I would like to know the meaning of d2P/dx2 = 0 which is the pressure boundary condition for oscillating machine i.e. GM Type pulse tube cryocooler. I want to know, how to deal with these pressure boundary conditions? If it is dP/dX=0, we can understand that pressure gradient is zero means pressure at a point and nearby point is same. But how to understand d2P/dx2 = 0?
The thesis I referred, tells, This d2P/dx2 = 0 boundary condition is same as that of fully developed flow of an incompressible fluid at the exit of the tube.
Can you help me to understand d2P/dx2 = 0?
For steady laminar fully developed ( rectilinear) flow in a tube, the pressure gradient in the axial direction (dp/dx) is everywhere a constant ( by definition of rectilinear flow) and therefore d^2p/dx^2, is therefore everywhere zero, including at the entrance and exit. If dp/dx is not a constant then d^2p/dx^2=0 implies that the the pressure profile p(x) has zero curvature at the exit. The zero curvature BC might be a valid if one assumes that the pressure field approaches a constant value ( say exponentially) and this constant value is not known or depends on some property of the flow field.
In the case of a pulse tube cryrocooler, the oscillatory nature of the piston would imply after all transients have died out that the pressure field in the regeneration zone would be time periodic, and of course in that case d^2p/dx^2 is never zero in the tube. Of course, there are locations in the tube were dp/dx is zero.
But I wonder whether the d^2p/dx^2 BC is being imposed after the orifice that ejects gas into the buffer or reservoir. It is conceivable if the orifice is designed appropriately then the pressure gradient in the tube downstream of the orifice may approximate d^2p/dx^2 =0 sufficiently far down stream of the orifice. Please note my understanding of PTC is limited. Perhaps others can elaborate...
The best condition would be to prescribe your pressure OUTSIDE your computational domain(anything you like,uniform,periodic,oscillatory) and apply the penalty method.This will enable you to Compute your pressure near the outlet,such that you satisfy continuity.
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