Curvilinear Coordinates - Royal Institute of Technology Flipbook PDF

1 Curvilinear Coordinates x y z r z P(r; ;z) Figure 1: Cylindrical polar coordinates Cylindrical Polar Coordinates The c

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1

Curvilinear Coordinates z • P (r, θ, z)

z y r θ x Figure 1: Cylindrical polar coordinates

Cylindrical Polar Coordinates The cylindrical polar coordinates are (r, θ, z), where θ is the azimuthal angle, see figure 1. The velocity can be written as u = u r er + uθ eθ + uz ez ,

(1)

where the unit vectors are related to Cartesian coordinates as er = ex cos θ + ey sin θ, eθ = −ex sin θ + ey cos θ, ez = e z .

(2)

Non-zero derivatives of unit vectors ∂er = eθ , ∂θ

∂eθ = −er . ∂θ

(3)

Gradient of a scalar p ∂p 1 ∂p ∂p e + e + e . ∂r r r ∂θ θ ∂z z

(4)

  ∂p 1 ∂2p ∂2p 1 ∂ r + 2 2 + 2. r ∂r ∂r r ∂θ ∂z

(5)

∇p = Laplacian of a scalar p ∇2 p =

2 Divergence of a vector u ∇·u=

∂uz 1 ∂(rur ) 1 ∂uθ + + . r ∂r r ∂θ ∂z

(6)

Advective derivative of a scaler p (u · ∇)p = ur

∂p uθ ∂p ∂p + + uz . ∂r r ∂θ ∂z

(7)

Curl of a vector u       1 ∂(ruθ ) ∂ur ∂ur ∂uθ ∂uz 1 ∂uz er + eθ + ez . (8) − − − ∇×u= r ∂θ ∂z ∂z ∂r r ∂r ∂θ Incompressible Navier-Stokes equations with no body force   ∂ur u2 1 ∂p u 2 ∂u + (u · ∇)ur − θ = − + ν ∇2 ur − 2r − 2 θ , ∂t r ρ ∂r r r ∂θ   u u 1 ∂p 2 ∂u u ∂uθ + (u · ∇)uθ + r θ = − + ν ∇2 uθ + 2 r − 2θ , ∂t r ρ r ∂θ r ∂θ r ∂uz 1 ∂p + (u · ∇)uz = − + ν∇2 uz . ∂t ρ ∂z

(9)

(10) (11)

Spherical Polar Coordinates The spherical polar coordinates are (r, θ, ϕ), where ϕ is the azimuthal angle, see figure 2. The velocity can be written as u = u r er + uθ eθ + uϕ eϕ ,

(12)

where the unit vectors are related to Cartesian coordinates as er = ex sin θ cos ϕ + ey sin θ sin ϕ + ez cos θ, eθ = ex cos θ cos ϕ + ey cos θ sin ϕ − ez sin θ, eϕ = −ex sin ϕ + ey cos ϕ.

(13)

Non-zero derivatives of unit vectors ∂er = eθ , ∂θ

∂er = eϕ sin θ, ∂ϕ

∂eθ = −er , ∂θ

∂eθ = eϕ cos θ, ∂ϕ

∂eϕ = −er sin θ − eθ cos θ. ∂ϕ

(14)

(15)

Gradient of a scalar p ∇p =

1 ∂p 1 ∂p ∂p e + e + e . ∂r r r ∂θ θ r sin θ ∂ϕ ϕ

(16)

3 z • P (r, θ, ϕ) r

θ y

ϕ x Figure 2: Spherical polar coordinates Laplacian of a scalar p     1 ∂ 1 ∂p 1 ∂ ∂2p 2 2 ∂p ∇ p= 2 r + 2 sin θ + 2 2 . r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂ϕ2

(17)

Divergence of a vector u ∇·u=

1 ∂(uθ sin θ) 1 ∂uϕ 1 ∂(r2 ur ) + + . r2 ∂r r sin θ ∂θ r sin θ ∂ϕ

(18)

Advective derivative of a scaler p (u · ∇)p = ur

∂p uθ ∂p uϕ ∂p + + . ∂r r ∂θ r sin θ ∂ϕ

(19)

Curl of a vector u 1 ∇×u= r sin θ



   ∂(uϕ sin θ) ∂uθ ∂(ruϕ ) 1 1 ∂ur er + eθ − − ∂θ ∂ϕ r sin θ ∂ϕ ∂r   1 ∂(ruθ ) ∂ur + eϕ . − r ∂r ∂θ

(20)

4 Incompressible Navier-Stokes equations with no body force u2 + u2ϕ ∂ur + (u · ∇)ur − θ ∂t r =−

(21)

  2u 2 ∂(uθ sin θ) 2 ∂uϕ 1 ∂p + ν ∇2 ur − 2r − 2 − 2 , ρ ∂r r r sin θ ∂θ r sin θ ∂ϕ

u2ϕ cot θ u u ∂uθ + (u · ∇)uθ + r θ − ∂t r r   2 ∂u u 2 cos θ ∂uϕ 1 ∂p , + ν ∇2 u θ + 2 r − 2 θ 2 − 2 2 =− ρ r ∂θ r ∂θ r sin θ r sin θ ∂ϕ

(22)

∂uϕ ur uϕ u uϕ cot θ + (u · ∇)uϕ + + θ (23) ∂t r r   uϕ 2 ∂ur 2 cos θ ∂uθ ∂p 1 . + ν ∇2 u ϕ + 2 + 2 2 − 2 2 =− ρr sin θ ∂ϕ r sin θ ∂ϕ r sin θ ∂ϕ r sin θ