Double-averaged kinetic energy budgets in flows over mobile granular beds: insights from DNS data analysis

Figures & data

Figure 1. Spatial distributions of the local (left-side plot in each subfigure) and global (right-side plot in each subfigure) space-time porosity ${\phi }_{VT}$ for scenarios HP (a) and LP (b), showing key features of bed morphology based on the distributions of ${\phi }_{VT}$

Figure 2. Vertical distributions of the terms of the globally averaged DMKE balance of Eq. (3(3) $\begin{array}{rl}& \underset{\left(1\right):C_{DD}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{VT}⟨{\overline{u}}_i⟩⟨{\overline{u}}_i⟩⟨{\overline{u}}_j⟩)}}+\underset{\left(2\right):C_{DM}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_i⟩⟨{\overline{u}}_j⟩)}}\\ & =\underset{\left(3\right):G_D}{\underbrace{{\phi }_{VT}⟨{\overline{u}}_1⟩f_1}}-\underset{\left(4\right):-T_{DT}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩⟨{\overline{u}}_i⟩)}}\\ & -\underset{\left(5\right):-T_{DF}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩⟨{\overline{u}}_i⟩)}}-\underset{\left(6\right):-T_{DM}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_j⟩⟨{\overline{u}}_i⟩⟨{\overline{u}}_i⟩)}}\\ & -\underset{\left(7\right):-T_{DP}}{\underbrace{\frac{1}{{\rho }_f}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\phi }_{Vm}⟨{\phi }_T\overline{p}⟩⟨{\overline{u}}_i⟩)}}+\underset{\left(8\right):T_{DV}}{\underbrace{{\nu }_f\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}⟩⟨{\overline{u}}_i⟩)}}\\ & +\underset{\left(9\right):P_{DT}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}+\underset{\left(10\right):P_{DF}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & +\underset{\left(11\right):P_{DM}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_j⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & +\underset{\left(12\right):P_{DP}}{\underbrace{\frac{{\phi }_{Vm}⟨{\phi }_T\overline{p}⟩}{{\rho }_f}\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_i}}}-\underset{\left(13\right):-D_D}{\underbrace{{\nu }_f{\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & +\underset{\left(14\right):{\Phi }_D}{\underbrace{\frac{⟨{\overline{u}}_i⟩}{{\rho }_f{V}_0}{\overline{{\int }_{S_{int}}p{n}_i\mathrm{d}S}}^s-\frac{⟨{\overline{u}}_i⟩}{V_0}{\overline{{\int }_{S_{int}}{\nu }_f\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{n}_j\mathrm{d}S}}^s}}\end{array}$(3) ): DMKE mean convection $C_{DD}$ (term 1) and velocity-time porosity correlation $C_{DM}$ (term 2) for scenarios HP (a) and LP (b); turbulent $T_{DT}$ (term 4), form-induced $T_{DF}$ (term 5) and velocity-time porosity transport $T_{DM}$ (term 6) of DMKE for scenarios HP (c) and LP (d); pressure $T_{DP}$ (term 7) and viscous transport $T_{DV}$ of DMKE (term 8) for scenarios HP (e) and LP (f); the energy supply $G_D$ (term 3), pressure work against bulk strain rate $P_{DP}$ (term 12), viscous dissipation to heat $D_D$ (term 13), and the interfacial energy exchange ${\Phi }_D$ (term 14) for scenarios HP (g) and LP (h); the energy exchange with the TKE balance $P_{DT}$ (term 9) and FKE balance $P_{DF}$ (term 10), $P_{DM}$ (term 11) for scenarios HP (i) and LP (j). The shown term values are normalized on $f_1{U}_b$

