Free turbulent flows Among the simplest flows of significant engineering importance are those in

3.4.1 Free turbulent flows Among the simplest flows of significant engineering importance are those in

the category of free turbulent flows: mixing layers, jets and wakes. A mixing layer forms at the interface of two regions: one with fast and the other with slow moving fluid. In a jet a region of high-speed flow is completely sur- rounded by stationary fluid. A wake is formed behind an object in a flow, so here a slow moving region is surrounded by fast moving fluid. Figure 3.8 is

a sketch of the development of the mean velocity distribution in the stream- wise direction for these free turbulent flows.

Figure 3.8 Free turbulent flows It is clear that velocity changes across an initially thin layer are important

in all three flows. Transition to turbulence occurs after a very short distance in the flow direction from the point where the different streams initially meet; the turbulence causes vigorous mixing of adjacent fluid layers and rapid widening of the region across which the velocity changes take place.

Figure 3.9 shows a visualisation of a jet flow. It is immediately clear that the turbulent part of the flow contains a wide range of length scales. Large eddies with a size comparable to the width across the flow are occurring alongside eddies of very small size.

The visualisation correctly suggests that the flow inside the jet region is fully turbulent, but the flow in the outer region far away from the jet is smooth and largely unaffected by the turbulence. The position of the edge of the turbulent zone is determined by the (time-dependent) passage of indi-

54 CHAPTER 3 TURBULENCE AND ITS MODELLING

Figure 3.9 Visualisation of a jet flow Source : Van Dyke (1982)

3.4 CHARACTERISTICS OF SIMPLE TURBULENT FLOWS 55

the surrounding region. During the resulting bursts of turbulent activity in the outer region – called intermittency – fluid from the surroundings is drawn into the turbulent zone. This process is termed entrainment and is

the main cause of the spreading of turbulent flows (including wall boundary layers) in the flow direction.

Initially fast moving jet fluid will lose momentum to speed up the sta- tionary surrounding fluid. Due to the entrainment of surrounding fluid the velocity gradients decrease in magnitude in the flow direction. This causes the decrease of the mean speed of the jet at its centreline. Similarly the dif- ference between the speed of the wake fluid and its fast moving surroundings will decrease in the flow direction. In mixing layers the width of the layer containing the velocity change continues to increase in the flow direction but the overall velocity difference between the two outer regions is unaltered.

Experimental observations of many such turbulent flows show that after

a certain distance their structure becomes independent of the exact nature of the flow source. Only the local environment appears to control the turbu- lence in the flow. The appropriate length scale is the cross-stream layer width (or half width) b. We find that if y is the distance in the cross-stream direction

B E =g B E =h B E (3.14) max −U min

C b F for mixing layers

C b F U max

C b F U max −U min

for wakes In these formulae U max and U min represent the maximum and minimum

for jets

mean velocity at a distance x downstream of the source (see Figure 3.8). Hence, if these local mean velocity scales are chosen and x is large enough, the functions f, g and h are independent of distance x in the flow direction.

Such flows are called self-preserving. The turbulence structure also reaches a self-preserving state, albeit after

a greater distance from the flow source than the mean velocity. Then u ′ 2 A y D v ′ 2 A y D w ′ 2 A y D u ′v′

2 =f U 1 B E 2 =f b 2 U B b E U 2 =f 3 B E 2 =f 4 B E (3.15)

C b F The velocity scale U ref is, as above, (U max −U min ) for a mixing layer and wakes

and U max for jets. The precise form of functions f, g, h and f i varies from flow to flow. Figure 3.10 gives mean velocity and turbulence data for a mixing layer (Champagne et al., 1976), a jet (Gutmark and Wygnanski, 1976) and a wake flow (Wygnanski et al., 1986).

The largest values of u ′ 2 , v ′ 2 , w ′ 2 and − u ′v′ are found in the region where the mean velocity gradient ∂U/∂y is largest, highlighting the intimate con- nection between turbulence production and sheared mean flows. In the flows shown above the component u ′ gives the largest of the normal stresses; its r.m.s. value has a maximum of 15 – 40% of the local maximum mean flow velocity. The fact that the fluctuating velocities are not equal implies an anisotropic structure of the turbulence.

As |y/b| increases above unity the mean velocity gradients and the veloc- ity fluctuations all tend to zero. It should also be noted that the turbulence

56 CHAPTER 3 TURBULENCE AND ITS MODELLING

Figure 3.10 Distribution of mean velocity and second

moments , , and u ′ 2 v ′ 2 w ′ 2 − u ′v′ for incompressible mixing layer, jet and wake

3.4 CHARACTERISTICS OF SIMPLE TURBULENT FLOWS 57

properties become more isotropic. The absence of shear means that turbu- lence cannot be sustained in this region.

The mean velocity gradient is also zero at the centreline of jets and wakes and hence no turbulence is produced here. Nevertheless, the values of u ′ 2 , v ′ 2 and w ′ 2 do not decrease very much because vigorous eddy mixing trans- ports turbulent fluid from nearby regions of high turbulence production towards and across the centreline. The value of − u ′v′ has to become zero at the centreline of jet and wake flows since it must change sign here by symmetry.