2.1. Helicity

The mean helicity $H$, fluctuating helicity $\langle h \rangle$ and their components ($H_i$ and $\langle h_i \rangle$) are defined as

(2.2a,b)\begin{equation} H=\sum_{i=1}^3 H_i=\sum_{i=1}^3\left\langle U_{\underline{i}}\right\rangle \left\langle W_{\underline{i}} \right\rangle, \quad\left\langle h \right\rangle=\sum_{i=1}^3\left\langle h_i \right\rangle =\sum_{i=1}^3\left\langle u_{\underline{i}}\omega_{\underline{i}} \right\rangle, \end{equation}

where $W_i=\epsilon _{ijk}\partial U_k/ \partial x_j$ is the vorticity, ${u_i}$ is the fluctuating velocity, ${\omega _i}=\epsilon _{ijk} \partial u_k/\partial x_j$ is the fluctuating vorticity, the underlines in the subscript represent no contraction, and $\langle {\cdot } \rangle$ represents the average on the $x_1$ and $x_3$ direction. The mean velocities and vorticities are given in Appendix A.1 for reference.

Figure 1(a,b) shows the mean helicity $H$ and its decomposition $H_i$, respectively. As shown in figure 1(a), the mean helicity is positive around the wall but negative around the channel centre. However, with increasing rotation, the mean helicity is extended to the vicinity of the wall. In contrast, as $Re_\tau$ increases, the mean helicity is reduced. The decomposition in figure 1(b) does not include the wall-normal component, because $H_2=\langle U_2 \rangle \langle W_2 \rangle =0$. In addition, when $x_2^+\lessapprox 4$, $H_1\approx -H_3$, which is because

(2.3)\begin{equation} \left.\begin{gathered} \left\langle U_1 \right\rangle\left\langle W_1 \right\rangle=\left\langle U_1 \right\rangle \left\langle \frac{\partial U_3}{\partial x_2} \right\rangle \approx \left\langle \frac{\partial U_1}{\partial x_2} \right\rangle\left\langle \frac{\partial U_3}{\partial x_2} \right\rangle x_2,\\ \left\langle U_3 \right\rangle \left\langle W_3 \right\rangle={-}\left\langle U_3 \right\rangle \left\langle \frac{\partial U_1}{\partial x_2} \right\rangle \approx{-}\left\langle \frac{\partial U_1}{\partial x_2} \right\rangle\left\langle \frac{\partial U_3}{\partial x_2} \right\rangle x_2. \end{gathered}\right\} \end{equation}

Figure 1.

Profile of mean helicity: (a) overall helicity $H^+$; (b) decomposed helicity $H_i^+$ of ST30 and the grey filled region shows the error bar of $H^+$.

Figure 2 gives the fluctuating helicity $\langle h \rangle$. The distribution of $\langle h \rangle$ is similar to that of $H$ but has a shorter positive range. With rotation becoming stronger, the maxima of the helicity shift towards lower locations. In contrast, as $Re_\tau$ increases, the fluctuating helicity is reduced. The decomposed fluctuating helicity $\langle h_i \rangle$ of ST30 is shown in figure 2(b). When $x_2\lessapprox 4$, $\langle h_1 \rangle \approx -\langle h_3 \rangle$, which can be deduced in a way similar to (2.3). Furthermore, $\langle h_2 \rangle$ is one order less than the other two components. Above the buffer layer, the three components are all negative.

Figure 2.

Profile of fluctuating helicity: (a) overall helicity $\langle h \rangle$; (b) decomposed helicity $\langle h_i \rangle$ of ST30 and the grey filled region shows the error bar of $\langle h \rangle ^+$.

Figure 3 shows the relation between the peak position $x_2^+ |_{max{[\langle h \rangle ^+]}}$ and the parameter $Re_\tau /Ro_\tau$. As $Ro_\tau$ increases or $Re_\tau$ decreases, the peaks approach the wall. Especially, ST07 and ST15R have the same $Re_\tau /Ro_\tau$, and their peaks almost overlap with each other. The relation between the peak positions and the parameter $Re_\tau /Ro_\tau$ is fitted using a sigmoid function, leading to the following expression:

(2.4)\begin{equation} x_2^+ |_{\max{[\left\langle h \right\rangle ^+]}}=2.061/(0.1316+\exp({-}0.1218\, Re_\tau/Ro_\tau)). \end{equation}

In terms of the underlying mechanisms, as shown by the N–S equations (2.1), the Coriolis effects is proportional to the rotation rates $\varOmega$. As $Ro_\tau$ increases, rotation effects could penetrate deeper regions within the boundary layer. Considering the effects of $Re_\tau$, for the near-wall inclined vortex structures, the streamwise velocity and the rotation-induced spanwise velocity are two opposing effects (Hu et al. Reference Hu, Li and Yu2023), which could be the same for the helicity. The streamwise velocity effects can be reflected through $Re_\tau$. In the following analysis, the mechanism will be further discussed using the helicity budgets and structure functions.

Figure 3.

Relation between the peak $x_2^+ |_{max{[\langle h \rangle ^+]}}$ and the parameter $Re_\tau /Ro_\tau$. The black solid line is the fitted sigmoid function (2.4) and the black dashed line serves as a reference for the linear law.

