5.1. Effect of Saffman lift on particle motion

Firstly, the effect of Saffman lift on particle motion is investigated for particles of Stokes number $St=3$. Three different cases are compared with one another, i.e. single-phase turbulent channel flow (denoted as SP), particle-laden turbulent channel flow with Saffman lift included in particle motion equation (denoted as SL) and particle-laden turbulent channel flow without Saffman lift (denoted as no SL).

Wall accumulation of particles is a typical flow pattern in particle-laden wall-bounded turbulent flows. Shown in figure 6 are spatial-temporal evolutions of the particle number for cases no SL and SL, respectively. The spatial-temporal contours of particle number are plotted in both the inner scale and outer scale of the channel for each case. The particles are uniformly injected into the flow at $t=0$. When the Saffman lift is absent, the particles are observed to be constantly transported to the near-wall regions. As is seen in figure 6(a), the particles tend to accumulate in the region of $y^+<1$ (viscous sublayer) with the advance of time. There are very few particles above the viscous sublayer, as depicted in figure 6(c). However, the numerical results show a totally different trend when the Saffman lift is present. Unlike case no SL, the particles do not exhibit successive movement toward the wall. Meanwhile, the profile of particle number reaches a statistically steady regime within a very short period. By contrast, the case without Saffman lift requires much more time to achieve a statistically steady state. It should be mentioned here that the statistical results presented in what follows are obtained after the case no SL reaches the statistically steady state. The particle number in the viscous sublayer of case SL (figure 6b) is much smaller than that of case no SL. Figure 6(d) allows a better observation of the particle distribution for case SL because the maximum particle number is located at the channel centre. When the Saffman lift is present, the difference in particle numbers within different layers of the channel is not very large along the wall-normal direction. By contrast, the particle number in the viscous layer is more than 5 times that at the centre of the channel in case no SL. Therefore, it is believed that the Saffman lift should significantly reduce the wall accumulation of particles, and rapidly lead the near-wall particle distribution to a statistically steady state. It needs to be emphasized that such an obvious difference in the particle distribution caused by the Saffman lift along the wall-normal direction accounts for all the discrepancies in turbulence modulation, which will be discussed in § 5.2.

Figure 6.

Contours of spatial-temporal evolution of particle numbers normalized by the total number of particles in the channel: (a) case no SL in inner coordinates, (b) case SL in inner coordinates, (c) case no SL in outer coordinates and (d) case SL in outer coordinates.

Then, the effects of Saffman lift on the mean particle streamwise velocity and the fluctuating particle velocities are examined as they are very important in understanding and predicting the wall erosion caused by particles. Shown in figure 7 are the mean streamwise velocities for the fluid and particles normalized by the friction velocity $u_{\tau }$. The numerical results confirm the experimental observation that the particles have little influence on the mean flow as the particles hardly change $\langle u \rangle ^+$ (Kulick et al. Reference Kulick, Fessler and Eaton1994). Whether the Saffman lift is present or not, the mean streamwise fluid velocity agrees well with the results of single-phase flow, which obeys the law of the wall. However, the Saffman lift alters significantly $\langle u^p \rangle ^+$ in the near-wall region. In a statistical sense, particles under the action of Saffman lift tend to move faster along the streamwise direction than those in the absence of Saffman lift under the buffer layer ($\kern0.7pt y^+<10$). This is consistent with the experimental results reported by Kulick et al. (Reference Kulick, Fessler and Eaton1994) for large Stokes numbers, i.e. the mean streamwise velocity profile of particles is flatter than that of the fluid phase in the near-wall region and the corresponding slip velocity between the particle and fluid is substantial. This proves that the particle velocities given by PP DNS with Saffman lift are closer to reality than those obtained by PP DNS without Saffman lift. It should be stressed, however, that the effect of gravity needs to be included in numerical simulations in order to fully explain the experimental observations.

Figure 7.

Mean streamwise velocity profiles for fluid phase (lines) and particle phase (symbols). The grey lines are theoretical solutions of incompressible turbulent channel flow.

