We begin our discussion by noting that there are three quite general and distinct types of turbulent flows: namely, Homogeneous and/or isotropic turbulence, free shear flows and wall-bounded shear flows.
Homogeneous and/or Isotropic Turbulence
Homogeneous turbulence is such that statistics are invariant under spatial translations, while isotropic turbulence is invariant under rotations and reflections. As rotations and reflections can always be constructed as combinations of translations, we might be tempted to assume that homogeneity implies isotropy.
The requirement that statistical properties remain invariant under arbitrary translations implies as in homogeneity implies for example that:
This does not mean however that:
In contrast, as isotropy requires invariance of statistical quantities under rotations and reflections of the coordinate system, then if we decide to rotate the coordinates in 90° then (up to a sign) what had been the u component in physics will now be the v component, hence for statistical invariant under rotation it must be that in the entire domain of isotropy:
Furthermore, for the invariance under rotation to be valid under arbitrary rotation an additional requirement that the derivatives in the normal coordinate direction relative to each of the above quantities should also be invariant, i.e it must also hold that:
From the above and the relationship between the strain rate and the velocity gradient tensor:
We can see from the relation that a uniform strain in one direction on homogeneous isotropic turbulence will not allow isotropy but not homogeneity. However, in physical shear flows the strain is rarely uniform nor is it aligned with a specific coordinate so generally a strained flow will eliminate both. In other words, physically realizable turbulent flows are globally inhomogeneous and anisotropic.
Nevertheless, for very high wavenumbers (smallest scales) the viscous terms dominate the advective (and pressure gradient terms), coupling between momentum equations disappears, they retain a similar form, so same solution may be expected, i.e. small-scale behavior is equivalent to local isotropy.
Anecdote: Andrey Kolmogorov’s 4/5 law, directly derivable from the N.–S. equations under the hypotheses of high Reynolds homogeneous isotropic turbulence is the only exact (without adjustable constants), nontrivial and in good agreement with experimental data known up to the present time.
Free Shear Turbulent Flows
A flow is termed free if it is not bounded by solid surfaces. Three generic types of free shear flows are: the far wake, the mixing layer, and the jet (see figure below).
- A wake forms downstream of any object placed in a stream of fluid.
- A mixing layer occurs between two parallel streams moving at different speeds.
- A jet occurs when fluid is ejected from a nozzle or orifice.
Far enough downstream the three shear flows types approach what is termed self-similarity such that details of the flow near the y axis become unimportant. As the velocity profile can be expressed in the for:
The self-similarity amounts to saying that two velocity profiles at different x-locations far enough downstream have the same shape in a scaled plot according to:
Flows exhibiting a single length scale include the free shear flows such as the three generic types presented above lend themselves quite successfully to the Prandtl’s mixing length hypothesis (albeit with a different length for each type of flow), the basis of zero-equation (algebraic) turbulence models, when the mixing length is typically expressed as:
C1 being the closure constant determined such that computed mean velocity profiles match measurements:
Law of the Wall
The law of the wall is a semi-empirical expression relating velocity to distance from the wall in a turbulent wall-bounded flow.
Most flows of engineering value involve the interaction of turbulent with solid boundaries. wall-bounded turbulent shear flows involve a few length scales (the number of which depends on how one counts) to represent its physical behavior.
In what follows I shall present a fairly heuristic description of the most widely-used concepts from the classical theory of turbulence
The velocity profile for a fully-developed (time mean) turbulent flow in a duct is very different from the laminar profile:
Shear stress at the wall is calculated as:
It can be seen that the velocity gradient at the wall, and hence also the wall shear stress, corresponding to a fully-developed Poiseuille flow in a duct is much larger for the turbulent case than for the laminar one.
It can also be understood just from looking at the turbulent flow velocity profile that there must be at least two length scales associated with this flow.
The first corresponds to the rather thin region adjacent to the walls in which the velocity profile is nonuniform, with large gradients (keep in mind the non-slip boundary condition), and exhibits a near linear velocity profile. This layer, dominated by viscosity effects is termed the viscous sublayer. In light of the dominance of the viscous effects it is often associated with the Kolmogorov scale (These scales were predicted on the basis of dimensional analysis as part of the K41 theory, corresponding to small length scales large wavenumbers where kinetic energy of fluid motion is converted to thermal energy – mind that such association is quite subtle). The outer region shows a near constant velocity with distance from the wall.
I will note for some rigor motivating the more heuristic treatment that relying on perturbation analysis (a powerful mathematical tool intended on matching solutions in the form of an asymptotic expansions in our case) it could be shown that the inner profile (nearly linear) which satisfies the no-slip boundary condition at the wall does not correctly asymptote the outer (nearly uniform) solution which at the same time cannot satisfy the no-slip boundary condition, hence, a third solution is to match them (derived by means of asymptotic expansion). For the heuristic treatment, somewhat following H. Tennekes and J. L. Lumley. A First Course in Turbulence the above formalism is not required.
and we denote a velocity scale for the inner region as:
The first is associated with a large advective length scale (as the hight of the duct):
and the latter with small viscous length scale which by pure dimensional analysis we can express through:
In order for an intermediate scale to be realizable, the ratio of the two length scales must be large:
Now we identify a range distance y from the wall for which:
within this range the heuristic argument claims that ν/uτ is to small enough and h is large enough to not be able control the flow dynamics for the first nor result in direct interaction for the latter. In other words we claim that y is the only length scale corresponding for this region but there are two velocity scales:
Basing only on dimensional analysis we relate the two according to:
with the constant determined from experimental data.
We define dimensionless quantities:
so the above relation between the two velocities becomes:
Now we integrate to achieve the following:
This is the well-know “log-law” matching the inner and outer layer (the second constant added is an integration constant also to be evaluated experimentally).
The friction velocity chosen earlier is defined as:
Below the law of the wall is finally displayed clearly and we can see four different length scales:
- Viscous sublayer – (explained above).
- buffer layer – smoothly connects the viscous sublayer to the inertial sublayer and might be thought of as a region of which the dissipation and inertial effects are somewhat balanced.
- Inertial sublayer – where the log profile holds.
- Defect sublayer – the outermost, furthest from the walls region.
Finally, besides its enormous descriptive power of the (fully developed – self-similar) turbulent boundary the law of the wall may serve during CFD analysis in prior estimation of grid requirement as explained below: