Barbara is designing a conical diffuser for the axisymmetric wind tunnel of Prob. 15-33. She needs to achieve at least 40 Pa of pressure recovery through the diffuser, while keeping the diffuser length as small as possible. Barbara decides to use CFD to compare the performance of diffusers of various half-angles ( $5^{\circ} \leq \theta \leq 90^{\circ}$ ) (see Fig. P15-33 for the definition of $\theta$ and other parameters in the problem). In all cases, the diameter doubles through the dif-fuser-the inlet and outlet diameters are $D_1=0.50 \mathrm{~m}$ and $D_2$ $=1.0 \mathrm{~m}$, respectively. The inlet velocity is nearly uniform at $V=10.0 \mathrm{~m} / \mathrm{s}$. The axial distance upstream of the diffuser is $L_1=1.50 \mathrm{~m}$, and the axial distance from the start of the diffuser to the outlet is $L_2=8.00 \mathrm{~m}$. (The overall length of the computational domain is 9.50 m in all cases.)
Run FlowLab with template Diffuser_angle. In addition to the axis and wall boundary conditions labeled in Fig. P15-33, the inlet is specified as a velocity inlet and the outlet is specified as a pressure outlet with $P_{\mathrm{cot}}=0$ gage pressure for all cases. The fluid is air at default conditions, and turbulent flow is assumed.
(a) Generate CFD solutions for half-angle $\theta=5,7.5,10$, $12.5,15,17.5,20,25,30,45,60$, and $90^{\circ}$. Plot streamlines for each case. Describe how the flow field changes with the diffuser half-angle, paying particular attention to flow separation on the diffuser wall. How small must $\theta$ be to avoid flow separation?
(b) For each case, calculate and record $\Delta P=P_{\text {in }}-P_{\text {out }}$. Plot $\Delta P$ as a function of $\theta$ and discuss your results. What is the maximum value of $\theta$ that achieves Barbara's design objectives?