streamlines.jl

### A Pluto.jl notebook ###
# v0.17.3

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ ee78fc07-05ef-4acf-be5a-225a1c147fe4
begin
	using PlutoUI
	using Plots
	using NLsolve
	TableOfContents()
end

# ╔═╡ 6b8caa29-e803-4b13-af95-d59b38d2b4d1
md"""
!!! danger "Potential flow theory"

	_by [Jean-François Barthélémy](https://jfbarthelemy.github.io/)_

	This tutorial provides tool to visualize 2D streamlines of incompressible fluid around an obstacle.
"""

# ╔═╡ 4e62378a-cb80-4f4d-8f0a-2aa47005f388
html"<button onclick='present()'>Presentation mode</button>"

# ╔═╡ 1b444397-1ac1-4465-ad58-7d71e0188627
md"""
# Potential flow theory
"""

# ╔═╡ 6942d945-fb01-42f5-b62c-eb9fa245a672
begin
	myasin(x) = abs(x)<=1 ? asin(x) : NaN
	θ₁(r,ψ=0,R=1,m=0,U=1) = myasin((ψ+m/2π*log(r/R))/(U*r*(1-R^2/r^2))) ;
	θ₂(r,ψ=0,R=1,m=0,U=1) = π - θ₁(r,ψ,R,m,U) ;
	h(Z,c) = Z+c^2/Z
	Z₀(Z,α) = Z*exp(im*α)
	ζ(Z,c,R,α,β) = c-R*exp(-im*β)+Z*exp(im*α)
	reim(tab) = real.(tab), imag.(tab)
	cercle(R=1,n=100) = [R*exp(im*θ) for θ ∈ range(0,2π,length=n)]
	# streamline(θᵢ,lr,ψ,R,m,U) = [r*exp(im*θᵢ(r,ψ,R,m,U)) for r ∈ lr]
	streamline01(lr,R,m,U) = [r*exp(im*θ₁(r,0,R,m,U)) for r ∈ reverse(lr)]
	streamline02(lr,R,m,U) = [r*exp(im*θ₂(r,0,R,m,U)) for r ∈ lr]
	streamline(lr,ψ,R,m,U) = [[r*exp(im*θ₁(r,ψ,R,m,U)) for r ∈ reverse(lr)]...,[r*exp(im*θ₂(r,ψ,R,m,U)) for r ∈ lr]...]
	function solve_rmin(ψ,UR,m)
		ϵ = ψ>=0 ? 1 : -1
		f(x) = ψ-ϵ*UR*(x[1]-1/x[1])+m/2π*log(x[1])
		return nlsolve(f, [1.,], autodiff=:forward).zero[1]
	end
end ;

# ╔═╡ 18352e0c-3949-4b49-8979-f2acceede77b
begin
md"""
## Acyclic flow around a circular cylinder
- Remote velocity $\underline{U}=U\underline{e}_x$
- Complex potential $F(Z)=U\left(Z+\frac{R^2}{Z}\right)$
- Stream function $ψ(r,θ)=U\left(r-\frac{R^2}{r}\right)\sin{θ}$
"""
end

# ╔═╡ bb4ae5f5-8b88-4fe0-8db2-4f30df85d3d9
md"""`𝑅` = $(@bind R_acyc Slider(0:0.1:2, default=1,show_value=true)) ;  Visible obstacle $(@bind cont_acyc CheckBox(default=true)) """

# ╔═╡ df2be894-7ac5-493d-b455-54258022a015
md"""`𝑈` = $(@bind U_acyc Slider(0:0.1:2, default=1,show_value=true))"""

# ╔═╡ 37d7e954-1890-4d1e-824c-4fa87efa68f1
md"""
Streamlines $(@bind lig_acyc CheckBox(default=true)), 
nb = $(@bind n_acyc Scrubbable(3 : 21; default=21))
`ψₘᵢₙ = ` $(@bind ψₘᵢₙ_acyc Slider(-2:0.05:0, default=-2,show_value=true))
`ψₘₐₓ = ` $(@bind ψₘₐₓ_acyc Slider(0:0.05:2, default=2,show_value=true))
"""

