2 Literature review

The interaction of the centrifugal force and axial momentum is less investigated at the inlet of a rotating pipe than in the fully developed region. At the inlet of a rotating pipe, the boundary layers develop and an axial gradient of the velocity field occurs. At the fully developed region, this gradient vanishes and the boundary layers reach the centreline of the pipe. For a turbulent flow, the boundary layers reach the centreline for $z>10^{2}$ depending on flow number and Reynolds number (Nishibori, Kikuyama & Murakami Reference Nishibori, Kikuyama and Murakami1987). In the following, the influence of the swirl on friction, axial and circumferential velocity profile and flow separation is analysed depending on the flow, e.g. turbulent versus laminar, and region, e.g. fully developed region versus inlet of a rotating pipe.

In contrast to Blasius’ investigations of friction losses in non-rotating smooth pipes by a turbulent flow (Blasius Reference Blasius1913), the hydraulic losses decrease with decreasing flow number in a rotating pipe (Levy Reference Levy1927; White Reference White1964). This is due to the turbulence damping by the swirl, which stabilizes the flow. The damping has a stronger influence close to the wall than in the core. Consequently, more low-frequency than high-frequency vibrations are damped (Borisenko, Kostikov & Chumachenko Reference Borisenko, Kostikov and Chumachenko1973). At the inlet of the rotating pipe for $\unicode[STIX]{x1D711}>1$ the turbulence is stimulated due to a sudden increase of the swirl for $z<20$ , but further downstream the damping due to centrifugal force predominates (Nagib et al. Reference Nagib, Lavan, Fejer and Wolf1973; Bissonnette & Mellor Reference Bissonnette and Mellor1974). For a laminar flow with small Reynolds number, the swirl increases the friction coefficient and stimulates the turbulence. Thus, the transition point to a turbulence flow is further upstream than in a non-rotating pipe (White Reference White1964).

At the inlet of a rotating pipe, a complex transformation of the axial velocity profile occurs due to the interaction of the increased centrifugal force and the stimulating turbulence burst. The transformation depends on the inlet condition: when a fully developed, turbulent flow enters the rotating pipe, the turbulent profile transforms continuously into a laminar profile in the axial direction (Kikuyama et al. Reference Kikuyama, Murakami, Nishibori and Maeda1983b ). This effect is called ‘laminarization’ (Nishibori et al. Reference Nishibori, Kikuyama and Murakami1987; Weigand & Beer Reference Weigand and Beer1994; Imao, Itohi & Harada Reference Imao, Itohi and Harada1996). For higher flow number, the axial velocity profile is retransformed to a turbulent profile, being well described by Prandtl’s $1/7$ power law (Nishibori et al. Reference Nishibori, Kikuyama and Murakami1987), and for small flow number the ‘laminarized’ profile reaches the fully developed region. Weigand & Beer (Reference Weigand, Beer, Kim and Yang1992) model the transformation analytically and meet the experimental results qualitatively whereby the coupling effect between the swirl and axial flow is overestimated. Thus, the influence of the swirl on the axial momentum has so far not been sufficiently investigated at the inlet of a rotating pipe.

The evolution of the swirl is given by the swirl boundary layer with the circumferential velocity profile. At the fully developed region, the circumferential velocity profile is parabolic and follows $u_{\unicode[STIX]{x1D711}}=r^{2}$ when a turbulent flow enters a rotating pipe (Murakami & Kikuyama Reference Murakami and Kikuyama1980; Kikuyama, Murakami & Nishibori Reference Kikuyama, Murakami and Nishibori1983a ; Reich Reference Reich1988; Imao et al. Reference Imao, Itohi and Harada1996), and for a laminar flow it follows solid-body rotation $u_{\unicode[STIX]{x1D711}}=r$ (Reich Reference Reich1988). Both profiles are derived by Lie group analysis (Oberlack Reference Oberlack1999). In the transition region between laminar and turbulent flow, the circumferential velocity profile is between the parabolic and linear profiles (Imao, Zhang & Yamada Reference Imao, Zhang and Yamada1989). At the inlet of a rotating pipe, the circumferential velocity profile is less investigated. The experimental investigations with a thin laminar boundary layer at $\unicode[STIX]{x1D711}>0.71$ show a possible transformation of the circumferential velocity profile. Whether and where the transformation occurs depend on flow number and Reynolds number. The circumferential velocity profile transforms before an effect on the axial velocity can be perceived. After the transformation, both boundary layers become thicker (Nishibori et al. Reference Nishibori, Kikuyama and Murakami1987). Kikuyama et al. (Reference Kikuyama, Murakami and Nishibori1983a ) give a circumferential velocity profile scaled with the momentum thickness $\unicode[STIX]{x1D6FF}_{2}$ for high flow numbers. However, considerable deviations exist for $z<40$ . Also the profile presented by Weigand & Beer (Reference Weigand, Beer, Kim and Yang1992) is not sufficient because it is not independent of the axial coordinate with deviations for $z<20$ . Thus, there is a need to describe the circumferential velocity profile and the evolution of the swirl in more detail at the inlet of a rotating pipe.

For $\unicode[STIX]{x1D711}<\unicode[STIX]{x1D711}_{c}$ , flow separates and a recirculation bubble appears at the wall. The separation is caused by the sudden pressure increase due to swirl at the wall. At the inlet of a rotating pipe, the flow separation is analytically and numerically investigated by a laminar flow with $Re<10^{3}$ (Lavan, Nielsen & Fejer Reference Lavan, Nielsen and Fejer1969; Crane & Burley Reference Crane and Burley1976). For small Reynolds numbers $Re\ll 10^{3}$ , the critical flow number is a linear function of the Reynolds number (Lavan et al. Reference Lavan, Nielsen and Fejer1969; Crane & Burley Reference Crane and Burley1976), but for $Re>10^{2}$ , the linear relation loses its validity (Crane & Burley Reference Crane and Burley1976). Experimental investigations by Imao et al. (Reference Imao, Zhang and Yamada1989) show flow separation for $\unicode[STIX]{x1D711}<\unicode[STIX]{x1D711}_{c}\approx 0.33$ at $Re=3000$ . Up to now, flow separation for a turbulent flow at the inlet of a rotating pipe has not been sufficiently analysed.

Najafi et al. (Reference Najafi, Saidi, Sadeghipour and Souhar2005) investigate the complementary case in which a turbulent rotating flow enters a pipe at rest. Analysing this work, we find three limitations: first, the radial pressure gradient, which is the root cause of flow separation, is initially assumed to be constant and is later neglected in the analytic section; second, the governing equations are linearized, which is not justified in this paper; and finally, the results are ‘validated’ by Reynolds-averaged Navier–Stokes (RANS) simulations. The referenced RANS simulations, solved with a commercial solver, still do not predict turbulent rotating flows satisfyingly today. Thus, RANS simulations are not suitable to validate these kinds of flows.