2.2. Helicity budget

The budget equation for $\langle h \rangle$ is written as

(2.5) \begin{align} &-\! \left\langle \gamma_{i2} \right\rangle \frac{\mathrm{d} \left\langle U_i \right\rangle }{\mathrm{d}\kern 0.06em x_2} -\left\langle u_iu_2 \right\rangle \frac{\mathrm{d} \left\langle W_i \right\rangle}{\mathrm{d}\kern 0.06em x_2} -\frac{\mathrm{d} \left\langle hu_2 \right\rangle}{\mathrm{d}\kern 0.06em x_2} +\frac{1}{2}\frac{\mathrm{d} }{\mathrm{d}\kern 0.06em x_2}\left\langle u_iu_i\omega_2 \right\rangle -\frac{1}{\rho}\frac{\mathrm{d} \left\langle \omega_2 p_R\right\rangle }{\mathrm{d}\kern 0.06em x_2} -\frac{1}{\rho}\frac{\mathrm{d} \left\langle \omega_2 p_T\right\rangle}{\mathrm{d}\kern 0.06em x_2} \nonumber\\ &\quad +\nu\frac{\mathrm{d} ^2 \left\langle h \right\rangle}{\mathrm{d}\kern 0.06em x_2^2} -2\nu \left\langle \frac{\partial u_i}{\partial x_j}\frac{\partial \omega_i}{\partial x_j} \right\rangle +2\varOmega\epsilon_{ij1}\left\langle u_j \omega_i \right\rangle =0, \end{align}

where $\gamma _{ij} = \omega _iu_j-\omega _ju_i$ is the helical stress, and $p_R$ and $p_T$ are the decomposed pressure related to the Coriolis terms and the turbulent convection, respectively (Yang et al. Reference Yang, Deng, Wang and Shen2020a; Hu, Li & Yu Reference Hu, Li and Yu2022a).

For the helical stress $\langle \gamma _{ij} \rangle$, the following relations can be derived:

(2.6a,b)\begin{equation} \left\langle \gamma_{12} \right\rangle =\frac{\mathrm{d} }{\mathrm{d}\kern 0.06em x_2} \left\langle u_2 u_3 \right\rangle,\quad -\left\langle \gamma_{32} \right\rangle =\frac{\mathrm{d} }{\mathrm{d}\kern 0.06em x_2} \left\langle u_1 u_2 \right\rangle. \end{equation}

That is, partial components of $\langle \gamma _{ij} \rangle$ can be represented by the wall-normal gradients of the Reynolds stresses. According to (2.6a,b), the production and the Coriolis term can be simplified, and the fluctuating helicity budget equation (FHE) can be rewritten as

(2.7)\begin{align} &\underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{\mathrm{d} \left\langle u_2 u_3 \right\rangle }{\mathrm{d}\kern 0.06em x_2} \frac{\mathrm{d} \left\langle U_1 \right\rangle }{\mathrm{d}\kern 0.06em x_2} +\frac{\mathrm{d} \left\langle u_1 u_2 \right\rangle }{\mathrm{d}\kern 0.06em x_2} \frac{\mathrm{d} \left\langle U_3 \right\rangle }{\mathrm{d}\kern 0.06em x_2} -\left\langle u_iu_2 \right\rangle \frac{\mathrm{d} \left\langle W_i \right\rangle}{\mathrm{d}\kern 0.06em x_2} }_{ \left\langle\varPi\right\rangle } \underbrace{-\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle}\frac{\mathrm{d} \left\langle hu_2 \right\rangle}{\mathrm{d}\kern 0.06em x_2} +\frac{1}{2}\frac{\mathrm{d} }{\mathrm{d}\kern 0.06em x_2}\left\langle u_iu_i\omega_2 \right\rangle }_{\left\langle T \right\rangle} \nonumber\\ &\quad \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle}-\frac{1}{\rho}\frac{\mathrm{d} \left\langle \omega_2 p_R\right\rangle }{\mathrm{d}\kern 0.06em x_2} }_{\left\langle G_R \right\rangle} \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle}-\frac{1}{\rho}\frac{\mathrm{d} \left\langle \omega_2 p_T\right\rangle}{\mathrm{d}\kern 0.06em x_2} }_{\left\langle G_T \right\rangle} +\underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle}\nu\frac{\mathrm{d} ^2 \left\langle h \right\rangle}{\mathrm{d}\kern 0.06em x_2^2} }_{\left\langle D \right\rangle} \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle}-2\nu \left\langle \frac{\partial u_i}{\partial x_j}\frac{\partial \omega_i}{\partial x_j} \right\rangle}_{-\left\langle E \right\rangle} +\underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle}2\frac{\mathrm{d} \left\langle u_1 u_2 \right\rangle }{\mathrm{d}\kern 0.06em x_2}\varOmega}_{\left\langle C \right\rangle} =0, \end{align}

where $\langle \varPi \rangle$ is the production and represents the interaction between the mean and fluctuating fields, $\langle T \rangle$ is the spatial turbulent convection, $\langle G_R \rangle$ and $\langle G_T \rangle$ are the pressure transfer terms related to the rotation effect and turbulent convection, $\langle D \rangle$ is the spatial viscous diffusion, $\langle E \rangle$ is the pseudo-dissipation, and $\langle C \rangle$ is the Coriolis term. Here, $\langle G \rangle =\langle G_T \rangle +\langle G_R \rangle$ is the total pressure transfer term. As shown in the equation, the Coriolis force directly affects the helicity distribution. Specifically, $\int _{-1}^0 \langle C \rangle \,\textrm {d}x_2=0$. The Coriolis term is a transfer term similar to the turbulent convection. This is non-trivial, because in the budget equation of the turbulent kinematic energy (TKE) and the GKE, the Coriolis term is zero and only redistributes energy among three components of TKE (Yang & Wang Reference Yang and Wang2018). The direct effects of $\langle C \rangle$ on the fluctuating helicity imply that in addition to the TKE, $\langle h \rangle$ could be another important quantity in the dynamics of the streamwise-rotating channel turbulence.

