3.1 The general picture

Figure 3. Mean velocity profiles of all test cases.

Figure 4. Mean velocity profiles of all test cases in wall coordinates.

Figures 3 and 4 show the velocity profiles of all the test cases in linear global coordinates and semi-logarithmic wall coordinates, respectively. With the increase of the body force ( $F$ ), through either increasing the coverage or the amplitude, the general trends of the variations of the velocity profiles in each of the groups are similar to each other. We focus on group B to elucidate such trends but comment on the differences between the groups as appropriate. With a relatively low body force (cases B1 to B3), the velocity profile is distorted only slightly, becoming flattened in the core of the flow. The centreline velocity becomes lower than that of the reference case, and the velocity around say $y=0.3$ becomes higher. The velocity gradient near the wall is always lower than that of the reference flow, and decreasing with the increase of the body force. This is however not always the case in other groups. For example, the velocity gradient adjacent to the wall in A1 and C1 is higher than in the reference flow, and hence under such conditions the friction factor is increased in comparison with the reference flow even though the flow is partially laminarised (see later). Further increasing the body force, the velocity profiles suddenly switch to an M-shape, which is typical of mixed convection with a strong influence of buoyancy (e.g. Steiner Reference Steiner1971; Carr et al. Reference Carr, Connor and Buhr1973; You et al. Reference You, Yoo and Choi2003). With the increase of the body force, the velocity in the centre of the pipe reduces and the velocity gradient adjacent to the wall increases, and as a result, the friction factor also increases. The variation of the velocity profiles with increase of the body force follows a similar trend in group D, but in groups A and C, the velocity gradient near the wall remains largely the same in the various test cases. That is, the overall friction is insensitive to the increase of the imposed body force under such conditions.

Figure 4 shows how far the velocity profiles can still be described by the generic logarithmic distribution. For the weaker body-force-influenced cases (B1 to B3), the velocity is upshifted with increase of the body force, showing increasingly stronger laminarisation. There is always a region of logarithmic distribution (linear variation in the semi-log plot) in each case, but the gradient becomes steeper with increase of the body force. In consistent with the observation in figure 3, the profiles of cases A1 and C1 are actually shifted downwards as a result of the increase of the friction velocity. The plots of the M-shaped velocity profiles have little meaning in this style of presentation, except that it does show that for most cases, the profiles follow the ‘standard’ curve up to $y^{+}=15$ , and then either curve up or down first before going downwards in the core of the flow.

Figure 5. Distribution of turbulent shear stress in all test cases.

Figure 6. Distribution of turbulent kinetic energy in all test cases.

To illustrate the effect of the body force on turbulence, the turbulent shear stress ( $\overline{u_{z}^{\prime }u_{r}^{\prime }}$ ) and kinetic energy ( $k$ ) are shown in figures 5 and 6, respectively. When the body force is relatively weak, say, B1 to B3, both turbulent shear stress and kinetic energy reduce progressively with increase of the body force. In fact, the flow is effectively completely laminarised in B3 (and A3 and C2), where both $\overline{u_{z}^{\prime }u_{r}^{\prime }}$ and $k$ are negligibly small. With a further increase of the body force, negative shear stresses are generated in the core of the flow, and the magnitude of which increases with the increase of the body force. At the same time, positive shear stress also starts to appear in the wall region. The peak value increases with increase of the body force, and its location moves closer to the wall. The production of negative turbulent shear stress stems from the velocity gradient of the M-shaped profile in the core region, which has been observed by various researchers previously (e.g. Polyakov & Shindin Reference Polyakov and Shindin1988; Bae et al. Reference Bae, Yoo and Choi2005). Along with the increase of $\overline{u_{z}^{\prime }u_{r}^{\prime }}$ , turbulent kinetic energy also increases with increase of the body force. There are two peaks in the distribution of $k$ , one is clearly associated with the wall layer (the outer legs of the M) and the other is associated with the inner flank of the M-profile (the core of the flow), linked to the negative turbulent shear stress.

According to the effect on turbulence, the body-force-aided flows can be grouped into three categories (regimes), that is, partially laminarised (or, laminarising), fully laminarised or recovery flows. In a partially laminarised flow, turbulent kinetic energy and shear stress are both reduced in magnitude, and the reduction becomes stronger with the increase of the body force. The velocity profile is flattened in the core of the flow, but is overall only slightly distorted from that of the reference flow. In the recovery regime, the velocity takes an M-shape, where the inner flank of the M profile provides the reason for the generation of turbulence in the core of the flow. The stronger the body force, the stronger the turbulence becomes. In between the above two scenarios, turbulence completely or largely disappears. In such a fully laminarised flow, the velocity profile can take either of the above shapes.

The above results show a comprehensive picture of how the flow and turbulence vary with a systematically varied body force imposed through varying the amplitude or the coverage. It is interesting to note that a step-change body force can achieve a similar result as a linearly varying body force, and that increasing the amplitude and the coverage can also achieve similar results. The present results with systematically varied body force distributions of two distinct profiles provide an opportunity to gain new insights into how the body force modifies the flow and turbulence, which is discussed later in this paper. An inevitable limitation of previous studies on mixed convection is that the range of conditions covered is always limited. A complete picture such as that shown above has not previously been observed in a single study. The measurements of Shehata & McEligot (Reference Shehata and Mceligot1998) and DNS of Satake et al. (Reference Satake, Kunugi, Shehata and Mceligot2000) and Bae et al. (Reference Bae, Yoo, Choi and Mceligot2006) were limited to flows in the laminarising regime. The measurements of Carr et al. (Reference Carr, Connor and Buhr1973) and the DNS of You et al. (Reference You, Yoo and Choi2003) observed both types of the velocity profiles, but their strongest buoyancy flows saw only very small recovery of turbulence. In Bae et al.’s (Reference Bae, Yoo and Choi2005) DNS of mixed convection of $\text{CO}_{2}$ at supercritical pressure, the flow undergoes the process of first relaminarisation and then strong recovery along the length of the heated pipe, illustrating the various stages of buoyancy effect on the mean flow and turbulence as observed herein. Their study was however for only one flow condition (and a particular distribution of the body force). Polyakov & Shindin (Reference Polyakov and Shindin1988) carried out measurements of velocity and turbulence for a wide range of buoyancy values. A limitation of that work is however that the partially laminarising flow that they chose to study was actually close to be fully laminarised, and as a result, they observed the turbulence becoming more isotropic as the flow is laminarised, which is in contrast to the observations of all other studies.

Hereafter, we focus on the partially laminarised flow, and to a lesser extent the fully laminarised flow. The recovery flow will be further studied elsewhere.

Figure 7. Distribution of the root-mean-square of the turbulent velocities.

