1.

Consider the stencil. Assume a uniform grid. What is \(\dot{m_w} \phi_{wv}\) according to the second-order upwind scheme?(Note: \(\dot{m}\) and φ are the mass flow rate and flow variable).

A. \(\dot{m_w}\phi_w=(\frac{3}{2}\phi_P-\frac{1}{2}\phi_W)max⁡(\dot{m_w},0)+(\frac{3}{2}\phi_W-\frac{1}{2}\phi_{WW}) max⁡(-\dot{m_w},0)\)
B. \(\dot{m_w}\phi_w=(\frac{3}{2}\phi_P-\frac{1}{2}\phi_W)max⁡(\dot{m_w},0)-(\frac{3}{2}\phi_W-\frac{1}{2}\phi_{WW}) max⁡(-\dot{m_w},0)\)
C. \(\dot{m_w}\phi_w=(\frac{3}{2}\phi_P-\frac{1}{2}\phi_W)max⁡(-\dot{m_w},0)+(\frac{3}{2}\phi_W-\frac{1}{2}\phi_{WW}) max⁡(-\dot{m_w},0)\)
D. \(\dot{m_w}\phi_w=(\frac{3}{2}\phi_P-\frac{1}{2}\phi_W)max⁡(-\dot{m_w},0)-(\frac{3}{2}\phi_W-\frac{1}{2}\phi_{WW}) max⁡(-\dot{m_w},0)\)
E. .a) \(\dot{m_w}\phi_w=(\frac{3}{2}\phi_P-\frac{1}{2}\phi_W)max⁡(\dot{m_w},0)+(\frac{3}{2}\phi_W-\frac{1}{2}\phi_{WW}) max⁡(-\dot{m_w},0)\) b) \(\dot{m_w}\phi_w=(\frac{3}{2}\phi_P-\frac{1}{2}\phi_W)max⁡(\dot{m_w},0)-(\frac{3}{2}\phi_W-\frac{1}{2}\phi_{WW}) max⁡(-\dot{m_w},0)\) c) \(\dot{m_w}\phi_w=(\frac{3}{2}\phi_P-\frac{1}{2}\phi_W)max⁡(-\dot{m_w},0)+(\frac{3}{2}\phi_W-\frac{1}{2}\phi_{WW}) max⁡(-\dot{m_w},0)\) d) \(\dot{m_w}\phi_w=(\frac{3}{2}\phi_P-\frac{1}{2}\phi_W)max⁡(-\dot{m_w},0)-(\frac{3}{2}\phi_W-\frac{1}{2}\phi_{WW}) max⁡(-\dot{m_w},0)\)
Answer» C. \(\dot{m_w}\phi_w=(\frac{3}{2}\phi_P-\frac{1}{2}\phi_W)max⁡(-\dot{m_w},0)+(\frac{3}{2}\phi_W-\frac{1}{2}\phi_{WW}) max⁡(-\dot{m_w},0)\)


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