The results of the FHE are given in figure 4. The production $\langle \varPi \rangle$ is mainly negative. There are mainly two mechanisms: the coupling effects between the helical stresses and the mean velocity gradients, and those between the Reynolds stresses and the mean vorticity gradients. The near-wall behaviour analyses in Appendix B.1 indicate that the first mechanism is dominant in the viscous sublayer. In fact, numerical results suggest that the first mechanism ($- \langle \gamma _{i2} \rangle \,{\mathrm {d} \langle U_i \rangle }/{\mathrm {d} x_2}$) is always dominant, which is not shown here. The term in fact extracts positive fluctuating helicity to the mean helicity. The spatial turbulent convection $\langle T \rangle$ is mainly induced by convection ($-\mathrm {d} \langle h u_2 \rangle$) and vortex deformation ($\mathrm {d} \langle u_iu_i\omega _2 \rangle /\mathrm {d} x_2/2$). The term extracts positive helicity from the buffer layer to higher layers. Similarly, the viscous diffusion $\langle D \rangle$ transfers the positive helicity from the buffer layer and the high viscous sublayer towards the wall. Different from the pseudo-dissipation of energy, the helicity pseudo-dissipation $\langle E \rangle$ is not positive-definite. It is positive in the vicinity of the wall but negative at higher wall-normal positions. The Coriolis term $\langle C \rangle$ and the pressure term $\langle G \rangle$ are the two direct effects induced by rotation. In streamwise-rotating channel turbulence, the profile of the Reynolds stress $\langle u_1u_2 \rangle$ is approximately not affected by rotation (Yang & Wang Reference Yang and Wang2018). According to the definition of the Coriolis term and the profile of Reynolds stress $\langle u_1u_2 \rangle$, it could be inferred that the term is negative around the wall but positive around the channel centre. The term transfers positive helicity from the buffer layer toward the higher layers. In contrast, the pressure term $\langle G \rangle$ transfers positive helicity from the channel centre toward the wall. The term is induced by the turbulent convection $\langle G_T \rangle$ and the Coriolis force $\langle G_R \rangle$. According to the Green function of the pressure Poisson equation (Kim Reference Kim1989), the pressure always has the opposite values with its origin (convection or rotation) (Yang & Wang Reference Yang and Wang2018; Yang et al. Reference Yang, Deng, Wang and Shen2018). In fact, since there is no fluctuating helicity in non-rotating channel turbulence, the Coriolis term and the corresponding pressure transfer term are the direct reasons for the non-zero fluctuating helicity. However, the Coriolis term has the opposite sign with the fluctuating helicity, especially for ST07. The rotation-induced pressure transfer terms could be the main source for the fluctuating helicity.

Figure 4.

Fluctuating helicity budget: (a) ST07; (b) ST30. The black solid lines of $x_2^+=23.4$ and $x_2^+=7.8$ give the zeros points of the helicity distribution in figure 2(a).

Figure 5 shows the distribution of $\langle G_R \rangle$, $\langle C \rangle$ and $\langle T\rangle$ for different cases. As shown, the two terms $\langle G_R \rangle$ and $\langle C \rangle$ are both proportional to $Ro_\tau$, while their relationship with $Re_\tau$ remains less evident. In comparison, the turbulent convection $\langle T \rangle$ is increased by both $Ro_\tau$ and $Re_\tau$. Especially, the comparison between ST15 and ST15R indicates that the Reynolds number effects are more remarkable for $\langle T \rangle$. These findings suggest that the contrasting impacts of $Re_\tau$ and $Ro_\tau$ on the peak law (2.4) might be associated with the terms $\langle T \rangle$ and $\langle G_R \rangle +\langle C \rangle$, respectively.

Figure 5.

(a) $\langle G_R \rangle$, (b) $\langle C \rangle$ and (c) $\langle T \rangle$ of different cases in the helicity budget.

In addition, the near-wall behaviours of all terms in the helicity budget are estimated in Appendix B.1:

(2.8)\begin{equation} \left.\begin{gathered} |\left\langle\varPi\right\rangle^+|\sim x_2^{{+}2},\quad |\left\langle T\right\rangle^+|\sim x_2^{{+}2},\quad |\left\langle G \right\rangle^+ |\sim 1,\\ |\left\langle D \right\rangle^+ |\sim 1,\quad |\left\langle E \right\rangle^+ |\sim 1,\quad |\left\langle C \right\rangle^+ |\sim x_2^{{+}2}, \end{gathered}\right\} \end{equation}

which is verified in figure 6. Especially, on the wall, there is a relation between the pressure transfer and viscous effects:

(2.9)\begin{equation} \left.\left\langle G \right\rangle^+ \right|_{x_2={\pm} h}=\left.\left\langle D \right\rangle^+ \right|_{x_2={\pm} h}= \tfrac{1}{2}\left.\left\langle E \right\rangle^+ \right|_{x_2={\pm} h}, \end{equation}

which means that the total pressure transfer is equal to the viscous diffusion on the wall. The relation is proved in Appendix B.2.

Figure 6.

Near-wall behaviours of helicity budget of ST30.

