2.1. Experimental set-up

A large octagonal-shaped tank made of G10 fibreglass with a volume of approximately 2 m³ is used for the present water-based experiments. The tank is fitted with a polished 1.6 m diameter glass disk centred in the tank, as shown in figure 1(a) and was filled with the water-based MTV solution up to a height of 25 $\text{cm}$ . To the authors’ knowledge, this RDTBL facility is the largest of its kind and is 60 $\,\%$ larger in diameter compared with the air-based facilities previously used by Littell & Eaton (Reference Littell and Eaton1994) and Itoh & Hasegawa (Reference Itoh and Hasegawa1994). This tank has eight side windows located at a radial distance of 1.3 m from the centre of the tank. The disk is rotated via a belt-driven shaft that links it to a three-phase induction motor. The angular velocity of the disk can be set within a resolution of 0.02 revolutions per minute (RPM) by adjusting the operating frequency of the motor using its variable frequency drive. An optical tachometer is employed to measure the angular velocity of the disk with a resolution of 0.01 RPM. In the present experiments, as mentioned in table 1, we have tested the angular velocity up to 33.35 RPM corresponding to the friction Reynolds number of $\delta ^{+}\cong 3100$ . Here, $\delta ^{+}=u_{\tau _\theta } \delta /\nu$ , where $u_{\tau _\theta }=\sqrt {\tau _{w_\theta }/\rho }$ is the tangential friction velocity and $\delta$ represents the local thickness of the boundary layer. The boundary layer thickness $\delta$ is taken as the wall-normal distance where the mean tangential velocity is equal to 1 % of the local velocity of the disk ( $U_w$ ). In the following sections, the present measurement techniques and experimental challenges associated with implementing them in this study are described.

Figure 1.

(a) A schematic view of the rotating disk facility. (b) Configuration of cameras to simultaneously record instantaneous radial and tangential velocities in MTV experiments.

2.2. Measurement technique: two-view MTV system

Molecular tagging velocimetry is a laser diagnostic technique that is used to non-intrusively measure flow velocities (e.g. Gendrich & Koochesfahani Reference Gendrich and Koochesfahani1996). This method is similar to particle image velocimetry (PIV), in which particle constellations are tracked in space to measure fluid velocities. In the MTV technique, the fluid is a dilute aqueous chemical solution with a homogeneous concentration of molecular tracer. Hence, given its molecular-based nature, it overcomes some drawbacks of the PIV technique, including the insufficient seeding density of the particles in the flow and the inconsistency between the mass density of the particles and that of the main fluid (Hu & Koochesfahani Reference Hu and Koochesfahani2006). The water-soluble molecular tracers used in the present adaptation of the MTV technique exhibit phosphorescence once they are excited by an energy source such as a pulsed ultra-violet laser. This phosphorescence lasts for several milliseconds, and tagged molecules can be followed as tracers to determine the velocity of the flow (Hu & Koochesfahani Reference Hu and Koochesfahani2006). The excited molecules are imaged twice during their moderate lifetime, and consequently, the velocity of the flow is estimated by using the Lagrangian displacement of the tracers and knowing the time difference between the two successive images. The supramolecular complex used in the present MTV experiments consists of three components: (i) 1-bromonaphthalene (with a concentration of $10^{-5} \text{M}$ ) which is the lumophore producing the phosphorescent effect; (ii) maltosyl- $\beta$ -cyclodextrin ( $2\times 10^{-4} \text{M}$ ) which facilitates the solubility of the lumophore within water; and (iii) cyclohexanol ( $0.05\,\text{M}$ ), a alcohol that reduces the quenching problem caused by oxygen in water molecules (Hu & Koochesfahani Reference Hu and Koochesfahani2006). Due to the low concentration of the MTV solution, its influence on the mass density and viscosity of water is negligible.

A pulsed ultra-violet Excimer laser (130 mJ/pulse) provided the energy required to photoexcite the molecular tracers. The laser beam wavelength is 308 nm and the laser was fired with a frequency of 15 Hz during the MTV experiments. The beam exiting the laser aperture was passed through a circular pinhole with a diameter of 1 $\text{mm}$ and was directed using three mirrors to enter the MTV solution from the top surface, tagging a straight line perpendicular to the disk (see figure 1(b)). The relatively large pinhole was needed to produce a beam of energy sufficient to penetrate the full depth of the MTV solution. The radial location of the tagged line was adjusted using a linear slider on which the last mirror was mounted. The tagged line is undeformed upon excitation and then becomes deformed due to the fluid motion. As depicted in figure 1(b), two pco.dicam C1 intensified 16-bit sCMOS cameras with 1504 $\times$ 1304 pixels were employed in an orthogonal configuration to simultaneously record the undeformed and deformed images in the radial ( $r$ ) and tangential ( $\theta$ ) directions. These images yielded the instantaneous $r$ and $\theta$ velocity profiles. Note that each camera acquires two images (with exposure of a few ms) for every laser pulse (i.e. every 1/15 sec), i.e. 15 velocity profiles per second. Each instantaneous velocity profile consisted of approximately 1200 equally spaced data points, i.e. one data point per pixel along the tagged line shown in figure 1(b). The wall-normal spacing varies between 1.25 and 3.15 viscous units for the range of Reynolds number investigated in the present MTV measurements. It is noted that the friction velocity used for the inner normalisation of the turbulent statistics is obtained from the Clauser chart method (Clauser Reference Clauser1956), assuming a log-law equation with $\kappa =0.41$ and $B=5$ (Coles Reference Coles1968). This is due to the insufficient resolution of measuring mean velocities within the viscous sublayer to directly extract the tangential friction velocity from the gradient of mean tangential velocity in this region, particularly at higher Reynolds numbers. It is worth noting that there is a negligible change ( $ \lt 0.4$ %) in friction velocity values obtained from the Clauser chart method if we consider $\kappa =0.384$ and $B=4.17$ for the fitting log-law equation (Nagib et al. Reference Nagib, Chauhan and Monkewitz2007) rather than $\kappa =0.41$ and $B=5$ . This agreement will become apparent later in our discussion of figure 3, as the two log laws convincingly coalesce within the bounds of the inertial sublayer with $\kappa =0.384$ showing a more extended log layer and a more canonical (i.e. similar to ZPG-TBL) feature.

2.3. Measurement technique: two-component LDV system

The two–component LDV system allows us to measure instantaneous tangential and wall-normal velocity components. LDV is a technique by which the velocity at a point in a flow is non-invasively measured by leveraging the Doppler effect (e.g. Zhang Reference Zhang2010). The present two-component LDV system is from TSI Inc. A central component of the present LDV system is an air-cooled 500 mW continuous-wave argon ion laser. The laser beam is directed into a fibrelight multicolour beam separator unit. Inside this unit, a Bragg cell divides the beam into two separate beams. One of these beams has its frequency shifted by 40 MHz, allowing for the directional sensitivity of the measured velocity. Following the initial separation, a prism is used to divide the shifted and unshifted beams further into six distinct beams, consisting of two green, two blue and two violet components. In the current two-component LDV system, only blue (with a wavelength of 488 $\text{nm}$ ) and green (with a wavelength of 514.5 $\text{nm}$ ) beams are used to measure the tangential ( $\theta$ ) and wall-normal ( $y$ ) velocities in the beam-crossing volume, respectively. Optical fibres guide the four beams towards the transmitting fibreoptic probe, where they converge onto an ellipsoidal measurement volume. In the present study, a lens with a 60 $\text{mm}$ clear aperture and 762 $\text{mm}$ focal length was mounted on the fibreoptic transceiver probe, enabling velocity measurements over a wide range of radial distances above the rotating disk. This lens provides a mean fringe spacing of 7.65 ${\unicode{x03BC}}\text{m}$ , a beam-crossing half angle of 1.88 $^\circ$ and a mean measurement volume diameter of 185 ${\unicode{x03BC}}\text{m}$ for the green and blue beams (TSI manual (2005)), which equals 18 viscous units at the highest Reynolds number investigated.

