navier_stokes_2d_exact

navier_stokes_2d_exact, an Octave code which evaluates exact solutions to the incompressible time-dependent Navier-Stokes equations (NSE) over an arbitrary domain in 2D.

• cavity: steady, polynomial in space, zero velocity boundary conditions on sides and bottoms, variable velocity on "top". This is NOT the standard "driven cavity" example;
• exppoly: time dependent, polynomial in space, exponential growth in time, zero velocity boundary conditions on unit square;
• exptrig: time dependent, trigonometric in space, exponential growth in time, zero velocity boundary conditions on unit square; at t=1, this flow forms a spiral. At later times, the exponential growth seems to make the solution physically absurd and computationally intractable.
• GMS: time dependent, vortices do not decay to zero;
• Lukas: steady, zero pressure;
• Poiseuille: steady, zero vertical velocity, zero source term;
• Spiral: time dependent, zero velocity on the unit square;
• Taylor: time dependent, zero source term, solution decays exponentially.
• Vortex: steady, same velocity pattern as Taylor.

Licensing:

The computer code and data files made available on this web page are distributed under the MIT license

Languages:

navier_stokes_2d_exact is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

navier_stokes_2d_exact_test

navier_stokes_3d_exact, an Octave code which evaluates exact solutions to the incompressible time-dependent Navier-Stokes equations (NSE) over an arbitrary domain in 3D.

spiral_data, an Octave code which computes a velocity vector field that satisfies the continuity equation, writing the data to a file that can be plotted by gnuplot.

stokes_2d_exact, an Octave code which evaluates exact solutions to the incompressible steady Stokes equations over the unit square in 2D.

Reference:

1. Jean-Luc Guermand, Peter Minev, Jie Shen,
   An overview of projection methods for incompressible flows,
   Computer methods in applied mechanics and engineering,
   Volume 105, pages 6011-6045, 2006.
2. Xiaoli Li, Jie Shen,
   Error analysis of the SAC-MAC scheme for the Navier-Stokes equations,
   arXiv:1909.05131v1 [math.NA] 8 Sep 2019
3. Maxim Olshanskii, Leo Rebholz,
   Application of barycenter refined meshes in linear elasticity and incompressible fluid dynamics,
   ETNA: Electronic Transactions in Numerical Analysis,
   Volume 38, pages 258-274, 2011.
4. Tien-Mo Shih, C H Tan, B C Hwang,
   Effects of grid staggering on numerical schemes,
   International Journal for Numerical Methods of Fluids,
   Volume 9, Number 2, pages 193-212, February 1989.
5. Geoffrey Taylor,
   On the decay of vortices in a viscous fluid,
   Philosophical Magazine,
   Volume 46, 1923, pages 671-674.
6. Geoffrey Taylor, Albert Green,
   Mechanism for the production of small eddies from large ones,
   Proceedings of the Royal Society of London,
   Series A, Volume 158, 1937, pages 499-521.

Source Code:

• all_cavity.m, evaluates all the cavity variables.
• all_exppoly.m, evaluates all the exppoly variables.
• all_exptrig.m, evaluates all the exptrig variables.
• all_gms.m, evaluates all the GMS variables.
• all_spiral.m, evaluates all the spiral variables.
• grid_2d.m, returns a regular 2D grid.
• ns2de_gnuplot.m, writes the velocity vector field to files for GNUPLOT.
• r8vec_uniform_ab.m, returns a scaled pseudorandom R8VEC.
• resid_c.m, evaluates the cavity residuals.
• resid_exppoly.m, evaluates the exppoly residuals.
• resid_exptrig.m, evaluates the exptrig residuals.
• resid_gms.m, evaluates the GMS residuals.
• resid_lukas.m, evaluates the Lukas Bystricky residuals.
• resid_poiseuille.m, evaluates the Poiseuille residuals.
• resid_spiral.m, evaluates the spiral residuals.
• resid_taylor.m, evaluates the Taylor residuals.
• resid_vortex.m, evaluates the Vortex residuals.
• rhs_cavity.m, evaluates the cavity source terms.
• rhs_exppoly.m, evaluates the exppoly source terms.
• rhs_exptrig.m, evaluates the exptrig source terms.
• rhs_gms.m, evaluates the GMS source terms.
• rhs_lukas.m, evaluates the Lukas Bystricky source terms.
• rhs_poiseuille.m, evaluates the Poiseuille source terms.
• rhs_spiral.m, evaluates the spiral source terms.
• rhs_taylor.m, evaluates the Taylor source terms.
• rhs_vortex.m, evaluates the Vortex source terms.
• timestamp.m prints the YMDHMS date as a timestamp.
• uvp_cavity.m, evaluates the cavity flow variables.
• uvp_exppoly.m, evaluates the exppoly flow variables.
• uvp_exptrig.m, evaluates the exptrig flow variables.
• uvp_gms.m, evaluates the GMS flow variables.
• uvp_lukas.m, evaluates the Lukas Bystricky flow variables.
• uvp_poiseuille.m, evaluates the Poiseuille flow variables.
• uvp_spiral.m, evaluates the spiral flow variables.
• uvp_taylor.m, evaluates the Taylor flow variables.
• uvp_vortex.m, evaluates the Vortex flow variables.