3.1. Scale helicity distribution

First, we want to explain the relation between structure function and Fourier spectrum. Taking the scale energy for example (Davidson Reference Davidson2015), supposing the energy spectrum gives the exact definition of the energy at specific scale, one-dimensional Fourier transformation leads to the conclusion that the second-order structure function represents all energy in eddies of size $r$ or less plus a term related to the enstrophy in eddies of size $r$ or greater. However, since eddies of given size contribute to the energy spectrum across the full range of wavenumbers, the energy spectrum is not the exact definition of scale helicity. Therefore, for simplicity, one can also believe that the second-order structure function for energy (helicity) represents all energy (helicity) in eddies of size $r$ or less.

To get the helicity at a given scale, differentiation with respect to the scale $r$ is needed. Since the streamwise direction is strongly affected by rotation (Yang & Wang Reference Yang and Wang2018), in this paper, the focus is on the interscale dynamics in the $x_1$ direction. The scale helicity is then defined as

(3.1)\begin{equation} \left\langle \delta h \right\rangle ({X}_2,{r_1}) =\left\langle \partial_{r_1}( \delta {u}_i(\boldsymbol{X},{r_1}) \delta {\omega}_i(\boldsymbol{X},{r_1})) \right\rangle, \end{equation}

where $\delta$ means the increment of a quantity at two positions, $\boldsymbol {X}=(\boldsymbol {x}+\boldsymbol {x'})/2$ is the centre of the two positions, $r_1$ is the streamwise component of $\boldsymbol {r}$ and $\boldsymbol {r}=(\boldsymbol {x}-\boldsymbol {x'})$ is the scale vector. Specifically, the velocity and the vorticity increments between the two positions $\boldsymbol {x}$ and $\boldsymbol {x'}$ are defined as

(3.2a,b)\begin{equation} \delta {u}_i(\boldsymbol{X},{r_1})={u}_i(\boldsymbol{x})-{u}_i(\boldsymbol{x'}),\quad \delta {\omega}_i(\boldsymbol{X},{r_1})={\omega}_i(\boldsymbol{x})-{\omega}_i(\boldsymbol{x'}). \end{equation}

The streamwise scale helicity $\langle \delta h \rangle ({X}_2,r_1)$ is shortened as $\langle \delta h \rangle$ hereafter. When $r_1$ tends to infinity, the integral of the scale helicity tends to 2 times the fluctuating helicity:

(3.3)\begin{equation} \lim_{r_1\rightarrow\infty}\int_{0}^{r_1}\left\langle \delta h \right\rangle (X_2,l_1)\,\mathrm{d}l_1 = \lim_{r_1\rightarrow\infty}\left\langle (u_i-u_i') (\omega_i-\omega_i') \right\rangle = 2\left\langle u_i\omega_i \right\rangle, \end{equation}

which supports our definition about the scale helicity.

Figure 7(a) shows the scale helicity distribution of ST30 at different positions. As shown, the distribution at $x_2^+=10.5$ is negative at small scales ($r_1^+\lessapprox 10^2$) but positive at larger scales ($r_1^+\gtrapprox 10^2$), consistent with the observation of Yu et al. (Reference Yu, Hu, Yan and Li2022). It means that from the wall towards the channel centre, the small-scale helicity first changes from positive to negative. The change of large-scale helicity happens at a higher wall-normal position. The comparison of different cases in the buffer layer is shown in figure 7(b). As $Ro_\tau$ increases from 0 to 30, the small-scale helicity has a larger negative value. With continued intensification of rotation, the small-scale helicity slightly decreases and the large-scale positive helicity completely diminishes. It means that the scale helicity changes its signs at lower wall-normal positions. In contrast, as $Re_\tau$ increases, the scale helicity slightly decreases.

Figure 7.

Scale helicity distribution: (a) $\langle \delta h \rangle$ of ST30, and the pink filled region shows the error bar of $\langle \delta h \rangle$ in the buffer layer $x_2^+=10.5$; (b) $\langle \delta h \rangle$ in the buffer layer ($x_2^+=10.3$ for ST07R and ST15R, $x_2^+=10.0$ for ST07RR and $x_2^+=10.5$ for the other cases).

Similarly, the scale distribution of the pressure is defined as

(3.4)\begin{equation} \left\langle \delta p^2 \right\rangle ({X}_2,{r_1}) =\left\langle\partial_{r_1}( \delta p (\boldsymbol{X},{r_1}) \delta p (\boldsymbol{X},{r_1}))\right\rangle. \end{equation}

The decomposed scale pressures $\langle \delta p_R^2 \rangle$ and $\langle \delta p_T^2 \rangle$ are defined in the same way. Figure 8 displays the normalized scale pressure and its decomposition of ST30. As shown, $p_R$ has a larger streamwise length scale than $p$ and $p_T$, which can be inferred from the Poisson equation of the decomposed pressures (Yang et al. Reference Yang, Deng, Wang and Shen2018).

Figure 8.

Scale distribution of the pressure and its decomposition of ST30 non-dimensionalized using the corresponding mean-square value in the buffer layer $x_2^+=10.5$. The grey filled region shows the error bar of $\langle \delta p^{2} \rangle /2/\langle p^{2} \rangle$.

