$\displaystyle(-6)(-2)\times\Bigg{[}\frac{i}{2}\,\Big{(}Q^{\lambda,11}_{\frac{9}{2}}+i\sqrt{2}\,Q^{\lambda,12}_{\frac{9}{2}}-2\,Q^{\lambda,22}_{\frac{9}{2}}-2\,\bar{Q}^{\lambda,11}_{\frac{9}{2}}-i\sqrt{2}\,\bar{Q}^{\lambda,21}_{\frac{9}{2}}+\bar{Q}^{\lambda,22}_{\frac{9}{2}}\Big{)}\Bigg{]}\,,$
$\displaystyle\tilde{\Phi}^{(3),4}_{\frac{3}{2}}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(3),4}_{\frac{3}{2}}-\frac{1}{7}\,(1-4\lambda)\,\partial\,\Phi^{(3),4}_{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle(-6)(-2)\times\Bigg{[}\frac{1}{2}\,\Big{(}Q^{\lambda,11}_{\frac{9}{2}}+2\,Q^{\lambda,22}_{\frac{9}{2}}+2\,\bar{Q}^{\lambda,11}_{\frac{9}{2}}+\bar{Q}^{\lambda,22}_{\frac{9}{2}}\Big{)}\Bigg{]}\,,$
$\displaystyle\tilde{\Phi}^{(3)}_{2}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(3)}_{2}-\frac{1}{7}(1-4\lambda)\partial^{2}\Phi^{(3)}_{0}$
(F.2) $\displaystyle=$
$\displaystyle(-6)(-2)\times\Bigg{[}-2\Big{(}W^{\lambda,11}_{\mathrm{B},5}+W^{\lambda,22}_{\mathrm{B},5}+W^{\lambda,11}_{\mathrm{F},5}+W^{\lambda,22}_{\mathrm{F},5}\Big{)}\Bigg{]}.$

All of these in (F.2) are quasiprimary under the stress energy tensor (3.1).

##  Appendix G  The OPEs between the second ${\cal N}=4$ multiplet and itself

We summarize the super OPE between the second ${\cal N}=4$ multiplet and
itself as follows:

$\displaystyle{\bf\Phi}^{(2)}(Z_{1})\,{\bf\Phi}^{(2)}(Z_{2})=\frac{1}{z_{12}^{4}}\,\Bigg{[}\frac{128}{3}\,N\,(1-4\lambda)\,(1+2\lambda-4\lambda^{2})\Bigg{]}$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{6}}\,\Bigg{[}\,-\frac{512}{3}\,N\,(-1+10\lambda-80\lambda^{3}+80\lambda^{4})\,\Bigg{]}$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{5}}\,\Bigg{[}\,\frac{2048}{3}\,\lambda\,(1-\lambda)\,(1-2\lambda)\,(1+2\lambda)\,D^{i}{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{5}}\,\Bigg{[}\,\frac{4096}{3}\,\lambda\,(1-\lambda)\,(1-2\lambda)\,(1+2\lambda)\,\partial\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}^{4}}\,\Bigg{[}\,-\frac{256}{3}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,D^{k}\,D^{l}\,{\bf
J}$
$\displaystyle-\frac{128}{3}\,(1-\lambda)\,(1+2\lambda)\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,D^{m}\,D^{n}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{4}}\,\Bigg{[}\,\frac{4096}{3}\,\lambda\,(1-\lambda)\,(1-2\lambda)\,(1+2\lambda)\,\partial\,D^{i}{\bf
J}$
$\displaystyle+128\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial\,D^{i}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{4}}\,\Bigg{[}\,2048\,\lambda\,(1-\lambda)\,(1-2\lambda)\,(1+2\lambda)\,\partial^{2}\,{\bf
J}$
$\displaystyle-\frac{512}{3}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{2}\,{\bf J}\Big{)}$
$\displaystyle-32\,(-7-4\lambda+8\lambda^{2})\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}^{3}}\,\Bigg{[}\,-\frac{128}{3}\,(1-\lambda)\,(1+2\lambda)\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}\,{\bf
J}-(1-4\lambda)\,\partial\,D^{i}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}^{3}}\,\Bigg{[}\,-\frac{256}{3}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,\partial\,D^{k}\,D^{l}\,{\bf
J}$
$\displaystyle-\frac{128}{3}\,(1-\lambda)\,(1+2\lambda)\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,\partial\,D^{m}\,D^{n}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{3}}\,\Bigg{[}\,1024\,\lambda\,(1-\lambda)\,(1-2\lambda)\,(1+2\lambda)\,\partial^{2}\,D^{i}{\bf
J}$
$\displaystyle+\frac{1024}{9}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial^{2}\,D^{i}\,{\bf J}\Big{)}$
$\displaystyle-\frac{16}{3}\,(-11-2\lambda+4\lambda^{2})\,D^{i}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{3}}\,\Bigg{[}\,\frac{4096}{3}\,\lambda\,(1-\lambda)\,(1-2\lambda)\,(1+2\lambda)\,\partial^{3}\,{\bf
J}$
$\displaystyle+\frac{1792}{9}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,\partial\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{3}\,{\bf J}\Big{)}$
$\displaystyle-\frac{16}{3}\,(-43-16\lambda+32\lambda^{2})\,\partial\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{1}{z_{12}^{2}}\,\Bigg{[}\,-\frac{16}{3}\,(1-4\lambda)\,{\bf\Phi}^{(2)}$
$\displaystyle+\frac{256}{9}\,(1-\lambda)\,(1+2\lambda)\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,D^{i}\,D^{j}\,D^{k}\,D^{l}\,{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{2}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}^{2}}\,\Bigg{[}\,-\frac{256}{9}\,(1-\lambda)\,(1+2\lambda)\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial\,D^{j}\,D^{k}\,D^{l}\,{\bf
J}-(1-4\lambda)\,\partial^{2}\,D^{i}\,{\bf J}\Big{)}$
$\displaystyle-\frac{8}{3}\,(1-4\lambda)\,D^{i}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}^{2}}\,\Bigg{[}\,\frac{128}{3}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,\partial^{2}\,D^{k}\,D^{l}\,{\bf
J}$
$\displaystyle-\frac{64}{3}\,(1-\lambda)\,(1+2\lambda)\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,\partial^{2}\,D^{m}\,D^{n}\,{\bf
J}-4\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}$
$\displaystyle-6\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,D^{m}\,D^{n}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{2}}\,\Bigg{[}\,\frac{4096}{9}\,\lambda\,(1-\lambda)\,(1-2\lambda)\,(1+2\lambda)\,\partial^{3}\,D^{i}{\bf
J}$
$\displaystyle+\frac{160}{3}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial^{2}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial^{3}\,D^{i}\,{\bf J}\Big{)}$
$\displaystyle-\frac{2}{3}\,(-71-8\lambda+16\lambda^{2})\,\partial\,D^{i}\,{\bf\Phi}^{(2)}-\frac{14}{3}\,(1-4\lambda)\,\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{2}}\,\Bigg{[}\,\frac{5120}{9}\,\lambda\,(1-\lambda)\,(1-2\lambda)\,(1+2\lambda)\,\partial^{4}\,{\bf
J}$
$\displaystyle-\frac{512}{5}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,\partial^{2}\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{4}\,{\bf J}\Big{)}$
$\displaystyle-\frac{64}{21}\,(-38-11\lambda+22\lambda^{2})\,\partial^{2}\,{\bf\Phi}^{(2)}-\frac{64}{21}\,(1-4\lambda)\,\frac{1}{4!}\,\varepsilon^{ijkl}\,\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf\Phi}^{(2)}$
$\displaystyle+\frac{1}{3}\,{\bf\Phi}^{(4)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{1}{z_{12}}\,\Bigg{[}\,-\frac{8}{3}\,(1-4\lambda)\,\partial\,{\bf\Phi}^{(2)}$
$\displaystyle+\frac{128}{9}\,(1-\lambda)\,(1+2\lambda)\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,\partial\,D^{i}\,D^{j}\,D^{k}\,D^{l}\,{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{3}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}}\,\Bigg{[}\,-\frac{32}{3}\,(1-\lambda)\,(1+2\lambda)\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial^{2}\,D^{j}\,D^{k}\,D^{l}\,{\bf
J}-(1-4\lambda)\,\partial^{3}\,D^{i}\,{\bf J}\Big{)}$
$\displaystyle-2\,(1-4\lambda)\,\partial\,D^{i}\,{\bf\Phi}^{(2)}+2\,\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}}\,\Bigg{[}\,-\frac{128}{9}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,\partial^{3}\,D^{k}\,D^{l}\,{\bf
J}$
$\displaystyle-\frac{64}{9}\,(1-\lambda)\,(1+2\lambda)\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,\partial^{3}\,D^{m}\,D^{n}\,{\bf
J}+\frac{8}{3}\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,\partial\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}$
$\displaystyle+4\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,\partial\,D^{m}\,D^{n}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}}\,\Bigg{[}\,\frac{1280}{9}\,\lambda\,(1-\lambda)\,(1-2\lambda)\,(1+2\lambda)\,\partial^{4}\,D^{i}{\bf
J}$
$\displaystyle\frac{256}{15}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial^{3}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial^{4}\,D^{i}\,{\bf J}\Big{)}$
$\displaystyle-\frac{32}{21}\,(-13-\lambda+2\lambda^{2})\,\partial^{2}\,D^{i}\,{\bf\Phi}^{(2)}-\frac{64}{21}\,(1-4\lambda)\,\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial\,D^{j}\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}$
$\displaystyle+\frac{1}{24}\,D^{i}\,{\bf\Phi}^{(4)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}}\,\Bigg{[}\,\frac{512}{3}\,\lambda\,(1-\lambda)\,(1-2\lambda)\,(1+2\lambda)\,\partial^{5}\,{\bf
J}$
$\displaystyle-\frac{512}{15}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,\partial^{3}\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{5}\,{\bf J}\Big{)}$
$\displaystyle-\frac{67}{7}\,(-17-4\lambda+8\lambda^{2})\,\partial^{3}\,{\bf\Phi}^{(2)}-\frac{16}{7}\,(1-4\lambda)\,\frac{1}{4!}\,\varepsilon^{ijkl}\,\,\partial\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf\Phi}^{(2)}$
$\displaystyle+\frac{1}{4}\,\partial\,{\bf\Phi}^{(4)}\,\Bigg{]}(Z_{2})+\cdots.$
(G.1)