3.2 Turbulence in relaminarising flow – the conventional viewpoint

Figure 7 shows the three components of the turbulent normal stresses normalised using the centreline velocity of the laminar Poiseuille flow. The turbulent stresses reduce in all the cases though some of the reductions are stronger than others. More significantly, the reductions in the wall-normal and circumferential components are always much stronger than that in the streamwise component, which is consistent with the observation of laminarising flow in mixed convection (Kasagi & Nishimura Reference Kasagi and Nishimura1997; Bae et al. Reference Bae, Yoo, Choi and Mceligot2006). The cases with the strongest reductions in $u_{r}^{\prime }$ and $u_{\unicode[STIX]{x1D703}}^{\prime }$ are cases A2 and B2, where they reduce to approximately a half of that of the reference case. By contrast, the reduction in $u_{z}^{\prime }$ is only approximately 30 % and 10 % in cases A2 and B2 respectively. For case D1 the peak values of the $u_{r}^{\prime }$ and $u_{\unicode[STIX]{x1D703}}^{\prime }$ reduce by about 30 %, but the peak value of the $u_{z}^{\prime }$ remains more or less the same as that of the reference case near the wall, even though the values are clearly reduced in the core. As a result of such unequal reduction in the three components, the turbulence becomes strongly anisotropic which is known to be a characteristic of a laminarising flow (e.g. Piomelli, Balaras & Pascarelli Reference Piomelli, Balaras and Pascarelli2000; Lee & Choi Reference Lee and Choi2001).

In the above, the discussion of turbulence reduction is based on absolute values since the normalisation uses the same constant ( $U_{P}^{\ast }$ ) in all the cases. In figure 8, the same results are presented in wall coordinates, which enable us to study how far the turbulence in a laminarising flow deviates from the ‘generic’ distribution of an equilibrium turbulent flow. Again the turbulence quantities show various levels of reductions in all the cases studied. The reduction is stronger in $u_{r,rms}^{\prime +}$ and $u_{\unicode[STIX]{x1D703},rms}^{\prime +}$ and less so in $u_{z,rms}^{\prime +}$ . However, the percentage of reduction has changed more significantly in some cases than in others, leading to a change of the order of the severity of laminarisation of the flows. For example, the peak value of $u_{z,rms}^{\prime }$ is lower in B2 than in A1 but the opposite is true in $u_{z,rms}^{\prime +}$ presentation. Overall, the turbulence reduction can be stronger or less strong in the absolute terms or in wall units depending on the flow conditions, which makes the prediction more difficult. Additionally, the location of the peak $u_{z,rms}^{\prime +}$ has shifted away from the wall in the laminarising flows which has been observed in mixed convection under similar flow conditions (Carr et al. Reference Carr, Connor and Buhr1973; Bae et al. Reference Bae, Yoo and Choi2005).

Figure 8. Distribution of the root-mean-square of the turbulent velocities in wall units.

3.3 A new perspective on the relaminarisation of flow

Ordinarily, comparisons of flows under various conditions are based on equal Reynolds numbers, which is the basis of the discussion presented above. For a given flow rate, the turbulence in a flow subjected to a non-uniform body force may be significantly lower than that in the ‘undisturbed’ flow of the same Reynolds number where the flow is driven by only a pressure force (which is uniform on any cross-section of the pipe). Under such a condition, the turbulence intensity, $k^{\ast 1/2}/U^{\ast }$ , where $k^{\ast }$ is turbulent kinetic energy and $U^{\ast }$ the bulk velocity, is lower in the body-force-influenced flow. We therefore refer to such a flow to be (partially) laminarised, examples of which include, mixed convection in a vertical tube (Steiner Reference Steiner1971; Polyakov & Shindin Reference Polyakov and Shindin1988), accelerating flows (Escudier et al. Reference Escudier, Abdel-Hameed, Johnson and Sutcliffe1998; Piomelli et al. Reference Piomelli, Balaras and Pascarelli2000; Greenblatt & Moss Reference Greenblatt and Moss2004) and flows subject to certain control mechanisms (Lee & Choi Reference Lee and Choi2001; Xu et al. Reference Xu, Dong, Maxey and Karniadakis2007), as discussed in the introduction.

Herein, we establish a new interpretation of the flow phenomena, which is in strong contrast to the above conventional view. We show that the body-force-influenced flow behaves as a ‘standard’ undisturbed turbulent flow but at a lower apparent Reynolds number; and this apparent Reynolds number is directly associated with the pressure gradient of the body-force-influenced flow. The understanding is supported by the following hypothesis which is established from the observations of the results reported herein: the addition of a body force to a flow that is initially driven by a pressure gradient only does not change the turbulent mixing characteristics of the initial flow and, in particular, the turbulent viscosity remains largely unchanged.

In the remaining part of the paper, we provide evidence supporting the above statement, and apply the hypothesis to characterising the laminarising flow under the influence of a body force. Furthermore, we demonstrate the simplicity of flow analysis in the new framework, and reveal new fundamental characteristics of laminarising flows that are difficult to identify in the classical viewpoint.

Let us first illustrate the implications of the hypothesis based on the Reynolds-averaged analysis. The governing equation for a statistically steady flow of an incompressible fluid in a pipe, sufficiently away from the entrance is written as:

where the pressure gradient $-\text{d}P/\text{d}x$ is the main driving force being constant in time and space; and $f$ is an additional body force varying along the radius, but uniform axially and circumferentially, and constant in time. This body-force-influenced flow can be compared with two ‘standard’ undisturbed flows without any non-uniform body forces: a flow with the same flow rate or a flow with the same pressure gradient. Herein we refer to them as the equal flow rate (EFR) reference flow and the equal pressure gradient (EPG) reference flow, respectively. All cases except B21 are performed at the same flow rate/Reynolds number, $Re=5300$ , and hence share the same EFR reference flow, which is simply referred to as the base flow when there is no ambiguity, see table 1. In the literature, comparisons between flows are commonly made based on equal flow rate as noted earlier. Below we compare the body-force-influenced flows with their EPG reference flow; the latter is denoted with a subscript ‘ $p$ ’,

where $-(\text{d}P/\text{d}x)_{p}=-(\text{d}P/\text{d}x)$ , that is, the pressure gradient of this (EPG) reference flow is the same as that of the body-force-influenced flow to be compared with. Subtracting (3.2) from (3.1),

where

which represents a perturbation flow caused by imposing a non-uniform body force to a flow while maintaining the pressure gradient constant. This flow is denoted with a subscript ‘ $b$ ’. Introducing the eddy viscosity concept, i.e. $\overline{u_{z}^{\prime }u_{r}^{\prime }}=-(\unicode[STIX]{x1D708}_{t}/Re)(\unicode[STIX]{x2202}\overline{u}_{z}/\unicode[STIX]{x2202}r)$ , where $\unicode[STIX]{x1D708}_{t}=\unicode[STIX]{x1D708}_{t}^{\ast }/\unicode[STIX]{x1D708}^{\ast }$ and $\unicode[STIX]{x1D708}_{t}^{\ast }$ is the turbulent viscosity, the above equation can be written as:

