2.1 Why use $u_{0}$ as the normalizing velocity?

It should be noted that it is a little unusual to use the centreline velocity, $u_{0}$ , as the normalizing velocity scale. More commonly one uses the area averaged, or bulk, velocity

(2.12) $$\begin{eqnarray}\displaystyle \bar{u}=\frac{2}{\unicode[STIX]{x1D6FF}^{2}}\int _{0}^{\unicode[STIX]{x1D6FF}}ur\,\text{d}r & & \displaystyle\end{eqnarray}$$

while the pipe resistance is expressed in terms of the friction factor

(2.13) $$\begin{eqnarray}\displaystyle f=8\left(\frac{u_{\unicode[STIX]{x1D70F}}}{\bar{u}}\right)^{2}=4\overline{C_{f}}, & & \displaystyle\end{eqnarray}$$

where $\overline{C_{f}}$ is the friction coefficient normalized by $\bar{u}$ . The most easily accessible information about a pipe flow is the mass flow rate ${\dot{m}}=\unicode[STIX]{x1D70C}\bar{u}A$ , which is usually specified, and the pressure gradient which determines the wall friction, $\unicode[STIX]{x1D70F}_{w}=(\unicode[STIX]{x1D6FF}/2)\text{d}p_{w}/\text{d}x$ , and so it makes practical sense to use $\bar{u}$ to normalize everything. Also, to a surprising degree of accuracy, the law of the wall

(2.14) $$\begin{eqnarray}\displaystyle u^{+}=\frac{1}{k}\ln (y^{+})+C & & \displaystyle\end{eqnarray}$$

can be used as an approximation to the flow over the whole pipe. This is despite the singularity in the logarithm at the wall and the finite derivative of the log at the pipe centreline. Using (2.14) the bulk velocity integrates to

(2.15) $$\begin{eqnarray}\displaystyle \frac{\bar{u}}{u_{\unicode[STIX]{x1D70F}}}=\frac{2}{R_{\unicode[STIX]{x1D70F}}}\int _{0}^{R_{\unicode[STIX]{x1D70F}}}\left(\frac{1}{k}\ln (y^{+})+C\right)\left(1-\frac{y^{+}}{R_{\unicode[STIX]{x1D70F}}}\right)\text{d}y^{+}=\frac{1}{k}\ln (R_{\unicode[STIX]{x1D70F}})+C-\frac{3}{2k}. & & \displaystyle\end{eqnarray}$$

This simple result provides a very attractive direct connection to the law of the wall.

Data from PSP surveys 1 to 26 for $\bar{u}/u_{0}$ are shown in figure 2(a), and friction data, $u_{0}/u_{\unicode[STIX]{x1D70F}}$ and $\bar{u}/u_{\unicode[STIX]{x1D70F}}$ versus $R_{\unicode[STIX]{x1D70F}}$ , are presented in figure 2(b). The solid lines are generated using the functional form of (2.14) and (2.15) with the constants $k$ and $C$ selected to give least squares log–linear fits to the data. At first one might expect that $k_{u_{0}/u_{\unicode[STIX]{x1D70F}}}>k_{\bar{u}/u_{\unicode[STIX]{x1D70F}}}$ , as is the case in figure 2(b), because of the general tendency for $\bar{u}/u_{0}$ to increase slowly toward one with increasing Reynolds number. But there is no reason to expect the curves in figure 2(b) to eventually intersect. As long as the difference between the additive constants does not change, $\lim _{R_{\unicode[STIX]{x1D70F}}\rightarrow \infty }\bar{u}/u_{0}=1$ . Moreover, according to (2.15) the values of $k$ for $u_{0}/u_{\unicode[STIX]{x1D70F}}$ and $\bar{u}/u_{\unicode[STIX]{x1D70F}}$ versus $R_{\unicode[STIX]{x1D70F}}$ should be the same as the $k$ that appears in (2.14). Indeed, the empirically determined $k$ values in figure 2(b) are very close but the displacement between the curves is considerably more than the $3/2k$ that appears in (2.15). Clearly a better approximation to the velocity profile than (2.14) is needed.

Figure 2. (a) PSP survey data for $\bar{u}/u_{0}$ and (b) wall shear stress in terms of $\bar{u}/u_{\unicode[STIX]{x1D70F}}$ and $u_{0}/u_{\unicode[STIX]{x1D70F}}$ versus $R_{\unicode[STIX]{x1D70F}}$ . Open circles in (b) (○) are data for PSP surveys 1 to 26. Least square log–linear fits to the data for PSP surveys 6 to 26 are shown as solid lines. Root-mean-square error in the upper solid line is 0.112 in units of $u^{+}$ . Root-mean-square error in the lower solid line is 0.0975.

The goal in this paper is to approximate as accurately as possible the PSP velocity profiles, and for this it is necessary to integrate (2.1) where the outer boundary condition is $u/u_{0}=1$ for all $R_{\unicode[STIX]{x1D70F}}$ . This way the added complexity of the Reynolds number dependence of $\bar{u}/u_{0}$ is avoided. In the end we will see that the simple log–linear behaviour of $u_{0}/u_{\unicode[STIX]{x1D70F}}$ and $\bar{u}/u_{\unicode[STIX]{x1D70F}}$ , which dates back to Prandtl (Reference Prandtl and Durand1934a ), is generated rigorously by the universal model velocity profile derived in the next section.

3.1 The universal velocity profile

Integrate (3.3) from the wall to $y^{+}$ ,

(3.5) $$\begin{eqnarray}\displaystyle u^{+}(y^{+})=\int _{0}^{y^{+}}\left(-\frac{1}{2\unicode[STIX]{x1D706}(s)^{2}}+\frac{1}{2\unicode[STIX]{x1D706}(s)^{2}}\left(1+4\unicode[STIX]{x1D706}(s)^{2}\left(1-\frac{s}{R_{\unicode[STIX]{x1D70F}}}\right)\right)^{1/2}\right)\text{d}s. & & \displaystyle\end{eqnarray}$$

In the limit of small $R_{\unicode[STIX]{x1D70F}}$ , the velocity approaches the laminar profile (2.10), again, independent of the choice of $\unicode[STIX]{x1D706}(y^{+})$ .