Figure 3. Spatial distributions of the energy converting terms of the DMKE balance of Eq. (3(3) $\begin{array}{rl}& \underset{\left(1\right):C_{DD}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{VT}⟨{\overline{u}}_i⟩⟨{\overline{u}}_i⟩⟨{\overline{u}}_j⟩)}}+\underset{\left(2\right):C_{DM}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_i⟩⟨{\overline{u}}_j⟩)}}\\ & =\underset{\left(3\right):G_D}{\underbrace{{\phi }_{VT}⟨{\overline{u}}_1⟩f_1}}-\underset{\left(4\right):-T_{DT}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩⟨{\overline{u}}_i⟩)}}\\ & -\underset{\left(5\right):-T_{DF}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩⟨{\overline{u}}_i⟩)}}-\underset{\left(6\right):-T_{DM}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_j⟩⟨{\overline{u}}_i⟩⟨{\overline{u}}_i⟩)}}\\ & -\underset{\left(7\right):-T_{DP}}{\underbrace{\frac{1}{{\rho }_f}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\phi }_{Vm}⟨{\phi }_T\overline{p}⟩⟨{\overline{u}}_i⟩)}}+\underset{\left(8\right):T_{DV}}{\underbrace{{\nu }_f\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}⟩⟨{\overline{u}}_i⟩)}}\\ & +\underset{\left(9\right):P_{DT}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}+\underset{\left(10\right):P_{DF}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & +\underset{\left(11\right):P_{DM}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_j⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & +\underset{\left(12\right):P_{DP}}{\underbrace{\frac{{\phi }_{Vm}⟨{\phi }_T\overline{p}⟩}{{\rho }_f}\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_i}}}-\underset{\left(13\right):-D_D}{\underbrace{{\nu }_f{\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & +\underset{\left(14\right):{\Phi }_D}{\underbrace{\frac{⟨{\overline{u}}_i⟩}{{\rho }_f{V}_0}{\overline{{\int }_{S_{int}}p{n}_i\mathrm{d}S}}^s-\frac{⟨{\overline{u}}_i⟩}{V_0}{\overline{{\int }_{S_{int}}{\nu }_f\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{n}_j\mathrm{d}S}}^s}}\end{array}$(3) ): energy supply from the volume force work rate $G_D$ (term 3) for scenarios HP (a) and LP (b); turbulent conversion $P_{DT}$ (term 9) for scenarios HP (c) and LP (d); viscous dissipation $D_D$ (term 13) for scenarios HP (e) and LP (f). The term values are normalized on $f_1{U}_b$

Figure 4. Spatial distributions of the transport terms of the DMKE balance of Eq. (3(3) $\begin{array}{rl}& \underset{\left(1\right):C_{DD}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{VT}⟨{\overline{u}}_i⟩⟨{\overline{u}}_i⟩⟨{\overline{u}}_j⟩)}}+\underset{\left(2\right):C_{DM}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_i⟩⟨{\overline{u}}_j⟩)}}\\ & =\underset{\left(3\right):G_D}{\underbrace{{\phi }_{VT}⟨{\overline{u}}_1⟩f_1}}-\underset{\left(4\right):-T_{DT}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩⟨{\overline{u}}_i⟩)}}\\ & -\underset{\left(5\right):-T_{DF}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩⟨{\overline{u}}_i⟩)}}-\underset{\left(6\right):-T_{DM}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_j⟩⟨{\overline{u}}_i⟩⟨{\overline{u}}_i⟩)}}\\ & -\underset{\left(7\right):-T_{DP}}{\underbrace{\frac{1}{{\rho }_f}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\phi }_{Vm}⟨{\phi }_T\overline{p}⟩⟨{\overline{u}}_i⟩)}}+\underset{\left(8\right):T_{DV}}{\underbrace{{\nu }_f\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}⟩⟨{\overline{u}}_i⟩)}}\\ & +\underset{\left(9\right):P_{DT}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}+\underset{\left(10\right):P_{DF}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & +\underset{\left(11\right):P_{DM}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_j⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & +\underset{\left(12\right):P_{DP}}{\underbrace{\frac{{\phi }_{Vm}⟨{\phi }_T\overline{p}⟩}{{\rho }_f}\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_i}}}-\underset{\left(13\right):-D_D}{\underbrace{{\nu }_f{\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & +\underset{\left(14\right):{\Phi }_D}{\underbrace{\frac{⟨{\overline{u}}_i⟩}{{\rho }_f{V}_0}{\overline{{\int }_{S_{int}}p{n}_i\mathrm{d}S}}^s-\frac{⟨{\overline{u}}_i⟩}{V_0}{\overline{{\int }_{S_{int}}{\nu }_f\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{n}_j\mathrm{d}S}}^s}}\end{array}$(3) ): mean convection $C_{DD}$ (term 1) for scenarios HP (a) and LP (b); turbulent transport $T_{DT}$ (term 4) for scenarios HP (c) and LP (d); viscous transport $T_{DV}$ (term 8) for scenarios HP (e) and LP (f). The term values are normalized on $f_1{U}_b$

