# Introduction to turbulence/Free turbulent shear flows

## Introduction

Free shear flows are inhomogeneous flows with mean velocity gradients that develop in the absence of boundaries. Turbulent free shear flows are commonly found in natural and engineering environments. The jet of of air issuing from one's nostrils or mouth upon exhaling, the turbulent plume from a smoldering cigarette, and the buoyant jet issuing from an erupting volcano - all illustrate both the omnipresence of free turbulent shear flows and the range of scales of such flows in the natural environment. Examples of the multitude of engineering free shear flows are the wakes behind moving bodies and the exhausts from jet engines. Most combustion processes and many mixing processes involve turbulent free shear flows.

Free shear flows in the real world are most often turbulent. Even if generated as laminar flows, they tend to become turbulent much more rapidly than the wall-bounded flows which we will discuss later. This is because the three-dimensional vorticity necessary for the transition to turbulence can develop much more rapidly in the absence of walls that inhibit the qrowth velocity components normal to them.

The tendency of free shear flows to become and remain turbulent can be greatly modified by the presence of density gradients in the flow, especially if gravitational effects are also important. Why this is the case can easily be seen by examining the vorticity equation for such flows in the absence of viscosity,

 $\left[ \frac{ \partial \omega_{i} }{ \partial t } + \tilde{u_{j}} \frac{ \partial \tilde{\omega_{i} } }{ \partial x_{j}} \right] = \tilde{\omega_{j} } \frac{ \partial \tilde{u_{i}} }{ \partial x_{j} } + \epsilon_{ijk} \frac{ \partial \tilde{ \rho } }{ \partial x_{j} } \frac{ \partial \tilde{p} }{ \partial {x_{k} } }$ (1)

The last term can act to either increase or decrease vorticity production but only in non-barotropic flows. (Recall that a barotropic flow is one in which the gradients of density and pressure are co-linear, because the density is a function of the pressure only). For example, in the vertically-oriented buoyant plume generated by exhausting a lighter fluid into heavier one, the principal density gradient is across the flow and thus perpendicular to the gravitational force which is the principal contributor to the pressure gradient. As a consequence the turbulent buoyant plume develops much more quickly than its uniform density counterpart, the jet. On the other hand, horizontal free shear flows in a stably stratified environment (fluid density decreases with height) can be quickly suppressed since the density and pressure gradients are in opposite directions.

Free turbulent shear flows are distinctly different from the homogeneous shear flows. In a free turbulent shear flow, the vortical fluid is patially confined and is separated from the surrounding fluid by an interface, the turbulent-nonturbulent interface (also known as the ”Corrsin superlayer” after itself discoverer). The turbulent/non-turbulent interface has a thickness which is characterized by the Kolmogorov microscale, thus its characterization as an interface is appropriate. The actual shape of the interface is random and it is severely distorted by the energetic turbulent processes which take place below it, with the result that at any given location the turbulence can be highly intermittent. This means that at a given location, it is sometimes turbulent, sometimes not.

It should not be inferred from the above that the non-turbulent fluid outside the superlayer is quiescent. Quite the opposite is true since the motion of the fluid at the interface produces motions in the surrounding stream just as would the motions of a solid wall. Alternately, the flow outside the interface can be viewed as being “induced” by the vortical motions beneath it. It is easy to show that these induced motions are irrotational. Thus since these random motions of the outer flow have no vorticity, they can not be considered turbulent.

Figure 7.1 shows records of the velocity versus time at a number of locations in the mixing layer of a round jet. When turbulent fluid passes the probes, the velocity signals are characterized by bursts of activity. The smooth undulations between the bursts are the irrotational fluctuations induced by the turbulent vorticity on the other side of the interface. Note that near the center of the mixing layer where the shear is a maximum, the flow is nearly always turbulent while it becomes increasingly intermittent as one proceeds away from the region of maximum production of turbulence energy. This increasing intermittency toward the outer edge is a characteristic of all free shear flows, and is an indication of the fact that the turbulent/non-turbulent interface is constantly changing its position.

One of the most important features of free shear flows is that the amount of fluid which is turbulent is continuously increased by a process known as entrainment. No matter how little fluid is in the flow initially, the turbulent part of the flow will continue to capture new fluid by entrainment as it evolves. The photograph of an air jet in Figure 1.2 illustrates this phenomenon dramatically. The mass flow of the jet increases at each cross-section due to entrainment. Entrainment is not unique to turbulent flows, but is also an important haracteristic of laminar flow, even though the actual mechanism of entrainment is quite different.

There are several consequences of entrainment. The first and most obvious is that free shear flows continue to spread throughout their lifetime. (That such is the case for the air jet of Figure 7.1 is obvious). A second consequence of entrainment is that the fluid in the flow is being continuously diluted by the addition of fluid from outside it. This is the basis of many mixing processes, and without such entrainment our lives would be quite different. A third consequence is that it will never be possible to neglect the turbulent transport terms in the dynamical equations, at least in the directions in which the flow is spreading. This is because the dilution process has ensured that the flow can never reach homogeneity since it will continue to entrain and spread through its lifetime (Recall that the transport terms were identically zero in homogeneous flows). Thus in dealing with free shear flows, all of the types of terms encountered in the turbulence kinetic energy equation of Chapter 4 must be dealt with — advection, dissipation, production, and turbulent transport.