Now applying the hypothesis introduced above, i.e. the turbulent viscosity remains uninfluenced by the imposition of the body force ( $\unicode[STIX]{x1D708}_{t}=\unicode[STIX]{x1D708}_{t1}$ ), (3.5) becomes:

where $\unicode[STIX]{x1D708}_{t1}$ is the eddy viscosity of the EPG reference flow (i.e. from the solution of (3.2)) and the superscript ‘ $e$ ’ indicates that the values of $(\overline{u}_{z})_{b}$ computed are estimates or predictions based on the above assumption about the turbulent viscosity. In addition, we can estimate the turbulent shear stress due to the perturbation flow from

Consequently, with the availability of the solution of EPG reference flow (and hence $\unicode[STIX]{x1D708}_{t1}$ ), the perturbation flow due to the imposed body force can be estimated by simple integrations of (3.6). In essence, the hypothesis introduced earlier in this section implies: (i) $(\overline{u}_{z})_{b}\approx (\overline{u}_{z})_{b}^{e}$ or $\overline{u}_{z}\approx (\overline{u}_{z})_{p}+(\overline{u}_{z})_{b}^{e}$ , and, (ii) $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}\approx (\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ or $\overline{u_{z}^{\prime }u_{r}^{\prime }}\approx (\overline{u_{z}^{\prime }u_{r}^{\prime }})_{p}+(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ . That is, the body-force-influenced flow can be approximated by the sum of its corresponding EPG reference flow (i.e. an ‘undisturbed’ flow) and a perturbation flow due to the body force. The latter is readily computed from the body-force distribution and the EPG reference flow parameter $\unicode[STIX]{x1D708}_{t1}$ .

Figure 9. Total and wall shear stresses of case B2 and those of its corresponding EFR and EPG reference flows.

Figure 10. Distribution of turbulent viscosity in wall distance based on (a) conventional friction velocity and (b) apparent friction velocity.

Integrating equation (3.1) results in an expression for the force balance of the flow:

The first and second terms on the left-hand side are the contributions to the total shear stress from the pressure gradient and the body force, denoted as, $\unicode[STIX]{x1D70F}_{p}$ and $\unicode[STIX]{x1D70F}_{b}$ , respectively. Letting $r=1$ , we obtain the total wall shear stress to be $-(1/Re)(\unicode[STIX]{x2202}\overline{u}_{z}/\unicode[STIX]{x2202}r)_{r=1}=-(1/2)(\text{d}P/\text{d}x)+\int _{0}^{1}rf\,\text{d}r$ , that is, the total wall shear ( $\unicode[STIX]{x1D70F}_{w}$ ) is made of contributions from the pressure force ( $\unicode[STIX]{x1D70F}_{wp}$ ) and the additional non-uniform body force ( $\unicode[STIX]{x1D70F}_{wb}$ ). Taking case B2 as an example, the total shear stress ( $\unicode[STIX]{x1D70F}$ ) of the body-force-influenced flow together with those of its corresponding EFR and EPG reference flows are shown in figure 9 to illustrate a number of concepts introduced herein. In this particular case, the total wall shear ( $\unicode[STIX]{x1D70F}_{w}$ ) is smaller than that of the EFR reference case ( $\unicode[STIX]{x1D70F}_{wRe}$ ), but the opposite is true in some other cases. By definition, the wall shear stress of the EPG reference case and the contribution of the pressure gradient in the body-force-influenced case are equal, being $\unicode[STIX]{x1D70F}_{wp}$ . We define a friction velocity based on the pressure gradient component of the wall shear as $u_{\unicode[STIX]{x1D70F}p}^{\ast }=\sqrt{\unicode[STIX]{x1D70F}_{wp}^{\ast }/\unicode[STIX]{x1D70C}^{\ast }}$ , which is referred to as the apparent friction velocity, and similarly, $Re_{\unicode[STIX]{x1D70F}p}=u_{\unicode[STIX]{x1D70F}p}^{\ast }R^{\ast }/\unicode[STIX]{x1D708}^{\ast }$ as the apparent Reynolds number of the body-force-influenced flow. These are important parameters in the new framework of analysis. The values of the apparent Reynolds number for the various cases are shown in table 2, together with the averaged body force ( $F$ ) and the bulk velocity of the body-force-induced perturbation flow, $U_{bf}$ .

We now inspect the turbulent viscosity obtained for the various test cases studied herein, which are shown in figure 10(a,b) in wall coordinates based on the conventional friction velocity ( $u_{\unicode[STIX]{x1D70F}}^{\ast }$ ) and the apparent friction velocity ( $u_{\unicode[STIX]{x1D70F}p}^{\ast }$ ), i.e. $y^{+}=(\sqrt{\unicode[STIX]{x1D70F}_{w}^{\ast }/\unicode[STIX]{x1D70C}^{\ast }}y^{\ast })/\unicode[STIX]{x1D708}^{\ast }$ and $y^{+1}=(\sqrt{\unicode[STIX]{x1D70F}_{wp}^{\ast }/\unicode[STIX]{x1D70C}^{\ast }}y^{\ast })/\unicode[STIX]{x1D708}^{\ast }$ , respectively. It can be seen from figure 10(a) that the turbulent viscosity reduces in the body-force-influenced flows in comparison with the EFR reference flow. Generally speaking, the larger the body force, the stronger the reduction, which is consistent with the observations on turbulence quantities presented earlier. This is also in good accordance with the conventional theory, which explains flow laminarising phenomena (Launder & Jones Reference Launder and Jones1969; Sreenivasan Reference Sreenivasan1982; Piomelli & Yuan Reference Piomelli and Yuan2013).

