https://www.cfd-online.com/W/index.php?title=Special:Contributions/Callahance&feed=atom&limit=50&target=Callahance&year=&month=CFD-Wiki - User contributions [en]2018-02-20T08:07:54ZFrom CFD-WikiMediaWiki 1.16.5https://www.cfd-online.com/Wiki/Introduction_to_turbulence/Reynolds_averaged_equationsIntroduction to turbulence/Reynolds averaged equations2013-05-01T16:32:04Z<p>Callahance: /* Reynolds stress equations */</p>
<hr />
<div>{{Introduction to turbulence menu}}<br />
<br />
== Equations governing instantaneous fluid motion ==<br />
<br />
All fluid motions, whether turbulent or not, are governed by the dynamical equations for a fluid. These can be written using Cartesian tensor notation as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac{\partial \tilde{p}}{\partial x_i}+\frac{\partial \tilde{T}_{ij}^{(v)}}{\partial x_j}</math><br />
</td><td width="5%">(1)</td></tr></table><br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left[\frac{\partial \tilde{\rho}}{\partial t}+\tilde{u}_j\frac{\partial \tilde{\rho}}{\partial x_j}\right]+ \tilde{\rho}\frac{\partial \tilde{u}_j}{\partial x_j}= 0 </math><br />
</td><td width="5%">(2)</td></tr></table><br />
<br />
where <math>\tilde{u_i}(\vec{x},t)</math> represents the i-the component of the fluid velocity at a point in space,<math>[\vec{x}]_i=x_i</math>, and time,t. Also <br />
<math>\tilde{p}(\vec{x},t)</math> represents the static pressure, <math>\tilde{T}_{ij}^{(v)}(\vec{x},t)</math>, the viscous(or deviatoric) stresses, and <math>\tilde\rho</math> the fluid density. The tilde over the symbol indicates that an instantaneous quantity is being considered. Also the [[einstein summation convention]] has been employed.<br />
<br />
In equation 1, the subscript <math>i</math> is a free index which can take on the values 1,2 and 3. Thus equation 1 is in reality three separate equations. These three equations are just Newton's second law written for a continuum in a spatial(or Eulerian) reference frame. Together they relate the rate of change of momentum per unit mass <math>(\rho{u_i})</math>,a vector quantity, to the contact and body forces.<br />
<br />
Equation 2 is the equation for mass conservation in the absence of sources(or sinks) of mass. Almost all flows considered in this material will be incompressible, which implies that derivative of the density following the fluid material[the term in brackets] is zero. Thus for incompressible flows, the mass conservation equation reduces to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{D \tilde{\rho}}{Dt}=\frac{\partial \tilde{\rho}}{\partial t}+\tilde{u}_j\frac{\partial \tilde{\rho}}{\partial x_j}= 0</math><br />
</td><td width="5%">(3)</td></tr></table><br />
<br />
From equation 2 it follows that for incompressible flows,<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial \tilde{u}_j}{\partial x_j}= 0</math><br />
</td><td width="5%">(4)</td></tr></table><br />
<br />
The viscous stresses(the stress minus the mean normal stress) are represented by the tensor<math>\tilde{T}_{ij}^{(v)}</math>. From its definition,<math>\tilde{T}_{kk}^{(v)}</math>=0. In many flows of interest, the fluid behaves as a Newtonian fluid in which the viscous stress can be related to the fluid motion by a constitutive relation of the form.<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{T}_{ij}^{(v)}= 2\mu[\tilde{s}_{ij}-\frac{1}{3}\tilde{s}_{kk}\delta_{ij}] </math><br />
</td><td width="5%">(5)</td></tr></table><br />
<br />
The viscosity, <math>\mu</math>, is a property of the fluid that can be measured in an independent experiment. <math>\tilde s_{ij}</math> is the instantaneous strain rate tensor defined by<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{s}_{ij}= \frac{1}{2}\left[\frac{\partial \tilde u_i}{\partial x_j}+\frac{\partial \tilde u_j}{\partial x_i}\right] </math><br />
</td><td width="5%">(6)</td></tr></table><br />
<br />
From its definition, <math>\tilde s_{kk}=\frac{\partial \tilde u_k}{\partial x_k}</math>. If the flow is incompressible, <math>\tilde s_{kk}=0</math> and the Newtonian constitutive equation reduces to<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{T}_{ij}^{(v)}= 2\mu\tilde{s}_{ij}</math><br />
</td><td width="5%">(7)</td></tr></table><br />
<br />
Throughout this material, unless explicitly stated otherwise, the density <math>\tilde\rho=\rho</math> and the viscosity <math>\mu</math> will be assumed constant. With these assumptions, the instantaneous momentum equations for a Newtonian Fluid reduce to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac {1}{\tilde\rho}\frac{\partial \tilde{p}}{\partial x_i}+\nu\frac{\partial^2 {\tilde{u}_i}}{\partial x_j^2}</math><br />
</td><td width="5%">(8)</td></tr></table><br />
<br />
where the kinematic viscosity, <math>\nu</math>, has been defined as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\nu\equiv\frac{\mu}{\rho}</math><br />
</td><td width="5%">(9)</td></tr></table><br />
<br />
Note that since the density is assumed constant, the tilde is no longer necessary.<br />
<br />
Sometimes it will be more instructive and convenient to not explicitly include incompressibilty in the stress term, but to refer to the incompressible momentum equation in the following form:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac{\partial \tilde{p}}{\partial x_i}+\frac{\partial \tilde{T}_{ij}^{(v)}}{\partial x_j}</math><br />
</td><td width="5%">(10)</td></tr></table><br />
<br />
This form has the advantage that it is easier to keep track of the exact role of the viscous stresses.<br />
<br />
== Equations for the average velocity ==<br />
<br />
Although laminar solutions to the equations often exist that are consistent with the boundary conditions, perturbations to these solutions(sometimes even infinitesimal) can cause them to become turbulent. To see how this can happen, it is convenient to analyze the flow in two parts, a mean(or average) component and a fluctuating component. Thus the instantaneous velocity and stresses can be written as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\tilde {u}_i=U_i+u_i<br />
</math><br />
:<math><br />
\tilde p=P+p<br />
</math><br />
:<math><br />
\tilde T_{ij}^{(v)}=T_{ij}^{(v)}+\tau_{ij}^{(v)}<br />
</math><br />
</td><td width="5%">(11)</td></tr><br />
</table><br />
<br />
Where <math>U_i</math>, <math>P</math> and <math>T_{ij}^{(v)}</math> represent the mean motion, and <math>u_i</math>, <math>p</math> and <math>\tau_{ij}^{(v)}</math> the fluctuating motions. This technique for decomposing the instantaneous motion is referred to as the '''''Reynolds decomposition.''''' Note that if the averages are defined as ensemble means, they are, in general, time-dependent. For the remainder of this material unless other wise stated, the density will be assumed constant so<math>\tilde{\rho}\equiv\rho</math>,and its fluctuation is zero.<br />
<br />
Substitution of equations 11 into equations 10 yields<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial (U_i+u_i)}{\partial t}+(U_j+u_j)\frac{\partial (U_i+u_i)}{\partial x_j}\right] = -\frac{\partial (P+p)}{\partial x_i}+\frac{\partial (T_{ij}^{(v)}+\tau_{ij}^{(v)})}{\partial x_j}</math><br />
</td><td width="5%">(12)</td></tr></table><br />
<br />
This equation can now be averaged to yield an equation expressing momentum conservation for the averaged motion. Note that the operations of averaging and differentiation commute; i.e., the average of a derivative is the same as the derivative of the average. Also the average of a fluctuating quantity is zero. Thus the equation for the averaged motion reduces to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial U_i}{\partial t}+U_j\frac{\partial U_i}{\partial x_j}\right] = -\frac{\partial P}{\partial x_i}+\frac{\partial T_{ij}^{(v)}}{\partial x_j}-\rho\left \langle u_j\frac{\partial u_i }{\partial x_j} \right \rangle</math><br />
</td><td width="5%">(13)</td></tr></table><br />
<br />
where the remaining fluctuating product term has been moved to the right hand side of the equation. Whether or not the last term is zero like the other fluctuating term depends on the correlation of the terms in the product. In general, these correlations are not zero.<br />
<br />
The mass conservation equation can be similarly decomposed. In incompressible form, substitution of equations 11 into equation 4 yields:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial (U_j+u_j)}{\partial x_j}=0</math><br />
</td><td width="5%">(14)</td></tr></table><br />
<br />
of which average is <br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial U_j}{\partial x_j}=0</math><br />
</td><td width="5%">(15)</td></tr></table><br />
<br />
It is clear from equation 15 that the averaged motion satisfies the same form of the mass conservation equation as does the instantaneous motion at least for incompressible flows. How much simpler the turbulence problem would be if the same were true for the momentum! Unfortunately, as is easily seen from equation 13, such is not the case.<br />
<br />
Equation 15 can be subtracted from equation 14 to yield an equation for instantaneous motion alone; i.e,<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial u_j}{\partial x_j}=0</math><br />
</td><td width="5%">(16)</td></tr></table><br />
<br />
Again, like the mean, the form of the original instantaneous equation is seen to be preserved. The reason, of course, is obvious: the continuity equation is linear. The momentum equation , on the other hand, is not; hence the difference.<br />
<br />
Equation 16 can be used to rewrite the last term in equation 13 for the mean momentum. Multiplying equation 16 by <math>u_i</math> and averaging yields:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left \langle u_i\frac{\partial u_j}{\partial x_j}\right \rangle=0</math><br />
</td><td width="5%">(17)</td></tr></table><br />
<br />
Now add:<math>\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle</math> to both sides of eqaution 17, then using the product rule you obtain:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle +0=\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle+ \left \langle u_i\frac{\partial u_j}{\partial x_j}\right \rangle =\frac{ \partial}{\partial x_j}{\left \langle u_iu_j\right \rangle} </math><br />
</td><td width="5%">(18)</td></tr></table><br />
<br />
Where again the fact that arithmetic and averaging operations commute has been used.<br />
<br />
The equation for the averaged momentum, equation 13 can now be rewritten as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial U_i}{\partial t}+U_j\frac{\partial U_i}{\partial x_j}\right] = -\frac{\partial P}{\partial x_i}+\frac{\partial T_{ij}^{(v)}}{\partial x_j}-\frac{ \partial}{\partial x_j}{(\rho\left \langle u_iu_j\right \rangle)}</math><br />
</td><td width="5%">(19)</td></tr></table><br />
<br />
The last two terms on the right hand side are both divergence terms and can be combined; the result is:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial U_i}{\partial t}+U_j\frac{\partial U_i}{\partial x_j}\right] = -\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_j}[T_{ij}^{(v)}-{\rho\left \langle u_iu_j\right \rangle}]</math><br />
</td><td width="5%">(20)</td></tr></table><br />
<br />
Now the terms in square brackets on the right have the dimensions of stress. The first term is, in fact , the viscous stress. The second term, on the other hand, is not a stress at all, but simply a re-worked version of the fluctuating contribution to the non-linear acceleration terms. The fact that it can be written this way, however, indicates that at least as far as the motion is concerned, it acts as though it were a stress- hence its name, the '''Reynolds stress'''. In the succeeding sections the consequences of this difference will be examined.<br />
<br />
== The turbulence problem ==<br />
