Keywords

3.1 Richardson’s Energy Cascade

One important concept of fluid turbulence is the energy cascade introduced by Richardson [1]. Turbulent flows are dissipative thus energy input by external forces is required to maintain a turbulent fluid flow. The characteristic scale of this energy input is typically of the size of the system, known as the integral scale, denoted by \(l_0\). On the other hand, the characteristic scale of energy dissipation by viscous effects, known as the dissipative scale and, denoted by \(l_d\), is much smaller than the integral scale. As a result, there must be a transfer of energy from large to small scales. A transfer of energy between scales indicates an interaction between the Fourier modes of velocity of different wave numbers or scales. This is possible because of the nonlinear advection term \(\vec {U} \cdot \vec {\nabla } \vec {U}\) in the Navier–Stokes equation. Richardson pictured this energy transfer as a cascade process. Large eddies of the integral scale are produced by the external forces. They are unstable due to the nonlinearity of the dynamics and break up into eddies of smaller scale. These smaller eddies are themselves unstable and break up into eddies of even smaller scale. This process continues until dissipative effects due to viscosity are significant, and the turbulent kinetic energy is then dissipated into heat. This picture of energy cascade is succinctly summarized in Richardson’s famous rhyme [1]:

Big whorls have little whorls

Which feed on their velocity

And little whorls have lesser whorls

And so on to viscosity

(in the molecular sense)

An inherent feature of the cascade picture is that the energy transfer among scales is local, that is, the effective energy exchange between modes of different wave numbers decreases as the ratio of the wave numbers increases. It is expected that the statistics at the integral scale are determined by the mechanism of energy input and would vary from flow to flow. Because of the locality of the energy transfer, the statistics at small scales, scales further down the cascade and far from the integral scale, are not directly influenced by the mechanism of energy input. Thus this locality feature of the energy cascade allows for the possibility of universal characteristics for the statistics of small scales.

3.2 The Kolmogorov 1941 Theory

Based on Richardson’s energy cascade, Kolmorogov developed in 1941 a phenomenological theory (K41) [2] for the statistics of velocity difference,

$$\begin{aligned} \delta \vec {U}(\vec {r},\vec {l})\equiv \vec {U}(\vec {r}+\vec {l},t)-\vec {U}(\vec {r},t) \end{aligned}$$
(3.1)

measured at the same time and at two positions separated by a displacement vector \(\vec {l}\). There are several hypotheses in the K41 theory. We shall focus on two of them. The first one is an assumption of statistical homogeneity and isotropy of the small-scale turbulent motion when, the Reynolds number (Re) is sufficiently high and far from the boundaries. For statistically homogeneous fluctuations, \(\delta U(\vec {r},\vec {l})=\delta u(\vec {r},\vec {l})\) as \(\langle \vec {U}(\vec {r},t) \rangle =\langle \vec {U}(\vec {r}+\vec {l},t) \rangle \). Thus under this hypothesis, the statistics of \(\delta \vec {u}(\vec {r},\vec {l})\) do not depend on \(\vec {r}\) nor the direction of \(\vec {l}\) but depend only on \(l=|\vec {l}|\) for \(l \ll l_0\). The second assumption is that under the same conditions stated in the first assumption, there exists a range of intermediate length scales in which the statistics of \(\delta \vec {u}(l)\) are uniquely and universally determined by the mean energy transfer rate and \(l\). This range of intermediate length scales, \(l_d \ll l \ll l_0\), is known as the inertial range. The locality of energy cascade makes it possible that the statistics in the inertial range to be universal. The mean energy transfer rate is equal to the mean energy dissipation rate as well as to the mean energy input rate. The mean energy dissipation rate is given by \(\langle \epsilon \rangle \), where \(\epsilon (\vec {r},t)\) is defined by Eq. (1.29), and the ensemble average, can be taken as the spatial average in statistically homogeneous turbulent flows.

Using these two hypotheses and dimensional analysis, one therefore obtains

$$\begin{aligned} \langle \delta \vec {u}(l) \cdot \delta \vec {u}(l) \rangle = C \langle \epsilon \rangle ^{2/3} l^{2/3} \end{aligned}$$
(3.2)

where \(C\) is a universal constant. Since

$$\begin{aligned} \langle \delta \vec {u}(l) \cdot \delta \vec {u}(l) \rangle = 2 [\langle \vec {u} \cdot \vec {u} \rangle - \langle \vec {u}(\vec {r}+\vec {l},t)\cdot \vec {u}(\vec {r},t) \rangle ] \end{aligned}$$
(3.3)

we obtain

$$\begin{aligned} \langle \delta \vec {u}(l) \cdot \delta \vec {u}(l) \rangle = 4 \int \limits _0^\infty E(k) \left[ 1 - \left( \frac{\sin kl}{kl}\right) \right] dk \end{aligned}$$
(3.4)

using Eq. (2.55). As a result, Eq. (3.2) is equivalent to the result that the spatial energy spectrum \(E(k)\) follows a \(k^{-5/3}\) law over a suitable range of wave number \(k\). There is good experimental support for the \(-5/3\) power-law in the energy frequency spectrum \(E(f)\) (see [5] for details). Similar arguments can be applied to give higher-order statistics of \(\delta \vec {u}(l)\). The longitudinal velocity difference along the direction of the separation, denoted by \(\delta u_{\parallel }(l)\), is given by

$$\begin{aligned} \delta u_{\parallel }(l) \equiv \delta \vec {u}(l)\cdot \frac{\vec {l}}{l} \end{aligned}$$
(3.5)