# ╔═╡ b200a768-9f8f-4510-a67b-24e21076c5fa
begin
	lψ_acyc = range(ψₘᵢₙ_acyc,ψₘₐₓ_acyc,length=n_acyc)
	plot()
	if cont_acyc
		plot!(reim(cercle(R_acyc,200))...,fill=(0,:black),legend=false,aspect_ratio=:equal)
	end
	if lig_acyc
		for ψ ∈ lψ_acyc
			lr = exp10.(range(log10(abs(ψ)/2U_acyc+√((ψ/2U_acyc)^2+R_acyc^2))+0.0001,log10(5*R_acyc),length=200))
			if iszero(ψ)
				plot!(reim(streamline01(lr,R_acyc,0,U_acyc))...,legend=false,aspect_ratio=:equal)
				plot!(reim(streamline02(lr,R_acyc,0,U_acyc))...,legend=false,aspect_ratio=:equal)
				else
				plot!(reim(streamline(lr,ψ,R_acyc,0,U_acyc))...,legend=false,aspect_ratio=:equal)
			end
		end
	end
	plot!()
end

# ╔═╡ 8cedd431-ef0c-478e-83b8-48aef88e3d40
begin
md"""
## Cyclic flow around a circular cylinder
- Remote velocity $\underline{U}=U\underline{e}_x$
- Complex potential $F(Z)=U\left(Z+\frac{R^2}{Z}\right)-\frac{i\,m}{2π}\log{\frac{Z}{R}}$
- Stream function $ψ(r,θ)=U\left(r-\frac{R^2}{r}\right)\sin{θ}-\frac{m}{2π}\log{\frac{r}{R}}$
"""
end

# ╔═╡ 6931b900-a2f8-4181-97f7-0272c07e2b74
md"""`𝑅` = $(@bind R_cyc Slider(0:0.1:2, default=1,show_value=true)) ;  Visible obstacle $(@bind cont_cyc CheckBox(default=true)) """

# ╔═╡ 1ef4dd71-db13-429d-9ddc-b32f7cc397cc
md"""`𝑈` = $(@bind U_cyc Slider(0:0.1:2, default=1,show_value=true))"""

# ╔═╡ 3eeb0619-8e7e-4c27-9938-58bbdb2be42b
md"""`𝑚` = $(@bind m_cyc Slider(-20:0.1:20, default=-9,show_value=true))"""

# ╔═╡ f9a97ea9-d1a1-438f-954c-4196e5d1d254
md"""
Streamlines $(@bind lig_cyc CheckBox(default=true)), 
nb = $(@bind n_cyc Scrubbable(3 : 21; default=21))
`ψₘᵢₙ = ` $(@bind ψₘᵢₙ_cyc Slider(-2:0.05:0, default=-2,show_value=true))
`ψₘₐₓ = ` $(@bind ψₘₐₓ_cyc Slider(0:0.05:2, default=2,show_value=true))
"""

# ╔═╡ d557dd59-c3d5-40bc-ab94-c4b19a396fd7
begin
	lψ_cyc = range(ψₘᵢₙ_cyc,ψₘₐₓ_cyc,length=n_cyc)
	A_cyc = m_cyc/(4π*R_cyc*U_cyc)
	if abs(A_cyc)>1
		X₀ = abs(A_cyc)+√(A_cyc^2-1)
		ψ₀ = R_cyc*U_cyc*(X₀-1/X₀)*sign(A_cyc)-m_cyc/2π*log(X₀)
		lψ_cyc = push!(collect(lψ_cyc),ψ₀)
	end
	plot()
	if cont_cyc
		plot!(reim(cercle(R_cyc,200))...,fill=(0,:black),legend=false,aspect_ratio=:equal)
	end
	if lig_cyc
		for ψ ∈ lψ_cyc
			r_min_cyc = R_cyc
			try
				r_min_cyc = R_cyc*max(solve_rmin(ψ,U_cyc*R_cyc,m_cyc)*1.0001,1)
			catch e
				r_min_cyc = R_cyc
			end
			lr = exp10.(range(log10(r_min_cyc),log10(5*R_cyc),length=2000))
			if iszero(ψ)
				plot!(reim(streamline01(lr,R_cyc,m_cyc,U_cyc))...,legend=false,aspect_ratio=:equal)
				plot!(reim(streamline02(lr,R_cyc,m_cyc,U_cyc))...,legend=false,aspect_ratio=:equal)
			else
				plot!(reim(streamline(lr,ψ,R_cyc,m_cyc,U_cyc))...,legend=false,aspect_ratio=:equal)
			end
		end
	end
	plot!()
end

