A First Course in Turbulence - Chapter 2

Turbulent Transport of Momentum and Heat
In this chapter TL shows that turbulent flow must be treated by statistical methods. The parameters in the equations of motion are broken down into mean and fluctuating components. The momentum flux due to turbulence is treated like a stress in the flow.
==The Reynolds Equations==
The first part of this section simply introduces the governing equations of motion for an incompressible fluid that will be used in future sections. The continuity and momentum equations have been listed below using the notations given in the book.
*Continuity: \frac{\partial \tilde u_i}{\partial x_i} = 0
*Momentum: \frac{\partial \tilde u_i}{\partial t} + \tilde u_j \frac{\partial \tilde u_i}{\partial x_j} = \frac{1}{\rho}\frac{\partial}{\partial x_j} \tilde \sigma_{ij}
Where:
:\sigma_{ij} = - \tilde p \delta _{ij} +2\mu \tilde s_{ij}
and:
:\tilde s_{ij} = \frac{1}{2}\left( \frac{\partial \tilde u_i}{\partial x_j} + \frac{\partial \tilde u_j}{\partial x_i} \right) 
By substitution, the Navier-Stokes equations are obtained:
:\frac{\partial \tilde u_i}{\partial t} + \tilde u_j \frac{\partial \tilde u_i}{\partial x_j} = -\frac{1}{\rho}\frac{\partial \tilde p}{\partial x_i} + \nu \frac{\partial ^2 \tilde u_i}{\partial x_j \partial x_j}
It should be noted that all of the parameters with tildes represent instantaneous terms. This will be important in the next section.
===The Reynolds Decomposition===
The instantaneous velocity (along with other parameters like pressure, temperature, stress, strain, and any others) can be decomposed into mean and fluctuating terms. The mean component of velocity is the time average of the velocity over a long enough time such that the flow appears steady. The fluctuating component is simply the instantaneous velocity deviations from the mean.
The instantaneous velocity is given by:
:\tilde u_i = U_i + u_i
Note that for TL, the mean component is a capital letter, and the fluctuating component is a lowercase letter.
Mathematically, the mean component of velocity is:
:U_i = \lim_{T \to \infty} \frac{1}{T} \int_{t_0}^{t_0 + T} \tilde u_i dt
By definition, the time average of the fluctuating component must equal zero because the mean component is arrived at by averaging all of the fluctuations out.
: \bar u_i = \lim_{T \to \infty} \frac{1}{T} \int_{t_0}^{t_0 + T} (\tilde u_i - U_i) dt \equiv 0
For the time average to make sense, the mean velocity must be independent of time, which means that its partial with respect to time is zero.
:\frac{\partial U_i}{\partial t} = 0
An important thing to note is that taking the time average of a spatial derivative is the same as taking the derivative of the time averaged variable. This is due to the fact that the integration and derivative operations can be interchanged since they are over different independent variables.
TL then go on to show that pressure and stress can be decomposed into mean and fluctuating terms. This is done exactly like the velocity terms, so it will not be repeated here. However, the stress tensor will be shown for clarification. Just like the velocity, the stress tensor has mean and fluctuating terms.
:\tilde \sigma_{ij} = \Sigma_{ij} + \sigma_{ij}
The mean and fluctuating components of the stress tensor contain only mean and fluctuating subcomponents (pressure and strain), as shown below.
*Mean component: \Sigma_{ij} = -P \delta_{ij} + 2\mu S_{ij}
*Fluctuating component: \sigma_{ij} = -p \delta_{ij} + 2\mu s_{ij}
The strain-rate tensor is arrived at in a similar fashion, with the mean term only containing mean velocity components, and the fluctuating term only containing fluctuating velocity components.
===Correlated Variables===
For our purposes, two variables are said to be correlated when the mean of their product is non-zero. If the mean of their products is zero they are said to be uncorrelated.