By contrast, when the turbulent viscosity is plotted against $y^{+1}$ (figure 10 b), the spread of the data of the various cases is much smaller in the wall region, say, $y^{+1}<40$ , and much of the spread can be accounted for by the effect of the Reynolds number on the distribution of turbulent viscosity. To verify this idea, undisturbed flows close to the apparent Reynolds numbers of the various flow cases (and hence serving as the EPG reference flows) have been performed and the distributions of the turbulent viscosity of these cases are shown in figure 10(b) for direct comparison. It can be seen that the values of the turbulence viscosity in the various cases are largely bounded by those of the flows of $Re_{\unicode[STIX]{x1D70F}}=110$ and 180. A closer study of the data reveals that cases A1, A2, C1 and B21 agree closely with their corresponding EPG reference flows. The values of $\unicode[STIX]{x1D708}_{t}$ of B1, B2 and D1 overshoot those of their corresponding reference flows at some locations near the wall. These deviations are still relatively small considering the large range of flow conditions covered and the severity of the flow laminarisation observed. In conclusion, these results provide a good support for the hypothesis presented early: the values of the turbulent viscosity of the body-force-influenced flows is not significantly influenced by the body force and they agree well with those of their corresponding EPG reference flows. Consequently, the body-force-influenced flow in effect behaves as an ‘undisturbed’ flow at a lower apparent Reynolds number.

It is noted that in pursuing the development of semi-empirical correlations for buoyancy-influenced heat transfer, Jackson & Hall (Reference Jackson, Hall, Kakac and Spalding1979) introduced an assumption that the effect of buoyancy on the flow is to reduce the near-wall shear stress and hence turbulence production. They used the turbulent shear stress away from the buoyancy-influenced region to define a new equivalent flow, based on which heat transfer was calculated. This approach allowed them to develop a very successful, widely used, heat transfer correlation. The observation described herein, to some extent, explains the reasons behind the success.

Figure 11. Distribution of turbulent shear stresses, (i) DNS results $\overline{u_{z}^{\prime }u_{r}^{\prime }}$ (lines only in main figure), (ii) estimated turbulent shear stress due to body force calculated from (3.7),  $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ , (lines only in inset) and (iii) estimated overall turbulent shear stress using EPG reference flow data, $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{p}+(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ (lines with markers).

To further elucidate the idea, next we study the turbulent shear stresses. Figure 11 shows three sets of data: (i) DNS results $\overline{u_{z}^{\prime }u_{r}^{\prime }}$ , (ii) estimated turbulent shear stress resulting from body force calculated from (3.7), $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ and (iii) estimated overall turbulent shear stress using EPG reference flow data, $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{p}+(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ . It is noted that since the values of $Re_{\unicode[STIX]{x1D70F}p}$ of some cases are noticeably different from the closest $Re_{\unicode[STIX]{x1D70F}}$ of the available ‘standard’ cases, the EPG data used here are based on an interpolation of results with the closest $Re_{\unicode[STIX]{x1D70F}}$ . It can be seen that the body-force-induced turbulent shear stress, $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ , is only limited to the near-wall region, and the values vary significantly from case to case. Interestingly, the magnitude of $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ is not directly corresponding to the averaged body force ( $F$ ). For example, the body forces of A1 and B1 are both 0.35, but $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ is significantly higher in B1 than in A1. Clearly this difference stems from the different coverages of the body force in the two flows. The larger the coverage, the larger the value of $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ . In fact, the contribution of $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ to the total turbulent shear stress in A1 is so small that it can be neglected. This has an interesting implication, that is, the body-force-induced perturbation flow can be seen to be a laminar flow, and that the turbulent shear stress of the body-force-influenced flow is the same as that of the corresponding EPG reference flow. It is particularly interesting to note that the total body force does not have to be small; it only requires the coverage to be small. Flows that are in this category are A1, A2 and C1. Typically, the body-force-induced turbulence shear stress $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ will be negligible when the coverage of the body force is limited $y^{+}<20$ , which is the case in many mixed convection flows. To facilitate discussion, we refer to them as low $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ flow cases.

We now compare the turbulent shear stress directly calculated from the DNS results of the various cases ( $\overline{u_{z}^{\prime }u_{r}^{\prime }}$ ) and those estimated using the EPG reference flows, i.e. $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{p}+(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ . The agreement between the two sets of data is strikingly close in most cases. For the low $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}$ cases, this implies that the turbulent shear stress is exactly the same as that of the EPG reference flow and the body force does not change it in any way. For other cases, it can be inferred that the perturbation flow induced by the body force does result in an additional turbulent shear stress, but its value can be estimated with the turbulent viscosity of the EPG reference flow. Overall these results again provide evidence supporting the underlining assumption of the theory proposed herein, that is, the turbulence mixing characteristics are not modified by the body force.

Figure 12. Distribution of the root-mean-square turbulent velocities normalised by the apparent friction velocity.

Next we study the turbulent normal stresses in the new framework. The stresses normalised by the apparent frictional velocity $\sqrt{\unicode[STIX]{x1D70F}_{wp}^{\ast }/\unicode[STIX]{x1D70C}^{\ast }}$ are plotted against $y^{+1}$ in figure 12. Before studying the detailed behaviour, we first note that the apparent Reynolds numbers of most flows studied herein are rather low, and some of the flows are in the transitional flow regime, refer to table 2. Tsukahara et al. (Reference Tsukahara, Seki, Kawamura and Tochio2014) studied low Reynolds number turbulent and transitional flows in a channel with $Re_{\unicode[STIX]{x1D70F}}$ between 64 and 180. They have shown that the peak root-mean-square values of the $u_{r}^{\prime }$ and $u_{\unicode[STIX]{x1D703}}^{\prime }$ at low Reynolds numbers reduce significantly. The DNS of low Reynolds number flows performed herein ( $Re_{\unicode[STIX]{x1D70F}}=112,129,135,157$ and 180) clearly show this trend. These data serve as the EPG reference flows for the various flow cases (again refer to table 2). In strong contrast with figure 8 where the data are normalised with the conventional friction velocity, the spreads of $u_{r,rms}^{\prime +1}$ and $u_{\unicode[STIX]{x1D703},rms}^{\prime +1}$ between the different test cases are small, and most of these can be accounted for by the Reynolds number effect. Close to the wall ( $y^{+1}<30$ ), the profiles of $u_{r,rms}^{\prime +1}$ and $u_{\unicode[STIX]{x1D703},rms}^{\prime +1}$ follow very much what expected at their respective apparent Reynolds number in all the cases except for case D1, where the values are clearly higher than that of $Re_{\unicode[STIX]{x1D70F}}=135$ . The peak values also agree well with those of their corresponding EPG reference flows in most cases, except for cases B2 and D1, in which the peak values are higher than expected but are still within 10 %. These results demonstrate that, in contrast to the conclusions drawn earlier from the conventional approach using wall units based on the total flow, which show a strong reduction in turbulence, the radial and circumferential turbulence components are not significantly influenced by the imposed body force in the new framework.

