A First Course in Turbulence - Chapter 1

Tennekes and Lumley (subsequently written as TL) introduce the topic of tubulence in the book by pointing out that most flows in nature and engineering applications are turbulent. Laminar flows are the exception, not the rule. They can be found in flows with small dimensions and high viscosities, as in lubricating oil in bearings.
TL list a number of examples of turbulent flows in nature, including:
*in atmospheric sciences
**the boundary layer in the earth's atmosphere
**jet streams in the upper troposphere
**cumulus cloud formation
*in oceanic research, subsurface water currents like the Gulf Stream
*in astrophysics
**the photosphere of the sun and similar stars
**interstellar gas clouds
**wake of the earth in the solar wind
*in aerodynamics
**boundary layer on vehicles
**wake produced by vehicle
... and many more.
==The Nature of Turbulence==
Characteristics of turbulent flows include:
===Irregularity===
Turbulent flows are highly irregular. TL calls them random, but I believe they only have the appearance of being random, more resembling chaos. Their chaotic nature makes solving directly for the flow behavior an intractable problem because small changes in initial conditions can result in drastic changes in results. The irregular nature of turbulent flows leads to solutions based on statistical methods.
===Diffusivity===
A very important feature to turbulence is that it promotes rapid mixing and thus high rates of mass, momentum, and energy transfer within a flow. Whereas mixing in laminar flows is primarily done through viscosity, thermal conductivity, and molecular diffusion, mixing in turbulent flows is the result of vorticity and the spreading of velocity fluctuations in a flow, i.e. kinetic energy.
The diffusivity of turbulent flows is critical to many engineering applications because it accommodates larger adverse pressure gradients, and thus delays boundary layer separation. Vortex generators on aircraft wings and dimples on golf balls both exploit this beneficial property of turbulent flows.
===Large Reynolds numbers===
Turbulence always occurs at high Reynolds numbers, when small disturbances in laminar flows are not damped out by viscosity but are allowed to grow. The instabilities are the result of complex, nonlinear interactions between the viscous and intertial terms in the equations of motion. 
===Three-dimensional vorticity fluctuations===
Turbulent flows always exhibit high levels of fluctuating vorticity in three dimensions. An important mechanism to maintain vortical fluctuations is vortex stretching . This mechanism is three-dimensional; it does not exist for two dimensional flow.
===Dissipation===
Turbulence is, by its nature, dissipative. Viscous shear stresses work on the flow to convert a fluid's kinetic energy into internal energy. Turbulence must be given provided with energy to make up for this dissipation, or it will decay rapidly and relaminarize.
TL make the distinction between simple random wave motion (like acoustic noise and gravity waves) and turbulence. Random waves are nondissipative (viscous losses are insignificant), and so they are not turbulent.
===Continuum===
As with laminar flow, the continuum assumption is made for turbulent flows. It is a good assumption to make because the smallest scales of turbulence, at which kinetic energy is converted to internal energy, are much larger than any molecular length scale.
===Turbulent flows are flows, NOT fluids===
"Turbulence is not a feature of fluids but of fluid flows." There is no turbulent fluid property. This seems helpful because turbulent gases or liquids act similarly at high Reynolds numbers. Also, because it is a flow phenomenon, familes of similar flows can be studied independently. Examples of these familes of flows include boundary layers, jets, and wakes.
==Methods of Analysis==
No general solutions to the Navier-Stokes equations exist for turbulent flows. Quantitative results from these equations are arrived at through heavy use of emperical correlations. The problem with trying to get a purely analytical solution to turbulent flows is that there are many more unknowns than there are equations (closure problem). In order to get past the closure problem, an analyst has to make many assumptions that reduce the number of unknowns.
TL speak to the point that, although the equations are useful and necessary, they do not tell the whole story. Great aerodynamicists are able to utilize the equations in conjunction with their conceptual understanding of what is happening within the flow. This is why models are prevalent in turbulent flow analysis. The exact flow may not be solvable, but it can be modeled to better understand its properties and its effects on surfaces.