Figure 5. Vertical distributions of the terms of the globally averaged FKE balance of Eq. (4(4) $\begin{array}{rl}& \underset{\left(1\right):C_{FD}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_j⟩)}}\\ & =\underset{\left(2\right):G_F}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_1⟩f_1}}-\underset{\left(3\right):-P_{FF}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩\frac{⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & -\underset{\left(4\right):-P_{FM}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_j⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}-\underset{\left(5\right):-T_{FF}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩)}}\\ & -\underset{\left(6\right):-T_{FF}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}{\tilde{\overline{u}}}_i⟩)}}-\underset{\left(7\right):-T_{FP}}{\underbrace{\frac{1}{{\rho }_f}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\phi }_{Vm}⟨{\phi }_T\overline{p}{\tilde{\overline{u}}}_i⟩)}}\\ & +\underset{\left(8\right):T_{FV}}{\underbrace{{\nu }_f\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}{\tilde{\overline{u}}}_i⟩)}}+\underset{\left(9\right):P_{FT}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_j}⟩}}\\ & +\underset{\left(10\right):P_{FP}}{\underbrace{\frac{{\phi }_{Vm}}{{\rho }_f}⟨{\phi }_T\overline{p}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_i}⟩}}-\underset{\left(11\right):-D_F}{\underbrace{{\nu }_f{\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_j}⟩}}\\ & +\underset{\left(12\right):{\Phi }_F}{\underbrace{\frac{1}{{\rho }_f{V}_0}{\overline{{\int }_{S_{int}}{\tilde{\overline{u}}}_{i}p{n}_i\mathrm{d}S}}^s-\frac{1}{V_0}{\overline{{\int }_{S_{int}}{\nu }_f{\tilde{\overline{u}}}_i\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{n}_j\mathrm{d}S}}^s}}\end{array}$(4) ): mean $C_{FD}$ (term 1) and form-induced convection of FKE $T_{FF}$ (term 5) for scenarios HP (a) and LP (b); turbulent $T_{FT}$ (term 6), pressure $T_{FP}$ (term 7) and viscous stress transport $T_{FV}$ (term 8) for scenarios HP (c) and LP (d); energy supply $G_F$ (term 2) and interfacial energy exchange ${\Phi }_F$ (term 12) for scenarios HP (e) and LP (f); energy exchange with the DMKE balance $P_{FF}$ (term 3), $P_{FM}$ (term 4) and TKE balance $P_{FT}$ (term 9) for scenarios HP (g) and LP (h); the work of pressure on the form-induced strain rate $P_{FP}$ (term 10), viscous dissipation to heat $D_F$ (term 11) for the scenarios HP (i) and LP (j). The term values are normalized on $f_1{U}_b$

