Mathematics of advection Advection

1 mathematics of advection

1.1 advection equation
1.2 solving equation
1.3 treatment of advection operator in incompressible navier stokes equations

mathematics of advection

the advection equation partial differential equation governs motion of conserved scalar field advected known velocity vector field. derived using scalar field s conservation law, gauss s theorem, , taking infinitesimal limit.

one visualized example of advection transport of ink dumped river. river flows, ink move downstream in pulse via advection, water s movement transports ink. if added lake without significant bulk water flow, ink disperse outwards source in diffusive manner, not advection. note moves downstream, pulse of ink spread via diffusion. sum of these processes called convection.

the advection equation

in cartesian coordinates advection operator is

u

⋅
∇
=

u

x




∂

∂
x



+

u

y




∂

∂
y



+

u

z




∂

∂
z





{\displaystyle \mathbf {u} \cdot \nabla =u_{x}{\frac {\partial }{\partial x}}+u_{y}{\frac {\partial }{\partial y}}+u_{z}{\frac {\partial }{\partial z}}}

.

where




u

=
(

u

x


,

u

y


,

u

z


)


{\displaystyle \mathbf {u} =(u_{x},u_{y},u_{z})}

velocity field, ,



∇


{\displaystyle \nabla }

del operator (note cartesian coordinates used here).

the advection equation conserved quantity described scalar field



ψ


{\displaystyle \psi }

expressed mathematically continuity equation:

where



∇
⋅


{\displaystyle \nabla \cdot }

divergence operator , again




u



{\displaystyle \mathbf {u} }

velocity vector field. frequently, assumed flow incompressible, is, velocity field satisfies

∇
⋅


u


=
0


{\displaystyle \nabla \cdot {\mathbf {u}}=0}

and




u



{\displaystyle \mathbf {u} }

said solenoidal. if so, above equation can rewritten as

in particular, if flow steady, then

u


⋅
∇
ψ
=
0


{\displaystyle {\mathbf {u}}\cdot \nabla \psi =0}

which shows



ψ


{\displaystyle \psi }

constant along streamline. hence,



∂
ψ

/

∂
t
=
0
,


{\displaystyle \partial \psi /\partial t=0,}





ψ


{\displaystyle \psi }

doesn t vary in time.

if vector quantity




a



{\displaystyle \mathbf {a} }

(such magnetic field) being advected solenoidal velocity field




u



{\displaystyle \mathbf {u} }

, advection equation above becomes:

∂


a




∂
t



+

(


u


⋅
∇
)



a


=
0.


{\displaystyle {\frac {\partial {\mathbf {a}}}{\partial t}}+\left({\mathbf {u}}\cdot \nabla \right){\mathbf {a}}=0.}

here,




a



{\displaystyle \mathbf {a} }

vector field instead of scalar field



ψ


{\displaystyle \psi }

.

solving equation

a simulation of advection equation u = (sin t, cos t) solenoidal.

the advection equation not simple solve numerically: system hyperbolic partial differential equation, , interest typically centers on discontinuous shock solutions (which notoriously difficult numerical schemes handle).

even 1 space dimension , constant velocity field, system remains difficult simulate. equation becomes

∂
ψ


∂
t



+

u

x





∂
ψ


∂
x



=
0


{\displaystyle {\frac {\partial \psi }{\partial t}}+u_{x}{\frac {\partial \psi }{\partial x}}=0}

where



ψ
=
ψ
(
x
,
t
)


{\displaystyle \psi =\psi (x,t)}

scalar field being advected ,




u

x




{\displaystyle u_{x}}





x


{\displaystyle x}

component of vector




u

=
(

u

x


,
0
,
0
)


{\displaystyle \mathbf {u} =(u_{x},0,0)}

.

treatment of advection operator in incompressible navier stokes equations

according zang, numerical simulation can aided considering skew symmetric form advection operator.

1
2




u


⋅
∇


u


+


1
2


∇
(


u




u


)


{\displaystyle {\frac {1}{2}}{\mathbf {u}}\cdot \nabla {\mathbf {u}}+{\frac {1}{2}}\nabla ({\mathbf {u}}{\mathbf {u}})}

where

∇
(


u




u


)
=
[
∇
(


u



u

x


)
,
∇
(


u



u

y


)
,
∇
(


u



u

z


)
]


{\displaystyle \nabla ({\mathbf {u}}{\mathbf {u}})=[\nabla ({\mathbf {u}}u_{x}),\nabla ({\mathbf {u}}u_{y}),\nabla ({\mathbf {u}}u_{z})]}

and




u



{\displaystyle \mathbf {u} }

same above.

since skew symmetry implies imaginary eigenvalues, form reduces blow , spectral blocking experienced in numerical solutions sharp discontinuities (see boyd).

using vector calculus identities, these operators can expressed in other ways, available in more software packages more coordinate systems.

u

⋅
∇

u

=
∇

(



∥

u


∥

2



2


)

+

(
∇
×

u

)

×

u



{\displaystyle \mathbf {u} \cdot \nabla \mathbf {u} =\nabla \left({\frac {\|\mathbf {u} \|^{2}}{2}}\right)+\left(\nabla \times \mathbf {u} \right)\times \mathbf {u} }










1
2



u

⋅
∇

u

+


1
2


∇
(

u


u

)
=
∇

(



∥

u


∥

2



2


)

+

(
∇
×

u

)

×

u

+


1
2



u

(
∇
⋅

u

)


{\displaystyle {\frac {1}{2}}\mathbf {u} \cdot \nabla \mathbf {u} +{\frac {1}{2}}\nabla (\mathbf {u} \mathbf {u} )=\nabla \left({\frac {\|\mathbf {u} \|^{2}}{2}}\right)+\left(\nabla \times \mathbf {u} \right)\times \mathbf {u} +{\frac {1}{2}}\mathbf {u} (\nabla \cdot \mathbf {u} )}

this form makes visible skew symmetric operator introduces error when velocity field diverges. solving advection equation numerical methods challenging , there large scientific literature this.