3 Generalized von Kármán’s momentum equation

In the following we give an order-of-magnitude analysis of all relevant physical quantities. The order of magnitude of the time-averaged circumferential velocity $u_{\unicode[STIX]{x1D719}}$ is 1 and the order of magnitude of the time-averaged axial velocities $u,U$ follows $\unicode[STIX]{x1D711}$ . The order of magnitude of the wall coordinate $y$ follows $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}_{S}$ . For thin boundary layers $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}_{S}\ll 1$ , an order-of-magnitude analysis of the continuity equation within the boundary layer yields $u_{y}\sim \unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}$ ( ${\sim}$ represents ‘is of the order’) for the time-averaged radial velocity and $u_{y}^{\prime }\sim u^{\prime }\unicode[STIX]{x1D6FF}$ for the turbulent fluctuations since $z\sim 1$ . Here $u^{\prime }$ and $u_{y}^{\prime }$ are the fluctuations of the axial and radial velocity components, respectively. With the order-of-magnitude analysis given so far, the order of magnitude of $\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x2202}y$ is $1/\unicode[STIX]{x1D6FF}$ and  $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y\sim \unicode[STIX]{x1D711}/\unicode[STIX]{x1D6FF}$ . For $\unicode[STIX]{x1D711}\sim \unicode[STIX]{x1D6FF}\ll 1$ , $\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x2202}y$ dominates the time-averaged velocity gradient because all other components, especially $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y$ , are at least one order of magnitude smaller than $\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x2202}y$ . In contrast, for $\unicode[STIX]{x1D711}\sim 1/\unicode[STIX]{x1D6FF}\gg 1$ , $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y$ is the dominating component of the velocity gradient.

In a non-rotating pipe, the mixing length $l$ equals $\unicode[STIX]{x1D705}y$ near the smooth wall with the von Kármán constant $\unicode[STIX]{x1D705}=0.4$ (Nikuradse Reference Nikuradse1932), whereas in a rotating pipe $l\leqslant \unicode[STIX]{x1D705}y$ (Kikuyama et al. Reference Kikuyama, Murakami and Nishibori1983a ,Reference Kikuyama, Murakami, Nishibori and Maeda b ; Weigand & Beer Reference Weigand and Beer1994). For both cases, with $\unicode[STIX]{x1D705}=0.4$ and $y\sim \unicode[STIX]{x1D6FF}\sim 10^{-1}$ , the mixing length is of the order of magnitude $10^{-2}$ or equivalent to the order of magnitude $\unicode[STIX]{x1D6FF}^{2}$ . An order-of-magnitude analysis of the turbulence viscosity introduced by Smagorinsky (Reference Smagorinsky1963) yields

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D708}_{turb}=(\unicode[STIX]{x1D705}y)^{2}\left(2\unicode[STIX]{x1D626}_{ij}\unicode[STIX]{x1D626}_{ij}\right)^{1/2}\approx \underbrace{(\unicode[STIX]{x1D705}y)^{2}}_{{\sim}\unicode[STIX]{x1D6FF}^{4}}\underbrace{\sqrt{\left(\frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D719}}}{\unicode[STIX]{x2202}y}\right)^{2}+\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}\right)^{2}}}_{{\sim}1/\unicode[STIX]{x1D6FF}\sqrt{1+\unicode[STIX]{x1D711}^{2}}}\sim \unicode[STIX]{x1D6FF}^{3}\sqrt{1+\unicode[STIX]{x1D711}^{2}}\leqslant \unicode[STIX]{x1D6FF}^{3}\end{eqnarray}$$

for the present flow. Even though it is unusual to have an order of magnitude as a sum, we prefer to use this presentation for the distinction of cases $\unicode[STIX]{x1D711}\ll 1$ and $\unicode[STIX]{x1D711}\sim 1$ in the following. By doing so, the order-of-magnitude analysis keeps the flow number general. For the generic set-up considered here, flow separation is expected for $\unicode[STIX]{x1D711}\ll 1$ and is the focus of this paper. Hence, there is no need to consider the case $\unicode[STIX]{x1D711}\gg 1$ . Applying Boussinesq’s eddy viscosity concept (Pope Reference Pope2011), the Reynolds stress tensor is

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{ij,turb}=\overline{u_{i}^{\prime }u_{j}^{\prime }}={\textstyle \frac{1}{3}}\overline{u_{l}^{\prime }u_{l}^{\prime }}\unicode[STIX]{x1D6FF}_{ij}-\unicode[STIX]{x1D708}_{turb}\unicode[STIX]{x1D626}_{ij}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D6FF}_{ij}$ is the Kronecker tensor and $\unicode[STIX]{x1D626}_{ij}$ is the mean strain-rate tensor. In the following, (3.2) is exclusively applied to analyse the order of magnitude of the Reynolds stress tensor.

The normal components of the Reynolds stress tensor are implicit due to the turbulent kinetic energy $\overline{u_{i}^{\prime }u_{i}^{\prime }}/3$ . The order of magnitude of the normal components of the Reynolds stress tensor (3.2) is determined by eliminating the kinetic energy. Subtracting $\overline{u_{y}^{\prime }u_{y}^{\prime }}$ from $\overline{u^{\prime }u^{\prime }}$ yields

(3.3) $$\begin{eqnarray}\overline{u^{\prime }u^{\prime }}-\overline{u_{y}^{\prime }u_{y}^{\prime }}=\underbrace{\unicode[STIX]{x1D708}_{turb}}_{{\sim}\unicode[STIX]{x1D6FF}^{3}}\left(\underbrace{\frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}y}}_{{\sim}\unicode[STIX]{x1D711}}-\underbrace{\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}}_{{\sim}\unicode[STIX]{x1D711}}\right)\sim \unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{3}.\end{eqnarray}$$

With $u_{y}^{\prime }\sim u^{\prime }\unicode[STIX]{x1D6FF}$ it follows that $\overline{u_{y}^{\prime }u_{y}^{\prime }}\sim \overline{u^{\prime }u^{\prime }}\unicode[STIX]{x1D6FF}^{2}$ . Thus, $\overline{u^{\prime }u^{\prime }}$ is of the order of magnitude of $\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{3}$ . Hence, $\unicode[STIX]{x1D70F}_{zz,turb}=\overline{u^{\prime }u^{\prime }}\sim \unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{3}$ and $\unicode[STIX]{x1D70F}_{yy,turb}=\overline{u_{y}^{\prime }u_{y}^{\prime }}\sim \unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{5}$ . With this result, the order of magnitude of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}\unicode[STIX]{x1D719},turb}=\overline{u_{\unicode[STIX]{x1D719}}^{\prime }u_{\unicode[STIX]{x1D719}}^{\prime }}\sim \unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{3}$ is given.