Variable correlation comes into play with respect to the equations of motion when finding the average of the product of two instantaneous terms. For instance:
:\overline{\tilde u_i \tilde u_j} = \overline{(U_i + u_i)(U_j + u_j)} = U_i U_j + \overline{u_i u_j} + U_i \overline{u_j} + U_j \overline{u_i} = U_i U_j + \overline{u_i u_j}
For turbulent flows, the \overline{u_i u_j} component will not be zero, meaning that the different components of fluctuating velocity are correlated. This will be shown to be the "turbulent stress" in the flow.
TL points out that the root-mean-square (RMS) amplitude is denoted in their book by a prime, as in u'. This is used as a characteristic velocity, or "velocity scale," of turbulence.
===Equations for the Mean Flow===
Let us now apply the Reynolds decomposition to the continuity and momentum equations.
For the continuity equation:
:\frac{\partial \tilde u_i}{\partial x_i} = \frac{\partial}{\partial x_i} (U_i + u_i) = \frac{\partial U_i}{\partial x_i} + \frac{\partial u_i}{\partial x_i} = 0
If you take the time average of the continuity equation the fluctuating term vanishes revealing that the mean flow is incompressible:
:\frac{\partial U_i}{\partial x_i} = 0
This then shows that the fluctuating component is also incompressible:
:\frac{\partial u_i}{\partial x_i} = 0
For the momentum equation, if we take the time average of the previously described momentum equation, we get:
:U_j \frac{\partial U_i}{\partial x_j} + \overline{u_j \frac{\partial u_i}{\partial x_j}} = \frac{1}{\rho}\frac{\partial}{\partial x_j} \Sigma_{ij}
The nonlinear convective fluctuating velocity term on the LHS can be rewritten as:
:\overline{u_j \frac{\partial u_i}{\partial x_j}} = \frac{\partial}{\partial x_j} \overline{u_i u_j}
This is similar to the convective mean velocity term, but it represents the momentum transport through turbulent fluctuations, rather than the mean flow. If ui and uj were not correlated, this term would be zero, which would correspond to no turbulent motion in the flow. Experience shows that these terms must be correlated, and thus the convective turbulent component to the momentum equation is not zero.
Reynolds treated this term as the divergence of a stress on the flow, and considers turbulence as an agent that produces stresses within the flow. As such in the Reynolds Momentum Equation this term is brought to the RHS and grouped with the other stresses:
:U_j \frac{\partial U_i}{\partial x_j} = \frac{1}{\rho}\frac{\partial}{\partial x_j} (\Sigma_{ij} - \overline{\rho u_i u_j} ) 
-\overline{\rho u_i u_j} is known as the Reynolds stress.
The total mean stress in turbulent flow can then be written as:
:T_{ij} = \Sigma_{ij} - \overline{\rho u_i u_j} = -P \delta _{ij} + 2\mu S_{ij} - \overline{\rho u_i u_j}
===The Reynolds Stress===
The Reynolds stress is the turbulent contribution to the total stress in the flow. It is designated by:
:\tau_{ij} \equiv -\overline{\rho u_i u_j}
The Reynolds stress tensor is symmetric, therefore it has six independent components. The diagonal components (when i = j) are normal stresses, which are pressures. These contribute very little to the transport of momentum. The off-diagonal components are shear stresses, which are the driving mechanism for turbulent momentum transfer.
TL brings up again the closure problem with turbulent flows and their equations of motion. The Reynolds decomposition brings clarity to the physical problem, but it introduces six unknown stresses, along with the fluctuating unknowns for all of the other independent variables. TL tells us that some investigators attempt to link \tau_{ij} and S_{ij} to varying degrees of effectiveness. This will be investigated in a future section.
