velocity profile & grid | entrance flow | boundary-layer formation

pressure distribution | velocity development (vector) | velocity development (contour)

The geometry may be considered as two-dimensional; designating x as the horizontal, and y as the vertical, we neglect the z-axis, which comes out of the page and may be considered to be a plane of symmetry at the depth studied -- as if we considered only areas of the plates far from the edges which run parallel to the flow. We assume a uniform velocity field across the entrance, and give this value to Phoenics as an inlet boundary condition. The total problem specification includes:Gravity is neglected. This problem may be solved analytically through use of the continuity and Navier-Stokes equations. Assuming that velocity is a function of (x) alone, and assuming further that u is the only velocity component, we reach the following result: u(y) = (y*y - L*L) * 1/2 * 1/visc * dp/dx (where 2*L=plate separation, p=pressure) using u(y) = U = constant at inlet = average velocity everwhere (incompressible), we have: u,max = u(0) = -L*L/2 * 1/visc * dp/dx u,avg = -L*L/3 * 1/visc * dp/dx = 2/3 * u,max and, arranging everything in terms of U (which is given), we expect: dp/dx = -3*visc * 1/L*L * U and u(0) = 3/2 * U where u(0) is the velocity which develops on the centerline far downstream of the inlet plane. We need only select a Reynold's number suitable for laminar flow, use it to determine appropriate U and L, and give these numbers, along with the geometry, to Phoenics to see if the correct pressure gradient and centerline velocity are produced numerically. For the results which follow, I chose: U = .15 m/s , L = .005 m , visc = 1.8 E-5 (air) for a Reynold's number of Re = 2LU*dens./visc. = 100. With these inputs, analytical solutions are: dp/dx = .324 Pa/m u(0) = .225 m/s With a moderately dense grid (20 y-cells; 1 x-cell per cm), Phoenics found the following after approximately 15cm of plate length: dp/dx = .328 Pa/m u(0) = .224 m/s --> Both results are within 2% of their expected values. <--

- Inlet velocity, m/s
- Plate separation, m
- Fluid viscosity, kg/m*s
- Fluid density, kg/m^3

A variety of pictures are available from the Poiseuille test:

- Fully developed velocity profile, with view of grid structure.

- Split-level view of entrance flow, showing both u and v. Note: velocity in the v direction DOES exist where the boundary layers are FORMING. In this picture, a blue color indicates slow speeds for u, but a green color indicates slow speeds for v.

- Another view of v in the boundary-layer formation region -- showing the two "pools" of significant y-axis motion.

- A simple view of the uniform pressure distribution along a large length of the plates.

- An overall (vector) view of the velocity profile development

- As above, but with the velocity represented by contoured shading

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