The radial component of the momentum equation is

(3.4) $$\begin{eqnarray}\displaystyle & & \displaystyle -\underbrace{u_{y}\frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}y}}_{{\sim}\unicode[STIX]{x1D711}^{2}\unicode[STIX]{x1D6FF}}-\underbrace{u\frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}z}}_{{\sim}\unicode[STIX]{x1D711}^{2}\unicode[STIX]{x1D6FF}}-\underbrace{\frac{u_{\unicode[STIX]{x1D719}}^{2}}{1-y}}_{{\sim}1}=\underbrace{\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}y}}_{{\sim}1}+\underbrace{\frac{2}{Re}}_{{\sim}\unicode[STIX]{x1D6FF}^{2}}\left[-\underbrace{\frac{\unicode[STIX]{x2202}^{2}u_{y}}{\unicode[STIX]{x2202}y^{2}}}_{{\sim}\unicode[STIX]{x1D711}/\unicode[STIX]{x1D6FF}}+\underbrace{\frac{1}{1-y}\frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}y}}_{{\sim}\unicode[STIX]{x1D711}}+\underbrace{\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}z^{2}}}_{{\sim}\unicode[STIX]{x1D711}}+\,\underbrace{\frac{u_{y}}{\left(1-y\right)^{2}}}_{{\sim}\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\underbrace{\frac{\unicode[STIX]{x2202}\overline{u_{y}^{\prime }u_{y}^{\prime }}}{\unicode[STIX]{x2202}y}}_{{\sim}\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{4}}+\underbrace{\frac{\unicode[STIX]{x2202}\overline{u_{y}^{\prime }u^{\prime }}}{\unicode[STIX]{x2202}z}}_{{\sim}\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{2}}-\underbrace{\frac{\overline{u_{y}^{\prime }u_{y}^{\prime }}}{1-y}}_{{\sim}\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{5}}-\underbrace{\frac{\overline{u_{\unicode[STIX]{x1D719}}^{\prime }u_{\unicode[STIX]{x1D719}}^{\prime }}}{1-y}}_{{\sim}\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{3}}\end{eqnarray}$$

with the static pressure $p$ . An order-of-magnitude analysis of (3.4) yields that the viscous and Reynolds stresses are negligible for $\unicode[STIX]{x1D711}\ll 1$ , $\unicode[STIX]{x1D711}\sim 1$ and $\unicode[STIX]{x1D711}>1$ . The case $\unicode[STIX]{x1D711}>1$ is treated by Kikuyama et al. (Reference Kikuyama, Murakami and Nishibori1983a ), Kitoh (Reference Kitoh1991) and Imao et al. (Reference Imao, Itohi and Harada1996). The order of magnitude of the centrifugal force, i.e. the third term on the left-hand side of (3.4), is 1. The momentum fluxes, i.e. the first and second terms on the left-hand side of (3.4), are of the order of magnitude $\unicode[STIX]{x1D711}^{2}\unicode[STIX]{x1D6FF}$ . The order of magnitude of the viscous stress, i.e. the terms in the square brackets on the right-hand side of (3.4), are smaller than or equal to $\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}$ since $Re\sim 1/\unicode[STIX]{x1D6FF}^{2}$ for a laminar flow (Schlichting Reference Schlichting1970; Pope Reference Pope2011). For a turbulent flow, the Reynolds number is much larger. Hence, $Re\sim 1/\unicode[STIX]{x1D6FF}^{2}$ serves as an upper bound on the order of magnitude of the viscous stress for the order-of-magnitude analysis. Indeed, our measurements indicate that $Re\sim 10^{4}\approx 1/\unicode[STIX]{x1D6FF}^{4}$ for a turbulent flow. The Reynolds stress, i.e. the last four terms on the right-hand side of (3.4), is of the order of magnitude smaller than or equal to $\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{2}$ . Thus, the viscous and the Reynolds stresses are at least one order of magnitude smaller than the centrifugal force and are negligible, especially for $\unicode[STIX]{x1D711}\ll 1$ . Consequently, the radial pressure gradient has to balance the centrifugal force, and this gradient is of the order of magnitude $1$ . The radial component of the momentum equation (3.4) yields

(3.5) $$\begin{eqnarray}-\frac{u_{\unicode[STIX]{x1D719}}^{2}}{1-y}=\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}y}+O(\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF})\end{eqnarray}$$

$\text{for}~y<\unicode[STIX]{x1D6FF}_{S}$ . For $y>\unicode[STIX]{x1D6FF}_{S}$ , the circumferential velocity vanishes ( $u_{\unicode[STIX]{x1D719}}=0$ ) by definition and the static pressure is a function only of the axial coordinate $P(z)$ , satisfying the axial component of Euler’s equation

(3.6) $$\begin{eqnarray}\frac{\text{d}P}{\text{d}z}=-U\frac{\text{d}U}{\text{d}z}\sim \unicode[STIX]{x1D711}^{2},\end{eqnarray}$$

where $U$ is the axial flow velocity in the core region. Hence, the pressure distribution within the swirl boundary layer $y<\unicode[STIX]{x1D6FF}_{S}$ is

(3.7) $$\begin{eqnarray}p(y,z)=\int _{y}^{\unicode[STIX]{x1D6FF}_{S}}\frac{u_{\unicode[STIX]{x1D719}}^{2}}{1-y^{\prime }}\,\text{d}y^{\prime }+P(z)\sim \unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D711}^{2}.\end{eqnarray}$$

For the sketched control volume in figure 1 (broken line), the axial momentum balance reads

(3.8) $$\begin{eqnarray}\displaystyle & & \displaystyle \underbrace{\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)u^{2}\,\text{d}y}_{{\sim}\unicode[STIX]{x1D711}^{2}\unicode[STIX]{x1D6FF}}-\underbrace{U\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)u\,\text{d}y}_{{\sim}\unicode[STIX]{x1D711}^{2}\unicode[STIX]{x1D6FF}}\nonumber\\ \displaystyle & & \displaystyle \quad =\underbrace{\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)\unicode[STIX]{x1D70F}_{zz}(y,z)\,\text{d}y}_{{\sim}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D711}^{2})+\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{3}+\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{4}}-\underbrace{(1-\unicode[STIX]{x1D6FF})\unicode[STIX]{x1D70F}_{zz}(\unicode[STIX]{x1D6FF},z)\frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}z}}_{{\sim}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D711}^{2})+\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{3}+\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{4}}-\underbrace{\unicode[STIX]{x1D70F}_{yz,w}}_{{\sim}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D711}^{2})},\end{eqnarray}$$

where $u$ is the time-averaged axial velocity component inside the axial boundary layer, $\unicode[STIX]{x1D70F}_{yz,w}$ is the $yz$ shear-stress component at the wall and $\unicode[STIX]{x1D70F}_{zz}$ is the $zz$ shear-stress component. For a Newtonian fluid and a turbulent flow, $\unicode[STIX]{x1D70F}_{zz}=-p+\unicode[STIX]{x1D70F}_{zz,vis}+\unicode[STIX]{x1D70F}_{zz,turb}$ . Hereby, the order of magnitude of the viscous stress is

