A mathematical model for the scaling of turbulence

Derivation of the Model

We first present a heuristic derivation of an equation for the mean flow in the intermediate layer in a wall-bounded turbulent flow. This derivation is heuristic, but the analysis of the equation will be rigorous, and the heuristic derivation includes several points needed for the interpretation of the rigorous results.

Consider a turbulent flow bounded by a wall, and a point P a distance y above that wall (Fig. 1). We wish to determine the mean velocity profile u = u(y) parallel to the wall as a function of the distance y from the wall. We consider the flow in the region δ ≤ y ≤ d, i.e., outside a viscous sublayer of thickness δ near the wall where viscous effects are dominant; d is a length typical of the height of the intermediate layer. A prime example of a wall-bounded flow is flow in a pipe, where d is the diameter of the pipe.

It is widely believed that the mechanics of a turbulent flow are determined by a mixing length , a turbulence analog of a mean-free path in kinetic theory (19, 20). Following von Kármán (21) we assume . The simplest assumption about is (an irrelevant proportionality constant is omitted), and thus yu″ + u′ = 0, yu′ = C ₁, and (We denote by C_i constants whose value is immaterial and, whenever it leads to no confusion, set C_i = 1.) From this assumption we obtain u(d) = ln d + C ₂, u(δ) = ln δ + C ₂, and u(d) – u(δ) = ln(d/δ). Redefine the velocity u as u = u – u(d); then u(δ) = –ln(d/δ). Clearly, u′(δ) = 1/δ.

We now ask how these conclusions are modified when one takes into account the direct effect of viscosity on the flow. The analog of a Reynolds number here is d/δ, the ratio of the scale of the domain to a viscous length. The boundary conditions, should remain unchanged; they describe the mean velocity on the outer edge of the viscous sublayer 0 < y < δ whose formation is due to viscous effects and which is not modified by the inclusion of viscous effects in the description of the layer in which we are interested. The importance of coherent structures in turbulence has been shown in numerical computations (22) and in experiments (23), they are responsible for most of the turbulent shear (24), and their scale increases with the Reynolds number (22, 25). Thus, should increase; assume the increase is proportional to the small parameter 1/ln(d/δ); this is plausible since the quantity ln(d/δ) has already appeared in the analysis as the major large parameter (also see the discussion below). Thus, Then u″ = –(u′/y)(1 + (1/ln(d/δ))^–1 and neglecting terms quadratic in the small parameter 1/ln(d/δ) we find This is our model equation. The main assumption in this heuristic derivation is Eq. 4, which quantifies the impact of a viscous length scale on the mixing length.

Properties of the Model Equation

The derivation we have just given is heuristic, but there is nothing heuristic about the analysis we now give.

The solution of Eq. 5 with the boundary conditions of Eq. 3 is The quantity yu′(y) = (d/d ln y)u can be calculated, so that for each fixed y, its limit as δ → 0 is e = 2.71.... As δ → 0, u converges pointwise to the curve (or, in logarithmic coordinates, straight line) u = e · ln y. There is a separate function u for each value of δ/d; the envelope of these curves is u = ln y; thus, in the (ln y, u) plane the curves have this straight line as envelope and have a common asymptotic slope e (see Fig. 2).

Consider these facts from the point of view of the similarity theory presented above. The solution u is a function of y, δ, and d. The derivative du/dy must satisfy the relation where u is dimensionless, though we have omitted a division by a characteristic velocity scale, y/δ, d/δ are large and Φ is unknown. If one makes an assumption of complete similarity, Φ ∼ C, where C is a constant, one finds u′ = C/y, and from the boundary conditions u = ln y, which is the equation of the envelope. The same wrong result can be obtained by simply neglecting the small term 1/ln(d/δ) in Eq. 5. A substitution of the true expression for u yields relation which represents incomplete similarity in y/δ with anomalous exponent 1/ln(d/δ) and no similarity in d/δ.

The relation between the coefficient of the right hand side of Eq. 5 and the parameter δ is a critical relation in the following sense: Consider instead of Eq. 5 the more general equation where ψ = ψ(δ/d) is a monotonic function of δ/d, with dψ/dδ > 0, and with the boundary conditions of Eq. 3. The solution of Eq. 8 is One easily finds ydu _ψ/dy = (y/δ)^ψ and hence that the limit of ydu _ψ/dy as δ → 0 is In the first case the solution is asymptotically given by a complete similarity relation; the second case is the one we discussed; in the third case there is no similarity result. This conclusion affirms once more the close connection between criticality and incomplete similarity (see refs. 17 and 18). Note that in the first two cases the limit δ → 0 is approached at the a rate proportional to ψ, i.e., logarithmically in the critical case and faster in the subcritical (ψ ln(d/δ) → 0) case, as is consistent with the assumed form of the coefficient .

In summary, one of the following possibilities must hold for solutions of Eq. 8: (i) A faster than inverse logarithmic rate of convergence (in δ) of the profiles gradients to a limit, asymptotic complete similarity, and a logarithmic dependence of u on y; (ii) an inverse logarithmic rate of convergence of the profile gradients to a limit, incomplete similarity, and a power law profile; or (iii) no well defined limiting velocity profile.

The Scaling of Turbulence

Consider first wall-bounded turbulent flows, for example, flow in a pipe. In Fig. 3 we display mean velocity profiles obtained experimentally by Zagarola in a Princeton experiment (26). The flaws in the experimental procedure (see ref. 11) do not invalidate the qualitative shape of the curves.

