D Turbulence Equations

This appendix presents the development of the volume-averaged form of the turbulence energy, dissipation and Reynolds stress equations. The equation for the turbulence energy of the continuous phase in a fluid-

particle flow is derived using the same procedure as used for single phase flows. The procedure starts with the mechanical energy equation which is obtained by taking the vector product of the velocity with the momentum equation and can be expressed as (Equation 62 in Appendix C)

  (

 2 ) +

 

³  2

´ = −  + 

   +  (D.1)

This equation is volume averaged using the relationships developed in Appendix B for the spatial and temporal gradients. The velocities are then decomposed into the sum of a phase average and deviation,

 = hi+  Finally the product of the volume-averaged momentum equation, Equation 34 in Appendix C, and the volume-averaged velocity is subtracted from the volume-averaged mechanical energy equation to obtain an equation for the phase-averaged turbulence kinetic energy,

 = 1 Z 

 2

 = ¿

À (D.2)

The volume-averaged form of the rate of turbulence energy dissipation is

 =  ¿ 

À (D.3)

i

i

The analysis presented here is an extension of the temporal averaging approach for single phase flows (Bernard and Wallace, 2002) in which the dissipation is defined as1

 =   0

where 0 is the fluctuation velocity and the double overbar indicates time averaging. A transport equation for dissipation is developed by taking the time average of the product of the gradient of the Navier-Stokes equations, the gradient of the fluctuation velocity and twice the kinematic viscosityµ 2 0 

¶ . The notation  is the  component of the Navier-Stokes

equation. Arguments are presented for grouping terms together and modeling other terms. The same approach is used for volume averaging, starting with (2  ) and using similar arguments for term groupings and modeling. The volume averaged form of the incompressible Reynolds stress tensor is

 = hi (D.5) A transport equation for the Reynolds stress can be found by combining the products of the velocities and Navier-Stokes equations,  + , taking the volume average and then subtracting the corresponding form of the volume-averaged momentum equation. The analyses for the turbulence energy, dissipation and Reynolds stress

are based on the following assumptions:

1. Constant density fluid.