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{zz,vis}=\frac{4}{Re}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}\sim \unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{2},\end{eqnarray}$$

since employing the lower bound of the Reynolds number $Re\sim 1/\unicode[STIX]{x1D6FF}^{2}$ for a laminar flow yields an upper bound of the order of magnitude of the viscous stress. As stated earlier, with $Re\sim 1/\unicode[STIX]{x1D6FF}^{4}$ for a turbulent flow, this gives the lower bound for the viscous stress. As known from (3.3), $\unicode[STIX]{x1D70F}_{zz,turb}=\overline{u^{\prime }u^{\prime }}\sim \unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{3}$ . Thus, an order-of-magnitude analysis of the axial momentum balance (3.8) yields the following: the viscous and Reynolds stress of the $zz$ component are negligible. The axial momentum fluxes, i.e. the terms on the left-hand side of (3.8), are of the order of magnitude $\unicode[STIX]{x1D711}^{2}\unicode[STIX]{x1D6FF}$ . The order of magnitude of the pressure is $\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D711}^{2}$ as (3.6) and (3.7) indicate. Thus, the order of magnitude of the centrifugal force, i.e. the terms on the right-hand side of (3.8) including the pressure $p$ , is $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D711}^{2})$ . The wall shear stress, i.e. the third term on the right-hand side of (3.8), needs to balance the axial momentum fluxes and the centrifugal force. Thus, its order of magnitude is $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D711}^{2})$ . All terms with the viscous stress of the $zz$ component are of the order of magnitude $\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{3}$ and of the Reynolds stress are of the order of magnitude $\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{4}$ . The terms of the viscous and Reynolds stresses are at least one order of magnitude smaller than all other terms and are negligible. Hence, the axial momentum balance reads

(3.10) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)u^{2}\,\text{d}y-U\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)u\,\text{d}y\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)p(y,z)\,\text{d}y+(1-\unicode[STIX]{x1D6FF})p(\unicode[STIX]{x1D6FF},z)\frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}z}-\unicode[STIX]{x1D70F}_{yz,w}+O(\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}^{3}).\end{eqnarray}$$

For $\unicode[STIX]{x1D711}\ll 1$ , the axial momentum fluxes are an order of magnitude smaller than the centrifugal force. Hence, the centrifugal force and the wall shear stress dominate the axial momentum balance. The pressure $p(y>\unicode[STIX]{x1D6FF}_{S},z)$ equals the pressure of the core flow $P(z)$ .

Using the displacement thickness $\unicode[STIX]{x1D6FF}_{1}$ , defined as usual (Piquet Reference Piquet1999) as

(3.11) $$\begin{eqnarray}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)u\,\text{d}y:=\frac{1}{2}U\left[(1-\unicode[STIX]{x1D6FF}_{1})^{2}-(1-\unicode[STIX]{x1D6FF})^{2}\right]\end{eqnarray}$$

and the momentum thickness $\unicode[STIX]{x1D6FF}_{2}$ defined by

(3.12) $$\begin{eqnarray}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)u^{2}\,\text{d}y:=\frac{1}{2}U^{2}\left[(1-\unicode[STIX]{x1D6FF}_{2})^{2}+(1-\unicode[STIX]{x1D6FF}_{1})^{2}-(1-\unicode[STIX]{x1D6FF})^{2}-1\right],\end{eqnarray}$$

the axial momentum balance yields

(3.13) $$\begin{eqnarray}(1-\unicode[STIX]{x1D6FF}_{2})\frac{\text{d}\unicode[STIX]{x1D6FF}_{2}}{\text{d}z}+\left(2\unicode[STIX]{x1D6FF}_{2}-\unicode[STIX]{x1D6FF}_{2}^{2}+\unicode[STIX]{x1D6FF}_{1}-\frac{1}{2}\unicode[STIX]{x1D6FF}_{1}^{2}\right)\frac{1}{U}\frac{\text{d}U}{\text{d}z}+\frac{G}{U^{2}}=\frac{\unicode[STIX]{x1D70F}_{yz,w}}{U^{2}}+O\left(\frac{\unicode[STIX]{x1D6FF}^{3}}{\unicode[STIX]{x1D711}}\right),\end{eqnarray}$$

with the coupling term

(3.14) $$\begin{eqnarray}G:=-\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)\int _{y}^{\unicode[STIX]{x1D6FF}_{S}}\frac{u_{\unicode[STIX]{x1D719}}^{2}}{1-y^{\prime }}\,\text{d}y^{\prime }\,\text{d}y+(1-\unicode[STIX]{x1D6FF})\frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}z}\int _{\unicode[STIX]{x1D6FF}}^{\unicode[STIX]{x1D6FF}_{S}}\frac{u_{\unicode[STIX]{x1D719}}^{2}}{1-y}\,\text{d}y.\end{eqnarray}$$

The second term on the right-hand side of (3.14) vanishes for $\unicode[STIX]{x1D6FF}>\unicode[STIX]{x1D6FF}_{S}$ because $u_{\unicode[STIX]{x1D719}}(y>\unicode[STIX]{x1D6FF}_{S})=0$ . For swirl-free flow, the coupling term vanishes ( $G=0$ ) and (3.13) reduces to the von Kármán momentum equation. Hence, (3.13) is a generalization of the von Kármán momentum equation and considers the radial pressure distribution due to swirl. The angular momentum equation

(3.15) $$\begin{eqnarray}\underbrace{\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)^{2}u_{\unicode[STIX]{x1D719}}u\,\text{d}y}_{{\sim}\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}}+\underbrace{\frac{\text{d}}{\text{d}z}\int _{\unicode[STIX]{x1D6FF}}^{\unicode[STIX]{x1D6FF}_{S}}(1-y)^{2}u_{\unicode[STIX]{x1D719}}U\,\text{d}y}_{{\sim}\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}}=\underbrace{\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}_{S}}(1-y)^{2}\unicode[STIX]{x1D70F}_{z\unicode[STIX]{x1D719}}\,\text{d}y}_{{\sim}\unicode[STIX]{x1D6FF}^{3}+\unicode[STIX]{x1D6FF}^{4}}-\underbrace{\unicode[STIX]{x1D70F}_{y\unicode[STIX]{x1D719},w}}_{{\sim}\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}}\end{eqnarray}$$

is needed to determine $G$ . Here, $\unicode[STIX]{x1D70F}_{y\unicode[STIX]{x1D719},w}$ is the $y\unicode[STIX]{x1D719}$ shear-stress component at the wall and $\unicode[STIX]{x1D70F}_{z\unicode[STIX]{x1D719}}$ is the $z\unicode[STIX]{x1D719}$ shear-stress component. Equation (3.15) is valid for any case, either $\unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D6FF}_{S}$ or $\unicode[STIX]{x1D6FF}>\unicode[STIX]{x1D6FF}_{S}$ . For $\unicode[STIX]{x1D6FF}>\unicode[STIX]{x1D6FF}_{S}$ , the second term on the left-hand side of (3.15) vanishes because $u_{\unicode[STIX]{x1D719}}(y>\unicode[STIX]{x1D6FF}_{S})=0$ . For $\unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D6FF}_{S}$ , the axial velocity profile changes within the swirl boundary layer and this is considered by the second term on the left-hand side of (3.15). Furthermore, there is no convective angular momentum transport either into or out of the swirl boundary layer over $\unicode[STIX]{x1D6FF}_{S}$ , as there is with the convective axial momentum transport over $\unicode[STIX]{x1D6FF}$ (second term of the left-hand side of (3.8)). This is justified by the swirl-free flow outside the swirl boundary layer.

