Atomo

Páginas: 5 (1057 palabras) Publicado: 9 de mayo de 2012
Isothermal Spreading Gravity Currents
Linearised finite difference method to solve nonlinear diffusion equation
Ahmos Sansom
June 13, 2010
Abstract. The linearisation method is applied to the nonlinear diffusion equation governing the isothermal flow
of viscous gravity currents to derive a finite difference numerical scheme. The numerical approach can be validated
using similarity solutions.

1.Introduction
This document discusses the nonlinear diffusion equation noted below where a numerical algorithm using a finite difference approach is described.
1∂
∂h
=
∂t
3 ∂x

h3

∂h
∂x

.

(1)

The scheme can be validated by similarity solution as discussed in this document.
The diffusion equation can be derived from the Navier-Stokes equations assuming laminar flow,
small aspectratio and dominant viscous forces, see discussion in the Very viscous flow chapter in
Acheson [1] and derivation in [5] which discusses gravity currents with temperature dependent
viscosity.
The diffusion equation has been applied to lava dome growth by Huppert [3] where the resulting
similarity solutions was first discussed by Barenblatt [2].

2. Numerical Solution: Linearisation method
Theisothermal equation governing the free surface can be linearised as discussed for the general
nonlinear diffusion equation in Richtmyer and Morton[4] and is the inspiration for the numerical
solution discussed here. Rewritting the diffusion equation, such that
1 ∂ 2 h4
∂h
=
∂t
12 ∂x2

(2)

and discretising the above equation by denoting h((i − 1)∆x, n∆t) by hn using a Crank-Nicolson
iimplicit scheme; this gives,
(h4 )n+1 − 2(h4 )n+1 + (h4 )n+1 + (h4 )n 1 − 2(h4 )n + (h4 )n
hn+1 − hn
i−
i
i+1
i−1
i
i+1
i
i
=
.
∆t
24∆x2

(3)

The above results in a nonlinear system of simultaneous equations, however a linear scheme is
required. The solution is to approximate (h4 )n+1 by a linear scheme. This is obtained by time
i
marching using Taylor’s expansion of (h4 )n+1 ,such that
i
(h4 )n+1 = (h4 )n + ∆t
i
i
c 2010 .

∂ h4
∂t

n

+ O(∆t2 ).

(4)

i

.

IsothermalGravityCurrent.tex; 13/06/2010; 19:34; p.1

2

Ahmos Sansom

Note that
∂h4
∂h
= 4h3 ,
∂t
∂t
hence to O(∆t)
(h4 )n+1 = (h4 )n + 4(h3 )n hn+1 − hn .
i
i
i
i
i

(5)

Let wi = hn+1 − hn and substitute (5) in (3) giving
i
i
wi
= (h4 )n 1 + 4(h3 )n 1 wi−1 − 2(h4 )n− 8(h3 )n wi + (h4 )n
i−
i−
i
i
i+1
∆t
3n
4n
4n
4n
+4(h )i+1 wi+1 + (h )i−1 − 2(h )i + (h )i+1 /(24∆x2 ).
Rearranging gives,
−2rx (h3 )n 1 wi−1 + (1 + 4rx (h3 )n )wi − 2rx (h3 )n wi+1 = dn ,
i−
i
i+1
i

(6)

where
dn = rx (h4 )n 1 − 2(h4 )n + (h4 )n ,
i
i−
i
i+1

and rx = 12∆tx2 . Equation (6) can be solved for w using the Thomas algorithm taking the dn
i
term asknown. The relationship hn+1 = hn + wi completes the time step. Taking account of
i
i
symmetry in the problem at i = 1 gives hi+1 = hi−1 ( ∂h = 0) such that,
∂x

(1 + 4rx (h3 )n )wi − 4rx (h3 )n wi+1 = dn ,
i
i+1
i

(7)

where
dn = 2rx −(h4 )n + (h4 )n .
i
i
i+1
A similar boundary condition can be applied at the end of the discretised spatial domain.
To complete the numericaldescription, the initial condition for the height profile, at t = 0,
is h = (1 − x2 )+ + 10−6 , where a prewetting film of thickness 10−6 has been included to avoid
the difficulty of tracking sharp flow front.
The above nonlinear partial differential equation can also be solved by the method of lines
approach which reduces the problem to solving a system of ordinary differential equations in
NAG routineD03PCF.

3. Analytical Solution: Similarity method
Similarity methods, techniques based on invariance under continuous Lie group transformations,
can be used to simplify ordinary and partial differential equations. Similarity transformations reduce either the order of the ordinary differential equation, or the number of independent variables
in a partial differential equation. A particular...
Leer documento completo

Regístrate para leer el documento completo.

Estos documentos también te pueden resultar útiles

  • Atomo
  • Atomos
  • Atomo
  • Atomo
  • el atomo
  • Atomistas
  • Que Es El Atomo
  • Atomo

Conviértase en miembro formal de Buenas Tareas

INSCRÍBETE - ES GRATIS