In addition, Saffman lift will also increase the fluctuating velocities of particles in the near-wall region of $y^+<10$ in the streamwise, wall-normal and spanwise directions. Depicted in figure 8 are r.m.s. velocity fluctuations of particles normalized by the friction velocity $u_{\tau }$. It is manifested in figure 8(a) that Saffman lift will not only augment the mean streamwise particle velocity, but also increase the particle velocity fluctuations in the streamwise direction. Figure 8(b) shows the profile of r.m.s. particle velocity fluctuations in the wall-normal direction, which should naturally be altered by Saffman lift as it is applied in this direction. It is observed that the fluctuating velocity $v^{p\prime }$, i.e. the major contributor to the wall-normal particle velocity, tends to decrease from the centre of channel to the wall until it reaches the local minimum at approximately $y^+=10$. When the particles go through $y^+=10$, they will be accelerated toward the wall. These results correspond to the mean slip velocity $\langle u_s \rangle \equiv \langle u(\boldsymbol {x}^p)-u^p \rangle$ derived from figure 7, which decides the direction of Saffman lift. A positive $\langle u_s \rangle$ leads to a Saffman lift pointing to the centre of the channel, and a negative one orients the Saffman lift toward the wall. Therefore, there exists a ‘barrier’ at $y^+ \approx 10$ pushing the particles away from this position (as illustrated in figure 4). This ‘barrier’ corresponds to the unstable balance point of the particle velocity in the wall-normal direction. It is this ‘barrier’ that maintains the dynamic balance of particle number in the whole channel, unlike the case without Saffman lift, in which the particles are continuously transported toward the wall.

Figure 8.

The r.m.s. velocity fluctuations of particles normalized by friction velocity $u_{\tau }$ with (blue circles) and without (red circles) Saffman lift in the (a) streamwise direction, (b) wall-normal direction and (c) spanwise direction.

To better understand the impact of Saffman lift on particle motion, we show the joint probability density function (p.d.f.) of the particle velocity fluctuations $u^{p\prime }$ and $v^{p\prime }$ in figure 9. The results are classified into four types according to the position of particles in the channel, i.e. viscous sublayer, buffer layer, logarithmic region and wake region. Specifically, the joint p.d.f.s of case SL and case no SL are compared with each other. As can be seen, the Saffman lift influences a lot the joint p.d.f. in the viscous sublayer and buffer layer. With Saffman lift, one can observe two branches of the joint p.d.f. in the viscous sublayer, which intersect at $u^{p\prime }\approx -0.3$. The two branches represent the particle rebound caused by the wall condition. However, these two branches cannot be observed when Saffman lift is absent and the centre of the joint p.d.f. is located at the origin, which confirms that particles are captured by the viscous sublayer. Above the buffer layer, the p.d.f. patterns of these two cases are similar to each other in shape but not in strength due to the difference in particle numbers within these regions. In particular, the Q2 and Q4 patterns of particle motion in the buffer region are much more important when Saffman lift is considered. It reveals that Saffman lift will not only increase the near-wall particle velocities, but also change the near-wall motion patterns of particles.

Figure 9.

Joint probability distribution function of $u^{p\prime }$ and $v^{p\prime }$ in the viscous sublayer (a,b), buffer layer (c,d), logarithmic region (e, f) and wake region (g,h). Left and right columns are for case SL and case no SL, respectively. Eight contours between the minimum and maximum values of each region are plotted for different cases.

The present numerical results also show that Saffman lift has little effect on the motions of particles in the logarithmic and wake regions where the joint p.d.f.s of two cases are very similar to each other. The little difference in joint p.d.f.s in these two regions is mainly caused by the number of particles. As few particles are present in the logarithmic and wake regions for case no SL, the joint p.d.f. is very spotty due to the lack of sample points.

It should be pointed out that the aforementioned statistical results for the particle velocities are mostly described as being altered at a fixed location or time. To make the discussion more convincing to readers, it is necessary to look into the particle dynamics when it traverses the channel, especially under the buffer region where the Saffman lift is important. Figure 10 shows the position of a particle in the wall-normal direction as a function of its velocity components in the (a) streamwise and (b) wall-normal directions when it travels in the near-wall region. The instantaneous particle is coloured by its acceleration in the corresponding directions. Figure 10(a) confirms that this particle transfers high momentum towards the wall and low momentum away from wall because the particle is decelerated towards the wall and accelerated away from the wall, with the deceleration greater than the acceleration. Figure 10(b) shows how a particle moves in the $y$ direction within the near-wall region. When the particle travels downwards, the particle is first decelerated and then accelerated before it rebounds from the wall and moves upwards. During its upward movement, the particle velocity in the wall-normal direction starts to oscillate due to the interaction between the particle and turbulence. It is the Saffman lift that accelerates the particle when it approaches the wall, which gives the particle enough momentum to traverse the buffer region. Once the particle leaves the viscous sublayer, the particle velocity in the streamwise direction starts to be accelerated, as seen in figure 10(a).