An order-of-magnitude analysis of the angular momentum balance (3.15) yields that the viscous and Reynolds stresses of the $z\unicode[STIX]{x1D719}$ component are negligible. The angular momentum flux, i.e. the left-hand side of (3.15), is of the order of magnitude $\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}$ . The wall shear stress needs to balance the angular momentum fluxes, thus its order of magnitude is $\unicode[STIX]{x1D711}\unicode[STIX]{x1D6FF}$ as well. The torque due to the shear stress of the $z\unicode[STIX]{x1D719}$ component at the inlet and outlet of the control volume, i.e. the first term on the right-hand side of (3.15), is of the order of magnitude $\unicode[STIX]{x1D6FF}^{3}$ . This is due to the order of magnitude of the stress component $\unicode[STIX]{x1D70F}_{z\unicode[STIX]{x1D719}}$ . The upper bound of the order of magnitude of the viscous stress of this component is given by

(3.16) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{z\unicode[STIX]{x1D719},vis}=\frac{2}{Re}\frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D719}}}{\unicode[STIX]{x2202}z}\sim \unicode[STIX]{x1D6FF}^{2}.\end{eqnarray}$$

We employ (3.2) to approximate the order of magnitude of the Reynolds stress

(3.17) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{z\unicode[STIX]{x1D719},turb}=-\overline{u^{\prime }u_{\unicode[STIX]{x1D719}}^{\prime }}=\unicode[STIX]{x1D708}_{turb}\frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D719}}}{\unicode[STIX]{x2202}z}\sim \unicode[STIX]{x1D6FF}^{3}.\end{eqnarray}$$

Hence, at the inlet and outlet of the control volume, the order of magnitude of the viscous and Reynolds stresses $\unicode[STIX]{x1D70F}_{z\unicode[STIX]{x1D719}}$ is smaller than or even of the same order of magnitude as $\unicode[STIX]{x1D6FF}^{2}$ . Thus, only the torque integral on the right-hand side of (3.15) is of the order of magnitude $\unicode[STIX]{x1D6FF}^{3}$ and for $\unicode[STIX]{x1D711}\sim 1$ it is two orders of magnitude smaller than all other terms and one order of magnitude smaller for $\unicode[STIX]{x1D711}\ll 1$ . Hence, for the boundary layer approximation, (3.15) reduces to

(3.18) $$\begin{eqnarray}\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y)^{2}u_{\unicode[STIX]{x1D719}}u\,\text{d}y+\frac{\text{d}}{\text{d}z}\int _{\unicode[STIX]{x1D6FF}}^{\unicode[STIX]{x1D6FF}_{S}}(1-y)^{2}u_{\unicode[STIX]{x1D719}}U\,\text{d}y=-\unicode[STIX]{x1D70F}_{y\unicode[STIX]{x1D719},w}+O(\unicode[STIX]{x1D6FF}^{3}),\end{eqnarray}$$

and is valid for thin laminar and turbulent boundary layers. The system of equations (3.13) and (3.18) is completed by the continuity equation

(3.19) $$\begin{eqnarray}\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}}u(1-y)\,\text{d}y+\frac{\text{d}}{\text{d}z}\int _{\unicode[STIX]{x1D6FF}}^{1}U(1-y)\,\text{d}y=0,\end{eqnarray}$$

which describes the interdependence between the boundary layer and the core flow velocity. Substituting the first term on the left-hand side with the displacement thickness (3.11), the continuity equation yields

(3.20) $$\begin{eqnarray}\frac{1}{U}\frac{\text{d}U}{\text{d}z}=\frac{2}{1-\unicode[STIX]{x1D6FF}_{1}}\frac{\text{d}\unicode[STIX]{x1D6FF}_{1}}{\text{d}z}.\end{eqnarray}$$

By doing so, it becomes clear that the flow is accelerated due to the displacement thickness at the core region. Thus, the axial pressure gradient (3.6) must be negative at the core region, as (3.20) indicates.

To analyse the interdependence of the axial momentum balance and the angular momentum balance, (3.13), (3.18) and (3.20) have to be solved. All velocity profiles $u(y,z)$ and $u_{\unicode[STIX]{x1D719}}(y,z)$ must satisfy these conservation laws in integral form. In the following, we assume power laws for both axial and swirl velocity profiles with the corresponding wall shear stress and determine both boundary layer thicknesses $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6FF}_{S}$ and their interaction.

4 Solution of the generalized von Kármán momentum equation

The well-recognized advantage of the integral method is that neither detailed knowledge of the velocity distribution within the boundary layers nor a turbulence model is needed. As will be shown, the velocity profiles can be modelled by power laws, logarithmic, harmonic functions (Schlichting Reference Schlichting1970) or other functions that capture the asymptotic behaviour of the velocity profile. In other words, the method is robust with respect to the shape of the ansatz functions; the shape of the ansatz function has a very limited influence on the findings. This work employs power-law distributions as ansatz functions for the axial velocity profile

(4.1) $$\begin{eqnarray}\frac{u(y,z)}{U(z)}=\left(\frac{y}{\unicode[STIX]{x1D6FF}(z)}\right)^{n}\end{eqnarray}$$

and circumferential velocity profile

(4.2) $$\begin{eqnarray}u_{\unicode[STIX]{x1D719}}(y,z)=\left(1-\frac{y}{\unicode[STIX]{x1D6FF}_{S}(z)}\right)^{k}.\end{eqnarray}$$

Applying the integral method to flows with a turbulent boundary layer, it is known that the ansatz function like the chosen (4.1) has to be adjusted, since $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y\rightarrow \infty$ for $y\rightarrow 0$ . To adjust the ansatz function, it is common to use an empirical ansatz function in treating the turbulent boundary layer by integral methods (Schubauer & Tchen Reference Schubauer and Tchen1961; Schlichting Reference Schlichting1970). The integral method keeps the physical content of the flow even for seemingly crude assumptions. For a turbulent pipe flow, Schlichting (Reference Schlichting1970) models the wall shear stress with $\unicode[STIX]{x1D70F}_{yz,w}=\unicode[STIX]{x1D706}\unicode[STIX]{x1D711}^{2}/8$ using the Blasius law $\unicode[STIX]{x1D706}=0.3164(\unicode[STIX]{x1D711}Re)^{-1/4}$ (Blasius Reference Blasius1913), as shown in appendix A. The ansatz function (4.1) in combination with the Blasius law reads

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{yz,w}(z)=c^{-2/(n+1)}U(z)^{2/(n+1)}\left(\frac{2}{\unicode[STIX]{x1D6FF}(z)Re}\right)^{2n/(n+1)}.\end{eqnarray}$$

For the Reynolds number interval $Re=10^{4}{-}10^{6}$ , the exponent $n$ is $1/7$ and $c=8.74$ (Schlichting Reference Schlichting1970). Even the value of $c$ has a very limited influence on the findings, as shown in appendix A. This again shows the desired robustness of the chosen integral method.