Proper flow seeding with tracer particles is crucial for accurate LDV velocity measurements. For this purpose, 10 ${\unicode{x03BC}}\text{m}$ silver-coated hollow glass sphere particles made by Dantec Dynamics A/S were used. A single-axis BiSlide Velmex traverse controlled by a programmable stepper motor was used to vertically move the fibreoptic probe to facilitate pointwise velocity measurements across the boundary layer. The travel distance of this traverse is 508 $\text{mm}$ with a resolution of 2 $\text{mm}\,\text{rev}^{-1}$ corresponding to a wall-normal increment of 0.005 $\text{mm}$ per step.

2.4. Experimental challenges

Most previous studies of the RDTBL used air. Using water poses several challenges. One challenge is to mimic the RDTBL on an infinite disk using a confined-domain flow facility. To help attain conditions that approximate an infinite domain, the bottom wall of the facility was fitted with an annular perforated plate. This decreases the momentum of the radial outflow from the disk. As shown in figure 1(a), the facility was also fitted with double-sided stainless steel mesh screens placed at each of the eight corners. These diminish any bulk flow rotation beyond the upper edge of the boundary layer and minimise the creation of secondary flows. After implementing these adjustments, the measured mean radial and tangential velocities beyond the boundary layer edge approach zero (as shown in figure 4 and the inset of figure 2, respectively) and the mean wall-normal velocity asymptotes to a constant value (as shown in figure 6 a), similar to the trends observed in the von Kármán laminar flow similarity solution of the infinite RDBL (von Kármán Reference von Kármán1921).

The disk was made of polished glass to create a hydraulically smooth surface at all Reynolds numbers. However, the reflection of light in the MTV experiments adversely affected the detection of the wall location from the MTV images. This issue was effectively resolved by painting the disk surface black. In the MTV experiments, the camera used to measure the tangential velocity was positioned perpendicular to the side window. In contrast, the camera used to capture the radial velocity needed to be angled relative to the perpendicular direction of the window. This angled configuration resulted in considerable uncertainty in the radial velocity measurements due to the refraction of captured light while passing through the window from the MTV line by the angled camera. To address the issue of light refraction, a small glass prism was built and attached to the side window of the tank, creating a new window perpendicular to the radial camera. The prism was then filled with the MTV solution, effectively eliminating the refraction problem.

The precision of velocity measurements near the wall in MTV experiments is negatively impacted by factors such as spatial averaging and laser beam attenuation as it penetrates deeper into the MTV solution. The beam thickness, which is between 20 and 50 viscous units based on the half-width of the laser beam within the range of Reynolds number investigated, results in a spatial averaging effect (Elsnab et al. Reference Elsnab, Monty, White, Koochesfahani and Klewicki2017), leading to smaller magnitudes for the measured tangential and radial velocity fluctuations compared with the corresponding DNS data. Disk wobble, which is most pronounced at smaller RPMs ( $\sim$ 1.5 $\text{mm}\,\text{rev}^{-1}$ at $\delta ^{+} = 1200$ ), is another important reality that affected the accuracy of the measurements close to the wall, especially in the LDV experiments. Namely, in the MTV experiments, an instantaneous velocity profile is rectified at each instantaneous rotation angle, and thus, the disk wobble effect is almost entirely removed. In the LDV experiments, however, this effect cannot be prevented as the measurement volume is fixed and results in relatively less accurate mean velocity measurements closer to the disk surface. Furthermore, the LDV technique cannot accurately measure the velocity at wall-normal distances less than the diameter of the measurement volume. In viscous units, this diameter varies between 8 and 18 for the Reynolds number range investigated in this study.

A standard approach to approximate the measurement uncertainty in the MTV technique is by acquiring a series of MTV images of the tracer solution at rest under the same imaging conditions of the experiments (Ramesh et al. Reference Ramesh, Klewicki and Philip2024). Therefore, the measurement uncertainty can be calculated as the root mean square of tracer displacement at rest condition. This measurement uncertainty depends on attributes of the imaging system, such as lens magnification, exposure periods of the undeformed and deformed images, as well as the interframe time delay. For a fixed magnification, the measurement uncertainty decreases when the exposure period and/or interframe time delay decreases. Thus, in the present experiments, the largest measurement uncertainty is associated with the lowest Reynolds number investigated, where we had to consider longer exposure periods and interframe time delay to reach sufficient tracer displacement for velocity measurements. The measured displacement uncertainty at this Reynolds number is estimated as 0.4 pixel, which corresponds to $\Delta U_{\theta }=0.0092$ m s⁻¹ ( $\Delta U_{\theta }^+=0.22$ ). This leads to $\Delta U_{\theta }/U_w=0.0097$ at this Reynolds number, which shows less than 1 % measurement uncertainty for the mean tangential velocity compared with the disk velocity.

Estimating the mean velocity uncertainty is relatively straightforward, but estimating derivative errors is much more complicated. As demonstrated by Elsnab et al. (Reference Elsnab, Monty, White, Koochesfahani and Klewicki2017), in the study of a turbulent channel flow, the measured mean velocity, its derivatives (up to second order) and the associated RSS profiles through the single-component MTV technique exhibited an exceptional agreement with available DNS data for $y^+ \gt 20$ across a range of Reynolds numbers (see their figures 8 and 9). For the wall-normal velocity derivatives from present MTV experiments, we followed the same procedure as in Elsnab et al. (Reference Elsnab, Monty, White, Koochesfahani and Klewicki2017). As present wall-normal spacings are comparable with those in Elsnab et al. (Reference Elsnab, Monty, White, Koochesfahani and Klewicki2017), e.g. present $\Delta y^+= 2.4$ at $\delta ^+=2230$ compared with $\Delta y^+= 2.1$ at $\delta ^+=1800$ in their study, as well as using an improved-quality imaging system, we retain similar confidence in the validity of the measured wall-normal derivatives for the present MTV data. For the wall-normal derivative of the RSS, we simply follow the same third-order polynomial Savitzky–Golay derivative-taking technique. Furthermore, the uncertainty levels are well below those that can affect the final conclusions we make from the study. Please note that we do not draw any conclusions that require a high level of derivative accuracy. In this study, as the first mean momentum and stress balance analyses on the RDTBL, we are more focused on overall trends of various terms in the MMB equations to contrast with those in the ZPG-TBL.

