One-Dimensional Steady-State Convection and Diffusion

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Computational methodologies for forced convection
  1. One-Dimensional Steady-State Convection and Diffusion
    1. Central Difference Scheme
    2. Upwind Scheme
    3. Hybrid Scheme
    4. Exponential and Power Law Schemes
    5. A Generalized Expression of Discretization Schemes
  2. Multidimensional Convection and Diffusion Problems
  3. Numerical Solution of Flow Field
    1. Special Difficulties
    2. Staggered grid
    3. Pressure Correction Equation
    4. The SIMPLE Algorithm
  4. Numerical Simulation of Interfaces and Free Surfaces
  5. Application of Computational Methods

Exact Solution The objective of this subsection is to introduce various discretization schemes of the convection-diffusion terms through discussion of the one-dimensional steady state convection and diffusion problem. For a one-dimensional steady-state convection and diffusion problem, the governing equation is

\frac{d(\rho u\varphi )}{dx}=\frac{d}{dx}\left( \Gamma \frac{d\varphi }{dx} \right)

(1)

where the velocity, u, is assumed to be known. Both density, ρ, and diffusivity, Г, are assumed to be constants. The continuity equation for this one-dimensional problem is

\frac{d(\rho u)}{dx}=0

(2)

Equation (1) is subject to the following boundary conditions:

\varphi =\varphi _{0},\text{  }x=0

(3)

\varphi =\varphi _{L},\text{  }x=L

(4)

By introducing the following dimensionless variables

\Phi =\frac{\varphi -\varphi _{0}}{\varphi _{L}-\varphi _{0}},\text{  }X=\frac{x}{L}

(5)

the one-dimensional steady-state convection and diffusion problem can be nondimensionalized as

\text{Pe}\frac{d\Phi }{dX}=\frac{d^{2}\Phi }{dX^{2}}

(6)

\begin{matrix}{}\\\end{matrix}\Phi =0,\text{  X}=0

(7)

\begin{matrix}{}\\\end{matrix}\Phi =1,\text{  }X=1

(8)

where

\text{Pe}=\frac{\rho uL}{\Gamma }

(9)

is the Peclet number that reflects the relative level of convection and diffusion. Pe becomes zero for the case of pure diffusion and becomes infinite for the case of pure advection. The exact solution of eqs. (6) - (8) can be obtained as

\Phi =\frac{\varphi -\varphi _{0}}{\varphi _{L}-\varphi _{0}}=\frac{\exp (\text{Pe}X)-1}{\exp (\text{Pe})-1}

(10)

which will be used as a criterion to check the accuracy of various discretization schemes.

Central Difference Scheme
Upwind Scheme
Hybrid Scheme
Exponential and Power Law Schemes
A Generalized Expression of Discretization Schemes