vollume 145, pages 57-68 (2007):

"The Limiting Hole Pressure in Three Dimensions"

Vasileios Symeonidis and Bruce Caswell

Abstract: The three-dimensional analog of the Giesekus-Tanner (G-T) theorem [B. Caswell, Non-Newtonian flow at lowest order: The role of the Reiner-Rivlin stress, J. Non-Newtonian Fluid Mech., 133(1):1-13, 2006] allows the flow in the vicinity of a pressure tap to be analyzed with the solution of the inertialess flow of a Reiner-Rivlin fluid just as G-T was used [R.I. Tanner and A. Pipkin, Intrinsic Errors in Pressure-Hole Measurements, Trans. Soc. Rheol., 14:471, 1969] to analyze the corresponding two-dimensional flow with the Stokes solution. Later [E. A. Kearsley, Intrinsic Errors for Pressure Measurements in a slot along a flow, Trans. Soc. Rheol., 14(3):419-424, 1970] extended the Tanner-Pipkin analysis to the rectilinear flow in a channel with a parallel slot. In the two-dimensional case the Tanner-Pipkin term is the hole pressure relative to the Stokes value (usually neglected), and is determined solely by the first normal stress function evaluated at the wall shear rate of the undisturbed flow. It is thus ``intrinsic'' since dimensional analysis suggests the hole pressure should also depend on the hole geometry. Kearsley's result is independent of slot geometry only in the limit that the slot is deep and very narrow relative to the size of the main channel. In three-dimensions the Tanner-Pipkin term again contibutes an intrinsic pressure independent of hole geometry, but now it is now relative to the Reiner-Rivlin value which itself depends on the hole geometry and is not negligible. Hence the complete hole pressure in three-dimensions requires the solution of the Reiner-Rivlin problem which in general will be obtained numerically. The latter is formulated for numerical simulation so that the hole pressure can be read off from values of the pressure field without any post-processing calculation of velocity gradients. The numerical simulations were performed with a three-dimensional spectral element code [G.E. Karniadakis and S.J. Sherwin, Spectral/hp Element Methods for CFD, Oxford University Press, 1999] well suited to the efficient solution of flow problems in complex geometries. The limiting values of the Reiner-Rivlin hole pressure have been obtained for several hole/channel ratios. Numerical results for the larger holes are not in agreement with the H-P theory [K. Higashitani and W. G. Pritchard, A kinematic calculation of intrinsic errors in pressure measurements made with holes, Trans. Soc. Rheol., 16(4):687-696, 1972], and for such holes the Stokes hole pressure was found to be in the measurable range.