Poiseuille Flow Between Parallel Plates (Laminar)

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velocity profile & grid | entrance flow | boundary-layer formation
pressure distribution | velocity development (vector) | velocity development (contour)


A classic, and simple, problem in viscous, laminar flow involves the steady-state velocity and pressure distributions for a fluid moving laterally between two plates whose length and width is much greater than the distance separating them. The flow is driven by a pressure gradient in the direction of the flow, and is retarded by viscous drag along both plates, such that these forces are in balance.


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:

  1. Inlet velocity, m/s
  2. Plate separation, m
  3. Fluid viscosity, kg/m*s
  4. Fluid density, kg/m^3
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. <--


This experiment provided a solid verification of Phoenics laminar-flow operation, as well as confirmation that I was using the program correctly. Other tests, using higher Reynold's number and/or more complex geometry, are being developed.

A variety of pictures are available from the Poiseuille test:

(Note: geometry is sometimes not indicated, but is always the same as that stated here)

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