Figure 6. Spatial distributions of the energy converting terms of the FKE balance of Eq. (4(4) $\begin{array}{rl}& \underset{\left(1\right):C_{FD}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_j⟩)}}\\ & =\underset{\left(2\right):G_F}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_1⟩f_1}}-\underset{\left(3\right):-P_{FF}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩\frac{⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & -\underset{\left(4\right):-P_{FM}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_j⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}-\underset{\left(5\right):-T_{FF}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩)}}\\ & -\underset{\left(6\right):-T_{FF}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}{\tilde{\overline{u}}}_i⟩)}}-\underset{\left(7\right):-T_{FP}}{\underbrace{\frac{1}{{\rho }_f}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\phi }_{Vm}⟨{\phi }_T\overline{p}{\tilde{\overline{u}}}_i⟩)}}\\ & +\underset{\left(8\right):T_{FV}}{\underbrace{{\nu }_f\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}{\tilde{\overline{u}}}_i⟩)}}+\underset{\left(9\right):P_{FT}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_j}⟩}}\\ & +\underset{\left(10\right):P_{FP}}{\underbrace{\frac{{\phi }_{Vm}}{{\rho }_f}⟨{\phi }_T\overline{p}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_i}⟩}}-\underset{\left(11\right):-D_F}{\underbrace{{\nu }_f{\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_j}⟩}}\\ & +\underset{\left(12\right):{\Phi }_F}{\underbrace{\frac{1}{{\rho }_f{V}_0}{\overline{{\int }_{S_{int}}{\tilde{\overline{u}}}_{i}p{n}_i\mathrm{d}S}}^s-\frac{1}{V_0}{\overline{{\int }_{S_{int}}{\nu }_f{\tilde{\overline{u}}}_i\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{n}_j\mathrm{d}S}}^s}}\end{array}$(4) ): energy supply from the DMKE balance $P_{FF}$ (term 3) for scenarios HP (a) and LP (b); energy exchange with the TKE balance $P_{FT}$ (term 9) for scenarios HP (c) and LP (d); viscous dissipation $D_F$ (term 11) for the scenarios HP (e) and LP (f). The term values are normalized on $f_1{U}_b$

Figure 7. Spatial distributions of the transport terms of the FKE balance of Eq. (4(4) $\begin{array}{rl}& \underset{\left(1\right):C_{FD}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_j⟩)}}\\ & =\underset{\left(2\right):G_F}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_1⟩f_1}}-\underset{\left(3\right):-P_{FF}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩\frac{⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}\\ & -\underset{\left(4\right):-P_{FM}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i⟩⟨{\overline{u}}_j⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}-\underset{\left(5\right):-T_{FF}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_i{\tilde{\overline{u}}}_j⟩)}}\\ & -\underset{\left(6\right):-T_{FF}}{\underbrace{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}{\tilde{\overline{u}}}_i⟩)}}-\underset{\left(7\right):-T_{FP}}{\underbrace{\frac{1}{{\rho }_f}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\phi }_{Vm}⟨{\phi }_T\overline{p}{\tilde{\overline{u}}}_i⟩)}}\\ & +\underset{\left(8\right):T_{FV}}{\underbrace{{\nu }_f\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}{\tilde{\overline{u}}}_i⟩)}}+\underset{\left(9\right):P_{FT}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_j}⟩}}\\ & +\underset{\left(10\right):P_{FP}}{\underbrace{\frac{{\phi }_{Vm}}{{\rho }_f}⟨{\phi }_T\overline{p}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_i}⟩}}-\underset{\left(11\right):-D_F}{\underbrace{{\nu }_f{\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_j}⟩}}\\ & +\underset{\left(12\right):{\Phi }_F}{\underbrace{\frac{1}{{\rho }_f{V}_0}{\overline{{\int }_{S_{int}}{\tilde{\overline{u}}}_{i}p{n}_i\mathrm{d}S}}^s-\frac{1}{V_0}{\overline{{\int }_{S_{int}}{\nu }_f{\tilde{\overline{u}}}_i\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{n}_j\mathrm{d}S}}^s}}\end{array}$(4) ): mean convection $C_{FD}$ (term 1) for scenarios HP (a) and LP (b); form-induced convection $T_{FF}$ (term 5) for scenarios HP (c) and LP (d); turbulent transport $T_{FT}$ (term 6) for scenarios HP (e) and LP (f); viscous transport $T_{FV}$ (term 8) for the scenarios HP (g) and LP (h). The values are normalized on $f_1{U}_b$

