Numerics

The numerical approximation of the turbulence equations is in principle carried out as explained in section 3.1.2. One basic difference is however due to the fact that turbulent quantities are generally non-negative such that it is necessary that the discretised forms of the physical equations retain the principle of non-negativity. A typical model problem would be the following:

$$\dfrac{\partial {X}}{\partial {t}} =P-QX, \quad P,Q > 0$$ (95)

with $X$ denoting any non-negative quantity, $P$ a non-negative source term, $QX$ a non-negative linear sink term, and $t$ denoting time. $P$ and $Q$ may depend on $X$ and $t$. It can easily be shown that with (95), $X$ remains non-negative for any non-negative initial value $X_0$ and limited $Q$. For the $q^2 l$-equation and the $\epsilon$-equation (described in section 4.7.27 and section 4.7.28), $Q$ would be proportional to $q/l$ and $\epsilon /k¨$, repsectively.

A straight-forward, explicit discretisation in time of (95) can be written as

$$\frac{X^{n+1}-X^n}{\Delta t}=P^n-Q^nX^n$$ (96)

with the superscripts denoting the old ($n$) and the new ($n+1$) time level and $\Delta t$ denoting the time step. In this case, the numerical solution on the new time level would be

$$X^{n+1}_i= X^n_i(1-\Delta tQ^n_i)+\Delta t P_i^n \; , \quad$$ (97)

which is negative for negative right hand side of (96), provided that

$$\Delta t > \frac{X^n}{X^nQ^n-P^n} \quad .$$ (98)

Since it is computationally unreasonable to restrict the time step in such a way that (98) is avoided, a numerical procedure first published by Patankar (1980) is generally applied

$$\frac{X^{n+1}-X^n}{\Delta t}=P^n-Q^nX^{n+1} \; , \quad$$ (99)

which yields an always non-negative solution for $X^{n+1}$,

$$X^{n+1}= \frac{X^n+\Delta t P^n}{1+\Delta t Q^n} \quad .$$ (100)

Thus, the so-called quasi-implicit formulation (99) by Patankar (1980) is a sufficient condition for positivity applied in almost all numerical turbulence models.