  Simulating fluid dynamics on a quantum computer is intrinsically difficult
due to the nonlinear and non-Hamiltonian nature of the Navier-Stokes equation
(NSE). We propose a framework for quantum computing of fluid dynamics based on
the hydrodynamic Schr\"odinger equation (HSE), which can be promising in
simulating three-dimensional turbulent flows in various engineering
applications. The HSE is derived by generalizing the Madelung transform to
compressible/incompressible flows with finite vorticity and dissipation. Since
the HSE is expressed as a unitary operator on a two-component wave function, it
is more suitable than the NSE for quantum computing. The flow governed by the
HSE can resemble a turbulent flow consisting of tangled vortex tubes with the
five-thirds scaling of energy spectrum. We develop a prediction-correction
quantum algorithm to solve the HSE. This algorithm is implemented for simple
flows on the quantum simulator Qiskit with exponential speedup.
