Summary Our review of the characteristics of a number of two-dimensional turbulent

3.4.3 Summary Our review of the characteristics of a number of two-dimensional turbulent

flows revealed many common features. Turbulence is generated and main- tained by shear in the mean flow. Where shear is large the magnitudes of tur- bulence quantities such as the r.m.s. velocity fluctuations are high and their distribution is anisotropic with higher levels of fluctuations in the mean flow direction. Without shear, or an alternative agency to maintain it, turbulence decays and becomes more isotropic in the process. In spite of these common features, it was clear that, even in these relatively simple thin shear layers, the details of the turbulence structure are very much dependent on the flow itself. In regions close to solid walls the structure is dominated by shear due to wall friction and damping of turbulent velocity fluctuations perpendicular to the boundary. This results in a complex flow structure characterised by rapid changes in the mean and fluctuating velocity components concentrated within a very narrow region in the immediately vicinity of the wall. Since most engineering flows contain solid boundaries, the turbulence structure generated by them will be very geometry dependent. Engineering flow cal- culations must include sufficiently accurate and general descriptions of the turbulence that capture all the above effects and further interactions of turbulence and body forces.

3.5 The effect of turbulent

In this section we derive the flow equations governing the time-averaged fluctuations

properties of a turbulent flow, but before we do this we briefly examine the on properties of

physical basis of the effects resulting from the appearance of turbulent the mean flow

fluctuations.

In Figure 3.13 we consider a control volume in a two-dimensional turbu- lent shear flow parallel to the x-axis with a mean velocity gradient in the y-direction. The presence of vortical eddy motions creates strong mixing. Random currents that are associated with the passage of eddies near the boundaries of the control volume transport fluid across its boundaries. These recirculating fluid motions cannot create or destroy mass, but fluid parcels transported by the eddies will carry momentum and energy into and out of the control volume. Figure 3.13 shows that, because of the existence of the velocity gradient, fluctuations with a negative y-velocity will generally bring

Figure 3.13 Control volume within a two-dimensional turbulent shear flow

62 CHAPTER 3 TURBULENCE AND ITS MODELLING

fluid parcels with a higher x-momentum into the control volume across its top boundary and will also transport control volume fluid to a region of slower moving fluid across the bottom boundary. Similarly, positive y- velocity fluctuations will – on average – transport slower moving fluid into regions of higher velocity. The net result is momentum exchange due to

convective transport by the eddies, which causes the faster moving fluid layers to be decelerated and the slower moving layers to be accelerated. Consequently, the fluid layers experience additional turbulent shear stresses, which are known as the Reynolds stresses. In the presence of temperature

or concentration gradients the eddy motions will also generate turbulent

heat or species concentration fluxes across the control volume bound- aries. This discussion suggests that the equations for momentum and energy should be affected by the appearance of fluctuations.

Reynolds-averaged Navier---Stokes equations for incompressible flow

Next we examine the consequences of turbulent fluctuations for the mean flow equations for an incompressible flow with constant viscosity. These assumptions considerably simplify the algebra involved without detracting from the main messages. We begin by summarising the rules which govern time averages of fluctuating properties ϕ = Φ + ϕ′ and ψ = Ψ + ψ′ and their summation, derivatives and integrals:

∂s ∂s 冮 冮

ϕds = Φds (3.21)

ϕ+ψ =Φ+Ψ ϕψ = ΦΨ + ϕ′ψ ′ ϕΨ = ΦΨ ϕ′Ψ =0 These relationships can be easily verified by application of (3.2) and (3.3),

noting that the time-averaging operation is itself an integration. Thus, the order of time averaging and summation, further integration and/or differen-

tiation can be swapped or commuted, so this is called the commutative

property. Since div and grad are both differentiations, the above rules can be extended to a fluctuating vector quantity a = A + a′ and its combinations with a fluctuating scalar ϕ = Φ + ϕ′:

div a = div A; div( ϕa) = div( ϕa ) = div(ΦA) + div( ϕ′a′ ); div grad ϕ = div grad Φ

(3.22) To start with we consider the instantaneous continuity and Navier–Stokes

equations in a Cartesian co-ordinate system so that the velocity vector u has x-component u, y-component v and z-component w:

div u =0 (3.23) ∂u

1 ∂p

(3.24a) ∂t

+ div(uu) = −

+ ν div(grad(u))

(3.24b) ∂t

+ div(vu) = −

+ ν div(grad(v))

+ div(wu) = −

+ ν div(grad(w))

(3.24c)

3.5 EFFECT OF FLUCTUATIONS ON THE MEAN FLOW 63

This system of equations governs every turbulent flow, but we investigate the effects of fluctuations on the mean flow using the Reynolds decomposi- tion in equations (3.23) and (3.24a–c) and replace the flow variables u (hence

also u, v and w) and p by the sum of a mean and fluctuating component. Thus u = U + u′ u = U + u′ v = V + v′ w = W + w′ p = P + p′ Then the time average is taken, applying the rules stated in (3.21) – (3.22).

Considering the continuity equation (3.23), first we note that div u = div U. This yields the continuity equation for the mean flow:

div U =0 (3.25)

A similar process is now carried out on the x-momentum equation (3.24a). The time averages of the individual terms in this equation can be written as follows:

∂u ∂U =

div(u u) = div(UU) + div( u ′u′ ) ∂t ∂t

1 ∂P −

1 ∂p

=− ν div(grad(u)) = ν div(grad(U)) ρ ∂x

ρ ∂x