(3.6) $$\begin{eqnarray}\displaystyle \lim _{R_{\unicode[STIX]{x1D70F}}\rightarrow 0}\int _{0}^{y^{+}}\left(-\frac{1}{2\unicode[STIX]{x1D706}(s)^{2}}+\frac{1}{2\unicode[STIX]{x1D706}(s)^{2}}\left(1+4\unicode[STIX]{x1D706}(s)^{2}\left(1-\frac{s}{R_{\unicode[STIX]{x1D70F}}}\right)\right)^{1/2}\right)\text{d}s=y^{+}\left(1-\frac{y^{+}}{2R_{\unicode[STIX]{x1D70F}}}\right). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The model profile remains valid as the Reynolds number goes to zero and the small $R_{\unicode[STIX]{x1D70F}}$ limit of the friction coefficient is the laminar value (2.11).

(3.7) $$\begin{eqnarray}\displaystyle \lim _{R_{\unicode[STIX]{x1D70F}}\rightarrow 0}C_{f}=\frac{2}{\displaystyle \left(\lim _{R_{\unicode[STIX]{x1D70F}}\rightarrow 0}\int _{0}^{R_{\unicode[STIX]{x1D70F}}}\left(-\frac{1}{2\unicode[STIX]{x1D706}(s)^{2}}+\frac{1}{2\unicode[STIX]{x1D706}(s)^{2}}\left(1+4\unicode[STIX]{x1D706}(s)^{2}\left(1-\frac{s}{R_{\unicode[STIX]{x1D70F}}}\right)\right)^{1/2}\right)\text{d}s\right)^{2}}=\frac{8}{{R_{\unicode[STIX]{x1D70F}}}^{2}}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

3.2 A linear mixing length function

The simplest choice for $\unicode[STIX]{x1D706}(y^{+})$ is just a linear proportionality to the distance from the wall.

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}(y^{+})=ky^{+}, & & \displaystyle\end{eqnarray}$$

where $k$ is an empirically determined amplitude of the mixing length; essentially the Kármán constant. Using (3.8), (3.5) becomes

(3.9) $$\begin{eqnarray}\displaystyle u^{+}(y^{+})=\int _{0}^{y^{+}}\left(-\frac{1}{2(ks)^{2}}+\frac{1}{2(ks)^{2}}\left(1+4(ks)^{2}\left(1-\frac{s}{R_{\unicode[STIX]{x1D70F}}}\right)\right)^{1/2}\right)\text{d}s. & & \displaystyle\end{eqnarray}$$

Evaluate (3.9) at the pipe centreline.

(3.10) $$\begin{eqnarray}\displaystyle u^{+}(R_{\unicode[STIX]{x1D70F}})=\frac{u_{0}}{u_{\unicode[STIX]{x1D70F}}}=\int _{0}^{R_{\unicode[STIX]{x1D70F}}}\left(-\frac{1}{2(ks)^{2}}+\frac{1}{2(ks)^{2}}\left(1+4(ks)^{2}\left(1-\frac{s}{R_{\unicode[STIX]{x1D70F}}}\right)\right)^{1/2}\right)\text{d}s. & & \displaystyle\end{eqnarray}$$

Multiply both sides of (3.10) by $k$ and let $\unicode[STIX]{x1D6FC}=ks$ . In the limit of infinite Reynolds number

(3.11) $$\begin{eqnarray}\displaystyle \lim _{kR_{\unicode[STIX]{x1D70F}}\rightarrow \infty }\frac{ku_{0}}{u_{\unicode[STIX]{x1D70F}}}=\lim _{kR_{\unicode[STIX]{x1D70F}}\rightarrow \infty }\int _{0}^{kR_{\unicode[STIX]{x1D70F}}}\left(-\frac{1}{2\unicode[STIX]{x1D6FC}^{2}}+\frac{1}{2\unicode[STIX]{x1D6FC}^{2}}\left(1+4\unicode[STIX]{x1D6FC}^{2}\left(1-\frac{\unicode[STIX]{x1D6FC}}{kR_{\unicode[STIX]{x1D70F}}}\right)\right)^{1/2}\right)\text{d}\unicode[STIX]{x1D6FC}=\ln \left(\frac{2^{4}}{e^{3}}kR_{\unicode[STIX]{x1D70F}}\right). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The left- and right-hand sides of (3.11) are of the same form as (2.15), but in terms of the centreline velocity and with only one empirical constant, $k$ . That is, the additive constant in (3.11) is determined solely by $k$ . If we use $k=0.4320$ from figure 2(b) the friction law (3.11) becomes

(3.12) $$\begin{eqnarray}\displaystyle \frac{u_{0}}{u_{\unicode[STIX]{x1D70F}}}=\left(\frac{1}{0.4320}\right)\ln (R_{\unicode[STIX]{x1D70F}})-2.4693. & & \displaystyle\end{eqnarray}$$

The additive constant in (3.12) is quite different from the log–linear fit to the data shown in figure 2(b), even though the velocity profile (3.9) would seem to be an improvement since it lacks both the singularity of the law of the wall at $y^{+}=0$ , and the finite derivative of the log at the pipe centreline, both of which might be expected to throw off the accuracy of the law of the wall approximation. On the other hand, the disagreement between (3.12) and the data in figure 2 should not be too surprising since the additive constant in (3.12) is not freely selected to fit the data. The simple linear mixing length model (3.8) really precludes any sort of viscous wall layer, the thickness of which, is the primary determinant of the value of the additive constant $C$ . We need a choice for $\unicode[STIX]{x1D706}$ that can better approximate the complex shape of the velocity profile.

