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Note on two formulations of Crank-Nicolson method for Navier-Stokes equations

Journal Article · · Numerical Algorithms
 [1];  [2]
  1. Michigan Technological Univ., Houghton, MI (United States)
  2. Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
+ Show Author Affiliations
Here, we consider two formulations of the Crank-Nicolson (CN) method for the Navier-Stokes equations (NSE). The “natural” way of implementing CN for NSE is formally second order accurate in time for both velocity and pressure, whereas another formulation approximates pressure with only first order accuracy in time. Both versions of the method are applied to the benchmark problem of computing drag and lift in the flow around a cylinder. We show that the presumably more accurate version of the CN can create a solution with nonphysical oscillations and give incorrect predictions for the maximal drag coefficient, whereas the other formulation of the method predicts the drag and lift coefficients more accurately and does not introduce nonphysical oscillations. We locate the source of the issue and suggest several remedies.
Research Organization:
Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Sponsoring Organization:
USDOE
Grant/Contract Number:
AC05-00OR22725
OSTI ID:
2502172
Journal Information:
Numerical Algorithms, Journal Name: Numerical Algorithms; ISSN 1017-1398
Publisher:
SpringerCopyright Statement
Country of Publication:
United States
Language:
English

References (8)

Reference values for drag and lift of a two-dimensional time-dependent flow around a cylinder journal February 2004
A critique of the crank nicolson scheme strengths and weaknesses for financial instrument pricing journal July 2004
Benchmark Computations of Laminar Flow Around a Cylinder book January 1996
A Crank–Nicolson Leapfrog stabilization: Unconditional stability and two applications journal June 2015
A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type journal January 1947
On the Instability of the Crank Nicholson Formula Under Derivative Boundary Conditions journal May 1966
Finite-Element Approximation of the Nonstationary Navier–Stokes Problem. Part IV: Error Analysis for Second-Order Time Discretization journal April 1990
Approximate Deconvolution with Correction: A Member of a New Class of Models for High Reynolds Number Flows journal January 2020

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Related Subjects

Crank-Nicolson
Navier-Stokes
drag-lift coefficients
oscillations