Notes on Computational Fluid Dynamics: General Principles

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7.2 Initialisation of the k-epsilon model

Initial values and boundary conditions must be speciﬁed for $\text{[math]}$ and $\text{[math]}$ to solve their respective transport equations. The ideal speciﬁcation of boundary conditions for $\text{[math]}$ and $\text{[math]}$ follows those for $\text{[math]}$ described in Sec. 4.3 .

$PICT\relax \special {t4ht=$

Turbulence ﬁelds require: a ﬁxed value condition at inlets; zero gradient or inlet-outlet at outlets; and, a more complex speciﬁcation at solid walls, introduced in Sec. 7.7 .

Inlet values $\text{[math]}$ and $\text{[math]}$ must therefore be speciﬁed. There may be industry standards, published recommendations or measured data to help select these values for the speciﬁc problem being simulated.

But more often than not, $\text{[math]}$ and $\text{[math]}$ must be estimated. Inlet and initial estimates of $\text{[math]}$ are usually based on a Prandtl mixing length $\text{[math]}$ from the expression

(7.4)

This relation can be derived by considering a planar boundary layer. In the “log law” region (see Sec. 7.7 ) $\text{[math]}$ which can be combined with the mixing length Eq. (6.21 ) to give

(7.5)

Substituting $\text{[math]}$ from Eq. (6.31 ) yields Eq. (7.4).

A value for $\text{[math]}$ must then be speciﬁed in order to calculate inlet and initial values of $\text{[math]}$ from Eq. (7.4 ). Procedures to estimate $\text{[math]}$ are described in Sec. 7.3 .

Inlet and initial estimates for $\text{[math]}$ can be calculated by

kin = 3-(jujI)2; 2 \relax \special {t4ht=

(7.6)

from a speciﬁed turbulent intensity $\text{[math]}$ , the ratio of the root-mean-square (RMS) of turbulent ﬂuctuations $\text{[math]}$ to the mean ﬂow speed $\text{[math]}$ . The expression is derived from the deﬁnition $\text{[math]}$ .

A value for $\text{[math]}$ must then be speciﬁed in order to calculate the inlet and initial values of $\text{[math]}$ from Eq. (7.6 ). Procedures to estimate $\text{[math]}$ are also described in Sec. 7.3 . The values of $\text{[math]}$ and $\text{[math]}$ at inlet boundaries inﬂuence the solution throughout the CFD simulation, so should be estimated as accurately as possible.

The accuracy of the initial (internal) values is not so critical, since they do not inﬂuence the solution beyond a short period at the beginning of a simulation.

Initial values can, however, aﬀect stability during the early steps of a CFD simulation. The ﬂow boundary conditions generally cause sudden impulses which can generate large forces, causing ﬂuctuations in the solution of $\text{[math]}$ . Higher $\text{[math]}$ , based on initial $\text{[math]}$ and $\text{[math]}$ values, tends to cause larger ﬂuctuations, which may make the solution of the $\text{[math]}$ -equation unstable.

Notes on CFD: General Principles - 7.2 Initialisation of the k-epsilon model