3.2. DHGKE analysis

To analyse the interscale dynamics in channel turbulence, the GKE was used by Marati et al. (Reference Marati, Casciola and Piva2004) and Cimarelli et al. (Reference Cimarelli, De Angelis and Casciola2013, Reference Cimarelli, De Angelis, Schlatter, Brethouwer, Talamelli and Casciola2015, Reference Cimarelli, De Angelis, Jimenez and Casciola2016). In the present study, the interscale dynamics in the streamwise-rotating channel turbulence are investigated from the perspective of scale helicity through corresponding budgets, which is named as the differentiated generalized Kolmogorov equation for helicity (DHGKE) hereafter. Deduced from the N–S equation, with the assumption $\langle U_2 \rangle =\langle W_2 \rangle =0$, $\boldsymbol {\varOmega }=\varOmega \boldsymbol {e}_1$, DHGKE is written as

(3.5) \begin{align} \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} \frac{\partial }{\partial r_1} \frac{\partial}{\partial t} \delta u_i \delta \omega_i }_{\varDelta_t} &= \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{\partial}{\partial r_1} \delta u_i \delta u_j \frac{\mathrm{d}{} \left\langle W_i^\ast \right\rangle}{\mathrm{d}{} X_j} -\frac{\partial}{\partial r_1} (\delta \omega_i\delta u_j-\delta u_i \delta \omega_j) \frac{\mathrm{d}{} \left\langle U_i^\ast \right\rangle}{\mathrm{d}{} X_j} }_{\varPi^{S}}\nonumber\\ &\quad\underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{\partial}{\partial r_1}\frac{\partial }{\partial r_j}( \delta h \delta u_j ) +\frac{1}{2}\frac{\partial}{\partial r_1}\frac{\partial }{\partial r_j}( \delta \omega_j \delta u^2 ) }_{T^{SS}} \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{\partial}{\partial r_1}\frac{\partial }{\partial X_j}( u_j^\ast\delta h ) +\frac{1}{2}\frac{\partial}{\partial r_1}\frac{\partial }{\partial X_j}( \delta u^2 \omega_j^\ast ) }_{T^{SP}}\nonumber\\ &\quad \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{1}{\rho}\frac{\partial}{\partial r_1}\frac{\partial }{\partial X_j}( \delta \omega_j \delta p_T ) }_{G_T^{S}} \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{1}{\rho}\frac{\partial}{\partial r_1}\frac{\partial }{\partial X_j}( \delta \omega_j \delta p _R)}_{G_R^{S}} +\underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} 2\nu \frac{\partial}{\partial r_1}\frac{\partial^2 ( \delta h ) }{\partial r_j\partial r_j} }_{D^{SS}} +\underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle}\frac{\nu}{2}\frac{\partial}{\partial r_1}\frac{\partial ^2 ( \delta h )}{\partial X_j \partial X_j} }_{D^{SP}}\nonumber\\ &\quad \underbrace{\vphantom{\frac{\partial}{\partial r_1}\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle}-4 \frac{\partial }{\partial r_1}\epsilon^{H\ast}}_{{-}E^S} +\underbrace{\vphantom{\frac{\partial}{\partial r_1}\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} 2\varOmega \frac{\partial}{\partial r_1} \left(\delta \omega_2 \delta u_3 - \delta u_2 \delta \omega_3 +\delta u_k \delta \frac{\partial u_k}{\partial x_1}\right)}_{C^{S}}, \end{align}

where $\ast$ represents the average at the two positions $\boldsymbol {x}$ and $\boldsymbol {x'}$, $\varDelta _t$ is the time derivatives, $\varPi ^{S}$ is the production, $T^{SS}$ is the interscale turbulent convection, $T^{SP}$ is the spatial turbulent convection, $G_R^{S}$ and $G_T^{S}$ are the pressure transfer terms related to the rotation effects and turbulent convection, $D^{SS}$ is the interscale viscous diffusion, $D^{SP}$ is the spatial viscous diffusion, $E^{S}$ is the pseudo-dissipation, and $C^{S}$ is the Coriolis transfer term. Here, $G^{S} = G_T^{S} + G_R^{S}$ is the total pressure transfer term.

If averaging over the $x_1- x_3$ plane, using the relation $\partial /\partial X_1 \langle {\cdot } \rangle =\partial /\partial X_3 \langle {\cdot } \rangle =0$ and ${\partial }/{\partial t}=0$, DHGKE can be written as

