1, the basic form and significance of the equation
Naville-Stokes equation was first proposed by French mathematician Claude-Louis mary henle Naville and British physicist george gabriel stokes in the19th century.
It is used to describe the variation law of velocity field and pressure field of fluid. Equations include mass conservation equation (continuous equation) and momentum conservation equation. For incompressible fluids.
Continuous equation:? v =0
Momentum equation: rho (? v/？ t+v？ v)=-？ p+μv+f
Where V is the velocity field of the fluid, ρ is the fluid density, p is the pressure, μ is the dynamic viscosity, and f is the external force (such as gravity) field. Represents a gradient. /? T stands for partial derivative.
2. Physical meaning and application of the equation.
Navier-Stokes equation describes the motion state and variation law of fluid, which is very important for understanding fluid behavior and analyzing various natural and engineering phenomena.
These phenomena include but are not limited to aerodynamics, hydrodynamics, ocean circulation, meteorological phenomena, reservoir simulation, aircraft design, ship movement and so on. In scientific research and engineering practice, the velocity and pressure distribution of fluid can be predicted by solving Navier-Stokes equation, thus guiding related design and decision-making.
Main challenges of Naville-Stokes equation in practical numerical simulation;
1, numerical instability and convergence
The numerical simulation of Navier-Stokes equation is easily affected by numerical instability. In the process of discretization, numerical errors may accumulate gradually, resulting in divergent or inaccurate calculation results. Especially in high Reynolds number flow, there is turbulence in the fluid, which is highly unstable.
2. Grid dependence and geometric complexity
The numerical simulation of Navier-Stokes equation usually needs to divide the solution area into finite grid elements. However, the degree of grid refinement will have a significant impact on the simulation results, that is, grid dependence.