navier_stokes_2d_exact

navier_stokes_2d_exact, a MATLAB 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, 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;
• NoFlow: steady, zero velocity, nonzero 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 information on this web page is distributed under the MIT license.

Languages:

navier_stokes_2d_exact is available in a C version and a C++ version and a Fortran77 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

matlab_exact, a MATLAB code which evaluates exact solutions to a few selected examples of ordinary differential equations (ODE) and partial differential equations (PDE).

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.
38, pages 258-274, 2011.
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_lukas.m, evaluates all the lukas variables.
• all_noflow.m, evaluates all the noflow variables.
• all_poiseuille.m, evaluates all the poiseuille variables.
• all_spiral.m, evaluates all the spiral variables.
• all_taylor.m, evaluates all the taylor variables.
• all_vortex.m, evaluates all the vortex variables.
• grid_2d.m, returns a regular 2D grid.
• pressure_gnuplot.m, writes the pressure 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_noflow.m, evaluates the noflow 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_noflow.m, evaluates the noflow 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.
• 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_noflow.m, evaluates the noflow 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.
• velocity_gnuplot.m, writes the velocity vector field to files for GNUPLOT.