In contrast to the trends exhibited in $u_{r,rms}^{\prime +1}$ and $u_{\unicode[STIX]{x1D703},rms}^{\prime +1}$ (and that demonstrated in the conventional plots shown in figure 8), the peak values of the streamwise turbulent velocity are increased in all the test cases except for A1; in some cases, significantly. This is despite that the peak values of the corresponding EPG reference flows collapse on top of each other. We demonstrate later that the increase of $u_{z,rms}^{\prime +1}$ is associated with the enhancement of the streaks in the flow resulting from the imposition of the body force. Referring to the turbulence regeneration cycle (Kim Reference Kim2011), it appears that the body force enhances the linear processes of streak generation and the transient growth processes (the first and second legs) through the boundary layer that it induces. It does not seem to significantly influence the nonlinear vortex formation (the third leg). This will be further discussed later with reference to the flow structures.

Figure 13. The budget terms of the streamwise turbulent fluctuating velocity in conventional wall units. Solid lines: production; chain dotted lines: dissipation; short dashed lines: pressure strain; long dashed lines: turbulent transport; dotted lines: viscous diffusion. Thick lines: base case ( $Re_{\unicode[STIX]{x1D70F}}=180$ ); thin lines: body-force-influenced cases.

Figure 14. The budget terms of the wall-normal turbulent fluctuating velocity in conventional wall units. Solid lines with markers: pressure diffusion; chain dotted lines: dissipation; short dashed lines: pressure strain; long dashed lines: turbulent transport; dotted lines: viscous diffusion. Thick lines: base case ( $Re_{\unicode[STIX]{x1D70F}}=180$ ); thin lines: body-force-influenced cases.

Figure 15. The budget terms of the streamwise turbulent fluctuating velocity in wall units based on the apparent friction velocity. See figure 13 for legend.

Figure 16. The budget terms of the wall-normal turbulent fluctuating velocity in wall units based on the apparent friction velocity. See figure 14 for legend.

Figures 13 to 16 show the budget terms of the streamwise and wall-normal stresses normalised by $u_{\unicode[STIX]{x1D70F}}^{\ast 4}/\unicode[STIX]{x1D708}^{\ast }$ and $u_{\unicode[STIX]{x1D70F}p}^{\ast 4}/\unicode[STIX]{x1D708}^{\ast }$ , respectively. As expected, the terms for both $u_{z}^{\prime }$ and $u_{r}^{\prime }$ normalised based on $u_{\unicode[STIX]{x1D70F}}^{\ast }$ reduce in all the test cases, and in many cases, such as A2, B2 and C1, the reduction is very significant. This is consistent with the traditional view of flow laminarisation, demonstrating that turbulence is suppressed due to the presence of the body force. The picture is however very different when the terms are normalised based on $u_{\unicode[STIX]{x1D70F}p}^{\ast }$ . For the streamwise component, the budget terms in A1 and C1 agree very well with their corresponding reference values. For the rest of the cases, both the production and the dissipation are increased by various amounts. This trend is consistent with the observations in figure 12 where $u_{z,rms}^{\prime +1}$ is presented. The increase of the production is largely related to the generation of elongated streaks. For the wall-normal component, there is no direct production. The supply of energy comes from the pressure strain term, whereas the sink is the dissipation. In addition, the pressure diffusion term is also significant, especially close to the wall. It can be seen from figure 16 that, overall, the terms in the body-force-influenced flows agree very well with those of the base case, especially considering the large deviations shown in the conventional presentation (figures 13 and 14). The agreements between the main terms (pressure strain and dissipation) are particularly good for A1 and B1. This statement is also largely true for A2 and C1, though the pressure strain and dispersion terms show an interesting strong reduction near the wall ( $y^{+1}<8$ ). Both terms agree reasonably well with their respective reference data in the core region. Another observation is that the source term, pressure strain, increases somewhat in several cases, including B21 (not shown), B2 and D1, and to a lesser extent B1. These are the cases where the body-force-induced turbulent shear stress is relatively large. Overall however the agreement between the body-force-influenced cases and the reference cases is remarkable considering that in many cases the absolute values reduce by several folds.

Figure 17. Velocity profiles. (a) Body-force-induced perturbation flow; (b) total flow subtracting the body-force-induced flow, together with EPG reference flows; and (c) same as (b) but in wall coordinates.

Before leaving this section, we revisit some interesting features of the mean velocity field. Figure 17(a) shows the profiles of the mean velocity of the perturbation flow induced by the body force calculated from (3.6), $(\overline{u}_{z})_{b}^{e}$ ; and figure 17(b,c) show the velocity difference, $(\overline{u}_{z})^{\wedge }=\overline{u}_{z}-(\overline{u}_{z})_{b}^{e}$ , in outer scaling and wall scaling respectively. According to the theory established herein, $(\overline{u}_{z})^{\wedge }$ of any flow case is expected to be in good agreement with the velocities of its corresponding EPG reference flows $(\overline{u}_{z})_{p}$ , which are also shown in the figure comparison. It can be seen from figure 17(b) that this is indeed the case for most flow cases. Similar to the turbulence quantities presented above, relatively larger discrepancies are observed in cases B2 and D1, which are approximately 10 %. Next, it is interesting to note that the body-force perturbation flow is not necessarily directly proportional to the total body force ( $F$ ). The coverage has a significant effect. For example, A1 and B1 have the same total body force, but the perturbation flow of B1 is more than doubled that in A1. The next point to note is that the body-force-induced perturbation flow is very significant in some cases in comparison with the EPG reference flows. The ratios between the two flows are 13 % for A1, approximately 35 % for B1 and B21, approximately 50 % for C1 and D1 and then 70 % and 85 % for A2 and B2. Apparently, the addition of such a large perturbation flow does not seem to have a significant impact on the key turbulent characteristics of the flow as shown above. Finally it can be seen from the wall coordinates velocity plot that the profiles of $(\overline{u}_{z})^{\wedge }$ agree well with the EFR reference flow profile in most of the cases, clearly exhibiting a region of logarithmic behaviour. There are some cases where the velocity profile shows a relatively strong upshift, which is normally seen as a sign of laminarisation. This occurs in A2 and B2, which correspond to the lowest $(\overline{u}_{z})^{\wedge }$ and therefore the corresponding EPG reference flows have the lowest $Re_{\unicode[STIX]{x1D70F}}$ .