There exist the ${\cal N}=4$ stress energy tensor, the second ${\cal N}=4$
multiplet, the fourth ${\cal N}=4$ multiplet, as well as their descendants on
the right hand sides of (G.1).

The fourth ${\cal N}=4$ multiplet can be summarized by

$\displaystyle\Phi^{(4)}_{0}$ $\displaystyle=$
$\displaystyle\frac{768}{7}\,(-2+\lambda)\,(W^{\lambda,11}_{\mathrm{F},4}+W^{\lambda,22}_{\mathrm{F},4})+\frac{384}{7}\,(3+2\lambda)\,(W^{\lambda,11}_{\mathrm{B},4}+W^{\lambda,22}_{\mathrm{B},4})\,,$
$\displaystyle\Phi^{(4),1}_{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}\frac{1}{2}\,\Big{(}Q^{\lambda,11}_{\frac{9}{2}}+i\sqrt{2}\,Q^{\lambda,12}_{\frac{9}{2}}+2i\sqrt{2}\,Q^{\lambda,21}_{\frac{9}{2}}-2\,Q^{\lambda,22}_{\frac{9}{2}}$
$\displaystyle+$ $\displaystyle
2\,\bar{Q}^{\lambda,11}_{\frac{9}{2}}+2i\sqrt{2}\,\bar{Q}^{\lambda,12}_{\frac{9}{2}}+i\sqrt{2}\,\bar{Q}^{\lambda,21}_{\frac{9}{2}}-\bar{Q}^{\lambda,22}_{\frac{9}{2}}\Big{)}\Bigg{]}\,,$
$\displaystyle\Phi^{(4),2}_{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}-\frac{i}{2}\,\Big{(}Q^{\lambda,11}_{\frac{9}{2}}+2i\sqrt{2}\,Q^{\lambda,21}_{\frac{9}{2}}-2\,Q^{\lambda,22}_{\frac{9}{2}}+2\,\bar{Q}^{\lambda,11}_{\frac{9}{2}}+2i\sqrt{2}\,\bar{Q}^{\lambda,12}_{\frac{9}{2}}$
$\displaystyle-$
$\displaystyle\bar{Q}^{\lambda,22}_{\frac{9}{2}}\Big{)}\Bigg{]}\,,$
$\displaystyle\Phi^{(4),3}_{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}-\frac{i}{2}\,\Big{(}Q^{\lambda,11}_{\frac{9}{2}}+i\sqrt{2}\,Q^{\lambda,12}_{\frac{9}{2}}-2\,Q^{\lambda,22}_{\frac{9}{2}}+2\,\bar{Q}^{\lambda,11}_{\frac{9}{2}}+i\sqrt{2}\,\bar{Q}^{\lambda,21}_{\frac{9}{2}}$
$\displaystyle-$
$\displaystyle\bar{Q}^{\lambda,22}_{\frac{9}{2}}\Big{)}\Bigg{]}\,,$
$\displaystyle\Phi^{(4),4}_{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}-\frac{1}{2}\,Q^{\lambda,11}_{\frac{9}{2}}-Q^{\lambda,22}_{\frac{9}{2}}+\bar{Q}^{\lambda,11}_{\frac{9}{2}}+\frac{1}{2}\,\bar{Q}^{\lambda,22}_{\frac{9}{2}}\Bigg{]}\,,$
$\displaystyle\Phi^{(4),12}_{1}$ $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}2i\,W^{\lambda,11}_{\mathrm{B},5}-\sqrt{2}\,W^{\lambda,12}_{\mathrm{B},5}-2i\,\,W^{\lambda,22}_{\mathrm{B},5}+2i\,W^{\lambda,11}_{\mathrm{F},5}-2\sqrt{2}\,W^{\lambda,12}_{\mathrm{F},5}$
$\displaystyle-$ $\displaystyle 2i\,W^{\lambda,22}_{\mathrm{F},5}\Bigg{]}\,,$
$\displaystyle\Phi^{(4),13}_{1}$ $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}-2i\,W^{\lambda,11}_{\mathrm{B},5}+4\sqrt{2}\,W^{\lambda,21}_{\mathrm{B},5}+2i\,\,W^{\lambda,22}_{\mathrm{B},5}-2i\,W^{\lambda,11}_{\mathrm{F},5}+2\sqrt{2}\,W^{\lambda,21}_{\mathrm{F},5}$
$\displaystyle+$ $\displaystyle 2i\,W^{\lambda,22}_{\mathrm{F},5}\Bigg{]}\,,$
$\displaystyle\Phi^{(4),14}_{1}$ $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}2\,W^{\lambda,11}_{\mathrm{B},5}+i\sqrt{2}\,W^{\lambda,12}_{\mathrm{B},5}+4i\sqrt{2}\,\,W^{\lambda,21}_{\mathrm{B},5}-2\,W^{\lambda,22}_{\mathrm{B},5}-2\,W^{\lambda,11}_{\mathrm{F},5}$
$\displaystyle-$ $\displaystyle
2i\sqrt{2}\,W^{\lambda,12}_{\mathrm{F},5}-2i\sqrt{2}\,W^{\lambda,21}_{\mathrm{F},5}+2\,W^{\lambda,22}_{\mathrm{F},5}\Bigg{]}\,,$
$\displaystyle\Phi^{(4),23}_{1}$ $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}-2\,W^{\lambda,11}_{\mathrm{B},5}-i\sqrt{2}\,W^{\lambda,12}_{\mathrm{B},5}-4i\sqrt{2}\,\,W^{\lambda,21}_{\mathrm{B},5}+2\,W^{\lambda,22}_{\mathrm{B},5}-2\,W^{\lambda,11}_{\mathrm{F},5}$
$\displaystyle-$ $\displaystyle
2i\sqrt{2}\,W^{\lambda,12}_{\mathrm{F},5}-2i\sqrt{2}\,W^{\lambda,21}_{\mathrm{F},5}+2\,W^{\lambda,22}_{\mathrm{F},5}\Bigg{]}\,,$