(3.6) \begin{align} &\underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{\partial}{\partial r_1}\left\langle \delta u_i \delta u_2 \right\rangle \frac{\mathrm{d}{} \left\langle W_i^\ast \right\rangle}{\mathrm{d}{} X_2} -\frac{\partial}{\partial r_1}\left\langle \delta \omega_i\delta u_2-\delta u_i \delta \omega_2 \right\rangle \frac{\mathrm{d}{} \left\langle U_i^\ast \right\rangle}{\mathrm{d}{} X_2} }_{\left\langle \varPi^{S}\right\rangle} \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{\partial}{\partial r_1}\frac{\partial }{\partial r_j}\left\langle \delta h \delta u_j \right\rangle +\frac{1}{2}\frac{\partial}{\partial r_1}\frac{\partial }{\partial r_j}\left\langle \delta \omega_j \delta u^2 \right\rangle }_{\left\langle T^{SS} \right\rangle }\nonumber\\ &\quad \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{\partial}{\partial r_1}\frac{\partial }{\partial X_2}\left\langle u_2^\ast\delta h \right\rangle +\frac{1}{2}\frac{\partial}{\partial r_1}\frac{\partial }{\partial X_2}\left\langle \delta u^2 \omega_2^\ast \right\rangle }_{\left\langle T^{SP} \right\rangle } \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{1}{\rho}\frac{\partial}{\partial r_1}\frac{\partial }{\partial X_2}\left\langle \delta \omega_2 \delta p_T \right\rangle }_{\left\langle G_T^{S} \right\rangle } \underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} -\frac{1}{\rho}\frac{\partial}{\partial r_1}\frac{\partial }{\partial X_2}\left\langle \delta \omega_2 \delta p _R\right\rangle}_{\left\langle G_R^{S} \right\rangle }\nonumber\\ &\quad +\underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle} 2\nu \frac{\partial}{\partial r_1}\frac{\partial^2 \left\langle \delta h \right\rangle }{\partial r_j\partial r_j} }_{\left\langle D^{SS} \right\rangle } +\underbrace{\vphantom{\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle}\frac{\nu}{2}\frac{\partial}{\partial r_1}\frac{\partial ^2 \left\langle \delta h \right\rangle}{\partial X_2^2} }_{\left\langle D^{SP} \right\rangle } +\underbrace{\vphantom{\frac{\partial}{\partial r_1}\left\langle \frac{\partial^2}{\partial X_2^2}\right\rangle}2\varOmega\frac{\partial}{\partial r_1}\frac{\partial }{\partial X_2}\left\langle \delta u_2 \delta u_1 \right\rangle }_{\left\langle C^{S} \right\rangle }=0. \end{align}

Similar to the GKE results given by Marati et al. (Reference Marati, Casciola and Piva2004) and the limited behaviours in (3.3), when $r_1\rightarrow \infty$, there are also relations for the spatial and interscale transfer terms:

(3.7)\begin{equation} \left.\begin{gathered} \lim_{r_1\rightarrow\infty}\int_{0}^{r_1}\left\langle D^{SS} \right\rangle (X_2,l_1)\,\mathrm{d}l_1=\lim_{r_1\rightarrow\infty}\int_{0}^{r_1}\left\langle D^{SP} \right\rangle (X_2,l_1)\,\mathrm{d}l_1,\\ \lim_{r_1\rightarrow\infty}\int_{0}^{r_1}\left\langle T^{SS} \right\rangle (X_2,l_1)\,\mathrm{d}l_1=\lim_{r_1\rightarrow\infty}\int_{0}^{r_1}\left\langle T^{SP} \right\rangle (X_2,l_1)\,\mathrm{d}l_1. \end{gathered}\right\} \end{equation}

DHGKE results of ST07 and ST30 in the viscous sublayer ($x_2^+=3.5$) are shown in figures 9(a) and 9(b), respectively. As shown in figure 9(a), the pressure terms $\langle G^{S}_{R} \rangle$ and $\langle G^{S}_{T} \rangle$ are the main positive sources for the scale helicity. The interscale viscous diffusion $\langle D^{SS} \rangle$ transfers positive helicity from $r_1^+\sim 100$ to smaller scales ($r_1^+\sim 20$). In contrast, the spatial viscous diffusion $\langle D^{SP} \rangle$ is mainly negative and transfers positive scale helicity to higher wall-normal position. Consistent with the helicity budget in figure 4, the production $\langle \varPi ^{S}\rangle$ and the Coriolis term $\langle C^{S} \rangle$ are both negative here and restrain the scale helicity in the viscous sublayer. The spatial and interscale turbulent convections $\langle T^{SP} \rangle$ and $\langle T^{SS} \rangle$ are also negative. Generally, for energy transfer in streamwise-rotating channel turbulence (Yang & Wang Reference Yang and Wang2018), the pressure transfers are negligible. However, the pressure transfers are the main sources for the scale helicity here, while other terms except for $\langle D^{SS} \rangle$ suppress the development of the scale helicity. Figure 9(b) gives the DHGKE results of ST30. The amplitudes of all terms increase with rotation rates. Compared with other terms, the convection-induced pressure term $\langle G_T^{S} \rangle$ is strengthened. Additionally, the distribution of the interscale turbulent convection $\langle T^{SS} \rangle$ is completely changed. The term transfers positive helicity towards small scales. Relatively, $\langle D^{SS} \rangle$, $\langle D^{SP} \rangle$ and $\langle T^{SP} \rangle$ are weaker in ST30 than in ST07. It is because that $\langle C^{S} \rangle$ and $\langle G^{S}_R \rangle$ are defined to be proportional to the rotation rates, while the other turbulent processes are not.

Figure 9.

DHGKE in the viscous sublayer ($x^+_2=3.5$): (a) ST07; (b) ST30.