===Turbulent Transport of Heat===
Turbulence transports heat, and other contaminants, in much the same way as it transports momentum through the flow. TL use this subsection to decompose the energy equation in a similar fashion to the momentum equation. Starting with the energy equation, and assuming constant properties (density, isobaric specific heat, and thermal conductivity and diffusivity):
: \frac{\partial \tilde \theta}{\partial t} + \tilde u_j \frac{\partial \tilde \theta}{\partial x_j} = \gamma \frac{\partial ^2 \tilde \theta}{\partial x_j \partial x_j}
TL uses \gamma for thermal diffusivity, but most heat transfer books use \alpha, where:
: \gamma = \alpha \equiv \frac{k}{\rho c_p}
Just as before, the temperature can be broken into mean and fluctuating components, denoted by capital and lowercase letters, respectively:
: \tilde \theta = \Theta + \theta
When this is substituted into the energy equation, and the time average of all of the terms is taken, the result is the decomposed energy equation:
:U_j \frac{\partial \Theta}{\partial x_j} = \frac{\partial}{\partial x_j} \left( -\overline{\theta u_j} + \gamma \frac{\partial \Theta}{\partial x_j} \right)
or:
:U_j \frac{\partial \Theta}{\partial x_j} = \frac{1}{\rho c_p}\frac{\partial}{\partial x_j}\left( -\rho c_p \overline{\theta u_j} + k \frac{\partial \Theta}{\partial x_j} \right)
The similarity to the momentum equation is recognizable, with -\rho c_p \overline{\theta u_j} greatly resembling the Reynolds Stress.
By integrating both sides wrt x_j and multiplying by \rho c_p, we can get the mean heat flux per unit area per unit time in a turbulent flow:
:Q_j = \rho c_p \overline{\theta u_j} - k \frac{\partial \Theta}{\partial x_j}
The \rho c_p \overline{\theta u_j} component represents the turbulence contribution to heat flux while the k \partial \Theta / \partial x_j component represents the molecular diffusion contribution.
==Elements of the Kinetic Theory of Gases==
===Pure Shear Flow===
We start the discussion on the kinetic theory of gases by first looking at pure shear flow, where the flow is not varying in the x1 and x3 dimensions, only in the x2 dimension, the only mean velocity component is in x1 direction, U1. This is a typically how boundary layer flows are treated, with the x component of velocity the only one that is non-zero, and it only varying in the y direction. TL has a good figure of this on page 35 (Figure 2.2).
If the flow is laminar, the Reynolds stress is zero, and the only component of the viscous shear stress is (remembering that the stress tensor is symmetric):
:\sigma _{12} = \sigma _{21} = \mu \frac{\partial U_1}{\partial x_2}
The shear stress results from the transport of momentum from the x1 direction in the x2 direction. If we let v1 and v2 be the instantaneous velocity components in the x1 and x2 directions, respectively, then the momentum of a molecule with mass m in the x1 direction is mv1. The subsequent amount of momentum transport in the x2 direction is proportional to mv1v2, and if we consider not just a single molecule, but a collection of molecules (like a continuum), the momentum transport per unit area per unit time is proportional to Nm\overline{v_1 v_2}. The over-bar in this case represents the average momentum transport over N molecules per unit volume. The product Nm is the number of molecules per unit volume times the mass per molecule, which is the density, ρ.
The momentum flux per unit area per unit time can be equated with stress. In this case the stress is the shear stress in the x1 and x2 directions, so:
:\sigma _{12} = -\rho \overline{v_1 v_2}
The negative sign is required because a positive value of v2 results in a momentum deficit in the flow. This shear stress only exists if v1 and v2 are correlated, otherwise the momentum flux, and thus the resulting shear stress would be zero. For pure shear flow, like that described in the subsection by TL, this correlation must exist.
===Molecular Collisions===
TL utilizes \xi for the mean free path (the average distances a molecule travels between collisions) rather than the typical \lambda because \lambda is reserved for a specific turbulent length scale, yet to be described.
Referring to Figure 2.3 on page 36 of TL, if a molecule (named A) at an x2 position of -\xi traveling with a mean velocity in the x1 direction of U_1(-\xi) impacts a molecule (named O) at an x2 position of 0 traveling with a mean velocity in the x1 direction of U_1(0) and if molecule A then adjusts its momentum in the x1 direction to its new environment, meaning it accelerates from U_1(-\xi) to U_1(0), then the change in momentum of the environment at x2 = 0 is given by:
: M = m[U_1(0) - U_1(-\xi)]
The change in momentum of the environment is negative because the mean velocity is lower as x2 decreases. This can be thought of as a boundary layer where the mean velocity decreases as the flow gets closer to the wall.