3.4 Friction factor and the FIK identity analysis

Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002) introduced a simple but useful analysis of the contributions of different dynamic effects on the friction drag for turbulent flows. The local friction is decomposed into a laminar, turbulent, inhomogeneous and transient components. This method has been widely used in flow control and drag reduction investigations and is referred to as the FIK identity. For a flow in a pipe considered herein (which is described by (3.1)), the FIK expression is (Fukagata et al. Reference Fukagata, Iwamoto and Kasagi2002),

where $C_{f}=\unicode[STIX]{x1D70F}_{w}^{\ast }/(\unicode[STIX]{x1D70C}^{\ast }U^{\ast 2}/2)=8\unicode[STIX]{x1D70F}_{w}^{\ast }/(\unicode[STIX]{x1D70C}^{\ast }U_{P}^{\ast 2})$ , and $F=2\int _{0}^{1}fr\,\text{d}r$ , being the averaged body force. From left to right, the terms represent the contributions due to laminar and turbulent flows, and the body force. To better understand the contribution of the body force, the turbulent contribution is further split into (i) a component corresponding to the EPG reference flow, $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{p}$ , and (ii) one due to the perturbation flow induced by the body flow, which is calculated by subtracting the above component from the total turbulent shear stress, $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}=\overline{u_{z}^{\prime }u_{r}^{\prime }}-(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{p}$ .

The FIK analysis of the various laminarising cases is shown in figure 18. It can be seen that the total friction factor of the laminarising flows can be higher or lower than that of the EFR reference flow. The figure also illustrates that the contribution of the turbulence associated with the perturbation flow due to the body flow is negligible in A1, A2 and C1 and is substantial in the other test cases. This is consistent with earlier observations (refer to figure 11(a) where $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}$ is shown, for example). Comparing A1 with A2 (or B1 with B2), it is noted that the flow is more strongly laminarised as a result of a stronger body force, and hence a smaller contribution from $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{p}$ in the former, but the contribution from the body force itself increases. If a reduction of the friction factor is the objective, increasing the amplitude of the body force is more effective than increasing the coverage, comparing, B1 and B2 with A1 and A2.

Figure 18. Friction coefficient. (a) Contributions of the various components of FIK analysis, laminar ( $dp$ and $bf$ ) refer to the right-hand side terms 1 and 3 of (3.9), respectively; turbulent ( $dp$ and $bf$ ) refer to the right-hand side terms 1 and 2 of (3.10), respectively; (b) comparison of friction coefficients calculated from velocity gradient (DNS), FIK analysis and predictions based on EPG reference flows.

Next, we apply the theory established in § 3.3 to the prediction of the friction factors. We have established that the turbulent viscosity in the body-force-influenced flows remains little changed from the values of their corresponding EPG reference flows and that the turbulent shear stress due to the body-force-induced perturbation flow can be estimated using (3.7). Now, in (3.10), we replace $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}$ with $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}^{e}$ . The results are shown in figure 18(b) labelled ‘Predictions’. It can be seen that these predictions are very close to those calculated from the original FIK formulation (3.9). Both agree very well with the friction factor calculated directly using the velocity gradient (labelled DNS in the figure). Consequently, the friction factor of a laminarising flow can be accurately predicted simply using an undisturbed turbulent flow data (the EPG reference flow) and the profile of the imposed body force.

3.5 Full laminarisation

Consider a turbulent flow at a prescribed Reynolds number subjected to a non-uniformly distributed body force. From the results presented in § 3.1, we know that with the increase of the imposed body force while keeping the Reynolds number of the flow constant, the turbulence becomes progressively more strongly suppressed, that is, the flow is increasingly more severely laminarised. The terminology laminarisation refers to a comparison with the reference flow at the same Reynolds number. When the body force is sufficiently strong, the flow becomes fully laminarised, which is the case in A3 and B3. The above scenario occurs in many applications including mixed convection, for example, where the flow becomes progressively more severely laminarised with the increase of the buoyancy. The above process can be described using the new theory presented in § 3.3. The body-force-influenced flow can be decomposed into a pressure-driven flow (i.e. the EPG reference flow) and a perturbation flow due to the body force. The former can be approximated by an undisturbed turbulent pipe flow, whereas the latter is a flow described by (3.6), which is dependent on the imposed body force and the EPG reference flow (through $\unicode[STIX]{x1D708}_{t}$ ). The bulk velocity of the EPG reference flow can be written as the difference between those of the total flow and the perturbation flow due to the body force, that is,

With the increase of the body force ( $F$ ) while keeping the total flow constant, $U_{bf}$ increases and hence $U_{dp}$ reduces. There will be a point when the Reynolds number of the pressure driven flow is so low that it cannot sustain turbulence any longer, and hence the flow becomes laminar. This critical body force can be calculated using (3.11) given a critical Reynolds number of 2300, for example.

In the above discussion, the Reynolds number of the EPG reference flow plays an important role. Next we relate it to the imposed body force. Following the approach adopted by Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002), integrating equation (3.5) three times and making use of integration by parts, we can derive an expression for the mean velocity for the perturbation flow driven by the body force:

Noting that $Re_{dp}=(2(U^{\ast }-U_{bf}^{\ast })R^{\ast })/\unicode[STIX]{x1D708}^{\ast }=Re(1-2U_{bf})$ , we have

where $U_{bf}$ is calculated from (3.12). The above ratio of Reynolds numbers of the apparent and total flows effectively defines how far turbulence is reduced, or, the degree of laminarisation of the flow due to the imposition of the body force. This parameter is hence referred to as the laminarisation ratio. It is noted however that this parameter does not directly show how far the friction factor is varied, since the body force itself causes an additional friction to balance it, see discussion in § 3.4. The total friction is the sum of the friction from the apparent flow and the integral of the body force. In addition, the laminarisation ratio does not tell if a flow is fully laminarised either. The criterion for laminarisation is that the Reynolds number of the apparent flow reduces to a degree when it cannot sustain turbulence any more, that is, $Re_{dp}<2300$ . Combining this with (3.13), a criterion for full laminarisation is obtained,

It is noted here that even though the parameter that is evaluated is the Reynolds number of the pressure-driven flow, the effect of the body force is directly reflected in the resultant equations (3.11) and (3.14). After all, the body force is the reason for the changes in the flow. In the literature, criteria for flow laminarisation, even based on advanced mechanistic models, require experimental data to tune coefficients. By contrast, the above criterion is simply based on that the apparent Reynolds number of the flow is below a laminar–turbulent transition Reynolds number, which is the only constant used.

For the low- $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}$ cases, the second term of (3.12) is negligible and this normally occurs when the body force is limited to $y^{+}<20$ , which can happen in various real flows. For ease of reference, the laminar-flow-only expression is given by:

We now evaluate the body-force-induced flow based on (3.15) for the linear and step-change forces to establish some trends. Substituting (2.5) and (2.6) into (3.15), and integrating, the following are resulted for linear and step-change forces, respectively:

Substituting the above into (3.14), we obtain the criteria for full flow laminarisation for linear and step distributions of body forces, respectively, under the condition when the body-force-induced flow perturbation causes negligible turbulent shear stresses:

Note that $aRe^{2}=(a^{\ast }R^{\ast 3})/(\unicode[STIX]{x1D70C}^{\ast }\unicode[STIX]{x1D708}^{\ast 2})$ . Consequently, the left-hand side of the above equations, which basically is the Reynolds number of the body-force-induced flow ( $Re_{bf}$ ), is independent of the bulk velocity of the flow, $U^{\ast }$ . The above equation also implies that for both the linear and step distributions, the $Re_{bf}$ increases proportionally with the increase of the amplitude, whereas although increasing the coverage $r_{0}$ also increases $Re_{bf}$ , the relationship is nonlinear.