<br />
It is the appearance of the Reynolds stress which makes the turbulence problem so difficult - at least from the engineers perspective. Even though we can pretend it is a stress, the physics which give rise to it are very different from the viscous stress. The viscous stress can be related directly to the other flow properties by constitutive equations, which in turn depend only on the properties of the fluid (as in equation 5 for a Newtonian fluid). The reason this works is that when we make such closure approximations for a fluid, we are averaging over characteristic length and time scales much smaller than those of the flows we are interested in. Yet at the same time, these scales are much larger than the molecular length and time scales which characterize the molecular interactions that are actually causing the momentum transfer. (This is what the continuum approximation is all about).<br />
<br />
The '''''Reynolds stress''''', on the other hand, arises from the flow itself! Worse, the scales of the fluctuating motion which give rise to it are the scales we are interested in. This means that the closure ideas which worked so well for the viscous stress, should not be expected to work too well for the Reynolds stress. And as we shall see, they do not.<br />
<br />
This leaves us in a terrible position. Physics and engineering are all about writing equations(and boundary conditions) so we can solve them to make predictions. We don't want to have a build prototype airplanes first to see if they will they fall out of the sky. Instead we want to be able to analyze our designs before building, to save the cost in money and lives if our ideas are wrong. The same is true for dams and bridges and tunnels and automobiles. If we had confidence in our turbulence models, we could even build huge one-offs and expect them to work the first time. Unfortunately, even though turbulence models have improved to the point where we can use them in design, we still cannot trust them enough to eliminate expensive wind tunnel and model studies. And recent history is full of examples to prove this.<br />
<br />
The turbulence problem (from the engineers perspective) is then three-fold:<br />
<br />
* '''The averaged equations are not closed.''' Count the number of unknowns in equation 20 above. Then count the number of equations. Even with the continuity equation we have atleast six equations too few.<br />
<br />
* '''The simple ideas to provide the extra equations usually do not work.''' And even when we can fix them up for a particular class of flows (like the flow in a pipe, for example), they will most likely not be able to predict what happens in even a slightly different environment (like a bend).<br />
<br />
*'''Even the last resort of compiling engineering tables for design handbooks carries substantial risk.''' This is the last resort for the engineer who lacks equations or cannot trust them. Even when based on a wealth of experience, they require expensive model testing to see if they can be extrapolated to a particular situation. Often they cannot, so infinitely clever is Mother Nature in creating turbulence that is unique to a particular set of boundary conditions.<br />
<br />
'''Turbulent flows are indeed flows!'''. And that is the problem.<br />
<br />
== Origins of turbulence==<br />
<br />
Turbulent flows can often be observed to arise from laminar flows as the Reynolds number, (or someother relevant parameter) is increased. This happens because small disturbances to the flow are no longer damped by the flow, but begin to grow by taking energy from the original laminar flow. This natural process is easily visualized by watching the simple stream of water from a faucet (or even a pitcher). Turn the flow on very slow (or pour) so the stream is very smooth initially, at least near the outlet. Now slowly open the faucet (or pour faster) abd observe what happens, first far away, then closer to the spout. The surface begins to exhibit waves or ripples which appear to grow downstream . In fact, they are growing by extracting energy from the primary flow. Eventually they grow enough that the flow breaks into drops. These are capillary instabilities arisiing from surface tension, but regardless of the type of instability, the idea is the same -small (or infinitesimal ) disturbances have grown to disrupt the serenity (and simplicity) of laminar flow.<br />
<br />
The manner in which the instabilities grow naturally in a flow can be examined using the equations we have already developed above. We derived them by decomposing the motion into a mean and fluctuating part. But suppose instead we had decomposed the motion into a base flow part (the initial laminar part) and into a disturbance which represents a fluctuating part superimposed on the base flow. The result of substituting such a decomposition into the full Navier-Stokes equations and averaging is precisely that given by equations (13) and (15). But the very important difference is the additional restriction that what was previously identified as the mean (or averaged ) motion is now also the base or laminar flow.<br />
<br />
Now if the base flow is really laminar flow (which it must be by our original hypothesis), then our averaged equations governing the base flow must yield the same mean flow as the original laminar flow on which the disturbances was superimposed. But this can happen only if these new averaged equations reduce to '''exactly''' the same lamiane flow equations without any evidence of a disturbance. Clearly from equations 13 and 15, this can happen ''only if all the Reynolds stress terms vanish identically!'' Obviously this requires that the disturbances be infintesimal so the extra terms can be neglected - hence our interest in infinitesimal disturbances.<br />
<br />
So we hypothesized a base flow which was laminar and showed that it is unchanged even with the imposition of infintesimal disturbances on it - ''but only as long as the disturbances'' '''remain''' ''infinitesimal!'' What happens if the disturbance starts to grow? Obviously before we conclude that all laminar flows are laminar forever we better investigate whether or not these infinitesimal disturbances can grow to ''finite'' size. To do this we need an equation for the fluctuation itself.<br />
<br />
An equation for the fluctuation (which might be an imposed disturbance) can be obtained by subtracting the equation for the mean (or base) flow from that for the instantaneous motion. We already did this for the continuity equation. Now we will do it for the momentum equation. Subtracting equation 13 from equation 11 yields an equation for the fluctuation as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial u_{i}}{\partial t} + U_{j}\frac{\partial u_{i}}{\partial x_{j}} \right] = - \frac{\partial p}{\partial x_{i}} + \frac{\partial\tau^{(v)}_{ij}}{\partial x_{j}} - \rho \left[ u_{j}\frac{\partial U_{i}}{\partial x_{j}} \right] - \left\{ u_{j} \frac{\partial u_{i}}{ \partial x_{j}} - \rho \left\langle u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right\}<br />
</math><br />
</td><td width="5%">(21)</td></tr></table><br />
<br />
It is very important to note the type and character of the terms in this equation. First note that the left-hand side is the derivative of the ''fluctuating'' velocity following the ''mean'' motion. This is exactly like the term which appears on the left-hand side of the equation for the mean velocity, equation 13. The first two terms on the right-hand side are also like those in the mean motion, and represent the fluctuating pressure gradient and the fluctuating viscous stresses. The third term on the right-hand is is new, and will be seen later to represent the primary means by which fluctuations (and turbulence as well!) extract energy from the mean flow, the so-called ''production terms''. The last term is quadratic in the fluctuating velocity, unlike all the otherwhich are linear. Note that all of the terms vanish identically if the equation is averaged, the last because its mean is subtructed from it.<br />
<br />
Now we want to examine what happens if the disturbance is small. In the limit as the amplitude of the disturbance (or fluctuation) is ''infinitesmal'', the bracketed term in the equationfor the fluctuation vanishes (since it involves productsof infinitesimals) , and the remaining equation is ''linear in the disturbance''. The study of whether or not such infinitesmal disturbances can grow is called '''Linear Fluid Dynamic Stability Theory'''. These linearized equations are very different from those govering turbulence. Unlike the equations for disturbances of ''finite'' amplitude, the linearized equations are well-posed (or closed) since the Reynolds stress terms are gone.<br />
<br />
The absence of the non-linear terms, however, constrains the validity of the linear analysis to only the initial stage of disturbance growth. This is because as soon as the fluctuations begin to grow, their amplitudes can no longer be assumed infinitesmal and the Reynolds stress (or more properly, the non-linear fluctuating terms) become important. As a result the base flow equations begin to be modified so that the solution to them can no longer be identical to the laminar flow (or base flow) from which it arose. Thus while linear stability theory can predict ''when'' many flows become ''unstable'', it can say very little about ''transition to turbulence'' since this progress is highly non-linear.<br />
<br />
== Importance of non-linearity ==<br />
<br />
We saw in the preceding section that non-linearity was one of essential features of turbulence. When small disturbances grow large enough to interact ''with each other'', we enter a whole new world of complex behavior. Most of the rules we learned for linear system do not apply. Since most of your mathematical training has been for linear equations, most of your mathematical intuition therefore will not apply either. On the other hand, you may surprise yourself by discovering how much your ''non-mathematical'' intuition already recognizes non-linear behavior and accounts for it.<br />
<br />
Considering the following simple example. Take a long stick with one person holding each end and stand at the corner of a building. Now place the middle of against the building and let each person apply pressure in the same direction so as to bend the stick. If the applied force is small, the stick deflects (or bends) a small amount. Double the force, and the deflection is approximately doubled. Quadruple the force and the deflection is quadrupled. Now you don't need a Ph.D. in Engineering to know what is going to happen if you continue this process. '''The stick is going to break!'''<br />
<br />
But where in the equations for the deflection of the stick is there anything that predicts this can happen? Now if you are only like engineer, you are probably thinking: he's asking a stupid question. Of course you can't continue to increase the force because you will exceed first the yield stress, then the breaking limit, and of course the stick will break.<br />
<br />
But pretend I am the company president with nothing more than MBA. I don't know much about these things, but you have told me in the past that your computers have equations to predict everything. So I repeat: Where in the equations for the deflection of this stick does it tell me this going to happen?<br />
<br />
The answer is very simple: '''There is''' ''nothing'' '''in the equations that will predict this.''' And the reason is also quite simple: You lost the ability to predict catasrophes like breaking when you linearized the fundamental equations - which started out as Newton's Law too. In fact, before linearization, they were exactly the same as those for a fluid, only the constitutive equation was different.<br />