Figure 10.

Wall-normal particle position at different moments vs its velocity components in the (a) streamwise and (b) wall-normal directions. The instantaneous particle (solid circle) is coloured by its acceleration in the same direction. The arrows are oriented to the direction of particle movement.

Figure 11 shows the joint p.d.f. of the particle velocity in the wall-normal direction and particle accelerations under the buffer region at a fixed time. Figure 11(a,b) indicates that there are more particles accelerated and moving away from the wall than those decelerated and moving towards the wall in both the streamwise and wall-normal directions. Interestingly, very few particles are accelerated towards the wall. As a result, the number of particles moving away from the wall is larger than that of particles moving towards the wall. The particles moving towards the wall will transfer high momentum towards the wall and leave with less momentum. Therefore, the mean near-wall particle velocity is mainly determined by the particles moving upwards in a statistical sense. Again, this demonstrates that the streamwise particle velocity is affected by the component in the wall-normal direction.

Figure 11.

Joint p.d.f. of particle velocity in the wall-normal direction normalized by $u_{\tau }$ and particle acceleration normalized by $u_{\tau }/\tau ^+$ in the (a) streamwise and (b) wall-normal directions.

5.2. Effect of Saffman lift on turbulence modulation

This section is intended to investigate the effect of Saffman lift on turbulence modulation based on the same numerical simulations as in previous § 5.1, i.e. particle-laden turbulent channel flows with $St=3$. We first study the turbulence intensities, which are defined as the r.m.s. velocity fluctuations in three different directions as shown in figure 12. It can be observed that the turbulence attenuation caused by particles is very weak in all three directions when Saffman lift is absent. Particularly, the wall-normal and spanwise turbulence intensities in case no SL are very close to the profiles in the clean channel case. By contrast, obvious turbulence attenuation is present in case SL, which leads to the conclusion that Saffman lift will increase the turbulence attenuation. This is because the Saffman lift plays two roles. One is to prevent all particles from being captured by the viscous sublayer, and another is to result in a larger slip velocity in the near-wall region. The former guarantees the existence of enough particles in each layer along the wall-normal direction and the latter makes sure that the feedback force is not too small. These two functions of the Saffman lift finally cause a more important turbulence attenuation.

Figure 12.

Turbulence intensities normalized by friction velocity $u_{\tau }$ in the (a) streamwise direction, (b) wall-normal direction and (c) spanwise direction for cases SP (black solid line), no SL (red dashed line) and SL (blue dash-dotted line).

Another aspect to the study of turbulence modulation is the balance of total stress. Integrating the ensemble-averaged conservation equation of momentum and applying the 1-D statistically steady hypothesis of turbulent channel flow, one obtains

(5.1)\begin{equation} \tau(\kern0.7pt y) \equiv \frac{1}{Re} \left \langle \mu\frac{{\rm d} u}{{\rm d}y} \right \rangle- \langle \rho \rangle \{ u''v'' \} + \int_{0}^{y} \langle \varphi_{u1}\rangle (s) \,{\rm d} s = \tau_w \left( 1-\frac{1}{\delta} \int_{0}^{y} \frac{\langle \rho \rangle}{\rho_w}{\rm d} s \right), \end{equation}

in which $\{\}$ represents the density-weighted average (Favre average) defined as $\{{\cdot }\} = \langle \rho {\cdot } \rangle /\langle \rho \rangle$, and $''$ denotes the Favre fluctuation. The three terms in (5.1) represent the viscous stress $\tau _V$, Reynolds stress $\tau _R$ and particle-induced stress $\tau _P$, respectively. The numerical results of cases SP, no SL and SL are compared in figure 13. In comparison with the clean channel flow, particles cause a reduction of the Reynolds stress, which also serves as evidence for turbulence attenuation. Similar to the turbulence intensity, the reduction in Reynolds stress is more important when the Saffman lift is considered. Besides, the particle-induced stress in case SL is also greater than that in case no SL. These phenomena can be explained by the same argument as employed for the turbulence intensity.

