The fluid mechanics equation defined by the law of conservation of momentum, such as the Bernoulli equation, requires a flow that consists only of a single streamline to be solvable.
Consequently, the more general Navier-Stokes equation can more comprehensively model fluid flow within specific velocity or pressure fields. However, it’s important to note that the form of the Navier-Stokes equation is a non-linear partial differential equation, which, even as of the writing of this book, cannot be solved analytically and has smooth solutions under all conditions.
The difficulty in analytically modeling fluid flow equations arises from the existence of highly random and unsteady flow conditions known as turbulence.
Rather than attempting analytic solutions, researchers and engineers often simplify turbulent flow using specific modeling frameworks, known as turbulence modeling, enabling solutions through computer simulations (using CFD) without needing to compute detailed velocity and pressure fields.
Following is the general figure of turbulence modeling:
Figure 6.1. General Figure of Turbulence Models
The general rule is the higher the fidelity or accuracy of the model, the higher the computational effort. While methods like Direct Numerical Simulation (DNS) directly solve turbulent flows and can produce highly realistic and detailed solutions, DNS modeling is rarely encountered in general CFD programs due to the exceedingly high computational effort required, making it infeasible for everyday needs.
The Figure below shows the velocity distribution of a jet flow resulting from RANS and LES simulation. To capture the instantaneous velocity of a transient flow, RANS clearly doesn’t capture the detailed flow pattern compared to LES. But, the computational effort needed by the LES simulation is relatively higher. Although some LES modeling has been developed to reduce the effort, RANS is still more feasible for “daily-life” problems.
Figure 6.2. LES (top) vs RANS (bottom) instantaneus velocity result
Reference: