4.1. Joint p.d.f. of invariants and the volume fraction of flow topologies

Figure 4(a) shows the joint p.d.f. of invariants $Q$ and $R$ for the whole channel. Since the compressibility is relatively weak in the present case, the change of dilatation has not been considered here. It is seen that the contours of the joint p.d.f. of invariants present a self-similar ‘tear-drop’ shape, which is consistent with the previous results for different types of turbulence (Ooi et al. Reference Ooi, Martin, Soria and Chong1999; Tsinober Reference Tsinober2009; Meneveau Reference Meneveau2011). To study the invariants’ dynamic evolution, Martín et al. (Reference Martín, Ooi, Chong and Soria1998b) adopted the CMT in the space of $Q$–$R$ to describe the mean Lagrangian evolution of the invariants. Here, we also calculate the conditional mean of the material derivative of the invariant, and regard it as the conditional mean velocity of the invariant in the $Q$–$R$ plane:

(4.1)\begin{equation} \boldsymbol{v}=(v_R,v_Q) =\left\langle\left.\left(\frac{\mathcal{D}R}{\mathcal{D}t}, \frac{\mathcal{D}Q}{\mathcal{D}t}\right)\right\vert (R,Q )\right\rangle. \end{equation}

By integrating $\boldsymbol {v}$, the CMT of the invariant can be obtained.

Figure 4.

(a) The log scale of joint p.d.f. of invariants $Q$ and $R$ for the whole channel. (b) The CMTs of $Q$ and $R$, with the start points $(0.25,0),\ (1,0),\ (2,0)$ and $(3,0)$ denoted by square symbols and the end points denoted by triangular symbols. The vectors represent the conditional mean velocity $\boldsymbol {v}$ of invariants. Here $Q$ and $R$ are normalized by $Q_0=\int _{-h}^{h} Q_{rms}(y)\,\mathrm {d}y/2h$.

From the perspective of probability conservation, Lozano-Durán et al. (Reference Lozano-Durán, Holzner and Jiménez2015) and Lozano-Durán, Holzner & Jiménez (Reference Lozano-Durán, Holzner and Jiménez2016) claimed that the CMT must be a closed curve for the flow in a statistical steady and homogeneous turbulence or in a periodic domain, and the closure behaviour of the trajectory is significantly affected by the numerical accuracy. Figure 4(b) shows the conditional mean trajectories of the invariants for the whole channel. Four trajectories that start from four different positions on the axis $Q=0$ and arrive at positions near their start positions after a clockwise circle are shown, which form closed curves, suggesting that the present calculation's numerical accuracy is sufficient.

To understand the probability of different flow topologies arising in the channel, the volume fraction distribution of different flow topologies along the wall-normal direction ($y^+$) is shown in figure 5. It is found that when $y^+\gtrsim 150$ the proportion of flow topologies varies little with $y^+$, which is similar to the statistical results of homogeneous isotropic turbulence (Suman & Girimaji Reference Suman and Girimaji2010). In the inner layer of $y^+ \lesssim 150$, the proportion of different flow topologies varies significantly due to the increase of mean shear (Wang & Lu Reference Wang and Lu2012). The flow topology S4 has the largest volume fraction in the whole channel since the flow field is full of vortices. When $y^+ \gtrsim 100$, except for S4, the proportions of S2, S1 and S3 are also relatively high and decrease in turn. When $y^+ \lesssim 100$, with the decrease of $y^+$, the ratio of unstable focus and compressed topology S1 increases to the same probability as S4, and the probability of S2 decreases to the same probability as S3. The volume fraction of the flow topologies S5–S8 due to compressibility is also very low, which can be neglected in all other regions except very close to the wall. As a result, in the following discussions, these flow topologies (i.e. S5–S8) will be ignored. In contrast, the effects of the flow topologies S1–S4 on the statistical properties of the pressure-Hessian tensor will be discussed in detail.

Figure 5.

The distribution of volume fraction of different flow topologies versus $y^+$.

4.2. The geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor

Since the vorticity and strain rate are represented by the antisymmetric part and symmetric part of the VGT, understanding the geometrical relationship between the vorticity and the pressure-Hessian tensor is of great significance to explore the non-local effect of the pressure-Hessian tensor on the dynamics of the VGT (Suman & Girimaji Reference Suman and Girimaji2013). The analysis of incompressible isotropic turbulence reveals that the vorticity tends not only to be parallel to the second principal direction of the strain-rate tensor ($\boldsymbol {e}_2$) but also to be parallel to the second principal direction of the pressure-Hessian tensor. The first and third principal directions of the pressure-Hessian tensor tend to be in the plane $\boldsymbol {e}_1$–$\boldsymbol {e}_3$, and the preferential relative angles with $\boldsymbol {e}_1$ and $\boldsymbol {e}_3$ are about ${\rm \pi} /4$ (Ohkitani Reference Ohkitani1993; Kalelkar Reference Kalelkar2006). Here, the second principal direction corresponds to the second eigenvector when the eigenvalues are ordered in descending order, rather than the principal direction $\boldsymbol {f}_2$ defined in this study. These findings have also been confirmed to be valid in compressible isotropic decaying turbulence. Lawson & Dawson (Reference Lawson and Dawson2015) found that if the eigenvalues of the pressure-Hessian tensor (including the isotropic part) were sorted by their absolute values, the third principal direction appears to be aligned with the vorticity in the rotation-dominated regions (i.e. $Q>0$). In the strain-dominated regions, the third principal direction tends to be parallel to $\boldsymbol {e}_2$. Next, we will present the alignment statistics of the pressure-Hessian tensor with the strain-rate tensor/vorticity in the present wall turbulence, and the influence of different flow topologies is also considered.