$\displaystyle\Phi^{(4),24}_{1}$ $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}-2i\,W^{\lambda,11}_{\mathrm{B},5}+4\sqrt{2}\,W^{\lambda,21}_{\mathrm{B},5}+2i\,\,W^{\lambda,22}_{\mathrm{B},5}+2i\,W^{\lambda,11}_{\mathrm{F},5}-2\sqrt{2}\,W^{\lambda,21}_{\mathrm{F},5}$
$\displaystyle-$ $\displaystyle 2i\,W^{\lambda,22}_{\mathrm{F},5}\Bigg{]}\,,$
$\displaystyle\Phi^{(4),34}_{1}$ $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}-2i\,W^{\lambda,11}_{\mathrm{B},5}+\sqrt{2}\,W^{\lambda,12}_{\mathrm{B},5}+2i\,\,W^{\lambda,22}_{\mathrm{B},5}+2i\,W^{\lambda,11}_{\mathrm{F},5}-2\sqrt{2}\,W^{\lambda,12}_{\mathrm{F},5}$
$\displaystyle-$ $\displaystyle 2i\,W^{\lambda,22}_{\mathrm{F},5}\Bigg{]}\,,$
$\displaystyle\tilde{\Phi}^{(4),1}_{\frac{3}{2}}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(4),1}_{\frac{3}{2}}-\frac{1}{9}\,(1-4\lambda)\,\partial\,\Phi^{(4),1}_{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}-\frac{1}{2}\,\Big{(}Q^{\lambda,11}_{\frac{11}{2}}+i\sqrt{2}\,Q^{\lambda,12}_{\frac{11}{2}}+2i\sqrt{2}\,Q^{\lambda,21}_{\frac{11}{2}}-2\,Q^{\lambda,22}_{\frac{11}{2}}$
$\displaystyle-$ $\displaystyle
2\,\bar{Q}^{\lambda,11}_{\frac{11}{2}}-2i\sqrt{2}\,\bar{Q}^{\lambda,12}_{\frac{11}{2}}-i\sqrt{2}\,\bar{Q}^{\lambda,21}_{\frac{11}{2}}+\bar{Q}^{\lambda,22}_{\frac{11}{2}}\Big{)}\Bigg{]}\,,$
$\displaystyle\tilde{\Phi}^{(4),2}_{\frac{3}{2}}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(4),2}_{\frac{3}{2}}-\frac{1}{9}\,(1-4\lambda)\,\partial\,\Phi^{(4),2}_{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}\frac{i}{2}\,\Big{(}\,Q^{\lambda,11}_{\frac{11}{2}}+2i\sqrt{2}\,Q^{\lambda,21}_{\frac{11}{2}}-2\,Q^{\lambda,22}_{\frac{11}{2}}-2\,\bar{Q}^{\lambda,11}_{\frac{11}{2}}-2i\sqrt{2}\,\bar{Q}^{\lambda,12}_{\frac{11}{2}}$
$\displaystyle+$
$\displaystyle\bar{Q}^{\lambda,22}_{\frac{11}{2}}\Big{)}\Bigg{]}\,,$
$\displaystyle\tilde{\Phi}^{(4),3}_{\frac{3}{2}}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(4),3}_{\frac{3}{2}}-\frac{1}{9}\,(1-4\lambda)\,\partial\,\Phi^{(4),3}_{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}\frac{i}{2}\,\Big{(}Q^{\lambda,11}_{\frac{11}{2}}+i\sqrt{2}\,Q^{\lambda,12}_{\frac{11}{2}}-2\,Q^{\lambda,22}_{\frac{11}{2}}-2\,\bar{Q}^{\lambda,11}_{\frac{11}{2}}-i\sqrt{2}\,\bar{Q}^{\lambda,21}_{\frac{11}{2}}+\bar{Q}^{\lambda,22}_{\frac{11}{2}}\Big{)}\Bigg{]}\,,$
$\displaystyle\tilde{\Phi}^{(4),4}_{\frac{3}{2}}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(4),4}_{\frac{3}{2}}-\frac{1}{9}\,(1-4\lambda)\,\partial\,\Phi^{(4),4}_{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}\frac{1}{2}\,\Big{(}Q^{\lambda,11}_{\frac{11}{2}}+2\,Q^{\lambda,22}_{\frac{11}{2}}+2\,\bar{Q}^{\lambda,11}_{\frac{11}{2}}+\bar{Q}^{\lambda,22}_{\frac{11}{2}}\Big{)}\Bigg{]}\,,$
$\displaystyle\tilde{\Phi}^{(4)}_{2}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(4)}_{2}-\frac{1}{9}(1-4\lambda)\partial^{2}\Phi^{(4)}_{0}$
(G.2) $\displaystyle=$
$\displaystyle(-8)(-6)(-2)\times\Bigg{[}-2\Big{(}W^{\lambda,11}_{\mathrm{B},6}+W^{\lambda,22}_{\mathrm{B},6}+W^{\lambda,11}_{\mathrm{F},6}+W^{\lambda,22}_{\mathrm{F},6}\Big{)}\Bigg{]}.$

All of these are quasiprimary under the stress energy tensor (3.1). The
$\lambda$-dependence in the weight-$4$ operator in (G.2) can be obtained from
the factor $(4-2\lambda)$ appearing in $W_{B,4}^{\lambda,\bar{a}a}$ and the
factor $(3+2\lambda)$ appearing in $W_{F,4}^{\lambda,\bar{a}a}$ respectively.

This implies that for the $h$-th ${\cal N}=4$ multiplet, the
$\lambda$-dependence in the weight-$h$ operator can be obtained from the
factor $(h-2\lambda)$ appearing in $W_{F,h}^{\lambda,\bar{a}a}$ and the factor
$(h-1+2\lambda)$ appearing in $W_{B,h}^{\lambda,\bar{a}a}$ respsectively. For
the four weight-$(h+\frac{1}{2})$ operators, we simply take
$(-2h)\,\cdots\,(-8)\,(-6)\,(-2)$ multiplied by the quantities inside the
brackets after we replace $\frac{9}{2}$ with $(h+\frac{1}{2})$. For the six
weight-$(h+1)$ operators, we simply take $(-2h)\,\cdots\,(-8)\,(-6)\,(-2)$
multiplied by the quantities inside the brackets after we replace $5$ with
$(h+1)$. Similarly, for the four weight-$(h+\frac{3}{2})$ operators, we simply
take $(-2h)\,\cdots\,(-8)\,(-6)\,(-2)$ multiplied by the quantities inside the
brackets after we replace $\frac{11}{2}$ with $(h+\frac{3}{2})$. For the
weight-$(h+2)$ operator, we simply take $(-2h)\,\cdots\,(-8)\,(-6)\,(-2)$
multiplied by the quantity inside the bracket after we replace $6$ with
$(h+2)$. The $W_{F,h}^{\lambda,\bar{a}b}$ and the $W_{B,h}^{\lambda,\bar{a}b}$
can be written in terms of $\Phi_{0}^{(h)}$, $\Phi^{(h-1),ij}_{1}$ and
$\Phi_{2}^{(h-2)}$. Similarly, the $Q_{h+\frac{1}{2}}^{\lambda,\bar{a}b}$ and
the $\bar{Q}_{h+\frac{1}{2}}^{\lambda,a\bar{b}}$ can be written in terms of
$\Phi_{\frac{1}{2}}^{(h),i}$ and $\Phi_{\frac{3}{2}}^{(h-1),i}$.

##  Appendix H  The OPEs between the ${\cal N}=4$ stress energy tensor and the
third ${\cal N}=4$ multiplet

The OPEs between the ${\cal N}=4$ stress energy tensor and the third ${\cal
N}=4$ multiplet can be described by

$\displaystyle{\bf
J}(Z_{1})\,{\bf\Phi}^{(3)}(Z_{2})=-\frac{\theta_{12}^{4-0}}{z_{12}^{5}}\,\Bigg{[}-\frac{1536}{5}\,N\,\lambda\,(1-2\lambda)\,(1-4\lambda)\Bigg{]}$
$\displaystyle+\frac{1}{z_{12}^{3}}\,\Bigg{[}\,-\frac{192}{15}\,N\,(1-3\lambda+6\lambda^{2})\,\Bigg{]}$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{4}}\,\Bigg{[}\,\frac{1536}{5}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,D^{i}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}^{3}}\,\Bigg{[}\,-\frac{192}{5}\,(1-3\lambda+6\lambda^{2})\,D^{i}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}^{3}}\,\Bigg{[}\,\frac{192}{5}\,(1+2\lambda-4\lambda^{2})\,\frac{1}{2!}\,\varepsilon^{ijkl}\,D^{k}\,D^{l}\,{\bf
J}$
$\displaystyle-\frac{96}{5}\,(1-4\lambda)\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,D^{m}\,D^{n}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{4}}\,\Bigg{[}\frac{1152}{5}\,(1+2\lambda-4\lambda^{2})\,{\bf\Phi}^{(1)}-\frac{1536}{5}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,\partial\,{\bf
J}\Bigg{]}(Z_{2})$
$\displaystyle+\frac{1}{z_{12}^{2}}\,\Bigg{[}\,\frac{96}{5}\,(1-4\lambda)\,{\bf\Phi}^{(1)}-\frac{192}{5}\,(1-3\lambda+6\lambda^{2})\,\partial\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{3}}\,\Bigg{[}\,-\frac{1536}{5}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,\partial\,D^{i}\,{\bf
J}+\frac{576}{5}\,(1+2\lambda-4\lambda^{2})\,D^{i}\,{\bf\Phi}^{(1)}$
$\displaystyle+192\,\lambda\,(1-2\lambda)\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial\,D^{i}\,{\bf J}\Big{)}\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}^{2}}\,\Bigg{[}\,\frac{384}{5}\,(1-3\lambda+6\lambda^{2})\,\partial\,D^{i}\,{\bf
J}-\frac{96}{5}\,(1-4\lambda)\,D^{i}\,{\bf\Phi}^{(1)}$
$\displaystyle-\frac{96}{5}\,(1-4\lambda)\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial\,D^{i}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{1}{z_{12}}\,\Bigg{[}\,-\frac{96}{5}\,(1-4\lambda)\,\partial\,{\bf\Phi}^{(1)}+\frac{192}{5}\,(1-3\lambda+6\lambda^{2})\,\partial^{2}\,{\bf
J}$ $\displaystyle+\frac{64}{5}\,(1-4\lambda)\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,D^{i}\,D^{j}\,D^{k}\,D^{l}\,{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{2}\,{\bf
J}\Big{)}+24\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}^{2}}\,\Bigg{[}\,-\frac{96}{5}\,(1+2\lambda-4\lambda^{2})\,\frac{1}{2!}\,\varepsilon^{ijkl}\,\partial\,D^{k}\,D^{l}\,{\bf
J}$
$\displaystyle+\frac{48}{5}\,(1-4\lambda)\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,\partial\,D^{m}\,D^{n}\,{\bf
J}-\frac{48}{5}\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,D^{k}\,D^{l}\,{\bf\Phi}^{(1)}$
$\displaystyle-24\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,D^{m}\,D^{n}\,{\bf\Phi}^{(1)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}}\,\Bigg{[}\,-\frac{96}{5}\,(1-3\lambda+6\lambda^{2})\,\partial^{2}\,D^{i}\,{\bf
J}+\frac{12}{5}\,(1-4\lambda)\,\partial\,D^{i}\,{\bf\Phi}^{(1)}$
$\displaystyle+\frac{32}{5}\,(1-4\lambda)\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial^{2}\,D^{i}\,{\bf J}\Big{)}$
$\displaystyle+12\,\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}\,{\bf\Phi}^{(1)}-6\,D^{i}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{2}}\,\Bigg{[}\,6\,{\bf\Phi}^{(3)}\,\Bigg{]}(Z_{2})+\frac{\theta_{12}^{4-i}}{z_{12}}\,\Bigg{[}\,D^{i}\,{\bf\Phi}^{(3)}\,\Bigg{]}(Z_{2})+\frac{\theta_{12}^{4-0}}{z_{12}}\,\Bigg{[}\,2\,\partial\,{\bf\Phi}^{(3)}\,\Bigg{]}(Z_{2})+\cdots.$
(H.1)

Except the last three terms for the ${\cal N}=4$ primary condition [11], the
additional terms consisting of the ${\cal N}=4$ stress energy tensor, the
first and the second ${\cal N}=4$ multiplets (and their descendants) appear in
(H.1).

##  Appendix I  The OPEs between the ${\cal N}=4$ stress energy tensor and the
fourth ${\cal N}=4$ multiplet

The OPEs between the operators in (3.12) and the operators in (G.2) can be
summarized by

$\displaystyle{\bf
J}(Z_{1})\,{\bf\Phi}^{(4)}(Z_{2})=\frac{1}{z_{12}^{4}}\,\Bigg{[}-\frac{3072}{35}\,N\,(1-4\lambda)\,(3-2\lambda+4\lambda^{2})\Bigg{]}$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{6}}\,\Bigg{[}\,-\frac{147456}{7}\,N\,\lambda\,(1-2\lambda)\,(1-\lambda+2\lambda^{2})\,\Bigg{]}$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{5}}\,\Bigg{[}\,\frac{147456}{7}\,\lambda\,(1-2\lambda)\,(1-\lambda+2\lambda^{2})\,D^{i}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}^{4}}\,\Bigg{[}\,-\frac{12288}{35}\,(1-4\lambda)\,(3-2\lambda+4\lambda^{2})\,D^{i}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{5}}\,\Bigg{[}\,\frac{49152}{7}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,{\bf\Phi}^{(1)}-\frac{294912}{7}\,\lambda\,(1-2\lambda)\,(1-\lambda+2\lambda^{2})\,\partial\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}^{4}}\,\Bigg{[}\,\frac{18432}{35}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,D^{k}\,D^{l}\,{\bf
J}$
$\displaystyle-\frac{9216}{7}\,(1-\lambda+2\lambda^{2})\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,D^{m}\,D^{n}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{1}{z_{12}^{3}}\,\Bigg{[}\,\frac{6144}{7}\,(1-\lambda+2\lambda^{2})\,{\bf\Phi}^{(1)}-\frac{12288}{35}\,(1-4\lambda)\,(3-2\lambda+4\lambda^{2})\,\partial\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{4}}\,\Bigg{[}\,-\frac{294912}{7}\,\lambda\,(1-2\lambda)\,(1-\lambda+2\lambda^{2})\,\partial\,D^{i}\,{\bf
J}+\frac{24576}{7}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,D^{i}\,{\bf\Phi}^{(1)}$
$\displaystyle-\frac{36864}{35}\,(1-4\lambda)\,(2-3\lambda+6\lambda^{2})\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial\,D^{i}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}^{3}}\,\Bigg{[}\,\frac{36864}{35}\,(1-4\lambda)\,(3-2\lambda+4\lambda^{2})\,\partial\,D^{i}\,{\bf
J}-\frac{9216}{7}\,(1-\lambda+2\lambda^{2})\,D^{i}\,{\bf\Phi}^{(1)}$
$\displaystyle-\frac{9216}{7}\,(1-\lambda+2\lambda^{2})\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,D^{i}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{4}}\,\Bigg{[}\,-\frac{24576}{7}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,\partial\,{\bf\Phi}^{(1)}+\frac{147456}{7}\,\lambda\,(1-2\lambda)\,(1-\lambda+2\lambda^{2})\,\partial^{2}\,{\bf
J}$
$\displaystyle-\frac{9216}{7}\,(-4-3\lambda+6\lambda^{2})\,{\bf\Phi}^{(2)}$
$\displaystyle-\frac{24576}{35}\,(1-4\lambda)\,(2-3\lambda+6\lambda^{2})\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{2}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2}$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}^{3}}\,\Bigg{[}\,-\frac{18432}{35}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,\partial\,D^{k}\,D^{l}\,{\bf
J}$
$\displaystyle+\frac{9216}{7}\,(1-\lambda+2\lambda^{2})\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,\partial\,D^{m}\,D^{n}\,{\bf
J}$
$\displaystyle-\frac{1536}{7}\,(-3-4\lambda+8\lambda^{2})\frac{1}{2!}\,\varepsilon^{ijkl}\,D^{k}\,D^{l}\,{\bf\Phi}^{(1)}-\frac{2304}{7}\,(1-4\lambda)\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,D^{m}\,D^{n}\,{\bf\Phi}^{(1)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{1}{z_{12}^{2}}\,\Bigg{[}\,-\frac{9216}{7}\,(1-\lambda+2\lambda^{2})\,\partial\,{\bf\Phi}^{(1)}+\frac{18432}{35}\,(1-4\lambda)\,(3-2\lambda+4\lambda^{2})\,\partial^{2}\,{\bf
J}$
$\displaystyle+\frac{6144}{7}\,(1-\lambda+2\lambda^{2})\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{2}\,{\bf J}\Big{)}$
$\displaystyle+\frac{1152}{7}\,(1-4\lambda)\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{3}}\,\Bigg{[}\,\frac{73728}{7}\,\lambda\,(1-2\lambda)\,(1-\lambda+2\lambda^{2})\,\partial^{2}\,D^{i}\,{\bf
J}-\frac{4608}{7}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,\partial\,D^{i}\,{\bf\Phi}^{(1)}$
$\displaystyle+\frac{12288}{35}\,(1-4\lambda)\,(2-3\lambda+6\lambda^{2})\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial^{2}\,D^{i}\,{\bf J}\Big{)}$
$\displaystyle-1536\,\lambda\,(1-2\lambda)\,\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial\,D^{j}\,D^{k}\,D^{l}\,{\bf\Phi}^{(1)}-\frac{2304}{7}\,(-4-3\lambda+6\lambda^{2})\,D^{i}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}^{2}}\,\Bigg{[}\,-\frac{18432}{35}\,(1-4\lambda)\,(3-2\lambda+4\lambda^{2})\,\partial^{2}\,D^{i}\,{\bf
J}+\frac{384}{7}\,(15-8\lambda+16\lambda^{2})\,\partial\,D^{i}\,{\bf\Phi}^{(1)}$
$\displaystyle+\frac{6144}{7}\,(1-\lambda+2\lambda^{2})\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial^{2}\,D^{i}\,{\bf J}\Big{)}$
$\displaystyle+\frac{1152}{7}\,(1-4\lambda)\,\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}\,{\bf\Phi}^{(1)}-\frac{576}{7}\,(1-4\lambda)\,D^{i}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}^{2}}\,\Bigg{[}\,\frac{3072}{35}\,(1-\lambda)\,(1+2\lambda)\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,\partial^{2}\,D^{k}\,D^{l}\,{\bf
J}$
$\displaystyle-\frac{1536}{7}\,(1-\lambda+2\lambda^{2})\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,\partial^{2}\,D^{m}\,D^{n}\,{\bf
J}$
$\displaystyle+\frac{384}{7}\,(-3-4\lambda+8\lambda^{2})\frac{1}{2!}\,\varepsilon^{ijkl}\,\partial\,D^{k}\,D^{l}\,{\bf\Phi}^{(1)}+\frac{576}{7}\,(1-4\lambda)\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,\partial\,D^{m}\,D^{n}\,{\bf\Phi}^{(1)}$
$\displaystyle-\frac{288}{7}\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}-144\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,D^{m}\,D^{n}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{1}{z_{12}}\,\Bigg{[}\,\frac{384}{35}\,(39-32\lambda+64\lambda^{2})\,\partial^{2}\,{\bf\Phi}^{(1)}-\frac{6144}{35}\,(1-4\lambda)\,(3-2\lambda+4\lambda^{2})\,\partial^{3}\,{\bf
J}$
$\displaystyle-\frac{3072}{7}\,(1-\lambda+2\lambda^{2})\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,\partial\,D^{i}\,D^{j}\,D^{k}\,D^{l}\,{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{3}\,{\bf J}\Big{)}$
$\displaystyle-\frac{576}{7}\,(1-4\lambda)\,\partial\,{\bf\Phi}^{(2)}+48\,{\bf\Phi}^{(3)}+\frac{1152}{35}\,(1-4\lambda)\,\frac{1}{4!}\,\varepsilon^{ijkl}\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf\Phi}^{(1)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}}\,\Bigg{[}\,\frac{2048}{35}\,(1-4\lambda)\,(3-2\lambda+4\lambda^{2})\,\partial^{3}\,D^{i}\,{\bf
J}-\frac{384}{35}\,(9-2\lambda+4\lambda^{2})\,\partial^{2}\,D^{i}\,{\bf\Phi}^{(1)}$
$\displaystyle-\frac{768}{7}\,(1-\lambda+2\lambda^{2})\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial^{2}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial^{3}\,D^{i}\,{\bf J}\Big{)}$
$\displaystyle-\frac{1152}{35}\,(1-4\lambda)\,\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial\,D^{j}\,D^{k}\,D^{l}\,{\bf\Phi}^{(1)}+\frac{48}{7}\,(1-4\lambda)\,\partial\,D^{i}\,{\bf\Phi}^{(2)}$
$\displaystyle-8\,D^{i}\,{\bf\Phi}^{(3)}+48\,\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{2}}\,\Bigg{[}\,8\,{\bf\Phi}^{(4)}\,\Bigg{]}(Z_{2})+\frac{\theta_{12}^{4-i}}{z_{12}}\,\Bigg{[}\,D^{i}\,{\bf\Phi}^{(4)}\,\Bigg{]}(Z_{2})+\frac{\theta_{12}^{4-0}}{z_{12}}\,\Bigg{[}\,2\,\partial\,{\bf\Phi}^{(4)}\,\Bigg{]}(Z_{2})+\cdots.$
(I.1)

Except the last three terms for the ${\cal N}=4$ primary condition [11], the
additional terms consisting of the ${\cal N}=4$ stress energy tensor, the
first, the second and the third ${\cal N}=4$ multiplets (and their
descendants) appear in (I.1). We expect that the OPEs between the ${\cal N}=4$
stress energy tensor and the $h$-th ${\cal N}=4$ multiplet contain the first,
the second, $\cdots$ the $h$-th ${\cal N}=4$ multiplets.

##  Appendix J  The OPEs between the first ${\cal N}=4$ multiplet and the
third ${\cal N}=4$ multiplet

The OPEs between the operators in (3.32) and the operators in (F.2) can be
summarized by

$\displaystyle{\bf\Phi}^{(1)}(Z_{1})\,{\bf\Phi}^{(3)}(Z_{2})=\frac{1}{z_{12}^{4}}\,\Bigg{[}-\frac{768}{5}\,N\,\lambda\,(1-2\lambda)\,(1-4\lambda)\Bigg{]}$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{6}}\,\Bigg{[}\,3072\,N\,\lambda\,(-1+4\lambda-8\lambda^{2}+8\lambda^{3})\,\Bigg{]}$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{5}}\,\Bigg{[}\,\frac{6144}{5}\,\lambda\,(1-2\lambda)\,(2-\lambda+2\lambda^{2})\,D^{i}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}^{4}}\,\Bigg{[}\,-\frac{1536}{5}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,D^{i}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{5}}\,\Bigg{[}\,1536\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,{\bf\Phi}^{(1)}+\frac{6144}{5}\,\lambda\,(1-2\lambda)\,(1-4\lambda)^{2}\,\partial\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{1}{z_{12}^{3}}\,\Bigg{[}\,384\,\lambda\,(1-2\lambda)\,{\bf\Phi}^{(1)}-\frac{1536}{5}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,\partial\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}^{4}}\,\Bigg{[}\,-\frac{384}{5}\,(1-4\lambda)\,(3+2\lambda-4\lambda^{2})\,\frac{1}{2!}\,\varepsilon^{ijkl}\,D^{k}\,D^{l}\,{\bf
J}$
$\displaystyle-\frac{576}{5}\,(1+2\lambda-4\lambda^{2})\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,D^{m}\,D^{n}\,{\bf
J}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{4}}\,\Bigg{[}\,-\frac{1536}{5}\,\lambda\,(1-2\lambda)\,(1-4\lambda)^{2}\,\partial\,D^{i}\,{\bf
J}+192\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,D^{i}\,{\bf\Phi}^{(1)}$
$\displaystyle+\frac{576}{5}\,(1-4\lambda)\,(1+2\lambda-4\lambda^{2})\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial\,D^{i}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}^{3}}\,\Bigg{[}\,\frac{1536}{5}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,\partial\,D^{i}\,{\bf
J}-192\,\lambda\,(1-2\lambda)\,D^{i}\,{\bf\Phi}^{(1)}$
$\displaystyle-\frac{576}{5}\,(1+2\lambda-4\lambda^{2})\,\Big{(}-\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-(1-4\lambda)\,\partial\,D^{i}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{4}}\,\Bigg{[}\,-384\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,\partial\,{\bf\Phi}^{(1)}+\frac{1536}{5}\,\lambda\,(1-2\lambda)\,(1-4\lambda)^{2}\,\partial^{2}\,{\bf
J}$
$\displaystyle+\frac{768}{5}\,(1-4\lambda)\,(1+2\lambda-4\lambda^{2})\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{2}\,{\bf J}\Big{)}$
$\displaystyle+\frac{288}{5}\,(11+12\lambda-24\lambda^{2})\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{1}{z_{12}^{2}}\,\Bigg{[}\,-192\,\lambda\,(1-2\lambda)\,\partial\,{\bf\Phi}^{(1)}+\frac{768}{5}\,\lambda\,(1-2\lambda)\,(1-4\lambda)\,\partial^{2}\,{\bf
J}-\frac{144}{5}\,(1-4\lambda)\,{\bf\Phi}^{(2)}$
$\displaystyle-\frac{384}{5}\,(-1-2\lambda+4\lambda^{2})\,\Big{(}\frac{1}{2\cdot
4!}\,\varepsilon^{ijkl}\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf
J}-\frac{1}{2}\,(1-4\lambda)\,\partial^{2}\,{\bf J}\Big{)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{3}}\,\Bigg{[}\,\frac{288}{5}\,(1+\lambda)\,(3-2\lambda)\,D^{i}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}^{2}}\,\Bigg{[}\,-\frac{36}{5}\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}-18\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,D^{m}\,D^{n}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{3}}\,\Bigg{[}\,\frac{576}{5}\,(1+\lambda)\,(3-2\lambda)\,\partial\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}^{2}}\,\Bigg{[}\,\frac{6}{5}\,(59+8\lambda-16\lambda^{2})\,\partial\,D^{i}\,{\bf\Phi}^{(2)}-\frac{42}{5}\,(1-4\lambda)\,\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}}\,\Bigg{[}\,-\frac{6}{5}\,(1-4\lambda)\,\partial\,D^{i}\,{\bf\Phi}^{(2)}+6\,\frac{1}{3!}\,\varepsilon^{ijkl}\,D^{j}\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}^{2}}\,\Bigg{[}\,\frac{96}{35}\,(38+11\lambda-22\lambda^{2})\,\partial^{2}\,{\bf\Phi}^{(2)}-\frac{96}{35}\,(1-4\lambda)\,\frac{1}{4!}\,\varepsilon^{ijkl}\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf\Phi}^{(2)}+{\bf\Phi}^{(4)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-ij}}{z_{12}}\,\Bigg{[}\,-\frac{12}{5}\,(1-4\lambda)\,\frac{1}{2!}\,\varepsilon^{ijkl}\,\partial\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}-6\,\varepsilon^{ijkl}\,\frac{1}{2!}\,\varepsilon^{klmn}\,\partial\,D^{m}\,D^{n}\,{\bf\Phi}^{(2)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-i}}{z_{12}}\,\Bigg{[}\,\frac{48}{35}\,(13+\lambda-2\lambda^{2})\,\partial^{2}\,D^{i}\,{\bf\Phi}^{(2)}-\frac{96}{35}\,(1-4\lambda)\,\frac{1}{3!}\,\varepsilon^{ijkl}\,\partial\,D^{j}\,D^{k}\,D^{l}\,{\bf\Phi}^{(2)}$
$\displaystyle+\frac{1}{8}\,D^{i}\,{\bf\Phi}^{(4)}\,\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{4-0}}{z_{12}}\,\Bigg{[}\,\frac{48}{35}\,(17+4\lambda-8\lambda^{2})\,\partial^{3}\,{\bf\Phi}^{(2)}-\frac{48}{35}\,(1-4\lambda)\,\frac{1}{4!}\,\varepsilon^{ijkl}\,\partial\,D^{i}\,D^{j}\,D^{k}\,D^{l}{\bf\Phi}^{(2)}$
$\displaystyle+\frac{1}{2}\,\partial\,{\bf\Phi}^{(4)}\,\Bigg{]}(Z_{2})+\cdots.$
(J.1)

There are the ${\cal N}=4$ stress energy tensor, the second and the fourth
${\cal N}=4$ multiplets (and their descendants) in (J.1).

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# The merger of co-rotating vortices in dusty flows

Shuai Shuai 1 and Anubhab Roy2 and M. Houssem Kasbaoui 1
<EMAIL_ADDRESS>1School for Engineering of Matter, Transport and
Energy, Arizona State University, Tempe, AZ 85281, USA. 2Department of Applied
Mechanics, Indian Institute of Technology Madras, Chennai, 600036, India.

###### Abstract

We investigate the effect of particle inertia on the merger of co-rotating
dusty vortex pairs at semi-dilute concentrations. In a particle-free flow, the
merger is triggered once the ratio of vortex core size to vortex separation
reaches a critical value. The vortex pair separation then decreases
monotonically until the two cores merge together. Using Eulerian-Lagrangian
simulations of co-rotating particle-laden vortices, we show substantial
departure from the vortex dynamics previously established in particle-free
flows. Most strikingly, we find that disperse particles with moderate inertia
cause the vortex pair to push apart to a separation nearly twice as large as
the initial separation. During this stage, the drag force exerted by particles
ejected out of the vortex cores on the fluid results in a net repulsive force
that pushes the two cores apart. Eventually, the two dusty vortices merge into
a single vortex with most particles accumulating outside the core similar to
the dusty Lamb-Oseen vortex described in Shuai & Kasbaoui (_J. Fluid Mech._ ,
vol 936, 2022, A8) For weakly inertial particles, we find that the merger
dynamics follow the same mechanics as those of a single-phase flow, albeit
with a density that must be adjusted to match the mixture density. For highly
inertial particles, the feedback force exerted by the particles on the fluid
may stretch the two cores during the merger to a point where each core splits
into two, resulting in inner and outer vortex pairs. In this case, the merger
occurs in two stages where the inner vortices merge first, followed by the
outer ones.

###### keywords:

keyword 1, keyword 2, keyword 3

##  1 Introduction

The merger of vortices is relevant to many engineering problems and practical
applications including aeronautics, geophysical fluid dynamics, meteorology,
and astrophysics (Roberts & Christiansen, 1972; Rossow, 1977; Overman &
Zabusky, 1982). Much of the prior work on co-rotating vortices was motivated
by the observation of vortical structures in aircraft trailing wakes (Jacquin
et al., 2005; Chen et al., 1999), which represent a significant hazard to
following aircrafts during take-off and landing. Vortex merger is also
relevant in many geophysical flows where the large scale motions appear as
two-dimensional turbulence (Boffetta & Ecke, 2012). In such flows, vortex
merger is the primary mechanism for the evolution of the flow, as the merger
of small scale vortices produces larger vortices and leads to the transfer of
energy to larger scales, a well-known characteristic of two-dimensional
turbulence (Jiménez et al., 1996; Couder et al., 1989; McWilliams, 1990;
Hopfinger & Van Heijst, 1993). In three-dimensional turbulence, vortex
interaction occurs between coherent flow structures in the region of like-
signed vorticity (Vincent & Meneguzzi, 1991; Cadot et al., 1995).

The majority of prior work on vortex merger concerned single-phase flows. In
particular, the two-dimensional dynamics of an identical co-rotating vortex
pair in a particle-free flow have been studied extensively (Griffiths &
Hopfinger, 1987; Melander et al., 1988; Waugh, 1992; Dritschel, 1995; Meunier
et al., 2002; Cerretelli & Williamson, 2003; Brandt & Nomura, 2006; Orlandi,
2007). The current understanding is that two co-rotating vortices with equal
strength do not merge until the ratio of vortex core size $a$ to pair
separation $b$ exceeds a critical value. If the $a/b$ is below the critical
threshold $(a/b)_{\mathrm{crit}}$, the two vortices undergo a diffusive stage
during which their sizes grow by viscous diffusion as they rotate around one
another, but their separation remains approximately constant. This is
generally referred to as the _first diffusive_ stage (Cerretelli & Williamson,
2003; Meunier et al., 2002). The _convective stage_ starts once $(a/b)$
reaches the critical threshold $(a/b)_{\mathrm{crit}}$ during which the vortex
separation decreases significantly. In one of the earliest merger experiments,
Griffiths & Hopfinger (1987) found $(a/b)_{\mathrm{crit}}$ to be about $\sim
0.29-0.32$. Later experiments by Meunier & Leweke (2001) and Cerretelli &
Williamson (2003) showed that the threshold is closer to
$(a/b)_{\mathrm{crit}}=0.29$. For vortex pairs with nonuniform vorticity
distributions, Meunier et al. (2002) proposed a similar merger criterion based
on a refined definition of vortex core size. In addition to experimental
observation, linear stability analyses carried out by Dritschel (1995) and
Meunier et al. (2002) show that vortex pairs that are too close, with
$a/b>0.32$ in (Dritschel, 1995), become unstable with respect to infinitesimal
two-dimensional perturbations. Melander et al. (1988) proposed that the
formation of vortex filaments during the convective stage drives the merger of
the vortex pair. However, this notion was later disputed by Brandt & Nomura
(2006) and Orlandi (2007). Cerretelli & Williamson (2003) showed that the
dominant physical mechanism during the convective stage is controlled by the
antisymmetric vorticity field. The latter induces a velocity that pulls the
two cores together resulting in the two vortices becoming intertwined. They
further showed that a _second diffusive_ stage follows the convective stage.
During this final stage, viscous diffusion dominates once more and smoothens
the large vorticity gradients resulting from the merger.