# ╔═╡ 354e8eda-f515-4236-a2a4-cfd5c8afe1f0
begin
md"""
## Joukovski conformal map and angle of incidence
- Remote velocity $\underline{U}=U\left(\cos{α}\underline{e}_x+\sin{α}\underline{e}_y\right)$
- Joukovski conformal map $z=h(Z)=Z+\frac{c^2}{Z}$ avec $c\leq R$
"""
end

# ╔═╡ 568dbc30-06bc-4f2a-bda0-fbcc11751e08
md"""`𝑅` = $(@bind R_ell Slider(0:0.1:2, default=1,show_value=true)) ;  Visible obstacle $(@bind cont_ell CheckBox(default=true)) """

# ╔═╡ 88ac83e1-b2dc-4ea8-87c8-467ad54191d7
md"""`𝑐` = $(@bind c_ell Slider(0:0.1:R_ell, default=0.8,show_value=true))"""

# ╔═╡ 77e41dd2-1a18-4e87-a931-d5f7a59504e3
md"""`𝛼` = $(@bind α_ell Slider(0:π/24:π/2, default=π/12,show_value=true))"""

# ╔═╡ 96b875cc-c61e-441b-91de-5da491e3843e
md"""`𝑈` = $(@bind U_ell Slider(0:0.1:2, default=1,show_value=true))"""

# ╔═╡ 991ec9e8-00fd-4497-a219-a71339ca7d9d
md"""`𝑚` = $(@bind m_ell Slider(-20:0.1:20, default=-4π*U_ell*R_ell*sin(α_ell),show_value=true))"""

# ╔═╡ 65335f0e-bc56-4839-9b36-84d96a3e8198
md"""
Streamlines $(@bind lig_ell CheckBox(default=true)), 
nb = $(@bind n_ell Scrubbable(3 : 21; default=21))
`ψₘᵢₙ = ` $(@bind ψₘᵢₙ_ell Slider(-2:0.05:0, default=-2,show_value=true))
`ψₘₐₓ = ` $(@bind ψₘₐₓ_ell Slider(0:0.05:2, default=2,show_value=true))
"""

# ╔═╡ c263cc34-1293-449a-86f9-8142163de2c8
begin
	lψ_ell = range(ψₘᵢₙ_ell,ψₘₐₓ_ell,length=n_ell)
	A_ell = m_ell/(4π*R_ell*U_ell)
	if abs(A_ell)>1
		X₀_ell = abs(A_ell)+√(A_ell^2-1)
		ψ₀_ell = R_ell*U_ell*(X₀_ell-1/X₀_ell)*sign(A_ell)-m_ell/2π*log(X₀_ell)
		lψ_ell = push!(collect(lψ_ell),ψ₀_ell)
	end
	plot()
	if cont_ell
		plot!(reim(h.(Z₀.(cercle(R_ell,200),α_ell),c_ell))...,fill=(0,:black),legend=false,aspect_ratio=:equal)
	end
	if lig_ell
		for ψ ∈ lψ_ell
		    r_min_ell = R_ell
		    try
		        r_min_ell = R_ell*max(solve_rmin(ψ,U_ell*R_ell,m_ell)*1.0001,1)
		    catch e
		        r_min_ell = R_ell
		    end
		    lr = exp10.(range(log10(r_min_ell),log10(5*R_ell),length=200))
		    if iszero(ψ)
		        plot!(reim(h.(Z₀.(streamline01(lr,R_ell,m_ell,U_ell),α_ell),c_ell))...,legend=false,aspect_ratio=:equal)
		        plot!(reim(h.(Z₀.(streamline02(lr,R_ell,m_ell,U_ell),α_ell),c_ell))...,legend=false,aspect_ratio=:equal)
		    else
		        plot!(reim(h.(Z₀.(streamline(lr,ψ,R_ell,m_ell,U_ell),α_ell),c_ell))...,legend=false,aspect_ratio=:equal)
		    end
		end
	end
	plot!()
end

