Convection Diffusion Application

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The numerical solution of convection-diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth. This paper describes the Convection Diffusion Applications for solving this equation.




Under the assumption of incompressibility, the governing equations are given by

Problem description

  \rho C \frac{\partial T}{\partial t} + \rho C v \cdot \nabla T + \nabla
  \cdot q = 0 

  q = -\kappa \nabla \cdot T 

In the context of mass difussion within a fluid, (1) is is the mass conservation equation and (2) is a contitutive law proposed by Fourier. The notation is standard: ρ is the density, C the heat capacity, κ the thermal conductivity, T is the temperature, v is the velocity field and q is the diffusive flux per unit fluid density.

Remark: systen can be decoupled since we can plug (2) into (1) and solve the scalar equation

  \rho C \frac{\partial T}{\partial t} + \rho Cv \cdot \nabla T -
  \nabla \cdot (\kappa \nabla \cdot T) = 0 

Problem statement

Let us consider the transport by convection and diffusion in an open set Ω \subset \Re^d(d=2 or 3) \ with piecewise smooth boundary Γ, such that \Gamma = \Gamma_d \cup \Gamma_N. The unit outward normal vector to Γ is denoted n. The convection-diffusion initial-boundary value problem can be stated as follows: given a divergence-free velocity field a, the diffusion tensor κ and adequate initial and boundary conditions, find T : \bar{\Omega} \times \left[ 0, T \left] \rightarrow \Re \right. \right. such that

\rho C \frac{\partial T}{\partial t} + \rho Cv \cdot \nabla T -
   \nabla \cdot (\kappa \nabla T) = 0 in \Omega \times (0, T) 
(4) T(x,0) = T0(x)onΩ

T = T_D on \Gamma_D \times (0, T)

k (\nabla T) \cdot n = q on \Gamma_N \times (0, T)

Space discretization method

Multiplying Eq.(4) by a test function W and intehrating on the whole domain Ω the equation reads

\int_\Omega \rho C \frac{\partial T}{\partial t} W d V + \int_\Omega \rho Cv \cdot
   \nabla T W d V = \int_\Omega (\nabla \cdot (k \nabla T)) W d V 

Integratin by parts the right term of Eq.(5) leads to

 \int_\Omega \rho C \frac{\partial T}{\partial t} W d V + \int_\Omega v \cdot \nabla T W d
   V = - \int_\Omega k \nabla T \cdot \nabla W d V (6)

Finite element discretization

The temperature is discretized in the standard finite element method manner as 
 T = \sum N_i T_i 
(7) where Ni are the nodal shape functions. Substituting the finite element approximation (7) into the variational equation () and choosing a Galerling formulation (Wi = Ni) leads to the following equation:

 \rho C M \frac{\partial T}{\partial t} + \rho C S T 
 = - \kappa L T (7)

Time discretization method

Consider a first-order BDF (that is, the Euler implicit scheme)

\frac{\partial T}{\partial t}=\frac{(T^{n + 1} - T^n)}{\Delta t}

or a second-order BDF

\frac{\partial T}{\partial t}= \frac{1}{2} \frac{(3 \times T^{n + 1} - 4 \times T^n + T^{n-1})}{\Delta t}

How to analyse using the current application

We will use GiD as pre and post-processor link gid .


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