One way in which turbulent flows have been modeled is by assuming that the way that turbulent momentum and heat transfers are similar to molecular momentum and heat transfers, and to thus create a turbulence-generated "viscosity" term ("eddy viscosity") to act in the same way that molecular viscosity does, and a "mixing length" that is intended to resemble the mean free path in kinetic theory. Taylor, Prandtl, and others made use of these concepts, but they are imperfect at best. They will be discussed in chapter 2.
One of the main problems, conceptually, with this approach is that the creation of "eddy viscosity" and "mixing length" violate the last point on the description of turbulent flows: turbulent flows are flows not fluids. These newly created properties are fluid specific. However, for simple flows, this may be an acceptable compromise in order to get reasonable results.
===Dimensional Analysis===
Dimensional analysis is very useful in determining the relationships between dependent and independent variables in turbulent flows. Often times dimensional analysis can show that a turbulent flow structure may only depend, in large part, on a subset of independent variables, limiting the scope of the problem greatly. The concept of inertial subrange is a great example of this type of analysis being used.
===Asymptotic Invariance===
The concept of asymtotic invariance relates to how a flow behaves as a property or properties approach an asymptotic value. Since turbulent flows are characterized by very large Reynolds numbers, it makes sense to ensure that any descriptions of turbulence hold as Reynolds numbers approaches infinity.
A Reynolds number of infinity describes a flow in which the effects of molecular viscosity are vansishingly small, and thus overwhelmed by inertial effects in the flow. Turbulent flows, at all but the very smallest of length scales, turn out to be nearly independent of viscosity, which leads to the concept of "Reynolds-number similarity," where changes in Reynolds number lead to virtually no change in the flow.
===Local Invariance===
Local invariance, or "self-preservation," is the concept that turbulent motion is mainly controlled by its immediate environment. This means that, even if environment variables change gradually downstream, a flow may be assumed to be dynamically similar everywhere when it is nondimensionalized by its local length and time scales. In the case of wakes, the turbulent intensite should be nondimensionalized by \delta (local width of the wake) and \partial U / \partial y (average mean-velocity gradient across the wake).
TL describe that turbulence, with its large fluctuations governed by nonlinear equations, exhibits behaviors of limit cycles. Turbulence should be influenced by the dynamic behavior of the environment and constraints imposed on it, rather than the initial conditions of the system. If we consider that the structure of a specific type of turbulent flow is in dynamic equilibrium, where the local energy inputs are equal to the energy dissipation due to small scale turbulence, and further that the energy transfer mechanisms are fast enough that they can be considered to happen instantaneously (i.e. there are no transient effects where the past influences the present), this limit cycle equilibrium will be governed in large part by the local length and time scales.
The key point of this chapter is expressed here again. "Scaling laws are at the heart of turbulence research."
==The Origin of Turbulence==
Laminar flows transition to turbulence at high Reynolds numbers when instabilities arise in the flows that cannot be damped out. In pipe flow the critical Reynolds number is approximately 2000, based on mean velocity and pipe diameter. Transition to turbulent flow can be delayed to higher values of Reynolds number if great care is taken to avoid creating small disturbances in the flow that would then be amplified and trigger transition. The critical Reynolds number in boundary layers with zero pressure gradient is approximately 600, based on free-stream velocity and displacement thickness). Free shear flows transition to turbulence at much lower Reynolds numbers due to an inviscid instability mechanism that is not present in bounded shearing flows like boundary layers and pipe flow.
Turbulence depends on energy from the flow in order to persist. Common sources for this energy include shearing in the mean flow and buoyancy. Without a continuous energy source, the dissipative nature of turbulence will cause it to decay until the flow relaminarizes.
The transition to turbulence can be delayed and the relaminarization of turbulent flow can be achieved by consuming the turbulent kinetic energy in the flow in some other way. TL give the example of imposing a magnetic field on the flow. For typical engineering applications, like flow around a vehicle, this does not seem like a very practical solution.