3.3 A nonlinear mixing length function; the universal velocity profile

The linear function (3.8) just does not have the flexibility needed to reproduce the PSP velocity profiles accurately. We need to improve the mixing length model to account for damping at the wall and bulk mixing near the pipe centreline. From here on we will use a new combined wall–wake mixing length function,

(3.13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}(y^{+})=\frac{ky^{+}(1-\text{e}^{-(y^{+}/a)^{m}})}{\left(1+\left(\displaystyle \frac{y^{+}}{bR_{\unicode[STIX]{x1D70F}}}\right)^{n}\right)^{1/n}}. & & \displaystyle\end{eqnarray}$$

This form of $\unicode[STIX]{x1D706}$ has five free constants $k,a,m,b$ and $n$ that can be used to approximate the PSP data. They can be summarized as follows.

$k$ – At moderate to high Reynolds numbers, this coincides with the classical Kármán constant. At low to laminar Reynolds numbers, the velocity profile is insensitive to $k$ .

$a$ – This is a measure of the range of $y^{+}$ near the wall where the Reynolds shear stress is damped to zero. It can be regarded as a characteristic damping length scale. The exponential decay near the wall can be found in van Driest (Reference van Driest1956) but without the exponent $m$ .

$m$ – This exponent determines the shape of the damping region near the wall. A general analysis of an expansion of the three-dimensional (3-D) velocity field near the wall in the presence of a mean flow suggests that this exponent should be $1/2$ (She et al. Reference She, Chen and Hussain2017). This value would insure that $\overline{u^{\prime }v^{\prime }}\sim y^{3}$ near the wall, although a value larger than $1/2$ implying a faster decay of the near-wall shear stress is not ruled out.

The expression in the denominator of (3.13) is a transition function designed to cause the mixing length to tend to approach a constant value near the centreline of the pipe. This is required to insure that the velocity profile exhibits wake-like behaviour in this region. Similar functions are used by She et al. (Reference She, Chen and Hussain2017) along with symmetry analysis, to develop a multilayer theory of the velocity profile.

$b$ – This parameter takes effect well beyond the wall layer and represents a measure of the fraction of the pipe radius at which the wake function starts to kick in. It is essentially an outer flow length scale.

$n$ – This exponent determines how rapidly the wake-like behaviour of the velocity profile evolves outside the wall layer. Note that if $b$ is relatively small, of the order of $0.3$ or so, and $n$ is greater than one, the mixing length approaches a maximum value on the order of $\unicode[STIX]{x1D706}\simeq bkR_{\unicode[STIX]{x1D70F}}$ as the pipe centreline is approached.

The integral (3.5) combined with the wall–wake mixing length function (3.13) constitute the universal velocity profile referred to in the title of the paper.

It should be noted that the velocity integral (3.5) and the wall–wake mixing length function (3.13) can be used to determine the friction coefficient at any Reynolds number for which appropriate values of ( $k,a,m,b,n$ ) are known and, as was pointed out in connection with (3.7), the model profile is valid at all Reynolds numbers. As the Reynolds number is reduced the velocity profile and friction coefficient become increasingly insensitive to the values of ( $k,a,m,b,n$ ).

4.1 Correction of the mean velocity for the effect of Reynolds normal stress

The mean velocity measured by a pitot tube is inferred from the difference between the pitot (stagnation) pressure and the static pressure measured at a small hole, or tap, in the pipe wall. In a turbulent flow, the measured stagnation pressure includes the effective pressure exerted by the Reynolds normal stress, (Zagarola Reference Zagarola1996). To a reasonable approximation

(4.1) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle p_{t_{measured}}=p_{wall}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}((u_{corrected})^{2}+\overline{u^{\prime }u^{\prime }})\\ \displaystyle u_{corrected}=(2(p_{t_{measured}}-p_{wall})/\unicode[STIX]{x1D70C}-\overline{u^{\prime }u^{\prime }})^{1/2}\\ \displaystyle u_{corrected}=((u_{uncorrected})^{2}-\overline{u^{\prime }u^{\prime }})^{1/2}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

To determine the corrected velocity it is necessary to know the streamwise turbulent normal stress. The data presented in Hultmark (Reference Hultmark2012) were used to come up with the following purely empirical approximation to the normal stress:

(4.2) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle g_{1}(y^{+})=(\text{Ln}(0.15(5.9+y^{+})))^{2}\\ \displaystyle g_{2}(R_{\unicode[STIX]{x1D70F}},y^{+})=\text{Ln}\left(\frac{1+0.3y^{+}}{R_{\unicode[STIX]{x1D70F}}}\right)\\ \displaystyle g_{3}(y^{+})=g_{1}\text{e}^{-g_{1}}+\frac{g_{1}}{2(1+g_{1})}\\ \displaystyle g_{4}(y^{+})={\textstyle \frac{1}{2}}+{\textstyle \frac{1}{2}}\text{Erf}({\textstyle \frac{7}{8}}(1-g_{2}-10/1.4))\\ \displaystyle \frac{\overline{u^{\prime }u^{\prime }}}{(u_{\unicode[STIX]{x1D70F}})^{2}}=2.17g_{3}(1-g_{2}-g_{4}(1-g_{2}-10/1.4))-1.372\left(\frac{\text{Ln}(y^{+})}{\text{Ln}(R_{\unicode[STIX]{x1D70F}})}\right).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Figure 3. Approximation to the streamwise Reynolds normal stress, equation (4.2), plotted over the range of the PSP data.

Equation (4.2) is plotted in figure 3 for the twenty six PSP Reynolds numbers. The function captures the main features of the streamwise normal stress pretty well, including the peak near the wall and the smaller peak near the end of the log region. The Reynolds number independence of the wall region above $R_{\unicode[STIX]{x1D70F}}=3322$ ( $p_{d}=0.9~\text{mm}$ survey 6) is also captured. Reynolds numbers corresponding to cases 6 to 26 overlay each other quite closely near the wall in figure 3. The fit (4.2) also captures the inverse logarithmic dependence of the streamwise normal stress away from the wall discussed by Hultmark (Reference Hultmark2012). In general the correction due to Reynolds normal stress effects leads to a relatively small reduction of the measured mean velocity.

4.2 Correction of the mean velocity for the effect of static pressure errors

A second source of error in the measured mean velocity is in the measurement of the wall static pressure that appears in (4.1). Basically, the pressure measured at a wall static pressure tap is higher than the correct value due to curvature of the streamlines near the tap and associated circulatory motion in the fluid within the tap duct adjacent to the wall. This leads to a small positive correction to the velocity data. Measurements of this effect in the PSP facility are reported by McKeon & Smits (Reference McKeon and Smits2002) and McKeon et al. (Reference McKeon, Li, Jiang, Morrison and Smits2003). It is generally accepted that the static pressure error scales with the friction velocity

(4.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}p=\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D70C}{u_{\unicode[STIX]{x1D70F}}}^{2}). & & \displaystyle\end{eqnarray}$$

The function $\unicode[STIX]{x1D6F1}$ increases with both the pressure tap diameter Reynolds number and the pipe Reynolds number. When the static pressure effect is included in (4.1) the result is

(4.4) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}p_{t_{measured}}=p_{wall_{measured}}-\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D70C}{u_{\unicode[STIX]{x1D70F}}}^{2})+\frac{1}{2}\unicode[STIX]{x1D70C}((u_{corrected})^{2}+\overline{u^{\prime }u^{\prime }})\\ u_{corrected}=(2(p_{t_{measured}}-p_{wall_{measured}})/\unicode[STIX]{x1D70C}+2\unicode[STIX]{x1D6F1}{u_{\unicode[STIX]{x1D70F}}}^{2}-\overline{u^{\prime }u^{\prime }})^{1/2}\\ u_{corrected}=((u_{uncorrected})^{2}+2\unicode[STIX]{x1D6F1}{u_{\unicode[STIX]{x1D70F}}}^{2}-\overline{u^{\prime }u^{\prime }})^{1/2}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The problem is the selection of $\unicode[STIX]{x1D6F1}$ . Zagarola (Reference Zagarola1996) used values between $\unicode[STIX]{x1D6F1}=0.1$ and $\unicode[STIX]{x1D6F1}=3.3$ however McKeon & Smits (Reference McKeon and Smits2002) point out that this underestimates both the magnitude and complexity of the effect and that the effect becomes more pronounced as the Reynolds number is increased. In the end we decided to use $\unicode[STIX]{x1D6F1}=7(d^{+}/d_{max}^{+})$ where $d^{+}=du_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ and $d=0.79~\text{mm}$ , is the diameter of the pressure tap. The maximum pressure tap Reynolds number was $d_{max}^{+}=6486$ , ten times the thickness of the viscous wall layer, and the minimum was $d^{+}=10$ . The factor 7 is at the high end of the data presented by McKeon & Smits (Reference McKeon and Smits2002). We did not attempt to approximate the complexity of their figures 6 and 7 as we did with the Hultmark (Reference Hultmark2012) data mainly because it appears that this whole area is still a subject for continued research. If higher Reynolds number measurements are carried out in the future then the wall pressure will probably need to be measured using flush mounted sensors that do not break the wall surface. Both corrections to the velocity data are shown in figure 4.

Figure 4. (a) Corrections to $u^{+}$ due to Reynolds normal stress effects at each pitot tube position. (b) Corrections at each pitot tube position to $u^{+}$ due to wall static pressure effects. Total correction to the uncorrected $p_{d}=0.9~\text{mm}$ data is approximately the sum of the two effects. The horizontal coordinate is the index (1 to 52) of the position of the pitot tube above the wall.

4.3 Correction to the $y$ coordinate of the pitot tube position

The finite radius of the pitot tube ( $p_{r}=0.45~\text{mm}$ ) leads to a shift in the effective radial position of the measurement point due to the velocity gradient. Here, the correction scheme described by Chue (Reference Chue1975) and recommended by Zagarola (Reference Zagarola1996) was used. In wall units, the correction to the $y$ coordinate above the wall is

(4.5) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x0394}y^{+}}{p_{r}^{+}}=0.36\left(\frac{1}{u^{+}}\frac{\text{d}u^{+}}{\text{d}y^{+}}p_{r}^{+}-0.17\left(\frac{1}{u^{+}}\frac{\text{d}u^{+}}{\text{d}y^{+}}p_{r}^{+}\right)^{3}\right), & & \displaystyle\end{eqnarray}$$

where $p_{r}^{+}=p_{r}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ . The factor $0.36$ is the value suggested by Chue (Reference Chue1975) and used by Zagarola (Reference Zagarola1996). The main difficulty in applying (4.5) is that it requires the $y$ derivative of the velocity data, which is not known a priori. However, once the uncorrected velocity data were approximated by (3.5) and (3.13), that approximation was used in (4.5) to generate the velocity derivative and corrected $y^{+}$ values. The once-corrected approximation to $y^{+}$ was then used a second time to produce a once-improved derivative for (4.5) and a final set of corrected $y^{+}$ values. The final corrections applied to the position data are shown in figure 5.

Figure 5. (a) Values of the corrections in $y^{+}$ applied to the uncorrected $p_{d}=0.9~\text{mm}$ data at each pitot tube position. (b) Shows the correction close to the wall. The horizontal coordinate is the index (1 to 52) of the position of the pitot tube above the wall.

