17), formula (6.2.18b) or (6.2.20b) on page 150-151 of Peyret and Taylor.

(2) Use your program to compute the steady state solution at Reynold number 400 (n = .0025) of the problem described in section 6.6.1 of Peyret and Taylor.

(a) Compare your results with those presented in the table 6.6.1 on page 202 of the text by evaluating the functionals listed in Table 6.6.1.

(b) Compare your results with those you obtained using the vorticity stream-function formulation in Assignment 7. Which formulation produces more accurate answers?


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