tailieunhanh - KUNDU Fluid Mechanics 2 Episode 13

Tham khảo tài liệu 'kundu fluid mechanics 2 episode 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 514 Turbulence. and any tensor a Ui dxj is equal to the product of ũĩũj and symmetric part of a Ui namely Eij this is proved in Chapter 2 Section 11. If the mean flow is given by u y then ũịĩT àUi dxj uv du dy . We saw in the preceding section that MV is likely to be negative if du dy is positive. The fifth term MiMj ai 3xj is therefore likely to be negative in shear fiows. By analogy with the fourth term it must represent an energy loss to the agency that generates turbulent stress namely the fluctuating field. Indeed we shall see in the following section that this term appears on the right-hand side of an equation for the rate of change of turbulent kinetic energy but with the sign reversed. Therefore this term generally results in a loss of mean kinetic energy and a gain of turbulent kinetic energy. We shall call this term the shear production of turbulence by the interaction of Reynolds stresses and the mean shear. The sixth term represents the work done by gravity on the mean vertical motion. For example an upward mean motion results in a loss of mean kinetic energy which is accompanied by an increase in the potential energy of the mean field. The two viscous terms in Eq. namely the viscous transport 2v3 i i Etj dXj and the viscous dissipation 2vEijEij are small in a fully turbulent flow at high Reynolds numbers. Compare for example the viscous dissipation and the shear production terms 2vEij v U L 2 V 1 ũĩũ dUi dXj Úị u L UL where u is the scale for mean velocity L is a length scale for example the width of the boundary layer and Hrms is the rms value of the turbulent fluctuation we have also assumed that Mrms and u are of the same order since experiments show that Mrms is a substantial fraction of u. The direct influence of viscous terms is therefore negligible on the mean kinetic energy budget. We shall see in the following section that this is not true for the turbulent kinetic energy budget in which the viscous terms play a major role. What happens