The RHS can be expanded in a Taylor series as:
:M = m\xi \frac{\partial U_1}{\partial x_2} + \frac{1}{2}m\xi ^2 \frac{\partial ^2 U_1}{\partial x_2 ^2} + ...
The second order and higher terms can be neglected if:
:\frac{\partial U_1}{\partial x_2} \gg \frac{1}{2}\xi \frac{\partial ^2 U_1}{\partial x_2 ^2}
which is usually true. For future work and because it is convenient to do now, we then define a local length scale l as:
:l \equiv \frac{\partial U_1/\partial x_2}{\partial ^2 U_1/\partial x_2 ^2}
If the second order and higher terms can be neglected from the Taylor series expansion of the momentum loss, then this new local length scale can be shown to be much greater than the mean free path.
:l \gg \frac{1}{2} \xi
Back to the momentum loss equation, it can be very closely approximated by:
:M = m \xi \partial U_1/\partial x_2 
The quantity \xi \partial U_1/\partial x_2 represents the part of v1 that correlates with v2.
TL states that we can estimate the number of molecular collisions per unit area per unit time at x2 = 0 as Na where a is the speed of sound and N, like before, is the number of molecules per unit volume. They do not offer any explanation as to why this is a reasonable estimate for the number of molecular collisions, so we must just take this as truth until a better explanation can be found. a is a good representation of the RMS molecular velocity.
The momentum transfer per unit area per unit time is simply the momentum transfer per molecular collision times the estimate of the number of collisions that occur per unit area per unit time, or MNa. The shear stress is proportional to this momentum transfer:
:\sigma _{12} = \alpha MNa = \alpha Nma\xi\partial U_1/\partial x_2 
where α is a proportionality coefficient of order one. TL states that for air in "ordinary temperatures and pressures," α ≈ 2/3, thus:
:\sigma _{12} = \frac{2}{3} \rho a \xi \partial U_1/\partial x_2 = \rho \nu \partial U_1/\partial x_2
where we equated our shear stress from molecular collisions to the only non-vanishing component of shear stress from the equations of motion of laminar shearing flow. From this we can see that:
:\nu = \frac{2}{3} a \xi
And the Reynolds number formed from these length, velocity, and viscosity variables (\xi, a, and \nu) is:
: Re = \frac{\xi a}{\nu} = \frac{3}{2} = \frac{1}{\alpha}
This Reynolds number is not significant for engineering purposes because a gas is not at a continuum for length scales on the order of the mean free path, but it points out that the viscosity is defined on a molecular basis by the mean free path and speed of sound of the gas.
===Characteristic Times and Lengths===
The Knudsen number , Kn (or just K according to TL), is the ratio of \xi, the mean free path or the molecular length scale, to the local length scale, l. It is often used to as an indicator of whether or not the continuum assumption is valid for a given flow, where values of Kn << 1 would indicate that the flow can be expressed as a continuum.
In the previous subsection, we have already determined that for typical engineering applications,
:l \gg \frac{1}{2} \xi
But TL goes a bit further in this section. For air at typical temperatures and pressures:
:Kn = \frac{\xi}{l} = \frac{3}{2}\frac{\nu}{al} = \frac{3}{2}\frac{U}{a}\frac{\nu}{Ul} = \frac{3}{2}\frac{M}{Re}
where here M is the local Mach number of the flow. From this we see that Kn is proportional to the ratio of the Mach number to the Reynolds number. For typical engineering applications M << Re, which is another indication that for typically flows, the continuum assumption is valid.
We are also interested in how the molecular and flow time scales compare. The molecular time scale is the time between collisions, and it is given by \xi / a, which is typically on the order of 10-10 seconds (tremendously small!). The time scale of the shearing flow is proportional to the reciprocal of \partial U_1/\partial x_2, the velocity gradient in the shear layer flow. This time scale is typically less than 10-4 seconds.
