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^{1}, I. Goldman

^{1}and J. Chasnov

^{1}

### Abstract

A model for stationary, fully developed turbulence is presented. The physical model used to describe the nonlinear interactions provides an equation for the turbulent spectral energy function *F*(*k*) as a function of the time scale for the energy fed into the system, *n* ^{−1} _{ s }. The model makes quantitative predictions that are compared with the following available data of a different nature. (a) For *t* *u* *r* *b* *u* *l* *e* *n* *t* *c* *o* *n* *v* *e* *c* *t* *i* *o* *n*, in the case of a constant superadiabatic gradient and for σ≪1 (σ≡Prandtl number), the convective flux is computed and compared with the result of the mixing length theory (MLT). For the case of a variable superadiabatic gradient, and for arbitrary σ, as in the case of laboratory convection, the Nusselt number *N* versus Rayleigh number *R* relation is found to be *N*=*A* _{σ} *R* ^{1} ^{/} ^{3} as recently determined experimentally. The computed *A* _{σ} deviates 3% and 8% from recent laboratory data at high *R* for σ=6.6 and σ=2000. (b) The *K*–ε *a* *n* *d* *S* *m* *a* *g* *o* *r* *i* *n* *s* *k* *y* *r* *e* *l* *a* *t* *i* *o* *n* *s*. Four alternative expressions for the turbulent (eddy) viscosity are derived (the *K*–ε and Smagorinsky relations being two of them) and the numerical coefficients appearing in them are computed. They compare favorably with theoretical estimates (the direct interaction approximation and the renormalization group method), laboratory

data, and simulation studies. (c) The *s* *p* *e* *c* *t* *r* *a* *l* *f* *u* *n* *c* *t* *i* *o* *n*, *t* *r* *a* *n* *s* *f* *e* *r* *t* *e* *r* *m*, *a* *n* *d* *d* *i* *s* *s* *i* *p* *a* *t* *i* *o* *n* *t* *e* *r* *m*. The spectral energy function *F*(*k*), the transfer term *T*(*k*), and the dissipation term ν*k* ^{2} *F*(*k*) are computed and compared with

laboratory data on grid turbulence. (d) The *s* *k* *e* *w* *n* *e* *s* *s* *f* *a* *c* *t* *o* *r* *S*̄_{3} is computed and compared with laboratory data. The turbulence model is extended to treat temperature fluctuations characterized by a spectral function *G*(*k*). The main results are (e) when both temperature and velocity fluctuations are taken into account, the rate *n* _{ s }(*k*), that in the first part was taken to be given by the linear mode analysis, can be determined self‐consistently from the model itself; (f) in the *i* *n* *e* *r* *t* *i* *a* *l*‐*c* *o* *n* *v* *e* *c* *t* *i* *v* *e* *r* *a* *n* *g* *e*, the model predicts the well‐known result *G*(*k*)∼*k* ^{−} ^{5} ^{/} ^{3}; (g) the *K* *o* *l* *m* *o* *g* *o* *r* *o* *v* *a* *n* *d* *B* *a* *t* *c* *h* *e* *l* *o* *r* *c* *o* *n* *s* *t* *a* *n* *t* *s* are shown to be related by Ba=σ_{ t } Ko, where σ_{ t } is the turbulent Prandtl number; and (h) in the *i* *n* *e* *r* *t* *i* *a* *l*‐*c* *o* *n* *d* *u* *c* *t* *i* *v* *e* *r* *a* *n* *g* *e* the model predicts *G*(*k*)∼*k* ^{−} ^{1} ^{7} ^{/} ^{3} for thermally driven convection as well as for advection of a passive scalar, the difference being contained in the numerical coefficient in front. The predicted *G*(*k*) vs *k* compare favorably with experiments for air (σ=0.725), mercury (σ=0.018), and salt water (σ=9.2).

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