Determination of Critical Reynolds Numbers for Nonisothermal Flows by Using the Stochastic Theories of Turbulence and Equivalent Measures

Analytical Estimation of Velocity and Temperature Fields in a Circular Pipe on the Basis of Stochastic Equations and Equivalence of Measures

“…These well-known facts have not been explained theoretically up to now to the extent when, using a single theoretical formulation of the problem, it is possible to calculate a majority of turbulence parameters of interest for the practice and theory. The development of the DNS numerical methods has led to the necessity of "stochastization" of the Navier-Stokes equation or to its transformation to a form of the type of P. Langevin′s equation [8][9][10][11][12][13] by supplementing its right-hand side with a free term that determines perturbation, albeit without corresponding terms in the continuity and energy equations [16,[23][24][25][26][27][28][29][30].…”

“…Note that the stochastic equations, derived in [17][18][19], on their right-hand side include free terms of gradient and nongradient structures. For this purpose, the following space-time domains were determined in [17][18][19][20][21][22][23]: 1) the start of turbulence generation; 2) turbulence generation; 3) diffusion; 4) turbulence dissipation. With the use of the law of the equivalence of measures, each of the domains has its own system of stochastic equations within the framework of the general system of stochastic equations for mass, momentum, and energy.…”

“…With the use of the law of the equivalence of measures, each of the domains has its own system of stochastic equations within the framework of the general system of stochastic equations for mass, momentum, and energy. For a nonisothermal medium in domain 1, analytical expressions were presented in [21][22][23] for calculating the critical point and Reynolds number depending on the initial turbulence of fl ow in a pipe and on a fl at plate. In the present paper, analytical formulas for the indices of velocity and temperature profi les as functions of the parameters of initial turbulence and the Eckert and Prandtl numbers were obtained for domain 2 on the basis of the stochastic system of equations for energy, momentum, and mass.…”

“…The general system of stochastic conservation equations for isothermal and nonisothermal media was obtained in [17][18][19][20][21][22][23] …”

“…Here x 1 and x 2 are the coordinates directed along the wall and normally to it; Ec = (13), (19), and (20), i.e., the exponents n and n T can be determined. Just as in [21][22][23], according to the experimental conditions in [24][25][26][27][28] with Ec = -(0.1-0.010), Pr = 0.72, and Re st = 10-30, from (19) and (20) we have that n ~ 7 and from (13) that n T ~ 8.…”