In contrast to the axial velocity profile (4.1), the circumferential velocity profile (4.2) shows no singularity in the velocity gradient at the wall. Near the smooth wall, the Reynolds stress vanishes due to kinematic reasons. Hence, the viscous stress is

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{y\unicode[STIX]{x1D719},w}(z)=\frac{2a}{Re}\frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D719}}}{\unicode[STIX]{x2202}y}\bigg|_{w}=-\frac{2ak}{\unicode[STIX]{x1D6FF}_{S}(z)Re},\end{eqnarray}$$

with $a=1$ . Our experiments indicate $a>1$ to predict the turbulent swirl boundary layer thickness properly. For $k=2$ and $a=1.53$ , being independent of Reynolds number and flow number, the swirl boundary layer thickness is well predicted. The influence of the constant $a$ on the boundary layer thicknesses is plotted in appendix A for $1\leqslant a\leqslant 2$ . It has become clear so far that the Blasius constant $c=8.74$ and $a=1.53$ are empirical constants, as is the von Kármán constant $\unicode[STIX]{x1D705}=0.4$ . Despite these empirical constants, the findings are robust, since these constants are independent of flow parameters like Reynolds number and flow number.

The constant $a$ is determined by solving the generalized von Kármán equations (3.13) and (3.18) and the continuity equation (3.20) with the ansatz functions (4.1)–(4.4) and calibrates the swirl boundary layer thickness with the experimental measurements at $z=2$ , $\unicode[STIX]{x1D711}=0.40$ and $\log (Re)=4.41$ .

For those who are not familiar with the power of the integral method of the boundary layer theory approach of using ansatz functions, it may seem like fitting, but it is not. At the very end, the conservation laws in integral form are fulfilled to a very good degree of approximation as our experimental validation shows. The method and the outcomes, i.e. the boundary layer thicknesses, are very robust with respect to the ansatz function (von Kármán Reference von Kármán1921; Pohlhausen Reference Pohlhausen1921; Schlichting Reference Schlichting1970).

4.1 Axial and swirl boundary layer thickness

Solving the generalized von Kármán equation (3.13) and (3.18) and the continuity equation (3.20) with the velocity profiles (4.1) and (4.2) and the wall shear stresses (4.3) and (4.4) with $n=1/7$ and $k=2$ numerically, the swirl boundary layer thickness, depending on axial coordinate, Reynolds number and flow number, is given by the lines in figure 2. The solution of axial boundary layer thickness is shown in figure 3 and, due to swirl, the axial boundary layer is thickened. For $\unicode[STIX]{x1D711}>1$ , the swirl-free solution $\unicode[STIX]{x1D6FF}(G=0)$ (black lines in figure 3) is approached asymptotically. For $\unicode[STIX]{x1D711}<1$ , a clear influence of swirl is observed. As indicated by the order-of-magnitude analysis of (3.8), the centrifugal force balances the wall shear stress. A new asymptote appears and dominates the axial boundary layer thickness. The interdependence of the axial boundary layer and the swirl is represented by an increase of the axial boundary layer thickness $g:=\unicode[STIX]{x1D6FF}-\unicode[STIX]{x1D6FF}(G=0)$ .

Figure 2. Swirl boundary layer thickness versus axial coordinate, Reynolds number and flow number for $R_{z}=0.04\,\%$ . Experiments and solution of the boundary layer theory. Description of the experimental set-up is given by § 5.

Figure 3. Influence of Reynolds number and flow number on the axial boundary layer thickness. Solution of the boundary layer theory.

4.2 Power laws of the boundary layer thicknesses

Throughout this paper, only the empirical constants of von Kármán $\unicode[STIX]{x1D705}=0.4$ , Blasius constant $c=8.74$ and $a=1.53$ are the fundamental empirical constants. All other calibrations are employed only for a short representation of the solution of the generalized von Kármán momentum equation. Analysing the slope of the boundary layer solution in the double-logarithmic diagrams (figure 2) for each parameter far away from the singularity at $z=0$ , a power law for the swirl boundary layer thickness is given,

(4.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{S}=C_{S}Re^{m_{1}}\unicode[STIX]{x1D711}^{m_{2}}z^{m_{3}},\end{eqnarray}$$

with $m_{1}=-0.487$ , $m_{2}=-0.492$ and $m_{3}=0.507$ . The outcome of a ‘power law’ for the swirl boundary layer thickness is obvious. We calibrate the constant $C_{S}$ to the achieved swirl boundary layer thickness from the generalized von Kármán momentum equations (3.13), (3.18) and (3.20) with the ansatz functions (4.1)–(4.4), yielding $C_{S}=6.39$ . This constant and the exponents $m_{1}\ldots m_{3}$ are independent of the Reynolds number, flow number and axial coordinate. Therefore, the exponents and the constant are very robust concerning changes of the ansatz functions, cf. appendix A and von Kármán (Reference von Kármán1921), Pohlhausen (Reference Pohlhausen1921) and Schlichting (Reference Schlichting1970). Applying equation (4.5) by (4.4), we get an approximation for the circumferential wall shear stress for a turbulent, hydraulic smooth flow,

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{y\unicode[STIX]{x1D719},w}=-\frac{2ak}{C_{S}}Re^{-(m_{1}+1)}\unicode[STIX]{x1D711}^{-m_{2}}z^{-m_{3}},\end{eqnarray}$$

and also the dimensionless hydraulic torque, which is equal to the shaft power and non-dimensionalized with $\tilde{\unicode[STIX]{x1D70C}}\tilde{R}^{3}\tilde{\unicode[STIX]{x1D6FA}}^{2}$ , depending on the length of the pipe $L$ (see figure 1),

(4.7) $$\begin{eqnarray}T=2\unicode[STIX]{x03C0}\int _{0}^{L}\unicode[STIX]{x1D70F}_{y\unicode[STIX]{x1D719},w}\,\text{d}z=-\frac{4\unicode[STIX]{x03C0}}{1-m_{3}}\frac{ak}{C_{S}}Re^{-(m_{1}+1)}\unicode[STIX]{x1D711}^{-m_{2}}L^{1-m_{3}}.\end{eqnarray}$$

Applying the method once again for the axial boundary layer thickness as was done for the swirl boundary layer thickness (4.5), the axial boundary layer thickness follows $g\propto \unicode[STIX]{x1D711}^{m_{7}}Re^{m_{8}}z^{m_{9}}$ and we obtain

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}=C_{1}Re^{m_{4}}\unicode[STIX]{x1D711}^{m_{5}}z^{m_{6}}+C_{2}Re^{m_{7}}\unicode[STIX]{x1D711}^{m_{8}}z^{m_{9}},\end{eqnarray}$$

with $m_{4}=-0.198$ , $m_{5}=-0.198$ , $m_{6}=0.790$ , $m_{7}=-0.969$ , $m_{8}=-2.94$ and $m_{9}=0.974$ . The values for the constants $C_{1}=0.414$ and $C_{2}=11.520$ are analogous to $C_{S}$ . The value of $\unicode[STIX]{x1D6FF}(G=0)$ , i.e. the first term on the right-hand side of (4.8), is close to the turbulent boundary layer on a flat plate. Again, the constants in (4.5) and (4.8) are to a large degree independent of the ansatz function. The relative accuracy of the power-law equations (4.5) and (4.8) to the numerical solution of the generalized von Kármán momentum equation (figures 2 and 3) is less than 10 %. The advantage of the power laws is that the differential equation system need not be solved to predict the boundary layer thicknesses.