Figure 6 shows an overlay of all the $p_{d}=0.9~\text{mm}$ and $p_{d}=0.3~\text{mm}$ corrected data. At Reynolds numbers where both sets of data overlap, the agreement between the two measurements of the same velocity profile is generally excellent. The 19 Reynolds numbers where the $p_{d}=0.3~\text{mm}$ data are available provide valuable information near the wall which can be clearly seen as open circles to the left of the overlapping points.

Figure 6. Comparison between corrected $p_{d}=0.9~\text{mm}$ velocity data (filled circles ●) and corrected $p_{d}=0.3~\text{mm}$ velocity data (open circles ○) for all 26 PSP surveys. Each velocity profile is shifted vertically 3 units in $u^{+}$ in order to separate the profiles.

6.1 Velocity and friction using mean model parameters

The average value of $k$ in figure 8(b) over profiles 6 to 26 is $\bar{k}=0.4092$ and can be compared to the value $k=0.421$ measured by McKeon (Reference McKeon2003) and McKeon et al. (Reference McKeon, Zagarola and Smits2005) and $k=0.436$ reported by Zagarola (Reference Zagarola1996) and Zagarola & Smits (Reference Zagarola and Smits1998). The difference between 0.436 and 0.409 may not seem very large but to anyone fitting this kind of data to a logarithmic curve, the difference is quite noticeable and Zagarola’s value has generally been viewed as too large. But according to figure 2 the PSP survey data for $u_{0}/u_{\unicode[STIX]{x1D70F}}$ versus $R_{\unicode[STIX]{x1D70F}}$ are well approximated using $k=0.432$ , even though this value does not provide a good fit to any of the individual velocity profiles. We shall return to this point in § 9.3.

Table 1 shows the optimal values of ( $k,a,b,m,n$ ) determined for the 26 surveys. For PSP 1 to 5 the damping length scale $a$ is nearly constant around a value of approximately 25.7. At PSP 6, $a$ drops to 19.7, then increases slightly over PSP 6 to 21 and then levels off for PSP 22 to 26. The average value over PSP 6 to 26 is $\bar{a}=20.095$ . Over the entire data set the outer flow length scale, $b$ , shown in figure 10(b), is close to 0.3 except for a slight rise to approximately 0.34 to 0.35 between PSP 6 to 10. The average over PSP 6 to 26 is $\bar{b}=0.3195$ . The damping exponent, $m$ , shown in 10(c) shows similar behaviour; nearly constant at approximately 1.3 for PSP 1 to 5, then increasing to approximately 1.6 for PSP 6 to 26 with an average value $\bar{m}=1.621$ over these cases. Finally the wake exponent, $n$ , in figure 10(d) generally increases from 1.2 to 1.7 over PSP 1 to 26 with an average value over PSP 6 to 26 of $\bar{n}=1.619$ , remarkably close to $\bar{m}$ .

Figure 13. The PSP data and model profiles in this figure are identical to that shown in figure 7. The filled circles (●) are determined from the model profile (3.5) and (3.13) with all parameters fixed at $(\bar{k},\bar{a},\bar{m},\bar{b},\bar{n})=(0.4092,20.095,1.621,0.3195,1.619)$ .

Figure 14. (a) Value of $\bar{u}/u_{\unicode[STIX]{x1D70F}}$ versus $R_{\unicode[STIX]{x1D70F}}$ , (b) $\bar{u}/u_{\unicode[STIX]{x1D70F}}$ versus $\bar{R}_{e}$ . Open circles, (○) are the combined Oregon and Princeton data from McKeon et al. (Reference McKeon, Swanson, Zagarola, Donnelly and Smits2004b ). Filled circles (●) are computed from (3.5) and (3.13) using $(\bar{k},\bar{a},\bar{m},\bar{b},\bar{n})=(0.4092,20.095,1.621,0.3195,1.619)$ .

Figure 13 shows the 26 sets of PSP survey velocity data as open circles and the comparison model profiles as solid lines. These are the same data shown in figure 7. The filled circles overlaid on the data are derived from the model profile with all five parameters fixed at their average values, $(\bar{k},\bar{a},\bar{m},\bar{b},\bar{n})=(0.4092,20.095,1.621,0.3195,1.619)$ . The fit is quite good considering the three orders of magnitude in the Reynolds number between surveys 1 and 26. The largest error occurs at the highest Reynolds number where the filled circles tend to overestimate the velocity. This is mainly due to the use of $\bar{k}=0.4092$ rather than the somewhat larger optimal value $k=0.4190$ .

In figure 14, $\bar{u}/u_{\unicode[STIX]{x1D70F}}$ versus $R_{\unicode[STIX]{x1D70F}}$ and the bulk Reynolds number, $\bar{R_{e}}$ , for the combined Oregon and Princeton data listed in McKeon et al. (Reference McKeon, Swanson, Zagarola, Donnelly and Smits2004b ), is compared with data computed from the universal velocity profile (3.5), (3.13) with the model parameters fixed at their average values. Except for laminar–turbulent transition, the universal profile approximates the friction data quite well over the entire range of Reynolds numbers from $0$ to $10^{7}$ .

9.1 Centreline velocity friction law

Figure 24. (a) The pipe flow shape function, $\unicode[STIX]{x1D6F7}(ka,m,b,n,kR_{\unicode[STIX]{x1D70F}},kR_{\unicode[STIX]{x1D70F}})$ and $\bar{\unicode[STIX]{x1D6F7}}(ka,m,b,n,kR_{\unicode[STIX]{x1D70F}},kR_{\unicode[STIX]{x1D70F}})$ versus $kR_{\unicode[STIX]{x1D70F}}$ calculated using (8.3) and (9.1) with fixed parameters $(\bar{k}\bar{a},\bar{m},\bar{b},\bar{n})=(0.4092\times 20.095,1.621,0.3195,1.619)$ . (b) High Reynolds number limiting value, $\unicode[STIX]{x1D6F7}(ka,m,b,n,\infty ,\infty )$ and $\bar{\unicode[STIX]{x1D6F7}}(ka,m,b,n,\infty ,\infty )$ versus $ka$ with $(b,m,n)=(1.621,0.3195,1.619)$ . Also shown are linear fits to the limiting integral values. Root-mean-square error in $\unicode[STIX]{x1D6F7}$ and $\bar{\unicode[STIX]{x1D6F7}}$ for both linear fits is 0.0027.

