2 Basic setting of the hypothesis

Here is a summary of the attached-eddy hypothesis (Townsend Reference Townsend1976). The turbulence is taken to be homogeneous in the streamwise and spanwise directions. For the case of a boundary layer, we assume that it has been well developed and hence is negligibly dependent on the streamwise position $x$ . Since this hypothesis is for the constant-stress layer, the value of the kinematic viscosity $\unicode[STIX]{x1D708}$ is not essential. The limit $\unicode[STIX]{x1D708}\rightarrow 0$ is taken so as to ignore the viscous length $\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ with respect to the height $z$ . Then, a free-slip condition, i.e. $u\neq 0$ and $v\neq 0$ , is imposed on the wall at $z=0$ .

The attached eddies extend from the wall into the flow but are bounded in each of the directions. Regardless of the size of the eddies, they have an identical shape with a common characteristic velocity $u_{\unicode[STIX]{x1D70F}}$ . That is, if $\boldsymbol{x}_{e}=(x_{e},y_{e},h_{e})$ lies at the highest position of an eddy (see figure 1), its velocity field $\boldsymbol{v}_{e}$ is given for a position $\boldsymbol{x}=(x,y,z)$ as

We regard $h_{e}$ as the size of this eddy. The existence of the wall imposes $f_{w}=0$ at $z=0$ . As for the undermost layer at $z\ll h_{e}$ of such an eddy, it is enough to assume  $f_{w}\propto z/h_{e}$ . Furthermore, the free-slip wall condition imposes $f_{u}\neq 0$ and $f_{v}\neq 0$ at $z=0$ . These are summarized by using $\boldsymbol{g}$ as some function of $(x-x_{e})/h_{e}$ and $(y-y_{e})/h_{e}$ ,

The condition at $z>h_{e}$ is $f_{u}=f_{v}=f_{w}=0$ . No more assumptions are required about the functional form of $\boldsymbol{f}$ .

The sizes of the attached eddies are distributed continuously from $h_{e}=\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}\rightarrow 0$ to $h_{e}=\unicode[STIX]{x1D6FF}$ . Here $\unicode[STIX]{x1D6FF}$ corresponds to the height of the wall turbulence, i.e. pipe radius, channel half-width or boundary layer thickness. The asymptotic laws for $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ are regarded as those for the constant-stress layer. On the wall, the eddies are distributed randomly and independently. They could even overlap one another. We do not assume any more about the distribution of the eddies, although some previous studies have assumed a particular hierarchy in that distribution (e.g. Perry & Chong Reference Perry and Chong1982).

From the random and independent distribution of the attached eddies, it follows that the entire velocity field is a superposition of those of the individual eddies. Any two-point correlation over a streamwise distance $\boldsymbol{r}_{l}=\boldsymbol{r}_{x}=(r,0,0)$ at a height $z$ is the sum of those within the individual eddies of various sizes from $h_{e}=z$ to $h_{e}=\unicode[STIX]{x1D6FF}$ ,

The subscripts $i$ and $j$ denote either $x$ , $y$ or $z$ such that $v_{i}$ and $v_{j}$ denote either $u$ , $v$ or $w$ . As justified later, the number density of the attached eddies at size $h_{e}$ per unit area of the wall is $n_{e}(h_{e})=N_{e}h_{e}^{-3}$ , where $N_{e}$ is a constant. By using $\langle v_{i}(x+r,z)v_{j}(x,z)\rangle$ in place of $\langle v_{i}(\boldsymbol{x}+\boldsymbol{r}_{x})v_{j}(\boldsymbol{x})\rangle$ ,

with the correlation for eddies of size $h_{e}$ ,

Since the streamwise and spanwise sizes of each eddy have been taken as finite, the integral is always finite. Together with the wall condition of (2.1b–d ), which yields a condition for $I_{v_{i}v_{j}}(r/h_{e},z/h_{e})$ in the case of $r/h_{e}=0$ , the form of (2.3) serves as the basis of the attached-eddy hypothesis.

