  We investigate numerically the model proposed in Sahoo et al [Phys. Rev.
Lett. 118, 164501, (2017)] where a parameter $\lambda$ is introduced in the
Navier-Stokes equations such that the weight of homochiral to heterochiral
interactions is varied while preserving all original scaling symmetries and
inviscid invariants. Decreasing the value of $\lambda$ leads to a change in the
direction of the energy cascade at a critical value $\lambda_c \sim 0.3$. In
this work, we perform numerical simulations at varying $\lambda$ in the forward
energy cascade range and at changing the Reynolds number $\mathrm{Re}$. We show
that for a fixed injection rate, as $\lambda \to \lambda_c$, the kinetic energy
diverges with a scaling law $\mathcal{E} \propto (\lambda-\lambda_c)^{-2/3}$.
The energy spectrum is shown to display a larger bottleneck as $\lambda$ is
decreased. The forward heterochiral flux and the inverse homochiral flux both
increase in amplitude as $\lambda_c$ is approached while keeping their
difference fixed and equal to the injection rate. As a result, very close to
$\lambda_c$ a stationary state is reached where the two opposite fluxes are of
much higher amplitude than the mean flux and large fluctuations are observed.
Furthermore, we show that intermittency as $\lambda_c$ is approached is
reduced. The possibility of obtaining a statistical description of regular
Navier-Stokes turbulence as an expansion around this newly found critical point
is discussed.