To describe the vectors’ geometrical relationship, the azimuthal angle $\phi$ and polar angle $\theta$ are defined (Zamansky, Vinkovic & Gorokhovski Reference Zamansky, Vinkovic and Gorokhovski2011; Lawson & Dawson Reference Lawson and Dawson2015), as illustrated in figure 6. Figure 7 shows the statistical results of the joint p.d.f. $\mathcal {P}(\theta _\omega, \phi _\omega )/\mathcal {P}_{max}$ of the vorticity in the central plane of the channel ($y^+=532$) for the flow topologies S1–S4. As shown in figure 7(a,b), the statistical properties are generally similar to those of incompressible isotropic turbulence (Lawson & Dawson Reference Lawson and Dawson2015). In the S1 and S4 flow topologies with focus, the maximum value of p.d.f. in the plane $\theta _\omega$–$\phi _\omega$ occurs near $({\rm \pi} /2,{\rm \pi} /2)$. The vorticity is perpendicular to $\boldsymbol { f}_1$, $\boldsymbol {f}_2$, and aligned with $\boldsymbol {f}_3$. In the regions of flow topology S2, the large values of p.d.f. are relatively concentrated near $\phi _\omega ={\rm \pi} /2$, indicating that the vorticity is preferentially perpendicular to $\boldsymbol {f}_1$. The values of p.d.f. in the range of $\theta _\omega \in [-{\rm \pi} /2,{\rm \pi} /2]$ around $\phi _\omega ={\rm \pi} /2$ are relatively large, suggesting that the possible direction of the vorticity is distributed in the plane $\boldsymbol {f}_2$–$\boldsymbol { f}_3$. In the regions of flow topology S3, the p.d.f. has a similar distribution to that of S2, but the distribution is wider, centred at $\phi _\omega ={\rm \pi} /2$. These results show that in the channel's central regions, the flow has similar statistical properties to those of isotropic turbulence. The geometrical relationships between the vorticity and the principal directions of the pressure-Hessian tensor show similar properties in the strain-dominated regions and the rotation-dominated regions, regardless of whether the topologies belong to the stable topology or the unstable topology.

Figure 6.

(a) The polar angle $\theta _\omega$ and the azimuthal angle $\phi _\omega$ of the vorticity in the principal coordinate system of the pressure-Hessian tensor. (b) The polar angle $\theta _i$ and the azimuthal angle $\phi _i$ of $\boldsymbol {f}_i$ in the principal coordinate system of the strain-rate tensor. Since the orientation of the principal direction can differ by a negative sign, when $\phi _\omega <0$, the value can be set as $\phi _\omega +{\rm \pi}$. This point is the same for $\phi _i$.

Figure 7.

The joint p.d.f. $\mathcal {P}(\theta _\omega, \phi _\omega )/\mathcal {P}_{max}$ of the vorticity in the central plane of the channel ($y^+=532$) for different flow topologies: (a) S4, (b) S1, (c) S3 and (d) S2.

Figure 8 shows the joint p.d.f. $\mathcal {P}(\theta _\omega, \phi _\omega )/\mathcal {P}_{max}$ of the vorticity in the horizontal plane inside the buffer layer of $y^+=20.5$ for different flow topologies. It is seen that in the regions of flow topology S4, there are three pairs of angles with peak p.d.f. values, which are roughly $(\theta _\omega, \phi _\omega ) = ( {\rm \pi}/2, {\rm \pi}/2 )$, $(2{\rm \pi} /3, {\rm \pi}/8)$ and $({\rm \pi} /3,7{\rm \pi} /8)$. The pair $({\rm \pi} /2, {\rm \pi}/2)$ means that the vorticity is aligned with $\boldsymbol {f}_3$, the same as that in the central plane of the channel. Furthermore, by analysing the statistical results of more near-wall positions, it is found that, with the decrease of wall distance, the angle $\phi _\omega$ corresponding to the peak p.d.f. value gradually shifts from $\phi _\omega ={\rm \pi} /2$ to $\phi _{\omega } = 0$ or ${\rm \pi}$, and finally $(\theta _\omega, \phi _\omega )$ of the peak p.d.f. value reaches $({\rm \pi} /2,0 )$ or $( {\rm \pi}/2, {\rm \pi})$ in the viscous sublayer. This result shows that when approaching the wall, the vorticity gradually changes from being parallel to $\boldsymbol {f}_3$ to being aligned with $\boldsymbol {f}_1$ due to the effect of strong mean shear.

Figure 8.

(a) The joint p.d.f. $\mathcal {P}(\theta _\omega, \phi _\omega )/\mathcal {P}_{max}$ of the vorticity in the buffer layer ($y^+=20.5$) for different flow topologies: (a) S4, (b) S1, (c) S3 and (d) S2.

Meanwhile, in the regions of flow topology S1, the large p.d.f. values occur near $(\theta _\omega, \phi _\omega ) = ( {\rm \pi}/2,0)$, $({\rm \pi} /2, {\rm \pi})$ and $(0,0 \sim {\rm \pi})$ or $({\rm \pi}, 0 \sim {\rm \pi})$, that is, the vorticity is aligned with $\boldsymbol {f}_1$ or $\boldsymbol {f}_2$. The vorticity in the viscous sublayer (S1 region) is ultimately aligned with $\boldsymbol {f}_1$. Figure 8(c,d) shows that in the strain-dominated regions, i.e. the regions of flow topology S2 and S3, the distributions of joint p.d.f. are similar, the p.d.f. has a large value as $\phi _\omega =0,{\rm \pi}$ and $\theta _\omega =0,{\rm \pi}$, suggesting that the most possible direction of the vorticity lies in the plane $\boldsymbol {f}_1$–$\boldsymbol {f}_2$.

Figure 9 shows the most possible distribution sketch maps of the vorticity direction in the turbulent central regions and buffer layers in the principal coordinate system of the pressure-Hessian tensor. The above results show that in wall turbulence, the geometrical relationship between the vorticity and the pressure-Hessian tensor changes significantly near the wall due to mean shear, implying that the pressure-Hessian tensor is strongly affected by inhomogeneity and anisotropy. Based on (3.15), it is known that the change of the geometrical relationship will directly affect the contribution of the pressure-Hessian tensor to the generation of $R_\omega$.

Figure 9.

The most possible distribution sketch maps of the vorticity direction in the principal coordinate system of the pressure-Hessian tensor for different flow topologies at different vertical positions: (a) $y^+=532$ and (b) $y^+=20.5$. The thick red line with arrowhead indicates the vorticity direction, and the fine lines with arrows are the principal directions of the pressure-Hessian tensor.

