%0 Unpublished Work %1 mcbain2020primitive %A McBain, G. D. %B Proceedings of the 22nd Australasian Fluid Mechanics Conference %D 2020 %K 65f15-numerical-eigenvalues-eigenvectors 76e05-parallel-shear-flows 76m10-finite-element-methods-in-fluid-mechanics %T The primitive Orr--Sommerfeld equation and its solution by finite elements %X The linear stability of parallel shear flows of incompressible viscous fluids is classically described by the Orr--Sommerfeld equation in the disturbance streamfunction. This fourth-order equation is obtained from the second-order linearized Navier--Stokes equation by eliminating the pressure. Here we consider retaining the primitive velocity--pressure formulation as is required for the linear stability analysis in general multidimensional geometries for which the streamfunction is unavailable; this conveniently reduces the conceptual and notational step from one to higher dimensions. %% Further, multidimensional flow simulation in arbitrary geometry is generally based on the primitive Navier--Stokes equations; having the same formulation and discretization in the linear stability analysis simplifies the comparison, removing possible numerical causes of discrepancy. The Orr--Sommerfeld equation is here discretized using Python and scikit-fem, in classical and primitive forms with Hermite and Mini elements, respectively. The solutions for the standard test problem of plane Poiseuille flow show the primitive formulation to be simple, clear, well-conditioned, and very accurate.

@unpublished{mcbain2020primitive, abstract = {The linear stability of parallel shear flows of incompressible viscous fluids is classically described by the Orr--Sommerfeld equation in the disturbance streamfunction. This fourth-order equation is obtained from the second-order linearized Navier--Stokes equation by eliminating the pressure. Here we consider retaining the primitive velocity--pressure formulation as is required for the linear stability analysis in general multidimensional geometries for which the streamfunction is unavailable; this conveniently reduces the conceptual and notational step from one to higher dimensions. %% Further, multidimensional flow simulation in arbitrary geometry is generally based on the primitive Navier--Stokes equations; having the same formulation and discretization in the linear stability analysis simplifies the comparison, removing possible numerical causes of discrepancy. The Orr--Sommerfeld equation is here discretized using Python and scikit-fem, in classical and primitive forms with Hermite and Mini elements, respectively. The solutions for the standard test problem of plane Poiseuille flow show the primitive formulation to be simple, clear, well-conditioned, and very accurate.}, added-at = {2020-09-24T08:28:14.000+0200}, author = {McBain, G. D.}, biburl = {https://www.bibsonomy.org/bibtex/2e02b5d4f597984ef68a5f2b9458c8bbb/gdmcbain}, booktitle = {Proceedings of the 22nd Australasian Fluid Mechanics Conference}, eventdate = {dec}, interhash = {bedd495191d094aec5ea94bb50919be5}, intrahash = {e02b5d4f597984ef68a5f2b9458c8bbb}, keywords = {65f15-numerical-eigenvalues-eigenvectors 76e05-parallel-shear-flows 76m10-finite-element-methods-in-fluid-mechanics}, timestamp = {2020-11-18T03:32:57.000+0100}, title = {The primitive Orr--Sommerfeld equation and its solution by finite elements}, venue = {Brisbane}, year = 2020 }