The discussion revolves around solving the Navier-Stokes equations for a 2D rectangular cavity filled with liquid, focusing on the pressure gradient and velocity profiles under specific boundary conditions. Participants explore the implications of flow assumptions, pressure gradients, and integral constraints in the context of fluid dynamics.
Participants generally agree on the need to consider pressure gradients and the integral constraint, but there is disagreement on how these concepts apply under different assumptions about velocity components. The discussion remains unresolved regarding the best approach to set up the equations for the 2D case.
Participants express uncertainty about the assumptions made regarding velocity components and the implications of pressure gradients, indicating that the discussion is highly technical and dependent on specific fluid dynamics principles.
Chestermiller said:I'm having second thoughts on your equation for ##\omega##. Shouldn't it be a function of x times a function of y, not the sum of such functions?
cosh cos, cosh sin, sinh cos, sinh sin
No. Solve the equations involving both omega and psi. The only tricky part is numerically specifying the boundary conditions on omega. But I know how to do that.joshmccraney said:Shoot, you're totally right. Then ##\nabla \cdot \psi = \omega## is not separable. I can't think of a way to analytically solve. How would you do this numerically? Finite difference ##\nabla^4 \psi=0##?