This section presented the solution, i.e. the axial and swirl boundary layer thicknesses $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6FF}_{S}$ , of the differential equation system, (3.13), (3.18) and (3.20), using (4.1) and (4.2) for the velocity profiles with $n=1/7$ and $k=2$ and (4.3) and (4.4) for the wall shear stresses. In addition, this section gave power laws for $\unicode[STIX]{x1D6FF}$ and  $\unicode[STIX]{x1D6FF}_{S}$ .

5 Experiments and discussion

An apparatus is designed for the experimental validation of the swirl boundary layer thickness; see figure 4. With this apparatus, the swirl velocity distribution, the swirl boundary layer thickness and the flow separation depending on flow number, Reynolds number, axial position and surface roughness are investigated. Thereby, only the circumferential velocity component is measured by laser Doppler anemometry (LDA).

5.1 Experimental set-up

The air flow at room temperature and ambient pressure is provided by a radial fan, which increases pressure in a large plenum chamber with a flow resistance. The outlet of this chamber is followed by the first flow straightener, shown in figure 4 on the left side. Using this set-up, possible forced pulsations by the fan are minimized. The volume flow is measured by an orifice plate. Hence, the average flow velocity is given and is controlled by changing the rotation speed of the fan. The apparatus allows axial velocity up to $13~\text{m}~\text{s}^{-1}$ , resulting in a Mach number smaller than 0.1.

Figure 4. Configuration test rig.

The orifice plate is followed by a diffuser, the second flow straightener, three turbulence screens and a so-called Börger nozzle (Börger Reference Börger1973). Owing to the acceleration with the nozzle, the velocity profile is forced to be uniform across the inlet section of the rotating pipe. We measure the axial boundary layer of a thickness of $\tilde{\unicode[STIX]{x1D6FF}}/\tilde{R}=0.08{-}0.12\,\%$ at the inlet of the rotating pipe $(z=0)$ . For $\log (\unicode[STIX]{x1D711}Re)\geqslant 4.22$ the axial boundary layer is turbulent; the turbulence intensity has a value of approximately $Tu_{z}:=\tilde{u} _{rms}^{\prime }/\tilde{U} =1\,\%$ in the core region and $13\,\%$ in the boundary layer. The turbulence intensity slightly increases with increasing core velocity.

There is a small axial gap of $4\,\unicode[STIX]{x2030}\,\tilde{R}$ between the non-rotating and the rotating pipe. The rotating pipe with a diameter of 50 mm and a length of $5\tilde{R}$ is supported by sealed ball bearings and is driven by a belt. Thus, the gap is sealed as well. By doing so, a maximum angular speed of $\tilde{\unicode[STIX]{x1D6FA}}=1308~\text{s}^{-1}$ is reached and controlled. The rotating pipe is made of stainless steel with a surface roughness of $\tilde{R}_{z}/\tilde{R}=0.04\,\%$ . For the experiment we also coated the pipe with silicon carbide powder of different grain sizes to change the surface roughness. This resulted in three further relative roughnesses: $0.75\,\%$ , $1.24\,\%$ and $4.90\,\%$ . The air leaves the rotating pipe as a free jet. The advantage of this design is a convenient accessibility from downstream to the flow field within the rotating pipe.

This accessibility is used to measure the circumferential velocity component by one-dimensional LDA with frequency shift. The probe has a focus of $315$  mm and is located downstream of the rotating pipe at an angle of $12^{\circ }$ . By doing so, only the swirl component is measured and averaged over ${\geqslant}30$  s, weighted by the transit time and consisting of at least 500 velocity measurements. The measurement volume has a length of ${<}1.6\,\%\,\tilde{R}$ and a diameter of ${<}2\,\unicode[STIX]{x2030}\,\tilde{R}$ . The measurement volume can be moved in a two-dimensional plane by using a traverse table. The positioning accuracy of the measurement volume is approximated as $\pm 4\,\unicode[STIX]{x2030}\,\tilde{R}$ in the plane. With this LDA system, a measurement of the swirl velocity up to a wall distance of ${>}1.6\,\%\,\tilde{R}$ is possible. An aerosol of silicone oil as tracer particles is added to the air to enable LDA measurements; see figure 4.

Hence, all design and operation parameters are chosen so that typical Reynolds numbers $\log (Re)=4.1{-}5.1$ and flow numbers $\unicode[STIX]{x1D711}=0.05{-}1$ of the turbomachinery are met. The systematic error of the Reynolds number is less than $2\,\%$ and that of the flow number less than $3.6\,\%$ .

Figure 5. Measured self-similar swirl velocity profile versus axial coordinate, Reynolds number and flow number for attached flow for $R_{z}=0.04\,\%$ .

5.3 Surface roughness

On the basis of Nikuradse’s and Prandtl’s work we know that, when the Reynolds number of a non-swirling flow reaches a critical value, depending on surface roughness, both the velocity profile and also wall shear stress become independent of viscous friction, i.e. Reynolds number. Because up to this point we assumed hydraulically smooth walls, we examined the swirl boundary layer thickness at constant flow number $\unicode[STIX]{x1D711}=0.35$ , varying Reynolds number $\log (Re)=4.1{-}5.1$ for different pipe roughnesses. Figure 6 shows the measured swirl boundary layer thickness at $z=2$ . What is known from the axial boundary layer is also true for the swirl boundary layer: for Reynolds numbers below the critical number, the swirl boundary layer thickness is independent of roughness. When the critical value is exceeded, the swirl boundary layer thickness increases and ultimately becomes independent of the Reynolds number.

Figure 6. Influence of surface roughness on swirl boundary layer thickness versus Reynolds number.

Thus, the given analytical solutions with the chosen assumption of swirl velocity distribution and wall shear stress only represent the turbulent flow in a hydraulically smooth pipe. The extension of the presented method to rough surfaces is within the current research focus.

For safe operation of turbomachinery it is desirable to predict part load recirculation. To do so, the critical flow number $\unicode[STIX]{x1D711}_{c}$ is needed. The method validated so far can easily serve for Stratford’s criterion (Stratford Reference Stratford1959) to predict the stability limit $\unicode[STIX]{x1D711}_{c}(Re)$ . This is done in § 5.4.

Figure 7. Experimentally measured $\unicode[STIX]{x1D711}_{c}$ for fully separated flow at the inlet of a rotating pipe for the hydraulically smooth case.

