Non-Dimensionalization and Scaling of Helmholtz Equation and Schrodinger Equation, Which Reformulated for Fluid Dynamics

In fluid mechanics, the Reynolds number (Re) is a dimensionless number $R_e=\frac{F_int}{F_vis}$ I defined Reynolds number in a different situation, through the Helmholtz equation which represents the time-independent wave equation, $\nabla^2\psi+k^2\psi=0$ Now i consider wave vector $k$ as Reynolds number per a characteristic linear dimension so, $\nabla^2\psi+\frac{R_e^2}{L^2}\psi=0$ which led to the non-dimensionalization and scaling of Helmholtz equation, $L^2\nabla^2\psi+R_e^2\psi=0$ this equation is applicable to fluid dynamics. then I reformulate schrodinger equation, $-\frac{\mu^2}{2\rho}\nabla^2\psi +U_v\psi=E_v\psi$ where the $\mu$ denotes viscosity, $\rho$ is density, $U_v$ and $E_v$ are potential and total energy per unit volume. non-dimensionalization tricks: $\frac{\mu^2}{2\rho}=\frac{\mu Lv}{2R_e}$ where the $v$ is velocity, $L$ is linear dimension. now if we take factor of $\frac {\mu v}{L}$ from both side of equation, the Non-dimensionalization and scaling of schrodinger equation for fluid dynamics will be, $-\frac {L^2}{2R_e}\nabla^2\psi +U_v^*\psi=E_v^*\psi$