Figure 2.

Schematic of $I_{uu}$ , $I_{vv}$ (a), $-I_{uw}$ , $I_{\unicode[STIX]{x1D700}}$ (b) and $I_{ww}$ , $I_{\unicode[STIX]{x1D700}\unicode[STIX]{x1D700}}$ (c) for $r_{\ast }=r/h_{e}=0$ as a function of $z_{\ast }=z/h_{e}$ , where grey areas correspond to the undermost layers of the eddies.

To justify the distribution of the eddy size $n_{e}(h_{e})=N_{e}h_{e}^{-3}$ , the constant nature of $\langle -uw\rangle$ is reproduced in the limit $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ . That is, for $r=0$ in (2.3a ),

We have used $z_{\ast }=z/h_{e}$ and $\text{d}z_{\ast }/z_{\ast }=-\text{d}h_{e}/h_{e}$ . The wall condition of (2.1b–d ) yields $I_{uw}(0,z_{\ast })=a_{e}z_{\ast }$ with a constant $a_{e}$ at $z_{\ast }\ll 1$ . If $z/\unicode[STIX]{x1D6FF}$ lies in this range of $z_{\ast }$ as shown in figure 2(b), the integral is equal to $b_{e}-a_{e}z/\unicode[STIX]{x1D6FF}$ . The constant $b_{e}=\int _{0}^{1}(\text{d}z_{\ast }/z_{\ast })\,I_{uw}(0,z_{\ast })$ is dominant in the limit $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ . To fix the value as $\langle -uw\rangle /u_{\unicode[STIX]{x1D70F}}^{2}=1$ , we require $N_{e}=-1/b_{e}$ , i.e. $N_{e}\propto 1/|\boldsymbol{f}|^{2}$ in (2.3) for a given shape of $\boldsymbol{f}$ .

The velocity variances $\langle v_{i}^{2}(z)\rangle$ such as that in (1.1) are obtained in the same manner. As shown in figures 2(a) and 2(c), the wall condition of (2.1b–d ) yields

Here $d_{uu}$ and $d_{vv}$ are constants. The result for the constant-stress layer $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ is

Here $c_{uu}$ , $c_{vv}$ and $c_{ww}$ are constants. The law for $\langle u^{2}(z)\rangle$ has been confirmed with $c_{uu}\simeq$ 1.5–2.5 and $d_{uu}\simeq 1.3$ in a variety of wall turbulence (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). We rely on the other laws as well. They do hold in the wall turbulence, although we are not yet certain of the values of $c_{vv}$ , $c_{ww}$ and $d_{vv}$ (Sillero et al. Reference Sillero, Jiménez and Moser2013; Lee & Moser Reference Lee and Moser2015).

3 Velocity correlation along a wall-parallel line

Two-point velocity correlations along a wall-parallel line $\langle v_{i}(x+r,z)v_{i}(x,z)\rangle$ are studied at a height $z$ in the constant-stress layer. Although the line has been set streamwise, the same discussion is applicable to the case of a spanwise line. We make use of the correlation length

This is expected to be well defined in the attached-eddy hypothesis, where any correlation is determined by the individual eddies that are not dependent on one another and are of finite size. From (2.3a ) with $r_{\ast }=r/h_{e}$ and $z_{\ast }=z/h_{e}$ ,

