An estimation of the velocity profile for the laminar-turbulent transition in the plane jet on the basis of the theory of stochastic equations and equivalence of measures

Abstract

The theory of stochastic equations and the theory of equivalence of measures previously applied to flows in the boundary layer and in the pipe are considered to calculate the velocity profile of the flat jet. This theory previously made it possible to determine the critical Reynolds number and the critical point for the flow of the plane jet. Here based on these results the analytical dependence for the index of the velocity profile is derived. Velocity profiles are calculated for a laminar-turbulent transition in the jet. This formula reliably reflects an increase of the energy transferred from a deterministic state to a random one with an increase of the index of the velocity profile. Results show satisfactory agreement with the known experimental data for the velocity profile of the flat jet. Using obtained results it is possible to determine the location of technical devices for laminarization of the flow in the jet. This is important both for reducing friction in the flow around aerodynamic vehicles and for maintaining the jet profile if it is necessary to ensure the stability of the flow characteristics. Also the obtained relations can be useful for researching of the processes in combustion chambers, in the case of welding and in other technical devices.

Stochastic Equations of Hydrodynamic Theory of Plasma

“…Here, on the basis of stochastic equations and equivalence relations of measures [71][72][73] developed for the continuum, the new results for turbulent plasma motion are presented. The correlator D N,M was derived in [71][72][73][74][99][100][101][102][103][104]; see also [105,108] as the definition of equivalence of measures between laminar (deterministic) and turbulent (random) motions. The application of the correlator D N,M for solving equations for mass, motion, and energy leads to sets of stochastic equations for four space-time areas: (1) the onset of generation (subscript 1, 0, or 1); (2) the generation (subscript 1, 1); (3) the diffusion (subscript 1, 1, 1) and ( 4) the dissipation of the turbulent fields.…”

“…In the critical point r i → r c ; ∆τ i → τ c for the parameter m i → m c , and for each of four space-time regions of the correlator. The correlator D M,N r c ; m cj ; τ c = D 1,0 r c ; m cj ; τ c [72][73][74][75][100][101][102][103][104][105] for the pair (M,N) = (1,0) gives the following equations…”

“…In accordance with [72][73][74][75][100][101][102][103][104][105][106][107][108][109], for space-time area (1) of the onset of generation for a pair (N, M)(1, 0)r c0 (x c + ∆x 0 , τ c + ∆τ 0 ) − r c , the set of Equations ( 1)-( 3) can be written as follows (see also [105,108]):…”

“…The correlation dimension of the attractor in the boundary layer was calculated in [99,100]. The equation for the spectral function was shown in [101][102][103][104][105][106]. For non-isothermal flows, the Reynolds analogy was obtained in [104].…”

“…The equation for the spectral function was shown in [101][102][103][104][105][106]. For non-isothermal flows, the Reynolds analogy was obtained in [104]. Then the formulas for friction coefficients on the wall of a flat plate and in a round tube during a laminar-turbulent transition [107,108] were derived.…”