Figure 13.

Profiles of the total stress and its contribution components normalized by $\tau _w$: (a) case SP (symbols) and case no SL (lines); (b) case SP (symbols) and case SL (lines).

The evolution of turbulent kinetic energy (TKE, defined as $k\equiv \{u_i''u_i''\}/2$) can help us understand the interphase energy exchange. The equation of TKE reads

(5.2)\begin{align} \frac{\partial \langle\rho\rangle k}{\partial t} + \frac{\partial \langle\rho\rangle \{u_j\} k}{\partial x_j} + \frac{\partial T_j''}{\partial x_j} &={-}\langle\rho\rangle \{u_i''u_j''\} \frac{\partial \{u_i\}}{\partial x_j} - \frac{1}{Re} \left\langle \sigma_{ij} \frac{\partial u_i''}{\partial x_j} \right\rangle + \langle u_i''\varphi_{ui}\rangle \nonumber\\ &\quad +\frac{1}{\gamma Ma^2} \left\langle p'\frac{\partial u_j''}{\partial x_j}\right\rangle + \frac{1}{\gamma Ma^2} \langle u_j'' \rangle\frac{\partial \langle p \rangle}{\partial x_j}, \end{align}

with $T_j'' = \frac {1}{2}\langle \rho u_i''u_i''u_j'' \rangle - ({1}/{Re}) \langle \sigma _{ij}u_i'' \rangle +({1}/{\gamma Ma^2}) \langle p'u_j'' \rangle$ being the turbulent energy flux, $\mathcal {P} = -\langle \rho \rangle \{u_i''u_j''\}({\partial \{u_i\}}/{\partial x_j})$ the turbulent production term, $\varepsilon = ({1}/{Re}) \langle \sigma _{ij} ({\partial u_i''}/ {\partial x_j}) \rangle \approx 2 ({\langle \mu \rangle }/{Re}) \langle S_{ij}''S_{ij}'' \rangle$ the turbulent dissipation and $F_b = \langle u_i''\varphi _{ui}\rangle$ the particle feedback term. Here, $({1}/{\gamma Ma^2}) \langle p'({\partial u_j''}/{\partial x_j})\rangle$ and $({1}/{\gamma Ma^2}) \langle u_j'' \rangle ({\partial \langle p \rangle }/{\partial x_j})$ are the dilatation and compression by pressure and energy transport by the mean pressure gradient, respectively, which are found to be very small and negligible in compressible turbulent channel flows (Huang, Coleman & Bradshaw Reference Huang, Coleman and Bradshaw1995). In other words, the equation of TKE is dominated by turbulence production, turbulence dissipation and the particle feedback term. Figure 14(a,b) implies that the attenuation of turbulence production and turbulence dissipation is stronger in case SL than in case no SL compared with that in clean channel flow. It needs to be pointed out that the turbulence production and turbulence dissipation in case no SL almost collapse onto the corresponding profiles for clean channel flow. This agrees with our observation for turbulence intensity and Reynolds stress, and proves again that the Saffman lift will increase turbulence attenuation. It is interesting to notice that the particle feedback effect is nearly zero in case no SL. By contrast, the particle feedback term is positive in the viscous sublayer and then becomes negative in the buffer region in case SL. This indicates that the particle feedback force will increase TKE in the viscous sublayer and decrease TKE in the buffer region. However, the particle feedback term is much smaller compared with turbulence production and turbulence dissipation. Therefore, the main effect of the Saffman lift on TKE is the reduction of turbulence production and turbulence dissipation but not the particle feedback term $F_b$ itself.

Figure 14.

Profiles of turbulence production, turbulence dissipation and particle feedback term in the TKE equation normalized by $u_{\tau }^2/2$: (a) case SP (symbols) and case no SL (lines); (b) case SP (symbols) and case SL (lines).