Next, we analyse the distribution characteristics of the eigenvalues of the pressure-Hessian tensor. We arrange the eigenvalues $\eta _1$, $\eta _2$ and $\eta _3$ of $\tilde {\boldsymbol{\mathsf{H}}}$ in descending order, and mark them as $\eta _1'\geq \eta _2'\geq \eta _3 '$, and define the normalized eigenvalues as

(4.2)\begin{equation} \eta_i^*=\frac{\eta_i'}{\sqrt{\eta_1'^2+\eta_2'^2+\eta_3'^2}}. \end{equation}

Since the trace of $\tilde {\boldsymbol{\mathsf{H}}}$ is zero, i.e. $\eta _1'+\eta _2'+\eta _3'=0$, then the $\eta _i^*$ must lie in the ranges

(4.3a–c)\begin{equation} \eta_1^*\in[\sqrt{6}/6,\sqrt{6}/3],\quad \eta_2^*\in[-\sqrt{6}/6,\sqrt{6}/6],\quad \eta_3^*\in[-\sqrt{6}/3,-\sqrt{6}/6]. \end{equation}

Figure 10 shows the p.d.f. of normalized eigenvalues of the pressure-Hessian tensor. As shown in figure 10(a), in the central region of the channel, different flow topologies have a certain impact on the p.d.f. of eigenvalues. In the S1 and S4 flow topological regions, the maximum values of p.d.f. for $\eta _1^*$ and $\eta _3^*$ are $\eta _1^*=\sqrt {6}/3$ and $\eta _3^*=-\sqrt {6 }/3$, while in the S4 flow topological regions, $\eta _2^*$ reaches the peak value of p.d.f. near zero, and the peak value of p.d.f. for S1 topology corresponds to a positive $\eta _2^*$. Since the flow topology S4 is closely related to the vortex structure, it appears that the pressure-Hessian tensor tends to have the first and third eigenvalues with equal absolute value and the second eigenvalue with a value close to zero in the vortical regions. In the regions of strain-dominated flow topology, when the p.d.f. values reach their peaks, the preferred ratio of the corresponding eigenvalues is about $\eta _1^*:\eta _2^*:\eta _3^*=-5:1:4$. From figure 10(a,b), it can be found that there is no significant change in the p.d.f.s of the eigenvalues in the central plane and the buffer layer, which means that the eigenvalues of the pressure-Hessian tensor are not sensitive to the mean shear.

Figure 10.

The p.d.f. of normalized eigenvalues of the pressure-Hessian tensor at different positions: (a) the central plane, $y^+=532$, and (b) the buffer layer, $y^+=20.5$.

To analyse the geometrical relationship between the pressure-Hessian tensor and the strain-rate tensor, figure 11 shows the joint p.d.f. $\mathcal {P}(\theta _i,\phi _i)/\mathcal {P}_{max}$ of the polar and azimuthal angles of $\boldsymbol {f}_i$ in the principal coordinate system of the strain-rate tensor in the central channel region and the buffer layer. Since the strain rate generally has a large value in the strain-dominated region, only the statistical properties in the S2 and S3 topological regions are considered here. Figure 11(a,c,e) shows that in the central region of the channel, the most preferred angles of $\boldsymbol {f}_1$ are $( \theta _1, \phi _1 ) = ( {\rm \pi}/2,3{\rm \pi} / 4)$, and the most preferred angles of $\boldsymbol {f}_2$ are $(\theta _2,\phi _2)=(0,0 - {\rm \pi} )$ or $({\rm \pi},0 - {\rm \pi} )$. In short, as shown in the figure 12(a), $\boldsymbol {f}_3$ is aligned with $\boldsymbol {e}_2$ in the central region of the channel, $\boldsymbol {f}_1$ and $\boldsymbol {f}_2$ lie in the plane $\boldsymbol {e}_1$–$\boldsymbol {e}_3$ and the preferred angle with $\boldsymbol {e}_1$ is ${\rm \pi} /4$, which is consistent with the finding in isotropic turbulence (Lawson & Dawson Reference Lawson and Dawson2015).

Figure 11.

The joint p.d.f. of the polar and azimuthal angles, $\mathcal {P}(\theta _i,\phi _i)/\mathcal {P}_{max}$, for $\boldsymbol {f}_i$ in the principal coordinate system of the strain-rate tensor in the strain-dominated regions. Panels (a,b), (c,d) and (e, f) correspond to $\boldsymbol {f}_1$, $\boldsymbol {f}_2$ and $\boldsymbol {f}_3$, respectively; (a,c,e) show the central statistics at $y^+=532$ and (b,d, f) show the buffer layer statistics at $y^+=20.5$.

Figure 12.

Schematic diagrams of the geometrical relationships between $\boldsymbol {f}_i$ and $\boldsymbol {e}_i$ for S2 and S3 topological regions at different wall-normal positions: (a) $y^+=532$ and (b) $y^+=20.5$.

Figure 11(b,d, f) illustrates that, in the buffer layer, the preferred angles of $\boldsymbol {f}_1$ in the buffer layer are $(\theta _1,\phi _1)=({\rm \pi} /2,3{\rm \pi} /4)$, and the preferred angles of $\boldsymbol {f}_3$ are $(\theta _3,\phi _3)=({\rm \pi} /2,{\rm \pi} /4)$. The eigenvector $\boldsymbol {f}_2$ has multiple preferred directions, e.g. $(\theta _2,\phi _2)=(0,0 - {\rm \pi} )$, $({\rm \pi} /2,{\rm \pi} /4)$ and $({\rm \pi} /2,3{\rm \pi} /4)$. In short, the geometrical relationship between $\boldsymbol {f}$ and $\boldsymbol {e}$ is illustrated in figure 12(b), i.e. $\boldsymbol {f}_2$ is aligned with $\boldsymbol {e}_2$, $\boldsymbol {f}_1$ and $\boldsymbol {f}_3$ lie in the plane $\boldsymbol {e}_1$–$\boldsymbol {e}_3$ and the preferred angle with $\boldsymbol {e}_1$ is ${\rm \pi} /4$.

These geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor have an apparent statistical propensity, as seen by the above results. With the decrease of wall distance $y^+$, these geometrical relationships present near-wall statistical properties close to the wall that are significantly different from those in isotropic turbulence. This will be directly related to the pressure-Hessian tensor's contribution to the evolution of the invariant dynamics, which will be further analysed in the following subsection.

4.3. The effects of pressure-Hessian tensor on mean transport of invariants

Firstly, the mean transport equations of each invariant are investigated to understand better the dominant balance mechanisms, especially the role of the pressure-Hessian term. Figure 13 shows the wall-normal profiles of the contribution terms in (A3)–(A5) for $Q$, $Q_S$ and $Q_W$. When $y^+\lesssim 100$, the transport equation of $Q$ is mainly balanced by the pressure-Hessian term $QH$, the viscosity term $QV$ and the convection term $QC$, while the contribution of nonlinear term $QS$ is relatively small. The transport equation of the opposite value of mechanical energy dissipation $Q_S$ is dominated by the viscous term $Q_S V$, nonlinear term $Q_SS$ and convection term $Q_SC$, but the contribution of the pressure-Hessian term $Q_SH$ is close to zero. The transport equation of enstrophy $Q_W$ is balanced by the generation term $Q_W S$, the dissipation term $Q_W V$ and the convection term $Q_W C$, which does not directly contain the contribution of the pressure-Hessian term. The above phenomena are generally consistent with the statistical properties in other wall turbulence (Balint, Wallace & Vukoslavčević Reference Balint, Wallace and Vukoslavčević1991; Pirozzoli & Grasso Reference Pirozzoli and Grasso2004; Chu et al. Reference Chu, Wang and Lu2014). By partial integration, the contribution of the pressure-Hessian term to the evolution of dissipation can be expressed as (Tsinober Reference Tsinober2009)

(4.4)\begin{equation} Q_S H =QH={-}\frac{1}{\rho}\left[ \frac{\partial}{\partial x_i}\left({\mathsf{S}}_{ik}\frac{\partial p}{\partial x_k} -p\frac{\partial {\mathsf{S}}_{ki}}{\partial x_k} \right) +p\frac{\partial^2 P}{\partial x_k\partial x_k}\right] -\frac{1}{3}P{\mathsf{H}}_{kk}. \end{equation}

When the flow is homogeneous and incompressible (i.e. $P=0$ and the density is constant), then $\overline {Q_SH}= 0$. In other words, for the incompressible homogeneous flow, the mean contribution of the pressure-Hessian term to the generation of dissipation should be strictly zero. However, due to inhomogeneity and the small but non-zero compressibility, this mean contribution is not zero in the current turbulence, as seen in the wall-normal profile of $QH$. On average, the pressure-Hessian term has a significant contribution to the transport of different components of the VGT, i.e. it mainly redistributes the energy of the VGT between various components (Vreman & Kuerten Reference Vreman and Kuerten2014; Gerolymos & Vallet Reference Gerolymos and Vallet2016).

Figure 13.

The mean transport equations of invariants (a) $Q$, (b) $Q_S$ and (c) $Q_W$ given by (A3)–(A5) versus  $y^+$.

Figure 14 shows the profiles of each term in the mean transport equations of the third invariants ($R$, $R_S$) and $R_\omega$ versus $y^+$. It is seen that the transport of $R$ is balanced by the nonlinear generation term $RS$ and the pressure-Hessian term $RH$. When $y^+\lesssim 100$, the pressure-Hessian term $R_SH$, viscous term $R_SV$, nonlinear generation term $R_SS$ and convection term $R_SC$ are the dominant terms in the transport equation of $R_S$. On average, the pressure-Hessian term $R_SH$ has a negative contribution to the generation of $R_S$, the same as the viscous term $R_SV$. A similar balance mechanism is also found in the central region of the channel. For the transport of $R_\omega$ in the buffer layer and viscous sublayer ($y^+ \lesssim 30$), the dominant balance mechanism is the same as that of $R_S$, with the pressure-Hessian term showing a negative contribution and weakening $R_\omega$. In the outer layer ($y^+\gtrsim 30$), the mean effect of the pressure-Hessian term is positive, which is balanced by the sum of the viscous and nonlinear generation terms.

Figure 14.

The mean transport equations of invariants (a) $R$, (b) $R_S$ and (c) $R_\omega$ given by (A7)–(A9) versus  $y^+$.

4.4. The decomposition of pressure-Hessian effect

Owing to the strong intermittence of enstrophy, dissipation and their generation processes (Tsinober Reference Tsinober2009), statistical averaging may cover up the real physical insights and thus cannot reflect the turbulent dynamic process. Based on the pressure-Hessian decomposition equations (3.13)–(3.15), the detailed contribution of different components and flow topologies to the dynamic processes of invariants is further discussed. To compare the relative magnitudes of the terms in (3.13)–(3.15), we define

(4.5a)$$\begin{gather} \varphi_{Q_SH}=\frac{-Q_SH}{\sum_k|\lambda_k\hat{{\mathsf{H}}}_{kk}|},\quad \varphi_{i;Q_SH}=\frac{-\lambda_i\hat{{\mathsf{H}}}_{ii}}{\sum_k|\lambda_k\hat{{\mathsf{H}}}_{kk}|}, \end{gather}$$

(4.5b)$$\begin{gather}\varphi_{R_SH}=\frac{R_SH}{\sum_k|\lambda^2_k\hat{{\mathsf{H}}}_{kk}|},\quad \varphi_{i;R_SH}=\frac{\lambda^2_i\hat{{\mathsf{H}}}_{ii}}{\sum_k|\lambda^2_k\hat{{\mathsf{H}}}_{kk}|}, \end{gather}$$

(4.5c)$$\begin{gather}\varphi_{R_\omega H}=\frac{4 R_\omega H/\omega^2}{\sum_k| \eta_k\cos^2 (\boldsymbol{f}_k,\boldsymbol{\omega}) |},\quad \varphi_{i;R_\omega H}=\frac{-\eta_i\cos^2 (\boldsymbol{f}_i,\boldsymbol{\omega})} {\sum_k|\eta_k\cos^2 (\boldsymbol{f}_k,\boldsymbol{\omega})|}, \end{gather}$$

which reflect the relative proportions of the various terms of the pressure-Hessian term.