Fix ( $m,b,n$ ) at their mean values and evaluate (8.5) at the pipe centreline.

(9.1) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6F7}(ka,m,b,n,kR_{\unicode[STIX]{x1D70F}},kR_{\unicode[STIX]{x1D70F}})\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{0}^{kR_{\unicode[STIX]{x1D70F}}}\left(-\frac{1}{2\tilde{\unicode[STIX]{x1D706}}^{2}}+\frac{1}{2\tilde{\unicode[STIX]{x1D706}}^{2}}\left(1+4\tilde{\unicode[STIX]{x1D706}}^{2}\left(1-\frac{\unicode[STIX]{x1D6FC}}{kR_{\unicode[STIX]{x1D70F}}}\right)\right)^{1/2}\right)\text{d}\unicode[STIX]{x1D6FC}-\ln (kR_{\unicode[STIX]{x1D70F}}).\end{eqnarray}$$

The reason that, with ( $m,b,n$ ) fixed, $\unicode[STIX]{x1D6F7}$ is a function of both $kR_{\unicode[STIX]{x1D70F}}$ and $ka$ is because, while $a$ could be set at its average value, $\bar{a}=20.397$ , the product $ka$ still depends on $k$ . In fact $k$ and $a$ are really not fully independent quantities; $k$ and $a$ tend to increase or decrease together and this is captured in the optimal values of $k$ and $a$ in figure 9. Figure 24(a) shows (9.1) plotted using the mean values ( $(\bar{k}\bar{a},\bar{m},\bar{b},\bar{n})=(0.4092\times 20.095,1.621,0.3195,1.619)$ ). It is apparent from this figure that $\unicode[STIX]{x1D6F7}$ , at a fixed $y/\unicode[STIX]{x1D6FF}$ , approaches its asymptotic limit above the relatively low value $kR_{\unicode[STIX]{x1D70F}}=2000$ . Above this Reynolds number, the high Reynolds number shape function, $\unicode[STIX]{x1D719}$ , shown in figure 20 is established and only minute changes occur as the Reynolds number is further increased. This is about the same Reynolds number as surveys 6 and 7 where scale separation between the inner and outer flow begins to appear and the velocity profile begins to show a nearly logarithmic intermediate region.

Figure 24(b) shows that the high Reynolds number limit, $\unicode[STIX]{x1D6F7}(ka,m,b,n,\infty ,\infty )$ is to a high degree of accuracy linearly proportional to $ka$ in the range of the PSP data. Thus we can write the high Reynolds number smooth wall pipe flow centreline velocity friction law as

(9.2) $$\begin{eqnarray}\displaystyle \lim _{kR_{\unicode[STIX]{x1D70F}}\rightarrow \infty }\frac{ku_{0}}{u_{\unicode[STIX]{x1D70F}}}=\ln (R_{\unicode[STIX]{x1D70F}})+0.2915(ka)+\ln (k)+1.0407, & & \displaystyle\end{eqnarray}$$

where the limit is rapidly approached above $kR_{\unicode[STIX]{x1D70F}}=2000$ .

9.2 Bulk velocity friction law

A similar analysis holds for the friction law based on the bulk velocity.

(9.3) $$\begin{eqnarray}\displaystyle \frac{k\bar{u}}{u_{\unicode[STIX]{x1D70F}}}=\frac{2}{kR_{\unicode[STIX]{x1D70F}}}\int _{0}^{kR_{\unicode[STIX]{x1D70F}}}\left(\int _{0}^{\unicode[STIX]{x1D6FC}}\left(-\frac{1}{2\tilde{\unicode[STIX]{x1D706}}^{2}}+\frac{1}{2\tilde{\unicode[STIX]{x1D706}}^{2}}\left(1+4\tilde{\unicode[STIX]{x1D706}}^{2}\left(1-\frac{\hat{\unicode[STIX]{x1D6FC}}}{kR_{\unicode[STIX]{x1D70F}}}\right)\right)^{1/2}\right)\text{d}\hat{\unicode[STIX]{x1D6FC}}\right)\left(1-\frac{\unicode[STIX]{x1D6FC}}{kR_{\unicode[STIX]{x1D70F}}}\right)\text{d}\unicode[STIX]{x1D6FC}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The shape function is

(9.4) $$\begin{eqnarray}\displaystyle & & \displaystyle \bar{\unicode[STIX]{x1D6F7}}(ka,m,b,n,kR_{\unicode[STIX]{x1D70F}},kR_{\unicode[STIX]{x1D70F}})\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{2}{kR_{\unicode[STIX]{x1D70F}}}\int _{0}^{kR_{\unicode[STIX]{x1D70F}}}\int _{0}^{\unicode[STIX]{x1D6FC}}\left(-\frac{1}{2\tilde{\unicode[STIX]{x1D706}}^{2}}+\frac{1}{2\tilde{\unicode[STIX]{x1D706}}^{2}}\left(1+4\tilde{\unicode[STIX]{x1D706}}^{2}\left(1-\frac{\hat{\unicode[STIX]{x1D6FC}}}{kR_{\unicode[STIX]{x1D70F}}}\right)\right)^{1/2}\right)\text{d}\hat{\unicode[STIX]{x1D6FC}}\left(1-\frac{\unicode[STIX]{x1D6FC}}{kR_{\unicode[STIX]{x1D70F}}}\right)\text{d}\unicode[STIX]{x1D6FC}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\ln (kR_{\unicode[STIX]{x1D70F}}).\end{eqnarray}$$

