Reynold's Equation: Navier-Stokes to Const. Density Flow

[mekrob]

Hi,

I've been asked to turn the Navier-Stokes Equation (incompressible flow, constant density) into the Reynold's Equation. However, I can't find the Reynold's equation in my notes or book. I don't need help solving the problem, I just need to know the end result, otherwise I have no idea what I'm working towards.

Thanks.

 

[arildno]

Reynold's equation is the time-averaged N-S equation when we explicitly regard the velocity field as random.
The main feature that will pop up will be "Reynolds' stress tensor", which is, in essence, the non-linear products of the random perturbations that affects the evolution of the avereged velocity field.

 

[charng]

Hi there,

The Reynold's Equation is a simplified version of the Navier-Stokes Equation, which describes the motion of a fluid. It is commonly used in fluid dynamics to analyze the flow of incompressible fluids at constant density. The equation is named after the British engineer and physicist, Osborne Reynolds, who first derived it in the 19th century.

The Reynold's Equation can be written as:

∂u/∂t + (u·∇)u = -∇p/ρ + ν∇^2u

where u is the fluid velocity, t is time, p is pressure, ρ is density, and ν is the kinematic viscosity of the fluid. This equation is essentially the same as the Navier-Stokes Equation, but it assumes that the fluid is incompressible, meaning that its density remains constant throughout the flow.

To arrive at this equation, the Navier-Stokes Equation is simplified by neglecting the convective term (u·∇)u and assuming that the density is constant. This results in a simplified form of the equation, which is easier to solve for certain types of flow problems.

I hope this helps clarify the Reynold's Equation for you. Let me know if you have any other questions. Good luck with your work!