On the base of modern probability approach closed equation for probability density function of coordinates and velocities of two particles in turbulent flow is obtained. The system of equations for balance of mass, averaged velocities and intensities of turbulent chaotic motion of particles with account of correlated motion of particles are deduced. The closed expressions for intensity of relative chaotic motion between particles are obtained on the base of probability density function of particles displacement with correlation effects. Spectral presentation of second velocity moments of gas phase is used for calculation of intensity of particles relative chaotic motion. Boundary condition taking into account coefficients of new particle formation and momentum restitution during two particles collision is found. Formula for calculation of turbulent coagulation kernel of particles in gravity field is gain. Influence of cloud turbulence and turbulence in a pipe flow on intensity of droplets coagulation is studied. Strong effects of relative turbulent diffusion between droplets, droplets inertia and droplets gravitational settling on intensity of coagulation are found out. Connection between internal structure of turbulence type and coagulation rate is illustrated. The calculation results are compared with data of large eddy simulations. The results of calculation intensity of droplets relative motion in atmospheric conditions are presented.

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