The spectral analysis may help one understand the effect of Saffman lift on turbulence structures. Here, the 1-D pre-multiplied spectral energy density is employed, which is defined as $\phi _{u'u'} = k_z E_{u'u'}(k_z)$, with $E_{u'u'}$ being the streamwise TKE spectrum and $k_z$ the spanwise wavenumber. Figure 15 displays contours of $\phi _{u'u'}$ in the plane of normalized spanwise wavelength $\lambda _z^+$ and wall-normal distance in wall units $y^+$. The TKE at the wavelength around $100\delta _{\nu }$ is massively reduced by particles in the buffer region when Saffman lift is considered. When the Saffman lift is absent, the pattern of pre-multiplied spectral energy density is very similar to that for single-phase flow in the buffer region, which suggests that the particles in the no SL case seem not to modulate the near-wall fluid streaks. Depicted in figure 16(a,b) are contours of $u'$ in the buffer layer ($\kern0.7pt y^+ \approx 10$) and all the particles located below this layer for cases SL and no SL, respectively. On the one hand, there exist clear small-scale particle streaks in the near-wall region while the Saffman lift is absent due to an intermediate particle inertia (Jie et al. Reference Jie, Cui, Xu and Zhao2022). The spanwise length scale of near-wall small-scale particle streaks is of the order $\sim 100 \delta _{\nu }$ (Sardina et al. Reference Sardina, Picano, Schlatter, Brandt and Casciola2011), which is the same as that of the near-wall fluid streaks in single-phase channel flow (Kim, Moin & Moser Reference Kim, Moin and Moser1987). Unlike the fluid streaks, the collaborative behaviour of the particles renders the particle streaks very long and straight (Sardina et al. Reference Sardina, Picano, Schlatter, Brandt and Casciola2011). These collective particle streaks are pretty much like long streamwise fins embedded on the walls, which tend to separate the near-wall fluid streaks, as observed in figure 16(b). On the other hand, Saffman lift reduces significantly the wall accumulation of particles when it is included in the particle motion equation. Consequently, Saffman lift prevents the particles from forming streak-like structures, as observed in figure 16(a).

Figure 15.

Contours of 1-D pre-multiplied spectral energy density $\phi _{u'u'}$ in the plane of spanwise wavelength and wall-normal distance in wall units: (a) case SP, (b) case SL and (c) case no SL. Eight contours are plotted between 0.1 and 0.8 times the maximum value based on the single-phase flow.

Figure 16.

Contours of instantaneous streamwise velocity fluctuation $u'$ in the normalized $x$–$z$ plane at $y^+ \approx 10$ overlaid on particles below this layer: (a) case SL and (b) case no SL.

The modulation of very-large-scale motion (VLSM) suggested by (Wang & Richter Reference Wang and Richter2019) is not observed in the present simulation due to the small Reynolds number. However, the large-scale motion (LSM), which is represented by the ‘handle-like’ structure located at $y^+ \approx 100$ (logarithmic region) of length scale $400<\lambda _z^+<500$ in figure 15(a), is modulated by inertial particles and shows different patterns in case no SL and case SL. This ‘handle’ bends to $y^+ \approx 40$ in case SL (figure 15b) and vanishes in case no SL (figure 15c). In case SL, the intense particle concentration in the logarithmic region (see figure 6b) prevents the LSM from penetrating into the outer region, which limits LSM under the upper edge of the buffer region. In case no SL, the particle accumulation in the near-wall region (see figure 6a) breaks the LSM structures. Therefore, It is believed that the inertial particles not only directly modulate the turbulence but also change the interaction among multi-scale turbulence structures due to particle concentration or clustering along the wall-normal direction.

5.3. Validation of theoretical criterion and models

In this section, the Saffman lift is included in all numerical simulations. The aim is to validate the criterion for neglecting Saffman lift given by (4.5) and the mean streamwise particle velocity model formulated as (4.22) by considering various Stokes numbers. Specifically, numerical simulations with mono-dispersed particles ($\rho _p = 1600$) are carried out by setting $St$ to $0.5$, $0.7$, $1$, $2$, $3$ and $5$.