The effect of the pressure-Hessian tensor on the transport of $Q_S$ can be represented by $Q_SH$, and figure 15 shows the profiles of $\varphi _{Q_SH}$ and its components versus $y^+$. When $y^+\gtrsim 100$, the components $\varphi _{i;Q_SH}$ basically vary little with $y^+$. In the buffer layer and the viscous sublayer, the $\varphi _{i;Q_SH}$ change significantly, indicating that the mean shear leads to changes in the action mechanism of the pressure-Hessian term on $Q_S$. The term $Q_SH$ is dominated by different mechanisms in different flow topologies. As shown in figure 15(a), for the S4 topology, the pressure-Hessian tensor can promote all three principal strain rates in the central region of the channel, increase the amplitude of the strain rate and enhance the dissipation, and the contribution to the third principal strain rate is dominant. With the decrease of $y^+$, the contribution of the pressure-Hessian tensor on the third principal strain rate gradually changes from promotion to inhibition, and it will counteract the contribution of $Q_SH$ on the first principal strain rate, resulting in the overall effect of the pressure-Hessian tensor on dissipation near the wall being zero on average.

Figure 15.

The profiles of $\varphi _{Q_SH}$, $\varphi _{1;Q_SH}$, $\varphi _{2;Q_SH}$ and $\varphi _{3;Q_SH}$ versus $y^+$ for different flow topologies: (a) S4, (b) S1, (c) S3 and (d) S2.

Figure 15(b) shows that for S1 topology, the promotion of the pressure-Hessian tensor on the first principal strain rate and its inhibition on the second principal strain rate cancel each other. Its contribution to dissipation is mainly reflected by its inhibition on the third principal strain rate in the channel's central region. In the vicinity of the channel wall, the pressure-Hessian tensor weakens the inhibition of the second and third principal strain rates. It enhances the promotion of the first principal strain rate, resulting in the weakening of the overall dissipation inhibition. Figure 15(c) shows that, for S3 topology, the pressure-Hessian tensor's effect on the first principal strain rate plays a dominant role, and its effect on the third principal strain rate is restrained, while its effect on the second principal strain rate is close to zero, which results in the promotion of overall dissipation in the channel's central region. Near the wall, the pressure-Hessian tensor's promotion effect on the first principal strain rate and the inhibition effect on the third principal strain rate are weakened. However, it promotes the generation of the second principal strain rate significantly, enhancing the dissipation in the viscous sublayer. Figure 15(d) shows that for the S2 topology, the effect of the pressure-Hessian tensor on the strain rate in the centre of the channel is consistent with that for the S1 topology. In the buffer layer and viscous sublayer, the pressure-Hessian tensor on the first principal strain rate changes from promotion to inhibition. Its effect on the third principal strain rate reaches the maximum relative inhibition at $y^+ \approx 20$, and then gradually becomes promotion in the viscous sublayer.

Figure 16 shows the mechanisms of the effect of the pressure-Hessian tensor on the processes of dissipation generation $R_S$ in different flow topological regions. Figure 16(a) shows that for the S4 topology, the promoting effect of the pressure-Hessian tensor on dissipation generation is mainly due to its action on the third principal strain rate, and its action on the first principal strain rate leads to the inhibition of dissipation generation, and the effect on the second principal strain rate has a relatively small mean contribution to dissipation generation. Near the wall, the pressure-Hessian tensor's suppression effect on the dissipation generation caused by the first principal strain rate's action increases relatively. However, the dissipation generation caused by its action on the third principal strain rate becomes suppressed, so the overall dissipation generation declines near the wall.

Figure 16.

The profiles of $\varphi _{R_SH}$, $\varphi _{1;R_SH}$, $\varphi _{2;R_SH}$ and $\varphi _{3;R_SH}$ versus $y^+$ for different flow topologies: (a) S4, (b) S1, (c) S3 and (d) S2.

As shown in figure 16(b,d), the influence of the pressure-Hessian tensor on dissipation generation in the outer layer of the channel for the S1 and S2 topologies is similar in the outer layer of the channel, which is controlled by the effect of the pressure-Hessian tensor on the third principal strain rate, inhibiting dissipation generation. Near the wall, in the S1 topological region, the inhibition effect of the pressure-Hessian tensor on the first principal strain rate enhances, while the inhibition effect on the third principal strain rate weakens. The effect of the pressure-Hessian tensor on the first and third principal strain rates leads to dissipation generation from inhibition to promotion in the S2 topological region. In the S3 topological region, the effect of the pressure-Hessian tensor on the first and third principal strain rates leads to the inhibition and dominance of dissipation generation, which leads to the inhibition of overall dissipation generation, as shown in figure 16(c).

Finally, the pressure-Hessian tensor's contribution to the enstrophy generation, $R_\omega H$, is investigated. Figure 17 shows the wall-normal profiles of $\varphi _{R_\omega H}$ and its components versus $y^+$. As shown in figure 17(a,b), for the focused topologies S1 and S4, the pressure-Hessian tensor plays a significant role in promoting the enstrophy generation in the central region of the channel, mainly contributed by the third eigenvalue of the pressure-Hessian tensor. The reason is that in the topologies S1 and S4, the vorticity tends to be parallel to $\boldsymbol {f}_3$ in the third principal direction of the pressure-Hessian tensor and perpendicular to other principal directions (Lawson & Dawson Reference Lawson and Dawson2015). For S1 and S4 flow topologies, the vorticity tends to be aligned with the principal direction $\boldsymbol {f}_2$ near the wall. Therefore, the second eigenvalue's contribution gradually dominates, and the overall effect of the pressure-Hessian tensor tends to inhibit the enstrophy generation.

Figure 17.

The profiles of $\varphi _{R_\omega H}$, $\varphi _{1;R_\omega H}$, $\varphi _{2;R_\omega H}$ and $\varphi _{3;R_\omega H}$ versus $y^+$ for different flow topologies: (a) S4, (b) S1, (c) S3 and (d) S2.