The molecular time scale is MUCH smaller than the flow time scale (as expected) which indicates that molecules collide with each other many, many times before the flow has progressed downstream appreciably. This is a very important characteristic!
A very important conclusion is given by TL in the next subsection, but I believe it actually belongs here. Since we have shown that the molecular time scale is much smaller than the flow time scale, we can state that "the dynamic equilibrium of the thermal motion of the molecules in shear flow of gases is, to a very close approximation, the same as the equilibrium state in a gas at rest." This means that the equation of state of the gas will not be upset unless M/Re is large, where the flow time scales are larger than the molecular time scales. Physically this would mean that the flow, rather than the chaotic motion of the individual molecules, was forcing collisions between molecules, which is a very atypical scenario.
===The Correlation between v1 and v2===
It is useful to get an idea of how well correlated v1 and v2 are, for future use. Earlier we showed that the part of v1 that is correlated to v2 is proportional to \xi \partial U_1/\partial x_2. TL gives a representative value for a "rapid shearing flow" in air of 7*10-4 m/s.
We define the correlation coefficient between v1 and v2 as:
:c \equiv \frac{\overline{v_1 v_2}}{(v_2')^2}\backsim \frac{\xi \partial U_1/\partial x_2}{v_2'}
TL states that v2' is of the same order of magnitude as a, so c is approximately 2*10-6, which indicates that there is a very poor correlation between v1 and v2.
If \partial U_1/\partial x_2 is estimated by U/l, which is the reciprocal of the time scale of the shearing flow, the correlation coefficient is shown to be on the order of M2/Re, which is extremely small for most flows.
It is important to remember that, at this point, we are investigating laminar pure shear flows. The result that, for typical flows, the correlation between v1 and v2 is very low should not come as a surprise for laminar flows, where the transport of momentum through the shear layer occurs due to molecular motion.
For turbulent flows, the analogous comparison made using the turbulent velocity fluctuations will not result in a poor correlation.
===Thermal Diffusivity===
This very short subsection very simply states that the molecular transport of heat in pure shearing flows is given by a similar equation to the transport of momentum, where the only surviving component is:
: Q_2 = -\rho c_p \gamma \partial \Theta/\partial x_2
TL then looks at the Prandtl number of air (without specifically calling it out as such) and indicates that, since it is less than one, the transport of heat occurs more readily than the transport of momentum.
==Estimates of the Reynolds Stress==
This section will deal with the turbulent transport of momentum and energy through a pure shear flow. The flow field of interest is shown in Figure 2.4 on page 40. The rates of turbulent momentum and heat transfer are given by:
:\tau_{12} = -\rho\overline{u_1 u_2}
:H_{2} = \rho c_p \overline{\theta u_2}
===Reynolds Stress and Vortex Stretching===
First we look specifically at the Reynolds stress and the concept of Vortex Stretching. Like the previous subsection, this subsection will investigate pure shear flows, but it will consider turbulent rather than laminar flows. As such we will be referring to velocity fluctuation terms (u1, u2, u3) rather than the instantaneous velocity of molecules (v1, v2)
Reynolds stress (-\rho \overline{u_1 u_2})is present in turbulent flows; without it, the turbulent transport of momentum would not exist. Because it must exist, the velocity fluctuations in the x1 and x2 directions, u1 and u2, must be correlated. For comparison, in laminar flow when instantaneous molecular velocity was discussed, v1 and v2 were very poorly correlated.
TL states that, for a shear flow where \partial U_1/\partial x_2 > 0, u1 and u2 are anti-correlated, meaning negative values of u1 are more likely with positive values of u2, and vice versa.
Again, in order to maintain turbulent eddies in flows, energy must be continually supplied to the eddies to make up for the energy lost to smaller and smaller eddies. In molecular diffusion this is not an issue because the collisions between molecules are elastic, and thus momentum is conserved. In the pure shear flow that we are considering, the energy is supplied to the eddies through shearing.