First of all, the Stokes number (or inertia) effect on particle motion is evaluated as shown in figure 17. The mean streamwise particle velocities at $St=0.5$, $0.7$ and $1$ are very small in the near-wall region, and nearly collapse onto the profile of the mean streamwise velocity of a clean channel flow (black solid line in figure 17a). This is very similar to the result of the case without Saffman lift at $St=3$ in § 5.1. When the particle diameter $d^+$ is larger than the critical value ($d_c^+=0.438$), the mean streamwise particle velocity behaves similarly to that in case SL, i.e. the near-wall particle velocity is larger than that of the background flow. A similar phenomenon can be observed for fluctuating particle velocities. All this evidence leads us to the conclusion that Saffman lift can be neglected for particles of $d^+< d_c^+$ (or equivalently $St< St_c$) in wall-bounded particle-laden turbulent flows. It is clearly seen from figure 17 that the Saffman lift cannot be neglected for the cases with $St>2$ (corresponding to the particle diameter $d^+ > 0.587$). In fact, the particle diameters ($d^+$) in these cases are all smaller than $1$ (as can be seen from table 1), which declares the failure of the criterion $d^+ < 1$ proposed by Costa et al. (Reference Costa, Brandt and Picano2020a). By contrast, the new theoretical criterion for neglecting Saffman lift agrees very well with the present numerical results.

Figure 17.

Profiles of (a) mean streamwise particle velocity and r.m.s. fluctuating particle velocities in the (b) streamwise, (c) wall-normal and (d) spanwise directions normalized by the wall friction velocity. Mean streamwise velocity of the background flow (black solid line) is also included in (a) for comparison purposes.

Another issue is the particle-to-fluid density ratio effect on turbulence modulation at the same $St$ when keeping the particle volume fraction or mass fraction the same. To this end, a series of simulations is conducted to compare the TKE in single-phase turbulent channel flow with that in turbulent channel flows laden with particles of different densities. The particle Stokes number and particle mass loading are fixed to $St=2$ and $\varPhi _m = 0.1$, respectively. According to the criterion for neglect of the Saffman lift in either its particle diameter version or its Stokes number version given in this manuscript, the Saffman lift is considered to be negligible for particles with density ratio $\rho _p/\rho > 1000St/0.7=2000/0.7 \approx 2860$, or equivalently a diameter normalized by the wall viscous length scale of less than $0.438$. Figure 18(a) depicts the TKE in case SP and cases with particles of different densities under the action of Saffman lift. The numerical results show that the TKE is attenuated by inertial particles and the level of turbulence attenuation decreases with the increasing particle density. When the particle-to-fluid density ratio equals $4500$ or $7800$, the attenuation of TKE is weaker than in other cases. However, it is hard to arrive at the conclusion that the Saffman lift plays very little role in turbulence modulation when the criterion is satisfied. It can be observed that, as long as Saffman lift is present, the near-wall turbulence is modified regardless of the particle-to-fluid density ratio. By contrast, figure 18(b) confirms again the conclusion that Saffman lift can be neglected for particle motion when the previous criterion is satisfied. Therefore, the newly proposed criterion is intended to describe the effect of shear-induced lift on particle motion, and the Saffman lift is probably non-negligible for turbulence modulation even when it may not be very important for particle motion.

Figure 18.

(a) The TKE normalized by $u_{\tau }^2$ of the background fluid and (b) the mean streamwise particle velocity in turbulent channel flow laden with particles at $St=2$ with different particle densities.

Then, the streamwise particle velocity $\langle u^p \rangle$ is found to be obviously associated with the Stokes number as shown in figure 17(a). Understanding and predicting the behaviour of $\langle u^p \rangle$ can be of importance for further study of the particle erosion dynamics. A second-order approximation of $\langle u^p \rangle$ is provided in § 4.2 as shown in (4.22). Both the present numerical results and theoretical result are plotted in figure 19 for different Stokes numbers. For the particles with $St$ smaller than the critical value, the theoretical model fits very well with the numerical simulations. For the particles with $St$ larger than the critical value, the theoretical model can recover major information of $\langle u^p \rangle$ down to the near-wall regions, except the region $y^+<1$. The reasonable explanations are twofold, i.e. the statistical aspect and the model aspect. There exist incomplete statistics due to the lack of particles in the very near-wall region, as shown in figure 6, which will introduce oscillations in the calculation of the gradient of $\langle k^p \rangle$. Besides, combining (3.1) and (3.2), one obtains that the total particle number decreases with the increase of Stokes number when the mass loadings for different cases are the same as one another. Therefore, the larger the Stokes number is, the fewer the particles existing in the near-wall region. In other words, the incompleteness of the statistics is more serious in the near-wall region for large Stokes number. Thus, the present theoretical model works worse for the region $y^+<1$ for large Stokes numbers (such as $St=5$) than for smaller ones. The second-order approximation may be another reason for the inaccurate prediction in the near-wall region. However, the theoretical estimation agrees pretty well with numerical simulations in a general view. It is believed that this model can be extended to various wall-bounded flows and inhomogeneous turbulence owing to the generality of (4.16).