As shown in figure 17(c,d), for the topologies S2 and S3, the effects of the first and second eigenvalues of the pressure-Hessian tensor on the generation term of enstrophy in the central region of the channel are opposite, and their amplitudes are approximately the same and cancel each other out. Finally, $\varphi _{R_\omega H}$ has a similar distribution versus $y^+$ with $\varphi _{3;R_\omega H}$, which is shown to inhibit the enstrophy generation. Here, the vorticity tends to be distributed in the plane $\boldsymbol {f}_2$–$\boldsymbol {f}_3$. However, because the absolute value of $\eta _1$ is usually significant, its effect on the enstrophy generation term is still relatively strong. In the buffer layer, the vorticity tends to be distributed in the plane $\boldsymbol {f}_1$–$\boldsymbol {f}_2$, the effect of the third eigenvalue is small, and the effect of the first eigenvalue is relatively strong and dominant.

4.5. The effects of pressure-Hessian tensor on dynamic evolution of invariants

To further understand the effect of the pressure-Hessian tensor on invariants’ dynamic evolution, we will analyse its effect on the mean trajectories of invariants in the plane $Q$–$R$. The conditional mean velocities of the invariants $Q$ and $R$ in the plane $Q$–$R$ can be split into

(4.6a)$$\begin{gather} \boldsymbol{v}=\boldsymbol{v}_S+\boldsymbol{v}_H+\boldsymbol{v}_V, \end{gather}$$

(4.6b)$$\begin{gather}\boldsymbol{v}_S=(v_{RS}, v_{QS})= \langle(RS,QS)\,| \,(R,Q)\rangle, \end{gather}$$

(4.6c)$$\begin{gather}\boldsymbol{v}_H=(v_{RH}, v_{QH})= \langle(RH,QH)\,| \,(R,Q)\rangle, \end{gather}$$

(4.6d)$$\begin{gather}\boldsymbol{v}_V=(v_{RV}, v_{QV})= \langle(RV,QV)\,| \,(R,Q)\rangle, \end{gather}$$

in which $\boldsymbol {v}_S$, $\boldsymbol {v}_H$ and $\boldsymbol {v}_V$ represent the contributions of nonlinear interaction, the pressure-Hessian tensor and viscosity to the conditional mean velocity, respectively.

Figure 18 shows the streamlines (i.e. CMTs) corresponding to conditional mean velocity vectors of invariants $Q$ and $R$. As shown in figure 18(a,b), the CMT of invariants given by $\boldsymbol {v}$ converges to the origin clockwise in the channel's central region and it diverges clockwise outwards in the buffer layer, similar to previous studies for wall turbulence (Elsinga & Marusic Reference Elsinga and Marusic2010; Atkinson et al. Reference Atkinson, Chumakov, Bermejo-Moreno and Soria2012; Chu & Lu Reference Chu and Lu2013; Bechlars & Sandberg Reference Bechlars and Sandberg2017). Figure 18(c,d) shows that the velocity $\boldsymbol {v}_S$ due to the nonlinear interaction of the invariants prompts the invariants to evolve from $(R,Q) = (-\infty, -\infty )$ to $(R,Q) = (+\infty, -\infty )$ along the curve $\varDelta =0$, similar to the property that the invariants tend to diverge in the RE equations. The viscosity term reflects the dissipation of the invariant, which makes the streamline converge to the origin, as shown in figure 18(g,h). It should be emphasized that the contribution of the pressure-Hessian term is opposite to that of the nonlinear interaction term, which makes the invariants evolve from $(R, Q)=(+\infty, -\infty )$ to $(R, Q)=(-\infty,-\infty )$ along the right branch of the curve $\varDelta = 0$ (see figure 18e, f). It is this behaviour of the pressure Hessian that resists the divergent effect of the RE equation on the invariants (Lüthi et al. Reference Lüthi, Holzner and Tsinober2009; Meneveau Reference Meneveau2011; Wilczek & Meneveau Reference Wilczek and Meneveau2014; Lawson & Dawson Reference Lawson and Dawson2015).

Figure 18.

The streamlines (i.e. CMTs) corresponding to conditional mean velocity vectors of invariants $Q$ and $R$: (a,b) $\boldsymbol {v}$; (c,d) $\boldsymbol {v}_S$; (e, f) $\boldsymbol {v}_H$; and (g,h) $\boldsymbol {v}_V$. Panels (a,c,e,g) show the statistical results in the central plane of the channel ($y^+=532$). Panels (b,d,f ,h) show the statistical results in the buffer layer ($y^+=20.5$). The background colours denote the amplitude of the corresponding vector, and the region of joint p.d.f. value less than $10^{-4} \max [\mathrm {p.d.f.} (R,Q) ]$ is omitted. Here, the physical quantities are normalized by the root mean square of fluctuating $Q$.

Figure 18(e) further shows that in the central region of the channel, there is a region of the S1 topology caused by the pressure-Hessian effect that drives the invariant to spin outwards anticlockwise, where the term $\boldsymbol {v}_H$ is very weak. This feature changes significantly in the buffer layer, as shown in figure 18( f). The anticlockwise spiral region caused by the pressure-Hessian tensor becomes divergent from the asymptote to the left-hand and right-hand sides, and the streamline is nearly parallel to the $R$-axis. The above results lead, in the nearby region of $R>0, Q=0$, to the pressure-Hessian tensor making the invariant $R$ tend to diverge, but its effect on $Q$ is relatively tiny.

To further understand the effects of the pressure-Hessian tensor on the CMTs of invariants, figure 19 shows the evolution of the pressure-Hessian's contribution along the mean trajectory of invariants (see figure 18a) in the channel's centre, in which the contribution has been decomposed by (3.13)–(3.15). Figure 19(a,b) shows that the effect of the pressure-Hessian tensor on the dynamic evolution of $Q_S$ and $R_S$ is mainly realized by acting on the first principal strain rate. The more the invariant evolves to the origin, the more dominant is the effect on the first principal strain rate. The pressure-Hessian tensor enhances the dissipation of mechanical energy (i.e. $-Q_S$) by promoting the first principal strain rate but inhibits the nonlinear generation rate of dissipation (i.e. $R_S$). As shown in figure 19(c), the first eigenvalue of the pressure-Hessian tensor suppresses $R_\omega$, the third eigenvalue promotes $R_\omega$, and the competition between them determines the effect of the pressure-Hessian tensor on the evolution of $R_\omega$, while the contribution of the second eigenvalue is relatively small. According to the definition of the eigenvalues, the pressure-Hessian tensor's first eigenvalue generally has a more significant absolute value. Thus, the corresponding term's contribution along the CMT is more productive than that of the third eigenvalue most of the time.