Then we have

$$\begin{aligned} S_p(l) \equiv \langle [\delta u_{\parallel } (l)]^p \rangle = C_p \langle \epsilon \rangle ^{p/3} l^{p/3} \end{aligned}$$
(3.6)

for arbitrary \(p > 0\). Here, \(S_p(l)\) is known as the longitudinal velocity structure function of order \(p\). A power-law dependence of \(S_p(l)\) on \(l\) indicates that the inertial-range turbulent statistics are scale-invariant and Eq. (3.6) is a statement of the K41 scaling. For \(p=3\), an exact result can be derived from the Navier–Stokes equation for statistically homogeneous and isotropic flows:

$$\begin{aligned} S_3(l) = - \frac{4}{5} \langle \epsilon \rangle l \end{aligned}$$
(3.7)

This exact result is known as the four-fifth law and was derived by Kolmogorov also in 1941 [3]. It is one of the very few exact results for turbulent flows, and will be discussed in details in the next section.

In the above discussion, the K41 scaling is obtained by dimensional analysis based on the requirement that in the inertial range \(S_p(l)\) depends only on \(\langle \epsilon \rangle \) and the scale \(l\). Thus we have the same scaling behavior for the velocity structure function using \(\delta u(l) = |\delta \vec {u}(l)|\):

$$\begin{aligned} S^{u}_p(l) \equiv \langle [\delta u(l)]^p \rangle \sim \langle \epsilon \rangle ^{p/3} l^{p/3} \end{aligned}$$
(3.8)

The K41 scaling can also be obtained by the requirement that the rate of energy transfer (per unit mass) is scale-independent in the inertial range. To see this, think of \(\delta u(l)\) as the velocity of a turbulent eddy of scale \(l\). The energy transfer rate (per unit mass) at scale \(l\) can be estimated as \([\delta u(l)]^2/t_l\). Here, \(t_l\), known as the eddy turnover time, is the typical time for the eddy of size \(l\) to deform or change in energy, and can be estimated as \(t_l= l/\delta u(l)\). Requiring the rate of energy transfer to be independent of scale \(l\) implies \([\delta u(l)]^3/{l} \sim const\) but the mean energy transfer rate has to be equal to the mean energy dissipation rate thus \(const = \langle \epsilon \rangle \), hence

$$\begin{aligned} \frac{[\delta u(l)]^3}{l} \sim \langle \epsilon \rangle \end{aligned}$$
(3.9)

This gives

$$\begin{aligned} \delta u(l) \sim \langle \epsilon \rangle ^{1/3} l^{1/3} \end{aligned}$$
(3.10)

which further implies Eq. (3.8).

The dissipative scale \(l_d\) can be estimated as the scale at which the rate of dissipation due to viscosity is comparable to \(\langle \epsilon \rangle \):

$$\begin{aligned} \nu \left[ \frac{\delta u(l_d)}{l_d}\right] ^2 \sim \langle \epsilon \rangle \end{aligned}$$
(3.11)

Then take \(l_d\) to be at the edge of the inertial range such that Eq. (3.10) holds at \(l=l_d\). Eliminating \(\delta u(l_d)\) from Eqs. (3.10) and (3.11) gives \( l_d \sim (\nu ^3/\langle \epsilon \rangle )^{1/4}\). The Kolmogorov dissipative scale \(\eta _K\) is defined as:

$$\begin{aligned} \eta _K \equiv \left( \frac{\nu ^3}{\langle \epsilon \rangle }\right) ^{1/4} \end{aligned}$$
(3.12)

and typically \(l_d\) is of the order of \(10 \eta _K\).

Denote the normalized velocity difference by \(Y_l\):

$$\begin{aligned} Y_l \equiv \frac{\delta u(l)}{\langle [\delta u(l)]^2 \rangle ^{1/2}} \end{aligned}$$
(3.13)

An important consequence of Eq. (3.8) is that all the moments of \(Y_l\) are independent of \(l\). This \(l\)-independence follows directly from the proportionality of the scaling exponents \(p/3\) of \(S^u_p(l)\) to \(p\), and further implies that the PDF of \(Y_l\) is independent of \(l\). Thus the K41 theory predicts that the statistics of inertial-range turbulent velocity fluctuations are scale-independent and thus self-similar. Experiments confirmed the power-law dependence or scaling of \(S^u_p(l)\) but show that the scaling exponents, defined by \(S^u_p(l) \sim l^{\zeta (p)}\), depend on \(p\) in a nonlinear fashion. This deviation of the scaling behavior from the K41 prediction is known as anomalous scaling. Furthermore, the discrepancy between the observed scaling exponents \(\zeta (p)\) and the predicted values of \(p/3\) is known as intermittency corrections as the origin of the correction is believed to be due to the intermittent nature of turbulent fluctuations. The problem of anomalous scaling is a longstanding problem of turbulence and is remained to be solved. We shall discuss one particular idea, the refined similarity hypothesis, proposed by Kolmogorov and Obukhov in Sect. 3.7.