<br />
If we had NOT linearized these equations and had constituve equations that were more general, then we possibly could apply these equation right to and past the limit. The point of fracture would be a bifurcation point for the solution.<br />
<br />
Now the good news is that for things like reasonable deflections of beams linearization work woderfully since we hope most things we build don't deflect too much. Unfortunately, as we noted above, for fluids the disturbances tend to quickly become dominated by non-linear terms. This, of course, means our linear analytical techniques are pretty useless for fluid mechanics, and especially turbulence.<br />
<br />
But all is not lost. Just as we learned to train ourselves to anticipate when sticks break, we have to train ourselves to anticipate how non-linear fluid phenomena behave. Toward that end we will consider two simple examples: one from algebra - the logistic map, and one from fluid mechanics - simple vortex streching.<br />
<br />
'''Example 1:''' An experiment with the logistic map. <br />
<br />
Consider the behavior of the simple equation:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
y_{n+1}= r y_{n} \left(1- y_{n} \right)<br />
</math><br />
</td><td width="5%">(22)</td></tr></table><br />
<br />
where <math> n= 1,2..., 0<y < 1 </math> and <math> r > 0 </math>. The idea is that you pick any value for <math> y_{1}</math>, use the equation to find <math> y_{2}</math>, then insert that value on the right-hand side to find <math> y_{3}</math>, and just continue the process as long as you like. Make sure you note any dependece of the final result on the initial value for <math> y </math>.<br />
<br />
*First notice what happens if you linearize this equation by disregarding the term in parentheses; i.e., comsider the simpler equation <math> y_{1+1} = r y_{n}</math>. My guess is that you won't find this too eciting - unless, of course, you are one of those rare individuals who likes watching grass grow.<br />
<br />
* Now consider the full equation and note what happens for <math> r<3 </math> , and especially what happens for very small values of <math> r </math>. Run as many iterations as necessary to make sure your answer has converged. Do NOT try to take short-cuts by programming all the steps at once. Do them one at time so you can see what is happening. It will be much easier this way in the long run.<br />
<br />
* Now research carefully what happens when <math> r=3.1, 3.5, </math> and <math> 3.8 </math>. Can you recognize any patterns.<br />
<br />
* Vary <math> r </math> between 3 and 4 to see if you can find the boundaries for what you are observing. <br />
<br />
* Now try values of <math> r>4 </math>. How do you explain this<br />
<br />
'''Example 2:''' Stretching of a simple vortex.<br />
<br />
Imagine a simple vortex filament that looks about like a strand of spaghetti. Now suppose it is in otherwise steady inviscid incompressible flow. Use the vorticity equation to examine the following: <br />
<br />
* Examine first what happens to it in two-dimensional velocity field. Note particularly whether any new vorticity can be produced; i.e., can the material derivative of the vorticity ever be greater than zero? (Hint: look at the <math> \omega_{j} \partial u_{i}/ \partial x_{j} <br />
</math> - term)<br />
<br />
* Now consider the same vortex filament in a three-dimensional flow. Note particularly the various ways new vorticity can be produced - if you have some to start with! Does all this have anything to do with non-linearities?<br />
<br />
== Turbulence closure problem and eddy viscosity ==<br />
<br />
From the point of view of the averaged motion, at least, the problem with the non-linearity of the instaneous equations is that they introduce new unknowns, the Reynolds stress into the averaged equations. There are six individual stress components we must deal with to be exact: <math> \left\langle u^{2}_{1} \right\rangle </math> , <math> \left\langle u^{2}_{2} \right\rangle </math> , <math> \left\langle u^{2}_{3} \right\rangle </math> , <math> \left\langle u_{1}u_{2} \right\rangle </math> , <math> \left\langle u_{1}u_{3} \right\rangle </math> , and <math> \left\langle u_{2}u_{3} \right\rangle </math>. These have to be related to the mean motion itself before the equations can be solved, since the number of unknows and number of equations must be equal. The absence of these additional equations is often reffered to as '''the Turbulence Closure Problem'''.<br />
<br />
A similar problem arose when the instantaneous equations were written (equations 1 2), since relations had to be introduced to relate the stresses (in particular, the viscous stresses) to the motion itself. These relations (or constitutive equations) ''depended only on the properties of the fluid material, and not on the flow itself''. Because of this fact, it is possible to carry out independent experiments, called viscometric experiments, in which these fluid properties can be determinded once and for all. Equation 5 provides an example just such a constitutive relation, the viscosity, <math>\mu </math>, depending only in the choice of the material. For example, once the viscosity of water at given temperature is determined, this value can be used in all flows at that temperature, not just the one in which the evaluation was made.<br />
<br />
It is tempting to try such an approach for the turbulence Reynolds stresses (even though we know the underlying requirements of scale separation are not satisfied). For example, a Newtonian type closure for the Reynolds stresses, often referred to as an "eddy" or "turbulent" viscosity model, looks like:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{i} u_{j} \right\rangle = 2 \mu_{t} \left[ S_{ij} - \frac{1}{3} S_{kk} \delta_{ij} \right]<br />
</math><br />
</td><td width="5%">(23)</td></tr></table><br />
<br />
where <math>\mu_{t} </math> is the turbulence "viscosity" (also called the eddy viscosity) and <math>S_{ij} </math> is the ''mean'' strain rate defined by:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
S_{ij}= \frac{1}{2} \left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right]<br />
</math><br />
</td><td width="5%">(24)</td></tr></table><br />
<br />
The second term of course, vanishes identically for incompressible flow. For the simple case of a two-dimensional shear flow, equation 3.23 for the Reynolds shear stress reduces to<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{1} u_{2} \right\rangle = \mu_{t} \frac{\partial U_{1}}{\partial x_{2}} <br />
</math><br />
</td><td width="5%">(25)</td></tr></table><br />
<br />
That such a simple model can adequately describe the mean motion in at least one flow is illustrated by the axisymmetric buoyant plume sketched in <font color="orange">Figure 3.1. Figures 3.2 and 3.3</font> show the calculation of the mean velocity and temperature profiles respectively. Obviously the mean velocity and temperature are reasonably accurately computed, as are the Reynolds shear stress and lateral turbulent heat flux shown in <font color ="orange">Figures 3.4 and 3.5</font>.<br />
<br />
The succes of the eddy viscosity in the preceding example is more apparent than real, however, since the value of the eddy viscosity and eddy diffusivity (for turbulent heat flux) have been chosen to give the best possible agreement with the data. This, in itself, would not be a problem if that chosen values could have been obtained in advance of the computation, or even if they could be used to successfully predict other flows. In fact, the values used work only for this flow, thus the computation is ''not a prediction at all, but a'' '''postdiction''' ''or'' '''hindcast''' from which no extrapolation to the future can be made. In other words, our turbulence "model" is about as useful as having a program to predict yesterday's weather. Thus the closure problem still very much remains.<br />
<br />
<font color="orange" size="3">Figure 3.1 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.2 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.3 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.4 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.5 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.6 not uploaded yet</font><br />
<br />
Another problem with the eddy viscosity in the example above is that it fails to calculate the vertical components of the Reynolds stress and turbulent heat flux. An attempt at such a computation is shown in <font color ="orange">Figure 3.6</font> where the vertical turbulent heat flux is shown to be severely underestimated. Clearly the value of the in the vertical direction must be different than in the radial direction. In other words, the turbulence for which a constituve equation is being written is ''not an isotropic "medium" ''. In fact, in this specific example the problem is that the vertical component of the heat flux is produced more by the interaction of buoyancy and the turbulence, than it is by the working of turbulence against mean gradient in the flow. We will discuss this in more detail in the next chapter when we consider the turbulence energy balances, but note for now that simple gradient closure models never work unless gradient production dominates. This rules out many flows involving buoyancy, and also many involving recirculations or separation where the local turbulence is convected in from somewhere else. <br />
<br />
A more general form of constitutive equation which would allow for the nonisotropic nature of the "medium" (in this case the turbulence itself) would be<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{i} u_{j} \right\rangle = \mu_{ijkl} \left[ S_{kl} - \frac{1}{3} S_{mm} \delta_{kl} \right]<br />
</math><br />
</td><td width="5%">(26)</td></tr></table><br />
<br />
This closure relation allows each component of the Reynolds stress to have its own unique value of the eddy viscosity. It is easy to see that it is unlikely this will solve the closure problem since the original six unknowns the <math>\left\langle u_{i} u_{j} \right\rangle </math> have been traded for eighty-one new ones, <math> \mu_{ijkl} </math>. Even if some can be removed by symmetries, the remaining number is still formidable. More important than the number of unknowns, however, is that there is no independent or general means for selecting them without considering a particular flow. This is because ''turbulence is indeed a property of the flow, not of the fluid''.<br />
<br />
== Reynolds stress equations ==<br />
<br />
It is clear from the preceding section that the simple idea of an eddy viscosity might not be the best way to approach the problem of relating the Reynolds stress to the mean motion. An alternative approach is to try to derive dynamical equations for the Reynolds stresses from the equations governing the fluctuations themselves. Such an approach recognizes that the Reynolds stress is really a functional of the velocity; that is, the stress at a point depends on the velocity everywhere and for all past times, not just at the point in question and at a particular instant in time.<br />
<br />
The analysis begins with the equation for the instantaneous fluctuating velocity, equation 21. This can be rewritten for a Newtonian fluid with constant viscosity as: <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\rho \left[ \frac{\partial u_{i}}{\partial t} + U_{j} \frac{\partial u_{i}}{\partial x_{j}} \right] = -\frac{\partial p}{\partial x_{i}} + \frac{\partial \tau ^{ ( v ) }_{ij}}{ \partial x_{j}} - \rho \left[ u_{j} \frac{\partial U_{i}}{ \partial x_{j} } \right] - \rho \left\{ u_{j} \frac{\partial u_{i}}{ \partial x_{j} } - \left\langle u_{j} \frac{\partial u_{i} }{\partial x_{j} } \right\rangle \right\}<br />