The correlation length is also defined in an eddy of a particular size $h_{e}$ ,

While $L_{x:v_{i}v_{i}}/z$ is dependent on $z/\unicode[STIX]{x1D6FF}$ alone, $\unicode[STIX]{x1D6EC}_{x:v_{i}v_{i}}/h_{e}$ is dependent on $z/h_{e}$ alone. That is, $L_{x:v_{i}v_{i}}$ and $\unicode[STIX]{x1D6EC}_{x:v_{i}v_{i}}$ are to be determined respectively by $z/\unicode[STIX]{x1D6FF}$ and $z/h_{e}$ through $L_{x:v_{i}v_{i}}/z$ and $\unicode[STIX]{x1D6EC}_{x:v_{i}v_{i}}/h_{e}$ . Although (3.2a ) could be rewritten so that $L_{x:v_{i}v_{i}}/\unicode[STIX]{x1D6FF}$ is another function of $z/\unicode[STIX]{x1D6FF}$ , it is not considered here because $\unicode[STIX]{x1D6FF}$ is not a characteristic of the constant-stress layer.

We require $L_{x:v_{i}v_{i}}/z$ to be finite. If $L_{x:v_{i}v_{i}}/z$ were not finite, it would follow that $L_{x:v_{i}v_{i}}$ is not determined and is not well defined in the attached-eddy hypothesis. The finiteness is also required for $\unicode[STIX]{x1D6EC}_{x:v_{i}v_{i}}/h_{e}$ because the streamwise size of an eddy is finite compared with its wall-normal size. For the undermost layer of such an eddy $z_{\ast }=z/h_{e}\ll 1$ , there is a constant $\unicode[STIX]{x1D6FC}\geqslant 0$ in relation to (3.2b ),

The asymptotic form of $I_{v_{i}v_{i}}(0,z_{\ast })$ is given in (2.5). By substituting the resultant form of $\int _{0}^{\infty }\,\text{d}r_{\ast }\,I_{v_{i}v_{i}}(r_{\ast },z_{\ast })$ into (3.2a ), along with $\int _{z/\unicode[STIX]{x1D6FF}}^{1}(\text{d}z_{\ast }/z_{\ast })\,I_{v_{i}v_{i}}(0,z_{\ast })$ given in (2.6), we obtain $L_{x:v_{i}v_{i}}/z$ in the limit $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ . The minimum $\unicode[STIX]{x1D6FC}$ value for $L_{x:v_{i}v_{i}}/z$ to be finite is used as the leading order to determine the asymptotic form of $I_{v_{i}v_{i}}(r_{\ast },z_{\ast })$ in (3.3). It determines the asymptotic form of $\langle v_{i}(x+r,z)v_{i}(x,z)\rangle$ in the constant-stress layer $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ in a manner similar to that for the variance $\langle v_{i}^{2}(z)\rangle$ in (2.6).

The individual cases of $v_{i}=u$ , $v$ and $w$ are as follows. Only leading-order behaviour for $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ is studied about $\int _{z/\unicode[STIX]{x1D6FF}}^{1}(\text{d}z_{\ast }/z_{\ast }^{2})\int _{0}^{\infty }\,\text{d}r_{\ast }\,I_{v_{i}v_{i}}(r_{\ast },z_{\ast })$ and $\int _{z/\unicode[STIX]{x1D6FF}}^{1}(\text{d}z_{\ast }/z_{\ast })\,I_{v_{i}v_{i}}(0,z_{\ast })$ , e.g. $\ln (\unicode[STIX]{x1D6FF}/z)$ about $c_{uu}+d_{uu}\ln (\unicode[STIX]{x1D6FF}/z)$ in (2.6a ).

For the streamwise velocity $u$ , we take $\unicode[STIX]{x1D6FC}=1$ in (3.3) and use (2.5) to obtain $\int _{0}^{\infty }\,\text{d}r_{\ast }\,I_{uu}(r_{\ast },z_{\ast })\propto z_{\ast }$ in the limit $z_{\ast }\rightarrow 0$ . This is applied for $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ as