3.3 The Four-Fifth Law

We show in detail the derivation of the exact four-fifth law. We shall follow the treatment in [4] and [5]. In this subsection, we denote \(\partial /\partial t\) and \(\partial /\partial r_i\) by \(\partial _t\) and \(\partial _i\), and adopt the Einstein notation of summation over repeated indices. We write the Navier–Stokes equation Eq. (1.1) with an external force in component form:

$$\begin{aligned} \partial _t U_i + U_k \partial _k U_i = - \frac{1}{\rho } \partial _i p + \nu \partial _k^2 U_i + f_i \end{aligned}$$
(3.14)

Here, \(\rho \vec {f}\) is the external force per unit volume. Denote quantities evaluated at \(\vec {r} \ '= \vec {r}+\vec {l}\) by the same notations with a prime, e.g., \(U_i'\equiv U_i(\vec {r} \ ',t)\), and \(\partial /\partial r'_i\) by \(\partial '_i\). Then taking the ensemble average of the product of \(U_j'\) with Eq. (3.14) and \(U_i\) with Eq. (3.14) for \(U_j'\), we have

$$\begin{aligned} \partial _t \langle U_i U_j' \rangle&= -\partial _k \langle U_k U_i U'_j \rangle -\partial '_k \langle U'_k U'_j U_i \rangle - \frac{1}{\rho } \partial _i \langle U'_j p \rangle - \frac{1}{\rho } \partial '_j \langle U_i {p'}\rangle \nonumber \\&\quad + \nu (\partial _k^2 + \partial _k'^2) \langle U_i U_j'\rangle + \langle U_j'f_i \rangle + \langle U_i f_j' \rangle \end{aligned}$$
(3.15)

Here, we have used the interchangeability of taking derivative and ensemble average, incompressibility, and the derivative of primed quantities with respect to unprimed coordinates vanishes.

Consider turbulent flows that are statistically homogeneous and isotropic. Because of homogeneity and isotropy, the averages of the product of primed and unprimed quantities depend only on \(l=|\vec {r} \ '-\vec {r}|\). Therefore,

$$\begin{aligned} \partial _i \langle \cdot \rangle = - \partial _i' \langle \cdot \rangle = - \partial _{l_i} \langle \cdot \rangle \end{aligned}$$
(3.16)

Thus

$$\begin{aligned} \langle U'_j p \rangle = g(l) n_j \end{aligned}$$
(3.17)

for some function \(g(l)\) and \(n_j \equiv l_j/l\). Incompressibility implies \(0=\partial '_j \langle U'_j p \rangle =\partial _{l_j} \langle U'_j p \rangle \), and using

$$\begin{aligned} \partial _{l_i} = \frac{\partial l}{\partial l_i} \partial _l = n_i \partial _l , \quad \partial _{l_i} = \frac{\partial n_k}{\partial l_i} \partial _{n_k} = \frac{1}{l} (\delta _{ik}-n_i n_k) \partial _{n_k} \end{aligned}$$
(3.18)

we get

$$\begin{aligned} \frac{dg(l)}{dl} + \frac{2}{l} g(l) = 0 \Rightarrow g(l) = \frac{const}{l^2} \end{aligned}$$
(3.19)

But \(g(l)\) has to be finite at \(l=0\) thus \(const=0\) giving \(g(l)=0\) or \(\langle U'_j p \rangle =0\). Similarly, \(\langle U_i {p'}\rangle =0\). Define the velocity correlation and structure functions as follows.

$$\begin{aligned} b_{i,j}&= \langle U_i U'_j \rangle \end{aligned}$$
(3.20)
$$\begin{aligned} B_{ij}&= \langle (U'_i-U_i)(U'_j-U_j) \rangle \end{aligned}$$
(3.21)
$$\begin{aligned} b_{ij,m}&= \langle U_i U_j U'_m\rangle \end{aligned}$$
(3.22)
$$\begin{aligned} B_{ijm}&= \langle (U'_i-U_i)(U'_j-U_j)(U'_m-U_m) \rangle \end{aligned}$$
(3.23)