</math><br />
</td><td width="5%">(27)</td></tr></table><br />
<br />
Note that the free index in this equation is <math> i </math>. Also, since we are talking about turbulence again, the capital letters represent mean or averaged quantities. <br />
<br />
Multiplying equation 27 by <math> u_{k} </math> and averaging yields:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\rho \left[ \left\langle u_{k} \frac{\partial u_{i} }{\partial t} \right\rangle + U_{j} \left\langle u_{k} \frac{\partial u_{i} }{\partial x_{j}} \right\rangle \right] = & - & \left\langle u_{k} \frac{\partial p }{\partial x_{i} } \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij} }{\partial x_{j}} \right\rangle \\<br />
& - & \rho \left[ \left\langle u_{k}u_{j} \right\rangle \frac{\partial U_{i} }{\partial x_{j} } \right] - \rho \left\{ \left\langle u_{k}u_{j} \frac{\partial u_{i} }{\partial x_{j} } \right\rangle \right\} \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(28)</td></tr></table><br />
<br />
Now since both <math> i </math> and <math> k </math> are free indices they can be interchanged to yield a second equation given by (alternatively equation 21 can be rewritten with free index <math> k </math>, then multiplied by <math> u_{i} </math> and averaged):<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\rho \left[ \left\langle u_{i} \frac{\partial u_{k} }{\partial t} \right\rangle + U_{j} \left\langle u_{i} \frac{\partial u_{k} }{\partial x_{j}} \right\rangle \right] = & - & \left\langle u_{i} \frac{\partial p }{\partial x_{k} } \right\rangle + \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj} }{\partial x_{j}} \right\rangle \\<br />
& - & \rho \left[ \left\langle u_{i}u_{j} \right\rangle \frac{\partial U_{k} }{\partial x_{j} } \right] - \rho \left\{ \left\langle u_{i}u_{j} \frac{\partial u_{k} }{\partial x_{j} } \right\rangle \right\} \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(29)</td></tr></table><br />
<br />
Equations 28 and 29 can be added together to yield an equation for the Reynolds stress<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\frac{\partial\left\langle u_{i}u_{k} \right\rangle}{\partial t} + U_{j}\frac{\partial\left\langle u_{i}u_{k} \right\rangle}{\partial x_{j}} <br />
& = & - \frac{1}{\rho} \left[ \left\langle u_{i} \frac{\partial p}{\partial x_{k}} \right\rangle + \left\langle u_{k} \frac{\partial p}{\partial x_{i}} \right\rangle \right] \\<br />
& - & \left[ \left\langle u_{i}u_{j} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle u_{k}u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
& + & \frac{1}{\rho} \left[ \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj} }{\partial x_{j}} \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij} }{\partial x_{j}} \right\rangle \right] \\<br />
& - & \left[ \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{k}}{\partial x_{j}} + \left\langle u_{k} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(30)</td></tr></table><br />
<br />
It is customary to rearrange the first term on the right hand side in the following way:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \left\langle u_{i} \frac{\partial p}{\partial x_{k}} \right\rangle + \left\langle u_{k} \frac{\partial p}{\partial x_{i}} \right\rangle \right] = & - & \left\langle p \left[ \frac{\partial u_{i}}{\partial x_{k}} + \frac{\partial u_{k}}{\partial x_{i}} \right] \right\rangle \\<br />
& + & \frac{\partial}{\partial x_{j}} \left[ \left\langle pu_{i} \right\rangle \delta_{kj} + \left\langle pu_{k} \right\rangle \delta_{ij} \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(31)</td></tr></table><br />
<br />
The first term on the right is generally referred to as the ''pressure strain-rate'' term. The second term is written as a divergence term, and generally referred to as the ''pressure diffusion'' term. We shal see later that divergence terms can never create nor destroy anything; they can simple move it around from one place to another. <br />
<br />
The third term on the right-hand side of equation 30 can similarly be rewritten as:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj}}{\partial x_{j}} \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij}}{\partial x_{j}} \right\rangle \right] = & - & \left[ \left\langle \tau^{(v)}_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle \tau^{(v)}_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
& + & \frac{\partial}{\partial x_{j}} \left[ \left\langle u_{i} \tau^{(v)}_{kj} \right\rangle + \left\langle u_{k} \tau^{(v)}_{ij} \right\rangle \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(32)</td></tr></table><br />
<br />
The first of these is also a divergence term. For a Newtonian fluid, the last is the so-called "dissipation of Reynolds stress" by the turbulence viscous stresses. This is easily seen by substituting the Newtonian constitutive relation to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left[ \left\langle \tau^{(v)}_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle \tau^{(v)}_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] = 2 \nu \left[ \left\langle s_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle s_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right]<br />
</math><br />
</td><td width="5%">(33)</td></tr></table><br />
<br />
It is not at all obvious what this has to do with dissipation, but it will become clear later on when we consider the trace of the Reynolds stress equation, which is the ''kinetic energy'' equation for the turbulence.<br />
<br />
Now if we use the same trick from before using the continuity equation, we can rewrite the third term in equation 30 to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left[ \left\langle u_{i}u_{j} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle u_{k}u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] = \frac{\partial}{\partial x_{j}} \left\langle u_{i} u_{k} u_{j} \right\rangle<br />
</math><br />
</td><td width="5%">(34)</td></tr></table><br />
<br />
This is also a divergence term.<br />
<br />
We can use all of the pieces we have developed above to rewrite equation 30 as <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\frac{\partial}{\partial t} \left\langle u_{i}u_{k} \right\rangle & + & U_{j} \frac{\partial}{\partial x_{j}} \left\langle u_{i}u_{k} \right\rangle = - \left\langle \frac{p}{\rho} \left[ \frac{\partial u_{i}}{\partial x_{k}} + \frac{\partial u_{i}}{\partial x_{k}} \right] \right\rangle \\<br />
& + & \frac{\partial}{\partial x_{j}} \left\{ - \left[ \left\langle pu_{k} \right\rangle \delta_{ij} + \left\langle pu_{i} \right\rangle \delta_{kj} \right] - \left\langle u_{i}u_{k}u_{j} \right\rangle + 2\nu \left[ \left\langle s_{ij}u_{k} \right\rangle + \left\langle s_{ij}u_{k} \right\rangle \right] \right\}\\<br />
& - & \left[ \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{k}}{\partial x_{j}} + \left\langle u_{k} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} \right] \\<br />
& - & 2\nu \left[ \left\langle s_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle s_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(35)</td></tr></table><br />
<br />
This is the so-called '''Reynolds Stress Equation''' which has been the primary vehicle for much of the turbulence modelling efforts of the past few decades.<br />
<br />
The left hand side of the Reynolds Stress Equation can easily be recognized as the rate of change of the Reynolds stress following the mean motion. It seems to provide exactly what we need: nine new equations for the nine unknowns we can not account for. The problems are all on the right-hand side. These terms are referred to respectively as<br />
<br />
* the pressure-strain rate term<br />
* the turbulence transport (or divergence) term<br />
* the "production" term, and<br />
* the "dissipation" term.<br />
<br />
Obviously these equations do not involve only <math> U_{i} </math> and <math> \left\langle u_{i} u_{j} \right\rangle </math>, but depend on many more new unknowns.<br />
<br />
It is clear that, contrary to our hopes, we have not derived a single equation relating the Reynolds stress to the mean motion. Instead, our Reynolds stress transport transport equation is exceedingly complex. Whereas the process of averaging the equation for the mean motion introduced only six new, independent unknowns, the Reynolds stress, <math> \left\langle u_{i} u_{j} \right\rangle </math>, the search for a transport equation which will relate these to the mean motion has produced many more unknowns. They are<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle p u_{i} \right\rangle - 3<br />
</math><br />
</td><td width="5%">(36)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle u_{i} s_{jk} \right\rangle - 27<br />
</math><br />
</td><td width="5%">(37)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle s_{ij}s_{jk} \right\rangle - 9<br />
</math><br />
</td><td width="5%">(38)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle u_{i}u_{k}u_{j} \right\rangle - 27<br />
</math><br />
</td><td width="5%">(39)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle p\frac{\partial u_{i}}{\partial x_{j}} \right\rangle - 9<br />
</math><br />
</td><td width="5%">(40)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
TOTAL - 75<br />
</math><br />
</td><td width="5%">(41)</td></tr></table><br />
<br />
Not at all of these are independent, since some can be derived from others. Even so, our goal of reducing the number of unknowns has clearly not been met.<br />
<br />
Equations governing each of these new quantities can be derived from the original dynamical equations, just as we did for the Reynolds stress. Unfortunately new quantities continue to be introduced with each new equation, and at a faster rate than the increase in the number of equations. Now the full implications of the closure problem introduced by the Reynolds decomposition and averaging has become apparent. No matter how many new equations are derived, the number of new unknown quantities introduced will always increase more rapidly.<br />
<br />
Our attempt to solve the turbulence problem by considering averages illustrates a general principle. Any time we try to fool Mother Nature by averaging out her details, she gets her revenge by leaving us with closure problem - more equations than unknowns. In thermodynamics, we tried to simplify the consideration of molecules by averaging over them, and were left with the need for an equation of state. In heat transfer, we tried to simplify considerations by which molecules transfer their kinetic energy, and found we were lacking a relation between the heat flux and the temperature field. And in fluid mechanics, we tried to simplify consideration of the "mean" motionof molecules and ended up with viscous stress. In all of these cases we were able to make simple physical models which worked at least some of the time; e.g., ideal gas, Fourier-Newtonian fluid. And these models all worked because we were able to make assumptions about the underlying molecular processes and assume them to be independent of the macroscopic flows of interest. Unfortunately such assumptions are rarely satisfied in turbulence. <br />
<br />
It should be obvious by now that the turbulence closure problem will not be solved by the straight-forward derivation of new equations, nor by direct analogy with viscous stresses. Rather, ''closure attempts will have to depend on an intimate knowledge of the dynamics of the turbulence itself''. Only by understanding how the turbulence behaves can one hope to ''guess'' an appropriate set of constitutive equations '''AND''' ''understand the limits of them''. This is, of course, another consequence of the fact that the ''turbulence is a property of the flow itself, and not of the fluid''!<br />
<br />
