Introduction to turbulence/Free turbulent shear flows

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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 reduces to:

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

(4)

Order of magnitude estimates

Now let's make an order of magnitude estimate for each of the terms. This procedure may seem trivial to some, or hopeless hand-waving to others. The fact is that if you fall either of these groups, you have missed something important. Learning to make good order-of-magnitude arguments and knowing when to use them (and when not to use them) are two of the most important skills in fluid mechanics, and especially turbulence. To do this right we will have to be very careful to make sure our estimates accurately characterize the terms we are making them for.

Naturally we should not expect changes of anything in the $x$ - direction to scale the same as changes in the $y$ -direction, especially in view of the above. (This, after all, is the whole idea of "thin" shear flow.) So let's agree that we will pick a length scale, say $L$ , characteristic of changes or mean quantities in the $x$ -direction; i.e.

$\frac{\partial}{\partial x} \sim \frac{1}{L}$

(5)

where for now ' $\sim$ ' means "of order of magnitude of". And we can do the same thing for changes of mean quantities in the $y$ -direction by defining a second lenght scale, say $\delta$ , to mean:

$\frac{\partial}{\partial y } \sim \frac{1}{\delta }$

(6)

Note that both these scales will vary with the streamwise position where we evaluate them. A good choice for $\delta$ might be proportional to the local lateral extent of the flow (or its "width"), while $L$ is related to the distance from the source.

Consider the mean velocity in equation 10 in section "Wall-bounded turbulent flows". It occurs in five different terms: $U$ alone; twice with $x$ -derivatives, $\partial U/ \partial x$ and $\partial^{2} U / \partial x^2$ ; and twice with $y$ -derivatives, $\partial U/ \partial y$ and $\partial^{2} U / \partial y^2$ . Now it would be tempting to simply pick a scale velocity for $U$ say $U_s$ , and use it to estimate all five terms, say as: $U_s$ , $U_s/L$ , $U_s/L^2$ , $U_s/ \delta$ , and $U_s/ \delta^2$ . But this is much too naive, and fails to appreciate the true role of the terms we are evaluating.

Look at the examples shown in Figure ??. Our simple approach would provide an appropriate estimate for a jet if we took our velocity scale equal to the mean centerline velocity at a given streamwise location; i.e., $U_s \left( x \right) = U_{cl} \left( x \right)$ . This is because both the mean velocity and the changes in the mean velocity across the flow are characterized by the centerline velocity. But by contrast, look at the wake. Even relatively close to the wake generator, the wake deficit, $\Delta U_{cl} = U_{0} - U_{cl}$ is small compared to the free stream velocity (i.e. $\Delta U_{cl} << U_{0}$ ). So the obvious choice to scale $U \left( x,y \right)$ is $U_{0}$ On the other hand, an estimate for the velocity gradient across the flow of $U_{0}/\delta$ would be much too big, again because the deficit is so small. Obviously a better choice would be to use the centerline mean velocity deficit $\Delta U_{cl}$ ; i.e.,

$\frac{\partial U}{\partial y} \sim \frac{\Delta U_{cl}}{\delta}$

(7)

In the order of magnitude analysis below, we shall try to keep the discussion as general as possible by using $U_{s}$ to characterize the mean velocity when it appears by itself, and $\Delta U_{s}$ to represent changes in the mean velocity. For the jet example of the preceding paragraph, both $U_{s}$ and $\Delta U_{s}$ are the same; i.e. $U_{s} = U_{cl}$ and $\Delta U_{s} = U_{cl}$ . But for the wake they are different because of the external stream; i.e., $U_{s} = U_{0}$ and $\Delta U_{s} = \Delta U_{cl}$ . If you can keep in mind why these differences exist among the various flows, it will be a lot easier to both understand the results and not confuse them.

Now we could distinguish changes of velocity in the $x$ -direction from those in the $y$ - direction. But this level of complexity is not necessary (at least for the examples considered here), especially since we have left the precise defenition of $L$ rather nebulous. What we can do is to use the same estimate for changes in the velocity as for the $y$ - direction, and define our length scale $L$ to make the estimate based on both correct; i.e. $\partial U / \partial x \sim \Delta U_{s} / L$ . To see why this make sence physically and can be reasoned (as opposed to guessed), let's look at the wake. Pick a spot outside, near the edge of the wake fairly close the generator (say point A). Now proceed at constant $y$ far downstream in $x$ . Eventually the wake will have spread past you and you will be close enough to the centerline so the local mean velocity will be closer to $U_{cl}$ than $U_{0}$ . Obviously we have simply traveled far enough at constant $y$ to ensure that $\Delta U_{cl}$ is the proper scale for the changes in velocity in the $x$ - direction. If the distance downstream over which this change occured is taken as $L$ , then proper estimate is easily seen to be:

$\frac{\partial U}{\partial x} \sim \frac{\Delta U_{cl}}{L}$

(8)

But this is exactly what we would have gotten by taking $U_{s} = U_{cl}$ as we agreed above. We simply have absorbed any differences into our choice of $L$ . When considering specific problems and applying similarity techniques, the seemingly arbitrary choices here become quite precise constraints (as we shall see).

