Model Solution Library

  CSDMS Model Solution Library

This is a collection of analytic or closed-form solutions to a variety of different mathematical models that are used in the realm of surface process dynamics. It is provided for the purpose of model validation by the CSDMS Cyber-informatics and Numerics Working Group. In the near future, this will include links to pages or papers where the solutions are given and described.

Navier-Stokes Equation

Batchelor vortex (See: Batchelor vortex, approximate solution.)

Burgers vortex (See: Burgers vortex.)

Couette flow (See: Couette flow.)

Hill spherical vortex (See: Vortex ring.)

Lamb-Chaplygin Dipole vortex

Lamb-Oseen vortex (See: Lamb vortex.)

Rankine vortex (See: Rankine vortex. This is a model, not an actual solution.)

Taylor-Couette flow (See: Taylor-Couette flow.)

Taylor-Dean flow

Shallow Water Equations

Inclined plane solution

Dam Break Characteristic Solution

Similarity solutions

Glacier Flow Equations

Halfar (1983) radially-symmetric (Glen Law) similarity solution

Bueler et al. (****) similarity solution

Potential Flow, 2D

• See: Potential flow for an overview.

Point source or sink

Free vortex (inviscid flow)

Flow around a semi-infinite plate (power-law conformal map: n=1/2)

Flow around a right-angle corner (power-law conformal map: n=2/3)

Uniform flow (power-law conformal map: n = 1)

Power-law conformal map: n=3/2

Flow through a right-angle corner or at a stagnation point (power-law conformal map: n=2)

Flow into a 60-degree corner (power-law conformal map: n=3)

Doublet solution (source-sink pair; power-law conformal map: n=-1)

Quadrupole solution (power-law conformal map: n=-2)

Joukowski airfoil solution

Darcy flow solutions

Jet, Wake and Mixing Layer Solutions

Albertson 2D turbulent jet approximation

Goertler 2D turbulent jet solution

Peckham 2D turbulent jet solution (Peckham, 2008; Goertler and Tollmien are special cases.)

Tollmien 2D turbulent jet solution

Channel and Pipe Flow Solutions

Poisseuille Flow

Driven Cavity Solutions

Lid-driven or buoyancy-driven, etc. ?

Numeric solution

Analytic solution

Boundary Layer Equation

Blasius

Falkners-Skan solution

Stokes First Problem

Stokes Second Problem

River Meandering Model

G. Seminara analytic solutions ??

Landscape Evolution Equations

• See: Landscape evolution model.

Steady-state, slope vs. area solution

Terry Smith longitudinal profile solutions

Peckham steady-state, uniform-rainrate solutions to "ideal landform equation"

Stratigraphic Evolution Equations

Peckham (2008) prograding solutions that use Laplace transform (including traveling-wave solutions)

Infiltration Theory

Broadsbent-Hammersley Solution

Wave Equation

• See: Wave equation.

d'Alembert solution

Heat Equation

• See: Heat equation.

Gaussian, radially-symmetric similarity solution

Laplace Equation

Many, via separation of variables method.

Poisson Equation

Many, from Poisson representation formula

Minimal Surface Equation

• See: Minimal surface equation.

Inclined plane solution

Helicoid solution (Meusnier, 1776) See: Helicoid.

Catenoid solution (Meusnier, 1776) See: Catenoid solution and Catenoid.

Scherk's first surface solution (Scherk, 1834) See: Scherk surface.

Scherk's second surface solution (Scherk, 1834)

Costa surface solution (Costa, 1984) See: Costa surface.

Burgers' Equation

Many, via Cole-Hopf transformation to Heat Equation

Soliton Solutions

• See: Soliton.