Figure 19.

In the central plane of the channel ($y^+=532$), the pressure-Hessian's contribution to the invariant dynamics along the CMT of invariants in figure 18(a): (a) $Q_S H$, (b) $R_S H$ and (c) $R_\omega H$. The shadow colour from light to dark indicates the time period of the CMT in the four flow topologies of S1–S4.

Figure 20 shows the evolution of the contribution of the pressure-Hessian tensor along the mean trajectory of invariants in figure 18(b). As shown in figure 20(a), in the buffer layer, the pressure-Hessian tensor inhibits the first principal strain rate in the evolution process of invariants from topology S1 to topology S2 and promotes it in other topological regions, while it always restrains the third principal strain rate. Because of the competition between them, when the invariants are in the regions of flow topologies S1 and S2, the pressure-Hessian tensor suppresses the dissipation. It makes the invariant $Q$ tend to increase. When the invariants are in the regions of flow topologies S3 and S4, the pressure-Hessian tensor promotes the dissipation and makes the invariant $Q$ tend to decrease. The above results are consistent with the clockwise evolution of invariants. Figure 20(b) shows that the contribution of the pressure-Hessian tensor to the dynamics of $R_S$ is always negative, which is shown as the inhibition of dissipation generation. Figure 20(c) shows that the pressure-Hessian tensor also inhibits the generation of enstrophy, which is similar to the statistical results in the central region of the channel.

Figure 20.

In the buffer layer ($y^+= 20.5$), the pressure-Hessian's contribution to the invariant dynamics evolves along the CMT of invariants in figure 18(b): (a) $Q_S H$, (b) $R_S H$ and (c) $R_\omega H$. The shadow colour from light to dark indicates the time period of the CMT in the four flow topologies of S1–S4. The legends of the various lines are the same as those in figure 19.

It is noted that the mean contribution of the pressure-Hessian tensor to invariant dynamics in the different flow topologies shown in figures 15–17 is inconsistent with the contribution of the pressure-Hessian tensor to invariant dynamics along the mean trajectory of invariants shown in figures 19–20. This is because the former is the mean behaviour of all the invariants in the same flow topology and the p.d.f. of the invariants is centralized at the origin, resulting in the fact that it better reflects the contribution of the pressure-Hessian tensor when the magnitude of the invariants is small.

The results above reveal the significant effects of inhomogeneity and anisotropy on pressure-Hessian statistics, based on the analysis of the instantaneous pressure-Hessian tensor at different wall-normal positions. However, it is unclear whether the inhomogeneity and anisotropy originate directly from the mean shear flow or the inhomogeneous and anisotropic fluctuating flow. The method of pressure decomposition (Tang et al. Reference Tang, Zhao, Wan and Liu2020) provides a chance to achieve more physical understanding. In our previous study (Tang et al. Reference Tang, Zhao, Wan and Liu2020), with the purpose of investigating the statistical characteristics of pressure fluctuations in compressible channel turbulence, we decomposed the pressure fluctuations into the fast pressure $p_r$, slow pressure $p_s$, Stokes pressure $p_ \tau$ and compressible pressure $p_c$. The fast pressure represents the pressure fluctuations generated by the interaction between turbulent fluctuations and the mean flow. The slow pressure represents the fluctuations induced by the interactions between turbulent fluctuations. The wall boundary condition causes the Stokes pressure, and the compressible pressure is defined as the difference between the total pressure and the other pressure components (Sarkar Reference Sarkar1992).

To evaluate which pressure mechanism contributes the most to the pressure-Hessian terms, we also decompose the pressure-Hessian tensor. Firstly, the instantaneous pressure is split in terms of the mean pressure and the fluctuation. Then the fluctuating pressure is decomposed following the method in Tang et al. (Reference Tang, Zhao, Wan and Liu2020). In turn, the derivatives of each pressure part are computed to obtain the corresponding pressure-Hessian part. The decomposition of the pressure-Hessian tensor reads

(4.7)\begin{equation} \boldsymbol{\mathsf{H}} = \boldsymbol{\mathsf{H}}_{r} + \boldsymbol{\mathsf{H}}_{s} + \boldsymbol{\mathsf{H}}_{\tau} + \boldsymbol{\mathsf{H}}_{c} + { \boldsymbol{\mathsf{H}}}_m, \end{equation}

where the subscripts $r$, $s$, $\tau$, $c$ and $m$ represent the pressure-Hessian corresponding to the rapid, slow, Stokes, compressible and mean pressure. Since the pressure-Hessian tensor in (A3), (A5) and (A7)–(A9) is linear, their contributions to the evolution of invariants can also be decomposed into the corresponding part of (4.7), and the conditional mean vector $\boldsymbol {v}_H$ in the $Q$–$R$ plane becomes

(4.8)\begin{equation} \boldsymbol{v}_H=\boldsymbol{v}_{H_r} +\boldsymbol{v}_{H_s} +\boldsymbol{v}_{H_\tau}+\boldsymbol{v}_{H_c}+{\boldsymbol{v}}_{H_m}. \end{equation}

In the central region of the channel, the slow-pressure-related term is expected to play a dominant role due to the nearly zero mean shear (Gerolymos, Sénéchal & Vallet Reference Gerolymos, Sénéchal and Vallet2013). To investigate the relative importance of each component of the conditional mean vector $\boldsymbol {v}_H$ in the near-wall region, figure 21 shows the relative proportion of the amplitude of each term in (4.8) in the buffer layer, respectively. It is found that the relative amplitude of the conditional mean velocity for the slow-pressure-related term in the buffer layer is dominant in most areas of the $Q$–$R$ plane, and the proportion of the amplitude is above $70\,\%$. In the S1 topological region, there is a region where the proportion of the amplitude of $\boldsymbol {v}_{H_s}$ is between $50\,\%$ and $60\,\%$. The proportion of the amplitude of $\boldsymbol { v }_{H_r}$ can reach more than $30\,\%$, which also plays a certain role in the contribution of the overall pressure-Hessian tensor. The contribution of other pressure terms (e.g. wall boundary, compressibility and mean pressure gradient) is tiny, almost no more than $10\,\%$. Further comparison shows that the region with smaller proportion of the amplitude of $\boldsymbol {v}_{H_s}$ is consistent with the region with small amplitude of $\boldsymbol {v}_{H}$ in figure 18( f), which further illustrates the dominance of the slow-pressure-related term.