It is usually stated that the mean velocity profile in most of the pipe, at large Reynolds numbers R and away from the center and from the near vicinity of the wall, is logarithmic and R-independent; this is usually derived by a complete similarity argument (see ref. 19): the mean velocity u should depend on the viscosity ν, the fluid density ρ, the pipe diameter d, the shear stress at the wall τ, and the distance to the wall y. Dimensional analysis allows the relationship between these quantities to be written in the form where , δ = ν/u*, and d/δ = u d/ν is a function of the conventional Reynolds number R = ū d*/ν, with ū equal to the flux divided by the cross section of the pipe (compare with Eq. 6). This can be rewritten as where both arguments of the unknown function Φ are large. If one then makes a complete similarity assumption that the function Φ tends to a nonzero constant C as its arguments become large, one obtains (du/dy)/u* = C/y; one usually defines κ = “Kármán's constant” = 1/C so that this becomes (du/dy)/u* = (1/κ)/y. An integration, and an additional assumption about the integration constant, yields the “logarithmic law of the wall” u/u* = (1/κ) ln y + B, where B is implicitly claimed to be R-independent. This conclusion is consistent with Eq. 2.

However, though this derivation is often reproduced, the conclusion is untenable. The “constants” κ, B vary widely from experiment to experiment, and the data (see Fig. 3) do not show a single R-independent line in the (ln y, u) plane but rather a series of curves, one per Reynolds number, with a straight envelope and a common asymptotic slope, much like Fig. 2. It is therefore more sensible to assume (see Eq. 7) (incomplete similarity in y/δ and no similarity in R); α(R) and A(R) are functions of R. We further expand A(R), α(R) in powers of 1/ln R, consistent with Eq. 4. One can show directly that this dependence of A, α on R is critical. This yields We then keep only the two leading terms in the series. To have a finite limit in the subcritical case one must have α₀ = 0; this leaves three constants, α₁, A ₀, and A ₁, to determine. We determined them by comparison with experimental data (9, 4, 6, 12). The results agree extremely well with experiment, and even more important, the constants do not move as one processes all pipe and boundary layer flows for which data are available (12). The logarithmic law of the wall corresponds to the wrong logarithmic solution in the analysis of the previous section.

A significant conclusion one can now draw is that as the Reynolds number increases, the first moments of the velocity tend to a limit, and do so at an inverse logarithmic rate.

One may ask next how higher moments of the velocity field behave as the Reynolds number increases. If the first moments tend to a limit at an inverse logarithmic rate, one expects higher moments to converge to a limit at the same rate or at a lower rate; they can also fail to have a limit. Consider for example the structure functions for the velocity field in homogeneous isotropic turbulence (the objects of Kolmogorov's theory). Consider a homogeneous isotropic flow and the velocity u at two points x, x + r, and in particular the velocity component u_D in the direction of the line that joins these points. Then, drop the subscript D. The pth order structure function is s_p =〈(u(x + r) – u(x))^p〉, where the brackets denote an average. The function s_p depends on the Reynolds number R (for a suitable definition of R, see refs. 3 and 14), a bulk length scale L, the variable distance r = |r|, and the mean rate of energy dissipation ε. Dimensional analysis yields s_p = (εr)^2/3Φ_p(L/r, R), where Φ_p is an unknown dimensionless function of two large arguments. If one makes the complete similarity assumptions Φ_p ∼ C for large arguments, one finds the classical Kolmogorov scaling s_p ∼ (εr)^{p /3}, and in particular s ₂ ∼ (εR)^2/3, from which it follows that in the inertial range the energy spectrum E(k) is proportional to k ^–5/3, the famous Kolmogorov–Obukhov spectrum.

But it is not necessarily so. One should expect the Kolmogorov–Obukhov spectrum to be a manifestation of criticality (see ref. 15), and one should assume incomplete similarity (also see ref. 8). In the plethora of possible assumptions one may as well start with the form of Eqs. 7 and 10 so that for p = 2 one finds with A(R) = A ₀ + A ₁/ln R +..., assuming, of course, in addition that the critical dependence on R is inverse logarithmic as above. This yields a spectrum i.e., a viscous correction to the Kolmogorov spectrum that changes slowly with R. Such a correction could easily be mistaken for a constant correction if one is not too careful with the data, and it is perfectly consistent with at least some of the data (see ref. 11). However, the critical dependence on R may well be slower than inverse logarithmic, in which case it would be even easier to overlook it in the processing of the data, especially when one is unaware of the possibility of such dependence. It has long been speculated that the exponent in the spectrum of the inertial range was in some sense a “critical” exponent, but the possibility that it should be R-dependent had not been properly considered.

The same argument applies to the case p = 3, where it is believed that the Kolmogorov scaling is exact; this would simply mean α = 0 for p = 3. As one goes up in p, one should expect the critical dependence on R to slow down and even change sign, so that eventually the moments do not have an inviscid limit and the critical dependence becomes a dependence on a growing function of R. It is compatible with the data, and with some numerical simulations (7, 13), to assume that this turnover happens at p = 3. If this is so, it explains the divergence of the measured higher structure functions from the Kolmogorov predictions without in any way invalidating the Kolmogorov theory.