Figure 8. Vertical distributions of the terms of the TKE balance of Eq. (5(5) $\begin{array}{rl}& \underset{\left(1\right):C_{TD}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_i^{\mathrm{\prime }}}⟩⟨{\overline{u}}_j⟩)}}\\ & =-\underset{\left(2\right):-P_{TT}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}-\underset{\left(3\right):-P_{TF}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_j}⟩}}\\ & -\underset{\left(4\right):-T_{TF}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_i^{\mathrm{\prime }}}{\tilde{\overline{u}}}_j⟩)}}-\underset{\left(5\right):-T_{TT}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩)}}\\ & -\underset{\left(6\right):-T_{TP}}{\underbrace{\frac{1}{{\rho }_f}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{p}^{\mathrm{\prime }}}⟩)}}+\underset{\left(7\right):T_{TV}}{\underbrace{{\nu }_f\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_j}}⟩\right)}}\\ & +\underset{\left(8\right):P_{TP}}{\underbrace{\frac{{\phi }_{Vm}}{{\rho }_f}⟨{\phi }_T\overline{p^{\mathrm{\prime }}\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_i}}⟩}}-\underset{\left(9\right):-D_T}{\underbrace{{\nu }_f{\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_j}\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_j}}⟩}}\\ & +\underset{\left(10\right):{\Phi }_T}{\underbrace{\frac{1}{{\rho }_f{V}_0}{\overline{{\int }_{S_{int}}{u}_i^{\mathrm{\prime }}p{n}_i\mathrm{d}S}}^s-\frac{1}{V_0}{\overline{{\int }_{S_{int}}{\nu }_f{u}_i^{\mathrm{\prime }}\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{n}_j\mathrm{d}S}}^s}}\end{array}$(5) ): mean $C_{TD}$ (term 1) and form-induced $T_{TF}$ (term 4) convection of TKE for scenarios HP (a) and LP (b); turbulent $T_{TT}$ (term 5), pressure $T_{TP}$ (term 6) and viscous $T_{TV}$ (term 7) transport for scenarios HP (c) and LP (d); turbulent production $P_{TT}$ (term 2), energy exchange with the FKE balance $P_{TF}$ (term 3), pressure–strain rate correlation $P_{TP}$ (term 8), viscous dissipation $D_T$ (term 9) and the interfacial energy exchange ${\Phi }_T$ (term 10) for scenarios HP (e) and LP (f). The values of the TKE budget terms are normalized on $f_1{U}_b$

Figure 9. Spatial distributions of the energy-converting terms of the TKE balance of Eq. (5(5) $\begin{array}{rl}& \underset{\left(1\right):C_{TD}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_i^{\mathrm{\prime }}}⟩⟨{\overline{u}}_j⟩)}}\\ & =-\underset{\left(2\right):-P_{TT}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}-\underset{\left(3\right):-P_{TF}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_j}⟩}}\\ & -\underset{\left(4\right):-T_{TF}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_i^{\mathrm{\prime }}}{\tilde{\overline{u}}}_j⟩)}}-\underset{\left(5\right):-T_{TT}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩)}}\\ & -\underset{\left(6\right):-T_{TP}}{\underbrace{\frac{1}{{\rho }_f}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{p}^{\mathrm{\prime }}}⟩)}}+\underset{\left(7\right):T_{TV}}{\underbrace{{\nu }_f\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_j}}⟩\right)}}\\ & +\underset{\left(8\right):P_{TP}}{\underbrace{\frac{{\phi }_{Vm}}{{\rho }_f}⟨{\phi }_T\overline{p^{\mathrm{\prime }}\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_i}}⟩}}-\underset{\left(9\right):-D_T}{\underbrace{{\nu }_f{\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_j}\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_j}}⟩}}\\ & +\underset{\left(10\right):{\Phi }_T}{\underbrace{\frac{1}{{\rho }_f{V}_0}{\overline{{\int }_{S_{int}}{u}_i^{\mathrm{\prime }}p{n}_i\mathrm{d}S}}^s-\frac{1}{V_0}{\overline{{\int }_{S_{int}}{\nu }_f{u}_i^{\mathrm{\prime }}\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{n}_j\mathrm{d}S}}^s}}\end{array}$(5) ): turbulent production $P_{TT}$ (term 2) for scenarios HP (a) and LP (b); viscous dissipation $D_T$ (term 9) for scenarios HP (c) and LP (d). The values of the TKE budget terms are normalized on $f_1{U}_b$