5.4 Flow separation

In the following, we restrict ourselves to turbulent flow in a hydraulically smooth pipe. From the velocity measurements, we derive the desired stability limit $\unicode[STIX]{x1D711}_{c}=\unicode[STIX]{x1D711}_{c}(Re)$ . In parallel with our experiments, we identify the critical flow number $\unicode[STIX]{x1D711}_{c}$ for the fully separated flow, e.g. part load recirculation at the inlet of the rotating pipe. The stability map is shown in figure 7. In the core region, the fluid is accelerated, hence $\text{d}P/\text{d}z<0$ , whereas at the wall, the axial component of the wall pressure gradient $\text{d}p_{w}/\text{d}z$ can be positive due to swirl. From (3.7) for $y=0$ with (3.6) and (3.20), we obtain

(5.1) $$\begin{eqnarray}\frac{\text{d}p_{w}}{\text{d}z}=\frac{\text{d}}{\text{d}z}\int _{0}^{\unicode[STIX]{x1D6FF}_{S}}\frac{u_{\unicode[STIX]{x1D711}}^{2}}{1-y^{\prime }}\,\text{d}y^{\prime }-\frac{2U^{2}}{1-\unicode[STIX]{x1D6FF}_{1}}\frac{\text{d}\unicode[STIX]{x1D6FF}_{1}}{\text{d}z}.\end{eqnarray}$$

Hence, the swirl can cause a positive pressure gradient at the wall and also aids the growth of the axial boundary layer. Both effects provoke a separation of the axial boundary layer when a critical flow number $\unicode[STIX]{x1D711}_{c}(Re)$ is reached. To indicate an incipient separation, we assume the swirl-induced pressure slope as a superimposed pressure field for an axial boundary layer in a non-rotating pipe $G(=0)$ . Then, we use the criterion of Stratford (Reference Stratford1959) to determine the point of zero wall friction. Starting from equation (16)

(5.2) $$\begin{eqnarray}\left(\frac{y^{\ast }}{\unicode[STIX]{x1D6FF}^{\ast }}\right)^{2-4n}=\frac{3(n\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FD})^{4}}{n(n+1)\left(\unicode[STIX]{x1D6FF}^{\ast }{\displaystyle \frac{\text{d}C_{p}}{\text{d}z}}\right)^{2}},\end{eqnarray}$$

with equation (17)

(5.3) $$\begin{eqnarray}\left(\frac{u}{u^{\ast }}\right)^{2}=\frac{3n}{1+n},\end{eqnarray}$$

and equation (18) of Stratford (Reference Stratford1959)

(5.4) $$\begin{eqnarray}C_{p}=\left[1-\left(\frac{u}{u^{\ast }}\right)^{2}\right]\left(\frac{y^{\ast }}{\unicode[STIX]{x1D6FF}^{\ast }}\right)^{2n},\end{eqnarray}$$

where $y^{\ast }=y$ , $\unicode[STIX]{x1D6FF}^{\ast }=\unicode[STIX]{x1D6FF}(G=0)$ and $C_{p}=2(p^{\ast }-p_{0}(G=0))/\unicode[STIX]{x1D711}^{2}$ with

(5.5) $$\begin{eqnarray}p^{\ast }=\frac{1}{\unicode[STIX]{x1D6FF}_{S}}\int _{0}^{\unicode[STIX]{x1D6FF}_{S}}p(y,z)\,\text{d}y,\end{eqnarray}$$

$p$ is given by (3.7) and $p_{0}(G=0)=\text{d}p_{0}(G=0)/\text{d}z=0$ . By doing so, an alternative, equivalent pressure distribution within the swirl boundary layer is defined by (5.5), which is independent of the wall coordinate as is necessary for using Stratford’s criterion. To avoid overestimating the influence of the centrifugal force, the pressure distribution is area averaged within the swirl boundary layer by (5.5). Here $p^{\ast }$ represents the pressure within the outer layer, and the pressure gradient in (5.2) represents the pressure distribution within the inner layer. Therefore, the wall pressure gradient (5.1) is used and not the pressure gradient of $p^{\ast }$ . To balance the omitted higher-order term, Stratford introduced the empirical parameter $\unicode[STIX]{x1D6FD}$ . The wall shear stress $\unicode[STIX]{x1D70F}_{yz,w}$ vanishes for

(5.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}(G=0)}{\unicode[STIX]{x1D711}^{2}}\frac{\text{d}p_{w}}{\text{d}z}\left(\frac{p^{\ast }}{\unicode[STIX]{x1D711}^{2}}\right)^{(1-2n)/2n}=\frac{1}{2}(n\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FD})^{2}\left(\frac{1-2n}{2+2n}\right)^{(1-2n)/2n}\sqrt{\frac{3}{n(n+1)}}.\end{eqnarray}$$

The von Kármán constant appears because Stratford describes the Reynolds stresses with Prandtl’s mixing model to consider the profile transformation within the inner layer due to the pressure rise; see (11) of Stratford (Reference Stratford1959). With $\unicode[STIX]{x1D705}=0.4$ , $n=1/7$ and $\unicode[STIX]{x1D6FD}=0.66$ , this simplifies to

(5.7) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{c}=3.442\left(\unicode[STIX]{x1D6FF}(G=0)\frac{\text{d}p_{w}}{\text{d}z}p^{\ast 5/2}\right)^{1/7}.\end{eqnarray}$$

With equation (4.2) and $k=2$ , $\text{d}p_{w}/\text{d}z$ is given by (5.1), $p^{\ast }$ is given by (5.5) and $\unicode[STIX]{x1D6FF}(G=0)$ . When the zero wall shear stress condition is fulfilled, an incipient separation is predicted. In figure 8, the isoline for zero wall shear stress (5.7) is plotted versus the axial coordinate at $\log (Re)=4.7$ .

Figure 8. Zero wall shear stress at $\unicode[STIX]{x1D711}_{c}$ and measured separation.

One must be aware that the criterion $\unicode[STIX]{x1D70F}_{yz,w}=0$ indicates the beginning of flow separation. Hence, the associated $\unicode[STIX]{x1D711}_{c}$ given by (5.7) indicates the incipient separation. For a validation of separation criterion (5.7), we observe the swirl velocity profile as an indicator of the beginning of flow separation. When separation occurs, the swirl velocity profile within the swirl boundary layer differs from the self-similar swirl velocity profile of an attached flow (broken lines in figure 9) and a new distribution develops. At different axial positions, the swirl velocity profile is measured by changing the flow number. Hence, we determine the separation with the swirl velocity profile as an indicator depending on the axial coordinate, as shown in figure 9. We define separation when the measured swirl velocity profile differs on average by more than $15\,\%$ from the attached swirl velocity profile.

Figure 9. Measured swirl velocity profile depending on axial coordinate and flow number to indicate separation.

By reducing the flow number, the measurements show a transformation of the swirl velocity profile beginning further downstream. This transformation moves upstream as the flow number is reduced, until the swirl reaches the non-rotating pipe and indicates the fully developed separation at the critical flow number. The comparison of the separation criterion (5.7) is in qualitative agreement with the experimental findings, as figure 8 indicates.