Figure 21.

The conditional mean vector $\boldsymbol {v}_H$ of the pressure-Hessian's contribution to the invariant dynamics in the buffer layer ($y^+=20.5$), which is decomposed by (4.8). The proportion of the amplitude of each part is: (a) $|\boldsymbol {v}_{H_s}|/ \sum |\boldsymbol {v}_i |$, (b) $| \boldsymbol {v}_{H_r} | / \sum | \boldsymbol {v}_i |$ and (c) $(|\boldsymbol {v}_{H_\tau }|+|\boldsymbol {v}_{H_c}| +|\boldsymbol {v}_{H_m}|)/ \sum | \boldsymbol {v}_i |$, where $i=H_r, H_s, H_c, H_\tau, H_m$.

Since the VGT represents the small-scale properties of flow (Meneveau Reference Meneveau2011), the pressure-Hessian term in the evolution equation of invariants directly affected by the large-scale mean shear is limited. Nevertheless, by analysing the geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor, and its contribution to invariant dynamics, it is shown that the statistical properties of the pressure-Hessian tensor and its effect on the evolution of invariants are still affected by inhomogeneity and anisotropy. Combined with the analysis of pressure decomposition, this proves that the influence of inhomogeneity and anisotropy on the pressure-Hessian is mainly due to the inhomogeneity and anisotropy of velocity fluctuations near the wall. As far as the VGT modelling is concerned, the present results indicate that keeping only the pressure-Hessian contributions associated with the slow pressure may be adequate to capture the main characters of inhomogeneity and anisotropy, i.e. the pressure-Hessian tensor in (3.13)–(3.15) can be further approximated by the component corresponding to the slow pressure.

4.6. Modelling implications for VGT dynamics

The main findings of the present work focus on the characteristics of the pressure-Hessian contributions to the dynamical evolution of VGT invariants in wall turbulence. Instructive implications can be obtained as far as the modelling of VGT dynamics is concerned.

The conventional works for models of VGT dynamics are usually done in the fixed Cartesian coordinate system (Girimaji & Pope Reference Girimaji and Pope1990; Chevillard & Meneveau Reference Chevillard and Meneveau2006; Suman & Girimaji Reference Suman and Girimaji2009; Meneveau Reference Meneveau2011; Wilczek & Meneveau Reference Wilczek and Meneveau2014; Johnson & Meneveau Reference Johnson and Meneveau2016; Pereira, Moriconi & Chevillard Reference Pereira, Moriconi and Chevillard2018), where all the terms are tensorial (see (3.9)). Although significant improvements in the understanding and modelling of VGT dynamics have been obtained in the past decades, modelling work in wall turbulence is rare. Based on the present results, we think that developing a VGT model in the principal coordinate system of the strain-rate tensor may be more tractable than that in the conventional fixed coordinate system. The starting point should be the dynamics equations of the VGT in the principal coordinates (Lawson & Dawson Reference Lawson and Dawson2015) as shown by (A11)–(A13). The unclosed terms for an autonomous system of a dynamical model of the VGT are the pressure-Hessian tensor and viscous terms. The modelling of the former is still the main story, while the latter may be modelled adequately by the damping effects (Meneveau Reference Meneveau2011).

It may be feasible to build a model of the VGT that can faithfully capture the dynamics of invariants by properly simplifying (A11)–(A13). Based on the statistics of the geometric relationship between the pressure-Hessian tensor and the strain-rate tensor, it is reasonable to prescribe the eigenvectors of the pressure Hessian, $\boldsymbol {f}_i$, to be random distributions centred at the configuration with the highest probability (see figure 12). A Gaussian random distribution might be a good choice, by which the pressure-Hessian tensor can be represented as an approximation comprising the eigenvalues of the pressure-Hessian tensor, $\eta _i$, and the eigenvectors of the strain-rate tensor, $\boldsymbol {e}_i$. Similarly, the product of pressure Hessian and vorticity in (A13) can also be represented approximately by $\eta _i$, $\boldsymbol {e}_i$ and the amplitude of vorticity, based on the geometric relationship between the pressure-Hessian tensor and the vorticity vector. As for the eigenvalues of the pressure-Hessian tensor, it is seen that $\eta _i$ are not sensitive to the mean shear (see figure 10). This implies that a model of $\eta _i$ from isotropic turbulence may be adequate to reproduce the main features of VGT dynamics. To the best of our knowledge, the theoretical frame based on Gaussian random velocity fields proposed by Wilczek & Meneveau (Reference Wilczek and Meneveau2014) is promising to perform the modelling of $\eta _i$.

It should be mentioned that a transition model connecting the near-wall (inhomogeneous and anisotropic) statistics and the nearly homogeneous and isotropic statistics in the central channel is still needed for a practical model of VGT dynamics. The fluid particles may transfer from the near-wall region to the central region of the channel and vice versa. The present results illustrate that the statistics of the pressure-Hessian tensor in the near-wall region may be significantly different from those in the central channel domain, i.e. the degree of inhomogeneity and anisotropy varies along the wall-normal direction. Thus, a transition model mimicking the quantitative dependence of inhomogeneity and anisotropy of the pressure-Hessian tensor on $y^+$ is desired. However, the development of such a transition model is definitely a tough task, which requires a lot of efforts in the future.