On the other hand, from (2.6a ),

By substituting these into (3.2a ), we confirm that $L_{x:uu}/z$ is finite. If $\unicode[STIX]{x1D6FC}<1$ , it would diverge as $(\unicode[STIX]{x1D6FF}/z)^{1-\unicode[STIX]{x1D6FC}}/\ln (\unicode[STIX]{x1D6FF}/z)$ . The above asymptotic form of $\int _{0}^{\infty }\,\text{d}r_{\ast }\,I_{uu}(r_{\ast },z_{\ast })\propto z_{\ast }$ is equivalent through $r_{\ast }=r/h_{e}$ and $z_{\ast }=z/h_{e}$ to

Thus, $I_{uu}(r/h_{e},z/h_{e})$ in this limit $z/h_{e}\rightarrow 0$ is independent of $h_{e}$ but is only a function of $r/z$ , say, $D_{x:uu}(r/z)/N_{e}$ . It is independent of $z_{\ast }=z/h_{e}$ if $r/z$ is fixed as $I_{uu}(r/h_{e},z/h_{e})=I_{uu}(rz_{\ast }/z,z_{\ast })$ . The correlation function in the constant-stress layer $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ is thereby derived from

The result is

Here $C_{x:uu}(r/z)$ is a function of $r/z$ . The correlation function of the spanwise velocity $v$ is derived in the same manner,

For the wall-normal velocity $w$ , we take $\unicode[STIX]{x1D6FC}=0$ in (3.3) and use (2.5) to obtain $\int _{0}^{\infty }\,\text{d}r_{\ast }\,I_{ww}(r_{\ast },z_{\ast })\propto z_{\ast }^{2}$ in the limit $z_{\ast }\rightarrow 0$ . This is applied for $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ as

On the other hand, from (2.6c ),

By substituting these into (3.2a ), we confirm that $L_{x:ww}/z$ is finite. The above form of $\int _{0}^{\infty }\,\text{d}r_{\ast }\,I_{ww}(r_{\ast },z_{\ast })\propto z_{\ast }^{2}$ in the limit $z_{\ast }\rightarrow 0$ leads to $\unicode[STIX]{x1D6FD}\geqslant 2$ for $I_{ww}(r_{\ast },z_{\ast })\propto z_{\ast }^{\unicode[STIX]{x1D6FD}}$ at each $r_{\ast }$ in this limit. Also, as for $I_{ww}(r_{\ast },z_{\ast })\propto r_{\ast }^{\unicode[STIX]{x1D6FE}}$ at each $z_{\ast }$ in the limit $r_{\ast }\rightarrow 0$ , we require $\unicode[STIX]{x1D6FE}\geqslant 0$ because $I_{ww}(0,z_{\ast })$ is finite. With use of $I_{ww}(rz_{\ast }/z,z_{\ast })\propto z_{\ast }^{\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FE}}$ in the case of $z_{\ast }\ll 1$ ,

and hence

To relate these correlations continuously to the variances $\langle v_{i}^{2}(z)\rangle$ in (2.6), we require $C_{x:v_{i}v_{i}}(0)=c_{v_{i}v_{i}}$ and $D_{x:v_{i}v_{i}}(0)=d_{v_{i}v_{i}}$ . Especially if $d_{v_{i}v_{i}}$ is existent and $\langle v_{i}^{2}(z)\rangle$ is dependent on $\ln (\unicode[STIX]{x1D6FF}/z)$ , then $D_{x:v_{i}v_{i}}(r/z)$ is existent and $\langle v_{i}(x+r,z)v_{i}(x,z)\rangle$ is dependent on $\ln (\unicode[STIX]{x1D6FF}/z)$ .

The functions $C_{x:v_{i}v_{i}}(r/z)$ and $D_{x:v_{i}v_{i}}(r/z)$ are to reflect the internal structures of the individual eddies. Since $D_{x:v_{i}v_{i}}(r/z)$ is due to eddies of sizes $h_{e}$ up to the height $\unicode[STIX]{x1D6FF}$ of the turbulence, it is multiplied by $\ln (\unicode[STIX]{x1D6FF}/z)$ and exists only for the wall-parallel velocities $u$ and $v$ that are not blocked by the wall. There is also $C_{x:v_{i}v_{i}}(r/z)$ due to eddies of sizes $h_{e}$ comparable to the height $z$ , existent for all the velocities $u$ , $v$ and $w$ .