Statistical homogeneity and isotropy imply that these functions depend only on \(l\). Moreover, \(\langle U_i(\vec {r},t) U_j(\vec {r}+\vec {l},t) \rangle = \langle U_i(\vec {r}-\vec {l},t) U_j(\vec {r},t) \rangle = \langle U_i(\vec {r}+\vec {l},t) U_j(\vec {r},t) \rangle \), thus \(b_{i,j}=b_{j,i}\) is symmetric in the indices \(i\) and \(j\). Therefore, the most general forms for \(b_{i,j}\) and \(b_{ij,m}\) are:

$$\begin{aligned} b_{i,j}&= A(l) \delta _{ij} + B(l) n_i n_j \end{aligned}$$
(3.24)
$$\begin{aligned} b_{ij,m}&= C(l) \delta _{ij}{n_m} + D(l) \left( \delta _{im}{n_j}+\delta _{jm}{n_i}\right) + F(l){n_i}{n_j}{n_m} \end{aligned}$$
(3.25)

The form in Eq. (3.25) takes into account the symmetry in the indices \(i\) and \(j\). Homogeneity implies \(\langle U'_j U'_k U_i \rangle = \langle U_j(\vec {r},t) U_k(\vec {r},t) U_i(\vec {r}-\vec {l},t)\), which is equal to \(-b_{jk,i}\) using Eq. (3.25). We write Eq. (3.15) for statistically homogeneous and isotropic turbulent flows:

$$\begin{aligned} \partial _t b_{i,j} = - \partial _k (b_{ki,j}+b_{kj,i}) + 2 \nu \partial _k^2 b_{i,j} + \langle U_j'f_i \rangle + \langle U_i f_j' \rangle \end{aligned}$$
(3.26)

Using Eq. (3.26), a relation between the second- and third-order longitudinal velocity structure functions can be derived and from this relation the four-fifth law follows.

As we are interested in the longitudinal structure functions, we let the \(x\)-axis be along the direction of \(\vec {l}\). Then we take \(i=j=1\) in Eq. (3.26) and obtain

$$\begin{aligned} \partial _t b_{\parallel ,\parallel } = -2 \partial _k b_{k\parallel ,\parallel } + 2 \nu \partial _k^2 b_{\parallel ,\parallel } + \frac{2}{3} \langle \vec {f} \cdot \vec {U} \rangle - \langle \delta f_{\parallel } \delta u_{\parallel } \rangle \end{aligned}$$
(3.27)

where \(\delta f_i = f'_i-f_i\). The subscript \(\parallel \) denotes the component along the longitudinal direction along \(\vec {l}\) and there is no summation over this direction. Next, we relate \(\partial _k^2 b_{\parallel ,\parallel }\) to \(S_2(l)\). Now \(S_2(l)\) can be written as

$$\begin{aligned} S_2(l) = \langle \delta u^2_{\parallel } \rangle = B_{ij} n_i n_j = \frac{2}{3} \langle \vec {U} \cdot \vec {U} \rangle - 2b_{\parallel ,\parallel } = \frac{2}{3} \langle \vec {U} \cdot \vec {U} \rangle - 2(A + B) \end{aligned}$$
(3.28)

where we have used

$$\begin{aligned} \langle U'_i U'_j \rangle = \langle U_i U_j \rangle = \frac{1}{3} \langle \vec {U} \cdot \vec {U} \rangle \delta _{ij} \end{aligned}$$
(3.29)

Using Eqs. (3.16) and (3.18), we get

$$\begin{aligned} \partial _k^2 b_{i,j} = \left( \frac{d^2 A}{dl^2} +\frac{2}{l} \frac{d A}{dl} + \frac{2}{l^2} B \right) \delta _{ij} + \left( \frac{d^2 B}{dl^2}+\frac{2}{l}\frac{d B}{dl} -\frac{6}{l^2} B \right) n_i n_j \end{aligned}$$
(3.30)

The incompressibility condition gives \(0=\partial '_j b_{i,j}\), which implies

$$\begin{aligned} \frac{l}{2}\frac{d}{dl} (A+B) + B = 0 \end{aligned}$$
(3.31)

Thus

$$\begin{aligned} \frac{d S_2}{dl} = -2 \frac{d}{dl} (A+B) = \frac{4}{l} B(l) \end{aligned}$$
(3.32)

and

$$\begin{aligned} \partial _k^2 b_{i,j}&= -\left( \frac{d^2 B}{dl^2}+\frac{4}{l}\frac{d B}{dl}\right) \delta _{ij} + \left( \frac{d^2 B}{dl^2}+\frac{2}{l}\frac{d B}{dl}-\frac{6}{l^2} B \right) n_i n_j \nonumber \\ \Rightarrow \qquad \partial _k^2 b_{\parallel ,\parallel }&= -\frac{2}{l^4} \frac{d(l^3 B)}{dl} = -\frac{1}{2l^4} \ \displaystyle \frac{d}{dl} \left[ l^4 \frac{d S_2}{dl}\right] \qquad \end{aligned}$$
(3.33)

Then we relate \(\partial _k b_{k\parallel ,\parallel }\) to \(S_3(l)\). The incompressibility condition gives \(0=\partial '_m b_{ij,m}\). Using again Eqs. (3.16) and (3.18), we get

$$\begin{aligned} \left[ \frac{d C}{dl} + \frac{2}{l}(C+D)\right] \delta _{ij} + \left[ 2\frac{d D}{dl}+ \frac{d F}{dl}+ \frac{2}{l} (F-D)\right] n_i n_j = 0 \end{aligned}$$
(3.34)

Thus

$$\begin{aligned} \frac{d C}{dl} + \frac{2}{l}(C+D)&= 0 \end{aligned}$$
(3.35)
$$\begin{aligned} \frac{d (3C+2D+F)}{dl} + \frac{2}{l} (3C+2D +F)&= 0 \end{aligned}$$
(3.36)

Equation (3.36) is obtained by taking the trace of Eq. (3.34). The functions \(C\), \(D\), and \(F\) have to be finite at \(l=0\), thus

$$\begin{aligned} 3C + 2 D + F = 0 \end{aligned}$$
(3.37)

We can then express \(D\) and \(F\) in terms of \(C\) and \(dC/dl\):

$$\begin{aligned} D&= -C- \frac{l}{2}\frac{d C}{dl} \end{aligned}$$
(3.38)
$$\begin{aligned} F&= l \frac{d C}{dl}-C \end{aligned}$$
(3.39)

and obtain

$$\begin{aligned} b_{ij,m} = C \delta _{ij} n_m - \left( C+\frac{l}{2} \frac{d C}{dl}\right) (\delta _{im}n_j + \delta _{jm} n_i) + \left( l\frac{d C}{dl}-C \right) n_i n_j n_m \qquad \end{aligned}$$
(3.40)

Thus

$$\begin{aligned} S_3(l)&= \langle \delta u^3_{\parallel } \rangle = B_{ijm} n_i n_j n_m \end{aligned}$$
(3.41)
$$\begin{aligned}&= 2(b_{ij,m}+b_{im,j}+b_{jm,i})n_i n_j n_m = -12 C(l) \end{aligned}$$
(3.42)

where we have used \(\langle U'_i U'_j U'_m\rangle = \langle U_i U_j U_m\rangle \). Moreover,

$$\begin{aligned} -\partial _k b_{ki,j}&= \left( -\frac{2}{l}C+2\frac{d C}{dl}+\frac{l}{2}\frac{d^2 C}{dl^2}\right) n_i n_j - \left( \frac{2}{l} C+3\frac{d C}{dl}+\frac{l}{2}\frac{d^2 C}{dl^2} \right) \delta _{ij} \nonumber \\ \Rightarrow \qquad -\partial _k b_{k\parallel ,\parallel }&= -\frac{1}{l^4} \frac{d}{dl}(l^4 C) = -\frac{1}{12 l^4} \frac{d}{dl}(l^4 S_3) \end{aligned}$$
(3.43)

Putting all the results together, we finally obtain

$$\begin{aligned} \frac{1}{6 l^4} \frac{d}{dl}(l^4S_3) - \frac{\nu }{l^4} \frac{d}{dl}\left( l^4 \frac{d S_2}{dl}\right) = -\frac{2}{3} \langle \epsilon \rangle - \frac{1}{2} \partial _t S_2 + \langle \delta f_{\parallel } \delta u_{\parallel }\rangle \end{aligned}$$
(3.44)

Here, we have used

$$\begin{aligned} \frac{1}{2}\partial _t \langle \vec {U} \cdot \vec {U} \rangle - \langle \vec {f} \cdot \vec {U} \rangle = - \langle \epsilon \rangle \end{aligned}$$
(3.45)

which follows from Eq. (3.14). For decaying turbulence, \(\vec {f}=0\) and \(\partial _t S_2 \approx 0\) for \(l \ll l_0\). For stationary turbulence forced by \(\vec {f}\) that acts only at the largest scales, \(\partial _t S_2 = 0\) and \(\langle \delta f_{\parallel } \delta u_{\parallel }\rangle \approx 0\) for \(l \ll l_0\). Thus for both cases, we have

$$\begin{aligned} \frac{1}{6 l^4} \frac{d}{dl}(l^4S_3) - \frac{\nu }{l^4} \frac{d}{dl}\left( l^4 \frac{d S_2}{dl}\right) = -\frac{2}{3} \langle \epsilon \rangle \end{aligned}$$
(3.46)

for \(l \ll l_0\). In the limit of \(\nu \rightarrow 0\), the viscous term is negligible. On the other hand, it is assumed that \(\langle \epsilon \rangle \) remains finite in this limit. This implies that the velocity gradients \(\partial u_i/\partial r_j\) become unlimited as \(\nu \rightarrow 0\) or as Re \(\rightarrow \infty \), which further implies that vorticity is generated in turbulent flows and increases with Re. The result that the dissipation remains finite as Re \(\rightarrow \infty \) is generally referred to as the “dissipative anomaly”, and is well supported by experimental and numerical results. Thus in the limit of \(\nu \rightarrow 0\), integrating Eq. (3.46) gives the four-fifth law Eq. (3.7). We note that for stationary homogeneous and isotropic turbulence with \(\vec {f}\) acting over all scales, we have the more general result [6]

$$\begin{aligned} S_3(l) = -\frac{4}{5} \langle \epsilon \rangle l + \frac{6}{l^4} \int \limits _0^l l'^4 \langle \delta f_{\parallel }(l') \delta u_{\parallel }(l') \rangle d l' \end{aligned}$$
(3.47)

for \( l \ll l_0\).

3.4 The Obukhov–Corrsin Theory for Passive Scalar

Obukhov [7] and Corrsin [8] extended Kolmogorov’s 1941 theory to study temperature fluctuations in weakly-heated incompressible turbulent flows. The heating is so weak that the resulted temperature variations have no dynamical effect on the turbulent flow itself. As a result, the velocity field is still governed by the Navier–Stokes equation. In this case, the temperature is known as a passive scalar. The equations of motion are thus Eqs. (1.1) and (1.8). The Obukhov–Corrsin theory gives the statistics of the temperature difference, defined by

$$\begin{aligned} \delta T(\vec {r},\vec {l})\equiv T(\vec {r}+\vec {l},t)-T(\vec {r},t) \end{aligned}$$
(3.48)

which is taken to be statistically homogeneous and isotropic. Besides the cascade of turbulent energy, there is also a cascade of temperature variance from large to small scales. The mean temperature dissipation rate is given by \(\langle \chi \rangle \), where \(\chi (\vec {r},t)\) is defined in Eq. (1.30). In analogy to the K41 theory, the temperature variance transfer rate, estimated by \([\delta T(l)]^2/t_l\), is scale-independent and thus equals to \(\langle \chi \rangle \) in the intermediate inertial-convective range, the range of scales within the inertial range where buoyancy is insignificant. That is,

$$\begin{aligned} \frac{[\delta T(l)]^2 \delta u(l)}{l} \sim \langle \chi \rangle \end{aligned}$$
(3.49)

Together with Eq. (3.10) for \(\delta u(l)\), we obtain

$$\begin{aligned} \delta T(l) \sim \langle \epsilon \rangle ^{-1/6} \langle \chi \rangle ^{1/2} l^{1/3} \end{aligned}$$
(3.50)

and the Obukhov–Corrsin (OC) scaling for passive temperature fluctuations:

$$\begin{aligned} S^{\theta }_p(l) \equiv \langle [\delta T(l)]^p \rangle \sim \langle \epsilon \rangle ^{-p/6} \langle \chi \rangle ^{p/2} l^{p/3} \end{aligned}$$
(3.51)

Here, \(S^{\theta }_p\) is known as the \(p\)th order temperature structure functions. Experiments again confirm the power-law dependence but show that there are intermittency corrections to the OC scaling such that \(S^{\theta }_p(l) \sim l^{\xi (p)}\) and \(\xi (p)\) deviates from \(p/3\) [9].

3.5 The Bolgiano–Obukhov Scaling

In turbulent convection, temperature variations result in a buoyancy force that drives the fluid motion, and temperature is now an active scalar. The presence of buoyancy could affect and modify the scaling behavior. In several theoretical studies [6, 10–13], arguments were given that buoyancy would give rise to a different scaling behavior:

$$\begin{aligned} S^u_p(l) \rangle&\sim (\alpha g)^{2p/5} \langle \chi \rangle ^{p/5} l^{3p/5} \end{aligned}$$
(3.52)
$$\begin{aligned} S^{\theta }_p(l)&\sim (\alpha g)^{-p/5} \langle \chi \rangle ^{2p/5} l^{p/5} \end{aligned}$$
(3.53)

This type of scaling behavior, which is known as the Bolgaino–Obukhov (BO) scaling, was originally proposed by Bolgiano [14] and Obukhov [15] for stably stratified flows (see also discussions in [16]) based on dimensional analysis and the argument that the velocity and temperature structure functions would depend only on \(\alpha g\), \(\langle \chi \rangle \) and \(l\). Here, \(\alpha g\) is the additional parameter that describes the strength of buoyant coupling when buoyancy is significant. In turbulent Rayleigh–Bénard convection, the BO scaling can be obtained based on a cascade of temperature variance (Eq. (3.49)) or a cascade of entropy flux  [12] (for \(\theta \ll T_0\), \(\int \theta ^2 d^3 x\) describes the entropy increase per unit mass and volume due to the temperature fluctuations [13]) together with the argument that the buoyant term dominates the dynamics and balances the nonlinear advection term:

$$\begin{aligned} \alpha g \delta T(l) \sim \frac{[\delta u(l)]^2}{l} \end{aligned}$$
(3.54)

Equations (3.49) and (3.54) imply

$$\begin{aligned} \delta u(l)&\sim (\alpha g)^{2/5} \langle \chi \rangle ^{1/ 5}l^{3/5} \end{aligned}$$
(3.55)
$$\begin{aligned} \delta T(l)&\sim (\alpha g)^{-1/5} \langle \chi \rangle ^{2/ 5} l^{1/5} \end{aligned}$$
(3.56)

Then Eqs. (3.52) and (3.53) follow directly.

3.6 Crossover in Scaling

The BO scaling would hold only when buoyancy is significant. When buoyancy is negligible, temperature behaves as a passive scalar and K41 and OC scaling would hold. The buoyant term, estimated by \(\alpha g \delta T(l) \delta u(l)\), increases with \(l\). Thus one expects a crossover from the K41-OC scaling to the BO scaling to occur at the crossover scale \(l_c\) when

$$\begin{aligned} \langle \delta u(l_c)^{BO} \rangle = \langle \delta u(l_c)^{K41} \rangle \end{aligned}$$
(3.57)

Using Eqs. (3.10) and (3.55), we get

$$\begin{aligned} l_c = \frac{\langle \epsilon \rangle ^{5/4}}{{(\alpha g)^{3/2} \langle \chi \rangle ^{3/4}}} \equiv L_B \end{aligned}$$
(3.58)

Thus the crossover scale is given by \(L_B\), which is known as the Bolgiano length and is the length scale above which buoyancy is important. The Bolgiano length was first defined in terms of \(\alpha g\), \(\langle \epsilon \rangle \), and \(\langle \chi \rangle \) using dimensional analysis [16], Furthermore, we have

$$\begin{aligned} \alpha g \langle \delta u(l) \delta T(l)\rangle \ge \langle \epsilon \rangle \qquad \mathrm{for} \ l \ge L_B \end{aligned}$$
(3.59)

therefore \(L_B\) is also the scale at which the power injected into the flow due to buoyancy is equal to the mean energy dissipation rate [17]. Using the exact relations Eqs. (1.41) and (1.42), \(L_B\) can be related to Nu and Ra:

$$\begin{aligned} L_B = \frac{\mathrm{Nu}^{1/2}}{(\mathrm{Pr Ra})^{1/4}} \ H \end{aligned}$$
(3.60)

Hence, the picture emerging from these scaling theories is that the BO scaling is expected to hold in the buoyancy subrange, \(l_0 \gg l > L_B\), while the K41-OC scaling is expected to hold in the inertial-convective subrange, \(l_d \ll l< L_B \). If \(L_B\) is of the order of \(l_0 \approx H\) or even larger, then only K41-OC scaling will be observed. On the other hand, if \(L_B\) is of the order of \(l_d\) or even smaller, then only the BO scaling would be observed [13]. However, there are two complications. The first complication is that turbulent Rayleigh–Bénard convection is inhomogeneous. Thus it is more appropriate to define a local crossover or Bolgiano length using the energy and thermal dissipation rates averaged over the local region of interest. As a result, it is possible that different scaling behavior is observed in different regions of the cell. This will be discussed in Chap. 4 when we examine the scaling behavior observed in experiments and numerical calculations. The second complication is the existence of intermittency corrections to the scaling behavior. In the next Section, we shall discuss one particular idea, the refined similarity hypothesis, which was proposed to account for the intermittency corrections.

3.7 Refined Similarity Hypothesis

To account for the intermittency corrections of velocity fluctuations, Kolmogorov proposed in 1962 [18] to refine his second hypothesis by replacing the mean energy dissipation rate \(\langle \epsilon \rangle \) with a locally-averaged energy dissipation rate over a scale \(l\), defined as

$$\begin{aligned} \epsilon _l(\vec {r},t) \equiv \frac{3}{4 \pi l^3} \int \limits _{|\vec {y}|\le l} \epsilon (\vec {x}+\vec {y},t) d \vec {y} \end{aligned}$$
(3.61)

Similar ideas were also proposed independently by Obukhov [19]. As a result of this refinement, which is known as the refined similarity hypothesis (RSH), Eq. (3.10) is modified to

$$\begin{aligned} \delta u(l)^{K41}&\sim \epsilon _l^{1/3} l^{1/3} \end{aligned}$$
(3.62)
$$\begin{aligned} \Rightarrow \qquad S^u_p(l)&\sim \langle \epsilon _l^{p/3} \rangle l^{p/3} \end{aligned}$$
(3.63)

Corrections to the K41 scaling can thus be resulted from the \(l\)-dependence of the moments of \(\epsilon _l\). In particular, let

$$\begin{aligned} \langle \epsilon _l^{q} \rangle \sim l^{\tau (q)} \end{aligned}$$
(3.64)

then

$$\begin{aligned} \zeta (p) = \tau \left( \frac{p}{3}\right) + \frac{p}{3} \end{aligned}$$
(3.65)

Different intermittency models have been proposed which give different results for \(\tau (q)\).

A direct implication of Eq. (3.62) is

$$\begin{aligned} \langle [\delta u(l)]^p \ \big | \ \epsilon _l = x \rangle \sim x^{p/3} l^{p/3} \end{aligned}$$
(3.66)

where \(\langle [\delta u(l)]^p \ \big | \ \epsilon _l = x \rangle \) is the conditional velocity structure function of order \(p\) when the value of \(\epsilon _l\) is fixed at a small range about \(x\). Thus \(\langle [\delta u(l)]^p \ \big | \ \epsilon _l = x \rangle \sim l^{p/3}\) exhibits the K41 scaling. Support for Eq. (3.66) has been found in both experiments [20] as well as in direct numerical simulations and large-eddy simulations [21].

The refined similarity hypothesis has been extended to temperature fluctuations by replacing also \(\chi \) by the locally averaged \(\chi _l(\vec {r},t)\), which is similarly defined:

$$\begin{aligned} \chi _l(\vec {r},t) = \frac{3}{4 \pi l^3} \int \limits _{|\vec {y}|\le l} \chi (\vec {x}+\vec {y},t) d^3 y \end{aligned}$$
(3.67)

For passive temperature fluctuations, Eq. (3.50) becomes [22, 23]:

$$\begin{aligned} \delta T(l)^{OC} \sim \epsilon _l^{-1/6} \chi _l^{1/2} l^{1/3} \end{aligned}$$
(3.68)

and for the BO scaling, Eqs. (3.55) and (3.56) become [24]:

$$\begin{aligned} \delta u(l)^{BO}&\sim (\alpha g)^{2/5} \chi _l^{1/5} l^{3/5} \end{aligned}$$
(3.69)
$$\begin{aligned} \delta T(l)^{BO}&\sim (\alpha g)^{-1/5} \chi _l^{2/5} l^{1/5} \end{aligned}$$
(3.70)

3.8 Conditional Structure Functions

We note the interesting observation that the dependence on \(\chi _l\) is different for the two scaling behaviors, K41-OC and BO, as shown in Eqs. (3.62), (3.68), (3.69) and (3.70). This difference can be clearly spelled out by studying the conditional velocity and temperature structure functions evaluated at fixed values of \(\chi _l\):

$$\begin{aligned} \tilde{S}^u_p(l,x)&\equiv \langle [\delta u(l)]^p \ \big | \ \chi _l=x \rangle \end{aligned}$$
(3.71)
$$\begin{aligned} \tilde{S}^{\theta }_p(l,x)&\equiv \langle [\delta T(l)]^p \ \big | \ \chi _l=x \rangle \end{aligned}$$
(3.72)

We have used these conditional structure functions [24] and similar conditional structure functions evaluated at given values of local temperature variance transfer rate [25] to examine the validity of refined similarity hypothesis in turbulent Rayleigh–Bénard convection.

To evaluate \(\tilde{S}^u_p(l,x)\) and \(\tilde{S}^{\theta }_p(l,x)\) from Eqs. (3.62) and (3.68) in the case of the K41-OC scaling, we need to evaluate the conditional average \(\langle \epsilon _l^q \ \big | \ \chi _l=x \rangle \) for various values of \(q\). In this case, temperature is a passive scalar so we make use of the measured approximate statistical independence of \(\epsilon _l\) and \(\chi _l\) for passive scalar fluctuations [26] to approximate:

$$\begin{aligned} \langle \epsilon _l^q \ \big | \ \chi _l=x \rangle \approx \langle \epsilon _l^q \rangle \qquad \mathrm{K41-OC} \end{aligned}$$
(3.73)

As a result, we obtain

$$\begin{aligned} \tilde{S}^u_p(l,x)&\sim {\left\{ \begin{array}{ll} \langle \epsilon _l^{p/3} \rangle l^{p/3} &{} \mathrm{K41} \\ (\alpha g)^{2p/5} x^{p/5} l^{3p/5} &{} \mathrm{BO} \end{array}\right. } \end{aligned}$$
(3.74)
$$\begin{aligned} \tilde{S}^\theta _p(l,x)&\sim {\left\{ \begin{array}{ll} \langle \epsilon _l^{-p/6} \rangle x^{p/2} l^{p/3} &{} \mathrm{OC} \\ (\alpha g)^{-p/5} x^{2p/5} l^{p/5} &{} \mathrm{BO} \end{array}\right. } \end{aligned}$$
(3.75)

From Eqs. (3.74) and (3.75), we see the different \(x\)-dependence of \(\tilde{S}^u_p\) and \(\tilde{S}^\theta _p\) for the two different scaling behaviors: \(\tilde{S}^u_p\) is independent of \(x\) for the K41 scaling but has a power-law dependence of \(x^{p/5}\) for the BO scaling. Similarly, \(\tilde{S}^\theta _p\) has the power-law dependence of \(x^{p/2}\) for the OC scaling but a different dependence of \(x^{2p/5}\) for the BO scaling. Hence it is possible to reveal the two different scaling behaviors by studying the \(x\)-dependence of \(\tilde{S}^u_p(l,x)\) and \(\tilde{S}^\theta _p(l,x)\). This method is particularly useful because the unknown intermittency corrections might hinder direct revelation of the scaling behavior. Details about this method will be discussed in Chap. 4.