Exponential and Power Law Schemes

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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

Since the exact solution of eq. (4.201) exists, one can reasonably expect that an accurate scheme can be derived if the result of the exact solution, eq. (4.210), is utilized. Equation (4.201) can be rewritten as

\frac{d}{dx}\left( \rho u\varphi -\Gamma \frac{d\varphi }{dx} \right)=0

(1)

Defining the total flux of \varphi due to convection and diffusion

J=\rho u\varphi -\Gamma \frac{d\varphi }{dx}

(2)

eq. (4.229) becomes

\frac{dJ}{dx}=0

(3)

Integrating eq. (4.231) over the control volume P (shaded area in Fig. 4.17), yields

\begin{matrix}{}\\\end{matrix}J_{e}=J_{w}

(4)

Instead of assuming piecewise linear distribution of \varphi as with central difference scheme or assuming \varphi at the face of the control volume is equal to the value of \varphi at the grid point on the upwind side in the upwind scheme, the distribution of \varphi between grid points can be taken as that obtained from the exact solution, eq. (4.210). Applying eq. (4.210) between grid points E and P, we have

\frac{\varphi (x)-\varphi _{P}}{\varphi _{E}-\varphi _{P}}=\frac{\exp [\text{Pe}_{\Delta \text{e}}(x-x_{P})/(\delta x)_{e}]-1}{\exp (\text{Pe}_{\Delta \text{e}})-1}

(5)

Substituting eq. (4.233) into eq. (4.230) and evaluating the result at x = xe, the total flux of \varphi at the face of control volume becomes

J_{e}=F_{e}\left[ \varphi _{P}+\frac{\varphi _{P}-\varphi _{E}}{\exp (\text{Pe}_{\Delta e})-1} \right]

(6)

Similarly, the total flux at the west face of the control volume is

J_{w}=F_{w}\left[ \varphi _{W}+\frac{\varphi _{W}-\varphi _{P}}{\exp (\text{Pe}_{\Delta w})-1} \right]

(7)

Substituting eqs. (4.234) and (4.235) into eq. (4.232) and rearranging the resulting equation yields

a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}

(8)

where

a_{E}=\frac{F_{e}}{\exp (\text{Pe}_{\Delta e})-1}

(9)

a_{W}=\frac{F_{w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}

(10)

\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})

(11)

Equations (4.237) and (4.238) can be rewritten in a format similar to that of eqs. (4.225) – (4.228), i.e.,

a_{E}/D_{e}=\frac{\text{Pe}_{\Delta e}}{\exp (\text{Pe}_{\Delta e})-1}

(12)

a_{W}/D_{w}=\frac{\text{Pe}_{\Delta w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}

(13)

The comparison of aE / De for different schemes is shown in Fig. 1. It can be seen that the hybrid scheme can be viewed as an envelope of the exponential scheme. The hybrid scheme is a good approximation if the absolute value of the grid Peclet number is either very large or near zero.

Comparison among different schemes
Figure 1: Comparison among different schemes.

While the exponential scheme is accurate, the computational time is much longer than for the central difference, upwind or hybrid schemes. Patankar (1981) proposed a power law scheme that has almost the same accuracy as the exponential scheme but a substantially shorter computational time. The coefficient of the neighbor grid point on the east side can be obtained by


a_{E}/D_{e}=\left\{ \begin{matrix}
   -\text{Pe}_{\Delta e}\text{                        Pe}_{\Delta e}<-10  \\
   (1+0.1\text{Pe}_{\Delta e})^{5}-\text{Pe}_{\Delta e}\text{  }-1\text{0}\le \text{Pe}_{\Delta e}<0  \\
   (1-0.1\text{Pe}_{\Delta e})^{5}\text{             0}\le \text{Pe}_{\Delta e}\le 10  \\
   0\text{                            Pe}_{\Delta e}>10  \\
\end{matrix} \right.

which can be rewritten in the following compact form

a_{E}/D_{e}=\left[\!\left[ 0,\left( 1-0.1\left| \text{Pe}_{\Delta e} \right| \right)^{5} \right]\!\right]+\left[\!\left[ 0,-\text{Pe}_{\Delta e} \right]\!\right]

(14)