We still haven't talked about how to estimate the velocity scale for $V$ , the cross-stream mean velocity component. From the continuity equation we know that:

$\frac{\partial V}{\partial y} = - \frac{\partial U}{\partial x}$

(9)

From our considerations above, we know that:

$\frac{\partial U}{\partial x} \sim \frac{\Delta U_{s}}{L}$

(10)

If there is no mean cross flow in the external stream, then the scale for $V$ is the same as the scale for changes in $V$ . Therefore,

$\frac{\partial V}{\partial y} \sim \frac{V_{s}}{\delta}$

(11)

It follows immediately that the order of magnitude of the cross-stream velocity is:

$V_{s} \sim \Delta U_{s} \frac{\delta}{L}$

(12)

We might have expected something like this if we had thought about it. If the $V$ -velocity were of the same order as the $U$ - velocity, how could the flow in any sense be a "thin shear flow". On the other hand, it also makes sense that $V_{s}/U_{s} \propto \delta/L$ since both are some measure of how the flow spreads/ Note that equation 12 would not be correct estimate for $V_{s}$ if there were an imposed cross-flow, since then we would have to consider $V$ and changes in $V$ separately (exactly as for $U$ ).

The mean pressure gradient term is always a problem to estimate at the outset. Therefore it is better to simply leave this problem alone, and see what is left at the end. In the estimates below you will see a question mark, which simply means we are postponing judgement until we have more information. Sometimes we will have to keep the term simply because we don't know enough to throw it away. Other times it will be obvious that it must remain because there is only one term left that must be balanced by something.

Now we have figured out how to estimate the order of magnitude of all the terms except the turbulence terms. For most problems this turns out to be pretty straight-forward if you remember our discussion of the pressure strain-rate terms. They so effectively distribute the energy that even that the three components of velocity are usually about the same order of magnitude. So if we choose a turbulence scale as simply $u$ , then $\left\langle u^{2} \right\rangle \sim u^{2}$ , $\left\langle v^{2} \right\rangle \sim u^{2}$ , $\left\langle w^{2} \right\rangle \sim u^{2}$ . But what about the Reynolds shear stress components like $\left\langle uv \right\rangle$ ? When acting against the mean shear to produce energy, they tend to be well-correlated and the maximum value of $\left\langle uv \right\rangle / u_{rms}v_{rms} \approx 0.4$ . Obviously the right choice though since some cross-stress like $\left\langle uw \right\rangle$ are identically zero in plane flows because of homogeniety. Allowing for the possibility of a Reynolds number dependent correlation coefficient will be sent to be especially important when the terms in the turbulence kinetic energy and dissipation equations are considered.

Streamwise momentum equation

$\begin{matrix} U \frac{\partial U}{\partial x} & + & V \frac{\partial U}{\partial y} \\ U_{s} \frac{\Delta U_{s}}{L} & & \left( \Delta U_{s} \frac{\delta}{L} \right) \frac{\Delta U_{s}}{ \delta} \\ \end{matrix}$

$\begin{matrix} = -\frac{1}{\rho} \frac{\partial P}{ \partial x}& - & \frac{ \partial \left\langle U^{2} \right\rangle}{ \partial x} & - & \frac{ \partial \left\langle uv \right\rangle}{ \partial y} & + & \nu \frac{\partial^{2}U}{\partial x^2} & + & \nu \frac{\partial^{2}U}{\partial y^2} \\ ? & & \frac{u^{2}}{L} & & \frac{u^{2}}{\delta} & & \nu \frac{\Delta U_{s}}{L^{2}}& & \nu \frac{\Delta U_{s}}{\delta^{2}} \end{matrix}$

Introduction to turbulence/Free turbulent shear flows

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Revision as of 16:17, 29 October 2010

Contents

Introduction

The averaged equations

The shear layer equations

Order of magnitude estimates

Streamwise momentum equation

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@@ Line 167: / Line 167: @@
 U \frac{\partial U}{\partial x} & + & V \frac{\partial U}{\partial y} \\
 U_{s} \frac{\Delta U_{s}}{L} & & \left( \Delta U_{s} \frac{\delta}{L} \right) \frac{\Delta U_{s}}{ \delta} \\
+\end{matrix}
+</math>
+</td><td width="5%"></td></tr></table>
+<table width="70%"><tr><td>
+:<math>
+\begin{matrix}
+= -\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}  \\
+? & & \frac{u^{2}}{L} & & \frac{u^{2}}{\delta} & & \nu \frac{\Delta U_{s}}{L^{2}}& &  \nu \frac{\Delta U_{s}}{\delta^{2}}
 \end{matrix}
 </math>
 </td><td width="5%"></td></tr></table>