# ╔═╡ 785ce092-6fc8-49c2-946e-f287e7e7a30d
begin
md"""
## Profile with cusp and camber effect
- Remote velocity $\underline{U}=U\left(\cos{α}\underline{e}_x+\sin{α}\underline{e}_y\right)$
- Eccentricity $b=c-R\,e^{-iβ}$
- Map $ζ=b+Z\,e^{iα}$
- Joukovski map $z=h(ζ)=ζ+\frac{c^2}{ζ}$ avec $c\leq R$
- Kutta-Joukovski condition $m=-4πRU\sin{(α+β)}$ (by default)
"""
end

# ╔═╡ 18a41216-ea0e-499d-bc4b-1bf492862781
md"""`𝑅` = $(@bind R_jou Slider(0:0.1:2, default=1,show_value=true)) ;  Visible obstacle $(@bind cont_jou CheckBox(default=true)) """

# ╔═╡ a10e9ecf-b887-4e95-8fe9-c7ac973858a2
md"""`𝑐` = $(@bind c_jou Slider(0:0.1:R_jou, default=0.8,show_value=true))"""

# ╔═╡ d16fa468-9ee1-4270-bd6d-80921c97b225
md"""`β` = $(@bind β_jou Slider(0:(acos(c_jou/R_jou)+π/1000)/10:acos(c_jou/R_jou)+π/1000, default=π/12,show_value=true))"""

# ╔═╡ 56ac2fe4-8ea5-4bbc-be0f-dff42dc0b943
md"""`𝛼` = $(@bind α_jou Slider(0:π/24:π/2, default=π/12,show_value=true))"""

# ╔═╡ 51299335-967a-48bc-9246-3a72d03bbe90
md"""`𝑈` = $(@bind U_jou Slider(0:0.1:2, default=1,show_value=true))"""

# ╔═╡ 6c1206ba-47bf-4757-b639-bbf52c7ebe3d
md"""`𝑚` = $(@bind m_jou Slider(-20:0.1:20, default=-4π*U_jou*R_jou*sin(α_jou+β_jou),show_value=true))"""

# ╔═╡ 70aaf81d-091a-474f-a3d7-f2d381193621
md"""
Streamlines $(@bind lig_jou CheckBox(default=true)), 
nb = $(@bind n_jou Scrubbable(3 : 21; default=21))
`ψₘᵢₙ = ` $(@bind ψₘᵢₙ_jou Slider(-2:0.05:0, default=-2,show_value=true))
`ψₘₐₓ = ` $(@bind ψₘₐₓ_jou Slider(0:0.05:2, default=2,show_value=true))
"""

# ╔═╡ 8cdae750-8e68-4119-9da5-5470757bfd43
begin
	lψ_jou = range(ψₘᵢₙ_jou,ψₘₐₓ_jou,length=n_jou)
	A_jou = m_jou/(4π*R_jou*U_jou)
	if abs(A_jou)>1
		X₀_jou = abs(A_jou)+√(A_jou^2-1)
		ψ₀_jou = R_jou*U_jou*(X₀_jou-1/X₀_jou)*sign(A_jou)-m_jou/2π*log(X₀_jou)
		lψ_jou = push!(collect(lψ_jou),ψ₀_jou)
	end
	plot()
	if cont_jou
		plot!(reim(h.(ζ.(cercle(R_jou,200),c_jou,R_jou,α_jou,β_jou),c_jou))...,fill=(0,:black),legend=false,aspect_ratio=:equal)
	end
	if lig_jou
		for ψ ∈ lψ_jou
		    r_min_jou = R_jou
		    try
		        r_min_jou = R_jou*max(solve_rmin(ψ,U_jou*R_jou,m_jou)*1.0001,1)
		    catch e
		        r_min_jou = R_jou
		    end
		    lr = exp10.(range(log10(r_min_jou),log10(5*R_jou),length=200))
		    if iszero(ψ)
		        plot!(reim(h.(ζ.(streamline01(lr,R_jou,m_jou,U_jou),c_jou,R_jou,α_jou,β_jou),c_jou))...,legend=false,aspect_ratio=:equal)
		        plot!(reim(h.(ζ.(streamline02(lr,R_jou,m_jou,U_jou),c_jou,R_jou,α_jou,β_jou),c_jou))...,legend=false,aspect_ratio=:equal)
		    else
		        plot!(reim(h.(ζ.(streamline(lr,ψ,R_jou,m_jou,U_jou),c_jou,R_jou,α_jou,β_jou),c_jou))...,legend=false,aspect_ratio=:equal)
		    end
		end
	end
	plot!()
end

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PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"

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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.1.1+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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