{{Turbulence credit wkgeorge}}<br />
<br />
{{Chapter navigation|Statistical analysis|Turbulence kinetic energy}}</div>Callahancehttps://www.cfd-online.com/Wiki/Introduction_to_turbulence/Reynolds_averaged_equationsIntroduction to turbulence/Reynolds averaged equations2013-04-27T22:00:02Z<p>Callahance: /* Reynolds stress equations */</p>
<hr />
<div>{{Introduction to turbulence menu}}<br />
<br />
== Equations governing instantaneous fluid motion ==<br />
<br />
All fluid motions, whether turbulent or not, are governed by the dynamical equations for a fluid. These can be written using Cartesian tensor notation as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac{\partial \tilde{p}}{\partial x_i}+\frac{\partial \tilde{T}_{ij}^{(v)}}{\partial x_j}</math><br />
</td><td width="5%">(1)</td></tr></table><br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left[\frac{\partial \tilde{\rho}}{\partial t}+\tilde{u}_j\frac{\partial \tilde{\rho}}{\partial x_j}\right]+ \tilde{\rho}\frac{\partial \tilde{u}_j}{\partial x_j}= 0 </math><br />
</td><td width="5%">(2)</td></tr></table><br />
<br />
where <math>\tilde{u_i}(\vec{x},t)</math> represents the i-the component of the fluid velocity at a point in space,<math>[\vec{x}]_i=x_i</math>, and time,t. Also <br />
<math>\tilde{p}(\vec{x},t)</math> represents the static pressure, <math>\tilde{T}_{ij}^{(v)}(\vec{x},t)</math>, the viscous(or deviatoric) stresses, and <math>\tilde\rho</math> the fluid density. The tilde over the symbol indicates that an instantaneous quantity is being considered. Also the [[einstein summation convention]] has been employed.<br />
<br />
In equation 1, the subscript <math>i</math> is a free index which can take on the values 1,2 and 3. Thus equation 1 is in reality three separate equations. These three equations are just Newton's second law written for a continuum in a spatial(or Eulerian) reference frame. Together they relate the rate of change of momentum per unit mass <math>(\rho{u_i})</math>,a vector quantity, to the contact and body forces.<br />
<br />
Equation 2 is the equation for mass conservation in the absence of sources(or sinks) of mass. Almost all flows considered in this material will be incompressible, which implies that derivative of the density following the fluid material[the term in brackets] is zero. Thus for incompressible flows, the mass conservation equation reduces to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{D \tilde{\rho}}{Dt}=\frac{\partial \tilde{\rho}}{\partial t}+\tilde{u}_j\frac{\partial \tilde{\rho}}{\partial x_j}= 0</math><br />
</td><td width="5%">(3)</td></tr></table><br />
<br />
From equation 2 it follows that for incompressible flows,<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial \tilde{u}_j}{\partial x_j}= 0</math><br />
</td><td width="5%">(4)</td></tr></table><br />
<br />
The viscous stresses(the stress minus the mean normal stress) are represented by the tensor<math>\tilde{T}_{ij}^{(v)}</math>. From its definition,<math>\tilde{T}_{kk}^{(v)}</math>=0. In many flows of interest, the fluid behaves as a Newtonian fluid in which the viscous stress can be related to the fluid motion by a constitutive relation of the form.<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{T}_{ij}^{(v)}= 2\mu[\tilde{s}_{ij}-\frac{1}{3}\tilde{s}_{kk}\delta_{ij}] </math><br />
</td><td width="5%">(5)</td></tr></table><br />
<br />
The viscosity, <math>\mu</math>, is a property of the fluid that can be measured in an independent experiment. <math>\tilde s_{ij}</math> is the instantaneous strain rate tensor defined by<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{s}_{ij}= \frac{1}{2}\left[\frac{\partial \tilde u_i}{\partial x_j}+\frac{\partial \tilde u_j}{\partial x_i}\right] </math><br />
</td><td width="5%">(6)</td></tr></table><br />
<br />
From its definition, <math>\tilde s_{kk}=\frac{\partial \tilde u_k}{\partial x_k}</math>. If the flow is incompressible, <math>\tilde s_{kk}=0</math> and the Newtonian constitutive equation reduces to<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{T}_{ij}^{(v)}= 2\mu\tilde{s}_{ij}</math><br />
</td><td width="5%">(7)</td></tr></table><br />
<br />
Throughout this material, unless explicitly stated otherwise, the density <math>\tilde\rho=\rho</math> and the viscosity <math>\mu</math> will be assumed constant. With these assumptions, the instantaneous momentum equations for a Newtonian Fluid reduce to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac {1}{\tilde\rho}\frac{\partial \tilde{p}}{\partial x_i}+\nu\frac{\partial^2 {\tilde{u}_i}}{\partial x_j^2}</math><br />
</td><td width="5%">(8)</td></tr></table><br />
<br />
where the kinematic viscosity, <math>\nu</math>, has been defined as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\nu\equiv\frac{\mu}{\rho}</math><br />
</td><td width="5%">(9)</td></tr></table><br />
<br />
Note that since the density is assumed constant, the tilde is no longer necessary.<br />
<br />
Sometimes it will be more instructive and convenient to not explicitly include incompressibilty in the stress term, but to refer to the incompressible momentum equation in the following form:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac{\partial \tilde{p}}{\partial x_i}+\frac{\partial \tilde{T}_{ij}^{(v)}}{\partial x_j}</math><br />
</td><td width="5%">(10)</td></tr></table><br />
<br />
This form has the advantage that it is easier to keep track of the exact role of the viscous stresses.<br />
<br />
== Equations for the average velocity ==<br />
<br />
Although laminar solutions to the equations often exist that are consistent with the boundary conditions, perturbations to these solutions(sometimes even infinitesimal) can cause them to become turbulent. To see how this can happen, it is convenient to analyze the flow in two parts, a mean(or average) component and a fluctuating component. Thus the instantaneous velocity and stresses can be written as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\tilde {u}_i=U_i+u_i<br />
</math><br />
:<math><br />
\tilde p=P+p<br />
</math><br />
:<math><br />
\tilde T_{ij}^{(v)}=T_{ij}^{(v)}+\tau_{ij}^{(v)}<br />
</math><br />
</td><td width="5%">(11)</td></tr><br />
</table><br />
<br />
Where <math>U_i</math>, <math>P</math> and <math>T_{ij}^{(v)}</math> represent the mean motion, and <math>u_i</math>, <math>p</math> and <math>\tau_{ij}^{(v)}</math> the fluctuating motions. This technique for decomposing the instantaneous motion is referred to as the '''''Reynolds decomposition.''''' Note that if the averages are defined as ensemble means, they are, in general, time-dependent. For the remainder of this material unless other wise stated, the density will be assumed constant so<math>\tilde{\rho}\equiv\rho</math>,and its fluctuation is zero.<br />
<br />
Substitution of equations 11 into equations 10 yields<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial (U_i+u_i)}{\partial t}+(U_j+u_j)\frac{\partial (U_i+u_i)}{\partial x_j}\right] = -\frac{\partial (P+p)}{\partial x_i}+\frac{\partial (T_{ij}^{(v)}+\tau_{ij}^{(v)})}{\partial x_j}</math><br />
</td><td width="5%">(12)</td></tr></table><br />
<br />
This equation can now be averaged to yield an equation expressing momentum conservation for the averaged motion. Note that the operations of averaging and differentiation commute; i.e., the average of a derivative is the same as the derivative of the average. Also the average of a fluctuating quantity is zero. Thus the equation for the averaged motion reduces to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial U_i}{\partial t}+U_j\frac{\partial U_i}{\partial x_j}\right] = -\frac{\partial P}{\partial x_i}+\frac{\partial T_{ij}^{(v)}}{\partial x_j}-\rho\left \langle u_j\frac{\partial u_i }{\partial x_j} \right \rangle</math><br />
</td><td width="5%">(13)</td></tr></table><br />
<br />
where the remaining fluctuating product term has been moved to the right hand side of the equation. Whether or not the last term is zero like the other fluctuating term depends on the correlation of the terms in the product. In general, these correlations are not zero.<br />
<br />
The mass conservation equation can be similarly decomposed. In incompressible form, substitution of equations 11 into equation 4 yields:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial (U_j+u_j)}{\partial x_j}=0</math><br />
</td><td width="5%">(14)</td></tr></table><br />
<br />
of which average is <br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial U_j}{\partial x_j}=0</math><br />
</td><td width="5%">(15)</td></tr></table><br />
<br />
It is clear from equation 15 that the averaged motion satisfies the same form of the mass conservation equation as does the instantaneous motion at least for incompressible flows. How much simpler the turbulence problem would be if the same were true for the momentum! Unfortunately, as is easily seen from equation 13, such is not the case.<br />
<br />
Equation 15 can be subtracted from equation 14 to yield an equation for instantaneous motion alone; i.e,<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial u_j}{\partial x_j}=0</math><br />
</td><td width="5%">(16)</td></tr></table><br />
<br />
Again, like the mean, the form of the original instantaneous equation is seen to be preserved. The reason, of course, is obvious: the continuity equation is linear. The momentum equation , on the other hand, is not; hence the difference.<br />
<br />
Equation 16 can be used to rewrite the last term in equation 13 for the mean momentum. Multiplying equation 16 by <math>u_i</math> and averaging yields:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left \langle u_i\frac{\partial u_j}{\partial x_j}\right \rangle=0</math><br />
</td><td width="5%">(17)</td></tr></table><br />
<br />
Now add:<math>\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle</math> to both sides of eqaution 17, then using the product rule you obtain:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle +0=\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle+ \left \langle u_i\frac{\partial u_j}{\partial x_j}\right \rangle =\frac{ \partial}{\partial x_j}{\left \langle u_iu_j\right \rangle} </math><br />
</td><td width="5%">(18)</td></tr></table><br />
<br />
Where again the fact that arithmetic and averaging operations commute has been used.<br />
<br />
The equation for the averaged momentum, equation 13 can now be rewritten as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial U_i}{\partial t}+U_j\frac{\partial U_i}{\partial x_j}\right] = -\frac{\partial P}{\partial x_i}+\frac{\partial T_{ij}^{(v)}}{\partial x_j}-\frac{ \partial}{\partial x_j}{(\rho\left \langle u_iu_j\right \rangle)}</math><br />
</td><td width="5%">(19)</td></tr></table><br />
<br />
The last two terms on the right hand side are both divergence terms and can be combined; the result is:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial U_i}{\partial t}+U_j\frac{\partial U_i}{\partial x_j}\right] = -\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_j}[T_{ij}^{(v)}-{\rho\left \langle u_iu_j\right \rangle}]</math><br />
</td><td width="5%">(20)</td></tr></table><br />
<br />
Now the terms in square brackets on the right have the dimensions of stress. The first term is, in fact , the viscous stress. The second term, on the other hand, is not a stress at all, but simply a re-worked version of the fluctuating contribution to the non-linear acceleration terms. The fact that it can be written this way, however, indicates that at least as far as the motion is concerned, it acts as though it were a stress- hence its name, the '''Reynolds stress'''. In the succeeding sections the consequences of this difference will be examined.<br />
<br />
== The turbulence problem ==<br />
<br />
