Stochastic Eulerian Lagrangian method

In computational fluid dynamics, the Stochastic Eulerian Lagrangian Method (SELM) is an approach to capture essential features of fluid-structure interactions subject to thermal fluctuations while introducing approximations which facilitate analysis and the development of tractable numerical methods. SELM is a hybrid approach utilizing an Eulerian description for the continuum hydrodynamic fields and a Lagrangian description for elastic structures. Thermal fluctuations are introduced through stochastic driving fields. Approaches also are introduced for the stochastic fields of the SPDEs to obtain numerical methods taking into account the numerical discretization artifacts to maintain statistical principles, such as fluctuation-dissipation balance and other properties in statistical mechanics.

The SELM fluid-structure equations typically used are

ρ

d

u

d

t

=

μ

Δ

u

−

∇

p

+

Λ

[

Υ

(

V

−

Γ

u

)

]

+

λ

+

f

t

h

m

(

x

,

t

)

{\displaystyle \rho {\frac {d{u}}{d{t}}}=\mu \,\Delta u-\nabla p+\Lambda [\Upsilon (V-\Gamma {u})]+\lambda +f_{\mathrm {thm} }(x,t)}

m

d

V

d

t

=

−

Υ

(

V

−

Γ

u

)

−

∇

Φ

[

X

]

+

ξ

+

F

t

h

m

{\displaystyle m{\frac {d{V}}{d{t}}}=-\Upsilon (V-\Gamma {u})-\nabla \Phi [X]+\xi +F_{\mathrm {thm} }}

d

X

d

t

=

V

.

{\displaystyle {\frac {d{X}}{d{t}}}=V.}

The pressure p is determined by the incompressibility condition for the fluid

∇

⋅

u

=

0.

{\displaystyle \nabla \cdot u=0.\,}

The

Γ

,

Λ

{\displaystyle \Gamma ,\Lambda }

operators couple the Eulerian and Lagrangian degrees of freedom. The

X

,

V

{\displaystyle X,V}

denote the composite vectors of the full set of Lagrangian coordinates for the structures. The

Φ

{\displaystyle \Phi }

is the potential energy for a configuration of the structures. The

f

t

h

m

,

F

t

h

m

{\displaystyle f_{\mathrm {thm} },F_{\mathrm {thm} }}

are stochastic driving fields accounting for thermal fluctuations.