Figure 10 shows the results in the buffer layer ($x^+_2=10.5$). For ST07, most terms have a similar distribution as those in the viscous sublayer. However, the spatial viscous diffusion $\langle D^{SP} \rangle$ is negligible in this layer. Here, $\langle T^{SS} \rangle$ is more important and transfers positive helicity towards smaller scales. Most importantly, the production term $\langle \varPi ^{S}\rangle$ is dominant in this layer and its peak locates on $r^+_1 \approx 30$. Notably, as shown in figure 7(b), the scale helicity in this layer is negative when $r^+_1 \lessapprox 100$ but positive when $r^+_1$ is larger, and its negative peaks also locate on $r_1^+\approx 30$. This implies that the production term is the main term inducing the sign change of the scale helicity. Similarly, $\langle T^{SP} \rangle$ is negative at small scales but positive at large scales, and could also be related to the scale discrepancy of the scale helicity distribution, even if its negative peak locates at a smaller scale ($r^+_1\approx 20$). Therefore, $\langle \varPi ^{S}\rangle$ and $\langle T^{SP} \rangle$ are the core effects of the mean flow and spatial turbulent convection on the small-scale negative helicity. In contrast, the interscale transfer terms $\langle D^{SS} \rangle$ and $\langle T^{SS} \rangle$ are positive at small scales but negative at large scales. These two terms cascade positive scale helicity to small scales and cancel the imbalance of chirality. The behaviours of $\langle T^{SS} \rangle$ are far different from the findings in homogeneous turbulence, where only the prevalence of a single chirality is considered (Mininni & Pouquet Reference Mininni and Pouquet2009, Reference Mininni and Pouquet2010; Mininni, Rosenberg & Pouquet Reference Mininni, Rosenberg and Pouquet2012; Hu, Li & Yu Reference Hu, Li and Yu2022b). DHGKE results of ST30 in figure 10(b) are more concise than those of ST07. Here, $\langle T^{SP} \rangle$, $\langle G^{S}_T \rangle$ and $\langle D^{SP} \rangle$ are negligible, while other terms have the same distribution as those of ST07. The results of ST30 highlight the effects of the mean flow gradients on the discrepancy between small- and large-scale helicity.

Figure 10.

DHGKE in the buffer layer ($x^+_2=10.5$): (a) ST07; (b) ST30.

To further discuss the mechanisms related to the peak laws (2.4), figure 11(a–d) shows the effects of $Re_\tau$ and $Ro_\tau$ on $\langle \varPi ^{S}\rangle$, $\langle T^{SS} \rangle$, $\langle G^{S}_R \rangle$ and $\langle C^S \rangle$, respectively, around the peaks ($x^+_2\approx 5.4$). As shown, the production $\langle \varPi ^S \rangle$, the Coriolis term $\langle C^S \rangle$ and corresponding pressure term $\langle G^{S}_R \rangle$ are remarkably proportional to the rotation rates, yet they remain unaffected by variations in the Reynolds number $Re_\tau$. In contrast, $Re_\tau$ has the opposing effects with $Ro_\tau$ on the interscale turbulent convection $\langle T^{SS} \rangle$. Specifically, at $r_1^+\sim 10$, $\langle T^{SS} \rangle$ exhibits a positive and amplified trend with the intensification of rotation, but becomes negative with increasing $Re_\tau$. When summing over $r_1^+$, the positive values at small scales induced by rotation are partially counteracted by the negative values at large scales, while $Re_\tau$ merely enhances the negative amplitudes of $\int \langle T^{SS} \rangle \,\mathrm {d}r_1$. This reveals the detailed opposing effects of $Re_\tau$ and $Ro_\tau$ on the peak laws (2.4) through the convection $\langle T^{SS} \rangle$ and rotation $\langle G^{S}_R \rangle +\langle C^S \rangle$.

Figure 11.

(a) $\langle \varPi ^{S}\rangle$, (b) $\langle T^{SS} \rangle ^+$, (c) $\langle G^{S}_R \rangle ^+$ and (d) $\langle C^S \rangle ^+$ for different cases around the peak of the fluctuating helicity ($x_2^+=5.3$ for ST07R and ST15R, $x_2^+=5.4$ for the other cases).

The DHGKE results in the log-law layer ($x_2^+=80.0$) are shown in figure 12. As shown, even if 81 time slices with a time interval of $1$ $h/u_\tau$ have been used to evaluate the data, the quality of results obtained from ST07 remains relatively poor. Additional error estimations are given in Appendix A.2. Since the results of ST30 are quite similar to those of ST07, the details of ST30 are discussed at first. As shown in figure 12(b), in this layer, similar to the TKE budget equation and GKE, the production $\langle \varPi ^{S}\rangle$ is negligible, attributed to the gradients of mean velocities and vorticities. For the spatial effects, traditionally, in the log-law layer, the spatial energy transfers are also negligible (Marati et al. Reference Marati, Casciola and Piva2004). The physical process in the log-law layer is usually believed to be isolated from the spatial effects and is closely related to the turbulent dynamics in homogeneous turbulence. However, for the DHGKE here, the Coriolis term $\langle C^{S} \rangle$ and the rotation-induced pressure transfer term $\langle G^{S}_R \rangle$ are both spatial transfers. It concretely shows that the turbulent structures are strongly influenced by $\langle C^{S} \rangle$ and $\langle G^{S}_R \rangle$. The two effects exhibit opposing tendencies, akin to those observed in the Reynolds stress budget provided by Yang & Wang (Reference Yang and Wang2018). The positive $\langle C^{S} \rangle$ can be readily deduced from its spatial transfer property and the negative $\langle G^S_R \rangle$ can be inferred from the pressure Poisson equation (Kim Reference Kim1989). In the log-law layer, $\langle T^{SS} \rangle$ cascades negative scale helicity to small scales. Since the main spatial effects locate on the largest scale and the scale helicity is always negative at this location, the interscale dynamics of helicity in the log-law layer are consistent with those of homogeneous turbulence. Similar to $\langle C^{S} \rangle$, $\langle T^{SP} \rangle$ is positive and transfers positive helicity from this layer towards the wall. The results of ST07 are similar to those of ST30, while the term $\langle G^{S}_T \rangle$ is not negligible. Figure 13 shows the Reynolds and rotation number effects on $\langle G^S_R \rangle$ and $\langle C^S \rangle$ in the log-law layer. As shown, with increasing rotation rates, the two terms usually become larger. In contrast, as $Re_\tau$ increases, the amplitudes of the two terms in the log-law layer decrease significantly, which can be linked to the vortex angles discussed in § 4.