It is the appearance of the Reynolds stress which makes the turbulence problem so difficult - at least from the engineers perspective. Even though we can pretend it is a stress, the physics which give rise to it are very different from the viscous stress. The viscous stress can be related directly to the other flow properties by constitutive equations, which in turn depend only on the properties of the fluid (as in equation 5 for a Newtonian fluid). The reason this works is that when we make such closure approximations for a fluid, we are averaging over characteristic length and time scales much smaller than those of the flows we are interested in. Yet at the same time, these scales are much larger than the molecular length and time scales which characterize the molecular interactions that are actually causing the momentum transfer. (This is what the continuum approximation is all about).<br />
<br />
The '''''Reynolds stress''''', on the other hand, arises from the flow itself! Worse, the scales of the fluctuating motion which give rise to it are the scales we are interested in. This means that the closure ideas which worked so well for the viscous stress, should not be expected to work too well for the Reynolds stress. And as we shall see, they do not.<br />
<br />
This leaves us in a terrible position. Physics and engineering are all about writing equations(and boundary conditions) so we can solve them to make predictions. We don't want to have a build prototype airplanes first to see if they will they fall out of the sky. Instead we want to be able to analyze our designs before building, to save the cost in money and lives if our ideas are wrong. The same is true for dams and bridges and tunnels and automobiles. If we had confidence in our turbulence models, we could even build huge one-offs and expect them to work the first time. Unfortunately, even though turbulence models have improved to the point where we can use them in design, we still cannot trust them enough to eliminate expensive wind tunnel and model studies. And recent history is full of examples to prove this.<br />
<br />
The turbulence problem (from the engineers perspective) is then three-fold:<br />
<br />
* '''The averaged equations are not closed.''' Count the number of unknowns in equation 20 above. Then count the number of equations. Even with the continuity equation we have atleast six equations too few.<br />
<br />
* '''The simple ideas to provide the extra equations usually do not work.''' And even when we can fix them up for a particular class of flows (like the flow in a pipe, for example), they will most likely not be able to predict what happens in even a slightly different environment (like a bend).<br />
<br />
*'''Even the last resort of compiling engineering tables for design handbooks carries substantial risk.''' This is the last resort for the engineer who lacks equations or cannot trust them. Even when based on a wealth of experience, they require expensive model testing to see if they can be extrapolated to a particular situation. Often they cannot, so infinitely clever is Mother Nature in creating turbulence that is unique to a particular set of boundary conditions.<br />
<br />
'''Turbulent flows are indeed flows!'''. And that is the problem.<br />
<br />
== Origins of turbulence==<br />
<br />
Turbulent flows can often be observed to arise from laminar flows as the Reynolds number, (or someother relevant parameter) is increased. This happens because small disturbances to the flow are no longer damped by the flow, but begin to grow by taking energy from the original laminar flow. This natural process is easily visualized by watching the simple stream of water from a faucet (or even a pitcher). Turn the flow on very slow (or pour) so the stream is very smooth initially, at least near the outlet. Now slowly open the faucet (or pour faster) abd observe what happens, first far away, then closer to the spout. The surface begins to exhibit waves or ripples which appear to grow downstream . In fact, they are growing by extracting energy from the primary flow. Eventually they grow enough that the flow breaks into drops. These are capillary instabilities arisiing from surface tension, but regardless of the type of instability, the idea is the same -small (or infinitesimal ) disturbances have grown to disrupt the serenity (and simplicity) of laminar flow.<br />
<br />
The manner in which the instabilities grow naturally in a flow can be examined using the equations we have already developed above. We derived them by decomposing the motion into a mean and fluctuating part. But suppose instead we had decomposed the motion into a base flow part (the initial laminar part) and into a disturbance which represents a fluctuating part superimposed on the base flow. The result of substituting such a decomposition into the full Navier-Stokes equations and averaging is precisely that given by equations (13) and (15). But the very important difference is the additional restriction that what was previously identified as the mean (or averaged ) motion is now also the base or laminar flow.<br />
<br />
Now if the base flow is really laminar flow (which it must be by our original hypothesis), then our averaged equations governing the base flow must yield the same mean flow as the original laminar flow on which the disturbances was superimposed. But this can happen only if these new averaged equations reduce to '''exactly''' the same lamiane flow equations without any evidence of a disturbance. Clearly from equations 13 and 15, this can happen ''only if all the Reynolds stress terms vanish identically!'' Obviously this requires that the disturbances be infintesimal so the extra terms can be neglected - hence our interest in infinitesimal disturbances.<br />
<br />
So we hypothesized a base flow which was laminar and showed that it is unchanged even with the imposition of infintesimal disturbances on it - ''but only as long as the disturbances'' '''remain''' ''infinitesimal!'' What happens if the disturbance starts to grow? Obviously before we conclude that all laminar flows are laminar forever we better investigate whether or not these infinitesimal disturbances can grow to ''finite'' size. To do this we need an equation for the fluctuation itself.<br />
<br />
An equation for the fluctuation (which might be an imposed disturbance) can be obtained by subtracting the equation for the mean (or base) flow from that for the instantaneous motion. We already did this for the continuity equation. Now we will do it for the momentum equation. Subtracting equation 13 from equation 11 yields an equation for the fluctuation as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial u_{i}}{\partial t} + U_{j}\frac{\partial u_{i}}{\partial x_{j}} \right] = - \frac{\partial p}{\partial x_{i}} + \frac{\partial\tau^{(v)}_{ij}}{\partial x_{j}} - \rho \left[ u_{j}\frac{\partial U_{i}}{\partial x_{j}} \right] - \left\{ u_{j} \frac{\partial u_{i}}{ \partial x_{j}} - \rho \left\langle u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right\}<br />
</math><br />
</td><td width="5%">(21)</td></tr></table><br />
<br />
It is very important to note the type and character of the terms in this equation. First note that the left-hand side is the derivative of the ''fluctuating'' velocity following the ''mean'' motion. This is exactly like the term which appears on the left-hand side of the equation for the mean velocity, equation 13. The first two terms on the right-hand side are also like those in the mean motion, and represent the fluctuating pressure gradient and the fluctuating viscous stresses. The third term on the right-hand is is new, and will be seen later to represent the primary means by which fluctuations (and turbulence as well!) extract energy from the mean flow, the so-called ''production terms''. The last term is quadratic in the fluctuating velocity, unlike all the otherwhich are linear. Note that all of the terms vanish identically if the equation is averaged, the last because its mean is subtructed from it.<br />
<br />
Now we want to examine what happens if the disturbance is small. In the limit as the amplitude of the disturbance (or fluctuation) is ''infinitesmal'', the bracketed term in the equationfor the fluctuation vanishes (since it involves productsof infinitesimals) , and the remaining equation is ''linear in the disturbance''. The study of whether or not such infinitesmal disturbances can grow is called '''Linear Fluid Dynamic Stability Theory'''. These linearized equations are very different from those govering turbulence. Unlike the equations for disturbances of ''finite'' amplitude, the linearized equations are well-posed (or closed) since the Reynolds stress terms are gone.<br />
<br />
The absence of the non-linear terms, however, constrains the validity of the linear analysis to only the initial stage of disturbance growth. This is because as soon as the fluctuations begin to grow, their amplitudes can no longer be assumed infinitesmal and the Reynolds stress (or more properly, the non-linear fluctuating terms) become important. As a result the base flow equations begin to be modified so that the solution to them can no longer be identical to the laminar flow (or base flow) from which it arose. Thus while linear stability theory can predict ''when'' many flows become ''unstable'', it can say very little about ''transition to turbulence'' since this progress is highly non-linear.<br />
<br />
== Importance of non-linearity ==<br />
<br />
We saw in the preceding section that non-linearity was one of essential features of turbulence. When small disturbances grow large enough to interact ''with each other'', we enter a whole new world of complex behavior. Most of the rules we learned for linear system do not apply. Since most of your mathematical training has been for linear equations, most of your mathematical intuition therefore will not apply either. On the other hand, you may surprise yourself by discovering how much your ''non-mathematical'' intuition already recognizes non-linear behavior and accounts for it.<br />
<br />
Considering the following simple example. Take a long stick with one person holding each end and stand at the corner of a building. Now place the middle of against the building and let each person apply pressure in the same direction so as to bend the stick. If the applied force is small, the stick deflects (or bends) a small amount. Double the force, and the deflection is approximately doubled. Quadruple the force and the deflection is quadrupled. Now you don't need a Ph.D. in Engineering to know what is going to happen if you continue this process. '''The stick is going to break!'''<br />
<br />
But where in the equations for the deflection of the stick is there anything that predicts this can happen? Now if you are only like engineer, you are probably thinking: he's asking a stupid question. Of course you can't continue to increase the force because you will exceed first the yield stress, then the breaking limit, and of course the stick will break.<br />
<br />
But pretend I am the company president with nothing more than MBA. I don't know much about these things, but you have told me in the past that your computers have equations to predict everything. So I repeat: Where in the equations for the deflection of this stick does it tell me this going to happen?<br />
<br />
The answer is very simple: '''There is''' ''nothing'' '''in the equations that will predict this.''' And the reason is also quite simple: You lost the ability to predict catasrophes like breaking when you linearized the fundamental equations - which started out as Newton's Law too. In fact, before linearization, they were exactly the same as those for a fluid, only the constitutive equation was different.<br />
<br />