4.1. Similarity of radial gradients

To help clarify structural differences between the RDTBL and 2-D ZPG flow, as well as to calculate the MI terms in (4.3) and (4.4), as discussed above, we now seek their alternative formulations in the tangential and radial MMB equations. By applying the product rule, the MI $_{\theta _2}$ and MI $_{r_2}$ sub-terms can be decomposed into three components as

(4.5) \begin{align} \text{MI}_{\theta _2}=U_r^{+}\frac {\partial U_{\theta }^{+}}{\partial r^{+}}+U_{\theta }^{+}\frac {\partial U_{r}^{+}}{\partial r^{+}}+\frac { 2\,U_r^{+}U_{\theta }^{+}}{ r^{+}}, \\[-28pt] \nonumber \end{align}

(4.6) \begin{align} \text{MI}_{r_2}=-2\,U_r^{+}\frac {\partial U_r^{+}}{\partial r^{+}}-\frac {{U_{r}^{+}}^2}{r^{+}}+\frac {{U_{\theta }^{+}}^2}{r^{+}}. \\[0pt] \nonumber \end{align}

To estimate these sub-terms, we use the following similarity approximations for the derivatives with respect to the radial coordinate:

(4.7a) \begin{align} \frac {\partial U_{r}^{+}}{\partial r^{+}}=a\,\frac {U_{r}^{+}}{r^{+}} , \\[-28pt] \nonumber \end{align}

(4.7b) \begin{align} \frac {\partial U_{\theta }^{+}}{\partial r^{+}}=b\,\frac {U_{\theta }^{+}}{r^{+}}, \\[-12pt] \nonumber \end{align}

which are similar to the von Kármán similarity transformations for the rotating disk in the laminar regime where $a=b=1$ . It is noted that even in a turbulent flow over a rotating disk $b=1$ at the wall, where $U_{\theta }=r\,\varOmega$ by definition. The rotating disk flow is often called the von Kármán pump. The basic mechanics of the flow remain the same in the laminar and turbulent regimes. Namely, it is a fundamental property of the flow that the spinning disk draws fluid toward the wall and then ejects it radially. In the laminar regime $a=b=1$ . In the turbulent flow, the same basic structure exists, but it is modified. The change in the magnitude of $a$ and $b$ is a reflection of this modification in turbulent flow.

To simplify present MMB and stress analyses, we assume here that $a=b$ for the RDTBL away from the wall and its magnitude varies only minimally across the boundary layer, i.e. allowing $a$ and $b$ to be independent of $y^+$ . The justifications for the above approximations are examined below. These approximations stem from the inherent radial acceleration due to the presence of a dominating CF. Hence, substituting the radial gradients from (4.7a ) and (4.7b ) into (4.5) and (4.6), new formulations for the MI $_{\theta _2}$ and MI $_{r_2}$ sub-terms are derived as

(4.8) \begin{align} \text{MI}_{\theta _2}=\frac { 2\,(a+1)}{ r^{+}}\,U_r^{+}U_{\theta }^{+}, \\[-28pt] \nonumber \end{align}

(4.9) \begin{align} \text{MI}_{r_2}=\frac {{U_{\theta }^{+}}^2-(2a+1)\,{U_{r}^{+}}^2}{r^{+}}. \\[0pt] \nonumber \end{align}

To test the accuracy of the similarity assumptions in (4.7) and to estimate the value of $a$ , we substitute the $\partial U_{r}^+/\partial r^+$ approximation from (4.7a ) into the continuity equation for the RDTBL

(4.10) \begin{align} \frac {\partial U_y^{+}}{\partial y^{+}}+\frac {\partial U_r^{+}}{\partial r^{+}}+\frac {U_r^{+}}{r^{+}}=0. \end{align}

The mean inner-normalised form of the wall-normal velocity component can then be analytically estimated by

(4.11) \begin{align} U_{y}^{+}(y^+)= -\int _{0}^{y^+} \frac {(a+1)}{r^+} \, U_{r}^{+}(y')\,\text{d}y', \end{align}

where $y'$ is an arbitrary variable of integration and the negative sign denotes the wallward direction of the mean wall-normal flow. That is, in the RDBL, the CF ‘pumps’ the flow in the radial direction, which is coupled to the wall-normal flow towards the wall. Adjusting $a$ to equal 1.5 reveals that the integration of (4.11) closely resembles the mean wall-normal velocity profile measured via LDV at a similar Reynolds number as shown in figure 6(a) using solid lines. Note here that the mean radial velocity profile from the MTV measurements is used in (4.11) to extract the $U_y$ profile. The estimation (4.11) also matches the DNS data in figure 6(a) quite closely.

The integration of (4.7a ) with respect to $r^+$ yields the inner-normalised form of the mean radial velocity, as a function of $r^+$ and $y^+$

(4.12) \begin{align} {U_r}^{+}(y^+)=C(y^{+})\boldsymbol{\cdot }{r^{+}}^{\,a}. \end{align}

As shown in figure 4, the ${U_r}^+$ profile from MTV experiments at $Re=1047$ ( $\delta ^{+}=1870$ ) closely resembles the data of Itoh & Hasegawa (Reference Itoh and Hasegawa1994) at $Re=1000$ ( $\delta ^{+}\approx 1500$ ), even though they are measured at different radial distances, angular velocities and in different fluids. In Itoh & Hasegawa (Reference Itoh and Hasegawa1994), $r = 450$ $\text{mm}$ and $\varOmega = 720$ RPM, while $r=750$ $\text{mm}$ and $\varOmega = 20.13$ RPM for the present MTV case. Calculating $r^{+} = r\,u_{\tau _{\theta }}/\nu$ values for these two cases, we reach almost similar $r^{+}$ values of $4.35\times 10^{4}$ and $4.61\times 10^{4}$ , respectively. The small difference between $r^{+}$ values results from a slightly larger $Re$ value for the MTV case. Thus, these results suggest that $\delta ^{+}$ is a parameter needed to achieve a self-similar form of (4.12) as

(4.13) \begin{align} {U_r}^{+}(y^+)=C(y^{+},\delta ^{+})\boldsymbol{\cdot }{r^{+}}^{\,a}. \end{align}

To further investigate the validity of similarity assumptions (4.7) and expand its applicability, we now derive analytical estimations for tangential and radial friction velocities, as well as for the flow angle at the surface.

4.1.1. Estimation of tangential and radial friction velocities from MMB and similarity assumptions

By integrating the dimensional forms of MMB equations (4.3) and (4.4) from $y=0$ to $y\gt \delta$ (see also (5.2) and (5.3)) and using similarity assumptions in (4.7), analytical approximations for the tangential and radial friction velocities are derived as

(4.14) \begin{align} u_{\tau _{\theta }}^{2}= \frac {2\,(a+1)}{r}\int _{0}^{\infty }U_{r}\,U_{\theta }\,\text{d}y, \\[-28pt] \nonumber \end{align}

(4.15) \begin{align} u_{\tau _{r}}^{2}= \frac {1}{r}\int _{0}^{\infty }\left [U_{\theta }^2-(2a+1)\,U_{r}^2\right ]\,\text{d}y. \\[0pt] \nonumber \end{align}

As presented in table 2, the difference between the tangential friction velocities from the Clauser chart and the $\theta$ -MMB equation with similarity assumption (4.14) (i.e. the use of $a=1.5$ ) is within approximately 3 %. This finding further reinforces the validity of the similarity hypotheses of (4.7).