The functional forms of (3.6b ), (3.6c ) and (3.8b ) hold for distances $r$ down to those in the inertial range. Below this range lies the dissipation range where the kinematic viscosity $\unicode[STIX]{x1D708}$ is important. Since we have taken the limit $\unicode[STIX]{x1D708}\rightarrow 0$ , those forms do not hold. The velocity fluctuations in the dissipation range are regarded to have been coarse grained at a length scale in the inertial range (see also § 6.2).

Lastly, we reconsider the correlation lengths in the constant-stress layer $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ , by substituting (3.6b ), (3.6c ) and (3.8b ) into (3.1) with $r_{\star }=r/z$ ,

These tend to increase with an increase in the height $z$ . However, $L_{x:uu}(z)$ and $L_{x:vv}(z)$ are not proportional to the height $z$ because of the factor $\ln (\unicode[STIX]{x1D6FF}/z)$ . This is not the case for $L_{x:ww}(z)$ , which is exactly proportional to the height $z$ .

4 Velocity correlation along a wall-normal line

Two-point velocity correlations are also studied over a wall-normal distance $\boldsymbol{r}_{l}=\boldsymbol{r}_{z}=(0,0,-r)$ in the range of $0\leqslant r<z$ . According to the attached-eddy hypothesis (see § 2), the correlation function is given by

with the correlation for eddies of size $h_{e}$ ,

Here $J_{v_{i}v_{j}}(0,z/h_{e})$ is identical to $I_{v_{i}v_{j}}(0,z/h_{e})$ . The correlation length is defined by using an integration from $r=0$ to $r=z$ as

From (4.1a ) with $r_{\ast }=r/h_{e}$ and $z_{\ast }=z/h_{e}$ ,

The correlation length is also defined in an eddy of a particular size $h_{e}$ ,

We require $L_{z:v_{i}v_{i}}/z$ and $\unicode[STIX]{x1D6EC}_{z:v_{i}v_{i}}/h_{e}$ to be finite (see § 3). Then, since $\unicode[STIX]{x1D6EC}_{z:v_{i}v_{i}}/h_{e}$ is finite in the limit $z_{\ast }=z/h_{e}\rightarrow 0$ , its asymptotic form is identical to that of $\unicode[STIX]{x1D6EC}_{x:v_{i}v_{i}}/h_{e}$ in (3.3). For the constant-stress layer $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ , the correlation function is derived in the same manner as for those in (3.6a ) and (3.8a )

The result is

Here $C_{z:v_{i}v_{i}}$ and $D_{z:v_{i}v_{i}}$ are functions of $r/z$ with $C_{z:v_{i}v_{i}}(0)=C_{x:v_{i}v_{i}}(0)=c_{v_{i}v_{i}}$ or with $D_{z:v_{i}v_{i}}(0)=D_{x:v_{i}v_{i}}(0)=d_{v_{i}v_{i}}$ . The functional forms of these correlations are summarized in table 1.

Table 1.

Correlations over the distance $\boldsymbol{r}_{l}=\boldsymbol{r}_{x}=(r,0,0)$ , $\boldsymbol{r}_{y}=(0,r,0)$ or $\boldsymbol{r}_{z}=(0,0,-r)$ according to the attached-eddy hypothesis.