Evidence suggests that eddies whose principle axis of vorticity is aligned with the mean strain rate are the most efficient at extracting turbulent energy from the mean flow. TL shows an graphical depiction of this in Figure 2.5 on page 41. Note that the vortices are rotating about an axis that is flat on the page and thus are rotating into and out of the page (in the x3 direction), further showing that turbulence is a three dimensional process. Two-dimensional eddies can form, but they are typically very inefficient at puling energy from the mean flow, and thus dissipate quickly.
The turbulent energy extraction by these "most efficient" eddies from the main flow is believed to be related to vortex stretching. TL states:
:As the eddies in Figure 2.5 are being strained by the shear, conservation of angular momentum tends to maintain the good correlation between u1 and u2, thus allowing efficient energy transfer.
Essentially the strain in the flow along the principle axis of vorticity stretches the vortex, making it longer and thinner. The vorticity (or rotation) in the thinner vortex must increase in order to conserve angular momentum.
All of this is to say that the interactions between eddies and the mean flow cannot be adequately represented by a simple model that extends from the kinetic theory of gases. With that being said, let's investigate a model that does just that to look at where it breaks down!
===The Mixing-Length Model===
The will be referring to Figure 2.6 in TL on page 42, which depicts particle motion in a pure shear flow. The momentum deficit in the x1 of a particle traveling from position 0 at time 0 to position x2 at time t is given by:
:\Delta M = \rho [\tilde u_1(x_2, t) - \tilde u_1(0,0)]
Which can then be decomposed into:
:\Delta M = \rho ([U_1(x_2) - U_1(0)] + [u_1(x_2, t) - u_1(0,0)])
Note that the mean flow is independent of time.
If the contribution of turbulence to the momentum deficit can be neglected (spoiler alert: it can't) and the length x2 in the x2 direction is small enough that the difference between the mean flows can be approximated by x_2 \partial U_1/\partial x_2 (where the gradient is taken at position 0), the momentum deficit can be approximated by:
: \Delta M = \rho x_2 \frac{\partial U_1}{\partial x_2}
(Spoiler alert: this approximation is bad) The momentum deficit transport in the x2 direction is:
:\tau_{12} = \Delta M \tilde u_2 = \rho x_2 \frac{\partial U_1}{\partial x_2}\tilde u_2 = \rho \frac{\partial U_1}{\partial x_2} \overline{x_2 u_2}
Where the over line is used to denote the average of many particles, not the time average. The term \overline{x_2 u_2} is known as the dispersion rate. If the fluid did not exchange momentum with its surroundings, u2 would remain constant and the \overline{x_2 u_2} would continue to increase as the fluid particle traveled in the x2 direction. This is not realistic. As such, we expect the correlation between x2 and u2 to increase as the particle travels. We assume that after some length l the two variables become uncorrelated, and thus their product is zero. This length is referred to as the mixing length. TL notes that our estimate of \overline{x_2 u_2} implies that momentum is NOT conserved in the x2 direction, which is a very dubious claim.
\overline{x_2 u_2} is of order u_2' l, so the momentum deficit transport in the x2 direction can be written as:
: \tau_{12} = c_1 \rho u_2' l \frac{\partial U_1}{\partial x_2}
Where c1 is an unknown correlation coefficient between \overline{x_2 u_2} and u_2' l.
If we defined an eddy viscosity term similar to the kinematic viscosity (thus treating turbulence as a fluid property rather than a property of the flow) the momentum deficit transport in the x2 direction can be written as:
:\tau_{12} \equiv \rho\nu_T \frac{\partial U_1}{\partial x_2}
Where the eddy viscosity is defined as:
:\nu_t = c_1 u_2' l
If u2' and l were known throughout the flow then the model work and the closure problem would be solve, but that is not the case, because u2 and l are properties of the flow and not of the fluid. The mixing-length model of turbulence is only really applicable for a very restrictive set of conditions, where u2 and l remain approximately constant through the flow.