If we had NOT linearized these equations and had constituve equations that were more general, then we possibly could apply these equation right to and past the limit. The point of fracture would be a bifurcation point for the solution.<br />
<br />
Now the good news is that for things like reasonable deflections of beams linearization work woderfully since we hope most things we build don't deflect too much. Unfortunately, as we noted above, for fluids the disturbances tend to quickly become dominated by non-linear terms. This, of course, means our linear analytical techniques are pretty useless for fluid mechanics, and especially turbulence.<br />
<br />
But all is not lost. Just as we learned to train ourselves to anticipate when sticks break, we have to train ourselves to anticipate how non-linear fluid phenomena behave. Toward that end we will consider two simple examples: one from algebra - the logistic map, and one from fluid mechanics - simple vortex streching.<br />
<br />
'''Example 1:''' An experiment with the logistic map. <br />
<br />
Consider the behavior of the simple equation:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
y_{n+1}= r y_{n} \left(1- y_{n} \right)<br />
</math><br />
</td><td width="5%">(22)</td></tr></table><br />
<br />
where <math> n= 1,2..., 0<y < 1 </math> and <math> r > 0 </math>. The idea is that you pick any value for <math> y_{1}</math>, use the equation to find <math> y_{2}</math>, then insert that value on the right-hand side to find <math> y_{3}</math>, and just continue the process as long as you like. Make sure you note any dependece of the final result on the initial value for <math> y </math>.<br />
<br />
*First notice what happens if you linearize this equation by disregarding the term in parentheses; i.e., comsider the simpler equation <math> y_{1+1} = r y_{n}</math>. My guess is that you won't find this too eciting - unless, of course, you are one of those rare individuals who likes watching grass grow.<br />
<br />
* Now consider the full equation and note what happens for <math> r<3 </math> , and especially what happens for very small values of <math> r </math>. Run as many iterations as necessary to make sure your answer has converged. Do NOT try to take short-cuts by programming all the steps at once. Do them one at time so you can see what is happening. It will be much easier this way in the long run.<br />
<br />
* Now research carefully what happens when <math> r=3.1, 3.5, </math> and <math> 3.8 </math>. Can you recognize any patterns.<br />
<br />
* Vary <math> r </math> between 3 and 4 to see if you can find the boundaries for what you are observing. <br />
<br />
* Now try values of <math> r>4 </math>. How do you explain this<br />
<br />
'''Example 2:''' Stretching of a simple vortex.<br />
<br />
Imagine a simple vortex filament that looks about like a strand of spaghetti. Now suppose it is in otherwise steady inviscid incompressible flow. Use the vorticity equation to examine the following: <br />
<br />
* Examine first what happens to it in two-dimensional velocity field. Note particularly whether any new vorticity can be produced; i.e., can the material derivative of the vorticity ever be greater than zero? (Hint: look at the <math> \omega_{j} \partial u_{i}/ \partial x_{j} <br />
</math> - term)<br />
<br />
* Now consider the same vortex filament in a three-dimensional flow. Note particularly the various ways new vorticity can be produced - if you have some to start with! Does all this have anything to do with non-linearities?<br />
<br />
== Turbulence closure problem and eddy viscosity ==<br />
<br />
From the point of view of the averaged motion, at least, the problem with the non-linearity of the instaneous equations is that they introduce new unknowns, the Reynolds stress into the averaged equations. There are six individual stress components we must deal with to be exact: <math> \left\langle u^{2}_{1} \right\rangle </math> , <math> \left\langle u^{2}_{2} \right\rangle </math> , <math> \left\langle u^{2}_{3} \right\rangle </math> , <math> \left\langle u_{1}u_{2} \right\rangle </math> , <math> \left\langle u_{1}u_{3} \right\rangle </math> , and <math> \left\langle u_{2}u_{3} \right\rangle </math>. These have to be related to the mean motion itself before the equations can be solved, since the number of unknows and number of equations must be equal. The absence of these additional equations is often reffered to as '''the Turbulence Closure Problem'''.<br />
<br />
A similar problem arose when the instantaneous equations were written (equations 1 2), since relations had to be introduced to relate the stresses (in particular, the viscous stresses) to the motion itself. These relations (or constitutive equations) ''depended only on the properties of the fluid material, and not on the flow itself''. Because of this fact, it is possible to carry out independent experiments, called viscometric experiments, in which these fluid properties can be determinded once and for all. Equation 5 provides an example just such a constitutive relation, the viscosity, <math>\mu </math>, depending only in the choice of the material. For example, once the viscosity of water at given temperature is determined, this value can be used in all flows at that temperature, not just the one in which the evaluation was made.<br />
<br />
It is tempting to try such an approach for the turbulence Reynolds stresses (even though we know the underlying requirements of scale separation are not satisfied). For example, a Newtonian type closure for the Reynolds stresses, often referred to as an "eddy" or "turbulent" viscosity model, looks like:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{i} u_{j} \right\rangle = 2 \mu_{t} \left[ S_{ij} - \frac{1}{3} S_{kk} \delta_{ij} \right]<br />
</math><br />
</td><td width="5%">(23)</td></tr></table><br />
<br />
where <math>\mu_{t} </math> is the turbulence "viscosity" (also called the eddy viscosity) and <math>S_{ij} </math> is the ''mean'' strain rate defined by:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
S_{ij}= \frac{1}{2} \left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right]<br />
</math><br />
</td><td width="5%">(24)</td></tr></table><br />
<br />
The second term of course, vanishes identically for incompressible flow. For the simple case of a two-dimensional shear flow, equation 3.23 for the Reynolds shear stress reduces to<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{1} u_{2} \right\rangle = \mu_{t} \frac{\partial U_{1}}{\partial x_{2}} <br />
</math><br />
</td><td width="5%">(25)</td></tr></table><br />
<br />
That such a simple model can adequately describe the mean motion in at least one flow is illustrated by the axisymmetric buoyant plume sketched in <font color="orange">Figure 3.1. Figures 3.2 and 3.3</font> show the calculation of the mean velocity and temperature profiles respectively. Obviously the mean velocity and temperature are reasonably accurately computed, as are the Reynolds shear stress and lateral turbulent heat flux shown in <font color ="orange">Figures 3.4 and 3.5</font>.<br />
<br />
The succes of the eddy viscosity in the preceding example is more apparent than real, however, since the value of the eddy viscosity and eddy diffusivity (for turbulent heat flux) have been chosen to give the best possible agreement with the data. This, in itself, would not be a problem if that chosen values could have been obtained in advance of the computation, or even if they could be used to successfully predict other flows. In fact, the values used work only for this flow, thus the computation is ''not a prediction at all, but a'' '''postdiction''' ''or'' '''hindcast''' from which no extrapolation to the future can be made. In other words, our turbulence "model" is about as useful as having a program to predict yesterday's weather. Thus the closure problem still very much remains.<br />
<br />
<font color="orange" size="3">Figure 3.1 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.2 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.3 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.4 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.5 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.6 not uploaded yet</font><br />
<br />
Another problem with the eddy viscosity in the example above is that it fails to calculate the vertical components of the Reynolds stress and turbulent heat flux. An attempt at such a computation is shown in <font color ="orange">Figure 3.6</font> where the vertical turbulent heat flux is shown to be severely underestimated. Clearly the value of the in the vertical direction must be different than in the radial direction. In other words, the turbulence for which a constituve equation is being written is ''not an isotropic "medium" ''. In fact, in this specific example the problem is that the vertical component of the heat flux is produced more by the interaction of buoyancy and the turbulence, than it is by the working of turbulence against mean gradient in the flow. We will discuss this in more detail in the next chapter when we consider the turbulence energy balances, but note for now that simple gradient closure models never work unless gradient production dominates. This rules out many flows involving buoyancy, and also many involving recirculations or separation where the local turbulence is convected in from somewhere else. <br />
<br />
A more general form of constitutive equation which would allow for the nonisotropic nature of the "medium" (in this case the turbulence itself) would be<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{i} u_{j} \right\rangle = \mu_{ijkl} \left[ S_{kl} - \frac{1}{3} S_{mm} \delta_{kl} \right]<br />
</math><br />
</td><td width="5%">(26)</td></tr></table><br />
<br />
This closure relation allows each component of the Reynolds stress to have its own unique value of the eddy viscosity. It is easy to see that it is unlikely this will solve the closure problem since the original six unknowns the <math>\left\langle u_{i} u_{j} \right\rangle </math> have been traded for eighty-one new ones, <math> \mu_{ijkl} </math>. Even if some can be removed by symmetries, the remaining number is still formidable. More important than the number of unknowns, however, is that there is no independent or general means for selecting them without considering a particular flow. This is because ''turbulence is indeed a property of the flow, not of the fluid''.<br />
<br />
== Reynolds stress equations ==<br />
<br />
It is clear from the preceding section that the simple idea of an eddy viscosity might not be the best way to approach the problem of relating the Reynolds stress to the mean motion. An alternative approach is to try to derive dynamical equations for the Reynolds stresses from the equations governing the fluctuations themselves. Such an approach recognizes that the Reynolds stress is really a functional of the velocity; that is, the stress at a point depends on the velocity everywhere and for all past times, not just at the point in question and at a particular instant in time.<br />
<br />
The analysis begins with the equation for the instantaneous fluctuating velocity, equation 21. This can be rewritten for a Newtonian fluid with constant viscosity as: <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\rho \left[ \frac{\partial u_{i}}{\partial t} + U_{j} \frac{\partial u_{i}}{\partial x_{j}} \right] = -\frac{\partial p}{\partial x_{i}} + \frac{\partial \tau ^{ ( v ) }_{ij}}{ \partial x_{j}} - \rho \left[ u_{j} \frac{\partial U_{i}}{ \partial x_{j} } \right] - \rho \left\{ u_{j} \frac{\partial u_{i}}{ \partial x_{j} } - \left\langle u_{j} \frac{\partial u_{i} }{\partial x_{j} } \right\rangle \right\}<br />
</math><br />
</td><td width="5%">(27)</td></tr></table><br />
