Model Solution Library: Difference between revisions

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=   '''CSDMS Model Solution Library''' =
=   '''CSDMS Model Solution Library''' =


: This is a collection of analytic or closed-form solutions to a variety of different mathematical models 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 where the solutions are given and described.
: 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.


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== {{ Bar Heading| text=Navier-Stokes Equation}} ==
== {{ Bar Heading| text=Navier-Stokes Equation}} ==
: Batchelor vortex
: Batchelor vortex (See: [http://en.wikipedia.org/wiki/Batchelor_vortex Batchelor vortex], approximate solution.)
: Burgers vortex
: Burgers vortex   (See: [http://en.wikipedia.org/wiki/Burgers_vortex Burgers vortex].)
: Hill spherical vortex
: Couette flow  (See: [http://en.wikipedia.org/wiki/Couette_flow Couette flow].)
: Lamb vortex
: Hill spherical vortex (See: [http://en.wikipedia.org/wiki/Vortex_ring Vortex ring].)
: Lamb-Chaplygin Dipole vortex
: Lamb-Chaplygin Dipole vortex
: Lamb-Oseen vortex
: Lamb-Oseen vortex   (See: [http://en.wikipedia.org/wiki/Lamb_vortex Lamb vortex].)
: Rankine vortex (not a solution, though)
: Rankine vortex (See: [http://en.wikipedia.org/wiki/Rankine_vortex Rankine vortex].  This is a model, not an actual solution.)
: Taylor-Couette flow (and Taylor-Dean flow ??)
: Taylor-Couette flow   (See: [http://en.wikipedia.org/wiki/Taylor–Couette_flow Taylor-Couette flow].)
: Vortex ring solution (???)
: Taylor-Dean flow