Table 2.

Estimation of friction velocities using mean tangential and radial velocity profiles from MTV experiments at $r=750$ $\text{mm}$ . The percentage difference $\Delta u_{\tau _{\theta }}$ is defined as the value obtained using the Clauser chart method minus the one from (4.14) and normalised by the Clauser chart value.

Table 2. Long description

A table with six rows and six columns comparing friction velocities using mean tangential and radial velocity profiles from MTV experiments. The columns are labeled with delta plus, u tau from Clauser chart in meters per second, u tau in meters per second, delta u tau in percent, u r in meters per second, and tan inverse of the ratio of u r squared to u tau squared in degrees. The rows are color-coded and labeled with different values of delta plus. Each row presents the values of u tau from Clauser chart, u tau, the percentage difference delta u tau, u r, and the surface flow angle beta calculated using the given formula.

One can also obtain the angle between the surface streamline and the tangential direction (i.e. $\beta$ ) using the radial and tangential friction velocities listed in table 2. The flow angle, tabulated in table 2 and presented in figure 5(b), is observed to decrease as the Reynolds number increases, consistent with the trend reported in the literature (e.g. Itoh & Hasegawa (Reference Itoh and Hasegawa1994), Littell & Eaton (Reference Littell and Eaton1994), Appelquist et al. (Reference Appelquist, Schlatter, Alfredsson and Lingwood2018b )). The present flow angles at $\delta ^+=1200$ and $1870$ are in good agreement with the experimental value of around $11^\circ$ at $\delta ^+\approx 1500$ reported by Itoh & Hasegawa (Reference Itoh and Hasegawa1994). The decreasing trend in flow angle is likely to be associated with the increasing flux of $\varOmega _{\theta }$ with increasing $r$ (i.e. with increasing $\delta ^+$ ) at a fixed $\varOmega$ , which will be discussed later in § 4.3. Furthermore, the use of the similarity assumption (4.7b ) correctly predicts that the mean inertia stress in the tangential direction approaches unity at $y=\delta$ , which will be shown in § 5.1. These empirical findings provide confidence in the similarity hypothesis.

4.2. Mean momentum balance structure in the tangential direction

The MI $_x$ and MI $_{\theta }$ terms, as seen respectively in (4.2) and (4.3), consist of two sub-terms. The contribution of each sub-term is indicated in figure 7, where dashed lines represent the contribution of the MI $_{x_1}$ and MI $_{\theta _1}$ sub-terms, dash-dotted lines denote MI $_{x_2}$ and MI $_{\theta _2}$ sub-terms and solid lines are the net MI contribution. Recall that in RDTBL (2-D TBL), the mean streamwise velocity is maximal (minimal) at the wall, and its magnitude decreases (increases) with increasing $y$ . As displayed in figure 7, the apparent difference between the MI distributions in the two flows is associated with the location of the peak in the MI profiles, which is in the outer and inner regions of the flat plate and RDTBLs, respectively. Furthermore, for the present range of Reynolds numbers, the amplitude of the MI peak in the RDTBL is nearly 3.5 times that of the ZPG-TBL. This results from the positive contribution of the mean radial velocity component in MI $_{\theta _2}$ distribution across the boundary layer, particularly close to the wall where the peak in the mean cross-flow velocity profile is located.

Figure 7.

The contribution of MI term and corresponding sub-terms to the streamwise/ tangential MMB structure of the 2-D ZPG and RDTBLs. Dashed and dash-dotted lines represent the contribution of MI $x_1$ (MI $_{\theta _1}$ ) and MI $_{x_2}$ (MI $_{\theta _2}$ ) sub-terms, respectively, as seen in (4.2) and (4.3), and solid lines are the net MI contribution.

It is seen that MI $_{x_1}$ is always negative in the ZPG flow and, hence, its net contribution $\int _{0}^{\infty } \text{MI}_{x_1}\,\text{d}y\neq 0$ . The situation is quite different in RDTBL, where $\int _{0}^{\infty } \text{MI}_{\theta _1}\,\text{d}y=0$ because $U_\theta$ is zero outside the boundary layer whereas $U_y=0$ at the wall. The zero integral condition for MI $_{\theta _1}$ implies that it must change sign within the boundary layer, which is observed in figure 7. Hence, the MI $_{\theta _1}$ acts as a sink of momentum close to the wall, while it plays as a momentum source after the peak location in $U_{\theta }^+U_{y}^+$ . Therefore, in the RDTBL, MI $_{\theta _1}$ has a momentum source–sink structure, which is not the case in the ZPG-TBL.

The contributions of the MI, VF and TI terms in (4.1) for the 2-D ZPG flow and (4.3) for the RDTBL are presented in figure 8(a). Overall, the VF and TI terms in the RDTBL exhibit trends similar to those observed in the 2-D ZPG flow, although their peak magnitudes in the inner region are slightly reduced. The primary distinction between the two flows lies in the MI term, which remains non-negligible in the inner region of the RDTBL, indicating the active role of the mean cross-flow velocity. Despite this difference, the dominant balance in the inner region of the boundary layer remains between the TI and VF terms, consistent with the ZPG-TBL. In the outer region, the balance shifts, with the MI and TI terms approximately counteracting each other. As shown in the inset of figure 8(a), both the magnitude of the MI term and the outer peak of the TI distribution in the RDTBL decrease with increasing $\delta ^{+}$ . This behaviour is consistent with the Reynolds number trend observed for the corresponding outer peaks in the MI and TI profiles of the ZPG-TBL.

Figure 8.

(a) Mean momentum balance structure in the streamwise direction. Solid lines denote MI (in red), TI (in black) and VF (in blue) terms in the RDTBL. Dashed, dash-dotted and dotted black lines represent the corresponding terms in the 2-D ZPG boundary layer derived from the DNS study of Sillero et al. (Reference Sillero, Jiménez and Moser2013) at $\delta ^{+}=1307$ . Plots with light colours show the experimental data, while dark lines represent DNS data. The vertical dash-dotted and dashed lines mark the bounds of layer III, as defined by Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005), for the RDTBL (in blue) and ZPG boundary layer (in black), respectively. (b) The distribution of $\textit{VF}/\textit{TI}$ vs. $y^+$ derived from the DNS data of Appelquist et al. (Reference Appelquist, Schlatter, Alfredsson and Lingwood2018b ) for RDTBL (blue symbols) at $\delta ^{+}\approx 900$ and Sillero et al. (Reference Sillero, Jiménez and Moser2013) for a 2-D ZPG boundary layer (black symbols) at $\delta ^{+}=1307$ .

Figure 8(b) depicts the layer structure deduced for the canonical 2-D wall flows (Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005), by plotting the distribution of $\textit{VF}/\textit{TI}$ as a function of $y^{+}$ for the 2-D ZPG flow at $\delta ^{+}=1307$ and the RDTBL at $\delta ^{+}\approx 900$ . It has been suggested that, with increasing Reynolds number, the extent of layer I diminishes, while layer II, defined by $|VF|\approx |TI|$ , progressively extends toward the wall (e.g. Klewicki et al. (Reference Klewicki, Ebner and Wu2011), Klewicki (Reference Klewicki2021)). Consistent with this expectation, no clear evidence of layer I is observed in either flow at the present Reynolds numbers.