The current understanding of transition from laminar to turbulent flow comes mainly from emperical data rather than mathematics. Most of the mathematical theory of turbulence is only applicable at very high Reynolds numbers (asymptotic theory) and does not hold for turbulence that cannot sustain itself. TL then give a high level physical description of transition. The picture at right is meant to supplement their description. Turbulence can be initiated by 2-dimensional instabilities called Tollmien-Schlichting waves. These lead to 3-dimensional instabilities and the formation of fully turbulent spots in the flow. These spots grow and merge with one another to become fully turbulent flow.
==Diffusivity of Turbulence==
As was previously stated, turbulence's diffusivity is one of its most important qualities. Turbulent motion can mix flows several orders of magnitude faster than molecular diffusion. A place where this is of great use is in combustors where quick energy release is key.
===Diffusion in a Problem with an Imposed Length Scale===
TL go though a fairly useful exercise to point out the order of magnitude differences in time scales between molecular and turbulent diffusion of heat in a long room. Setting up the problem, imagine a room 5 meters long with a radiator at one end. The characteristic length scale for the room is 5 meters.
First let us look at the time scale for molecular diffusion. For this we utilize the heat equation, shown below, in which \theta is temperature and \gamma is the thermal diffusivity, which is assumed to be constant. This ignores all effects due to buoyancy and essentially treats the air in the room as if it were a very poor solid thermal conductor.
:\frac{\partial \theta}{\partial t} = \gamma \frac{\partial ^2 \theta}{\partial \bar x^2}
Since we are only interested in comparing the magnitudes of time scales of the two methods for diffusion, we can simply replace the derivatives with discrete differences.
:\frac{\Delta \theta}{t_{mol}} \backsim \gamma \frac{\Delta \theta}{L^2}
Solving for tmol:
:t_{mol} \backsim \frac{L^2}{\gamma}
By comparison the turbulent time scale, tturb, is simply:
:t_{turb} \backsim \frac{L}{u}
Where u is the characteristic velocity (which could be thought of as the RMS amplitude of the velocity fluctuations).
For reasonable values of u and γ, tmol and tturb are approximately 106 seconds and 100 seconds, respectively.
One important point that TL brings up is that turbulent motion creates eddies that mix the flow quickly, but they rely on molecular diffusion to even out small scale irregularities. These eddies produce a tremendous amount of surface area over which the molecular diffusion acts, making the process very quick.
If we look at the ratio of the turbulent to molecular diffusion time scales it ends up being the inverse of the Péclet number, which is the ratio of the advective transport rate to diffusive transport rate. This makes sense because the time scales are proportional to the inverse of their resepective transport rates.
:\frac{t_{turb}}{t_{mol}} \backsim \frac{L \gamma}{u L^2} = \frac{\gamma}{uL}
Since this is an order of magnitude analysis, and for gases the heat conductivity is of the same order as the kinematic viscosity, the two can be swapped, revealing that the ratio of the time scales is approximated by the inverse of the Reynolds number of flow.
TL makes an important point at the end of this subsection. It is that thinking of Reynolds number as the ratio of inertial to viscous terms can be misleading for turbulent flows because, at high Reynolds numbers, viscous effects tend to operate at much smaller length scales than inertial effects. High Reynolds numbers would typically be thought of as inertial effects completely dominating viscous effects. Once again pointing out that length scales are very important in the analysis of turbulent flow!
===Eddy Diffusivity===
It would seem logical that an effective diffusivity could be used in the heat equation to model heat transfer due to turbulence. Doing so, however, would mask any understanding of what is actually occurring within the turbulent flow because it, in a way, treats turbulence as a fluid property. I say "in a way" because to use this method properly, the effective diffusivity will still be reliant on flow information, like the RMS amplitude of flow fluctuations. The creation of this term makes the math much easier, but it is dangerous because, if applied improperly to other systems, it would likely give non-physical results.
We again look at the heat equation, but with an effective turbulent diffusivity, K,  named "eddy diffusivity," replacing the thermal diffusivity based on molecular diffusion:
:\frac{\partial \theta}{\partial t} = K \frac{\partial ^2 \theta}{\partial \bar x^2}
To make this equation effective for a particular flow K will have to be chosen so that it approximates the turbulent time scale.