5 Energy spectrum along a wall-parallel line

From the velocity correlation along a wall-parallel line $\langle v_{i}(x+r,z)v_{i}(x,z)\rangle$ , the energy spectrum $\unicode[STIX]{x1D6F7}_{x:v_{i}v_{i}}(k,z)$ at a streamwise wavenumber $k$ is obtained as

Substitution of (3.6b ), (3.6c ) and (3.8b ) into (5.1) leads to the energy spectra of the attached-eddy hypothesis in the constant-stress layer $z/\unicode[STIX]{x1D6FF}\rightarrow 0$ ,

Here, we have defined $\tilde{C}_{x:v_{i}v_{i}}(kz)=(2/\unicode[STIX]{x03C0})\int _{0}^{\infty }\,\text{d}r_{\star }\,C_{x:v_{i}v_{i}}(r_{\star })\cos (kzr_{\star })$ and $\tilde{D}_{x:v_{i}v_{i}}(kz)=(2/\unicode[STIX]{x03C0})\int _{0}^{\infty }\,\text{d}r_{\star }\,D_{x:v_{i}v_{i}}(r_{\star })\cos (kzr_{\star })$ with $r_{\star }=r/z$ . The functional forms are the same even if the wavenumber is spanwise.

There is another model for the energy spectrum (Perry & Abell Reference Perry and Abell1977; Perry & Chong Reference Perry and Chong1982). Within a range of wavenumbers $k$ , some similarity is assumed among eddies of streamwise size $1/k$ so that $\unicode[STIX]{x1D6F7}_{x:uu}(k,z)$ depends only on $u_{\unicode[STIX]{x1D70F}}$ and $k$ , i.e. $\unicode[STIX]{x1D6F7}_{x:uu}(k,z)\propto u_{\unicode[STIX]{x1D70F}}^{2}/k$ . If the range of the similarity is from $k=a_{s}/\unicode[STIX]{x1D6FF}$ to $k=b_{s}/z$ , where $a_{s}$ and $b_{s}$ are constants, it follows that $\langle u^{2}(z)\rangle =\int _{0}^{\infty }\text{d}k\,\unicode[STIX]{x1D6F7}_{x:uu}(k,z)$ is dependent on $\ln (\unicode[STIX]{x1D6FF}/z)$ as in the case of (2.6a ). The same law of $u_{\unicode[STIX]{x1D70F}}^{2}/k$ is assumed for $\unicode[STIX]{x1D6F7}_{x:vv}(k,z)$ .

Since (2.6) originates in the attached-eddy hypothesis, the energy spectra of the hypothesis have been related to the law of $u_{\unicode[STIX]{x1D70F}}^{2}/k$ (Perry & Chong Reference Perry and Chong1982). This law is also used to constrain the functional forms of the correlations of the hypothesis (Davidson et al. Reference Davidson, Nickels and Krogstad2006). However, the existence of such a law remains controversial (e.g. Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015). The law of $u_{\unicode[STIX]{x1D70F}}^{2}/k$ is in fact not consistent with the attached-eddy hypothesis. At any height $z\ll \unicode[STIX]{x1D6FF}$ , we find in (5.2) that $\unicode[STIX]{x1D6F7}_{x:uu}(k,z)$ and $\unicode[STIX]{x1D6F7}_{x:vv}(k,z)$ are dominated by $\ln (\unicode[STIX]{x1D6FF}/z)\tilde{D}_{x:uu}(kz)$ and $\ln (\unicode[STIX]{x1D6FF}/z)\tilde{D}_{x:vv}(kz)$ . Their spectral shapes are not allowed to yield another factor of $\ln (\unicode[STIX]{x1D6FF}/z)$ for the variances $\langle u^{2}(z)\rangle$ and $\langle v^{2}(z)\rangle$ . Furthermore, the eddies of streamwise size $1/k$ are not identical to the attached eddies, which have various sizes $h_{e}$ but could contribute to the same streamwise wavenumber $k$ at each height $z$ . It is coincidental for (2.6) to be derived from these two.

