Question 9.2: The overall length of a prototype submarine is 2.25 m and it......
The overall length of a prototype submarine is 2.25 m and it can travel at a speed of 0.55 m/s. The temperature of water is 15°C (ρ_w = 999.1 kg/m³ and μ_w = 1.16 × 10^{-3} Pa-s). The design is to be tested in a wind tunnel making use of a 1/8 scale model. The air temperature in the wind tunnel is 25°C and the pressure is nearly atmospheric (ρ_{air} = 1.225 kg/m³ and μ_{air} = 1.98 × 10^{-5} Pa-s). What should be the air speed to achieve dynamic similarity? If a drag force of 2.5 N is measured in the test, find the prototype drag force.
Given
L_{r}=\frac{1}{8},\rho_{r}=\frac{1.225}{999.1}=1.226\times10^{-3},\,\mu_{r}=\frac{1.98\times10^{-5}}{1.16\times10^{-3}}=1.70\times10^{-2};V_{p}=0.55\,{\mathrm{m/s}}Find the V_r and force ratio F_r.
Since there is no free surface effect, the test will be conducted by maintaining the same Reynolds number in the model and prototype.
Thus (R_{e})_{r}=1;\left\lgroup{\frac{V L}{\frac{\mu}{\rho}}}\right\rgroup_{r}=1
or V_{r}=\left\lgroup{\frac{\mu}{\rho}}\times{\frac{1}{L}}\right\rgroup_{r}={\frac{1.70\times10^{-2}}{1.226\times10^{-3}}}\times{\frac{1}{\frac{1}{8}}}=110.92
Hence V_{m}=110.92\times V_{p}=110.92\times0.55=61.0\mathrm{~m/s}
Since dynamic similarity is achieved, E_{r}=1;\left\lgroup{\frac{\Delta p}{\rho V^{2}}}\right\rgroup_{r}=1;\ \mathrm{or}\ \ \Delta p_r \,= \, \rho_r V_r^2
The force ratio, (F)_{r}=A_{r}\,\Delta p_{r}=L_{r}^{2}\,\rho_{r}\,V_{r}^{2} \\ =\left\lgroup\frac{1}{8}\right\rgroup^{2}\times(1.226\times10^{-3})\times(110.92)^{2}=0.2357 \\ F_{p}={\frac{F_{m}}{0.2357}}={\frac{2.5}{0.2357}}=10.6\ \mathrm{N}
In this example, the Reynolds number in the model was the same as that of the prototype. Ideally, it ensures that there is dynamic similarity in the absence of the gravitational effects.