In this investigation, the transports of kinetic energy and scalar variance in the turbulence driven by a multiscale force $\mathit{M}{\nabla }^{\beta }{s}^{\prime }$ (which is associated with scalar fluctuations $s^{\prime }$) are theoretically studied. Although the velocity and scalar fields are strongly coupled, a universal flux conservation equation has been established in wave-number space and exhibits three different solutions, including one real solution and two complex solutions. The numerical analyses show that the turbulence generated under the multiscale force can possess four different cascade processes, including inertial subrange (constant fluxes of kinetic energy and scalar variance), constant-${\mathrm{\Pi }}_u$ subrange (quasiconstant flux of kinetic energy), constant-${\mathrm{\Pi }}_s$ subrange (quasiconstant flux of scalar variance), and a new subrange with both nonconstant fluxes of kinetic energy and scalar variance in addition to the dissipation subrange. Therefore, much richer information on the cascade processes and scaling exponents can be predicted in such a turbulent system, relative to the well-known Richardson cascade in conventional hydrodynamic turbulence. $\beta$ is a key to controlling the cascade processes and the scaling exponents. In the constant-${\mathrm{\Pi }}_u$ subrange, the slope of velocity spectrum is always ${\xi }_u=-5/3$, while the slope of the scalar spectrum is ${\xi }_s=-\left(6\beta +1\right)/3$. In the constant-${\mathrm{\Pi }}_s$ subrange, ${\xi }_u=\left(4\beta -11\right)/5$ and ${\xi }_s=-\left(2\beta +7\right)/5$. Relying on $\beta$, for the real solution, the transport of kinetic energy and scalar variance can be distinguished as four cases. (1) When $\beta <3/2$ (except $\beta =2/3$), the constant-${\mathrm{\Pi }}_u$ and constant-${\mathrm{\Pi }}_s$ subranges could be coexisted, with the former located on the lower wave-number side of the latter. At $\beta =2/3$, a new inertial subrange with both ${\xi }_u$ and ${\xi }_s$ equal to −5/3 is present in the multiscale-force dominated subrange. (2) When $3/2\le \beta <2$, only the constant-${\mathrm{\Pi }}_u$ subrange is predicted. (3) At $\beta =2$, special and singular exponents of ${\xi }_u=-1$, ${\xi }_s=-3$, ${\lambda }_u=1$ (the slope of kinetic energy flux), and ${\lambda }_s=-1$ (the slope of scalar variance flux) can be found, if $MN=1$. Otherwise, a constant-${\mathrm{\Pi }}_s$ subrange is predicted. (4) When $2<\beta \le 4$, only the constant-${\mathrm{\Pi }}_s$ subrange is predicted. For the second solution (one complex solution), when $\beta <3/2$, there are no constant-${\mathrm{\Pi }}_u$ and constant-${\mathrm{\Pi }}_s$ subranges, while for $\beta \ge 3/2$, a single constant-${\mathrm{\Pi }}_u$ subrange is predicted. For the third solution (the other complex solution), there is always a constant-${\mathrm{\Pi }}_s$ subrange. Finally, a complete transport picture of both kinetic energy and scalar variance has been established for the type of forced turbulence, which may unify hydrodynamic turbulence, stratified turbulence, turbulent thermal convection and electrokinetic turbulence, etc.

• Received 25 May 2022
• Accepted 8 August 2022

DOI:https://doi.org/10.1103/PhysRevFluids.7.084607