Figure 19.

Profiles of mean streamwise particle velocity from the present numerical simulations (black solid lines) and theoretical model (4.22) (symbols) for different Stokes numbers: (a) $St=0.5$; (b) $St=0.7$; (c) $St=1$; (d) $St=2$; (e) $St=3$; ( f) $St=5$.

Furthermore, the success of this theoretical model also proves that the use of the particle velocity in the development of ${\rm d} u/{\rm d} t$, i.e. (4.16), is better than the fluid velocity seen at the particle position, as in Maxey's model, for large inertial particles. In Maxey's model, the wall-normal particle velocity is replaced by the wall-normal fluid velocity. Repeating the same derivation as in Appendix B and expanding $\langle u^p \rangle$ to the first order, one obtains

(5.3)\begin{equation} \langle u^p \rangle \approx \langle u \rangle - \left\langle v'\frac{{\rm d} u'}{{\rm d}y} \right\rangle \langle \tau_p \rangle. \end{equation}

The first-order term can be rewritten as

(5.4)\begin{equation} \langle u^p \rangle \approx \langle u \rangle - \left(\frac{{\rm d} \langle u'v' \rangle}{{\rm d}y}- \left\langle u'\frac{{\rm d} v'}{{\rm d}y} \right\rangle \right) \langle \tau_p \rangle. \end{equation}

Introducing the turbulent dilatation term $\theta ' \equiv {\partial u'}/{\partial x} + {\partial v'}/{\partial y} + {\partial w'}/{\partial z}$, (5.4) can be reorganized as

(5.5)\begin{equation} \langle u^p \rangle \approx \langle u \rangle - \left(\frac{{\rm d} \langle u'v' \rangle}{{\rm d}y}- \langle u'\theta' \rangle - \left\langle u'\frac{\partial u'}{\partial x} \right\rangle -\left\langle u'\frac{\partial w'}{\partial z} \right\rangle \right) \langle \tau_p \rangle. \end{equation}

Under the 1-D hypothesis of channel flow, the terms containing partial derivatives with respect to $x$ and $z$ are negligible. The Reynolds stress in the very near-wall region is also insignificant. The dilatation term is generally very small compared with other terms in compressible channel flow (Huang et al. Reference Huang, Coleman and Bradshaw1995). As a result, Maxey's model gives

(5.6)\begin{equation} \langle u^p \rangle (\kern0.7pt y^+ \to 0) \approx \langle u \rangle (\kern0.7pt y^+ \to 0) = 0, \end{equation}

which implies that $\langle u^p \rangle$ in the near-wall region is always zero regardless of the Stokes number and the particle trajectory along the wall-normal direction. This is obviously unreasonable and does not agree with the experimental and numerical results for particles of large Stokes number. Again, Maxey's model can be applied to the particles of Stokes number smaller than the critical value because Saffman lift does not play an important role, and therefore the particle velocity in the wall-normal direction is approximately equal to the fluid velocity in the wall-normal direction. However, Maxey's model is no longer valid when the Saffman lift effect is non-negligible. This manifests that the new model can be applied to a wider range of Stokes number and contains more physical information than Maxey's model.

In addition, it is necessary to discuss a little about the weakness and prospectives of the present theoretical model (4.22). To the authors’ knowledge, this is the first time a theoretical model has successfully predicted the mean streamwise particle velocity profile in turbulent channel flow. However, the determination of the dimensionless parameter $\beta$ in this model is still an open question. The present data-based method is evidently not good enough. What is more, the parameter $\beta$ may be dependent on the Reynolds number and Mach number, which needs to be evaluated in further studies. The authors believe that the closure of the present model can also be improved. It should be mentioned that the methodology developed in this paper can be extended to all types of wall-bounded turbulent flows. Therefore, it is encouraging to establish and test this methodology in other particle-laden wall-bounded flows such as boundary layer and pipe flows.