Derivation of Governing Equations for Turbulent Flow

For incompressible flows, the Navier-Stokes equations can be rearranged in the form (33.1a) (33.1b) (33.1c) and (33.2)

Express the velocity components and pressure in terms of time-mean values and corresponding fluctuations. In continuity equation, this substitution and subsequent time averaging will lead to or,                          Since,                      We can write                (33.3a) From Eqs (33.3a) and (33.2), we obtain (33.3b)

It is evident that the time-averaged velocity components and the fluctuating velocity components, each satisfy the continuity equation for incompressible flow.  Imagine a two-dimensional flow in which the turbulent components are independent of the z -direction. Eventually, Eq.(33.3b) tends to (33.4) On the basis of condition (33.4), it is postulated that if at an instant there is an increase in u' in the x -direction, it will be followed by an increase in v' in the negative y -direction. In other words,  is non-zero and negative. (see Figure 33.2) Fig 33.2 Each dot represents uν pair at an instant  Invoking the concepts of eqn. (32.8) into the equations of motion eqn (33.1 a, b, c), we obtain expressions in terms of mean and fluctuating components. Now, forming time averages and considering the rules of averaging we discern the following. The terms which are linear, such as  and  vanish when they are averaged [from (32.6)]. The same is true for the mixed terms like  , or  , but the quadratic terms in the fluctuating components remain in the equations. After averaging, they form  ,  etc. If we perform the aforesaid exercise on the x-momentum equation, we obtain using rules of time averages, We obtain Introducing simplifications arising out of continuity Eq. (33.3a), we shall obtain. Performing a similar treatment on y and z momentum equations, finally we obtain the momentum equations in the form. In x direction,       (33.5a) In y direction,     (33.5b) In z direction,   (33.5c) Comments on the governing equation :  The left hand side of Eqs (33.5a)-(33.5c) are essentially similar to the steady-state Navier-Stokes equations if the velocity components u, v and w are replaced by ,  and .  The same argument holds good for the first two terms on the right hand side of Eqs (33.5a)-(33.5c).  However, the equations contain some additional terms which depend on turbulent fluctuations of the stream. These additional terms can be interpreted as components of a stress tensor. Now, the resultant surface force per unit area due to these terms may be considered as In x direction,            (33.6a) In y direction,  (33.6b) In z direction,        (33.6c) Comparing Eqs (33.5) and (33.6), we can write            (33.7) It can be said that the mean velocity components of turbulent flow satisfy the same Navier-Stokes equations of laminar flow. However, for the turbulent flow, the laminar stresses must be increased by additional stresses which are given by the stress tensor (33.7). These additional stresses are known as apparent stresses of turbulent flow or Reynolds stresses . Since turbulence is considered as eddying motion and the aforesaid additional stresses are added to the viscous stresses due to mean motion in order to explain the complete stress field, it is often said that the apparent stresses are caused by eddy viscosity . The total stresses are now (33.8) and so on. The apparent stresses are much larger than the viscous components, and the viscous stresses can even be dropped in many actual calculations