It is worth noting that full flow laminarisation (reverse transition) is not an absolute, definitive event, but to some extent is dependent on the particular circumstances. The critical Reynolds number, 2300, used above is only a nominal value. In experiments, the flow can be laminar at a rather high Reynolds number if the experimental facility is free from disturbances and noise. In computer simulations, both the initialisation of the flow and the size of the computational domain influence the critical Reynolds number. In the study of Tsukahara et al. (Reference Tsukahara, Seki, Kawamura and Tochio2014), they have achieved turbulent/transitional flow in a channel at $Re_{\unicode[STIX]{x1D70F}}=64$ . In the present study, we have not attempted systematically increasing the body force to determine the critical flow. However, in groups A and B, the cases with the highest body force that still remain turbulent are cases A2 and B2. Their Reynolds numbers ( $Re$ ) are 3100 and 2862, (or for $Re_{\unicode[STIX]{x1D70F}}$ , 112 and 111), respectively. Cases A3 and B3 are evidently laminarised, but the corresponding body-force-driven flow is significantly greater than the critical value required to cause a full laminarisation.

3.6 Flow structures

The high- and low-speed streaks and the vortical cores of selected cases are visualised in figure 19 using isosurfaces of the fluctuating streamwise velocity $u^{\prime }=\pm 0.15$ and $\unicode[STIX]{x1D706}_{2}=-1$ , respectively. Comparing B1 and B2 with the EFR reference flow, it is clear that both the number of streaks and the number of the vortices reduce significantly due to the effect of the body force; and the stronger the body force, the more significant the reduction is. It is interesting to note however, that with the increase of the body force, the streaks appear to become longer. This is not surprising since, as we note before, the cases with a stronger body force corresponds to a EPG reference flow of a lower Reynolds number. It is known that streaks are longer in low Reynolds number flows. Next we note that as far as B21 is concerned, the base case ( $Re_{\unicode[STIX]{x1D70F}}=180$ ) is its EPG reference flow. It is apparent that the numbers of vortical structures are largely the same in the two flows, whereas there are clearly more streaks in B21 than in the base case. This interesting observation is in good accordance with the statistics discussed earlier. The former can be related to the observation that the pressure strain term in the body-force-influenced flows remains largely unchanged from their corresponding EPG reference flows, and in turn $u_{r}^{\prime }$ and $u_{\unicode[STIX]{x1D703}}^{\prime }$ are largely unchanged. On the other hand, the increased number of streaks explains the increase in $u_{z}^{\prime }$ , even though the vortical structures remain largely unchanged.

Figure 19. Turbulent structures in cases; base, B21, B1 and B2. (a–d) High (green) and low (blue) speed streaks illustrated with three-dimensional isosurfaces of $u_{z}^{\prime }=\pm 0.15$ and (e–h) bottom four panels: vortical structures illustrated with three-dimensional isosurfaces of $\unicode[STIX]{x1D706}_{2}=-1.0$ .

Figure 20. Schematic of a self-sustaining process of near-wall turbulence structures. (Reproduced from Kim Reference Kim2011 with permission.)

Figure 21. Root-mean-square of streamwise vorticity fluctuations normalised using the (a) conventional friction velocity (b) apparent friction velocity.

The concept of turbulence regeneration cycle has been established over the last couple of decades which successfully explains wall turbulence flow physics and, in some cases, the mechanisms of flow control and drag reduction. Figure 20 shows a much simplified sketch illustrating the main processes of the regeneration cycle reproduced from Kim (Reference Kim2011). In the first leg, streamwise vortices (primarily consisting of the wall-normal and spanwise velocities) interact with the mean shear ( $\unicode[STIX]{x2202}\overline{u}_{z}/\unicode[STIX]{x2202}y$ ) generating streaks. This process is often referred to as lift-up. The streaks generated grow as long as the shear is maintained (second leg). Later, the streaks break down due to instability. At the third leg, new vortices are formed again due to three dimensional disturbances. It is clear that the streamwise vorticity and streaks are most important factors in the turbulence regeneration cycle of a wall shear flow depicted in figure 20. Below we analyse the streamwise vorticity and the streaks in the various flows studied.

Figures 21(a) and (b) show the root-mean-square of the fluctuating streamwise vorticity in wall coordinates based on the total flow ( $u_{\unicode[STIX]{x1D70F}}^{\ast }$ ) and the apparent friction velocity ( $u_{\unicode[STIX]{x1D70F}1}^{\ast }$ ), respectively. It can be seen from figure 21(a) that the vorticity normalised with $u_{\unicode[STIX]{x1D70F}}^{\ast }$ reduces significantly in the various test cases. The reduction is most severe in A2, with the value reducing to less than a 1/4 at $y^{+}=1$ , and the peak away from the wall reducing to approximately 30 % of the reference value. Consequently, the turbulence activities are significantly reduced in the various cases in comparison with the EFR reference flow. This observation is consistent with the descriptions of laminarisation in various scenarios (Launder & Jones Reference Launder and Jones1969; Sreenivasan Reference Sreenivasan1982; Piomelli & Yuan Reference Piomelli and Yuan2013). Figure 21(b) however shows a contrasting picture. In most cases (except for D1 and B21), the distributions of the streamwise vorticity in the body-force-influenced cases agree reasonably well with their respective EPG reference flows. This implies that the imposition of the body forces does not seem to cause any significant changes in the generation of streamwise vorticity, an important stage in the turbulence regeneration cycle. The increase of the vorticity fluctuations in B21 and D1 can be related to fact that the body-force-generated turbulent shear stress is larger in those cases, which apparently causes some increase in streamwise vorticity activities.

Figure 22. Contours of the streamwise velocity correlations in a vertical plane in wall units based on the conventional friction velocity. The values of the contours reduce monotonically from the outer edge of the contours to the centre.

Figure 23. Contours of the streamwise velocity correlations in a vertical plane in wall units based on an apparent friction velocity. The values of the contours reduce monotonically from the outer edge of the contours to the centre.

