UnitfulBuckinghamPi

This package is for solving for the adimensional Pi groups (or Π groups) in a given list of parameters, according to the Buckingham-Pi Theorem.

We use the package Unitful.jl in order to facilitate the construction of the parameters and to easily handle the dimensions associated with each parameter.

This package is inspired by a similar package written in python: ian-r-rose/buckinghampy.

Let us consider a couple of examples.

We start with the period of a simple pendulum.

The parameters taken for consideration are the length of the rod, the mass of the bob, the acceleration of gravity, the angle of the rod with respect to the downwards vertical direction, and the period of the swinging pendulum.

We define these parameters as Unitful.FreeUnits. Except for the acceleration of gravity, which is a constant and is given as a Unitful.Quantity value, and for the period, for which we do not associate any unit, only a dimension, just for fun.

To tell UnitfulBuckinghamPi that these are the parameters to consider, we use the macro @setparameters. Then, we find the adimensional Π groups with the function pi_groups(), which returns the groups as a vector. It can either be a vector of strings, with pi_groups(:String), or of expressions, with pi_groups(:Expr), which is the default.

julia> using Unitful

julia> using UnitfulBuckinghamPi

julia> ℓ = u"m"
m

julia> g = 9.8u"m/s^2"
9.8 m s⁻²

julia> m = u"g"
g

julia> T = u"𝐓"
𝐓

julia> θ = u"NoDims"
NoDims

julia> @setparameters ℓ g m T θ
[ Info: Parameter(s) registered:
[ Info:  ℓ = m
[ Info:  g = 9.8 m s⁻²
[ Info:  m = g
[ Info:  T = 𝐓
[ Info:  θ = NoDims

julia> Π_str = pi_groups(:String)
2-element Vector{String}:
 "ℓ^(-1//2)*g^(1//2)*T^(1//1)"
 "θ^(1//1)"

julia> Π = pi_groups(:Expr)
2-element Vector{Expr}:
 :(ℓ ^ (-1 // 2) * g ^ (1 // 2) * T ^ (1 // 1))
 :(θ ^ (1 // 1)) 

There are two adimensional groups, Π[1] and Π[2].

One can use korsbo/Latexify.jl to display the groups in Latex format, simply with

julia> using Latexify

julia> latexify(Π_str[1])
L"$g^{\frac{1}{2}} \cdot \ell^{\frac{-1}{2}} \cdot T^{\frac{1}{1}}$"

This produces $g^{\frac{1}{2}} \cdot \ell^{\frac{-1}{2}} \cdot T^{\frac{1}{1}}$.

With the parameters above, one cannot evaluate the adimensional group since that would amount to multiplying Unitful.FreeUnits or Unitful.Quantities like the Unitful.Dimensions parameter T. That i not allowed by Unitful.jl. One can solve that, however, by substituting T with a unit. Then, we can either parse each element in the vector of strings returned by pi_groups_str() and evaluate that or we can use pi_groups() to obtain directly the corresponding expressions and evaluate the expressions.

julia> τ = u"s"
s

julia> @setparameters ℓ g m τ θ
[ Info: Parameter(s) registered:
[ Info:  ℓ = m
[ Info:  g = 9.8 m s⁻²
[ Info:  m = g
[ Info:  τ = s
[ Info:  θ = NoDims

julia> Π = pi_groups()
2-element Vector{Expr}:
 :(ℓ ^ (-1 // 2) * g ^ (1 // 2) * τ ^ (1 // 1))
 :(θ ^ (1 // 1))

julia> eval(Π[1])
3.1304951684997055

julia> eval(Π[2])
NoDims

As expected, both are adimensional.

Adimensional groups are not unique. Even if you find a single group, any power of it is also adimensional. If there is more than one adimensional group, then you can linearly combine them to find many others. They are associated to the span of the null space of a "parameter-to-dimension" matrix. The solver here will just pick one combination from the basis obtained for the null space.