Within layer II, $|\textit{VF}/\textit{TI}|\approx 1$ for the ZPG flow, indicating a balance between the VF and TI terms. In contrast, the magnitude of this ratio is slightly greater than unity for the RDTBL, suggesting that the excess contribution of the VF term is compensated by the MI term. The outer boundary of layer II is marked by the vertical dash-dotted blue (RDTBL) and black (ZPG) lines, corresponding to the location where $|\textit{VF}/\textit{TI}|$ first exceeds 2.

In layer III, all terms have substantive contributions, and the TI term crosses zero within this region, coinciding with the maximum RSS. This location shifts to a larger $y^{+}$ with increasing Reynolds number. The TI term acts as a momentum source before this zero crossing and as a sink thereafter (e.g. Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005), Klewicki et al. (Reference Klewicki, Fife, Wei and McMurtry2007)). The outer bound of layer III where $|\textit{VF}/\textit{TI}|$ falls below 0.5 is also Reynolds number dependent. This is indicated by the vertical dashed blue and black lines for the RDTBL and ZPG flows, respectively, following the criteria of Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005).

Finally, layer IV is characterised by a balance between the MI and TI terms in the outer region. The logarithmic region in $U_{\theta }^{+}$ is expected to reside within this inertial layer, whose extent scales with $\sqrt {\delta ^{+}}$ . This expectation is consistent with our data of $U_{\theta }^{+}$ in figure 3.

Note that, as expected, the data in the outer region of figure 8(a) do not collapse when plotted versus $y^+$ . Therefore, in Appendix A we present the data of figure 8 in outer units and plot against $y/\delta$ , and, as observed in figure 13, we do observe a collapse of data in the outer region.

4.3. Mean momentum balance structure in the radial direction

The distributions of the MI sub-terms in (4.4), after incorporating the similarity assumptions, are shown in figure 9(a). Similar to what is observed in MI $_{\theta _1}$ distribution, the MI $_{r_1}$ has also a momentum source–sink structure (i.e. $\int _{0}^{\infty } \text{MI}_{r_1}\,\text{d}y=0$ ). This is due to the mean zero radial velocity value beyond the boundary layer edge, similar to that in the mean tangential velocity profile. This feature generates a peak in the $-U_{r}^+U_{y}^+$ profile, and consequently leads to both positive (in the inner region) and negative (after the peak) contributions from MI $_{r_1}$ in the RDTBL, as seen in the inset of figure 9(a).

The CF ( $={U_{\theta }^{+}}^{2}/r^{+}$ ), which is a part of MI $_{r_2}$ , is the main contributor in the inner region, which is denoted by dashed lines in the plot. The CF maintains its dominance until nearly the wall-normal location where the two sub-terms intersect, which is within $y^{+}=100{-}200$ for both the DNS and MTV cases (see inset in figure 9(a)). Thereafter, MI $_{r_1}$ exceeds the MI $_{r_2}$ contribution to the MI $_r$ structure until the location where MI $_r$ approaches zero, which is around $y^{+}=300$ for the DNS plot. The MI $_{r_2}$ becomes negative slightly after the intersection point with MI $_{r_1}$ and remains negative for the remainder of the boundary layer. The negative MI $_{r_2}$ denotes that the contribution of the $-(1/r^{+})\partial (r^+{U_r^{+}}^2)/\partial r^{+}$ (i.e. radial variation of the radial momentum) is larger than the CF in this outer region (not surprisingly as the CF is due to the wall rotation that is only dominant in the near-wall region). The location where the MI $_r$ term crosses zero is seen to move outward as the Reynolds number increases from $\delta ^{+}\approx 900$ to $\delta ^{+}=2230$ . The distinct feature of the MI $_r$ term compared with MI $_\theta$ and MI $_x$ terms is that it plays as a sink of momentum in the outer region of the RDTBL.

Figure 9.

(a) Contribution of the MI term and its associated sub-terms to the radial MMB structure of the RDTBL. Dash-dotted and dotted lines represent MI $_{r_1}$ and MI $_{r_2}$ sub-terms, respectively, and solid lines are the net MI contribution. Dashed lines denote the CF contribution (i.e. ${U_{\theta }^{+}}^{2}/r^{+}$ ), which is a part of MI $_{r_2}$ . For more clarity, only profiles associated with the DNS data are shown in the inset. (b) The MMB structure of the RDTBL in the radial direction. Solid lines denote the MI (in red), TI (in black) and VF (in blue) terms in (4.4). The vertical dash-dotted and dashed black lines mark the bounds of layer II.

The MMB structure of the RDTBL in the radial direction is presented in figure 9(b). The MTV results at $\delta ^{+}=2230$ are shown alongside the corresponding terms extracted from the DNS study of Appelquist et al. (Reference Appelquist, Schlatter, Alfredsson and Lingwood2018b ) at $\delta ^{+}\approx 900$ . This MMB is classified into three layers based on the relative contributions of the terms in (4.4). In the inner region, the MI term is strongly positive and is expected to attain its maximum at the wall, where the CF is largest. As shown, the main balance is between the MI and VF terms in layer I. Although the TI term increases continuously from zero at the wall, it reaches a relatively small peak value of ${\sim} 0.006$ at $y^{+}\approx 6.5$ , where $|\textit{VF}/\textit{TI}|\approx 2.8$ . This peak is approximately one order of magnitude smaller than the corresponding TI peak in the streamwise MMB. It should be noted that the TI term in figure 9(b) is not obtained from experiments; therefore, its behaviour is inferred solely from available DNS data, while the MI and VF terms are experimentally measured.

In layer II, the balance evolves. In its inner part, the MI and TI terms together balance the VF term. The TI profile changes sign at approximately $y^{+}=20$ , beyond which it acts as a momentum sink alongside the VF term. In the outer part of this layer, both TI and VF balance the MI term. The outer boundary of layer II is defined as where $|\textit{VF}/\textit{TI}|=0.5$ , similar to the criterion used for the outer bound of layer III in the tangential MMB.

Layer III is characterised by a balance primarily between the MI and TI terms, while the VF term becomes negligible. In the initial portion of this layer, denoted as layer III $_a$ , the negative TI term balances the positive MI $_{r}$ term, similar to the behaviour observed in the $\theta$ -direction MMB. However, a distinct feature of the radial MMB is that the TI profile crosses zero again at approximately $y^{+}=270$ , where the MI and TI distributions intersect. Beyond this point, referred to as layer III $_b$ , the roles of the MI and TI terms reverse: the TI term acts as a momentum source, while the MI term becomes a sink, persisting up to the boundary layer edge. This behaviour differs from the conventional scenario in which turbulence predominantly drives the mean flow. The turbulent transport in the radial momentum (i.e. $d(-\overline {u_{r}u_{y}})/\text{d}y$ ) is more closely connected to the radial mean velocity $U_r$ , but forms only an indirect connection with $U_\theta$ (as described above via mass conservation, i.e. the ‘pumping effect’) and the reduction in $U_\theta$ wake. As discussed, centrifugal forcing and turbulent transport interact in layers III $_a$ and III $_b$ , and this mechanism is likely akin to that present in wall jets wherein the mean inertia owing to an inlet flow takes the role of the CF.