Figure 19 shows that the streaks are strengthened by the imposition of the body force. To further characterise the streaks, figure 22 shows the contours of the spanwise correlations of the streamwise velocity on the $y{-}r\unicode[STIX]{x1D703}$ plane in wall coordinates based on the total flow. The data plotted are the amplitude of the negative correlations with the positive correlations set to zero. The locations of the peak values (i.e. minimum correlation) provide the following information: $2\times (r\unicode[STIX]{x1D703})_{min}$ is the averaged spacing of the steaks and $y_{min}$ is the averaged wall distance from the centre of the streaks. It can be seen from figure 22 that the averaged spacing of the streaks in the base flow is around 110 ( $(r\unicode[STIX]{x1D703})_{min}^{+}=55$ ) and the averaged wall distance of the streaks is $y_{min}^{+}=14$ . The streaks spacing in most of the test cases is increased and in some cases, significantly. The largest spacing, occurring in A2, is around $(r\unicode[STIX]{x1D703})^{+}=170$ . The streaks’ centre also moves away from the wall, and the largest distance is around $y^{+}=20$ , occurring in A2 and B2. There are some exceptions to the trends described above; for example, both the streaks spacing and the location do not change significantly in B21 and D1. The same data shown in the above figure are shown again in figure 23 in the $y{-}r\unicode[STIX]{x1D703}$ plane in wall coordinates based on the apparent friction velocity. In most cases, both the averaged spacing of the steaks ( $2\times (r\unicode[STIX]{x1D703})_{min}$ ) and the wall distance of the streaks centres are now similar to those of the base flow. The exceptions are cases B2, D1 and B21, where the spacing is reduced to approximately 90, but the vertical locations of the streaks centres are still similar to that of the reference flow. As observed earlier, these are the cases where the turbulence deviates somewhat from their corresponding EPG reference flows.

Figure 24. Quadrant analysis. (a) Number of ejections (Q2) over the total number of events; (b) number of sweeps (Q4) over the total number of events; (c) contributions of ejection (Q2) events to $\overline{u_{z}^{\prime }u_{r}^{\prime }}$ ; (d) contributions of sweep (Q4) events to  $\overline{u_{z}^{\prime }u_{r}^{\prime }}$ .

The near-wall turbulence structures can be studied through inspecting the ejection and sweep events which are associated with the vortices and streaks (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967; Wallace, Eckelmann & Brodkey Reference Wallace, Eckelmann and Brodkey1972; Kim et al. Reference Kim, Moin and Moser1987; Robinson Reference Robinson1991). Using the definition of the hyperbolic hole introduced by Lu & Willmarth (Reference Lu and Willmarth1973), the contribution of each quadrant to $\overline{u_{z}^{\prime }u_{r}^{\prime }}$ is defined as

and the percentage number count is

where $Q=1,\ldots ,4$ is the quadrant and

The greater the value of $h$ , the stronger the events are. Figure 24(a,b) show the percentages of the numbers of ejections and sweeps of the total turbulent events that result in turbulent shear stress, and figure 24(c,d) show the values of the shear stress associated with these two activities. A strength of $h=1$ is used. It is most interesting to see that the number of the events in the body-force-influenced cases remains more or less unchanged from the base flow distribution. For ejections the agreement between them is very good, within a few per cent, whereas for sweeps the deviations are slightly larger but are still within approximately 10 % for most cases. The only exception is B2 where the number is significantly lower than in all other cases. The contributions to shear stress from the ejections and sweeps also remain largely unchanged in the low $(\overline{u_{z}^{\prime }u_{r}^{\prime }})_{b}$ flows, but they are significantly increased in the wall region in other cases as a result of the imposition of the body force. That is, the strength of each event in these cases is greater. This can be nicely linked to the observations discussed above on vorticity and streaks. It is apparent that the imposition of the body force, which induces an additional boundary layer near the wall, does not significantly influence the generation of vortices, but for each ejection or sweep event, a larger shear stress is resulted in due to the larger velocity gradient. Additionally it is interesting to note that the influence on the ejection events occurs in the region of $y^{+1}<60$ , where the effect on sweep is restricted to $y^{+1}<30$ .

Finally, we evaluate the turbulence anisotropy by analysing the anisotropy-invariant map (AIM) of the Reynolds stress anisotropy tensor ( $\unicode[STIX]{x1D623}_{ij}$ ):

where $\unicode[STIX]{x1D739}_{ij}$ is the Kronecker delta tensor. The AIM proposed by Lumley (Reference Lumley1978) is constructed using the second and third principal invariants ( $II=-(1/2)(b_{zz}^{2}+b_{rr}^{2}+b_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}^{2}+2b_{zr}^{2})$ and $III=(b_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}b_{zz}b_{rr}+b_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}b_{zr}^{2})$ ), which are independent of the coordinate choice. Lumley’s anisotropic map provides useful information on the streaky structures since they are inherently anisotropic. The AIMs for the various body-force-influenced cases are plotted on top of that of the EFR reference flow in figure 25. It can be seen clearly that in all cases the profile moves towards the top right vertex, indicating that the flow becomes more one-dimensional and further demonstrating the streaky structures are elongated under the influence of the body force. The stronger the body force, the stronger the one-dimensional feature of the flow. In fact, the profile nearly reaches the top-right vertex in A2 and B2, showing the existence of strong streaks in these flows (also refer to figure 19 to see the streaks in B2). The profile in the undisturbed turbulent flow often reaches the topmost point at around $y^{+}=9$ . It can be seen that this point is only shifted slightly away from the wall in the body-force-influenced cases even though the strength are significantly increased. When measured using the wall units based on the apparent friction velocity, the location actually lowers down slightly.

Figure 25. The anisotropy-invariant maps (AIMs) of a number of flow cases.

3.7 Applications and limitations

The present study is based on two types of idealised body-force distributions, i.e. linear and step changes with a systematic variation of the coverage and amplitude of the body forces. Strictly speaking, the conclusions drawn herein are only directly applicable to such situations. However, the principle conclusions stem from an observation that the non-uniform body forces, whether linear or step change, do not alter the turbulence mixing characteristics in any significant way. By nature, such conclusions are not dependent on the detailed profile of the body force, and are likely to be applicable to more complex distributions. Nevertheless, care should be taken when extrapolating the conclusions to new situations, and in particular, when the body force extends further away from the wall than considered herein.

The understanding developed herein potentially provides new insight into the laminarisation of flow due to buoyant forces in mixed convection in vertical channels. However, it should be noted that, in practice, the flow and thermal fields are closely coupled in mixed convection, and both fields may continue developing streamwise due to the temperature dependence of fluid properties. Such effects are not addressed herein. It would naturally be a useful and interesting development next to analyse mixed convection with coupled flow and thermal fields to understand the full picture of the physical problem.