A major problem with the model is that turbulence occurs any many different length scales throughout the flow. Larger eddies contribute more to momentum transfer than smaller eddies (smaller eddies have higher vorticity and transfer the turbulent kinetic energy to heat through viscosity). Because the mixing-length model deals with momentum transfer, it favors larger-scale motions, and the mixing length l can be taken as proportional to the size of the large eddies in the flow.
The next two subsections deal with problems in the approximations and assumptions made in the mixing-length model.
===The Length-Scale Problem===
This section deals with the notion that \rho [U_1(x_2) - U_1(0)] can be approximated by x_2 \partial U_1/\partial x_2 over distances l in the x2 direction. In order for the mixing-length model to be appropriate, the approximations made have to hold over the length that the flow travels, l.
If we create another local length scale for the mean flow, L, defined as:
:L \equiv \frac{\partial U_1/\partial x_2}{\partial^2 U_1/\partial x_2^2}
The x_2 \partial U_1/\partial x_2 approximation would only be valid if:
:L >> \frac{1}{2}l
The largest eddies in turbulent flows typically span the width of the flow, and consequently l is on the same order as L. TL speaks of a "turbulent Knudsen number" defined as l/L being of order one.
Since l is on the same order as L are of the same order, the approximation of x_2 \partial U_1/\partial x_2 will not hold over the mixing length l, and thus is not justified.
Again, this indicates that turbulence is not a fluid property, but a property of the flow. It is strongly dependent on the environment!
===A Neglected Transport Term===
The turbulent contribution to the momentum deficit (\rho[u_1(x_2, t) - u_1(0,0)] = \rho \Delta u_1) was neglected in the mixing-length theory. For small distances in the x2 direction, this may be valid, but for distances on the order of the span of the flow (the mixing length, l), this will not be valid. Since there is no reason to believe that \Delta u_1 \backsim 0 over the mixing length l, this just adds to the problems with the mixing-length model.
===The Mixing Length as an Integral Scale===
In the mixing-length model derivation, we stated that:
:\overline{x_2 u_2} = c_1 \rho u_2' l
The distance that a particle travels in the x2 direction as a function of time is:
:x_2(t) = \int_0^t u_2(t') dt'
Which means that the dispersion rate is:
:\overline{x_2 u_2} = \int_0^t \overline{u(t)u(t')} dt'
The overbar denotes averaging over a large number of particles. Since we are considering flows that are statistically steady, it does not matter when the time starts. What matters is the time differential. If we define the time differential as \tau = t - t', a correlation coefficient, c(τ), can be defined as:
:c(\tau) \equiv \frac{\overline{u_2(t)u_2(t-\tau)}}{\overline{u_2^2}}
Thus, the dispersion rate is:
:\overline{x_2 u_2} = \overline{u_2^2} \int_0^t c(\tau)d\tau
As was stated earlier, the correlation between x2 and u2 must decrease with time, so the correlation coefficient decreases with τ. At large enough values of τ, the fluid particle should lose all correlation between x2 and u2, and thus its ability to transport momentum is gone. This is the essence of the mixing-length theory. Figure 2.7 on TL page 46 shows a feasible plot of the correlation coefficient as a function of τ. The area under that curve is given by:
:T = \int_0^\infty c(\tau)d\tau
c must decrease rapidly enough that T is a finite value. The time value T is known as the Lagrangian integral scale. Recall from Chapter 1 that integral scales refer to larger scale flow, as opposed to the Kolmogorov scales which were the smallest scales in turbulent motion. The dispersion rate becomes:
:\overline{x_2 u_2} = \overline{u_2^2} T
We then define a Lagrangian length scale as the length that a fluid particle travels in time T at the RMS velocity in the x2 direction, u2.
:l_L \equiv u_2' T
And the dispersion rate can be defined using the Lagrangian length scale as:
:\overline{x_2 u_2} = u_2' l_L
At this point TL discusses the theoretical Lagrangian time and length scales, to think about how the time and length scales would be determined experimentally (bringing the theoretical scales into reality). The time scale T is a very difficult parameter to determine experimentally. A large enough sample of fluid particles would have tagged and tracked through the flow to determine the time at which the vertical motion of the particles is arrested.