The MI and VF profiles obtained from the present experiments (shown in light red and blue) are in good agreement with the trends observed in the DNS data. The reduced magnitude of the outer negative peak in the experimental MI profile is attributed to the higher Reynolds number. Similarly, the smaller magnitude of the MI term in the inner region is consistent with the reduced contribution of the centrifugal term $({U_{\theta }^{+}}^{2}/r^{+})$ , which decreases as $r^{+}$ increases while ${U_{\theta }}^{+}$ remains relatively unchanged. Finally, the apparent uncertainty (jitter) in the outer region of the profiles in figure 9(b) arises from the differentiation of the mean radial velocity. This is due to the use of extracted DNS data from published figures, as the original datasets were not available.

The $r$ -MMB (4.4) considered thus far has an interesting interpretation at $y^{+}=0$ , where $\partial ^2 {U_r}^{+}/\partial {y^+}^{2}=-\,{U_{\theta }^{+}}^2/r^{+}$ , meaning that the VF term identically balances the CF term at the disk surface. At the wall, the mean tangential vorticity ( $\varOmega _{\theta }$ ) equals $\partial U_r/\partial y$ , and evaluation of the $r$ -MMB equation at $y^{+}=0$ yields $\mu \,\partial \varOmega _{\theta } /\partial y=-\,\rho {U_{\theta }}^2/r$ . It, thus, can be said that the centrifugal acceleration at the surface results in a continuous flux of $\varOmega _{\theta }$ into the flow. Knowing that $U_\theta =r\,\varOmega$ at the wall and $\nu =\mu /\rho$ , the above equation can be rewritten as

(4.16) \begin{align} \nu \,\frac {\partial \varOmega _{\theta }}{\partial y} =-\,r\,\varOmega ^2. \end{align}

Equation (4.16) indicates that the flux of mean tangential vorticity at the wall is proportional to the radial distance and increases linearly with $r$ . The flux of mean tangential vorticity per unit length around a constant $r$ -circuit on the disk surface, however, remains fixed. In this regard, the CF in the RDTBL exhibits an accelerative mechanism similar to the effect of the pressure gradient in the streamwise direction of the pressure-gradient boundary layers. This is shown by rewriting (4.1) at $y^+=0$ as

(4.17) \begin{align} \underbrace {\frac {\partial ^2 U^{+}}{\partial {y^+}^{2}}}_{\text{VF}}=\underbrace {{-U^{+}_{\infty }}\frac {\partial {U^{+}_{\infty }}}{\partial x^{+}}}_{\text{PG}}, \text{or}\, \nu \,\frac {\partial \varOmega _{z}}{\partial y}=U_{\infty }\frac {\partial {U_{\infty }}}{\partial x}. \end{align}

We now draw some comparisons between alike terms in the radial and tangential MMB equations. In both equations, the VF term is dominant in the inner region, but becomes subdominant in the inertial layer. The VF term becomes subdominant at much smaller $y^+$ in the $r$ -MMB than in the $\theta$ -MMB. The peak in the VF profile of the $\theta$ -MMB equation is located at a $y^{+}$ position nearly coincident with that in the ZPG-TBL ( $y^{+}\approx 7$ ). In contrast, the VF term is maximal on the wall in the $r$ -MMB equation owing to its exact balance with the CF. The peak value of the VF term in the $\theta$ -MMB equation is nearly 2.5 times that in the $r$ -MMB equation. The MI term, however, shows a different trend compared with the VF term. Similar to the VF term, the MI term is also larger in the inner region of the $r$ -MMB. The peak value of the MI term in the radial direction is almost 5 times that in the tangential equation, which is due to the significant contribution of the CF in the $r$ -MMB equation. The CF also causes the MI profile to have its maximum on the wall, where the CF is highest. As mentioned earlier, the TI term is subdominant in the $r$ -MMB equation. Its peak is more than one order of magnitude smaller than that in the TI profile of the $\theta$ -MMB equation. Importantly, throughout the inertial layer (MI $\approx$ TI) of $\theta$ -MMB, the MI term is the source and the TI term is the sink. In the $r$ -MMB, after a relatively small region where MI/TI are source/sink, the behaviour of MI/TI switches. For most part of the outer region the TI now acts as a source and MI as the sink of radial momentum. Overall, the radial and tangential MMB equations both provide leading-order contributions to the mean dynamics of the RDTBL, and thus the variations of the terms in the radial MMB cannot be neglected.

5.1. Mean stress budget in tangential direction

We now compare the mean stress budgets of the 2-D ZPG and RDTBL flows in the streamwise direction. The once-integrated form of the streamwise MMB equation (4.1) for 2-D pressure-gradient TBLs from the wall to a given $y^+$ location is given by

(5.1) \begin{align} \underbrace {\frac {\partial U^{+}}{\partial y^+}}_{\text{VS}=\int \text{VF}}+\underbrace {[-\overline {uv}^{+}]}_{\text{RSS}=\int \text{TI}}+\underbrace {\overbrace {(-U^{+}V^{+})}^{\text{MIS}_{x_1}}+\overbrace {\left (-\frac {\partial }{\partial x^{+}}\int _{0}^{y^+}{U^{+}}^2\,\text{d}y'\right )}^{\text{MIS}_{x_2}}}_{\text{MIS}_x=\int {\text{MI}_x}}+\underbrace {{U^{+}_{\infty }}\frac {\partial {U^{+}_{\infty }}}{\partial x^{+}}\,y^+}_{\text{PS}=\int \text{PG}}=1, \end{align}

where $y'$ is an arbitrary variable of integration. The viscous stress (VS) term is the integrated form of the VF term and equals unity on the wall. This term monotonically decreases across the boundary layer and becomes zero at the edge of the boundary layer. The RSS term is positive across the boundary layer, becoming maximal at a wall-normal location, $y^{+}_m$ , that is, in general relatively close to the wall. In 2-D APG TBLs, however, the location of the RSS peak moves outward compared with that in ZPG and FPG TBLs (e.g. Romero et al. (Reference Romero, Zimmerman, Philip, White and Klewicki2022a ,Reference Romero, Zimmerman, Philip and Klewicki b )). The MIS term develops differently depending on the type of flow. This difference is investigated in this section. The pressure stress (PS) term increases linearly across the boundary layer in flows with a constant pressure gradient. It should be noted that, regardless of the type of flow, all the terms on the left side of (5.1) sum to unity at any wall-normal location within the boundary layer.