:t_{turb} \backsim t_K \backsim \frac{L^2}{K}
:\frac{L}{u} \backsim \frac{L^2}{K}
:K \backsim uL
This shows that K should be on the order of uL, not that it should equal uL.
The ratio of eddy diffusivity to kinematic viscosity is approximately Reynolds number, so another way that Reynolds number can be thought of is as the ratio of turbulent viscosity to molecular viscosity.
A very important point to remember is that eddy diffusivity is only something created to make the math of turbulence easier! It is not in any way a physical property!
===Diffusion in a Problem with an Imposed Time Scale===
Akin to the problem of diffusion with an imposed length scale is diffusion with an imposed time scale. TL use the atmospheric boundary layer as an example, where the imposed time scale is the reciprocal of angular velocity of the earth at a given latitude.
It can be shown through similar manipulation of the equations used in the imposed length scale problem that the ratio of the turbulent to molecular diffusion length scales is on the order of one divided by the square root of the Reynolds number.
To summarize the results of the imposed time and length scale dimensional analyses, in problems with imposed length scale, the Reynolds number is proportional to the ratio of the turbulent to molecular diffusion time scales. In problems with imposed time scales, the Reynolds number is proportional to the square of the ratio of the turbulent to molecular diffusion length scales.
"Since Reynolds number is of most flows are large, these relations clearly show that turbulence is a far more effective diffusion agent than molecular motion."
==Length Scales in Turbulent Flow==
A very important topic to discuss with respect to turbulence, and fluid dynamics in general, are the multitude of length scales that exist within any given flow. They range from the dimensions of the flow field on the large side, where bulk motion is occuring, down to the Kolmogorov scales on the small side, where viscosity is working to convert inertial motion to internal energy. Some of these scales have very specific roles to play in decribing flow features. TL points out that the large number of length scales is the reason why spectral analysis of turbulent motion is useful.
===Laminar Boundary Layers===
Two main length scales exist in laminar boundary layers:
*Longitudinal or convective length scale related to the downstream transport of momentum
*Transverse or diffusive length scale representing the thickness of the boundary layer which relates to the molecular diffusion of the momentum deficit away from the wall
Looking at the Navier-Stokes equations for steady flow of an incompressible fluid, they are typically written with the inertial terms are on the LHS and the viscous terms on the RHS.
:u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \frac{\partial^2 u_i}{\partial x_j ^2}
If the length scales for the viscous and inertial terms were the same, dimensional analysis would show that the ratio of the inertial to the viscous terms is the Reynolds number, and that as Reynolds number became very large, the viscous terms would be negligible.
Initial or boundary conditions may make it impossible to neglect viscosity. The no-slip boundary condition at a wall, for example, would be impossible to satisfy if the viscous terms were neglected. For this reason, the viscous terms require a length scale that forces them to be the same order of magnitude as the inertial terms. The inertial terms utilize the longitudinal length scale, L, and the viscous terms utilize the transverse length scale, l.
If we perform a dimensional analysis on the Navier-Stokes equations now:
:\frac{U^2}{L} \backsim \frac{\nu U}{l^2}
And the ratio of the viscous and inertial length scales is:
:\frac{l}{L} \backsim \sqrt{\frac{\nu}{UL}} = Re^{-1/2}
The wide separation between the convective and diffusive length scales in shear flows allows for simplifying approximations that make the mathematics in the equations of motion tractable.
===Turbulent Boundary Layers===
In order to compare turbulent and laminar shear flows, the turbulent length and velocity scales must be defined. They are depicted in Figure 1.3 on page 16 of TL.
*U: mean flow velocity
*u: magnitude of velocity fluctuations
*L: convective length scale
*l: boundary layer thickness
For turbulent boundary layer flows, the boundary layer thickness is typically a good large length scale because it is similar in size to the largest eddies in the flow.