Figure 10. Spatial distributions of the transport terms of the TKE balance of Eq. (5(5) $\begin{array}{rl}& \underset{\left(1\right):C_{TD}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_i^{\mathrm{\prime }}}⟩⟨{\overline{u}}_j⟩)}}\\ & =-\underset{\left(2\right):-P_{TT}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩\frac{\mathrm{\partial }⟨{\overline{u}}_i⟩}{\mathrm{\partial }{x}_j}}}-\underset{\left(3\right):-P_{TF}}{\underbrace{{\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}\frac{\mathrm{\partial }{\tilde{\overline{u}}}_i}{\mathrm{\partial }{x}_j}⟩}}\\ & -\underset{\left(4\right):-T_{TF}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_i^{\mathrm{\prime }}}{\tilde{\overline{u}}}_j⟩)}}-\underset{\left(5\right):-T_{TT}}{\underbrace{\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{u}_i^{\mathrm{\prime }}{u}_j^{\mathrm{\prime }}}⟩)}}\\ & -\underset{\left(6\right):-T_{TP}}{\underbrace{\frac{1}{{\rho }_f}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}{p}^{\mathrm{\prime }}}⟩)}}+\underset{\left(7\right):T_{TV}}{\underbrace{{\nu }_f\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left({\phi }_{Vm}⟨{\phi }_T\overline{u_i^{\mathrm{\prime }}\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_j}}⟩\right)}}\\ & +\underset{\left(8\right):P_{TP}}{\underbrace{\frac{{\phi }_{Vm}}{{\rho }_f}⟨{\phi }_T\overline{p^{\mathrm{\prime }}\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_i}}⟩}}-\underset{\left(9\right):-D_T}{\underbrace{{\nu }_f{\phi }_{Vm}⟨{\phi }_T\overline{\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_j}\frac{\mathrm{\partial }{u}_i^{\mathrm{\prime }}}{\mathrm{\partial }{x}_j}}⟩}}\\ & +\underset{\left(10\right):{\Phi }_T}{\underbrace{\frac{1}{{\rho }_f{V}_0}{\overline{{\int }_{S_{int}}{u}_i^{\mathrm{\prime }}p{n}_i\mathrm{d}S}}^s-\frac{1}{V_0}{\overline{{\int }_{S_{int}}{\nu }_f{u}_i^{\mathrm{\prime }}\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{n}_j\mathrm{d}S}}^s}}\end{array}$(5) ): mean convection $C_{TD}$ (term 1) for scenarios HP (a) and LP (b); turbulent convection $T_{TT}$ (term 5) for scenarios HP (c) and LP (d); pressure transport $T_{TP}$ (term 6) for scenarios HP (e) and LP (f); viscous transport $T_{TV}$ (term 7) for scenarios HP (g) and LP (h). The values of the TKE budget terms are normalized on $f_1{U}_b$

Figure 11. Schematic representation of the energy fluxes (arrows) between double-mean flow, form-induced flow, turbulent flow and mobile bed in the particle transporting flows