The behaviour of each term in (5.1) is somewhat predictable in the canonical TBLs, except for the MIS term. In this section, the behaviour of each sub-term of the MIS term is investigated for the 2-D ZPG and RDTBLs with a focus on the distinct role of the mean cross-flow velocity. This sheds some light on how the MIS modifies the development of the primary RSS. Similar to (5.1), the stress balance equation for the RDTBL (after integrating (4.3)) is given by

(5.2) \begin{align} \underbrace {-\frac {\partial U_{\theta }^{+}}{\partial y^+}}_{\text{VS}=\int \text{VF}}+\underbrace {[\overline {u_{\theta }u_{y}}^{+}]}_{\text{RSS}=\int \text{TI}}+\underbrace {\overbrace {(U_{\theta }^{+}U_{y}^{+})}^{\text{MIS}_{\theta _1}}+\overbrace {\frac {1}{{r^{+}}^2}\frac {\partial }{\partial r^{+}}\left [{r^{+}}^2\,\int _{0}^{y^+}U_r^{+}U_\theta ^{+}\,\text{d}y'\right ]}^{\text{MIS}_{\theta _2}}}_{\text{MIS}_{\theta }=\int \text{MI}_{\theta }}=1, \end{align}

where $y'$ is the variable of integration. Note that taking $y^+\rightarrow \infty$ in (5.2) reduces it to the von Kármán equation (1.1) as the first three terms in (5.2) vanish. The viscous-scaled distributions of the MIS term and its two sub-terms are presented in figure 10(a). The MIS $_{x_1}$ (in 2-D ZPG-TBL) and MIS $_{\theta _1}$ (in the RDTBL) sub-terms, represented by the dashed lines, have negative contributions to the overall MIS profile structure in these TBLs. The shapes of these profiles are, however, quite different. The MIS $_{x_1}$ profile undergoes a continuous increase in magnitude throughout the boundary layer. A peak is, on the other hand, observed in the outer region of the MIS $_{\theta _1}$ distribution. The amplitude of the peak does not appear to change with Reynolds number, but its location moves farther from the wall. The second sub-terms of MIS show continuous increase across the boundary layer in both ZPG and RDTBLs. Their magnitudes increase from zero at the wall and asymptote to constant values (unity in the RDTBL) at the edge of the boundary layer. The MIS $_{\theta _2}$ undergoes a sharp increase near the wall as both $U_r$ and $U_\theta$ are significant in this region. In contrast, MIS $_{x_2}$ experiences a sharper increase in the outer region. As shown in figure 10(a), near $y=\delta$ the MIS $_{\theta _1}$ sub-term approaches zero while the MIS $_{\theta _2}$ sub-term, which results from the net $\theta$ -momentum ( $\rho U_{\theta }$ ) transported by $U_r$ , is balanced by the total $\theta$ -drag on the wall (as illustrated by the von Kármán equation (1.1)). The differences noted in the distributions of the two sub-terms result in a larger value of the MIS term across the RDTBL compared with ZPG flow at comparable Reynolds numbers. Note that, as derived earlier, MIS $_{\theta _2}$ is evaluated using the similarity hypothesis, and the fact that it correctly approaches unity as $y^+\rightarrow \infty$ provides credence to the hypothesis.

Figure 10.

(a) The contribution of MIS term and corresponding sub-terms to the streamwise/tangential stress balance structure of the 2-D ZPG and RDTBLs. Dashed line and dashed line with asterisk symbols represent the contribution of MIS $_{x_1}$ (MIS $_{\theta _1}$ ) and MIS $_{x_2}$ (MIS $_{\theta _2}$ ) sub-terms, respectively, and solid lines are the net MIS contribution. (b) Stress balance structure in streamwise/tangential direction. Solid, dashed and dash–dotted lines denote the MIS, VS and RSS terms, respectively. The circle symbol represents the present RSS in the RDTBL measured via the LDV technique at $\delta ^{+}=2230$ . Black and red lines show the stress terms in the 2-D ZPG boundary layer derived from the DNS study of Sillero et al. (Reference Sillero, Jiménez and Moser2013) at $\delta ^{+}=1307$ and the DNS data of Appelquist et al. (Reference Appelquist, Schlatter, Alfredsson and Lingwood2018b ) for RDTBL at $\delta ^{+}\approx 900$ , respectively. The horizontal dashed blue line marks the upper bound of unity for the MIS term in both TBLs.

Figure 10(b) presents the inner-scaled variation of all three terms (VS, RSS and MIS) in the overall stress budget for both 2-D ZPG-TBL (5.1) and RDTBL (5.2). In either flow, the VS profile shows a highly similar trend with a maximum value of 1 at the wall and a continual decrease to zero at the boundary layer edge. The VS data close to the wall are missing for the RDTBL due to the unreliable measured velocities in the buffer region and the lack of access to the original DNS data of Appelquist et al. (Reference Appelquist, Schlatter, Alfredsson and Lingwood2018b ) at $\delta ^{+}\approx 900$ . We know, however, that by definition, VS $=1$ at $y^{+}=0$ . As discussed above, the MIS value in the RDTBL is larger than in the ZPG case at comparable Reynolds numbers, although its magnitude decreases as the Reynolds number increases. The larger MIS term in the RDTBL is accompanied by an altered structure of the RSS in this flow, resulting in the generation of a sharper but smaller magnitude peak in the RSS profile, see figure 10(b). This peak broadens as the Reynolds number increases. The RSS values beyond $y_m$ decrease more gradually compared with the RSS profile in the canonical TBL.

5.2. Mean stress budget in radial direction

The integration of (4.4) from the disk surface to a given wall-normal location $y^+$ yields the stress balance in the radial direction

(5.3) \begin{align} & \underbrace {\frac {\partial U_{r}^{+}}{\partial y^+}}_{\text{VS}\,=\,\int \text{VF}}+\underbrace {[-\overline {u_{r}u_{y}}^{+}]}_{\text{RSS}\,=\,\int \text{TI}} \nonumber \\ & +\underbrace {\overbrace {(-U_r^{+}U_y^{+})}^{\text{MIS}_{r_1}}+\overbrace {\left [-\frac {1}{r^+}\frac {\partial }{\partial r^{+}}\left (r^+\int _{0}^{y^+}{U_r^{+}}^2\,\text{d}y'\right )+\frac {1}{r^{+}}\int _{0}^{y^+}{U_{\theta }^{+}}^2\,\text{d}y'\right ]}^{\text{MIS}_{r_2}}}_{\text{MIS}_{r}=\int \text{MI}_{r}}=u^2_{\tau _{r}}/u^2_{\tau _{\theta }}, \end{align}

where all terms have been inner-normalised using the tangential friction velocity $u_{\tau _{\theta }}$ . The sum of terms on the left side of the above equation, therefore, equals $u^2_{\tau _{r}}/u^2_{\tau _{\theta }}$ (or unity in case of inner normalising using $u_{\tau _{r}}$ ) at any wall-normal location. It is noted that the radial stress balance (5.3) becomes the von Kármán equation (1.2) as $y^+\rightarrow \infty$ with the first three terms in (5.3) vanishing (see also Alfredsson et al. (Reference Alfredsson, Kato and Lingwood2024)). Figure 11 shows the distribution of terms in (5.3) across the boundary layer for the DNS data of Appelquist et al. (Reference Appelquist, Schlatter, Alfredsson and Lingwood2018b ). Due to the low resolution of the experimental data close to the wall and the lack of $-\overline {u_{r}u_{y}}^{+}$ data in the present experiments, only the DNS data of Appelquist et al. (Reference Appelquist, Schlatter, Alfredsson and Lingwood2018b ) are examined here. The radial VS term in (5.3) is maximal at the disk surface. It undergoes a decrease and becomes zero at the peak location of the mean radial velocity profile. This term then experiences small negative values and approaches zero beyond the boundary layer edge. As shown in figure 11, the cross-flow induced RSS (i.e. $-\overline {u_{r}u_{y}}^{+}$ ) changes sign nearly at the same location VS crosses zero, where MIS $\approx u^2_{\tau _{r}}/u^2_{\tau _{\theta }}$ . Itoh & Hasegawa (Reference Itoh and Hasegawa1994) also reported that this RSS component in the RDTBL changes sign where the gradient of the mean radial velocity profile becomes zero. It is evidenced here that the mean radial velocity governs the cross-flow induced RSS. A positive $-\overline {u_{r}u_{y}}^{+}$ is located near the wall where $\partial U_r/\partial y \gt 0$ and $-\overline {u_{r}u_{y}}^{+}\lt 0$ in the outer region corresponding to $\partial U_r/\partial y \lt 0$ . Hence, the stress balance in the radial direction can be divided into two sections based on the peak location in the mean radial velocity profile. Before the peak, all terms contribute to the mean dynamics of the flow in radial direction, whereas the MIS and RSS terms balance each other beyond the peak, and VS is nearly zero in this region, as seen in figure 11. The corresponding RSS component is zero across the ZPG-TBL.