The boundary layer thickness is presumed to increase at a rate roughly equal to the magnitude of the velocity fluctuations in the flow. This topic will be discussed again in section 5.5
: \frac{\text{d}l}{\text{d}t} \backsim u
If the convective time scale, L/U , and the turbulent time scale, l/u>, are equated several interesting results can be found.
: \frac{L}{U} \backsim \frac{l}{u} \Rightarrow \frac{l}{L} \backsim \frac{u}{U}
or:
: l \backsim \frac{uL}{U}
This shows that the turbulent boundary layer thickness grows at a rate proportional to the convective length and the ratio of the velocity fluctuations to mean flow velocity. For comparison a similar laminar length scale result is:
: l \backsim \sqrt{\nu t}
It should be noted that these are not equations to calculate the expansion of the turbulent boundary layer. They are simply assessments of length, time, and velocity scales within boundary layer flows for order of magnitude comparison purposes.
As a lead in to a future section, TL mentions that not all turbulence has a time scale that can be equated with the convective time scale. Smaller scale turbulence has much smaller time scales, which make it statistically independent of the mean flow.
TL points out an interesting implication of,
:\frac{l}{L} \backsim \frac{u}{U}
in the following section. I believe it is more appropriate to put here. In boundary layer and wake flows, as L approaches infinity u/U and l/L tend to zero. Jets and free shear layers, u/U and l/L approach finite values. This would imply that in boundary layers and wakes, the thickness grows very slowly at large convective distances. Jets and free shear layers grow linearly with downstream distance.
TL note that even with this knowledge, most turbulent shear flows spread slowly enough to make l/L → 0 a useful approximation.
===Laminar and Turbulent Friction===
TL starts the section by stating that the rapid growth of turbulent shear layers should correspond to higher drag than similar laminar flows without offering much in terms of explanation of their logical reasoning. They then go on to show this result mathematically.
Drag, D,  is equal to the total rate of loss of momentum in turbulent and laminar boundary layer flows. The drag coefficient per unit span, cd, is defined as:
:c_d \equiv \frac{2 D}{\rho U^2 L}
For laminar flow on a flat plate of length L, the momentum loss due to drag can be estimated as:
:D \backsim \rho U^2 l
The drag coefficient for laminar flow on a flat plate then becomes:
:c_d \backsim 2 \frac{l}{L}
The ratio of diffusive to convective length scales was previously found proportional to one over the square root of the Reynolds number, so:
:c_d \backsim 2 Re^{-1/2}
Turbulent flows are much different. The mass flow deficit at the end of the plate is proportional to \rho u l, which will be elaborated on in Chapter 5. The rate of loss of momentum is proportional to the mass flow deficit times the mean flow velocity:
:D \backsim \rho u U l
The drag coefficient for turbulent flow is then:
:c_d \backsim 2 \frac{u}{U} \frac{l}{L} \backsim 2 \left( \frac{u}{U} \right) ^2
Experimental results show that u/U grows very slowly with Reynolds number. This means that as Reynolds number increases, the drag coefficient of a laminar flow over a flat plat decreases more quickly than a turbulent flow. Figure 1.4 of page 18 shows depicts this.
By looking at boundary layer profiles for laminar flow and time averaged turbulent flow, one can qualitatively see that turbulent boundary layers result in larger wall shear stress, and thus more drag. Since:
:\tau_{wall} \propto \left( \frac{\partial u}{\partial y} \right) _{wall}
As the slope of the velocity profile at the the wall increases, so does drag, and the turbulent profile has a much steeper velocity profile at the wall than the laminar profile.
===Small Scales in Turbulence===
This section is possibly the most important one for work in future chapters. The turbulent length scales discussed to this point were those that corresponded to the largest eddies in the flow. The large eddies are best for transferring mass, momentum, and energy thought the flow (mixing in general, or dispersion), but they do not contribute to flow dissipation. To look at dissipation, and other aspects of the dynamics of turbulence, other length scales are needed.