The similarity law of $u_{\unicode[STIX]{x1D70F}}^{2}/k$ could yet approximate the internal structures of the attached eddies, i.e. $\tilde{C}_{x:v_{i}v_{i}}(kz)$ or $\tilde{D}_{x:v_{i}v_{i}}(kz)$ in (5.2), if the range of the similarity is not from $k=a_{s}/\unicode[STIX]{x1D6FF}$ but from $k=a_{s}/z$ . At the height $z\ll \unicode[STIX]{x1D6FF}$ , the premultiplied spectrum $k\unicode[STIX]{x1D6F7}_{x:uu}(k,z)$ is known to exhibit an energy-containing broad plateau (e.g. Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015). This could be approximated by the law of $u_{\unicode[STIX]{x1D70F}}^{2}/k$ through $\tilde{D}_{x:uu}(kz)\propto 1/kz$ . From (5.1),

There is an asymptotic relation for the cosine integral function,

If the distance $r$ is comparable to the height $z$ , our law of $u_{\unicode[STIX]{x1D70F}}^{2}/k$ yields a factor of $\ln (r/z)$ as

Thus, the correlation function $\langle u(x+r,z)u(x,z)\rangle$ could exhibit approximate dependence on $\ln (r/z)$ in a range of $r/z$ , which is actually seen in the wall turbulence (Davidson et al. Reference Davidson, Nickels and Krogstad2006; Chung et al. Reference Chung, Marusic, Monty, Vallikivi and Smits2015, see also Davidson & Krogstad Reference Davidson and Krogstad2014).

6 Other correlations

Having studied the velocity correlations in table 1 by using the minimum assumptions of the attached-eddy hypothesis, some other correlations are studied by adding assumptions that are consistent with the hypothesis. Their functional forms are summarized in table 2.

Table 2.

Correlations over the distance $\boldsymbol{r}_{l}=\boldsymbol{r}_{x}=(r,0,0)$ , $\boldsymbol{r}_{y}=(0,r,0)$ or $\boldsymbol{r}_{z}=(0,0,-r)$ according to the attached-eddy hypothesis with some additional assumptions.

7 Concluding discussion

Two-point velocity correlations have been studied theoretically for the constant-stress layer of wall turbulence on the basis of the attached-eddy hypothesis, i.e. a phenomenological model of a random superposition of energy-containing eddies that are attached to the wall (Townsend Reference Townsend1976). While the shapes of the eddies are taken to be identical to one another with a common characteristic velocity $u_{\unicode[STIX]{x1D70F}}$ , their sizes are distributed up to a finite value in each of the directions.

From the random distribution and the finite sizes of the attached eddies, it follows that the correlation lengths defined in (3.2) and (4.3) always exist. We have used this fact to derive the most general forms of the correlation functions in (3.6b ), (3.6c ), (3.8b ) and (4.4b – d ) and of the energy spectra in (5.2). The results are summarized in table 1.

The correlation functions $\langle v_{i}(\boldsymbol{x}+\boldsymbol{r}_{l})v_{i}(\boldsymbol{x})\rangle$ are made up of $C_{l:v_{i}v_{i}}(r/z)$ due to eddies of sizes $h_{e}$ comparable to the height $z$ and of $D_{l:v_{i}v_{i}}(r/z)$ due to eddies of sizes $h_{e}$ larger than the height $z$ . We have obtained $D_{l:v_{i}v_{i}}(r/z)$ only for the wall-parallel velocities $u$ and $v$ that are not blocked by the wall. It is multiplied by $\ln (\unicode[STIX]{x1D6FF}/z)$ because the size $h_{e}$ of the large eddies is up to the height $\unicode[STIX]{x1D6FF}$ of the turbulence.

These results are not consistent with the results of previous studies, e.g. the spectra in the form of $\unicode[STIX]{x1D6F7}_{x:v_{i}v_{i}}(k,z)\propto u_{\unicode[STIX]{x1D70F}}^{2}/k$ (Perry & Chong Reference Perry and Chong1982). While we have focused on the minimum assumptions of the attached-eddy hypothesis, the previous studies had invoked additional assumptions. They are in fact not consistent with the hypothesis, aside from which is more reliable as a model for the wall turbulence.