Finally, one can add parameters to a given set of registered parameters and solve for the new set.

julia> v = u"m/s"
m s⁻¹

julia> @addparameters v
[ Info: Parameter(s) registered:
[ Info:  ℓ = m
[ Info:  g = 9.8 m s⁻²
[ Info:  m = g
[ Info:  τ = s
[ Info:  θ = NoDims
[ Info:  v = m s⁻¹

julia> pi_groups()
3-element Vector{Expr}:
 :(ℓ ^ (-1 // 2) * g ^ (1 // 2) * τ ^ (1 // 1))
 :(θ ^ (1 // 1))
 :(ℓ ^ (-1 // 2) * g ^ (-1 // 2) * v ^ (1 // 1))

With korsbo/Latexify.jl, these yield

julia> latexify.(pi_groups())
3-element Vector{LaTeXStrings.LaTeXString}:
 L"$g^{\frac{1}{2}} \cdot \ell^{\frac{-1}{2}} \cdot \tau^{\frac{1}{1}}$"
 L"$\theta^{\frac{1}{1}}$"
 L"$g^{\frac{-1}{2}} \cdot \ell^{\frac{-1}{2}} \cdot v^{\frac{1}{1}}$"

which look like

• $g^{\frac{1}{2}} \cdot \ell^{\frac{-1}{2}} \cdot \tau^{\frac{1}{1}}$
• $\theta^{\frac{1}{1}}$
• $g^{\frac{-1}{2}} \cdot \ell^{\frac{-1}{2}} \cdot v^{\frac{1}{1}}$

Another classical example of adimensional group is the Reynolds number.

What could characterize how complicate a fluid flow is? We should certainly consider the parameters characterizing the fluid, such as density and viscosity. Then, there is the velocity the fluid moves. Less obvious is the lenght scale we consider, which we can consider as a length scale associated with the injection of energy into the system, such as the width of an obstacle, the distance between the walls of channel, the distance between the bars of a grids, and so on.

With these parameters, the only possible adimensional group is the Reynolds number (or powers of it).

julia> ρ = u"g/m^3"
g m⁻³

julia> μ = u"g/m/s"
g m⁻¹ s⁻¹

julia> u = u"m/s"
m s⁻¹

julia> ℓ = u"m"
m

julia> @setparameters ρ μ u ℓ
[ Info: Parameter(s) registered:
[ Info:  ρ = g m⁻³
[ Info:  μ = g m⁻¹ s⁻¹
[ Info:  u = m s⁻¹
[ Info:  ℓ = m

julia> pi_groups()
1-element Vector{String}:
 "ρ^(1//1)*μ^(-1//1)*u^(1//1)*ℓ^(1//1)"

Again, we can use korsbo/Latexify.jl to display the adimensional group in Latex format:

$\rho^{\frac{1}{1}} \cdot \mu^{\frac{-1}{1}} \cdot u^{\frac{1}{1}} \cdot \ell^{\frac{1}{1}}$

One can recognize this as the Reynolds number

$$ \mathrm{Re} = \frac{\rho u \ell} {\mu} $$

The Buckingham-Pi Theorem relies on the Rank-nulity Theorem. A "parameter-to-dimension" matrix is composed, in which the columns correpond to the parameters and the rows to the collection of dimensions involved in the parameters. Each element in row i and column j corresponds to the power of the dimension i in the parameter j.

The number of adimensional groups is the dimension of the kernel of the matrix. And the adimensional groups are obtained from a basis of the null space.

When the powers are integers or rational numbers, which is usually the case, it is desirable to keep the type of these parameters when composing the matrix and when finding the null space and the associated adimensional Π groups.

While ian-r-rose/buckinghampy uses SymPy for symbolic manipulation of the powers of the parameters, to retain these types, we simply rely on the strong type system of the Julia language.

The LinearALgebra.nullspace, however, uses the LinearAlgebra.svd factorization, which does not preserve the Rational type. The first version of our packaged used instead the ability of the LU decomposition in the LinearAlgebra standard package to retain the Rational eltype of the matrices. However LinearAlgebra.lu factorization implements only partial pivoting and fails with singular matrices.

For this reason, we implemented our own LU factorization algorithm UnitfulBuckinghamPi.lu_pq(), with full pivoting. Then, we find the null space from the U factor of the decomposition and by properly taking into account the column permutations used in the pivoting process.

This package is licensed under the MIT license (see file LICENSE in the root directory of the project).