In this section we attempt to find the smallest length scales in turbulent flow. At the smallest scales, turbulent kinetic energy is dissipated into heat through viscosity, and viscosity acts to smooth out the velocity fluctuations that arise from the nonlinear nature of the equations of motion. Viscosity also prevents infintesimally small scales of motion by converting that motion to heat.
In earlier discussions of Reynolds numbers and length scales, it was mentioned that, even at large Reynolds numbers, it is incorrect to assume viscous terms can be neglected for turbulent flows. There will always exist length scales in the flow at which viscosity terms cannot be neglected. The nonlinear terms in the Navier-Stokes equations create motion at these scales, which ensures that viscosity will always play a role in turbulent flow. TL describe it well:
: There seems to be no way of doing away with viscosity: as soon as the scale of the flow field becomes so large that viscosity effects could conceivably be neglected, the flow creates small-scale motion, thus keeping viscosity effects (in particular dissipation rates) at a finite level.
At the smallest scales of turbulence, the motion is statistically independent from the main flow due to very small time scales in comparison to the main flow. The smaller time scales mean that the small scale turbulence changes very quickly with time while the larger scale eddies change more slowly. The small scale turbulence is thus only dependent on kinematic viscosity and the rate at which it is supplied with kinetic energy from the main flow.
Kolmogorov's universal equilibrium theory of small-scale structures, also known as Kolmogorov's similarity hypothesis , states that the rate of energy dissipation should be equivalent to the rate at which energy is supplied to the small structures because rate at which kinetic energy is supplied to the small scale turbulence is related to the time scale of the main flow.
The governing parameters for small scale turbulence are:
*\epsilon: energy dissipation rate per unit mass (m2/s3)
*\nu: kinematic viscosity (m2/s)
Length, time, and velocity scales are formed from these:
*\eta \equiv \left(\nu^3/\epsilon\right) ^{1/4}
*\tau \equiv \left(\nu/\epsilon\right) ^{1/2}
*v \equiv \eta/\tau = \left(\nu^3/\epsilon\right) ^{1/4} * \left(\nu/\epsilon\right) ^{-1/2} = \left(\nu\epsilon\right) ^{1/4}
These are called the Kolmogorov microscales. In Russian literature they are referred to as "inner" scales. The largest scales of turbulence, corresponding to eddies in the bulk flow are referred to as "outer" scales. This book, and other non-Russian literature will refer to the largest scales as "integral" scales.
One very interesting result of the Kolmogorov microscales is that their corresponding Reynolds number is unity.
:Re = \eta v/\nu = \left(\nu^3/\epsilon\right) ^{1/4} \left(\nu\epsilon\right) ^{1/4} / \nu = 1
As was stated previously, when the bulk flow Reynolds number grows to a high level where it may seem reasonable to neglect viscosity, smaller scale turbulence is created via the nonlinear terms in the Navier-Stokes equations. This small scale motion results in Reynolds numbers as low as unity, where viscosity can act to dissipate turbulent kinetic energy. From the perspective of the bulk flow, the small scale turbulent structures are constantly adjusting to the amount of energy that is being provided to them, but from the perspective of the small scale turbulence, the rate at which energy changes slowly enough that the energy supply and dissipation can be thought of as in equilibrium.
===An Inviscid Estimate for the Dissipation Rate===
Kinetic energy is supplied to the small scale turbulence though the motion of large eddies in the bulk flow. Kinetic energy is proporitional to u2 and the time scale at which the energy is supplied is the integral time scale: l/u.
The rate at which energy is supplied to small scale turbulence, and thus the rate at which that energy is dissipated, is roughly proportional to the kinetic energy of the large eddies divided by the integral time scale, or:
:\epsilon \backsim u^2 \times u/l = u^3/l
Notice that the rate of energy supplied from the large scale turbulence does not depend on viscosity. This fits with standard interpretation of high Reynolds number flow; that is, the inertial terms are far more important to the overall flow, at integral length scales. This also shows that the energy dissipation due to turbulence is a passive process that relies on the inertial energy present in, relatively, large eddies.
