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CHMLTECH 501 Computational Fluid Dynamics

MacCormacks Method

MacCormacks method, first introduced in 1969. It is the most popular explicit finite-difference method for solving fluid flows. It is closely related to the Lax-Wendroff method,but is easier to apply. Let us use the same nozzle problem, figure (1).

Figure 1: Subsonic-supersonic isentropic flow through a convergent-divergent nozzle.

MacCormacks method, like the Lax-Wendroff method, is based on a Taylors series

expansion in time. Let us consider the density at grid point i,

t+ti =ti+

t

ave

t (1)

Equation (1) is a truncated Taylors series that looks first-order accurater, however,t

ave

is an average time derivative taken between time t and t+ t. this derivative is evaluatedin such a fashion that the calculation oft+ti from Equation (1) becomes second-order ac-curate. The average time derivative inEquation(1) is evaluated from a predictor-correctorphilosophy as follows.

Predictor stepContinuity equation,

t =

1

Au

A

x u

x

u

x (2)

In Equation (2), calculate the spatial derivatives from the known flow field values attime t using forward differences. That is, from Equation (2),

t

ti

= 1

Atiu

ti

Ai+1 Ai

x

uti

ti+1

ti

x

ti

uti+1 u

ti

x

(3)

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Obtain a predictedvalue of density, t+ti , from the first two terms of a Taylors series, asfollows

t+ti =ti+

t

ti

t (4)

In Equation (4), ti is known, andt

t

iis a known number from Equation (3); hence,

t+ti is readily obtained. ina similar fashion, from the momentum and energy eqautions,predicted values of the other flow variables such as ut+ti , e

t+ti , etc. are obtained.

Corrector step

Here, we first obtain a predicted value of the time derivative,t

t+ti

, by substituting the

predictedvalues ofut+ti , t+ti , etc. into Equation (2), usingrearward differences.

t

t+ti

=

1At+ti u

t+ti

Ai

Ai1

x

ut+ti

t+ti

t+ti1

x

t+ti

ut+ti u

t+ti1

x

(5)

Now calculate the average time derivative as the arithmetic mean between Equation (3)and (5), i.e.

t

ave

=12

t

t

i

+t

t+

t

i

(6)

where numbers for the two terms on the right-hand side of Equation (6) come from Equation(3) and (5) respectively. Finally, we obtain the correctedvalue oft+ti from Equation (1),repeated below:

t+ti =ti+

t

ave

t (7)

The above predictor-corrector approach is carried out for all grid points throughout thenozzle, and is applied simultaneously to the momentum and energy equations in order to

generateut+ti ,et+ti . In this fashion, the flow field through the entire nozzle at timet+ tis calculated. This is repeated for a large number of time steps until the steady state isachieved.

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