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== {{ Bar Heading| text=Potential Flow, 2D}} ==
== {{ Bar Heading| text=Potential Flow, 2D}} ==
* See: [http://en.wikipedia.org/wiki/Potential_flow Potential flow] for an overview.
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: Point source or sink
: Free vortex (inviscid flow)
: Flow around a semi-infinite plate (power-law conformal map: n=1/2)
: 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)
: Flow around a right-angle corner (power-law conformal map: n=2/3)
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== {{ Bar Heading| text=Jet, Wake and Mixing Layer Solutions}} ==
== {{ Bar Heading| text=Jet, Wake and Mixing Layer Solutions}} ==
: Albertson 2D turbulent jet solution
: Albertson 2D turbulent jet approximation
: Goertler 2D turbulent jet solution
: Goertler 2D turbulent jet solution
: Peckham 2D turbulent jet solution
: Peckham 2D turbulent jet solution (Peckham, 2008; Goertler and Tollmien are special cases.)
: Tollmien 2D turbulent jet solution
: Tollmien 2D turbulent jet solution
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== {{ Bar Heading| text=Channel and Pipe Flow Solutions}} ==
== {{ Bar Heading| text=Channel and Pipe Flow Solutions}} ==
: Poisseuille Flow
: Poisseuille Flow   (See: [http://en.wikipedia.org/wiki/Poiseuille_flow Poiseuille flow].)
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== {{ Bar Heading| text=Driven Cavity Solutions}} ==
== {{ Bar Heading| text=Driven Cavity Solutions}} ==
: Lid-driven or buoyancy-driven, etc. ?
: Numeric solution
: Numeric solution
: Analytic solution
: Analytic solution
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== {{ Bar Heading| text=Boundary Layer Equation}} ==
== {{ Bar Heading| text=Boundary Layer Equation}} ==
: Blasius
* See: [http://en.wikipedia.org/wiki/Boundary_layer Boundary layer].
: Falkners-Skan solution
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: Blasius   (See: [http://en.wikipedia.org/wiki/Blasius_boundary_layer Blasius boundary layer].)
: Falkners-Skan solution (See: [http://en.wikipedia.org/wiki/Blasius_boundary_layer Blasius boundary layer].)
: Stokes First Problem
: Stokes First Problem
: Stokes Second Problem
: Stokes Second Problem
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== {{ Bar Heading| text=Stokes Settling Solutions}} ==
== {{ Bar Heading| text=Stokes Settling Solutions}} ==
* See: [http://en.wikipedia.org/wiki/Stokes%27_law Stoke's Law].
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== {{ Bar Heading| text=Water Wave Solutions and Models}} ==
* See: [http://en.wikipedia.org/wiki/Waves Waves].
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: Airy waves
: Capillary waves
: Cnoidal waves
: Kelvin waves
: Roll waves
: Rossby waves
: Stokes wave
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== {{ Bar Heading| text=Ekman Spiral Solution}} ==
== {{ Bar Heading| text=Ekman Spiral Solution}} ==
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== {{ Bar Heading| text=River Meandering Model}} ==
: G. Seminara analytic solutions ??
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== {{ Bar Heading| text=Landscape Evolution Equations}} ==
== {{ Bar Heading| text=Landscape Evolution Equations}} ==
* See: [http://en.wikipedia.org/wiki/Landscape_evolution_model Landscape evolution model].
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: Steady-state, slope vs. area solution
: Steady-state, slope vs. area solution
: Terry Smith longitudinal profile solutions
: Terry Smith longitudinal profile solutions
: Peckham steady-state "ideal landform equation" solutions
: Peckham steady-state, uniform-rainrate solutions to "ideal landform equation"
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== {{ Bar Heading| text=Stratigraphic Evolution Equations}} ==
: Peckham (2008) prograding solutions (obtained using Laplace transforms, including traveling-wave solutions)
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== {{ Bar Heading| text=Coastline Evolution Equations}} ==
: Larson, M., H. Hanson, and N. C. Kraus (1987), Analytical solutions of the one‐line model of shoreline change, Tech. Rep. CERC‐87‐15, U.S. Army Waterw. Exp. Stn., Vicksburg, Miss.
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== {{ Bar Heading| text=Infiltration Theory}} ==
== {{ Bar Heading| text=Infiltration Theory}} ==
: Broadsbent-Hammersley Solution
* See: [http://en.wikipedia.org/wiki/Infiltration_(hydrology) Infiltration (hydrology)].
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: Solutions in:  Smith, R.E. (2002) Infiltration Theory for Hydrologic Applications, AGU monograph.
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== {{ Bar Heading| text=Wave Equation}} ==
== {{ Bar Heading| text=Wave Equation}} ==
: d'Alembert solution
* See: [http://en.wikipedia.org/wiki/Wave_equation Wave equation].
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: d'Alembert solution (See: [http://en.wikipedia.org/wiki/D%27Alembert%27s_formula d'Alembert formula].)
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== {{ Bar Heading| text=Heat Equation}} ==
== {{ Bar Heading| text=Heat Equation}} ==
* See: [http://en.wikipedia.org/wiki/Heat_equation Heat equation].
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: Gaussian, radially-symmetric similarity solution
: Gaussian, radially-symmetric similarity solution
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== {{ Bar Heading| text=Laplace Equation}} ==
== {{ Bar Heading| text=Laplace Equation}} ==
* See: [http://en.wikipedia.org/wiki/Laplace_equation Laplace equation].
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: Many, via separation of variables method.
: Many, via separation of variables method.
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== {{ Bar Heading| text=Poisson Equation}} ==
== {{ Bar Heading| text=Poisson Equation}} ==
: many, from Poisson representation formula
* See: [http://en.wikipedia.org/wiki/Poisson_equation Poisson equation].
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: Many, from Poisson representation formula
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== {{ Bar Heading| text=Minimal Surface Equation}} ==
== {{ Bar Heading| text=Minimal Surface Equation}} ==
* See: [http://en.wikipedia.org/wiki/Minimal_surface_equation Minimal surface equation].
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: Inclined plane solution
: Inclined plane solution
: Helicoid solution (Meusnier, 1776)
: Helicoid solution (Meusnier, 1776) See: [http://en.wikipedia.org/wiki/Helicoid Helicoid].
: Catenoid solution (Meusnier, 1776)
: Catenoid solution (Meusnier, 1776) See: [http://mathworld.wolfram.com/Catenoid.html Catenoid solution] and [http://en.wikipedia.org/wiki/Catenoid Catenoid].
: Scherk's first surface solution  (Scherk, 1834)
: Scherk's first surface solution  (Scherk, 1834) See: [http://en.wikipedia.org/wiki/Scherk_surface Scherk surface].
: Scherk's second surface solution (Scherk, 1834)
: Scherk's second surface solution (Scherk, 1834)
: Costa surface solution (Costa, 1984)
: Costa surface solution (Costa, 1984) See: [http://en.wikipedia.org/wiki/Costa_surface Costa surface].
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== {{ Bar Heading| text=Burgers' Equation}} ==
== {{ Bar Heading| text=Burgers' Equation}} ==
: many, via Cole-Hopf transformation to Heat Equation
* See: [http://en.wikipedia.org/wiki/Burgers_equation Burgers' Equation].)
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: Many, via the Cole-Hopf transformation to Heat Equation
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== {{ Bar Heading| text=Soliton Solutions}} ==
* See: [http://en.wikipedia.org/wiki/Soliton Soliton].
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Latest revision as of 09:06, 23 May 2014

  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

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 (See: Poiseuille flow.)


Driven Cavity Solutions

Lid-driven or buoyancy-driven, etc. ?
Numeric solution
Analytic solution


Boundary Layer Equation

Blasius (See: Blasius boundary layer.)
Falkners-Skan solution (See: Blasius boundary layer.)
Stokes First Problem
Stokes Second Problem


Stokes Settling Solutions


Water Wave Solutions and Models

Airy waves
Capillary waves
Cnoidal waves
Kelvin waves
Roll waves
Rossby waves
Stokes wave


Ekman Spiral Solution


River Meandering Model

G. Seminara analytic solutions ??


Landscape Evolution Equations

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 (obtained using Laplace transforms, including traveling-wave solutions)


Coastline Evolution Equations

Larson, M., H. Hanson, and N. C. Kraus (1987), Analytical solutions of the one‐line model of shoreline change, Tech. Rep. CERC‐87‐15, U.S. Army Waterw. Exp. Stn., Vicksburg, Miss.


Infiltration Theory

Solutions in: Smith, R.E. (2002) Infiltration Theory for Hydrologic Applications, AGU monograph.


Wave Equation

d'Alembert solution (See: d'Alembert formula.)


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

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 the Cole-Hopf transformation to Heat Equation


Soliton Solutions

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