One of the key points that TL brings up is that the energy dissipation can be interpreted as saying that large eddies lose a relatively large amount of turbulent energy within one "turnover" time, which is the amount of time equal to the integral time scale. TL says that "the nonlinear mechanism that makes small eddies out of larger ones is as 'dissipative' as its characteristic length permits," meaning that as turbulent energy increases, or the integral time scale decreases, the rate of kinetic energy dissipation increases, and thus the inner or micro length scale decreases. More energy dissipation results in smaller small-scale turbulence. The nonlinear terms in the Navier-Stokes equations create smaller and smaller scales in turbulence until the Kolmogorov microscales are reached where the viscous dissipation occurs instantaneously, relative to the bulk flow.
One final note in this section once again compares turbulence to molecular diffusion, this time with respect to energy dissipation. The time scale for the viscous dissipation of the integral scale eddies in flow field is on the order of l^2/\nu. Since \nu is very small, the resulting time scale is very large, and the corresponding rate of energy dissipation:
:\epsilon _{mol} \backsim \nu u^2/l^2
This is much smaller than u3/l, when Reynolds number is large.
===Scale Relations===
TL starts by substituting the previously estimated value of the energy dissipation rate into the Kolmogorov microscale relations to show a few new relationships.
*\eta /l \backsim (ul/\nu)^{-3/4} = Re^{-3/4}
*\tau u /l \backsim \tau /t \backsim (ul/\nu)^{-1/2} = Re^{-1/2}
*v/u \backsim (ul/\nu)^{-1/4} = Re^{-1/4}
Since turbulent flows are characterized by large Reynolds numbers, these relations give some perspective on how much smaller the micro length, time, and velocity scales are than the integral scales. This gap between the small and large scales in the flow becomes larger and larger as Reynolds number increases.
The main difference between two flows with the same integral scale but different Reynolds numbers, say two jet flows with the same fluid and exit diameter but different exit velocities, are the smallest length scales of turbulence. TL shows a good drawing of this in Figure 1.6 on page 22.
The lower Reynolds number, slower flow, produces microscale eddies that are larger than those in the higher Reynolds number flow. This gives the lower speed flow a 'coarser' texture to it.
The video shown here also shows how the microscale size also gives away subscale explosions that are intended to be passed off as full scale destruction in movies.
TL concludes the section by pointing out that vorticity is proportional to the reciprocal of the time scale. Since the micro time scale is much smaller than the integral time scale, the vorticity in the microscale is much larger than that in the integral scale. The opposite is true for kinetic energy. The ratio of the velocity scales given above shows that the micro velocity scale is much smaller than the integral scale, thus more kinetic energy resides in the integral scale than the micro scale.
: This is typical of all turbulence: most of the energy is associated with large-scale motions, most of the vorticity is associated with small-scale motions.
===Molecular and Turbulent Scales===
Finally, the molecular and Kolmogorov scales are compared to show that, although we are discussing very small scales in turbulence, they are still very large in comparsion to free molecular length scales (mean free path), and thus the continuum assumption still holds for all practical flows in engineering.
Small Kolmogorov length scales are associated with low values of kinematic viscosity and high values of energy dissipation. High energy dissipation is related to high values of u, \epsilon \backsim u^3/l. Gases will be more likely to have higher values for u than liquids, so TL chooses to make the comparison only with gases. Another reason why it makes no sense to make any comparisons to a liquid mean free path is because liquid molecules are so close together that the concept of a mean free path does not have the same meaning as it does in a gas.
The molecular length and velocity scales are:
*Mean free path (as previously stated), \xi
*Speed of sound, a
The ratio of the mean free path to the Kolmogorov length scale is:
:\xi / \eta \backsim M/Re^{1/4}
Where Re is the turbulent Reynolds number and M is the turbulent Mach number.
For turbulent and molecular length scales to be similar, the turbulent Mach number, that is the speed of the velocity fluctuations within the flow relative to the local speed of sound, would need to be high and the turbulent Reynolds number would need to be low. These situation is very unlikely. TL goes on to describe gas nebulae as an example of where this could occur. Since that situation is completely unrelated to engineering applications, I have not taken notes on it here.