<br />
Note that the free index in this equation is <math> i </math>. Also, since we are talking about turbulence again, the capital letters represent mean or averaged quantities. <br />
<br />
Multiplying equation 27 by <math> u_{k} </math> and averaging yields:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\rho \left[ \left\langle u_{k} \frac{\partial u_{i} }{\partial t} \right\rangle + U_{j} \left\langle u_{k} \frac{\partial u_{i} }{\partial x_{j}} \right\rangle \right] = & - & \left\langle u_{k} \frac{\partial p }{\partial x_{i} } \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij} }{\partial x_{j}} \right\rangle \\<br />
& - & \rho \left[ \left\langle u_{k}u_{j} \right\rangle \frac{\partial U_{i} }{\partial x_{j} } \right] - \rho \left\{ \left\langle u_{k}u_{j} \frac{\partial u_{i} }{\partial x_{j} } \right\rangle \right\} \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(28)</td></tr></table><br />
<br />
Now since both <math> i </math> and <math> k </math> are free indices they can be interchanged to yield a second equation given by (alternatively equation 21 can be rewritten with free index <math> k </math>, then multiplied by <math> u_{i} </math> and averaged):<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\rho \left[ \left\langle u_{i} \frac{\partial u_{k} }{\partial t} \right\rangle + U_{j} \left\langle u_{i} \frac{\partial u_{k} }{\partial x_{j}} \right\rangle \right] = & - & \left\langle u_{i} \frac{\partial p }{\partial x_{k} } \right\rangle + \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj} }{\partial x_{j}} \right\rangle \\<br />
& - & \rho \left[ \left\langle u_{i}u_{j} \right\rangle \frac{\partial U_{k} }{\partial x_{j} } \right] - \rho \left\{ \left\langle u_{i}u_{j} \frac{\partial u_{k} }{\partial x_{j} } \right\rangle \right\} \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(29)</td></tr></table><br />
<br />
Equations 28 and 29 can be added together to yield an equation for the Reynolds stress<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\frac{\partial\left\langle u_{i}u_{k} \right\rangle}{\partial t} + U_{j}\frac{\partial\left\langle u_{i}u_{k} \right\rangle}{\partial x_{j}} <br />
& = & - \frac{1}{\rho} \left[ \left\langle u_{i} \frac{\partial p}{\partial x_{k}} \right\rangle + \left\langle u_{k} \frac{\partial p}{\partial x_{i}} \right\rangle \right] \\<br />
& - & \left[ \left\langle u_{i}u_{j} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle u_{k}u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
& + & \frac{1}{\rho} \left[ \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj} }{\partial x_{j}} \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij} }{\partial x_{j}} \right\rangle \right] \\<br />
& - & \left[ \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{k}}{\partial x_{j}} + \left\langle u_{k} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(30)</td></tr></table><br />
<br />
It is customary to rearrange the first term on the right hand side in the following way:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \left\langle u_{i} \frac{\partial p}{\partial x_{k}} \right\rangle + \left\langle u_{k} \frac{\partial p}{\partial x_{i}} \right\rangle \right] = & & \left\langle p \left[ \frac{\partial u_{i}}{\partial x_{k}} + \frac{\partial u_{k}}{\partial x_{i}} \right] \right\rangle \\<br />
& - & \frac{\partial}{\partial x_{j}} \left[ \left\langle pu_{i} \right\rangle \delta_{kj} + \left\langle pu_{k} \right\rangle \delta_{ij} \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(31)</td></tr></table><br />
<br />
The first term on the right is generally referred to as the ''pressure strain-rate'' term. The second term is written as a divergence term, and generally referred to as the ''pressure diffusion'' term. We shal see later that divergence terms can never create nor destroy anything; they can simple move it around from one place to another. <br />
<br />
The third term on the right-hand side of equation 30 can similarly be rewritten as:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj}}{\partial x_{j}} \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij}}{\partial x_{j}} \right\rangle \right] = & - & \left[ \left\langle \tau^{(v)}_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle \tau^{(v)}_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
& + & \frac{\partial}{\partial x_{j}} \left[ \left\langle u_{i} \tau^{(v)}_{kj} \right\rangle + \left\langle u_{k} \tau^{(v)}_{ij} \right\rangle \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(32)</td></tr></table><br />
<br />
The first of these is also a divergence term. For a Newtonian fluid, the last is the so-called "dissipation of Reynolds stress" by the turbulence viscous stresses. This is easily seen by substituting the Newtonian constitutive relation to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left[ \left\langle \tau^{(v)}_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle \tau^{(v)}_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] = 2 \nu \left[ \left\langle s_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle s_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right]<br />
</math><br />
</td><td width="5%">(33)</td></tr></table><br />
<br />
It is not at all obvious what this has to do with dissipation, but it will become clear later on when we consider the trace of the Reynolds stress equation, which is the ''kinetic energy'' equation for the turbulence.<br />
<br />
Now if we use the same trick from before using the continuity equation, we can rewrite the third term in equation 30 to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left[ \left\langle u_{i}u_{j} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle u_{k}u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] = \frac{\partial}{\partial x_{j}} \left\langle u_{i} u_{k} u_{j} \right\rangle<br />
</math><br />
</td><td width="5%">(34)</td></tr></table><br />
<br />
This is also a divergence term.<br />
<br />
We can use all of the pieces we have developed above to rewrite equation 30 as <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\frac{\partial}{\partial t} \left\langle u_{i}u_{k} \right\rangle & + & U_{j} \frac{\partial}{\partial x_{j}} \left\langle u_{i}u_{k} \right\rangle = - \left\langle \frac{p}{\rho} \left[ \frac{\partial u_{i}}{\partial x_{k}} + \frac{\partial u_{i}}{\partial x_{k}} \right] \right\rangle \\<br />
& + & \frac{\partial}{\partial x_{j}} \left\{ - \left[ \left\langle pu_{k} \right\rangle \delta_{ij} + \left\langle pu_{i} \right\rangle \delta_{kj} \right] - \left\langle u_{i}u_{k}u_{j} \right\rangle + 2\nu \left[ \left\langle s_{ij}u_{k} \right\rangle + \left\langle s_{ij}u_{k} \right\rangle \right] \right\}\\<br />
& - & \left[ \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{k}}{\partial x_{j}} + \left\langle u_{k} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} \right] \\<br />
& - & 2\nu \left[ \left\langle s_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle s_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(35)</td></tr></table><br />
<br />
This is the so-called '''Reynolds Stress Equation''' which has been the primary vehicle for much of the turbulence modelling efforts of the past few decades.<br />
<br />
The left hand side of the Reynolds Stress Equation can easily be recognized as the rate of change of the Reynolds stress following the mean motion. It seems to provide exactly what we need: nine new equations for the nine unknowns we can not account for. The problems are all on the right-hand side. These terms are referred to respectively as<br />
<br />
* the pressure-strain rate term<br />
* the turbulence transport (or divergence) term<br />
* the "production" term, and<br />
* the "dissipation" term.<br />
<br />
Obviously these equations do not involve only <math> U_{i} </math> and <math> \left\langle u_{i} u_{j} \right\rangle </math>, but depend on many more new unknowns.<br />
<br />
It is clear that, contrary to our hopes, we have not derived a single equation relating the Reynolds stress to the mean motion. Instead, our Reynolds stress transport transport equation is exceedingly complex. Whereas the process of averaging the equation for the mean motion introduced only six new, independent unknowns, the Reynolds stress, <math> \left\langle u_{i} u_{j} \right\rangle </math>, the search for a transport equation which will relate these to the mean motion has produced many more unknowns. They are<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle p u_{i} \right\rangle - 3<br />
</math><br />
</td><td width="5%">(36)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle u_{i} s_{jk} \right\rangle - 27<br />
</math><br />
</td><td width="5%">(37)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle s_{ij}s_{jk} \right\rangle - 9<br />
</math><br />
</td><td width="5%">(38)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle u_{i}u_{k}u_{j} \right\rangle - 27<br />
</math><br />
</td><td width="5%">(39)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle p\frac{\partial u_{i}}{\partial x_{j}} \right\rangle - 9<br />
</math><br />
</td><td width="5%">(40)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
TOTAL - 75<br />
</math><br />
</td><td width="5%">(41)</td></tr></table><br />
<br />
Not at all of these are independent, since some can be derived from others. Even so, our goal of reducing the number of unknowns has clearly not been met.<br />
<br />
Equations governing each of these new quantities can be derived from the original dynamical equations, just as we did for the Reynolds stress. Unfortunately new quantities continue to be introduced with each new equation, and at a faster rate than the increase in the number of equations. Now the full implications of the closure problem introduced by the Reynolds decomposition and averaging has become apparent. No matter how many new equations are derived, the number of new unknown quantities introduced will always increase more rapidly.<br />
<br />
Our attempt to solve the turbulence problem by considering averages illustrates a general principle. Any time we try to fool Mother Nature by averaging out her details, she gets her revenge by leaving us with closure problem - more equations than unknowns. In thermodynamics, we tried to simplify the consideration of molecules by averaging over them, and were left with the need for an equation of state. In heat transfer, we tried to simplify considerations by which molecules transfer their kinetic energy, and found we were lacking a relation between the heat flux and the temperature field. And in fluid mechanics, we tried to simplify consideration of the "mean" motionof molecules and ended up with viscous stress. In all of these cases we were able to make simple physical models which worked at least some of the time; e.g., ideal gas, Fourier-Newtonian fluid. And these models all worked because we were able to make assumptions about the underlying molecular processes and assume them to be independent of the macroscopic flows of interest. Unfortunately such assumptions are rarely satisfied in turbulence. <br />
<br />
It should be obvious by now that the turbulence closure problem will not be solved by the straight-forward derivation of new equations, nor by direct analogy with viscous stresses. Rather, ''closure attempts will have to depend on an intimate knowledge of the dynamics of the turbulence itself''. Only by understanding how the turbulence behaves can one hope to ''guess'' an appropriate set of constitutive equations '''AND''' ''understand the limits of them''. This is, of course, another consequence of the fact that the ''turbulence is a property of the flow itself, and not of the fluid''!<br />
<br />
{{Turbulence credit wkgeorge}}<br />
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