The Lagrangian length scale is much easier to determine because it is believed to be comparable to the Eulerian length scale. The Eulerian length scale is determined by:
: l \equiv \frac{\int_0^\infty \overline{u_2(x_2) u_2(0)}dx_2}{\overline{u_2^2}}
To determine this Eulerian length scale, the parameters are time averaged over a long enough period of time to eliminate any time dependence (steady flow), and the velocities are taken at two separate x2 locations. This can be determined experimentally fairly simply. It does not require particle tracking.
===The Gradient-Transport Fallacy===
In this subsection TL describes that the mixing-length model is a "dimensional necessity in a turbulent shear flow dominated by a single velocity scale u2' and a single length scale l." What is meant by this is that the equations used to define the mixing-length model were not arrived at through rigorous analytical analyses (i.e. momentum conservation); rather, they were created through proportionality.
TL goes on to say that the mixing-length model can be useful if there are no other length or time scales in the flow, but that the successful use of the model does not imply this. The mixing-length model "should not be mistaken for a gradient-transport postulate."
===Recaptiulation===
This section was a long one, so TL added a recap at the end of it. We have shown that:
*In a shear flow with one characteristic length and one characteristic velocity, the time scale of the turbulence is that of the mean flow.
*Turbulence is not in equilibrium with the mean flow, it is constantly trying to adjust itself to the environment without succeeding.
*A theory for the Reynolds stress cannot be patterned after the kinetic theory of gases, and thus the mixing-length model must be rejected.
After stating that turbulence is not in equilibrium, TL hint at approximate equilibrium for shear flows with respect to its environment in the x1 direction.
==Turbulent Heat Transfer==
===Reynolds' Analogy===
The Reynolds' analogy relates the momentum transport in a flow to the heat transport. Turbulence transports heat as rapidly as it does momentum, and thus the turbulent Prandtl number, \nu_T/\gamma_t, is typically close to one.
The vertical heat flux in a flow is given by:
:H_2 = \rho c_p \overline{u_2 \theta} 
If we treat the heat flux similar to how we treated the momentum transfer in the analysis leading up to the mixing-length model, we can define an eddy diffusivity for heat as:
:\gamma_T \equiv -\frac{\overline{u_2 \theta}}{\partial \Theta/\partial x_2}
This is placed into the equation for vertical heat flux resulting in:
:H_2 = -\rho c_p \gamma_T \partial \Theta/\partial x_2
If we assume the turbulent Prandtl number is one, the eddy viscosity and eddy diffusivity of heat can be equated resulting in:
:-\frac{H_2}{\rho c_p \partial \Theta / \partial x_2} = \frac{\tau_{12}}{\rho \partial U_1 / \partial x_2}
Which, after some manipulation, results in:
:\frac{H_2}{c_p \tau_{12}} = -\frac{\partial \Theta / \partial x_2}{\partial U_1 / \partial x_2}
This is the Reynolds' analogy, and it can be used to determine the heat flux in a flow field if the shear stress, velocity, and temperature are known. TL points out that one key part of this is that it does not explicitly define eddy viscosity or eddy diffusivity, it only states that they are equivalent in turbulent flows.
===The Mixing-Length Model===
The mixing-length model can be similarly applied to heat flux:
:H_2 = -c \rho c_p u_2' l \partial \Theta / \partial x_2
Where c is another correlation coefficient of order one, this time between u_2 and \theta. In our fluid particle, heat is more easily conserved than momentum, thus the mixing-length model for heat transfer can be more reasonably applied. Like the momentum transfer counterpart, if there is only one characteristic velocity and characteristic length in the flow, then the mixing-length model can be applied without any problems. The previous equation is good if the correlation between u_2 and \theta is good and if \theta '/\ell \backsim \partial\Theta / \partial x_2
==Turbulent Shear Flow Near a Rigid Wall==
===A Flow with Constant Stress===
===Nonzero Mass Transfer===
===The Mixing-Length Approach===
===The Limitations of Mixing-Length Theory===