# Realisability and Schwarz' inequality

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Realisability is the minimum requirement to prevent a turbulence model generating non-physical results. For a model to be realisable the normal Reynolds stresses must be non-negative and the Schwarz' inequality must be satisfied between fluctuating quantities:

$\left\langle{u^'_\alpha u^'_\alpha}\right\rangle \geq 0$
$\frac{\left\langle{u^'_\alpha u^'_\beta}\right\rangle}{ \left\langle{u^'_\alpha u^'_\beta}\right\rangle \left\langle{u^'_\alpha u^'_\beta}\right\rangle} \leq 1$

where there is no summation over the indices. Some workers only apply the first inequality to satisfy realisability, or maintain non-negative vales of k and epsilon. This "weak" form of realisability is satisfied in non-linear models by setting $C_\mu=0.09$.

## References

Speziale, C.G. (1991), "Analytical methods for the development of Reynolds-stress closures in turbulence", Ann. Rev. Fluid Mechanics, Vol. 23, pp107-157.