Without any confounding effect of such additional assumptions, our results would offer an opportunity to assess the extent to which the attached-eddy hypothesis is consistent with the actual wall turbulence. However, at present, the direct numerical simulations are not at Reynolds numbers that are high enough to attain a layer of the strictly constant stress (Sillero et al. Reference Sillero, Jiménez and Moser2013; Lee & Moser Reference Lee and Moser2015). The experiments are so far based mostly on Taylor’s frozen-eddy hypothesis. Improvements are desirable for the simulations and for the experiments.

There are nevertheless some experimental implications. It is known that the value of $d_{uu}=D_{l:uu}(0)$ is common to a variety of flows (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). This commonness is expected for the entire function $D_{l:uu}(r/z)$ , which reflects the undermost layers of the attached eddies that are insensitive to the flow configuration. The value of $c_{uu}=C_{l:uu}(0)$ is not common, and so is not the entire function $C_{l:uu}(r/z)$ . We expect these implications to apply further to the other velocities $v$ and $w$ .

The two-point correlation of the wall-normal velocity $w$ is determined by the attached eddies of sizes $h_{e}$ comparable to the height $z$ of the two points. If such eddies are actually dominant in the wall turbulence (Adrian Reference Adrian2007), their internal structures would be clarified in that correlation, especially as to whether the shape of the eddy is independent of its size. This is not the case for the correlations of the wall-parallel velocities $u$ and $v$ , which are affected by the larger eddies.

By adding assumptions that are consistent with the attached-eddy hypothesis, we have derived the correlations of some other quantities. That is, if the velocity field is Gaussian, the correlation functions of the kinetic energies $u^{2}$ , $v^{2}$ and $w^{2}$ and that of the momentum transfer rate $-uw$ are given in (6.2) and (6.3). If the energy dissipation rate of an attached eddy $\unicode[STIX]{x1D700}_{e}$ is proportional to $u_{\unicode[STIX]{x1D70F}}^{3}/h_{e}$ , the correlation function of the entire rate $\unicode[STIX]{x1D700}$ is given in (6.9). The results are summarized in table 2.

A logarithmic law such as that in (1.1) has been found also for the variance of the pressure fluctuations $p$ (Jiménez & Hoyas Reference Jiménez and Hoyas2008; Sillero et al. Reference Sillero, Jiménez and Moser2013). If this law is due to the attached eddies, the two-point correlation $\langle p(\boldsymbol{x}+\boldsymbol{r}_{l})p(\boldsymbol{x})\rangle$ is expected to have a functional form that is identical to those of the wall-parallel velocities $u$ and $v$ given in (3.6b ) and (3.6c ) or (4.4b ) and (4.4c ).

Lastly, we note that these results would allow us to use the attached-eddy hypothesis in its most general form. An example is the coarse graining to characterize or model the wall turbulence, say, the atmospheric boundary layer in a particular class of numerical simulations (Wyngaard Reference Wyngaard2004). Over a streamwise length $R$ , we coarse grain a quantity $q$ as

Through a relation (e.g. Monin & Yaglom Reference Monin and Yaglom1971), the variance of $q_{R}$ is dependent on the two-point correlation of $q$ ,

and hence

By using (3.1) to define the correlation length $L_{x:qq}(z)$ ,

If the two-point correlation is a function of $r/z$ alone as for $q=w$ , $w^{2}$ and $\unicode[STIX]{x1D700}$ in tables 1 and 2, the variance is a function of $R/z$ alone. The remaining variances depend also on $\ln (\unicode[STIX]{x1D6FF}/z)$ . Such a constraint would in turn constrain the theory or model on that coarse graining. Thus, for this and various other studies of wall turbulence, we could make use of the general forms of the correlation functions derived here from the minimum assumptions of the attached-eddy hypothesis.