Figure 24(a) shows (9.4) plotted using the same mean values $(\bar{k}\bar{a},\bar{m},\bar{b},\bar{n})=(0.4092\times 20.095,1.621,0.3195,1.619)$ used to generate $\unicode[STIX]{x1D6F7}$ . It is apparent from this figure that $\bar{\unicode[STIX]{x1D6F7}}$ also approaches a limiting constant above $kR_{\unicode[STIX]{x1D70F}}=2000$ . Figure 24(b) shows that the high Reynolds number limit of $\bar{\unicode[STIX]{x1D6F7}}(ka,\infty )$ is also linearly proportional to $ka$ in the relevant range of Reynolds number. We can write the high Reynolds number smooth wall bulk velocity friction law as

(9.5) $$\begin{eqnarray}\displaystyle \lim _{kR_{\unicode[STIX]{x1D70F}}\rightarrow \infty }\frac{k\bar{u}}{u_{\unicode[STIX]{x1D70F}}}=\ln (R_{\unicode[STIX]{x1D70F}})+0.2915(ka)+\ln (k)-0.7638. & & \displaystyle\end{eqnarray}$$

Figure 23(b) shows $u_{0}/u_{\unicode[STIX]{x1D70F}}$ and $\bar{u}/u_{\unicode[STIX]{x1D70F}}$ calculated using (9.2) and (9.5), with optimal values of $(k,a)$ for each point and $(\bar{m},\bar{b},\bar{n})=(1.621,0.3195,1.619)$ , compared with the PSP friction data. Note that the poorest agreement between the PSP data and the high Reynolds number frictions laws in figure 23(b) is with PSP surveys 1 to 5 where the Reynolds number is below $kR_{\unicode[STIX]{x1D70F}}=2000$ and the shape function $\unicode[STIX]{x1D6F7}$ is not converged.

According to (9.2) and (9.5), the difference between $ku_{0}/u_{\unicode[STIX]{x1D70F}}$ and $k\bar{u}/u_{\unicode[STIX]{x1D70F}}$ at high Reynolds number is constant. Subtract (9.5) from (9.2). The result is

(9.6) $$\begin{eqnarray}\displaystyle \lim _{kR_{\unicode[STIX]{x1D70F}}\rightarrow \infty }\frac{k(u_{0}-\bar{u})}{u_{\unicode[STIX]{x1D70F}}}=1.8219. & & \displaystyle\end{eqnarray}$$

Given the relatively rapid approach to the limiting behaviour of $\unicode[STIX]{x1D6F7}$ and $\bar{\unicode[STIX]{x1D6F7}}$ seen in figure 24, the transition between PSP surveys 5 and 6 can now be understood as the onset of scale separation, and approximately logarithmic behaviour in the velocity profile, and transition to a true logarithmic friction law.

9.3 About $k$ and $a$

There is a point to be made about the model parameters $k$ and $a$ . The friction law, (9.16) is usually expressed in the form suggested by Prandtl (Reference Prandtl and Durand1934a ),

(9.7) $$\begin{eqnarray}\displaystyle \frac{1}{\sqrt{f}}=C_{1}\ln (\bar{R}_{e}\sqrt{f})+C_{2}, & & \displaystyle\end{eqnarray}$$

where $f$ is the friction factor defined in (2.13). If we use the mean values $\bar{k}=0.4092$ and $\bar{a}=20.397$ , the Prandtl friction law would be

(9.8) $$\begin{eqnarray}\displaystyle \frac{1}{\sqrt{f}}=0.8640\ln (\bar{R}_{e}\sqrt{f})-0.2282. & & \displaystyle\end{eqnarray}$$

Equation (9.8) does not produce a particularly good fit to the friction data. A much better fit is generated using the Kármán constant, $k=0.4302$ , and the additive constant, $3.0420$ , that best approximate the $\bar{u}/u_{\unicode[STIX]{x1D70F}}$ survey data in figure 2(b). Using these values, the Prandtl formula becomes

(9.9) $$\begin{eqnarray}\displaystyle \frac{1}{\sqrt{f}}=0.8218\ln (\bar{R}_{e}\sqrt{f})+0.221. & & \displaystyle\end{eqnarray}$$

This result can be compared with equation (36) in Zagarola & Smits (Reference Zagarola and Smits1998). The comparison requires the Reynolds number based on radius in (9.9) to be doubled and the natural log to be converted to log base 10. Equation (9.9) becomes

(9.10) $$\begin{eqnarray}\displaystyle \frac{1}{\sqrt{f}}=1.892\log (2\bar{R}_{e}\sqrt{f})-0.349 & & \displaystyle\end{eqnarray}$$

compared to

(9.11) $$\begin{eqnarray}\displaystyle \frac{1}{\sqrt{f}}=1.884\log (2\bar{R}_{e}\sqrt{f})-0.331 & & \displaystyle\end{eqnarray}$$

from Zagarola & Smits (Reference Zagarola and Smits1998). In fact, the value $k=0.4302$ from figure 2(b) used to generate (9.9) is only slightly less than the value $k=0.436$ originally reported by Zagarola (Reference Zagarola1996).

Mean values of $k$ and the value of $a$ that would generate the additive constants in figure 2(b) are

(9.12) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}k_{\unicode[STIX]{x1D70F}}=(0.4320+0.4302)/2=0.4311\\ a_{\unicode[STIX]{x1D70F}}=(23.9649+23.2528)/2=23.6088.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The reason for the subscript, $\unicode[STIX]{x1D70F}$ will be explained shortly. The constants (9.12) provide a very good fit to virtually all the friction data, whether it is survey data in figure 2(b), data generated using the model velocity profile with optimal parameter values in figure 23(a) or data in figure 23(b) generated from the high Reynolds number friction laws, equations (9.2) and (9.5), with $(\bar{m},\bar{b},\bar{n})=(1.609,0.3158,1.605)$ .