Figure 11.

Stress budget in the radial direction. Solid lines represent the MIS term (in red), VS term (in blue) and RSS term $-\overline {u_{r}u_{y}}^{+}$ (in black) derived from the DNS study of Appelquist et al. (Reference Appelquist, Schlatter, Alfredsson and Lingwood2018b ) at $\delta ^{+}\approx 900$ . The dashed lines indicate the extrapolated portion of the data, extending to the bounds of the boundary layer. The horizontal blue line denotes $u^2_{\tau _{r}}/u^2_{\tau _{\theta }}$ ratio which equals the tangent of the flow angle ( $\beta =17^\circ$ ) at the disk surface reported by Appelquist et al. (Reference Appelquist, Schlatter, Alfredsson and Lingwood2018b ) at $\delta ^{+}\approx 900$ .

5.3. Quadrant analysis

As described in § 5.1, the primary RSS ( $\overline {u_{\theta }u_{y}}$ ) profile in the RDTBL differs from that in the 2-D TBL ( $-\overline {uv}$ ), especially regarding the peak magnitude and the profile shape in the outer region. This motivates a closer look at the RSS-producing events via quadrant analysis. The $Q2(-u,+v)$ and $Q4(+u,-v)$ quadrants are responsible for negative-RSS-producing events associated with the so-called ejection and sweep events (e.g. Corino & Brodkey (Reference Corino and Brodkey1969), Wallace et al. (Reference Wallace, Eckelmann and Brodkey1972), Wallace (Reference Wallace2016)). These two quadrants are the main contributors to the structure of the primary RSS in canonical turbulent flows. For the RDTBL flow in laboratory coordinates, the mean streamwise velocity is maximal on the wall, in contrast to its zero value in 2-D TBLs. Hence, plotting the velocity fluctuations in a typical way with $u_{\theta }$ on the abscissa and $u_y$ on the ordinate, ejection and sweep events are associated with $Q1$ and $Q3$ quadrants in RDTBL flow, as similarly noted by Littell & Eaton (Reference Littell and Eaton1994).

Figure 12.

(a) Inner-scaled and (b) fractional quadrant contributions to the primary RSS ( $\overline {u_{\theta }u_{y}}$ ) (or $-\overline {uv}$ in canonical flows) against the viscous-scaled wall-normal distance. Blue and black symbols represent the RDTBL data from the LDV experiment at $\delta ^{+}=1930$ and 2-D ZPG data from the hot-wire study conducted by Morrill-Winter et al. (Reference Morrill-Winter, Philip and Klewicki2017) at $\delta ^{+}=2400$ . Solid red line represents the logarithmic fit to the RSS data in the RDTBL and is given by $\overline {u_{\theta }u_{y}}=-1/4.54 \,\text{ln}\,(y^+)+1.76$ . Dashed lines represent DNS data of turbulent channel flow from Moser et al. (Reference Moser, Kim and Mansour1999) at $\delta ^{+}=590$ and the vertical dashed magenta line denotes the location of the peak in the mean radial velocity profile from MTV experiments of RDTBL at a similar Reynolds number.

Figure 12 presents the inner-normalised and fractional quadrant contributions to the primary RSS ( $\overline {u_{\theta }u_{y}}$ ) of the RDTBL from LDV experiments at $\delta ^{+}=1930$ . These are compared with the corresponding RSS ( $-\overline {uv}$ ) data of Morrill-Winter et al. (Reference Morrill-Winter, Philip and Klewicki2017) in a 2-D ZPG boundary layer at $\delta ^{+}=2400$ . The sum of the four quadrant contributions is equal to the total RSS ( $\overline {u_{\theta }u_{y}}$ or $-\overline {uv}$ ). The vertical dashed magenta line in figure 12 marks the location of the peak in the mean radial velocity profile. Consistent with the overall lower value of the primary RSS, it is seen in figure 12(a) that the contribution of each quadrant in the RDTBL is numerically smaller than the corresponding quadrant in the ZPG-TBL. The distribution of the negative-RSS-producing events (i.e. $Q2$ and $Q4$ ) in the ZPG flow undergoes a plateau-like behaviour beyond $y^+_m$ , while a steeper decrease is observed for the corresponding events (i.e. $Q1$ and $Q3$ ) beyond their peaks in the RDTBL. Similar behaviours are observed for the positive-RSS-producing events. The approximate logarithmic decay beyond the RSS peak in the RDTBL has similarities to what is observed in the turbulent sink flow as reported by Araya et al. (Reference Araya, Castillo and Hussain2015). This fundamental difference results in creating a RSS profile with a sharper peak in the RDTBL compared with the ZPG, which has a flatter peak that spans a considerable wall-normal distance. Also, it is the relatively larger reduction in sweeps and ejections (compared with the other two quadrants) in the RDTBL that results in a lower RSS compared with the 2-D ZPG-TBL.

As shown in figure 12(b), however, the fractional contributions to each quadrant in the RDTBL closely resemble the corresponding fractional contributions in the 2-D ZPG flow. It can be implied that 2-D ZPG and RDTBLs are similar in terms of having nearly equivalent probabilities of observing sweep and ejection events, but these events are modified in the RDTBL. Here, the inherent three-dimensionality (from $U_r$ ) of this flow is the likely cause (see figure 10(b)). The fractional quadrant contributions in the RDTBL are also compared with the DNS data of the turbulent channel flow performed by Moser et al. (Reference Moser, Kim and Mansour1999) at $\delta ^{+}=590$ , shown with dashed lines in figure 12(b), which extend close to the wall. The DNS results show that for $y^{+}\lt 15$ , the contribution of $Q4$ (sweep events) is noticeably larger than that of $Q2$ (ejection events), and for $y^{+}\gt 15$ , the opposite is true. A similar behaviour is seen in the RDTBL, but the intersection of $Q1$ and $Q3$ (equivalent to $Q2$ and $Q4$ in canonical wall flows) is located around $y^{+}=30$ , slightly lower than the peak location in the mean radial velocity profile. Subsequently, the magnitude of both $Q1$ and $Q3$ starts to decrease, which corresponds to the plateau seen in the fractional quadrant contribution profile in figure 12(b) and extends to the edge of the boundary layer. The fractional contributions of the quadrants in the RDTBL closely match those of turbulent channel flow. It is noted that $Q2$ and $Q4$ contribute to RSS almost equally across the flow in both the channel and the RDTBL, except in the immediate vicinity of the wall.