Numerical solution of the convection–diffusion equation

The transient convection diffusion equation governs the combined fluid flow and heat transfer that mainly involve convection and diffusion components. The same equation can also be called as time dependent convection diffusion equation. Since much of the framework is similar to the steady state convection diffusion equation, only a general discontinuous finite element formulation is needed . The unsteady convection-diffusion problem is considered, at first the known temperature T is expanded into a taylor series with respect to time taking into account its three components. Next, using the convection diffusion equation and equation obtained from the differentiation of this equation.

Equation

General

The following convection diffusion equation is considered here

$c\rho [{\frac {\partial T(x,t)}{\partial t}}+\epsilon u{\frac {\partial T(x,t)}{\partial x}}]=\lambda {\frac {\partial ^{2}T(x,t)}{\partial x^{2}}}+Q(x,t)$

In the above equation, four terms represents transient, convection, diffusion and source term respectively

where

• T is the temperature in particular case of heat transfer otherwise it is the variable of interest

• t is time

• c is the specific heat

• u is velocity

• $\epsilon$ is porosity that is the ratio of liquid volume to the total volume

• $\rho$ is mass density

• $\lambda$ is thermal conductivity

• Q(x,t) is source term represents capacity of internal sources

above equation can be written in the form of:

${\frac {\partial T}{\partial t}}=a{\frac {\partial ^{2}T}{\partial x^{2}}}-\epsilon u{\frac {\partial T}{\partial x}}+{\frac {Q}{c/rho}}$

where $a={\frac {\lambda }{c\rho }}$ is the diffusion coefficient

Explicit scheme

Explicit scheme of FDM (Finite Difference Method) has been considered and stability criteria are formulated. In this scheme, temperature is totally dependent on the old temperature. Substitution of $\theta =0$ gives the explicit discretization of the unsteady conductive heat transfer equation.

where $\theta$ is the weighing parameter between 0 and 1.

${\frac {T_{i}^{f}-T_{i}^{f-1}}{\Delta t}}=a{\frac {T_{i-1}^{f-1}-2T_{i}^{f-1}+T_{i+1}^{f-1}}{h^{2}}}-\epsilon u{\frac {T_{i+1}^{f-1}-T_{i-1}^{f-1}}{2h}}+{\frac {Q_{i}^{f-1}}{c\rho }}$

where

• $\Delta t=t^{f}-t^{f-1}$

• h is uniform grid spacing (mesh step)

$T_{i}^{f}=(1-{\frac {2a\Delta t}{h^{2}}})T_{i}^{f-1}+({\frac {a\Delta t}{h^{2}}}+{\frac {\epsilon u\Delta t}{2h}})T_{i-1}^{f-1}+({\frac {a\Delta t}{h^{2}}}-{\frac {\epsilon u\Delta t}{2h}})T_{i+1}^{f-1}+{\frac {Q_{i}^{f-1}}{c\rho }}\Delta t$

Stability Criteria

$h<{\frac {2a}{\epsilon u}}$

$\Delta t<{\frac {h^{2}}{2a}}$

This inequality sets a stringent maximum limit to the time step size and represents a serious limitation for the explicit scheme. This method is not recommended for general transient problems because maximum possible time step has to be reduced as the square of h.

Variation of temperature with time in 3 different schemes

Implicit Scheme

In Implicit Scheme, the temperature is dependent at the new time level $t+\Delta t$. After using implicit scheme, it was found that all coefficients are positive. It makes the implicit scheme unconditionally stable for any size of time step. This scheme is preferred for general purpose transient calculations because of its robustness and unconditional stability .

Crank-Nikolson Scheme

In Crank-Nicolson method, the temperature is equally dependent on t and $t+\Delta t$. It is a second order method in time and this method is generally used in diffusion problems.

Stability Criteria

$\Delta t<{\frac {h^{2}}{a}}$

This time step limitation is less restricted than the explicit method. The Crank-Nicolson method is based on the central differencing and hence it is second order accurate in time.

Finite element solution to convection-diffusion problem

Unlike the conduction equation (finite element solution is used), a numerical solution for the convection-diffusion equation has to deal with the convection part of the governing equation in addition to diffusion. When peclet number (Pe) exceeds a critical value, the spurious oscillations result in space and this problem is not unique to finite elements as all other discretization techniques have the same difficulties. in a finite difference formulation, the spatial oscillations are reduced by a family of discretization schemes like upwind scheme . In this method, the basic shape function is modified to obtain the upwinding effect. This method is an extension of runge kutta discontinuous for a convection diffusion equation. For time dependent equations, a different kind of approach is followed. The finite difference scheme has an equivalent in the finite element method (Galerkin method). Another similar method is characteristic galerkin method (uses an implicit algorithm). For scalar variables, above two methods are identical.

See Also

• convection-diffusion equation
• Double diffusive convection
• An Album of Fluid Motion
• Lagrangian and Eulerian specification of the flow field
• Fluid simulation
• Finite volume method for unsteady flow