What is going on here? Optimal values of $k$ for four profiles (PSP surveys 6, 12, 18 and 26) are shown in figure 23(a). Note that the values of $k$ shown are all markedly lower than $k_{\unicode[STIX]{x1D70F}}$ . As a concrete example, consider a straight line that would connect PSP 6 and PSP 26 in figure 23(a). The data at these two points are the following.

(9.13) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}37.9002=\frac{1}{0.4190}\ln (530023)+6.4427\\ 26.1918=\frac{1}{0.4034}\ln (3327)+6.0881.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The slope of the line that would connect the two points is

(9.14) $$\begin{eqnarray}\displaystyle \frac{\left(\displaystyle \frac{1}{0.4190}\ln (530023)+6.4427\right)-\left(\displaystyle \frac{1}{0.4034}\ln (3327)+6.0881\right)}{\ln (530023)-\ln (3327)}=2.3090 & & \displaystyle\end{eqnarray}$$

corresponding to $k=0.4331$ . The point of this illustration is that the values of $k$ and $a$ that give the minimum error fit to the friction data cannot be assumed to be necessarily equal to, or even all that close to, the $k$ and $a$ that best fit any particular velocity profile at a given Reynolds number. The reason has to do with the dependence of the additive constant in the friction law on $k$ and $a$ , especially $k$ , and the weak Reynolds number dependence of both parameters. For this reason (9.8) is not a good approximation to the friction data. As long as $k$ and $a$ depend on the Reynolds number, the $k$ and $a$ for a given velocity profile and the $k_{\unicode[STIX]{x1D70F}}$ and $a_{\unicode[STIX]{x1D70F}}$ that best approximate the friction law have to be viewed as distinct quantities.

Using $k_{\unicode[STIX]{x1D70F}}$ and $a_{\unicode[STIX]{x1D70F}}$ from (9.12), the limiting friction laws (9.2) and (9.5) become

(9.15) $$\begin{eqnarray}\displaystyle \lim _{R_{\unicode[STIX]{x1D70F}}\rightarrow \infty }\frac{u_{0}}{u_{\unicode[STIX]{x1D70F}}}=\frac{1}{0.4311}\ln (R_{\unicode[STIX]{x1D70F}})+7.3442 & & \displaystyle\end{eqnarray}$$

and

(9.16) $$\begin{eqnarray}\displaystyle \lim _{R_{\unicode[STIX]{x1D70F}}\rightarrow \infty }\frac{\bar{u}}{u_{\unicode[STIX]{x1D70F}}}=\frac{1}{0.4311}\ln (R_{\unicode[STIX]{x1D70F}})+3.1584. & & \displaystyle\end{eqnarray}$$

To be clear, the constants $k$ and $a$ used in (9.2) and (9.5) can be the values specific to a particular profile at a given $R_{\unicode[STIX]{x1D70F}}$ , or one can use $k_{\unicode[STIX]{x1D70F}}$ and $a_{\unicode[STIX]{x1D70F}}$ . The resulting values of $u_{0}/u_{\unicode[STIX]{x1D70F}}$ and $\bar{u}/u_{\unicode[STIX]{x1D70F}}$ will be nearly the same.

Although the constants in (9.15) and (9.16) fit the PSP friction data quite well, the results shown in figure 24(b) and the evidence presented in figures 8(b), 10(a) and table 1 indicate that the $k$ and $a$ that provide the best fit to the velocity profiles are weakly increasing functions of $R_{\unicode[STIX]{x1D70F}}$ over the range of the PSP data and there is no reason to expect that trend to change at higher Reynolds numbers. One can therefore expect to see increases in both $k_{\unicode[STIX]{x1D70F}}$ and $a_{\unicode[STIX]{x1D70F}}$ when higher Reynolds number pipe measurements are made, but any friction law that is expected to fit Reynolds numbers substantially beyond $20\,088\,000$ will be weighted by the existing variations in $k$ and $a$ between PSP 6 and 26, and so changes in $k_{\unicode[STIX]{x1D70F}}$ and $a_{\unicode[STIX]{x1D70F}}$ that may be expected to occur as the Reynolds number is pushed higher would be expected to be quite small. So it is fair to say that, until extreme values of the pipe Reynolds number are reached, the values of $k_{\unicode[STIX]{x1D70F}}$ and $a_{\unicode[STIX]{x1D70F}}$ in (9.12) are for all practical purposes fully established.

9.4 Channel flow

An obvious follow-on to the present work would be to apply the same methodology to turbulent channel flow. However there are no measurements of the planar case at Reynolds numbers comparable to the Princeton Superpipe experiments. Aside from the engineering challenge of constructing a planar pressure vessel or a very large facility, end wall effects and secondary flow limit the region of fully developed planar flow that can be achieved making pipe flow the obvious choice for an experimental campaign designed to study very high Reynolds number wall turbulence.

At moderate Reynolds numbers, a combination of experimental and simulation channel flow data would be the logical place to start, and the paper by Monty & Chong (Reference Monty and Chong2009) provides a good description of the research issues involved, particularly near the wall. Their study suggests that there may be an opportunity to apply the approach presented in this paper to measurements that include data quite close to the wall from both simulations and high aspect ratio channel experiments at nearly the same Reynolds number. Direct numerical simulations of turbulent channel flow have reached Reynolds numbers where significant scale separation is beginning to be achieved and one might be able to directly connect the sort of change in the character of the mean flow seen between PSP 5 and 6 with changes in the underlying structure of the turbulence.