Turbulent free shear flows have another distinctive feature in that they very often give rise to easily recognizable large scale structures or eddies. Figure 1.2 also illustrates this phenomenon, and coherent patterns of a scale equal to the lateral extent of the flow are clearly visible. These large eddies appear to control the shape of the turbulent/non-turbulent interface and play an important role in the entrainment process. They may also be important to the processes by which the turbulence gains and distributes energy from the mean flow.

A feature which free shear flows have in common with the homogeneous flows discussed in Chapter 6 is that their scales continue to grow as long as the flow remains turbulent. The dynamical equations and boundary conditions for many free shear flows can be shown to admit to similarity solutions in which the number of independent variables is reduced by one. According to the equilibrium similarity principle set forth in this chapter, such flows might be expected to asymptotically achieve such a state; and this, in fact, occurs. In the limit of infinite Reynolds number, some such flows can even be characterized by a single time and length scale, thus satisfying the conditions under which the simplest closure models might be expected to work. Care must be taken not to infer too much from the ability of a given closure model to predict such a flow, since any model which has the proper scale relations should work.

Finally there is the important question of whether free shear flows become asymptotically independent of their initial conditions (or source conditions). The conventional wisdom until very recently has been that they do. If correct, this means that there is nothing that could be done to alter the far downstream flow. There is recent theoretical and experimental evidence, however, that this traditional view may well be wrong. If so, this opens up previously un-imagined possibilities for flow control at the source.

In the remainder of this chapter, the averaged equations of motion will be simplified, and similarity solutions for several ideal shear flows will be derived and discussed in detail. The role of the large eddies will be discussed, and mechanisms for turbulent entrainment will be examined. The energy balance of several turbulent free shear flows will be studied in some detail. Finally, the effects of confinement and non-homogeneous external boundary conditions will be considered.

## The averaged equations

### The shear layer equations

One of the most important ideas in the history of fluid mechanics is that of the boundary layer approximation. These approximations to the Navier-Stokes equations were originally proposed by Prandtl in his famous theory for wall boundary layers. By introducing a different length scale for changes perpendicular to the wall than for changes in the flow direction, he was able to explain how viscous stresses could survive near the wall at high Reynolds number. These allowed the no-slip condition at a surface to be satisfied, and resolved the paradox of how there could be drag in the limit of zero viscosity.

It may seem strange to be talking about Prandtl’s boundary layer idea in a section about free shear flows, but as we shall see below, the basic approximations can be applied to all “thin” (or slowly growing) shear flows with or without a surface. In this section, we shall show that free shear flows, for the most part, satisfy the conditions for these “boundary layer approximations”. Hence they belong to the general class of flows referred as “boundary layer flows”.

One important difference will lie in whether momentum is being added to the flow at the source (as in jets) or taken out (by drag, as in wakes). A related influence is the presence (or absence) of a free stream in which our free shear flow is imbedded. We shall see that stationary free shear flows fall into two general classes, those with external flow and those without. One easy way to see why this makes a difference is to remember that these flows all spread by entraining mass from the surrounding fluid. You don’t have to think very hard to see that the entrained mass is carrying its own momentum into the shear flow. You should expect (and find) that even a small free stream velocity can make a significant difference, since the momentum carrried in is mixed in with that of the fluid particles which are already part of the turbulence. The longer the flow develops (or the farther downstream one looks), the more these simple differences can make a difference in how the flow spreads. In view of this, it should be no surprise that the presence or absence of an external stream plays a major role in petermining which mean convection terms which must be retained in the governing equations.

We will consider only flows which are plane (or two-dimensional) in the mean (although similar considerations can be applied to flows that are axisymmetric in the mean). In effect, this is exactly the same as assuming the flow is homogeneous in the third direction. Also we shall restrict our attention to flows which are statistically stationary, so that time derivatives of averaged quantities can be neglected. And, of course, we have already agreed to confine our attention to Newtonian flows at constant density.

It will be easier to abandon tensor notation for the moment, and use the symbols $x, U, u$ for the streamwise direction, mean and fluctuating velicities respectively, and $y, V, v$ for cross-stream. Given all this, the mean momentum equations reduce to:

x-component:

 $U \frac{\partial U}{\partial x} + V \frac{\partial U}{\partial y} = - \frac{1}{\rho} \frac{\partial P}{\partial x} - \frac{\partial \left\langle u^{2}\right\rangle}{\partial x} - \frac{\partial \left\langle uv \right\rangle}{\partial y} + \nu \frac{\partial^{2} U}{\partial x^{2}} + \nu \frac{\partial^{2} U}{\partial y^{2}}$ (2)

y-component:

 $U \frac{\partial V}{\partial x} + V \frac{\partial V}{\partial y} = - \frac{1}{\rho} \frac{\partial P}{\partial y} - \frac{\partial \left\langle uv\right\rangle}{\partial x} - \frac{\partial \left\langle v^{2} \right\rangle }{\partial y} + \nu \frac{\partial^{2} V}{\partial x^{2}} + \nu \frac{\partial^{2} V}{\partial y^{2}}$ (3)

In addition, we have the two-dimensional mean continuity equation which redics to:

 $\frac{\partial U}{\partial x} + \frac{\partial V}{\partial y} =0$ (4)

Now