Figure 12.

DHGKE in the log-law layer ($x^+_2=80.0$): (a) ST07; (b) ST30.

Figure 13.

(a) $G^{S+}_R$ and (b) $C^{S+}$ for different cases in the log-law layer ($x_2^+=81.6$ for ST07R and ST15R, $x_2^+=80.3$ for ST07RR, $x_2^+=80.0$ for the other cases).

To further investigate the DHGKE results, the correlation factor for any two variables $\zeta$ and $\xi$ (Baj, Portela & Carter Reference Baj, Portela and Carter2022) is introduced:

(3.8)\begin{equation} \textrm{corr}(\zeta ,\xi )=\frac{\left\langle {\zeta \xi}\right\rangle-\left\langle {\zeta }\right\rangle\left\langle {\xi}\right\rangle}{\sqrt{\left\langle {\zeta^2}\right\rangle-\left\langle {\zeta}\right\rangle^2}\sqrt{\left\langle {\xi^2}\right\rangle-\left\langle {\xi}\right\rangle^2}}. \end{equation}

Figure 14 shows the correlation factor among the terms in the DHGKE of ST30 located at $x_2^+=3.5$ in figure 14(a), $x_2^+=10.5$ in figure 14(b,d) and $x_2^+=80.0$ in figure 14(c). Figure 14(a–c) is of the scale $r_1^+=47.1$, and figure 14(d) is of the scale $r_1^+=164.9$. The comparison between figures 14(b) and 14(d) shows that the results of $r_1^+=47.12$ are similar to those of $r_1^+=164.9$. In the viscous sublayer (figure 14a) and the buffer layer (figure 14b), the time derivative $\varDelta _t$ is mainly proportional to the production $\varPi ^{S}$, which is dominant in this layer. Additionally, in the log-law layer, the spatial and interscale turbulent convections $T^{SP}$ and $T^{SS}$ are the dominant terms. The term $T^{SP}$ can be explained as the large-scale sweep effects (Baj et al. Reference Baj, Portela and Carter2022). In addition, in the viscous sublayer (figure 14a) and the buffer layer (figure 14b), $T^{SP}$ and $T^{SS}$ are anti-correlated. In contrast, in the buffer (figure 14b,d) and log-law layer (figure 14c), $D^{SS}$ and $D^{SP}$ are correlated. In Appendix C, the four effects are expanded in two-point correlation. The comparison between the expansions of $D^{SP}$ and $D^{SS}$ (or $T^{SP}$ and $T^{SS}$) indicates that the two terms have the same sub-terms but different signs for every sub-term. Taking $D^{SP}$ and $D^{SS}$ for example, the positive correlation between the two terms can be explained by the prevalence of sub-terms with identical signs compared with those with opposing signs. Specifically, the sub-term $2 ({\partial u_i}/{\partial x_j}) ({\partial \omega _i'}/{\partial x_j'})+2 ({\partial u_i'}/{\partial x_j'}) ({\partial \omega _i}/{\partial x_j})$ have opposing signs in the expansion of $D^{SP}$ and $D^{SS}$, and could be related to the two-point correlation of the pesudo-dissipation. The strong positive correlation between $D^{SP}$ and $D^{SS}$ above the buffer layer means the two-point correlation of the pesudo-dissipation is negligible compared with other sub-terms. Additionally, the anti-correlations between the Coriolis term $C^{S}$ and corresponding pressure term $G^{S}_R$ are also confirmed (Kim Reference Kim1989; Yang et al. Reference Yang, Deng, Wang and Shen2018). Here, $C^{S}$ is also correlated with the production term $\varPi ^{S}$, owing to the fact that, on average, $C^{S}$ can be represented by the Reynolds stress $\delta u_1\delta u_2$ (2.6a,b), which also presents in $\varPi ^{S}$.

Figure 14.

Correlation factor of all terms in the DHGKE in the (a) viscous sublayer ($x_2^+=3.5$), (b,d) buffer layer ($x_2^+=10.5$) and (c) log-law layer ($x_2^+=80.0$). Panels (a–c) present the results with the scale $r_1^+=47.1$ and panel (d) presents the results with the scale $r_1^+=164.9$.

In conclusion, different from the scale energy dynamics, the two main effects for the scale helicity balance are the Coriolis term and corresponding pressure term. Therefore, helicity could directly reflect the effects of rotation. In the viscous sublayer, another main effect is the interscale viscous diffusion of the scale helicity towards small scales. In the buffer layer, the production and the spatial turbulent convection lead to the scale discrepancy of the scale helicity. The interscale turbulent convection reduces the discrepancy between small- and large-scale helicity. Further studies around the peaks of helicity shows that the opposing effects of $Re_\tau$ and $Ro_\tau$ are mainly related to the turbulent convection and rotation effects (Coriolis term and corresponding pressure term), respectively. In the log-law layer, the negative scale helicity is found at all scales. The interscale turbulent convection has opposite sign with that in the buffer layer and cascades negative helicity towards small scales. Finally, using the correlation analysis, the large-scale sweep effects (Baj et al. Reference Baj, Portela and Carter2022) and other basic results are confirmed. The consistency of the spatial and interscale effects are found and explained by the prevalence of different sub-terms in two-point correlation expansions.