Finite-wing-analogy formula for compressibility correction to pressure coefficient of an underwater vehicle model at low Mach number

Wind tunnels are usually used to investigate the flows and forces associated with underwater vehicles when free-surface effects can be ignored. However, because of the large differences between air and water in density, and viscosity, the freestream Mach number in a wind tunnel is much higher than that in a water tunnel or towing tank at the same Reynolds number. Therefore, compressibility correction is required for accurate measurement compatibility between wind tunnels and water tunnels or towing tanks. In the study reported here, the flows and forces associated with an underwater vehicle model at different Mach numbers were investigated by solving the Navier-Stokes equations for compressible flow numerically as virtual-wind-tunnel experiments. The freestream Mach number Ma varying from 0.004 to 0.5. The distribution of the pressure coefficient on the hull and the effects of Ma on the peaks of the pressure coefficient are discussed in detail. The performances of the Prandtl-Glauert rule, the Karman-Tsien rule, and the Laitone rule for compressibility correction to the pressure coefficient of underwater vehicles are assessed. Defining the average correction factor with larger values for better correction effect, the values for these three compressibility correction formulas are 0.51, 0.38, and 0.23, respectively. A finite-wing-analogy formula to improve the compressibility correction to the pressure coefficient at low Ma is proposed. Inspired by the finite-wing correction to the lift slope of airfoils, the proposed formula offers good convergence of the pressure coefficient and highly accurate compressibility correction with an average correction factor of 0.84.