Particle-free co-rotating vortices may also be subject to three-dimensional
instabilities. Jimenez (1975) showed that co-rotating vortices are stable to
the Crow instability, a long-wavelength instability that is known for
destabilizing counter-rotating vortex tubes (Crow, 1970). However, co-rotating
vortex tubes may be susceptible to a short-wavelength instability called the
elliptic instability (Tsai & Widnall, 1976). Meunier & Leweke (2001) showed
that the elliptic instability emerges at circulation Reynolds number
$\Rey_{\Gamma}=\rho_{f}\Gamma/\mu_{f}$ over $\sim 2000$. Orlandi (2007)
performed direct numerical simulations at $\Rey_{\Gamma}=3000$, mirroring the
experiments of Meunier & Leweke (2001), and showed that, depending on the
initial axial disturbance, the merger dynamics in 3D can be significantly more
complex than in 2D. Below, the critical threshold
$\Rey_{\Gamma,\mathrm{crit}}\sim 2000$ a pair of co-rotating vortex tubes
evolve in a two-dimensional way.

To the best of our knowledge, vortex merger in semi-dilute dusty flows has not
been investigated. Yet, the dynamics in these flows may deviate considerably
from those in particle-free flows.  This is especially true for dusty flows in
the semi-dilute regime, where the average particle volume fraction
$\phi_{p,0}$ is in the range $O(10^{-6})$ to $O(10^{-3})$. Due to large solid-
to-gas density ratio ($\rho_{p}/\rho_{f}=O(10^{3})$), the mass loading
$M=\rho_{p}\phi_{p,0}/\rho_{f}$ is $O(1)$ in semi-dilute dusty flows which
leads to significant feedback force from the particles on the fluid, i.e.,
two-way coupling. In this regime, the disperse phase may cause large flow
modulation. Several experiments and simulations of turbulent flows laden with
semi-dilute inertial particles attest to this effect (Ahmed & Elghobashi,
2000; Hwang & Eaton, 2006a, b; Meyer, 2012; Richter, 2015; Kasbaoui et al.,
2019; Peng et al., 2019; Costa et al., 2021; Brandt & Coletti, 2022; Dave &
Kasbaoui, 2023). However, there is a dearth of studies of particle-vortex
interaction in canonical vortical semi-dilute dusty flows. These
configurations are best suited to tease out the fundamental mechanisms and
help build a physical understanding and intuition for how semi-dilute inertial
particles modulate flow structures in more complex flows.

Recently, we have shown that the interaction between disperse inertial
particles and a single vortex alters the dynamics from what is commonly
understood from particle-free vortex dynamics (Shuai & Kasbaoui, 2022; Shuai
et al., 2022). For example, a two-way coupled dusty Lamb-Oseen vortex decays
significantly faster than a particle-free one (Shuai & Kasbaoui, 2022). This
enhanced decay is due to the ejection of inertial particles from the vortical
core. While the particles cluster into a ring surrounding the vortex, their
feedback force on the fluid leads to faster decay of the flow structure.
Perhaps an even more striking effect is the fact that two-way coupled inertial
particles dispersed in the core of a two-dimensional vortex trigger an
instability (Shuai & Kasbaoui, 2022). This is in contrast to the remarkable
stability of particle-free vortices to two-dimensional perturbations (Fung &
Kurzweg, 1975). We have shown in (Shuai & Kasbaoui, 2022) that the ejection of
the particles from the vortex core activates a centrifugal Rayleigh-Taylor
instability that persists even for non-inertial particles.

In light of these previous findings, it is expected that the merger dynamics
in two-way coupled dusty flows will be considerably different from those noted
in earlier studies of particle-free flows (Cerretelli & Williamson, 2003;
Melander et al., 1988; Meunier et al., 2002). In the present study, we revisit
the problem of co-rotating vortices with equal strength, augmented with mono-
disperse inertial particles. We use Eulerian-Lagrangian simulations to show
new merger mechanisms that depend on particle inertia. A surprising outcome is
that inertial particles may even temporarily push apart the two vortices.

This paper is organized as follows. Section §2 presents the governing
equations applied in Eulerian-Lagrangian method to simulate the two-way
coupled particle-laden vortices. The vortex merger process in particle-free
flow is shown in §3. In §4, we present the physical mechanism for merger of
dusty vortices, the simulation configuration and the numerical results in
particle-laden flow. Finally, we provide concluding remarks in §5.

##  2 Governing equations

We describe the dynamics of the gas-solid flow using the Eulerian-Lagrangian
formalism previously deployed in (Shuai & Kasbaoui, 2022) and (Shuai et al.,
2022) for the study of vortical dusty flows. For the sake of brevity, we
reproduce here only the highlights of the approach.

In the present formulation, the carrier phase is treated in the Eulerian
frame, whereas solid particles are tracked individually. The conservation
equations for the carrier phase are obtained by volume-filtering the point-
wise Navier-Stokes equations (Anderson & Jackson, 1967). In the semi-dilute
regime, the equations for mass and momentum conservation read

$\displaystyle\frac{\partial\phi_{f}}{\partial
t}+\nabla\cdot\left(\phi_{f}\mathbf{u}_{f}\right)$ $\displaystyle=$
$\displaystyle 0,$ (1)
$\displaystyle\rho_{f}\left(\frac{{\partial\phi_{f}}\mathbf{u}_{f}}{\partial
t}+\nabla\cdot\left(\phi_{f}\mathbf{u}_{f}\mathbf{u}_{f}\right)\right)$
$\displaystyle=$ $\displaystyle-\nabla
p+\mu_{f}\nabla^{2}\mathbf{u}_{f}+\mathbf{F}+\nabla\cdot\mathbf{R}_{\mu},$ (2)

where $\mathbf{u}_{f}$ is the fluid velocity, $p$ is the pressure,
$\phi_{f}=1-\phi_{p}$ is the local fluid volume fraction, $\phi_{p}$ is the
local particle volume fraction, $\rho_{f}$ is the fluid density, and $\mu_{f}$
is the fluid viscosity. The tensor $\mathbf{R}_{\mu}$ results from filtering
the point-wise fluid stress tensor and is responsible for the apparent
viscosity of the suspension at large particle concentrations
$\phi_{p}>10^{-2}$. This tensor vanishes in the semi-dilute regime considered
in this study since $\phi_{p}=O(10^{-4})$. For the same reason, effects due to
volume displacement by the particles are negligible in the semi-dilute regime
as $\phi_{f}\simeq 1$ in equations (1) and (2).

The term $\mathbf{F}$ in (2) represents the momentum exchange between
particles and gas. It is due to the feedback force exerted by the particles on
the gas and reads

$\mathbf{F}=-\phi_{p}\nabla\cdot\bm{\tau}|_{p}+\rho_{p}\phi_{p}\frac{(\mathbf{u}_{p}-\mathbf{u}_{f|p})}{\tau_{p}},$
(3)

where
$\bm{\tau}=-p\bm{I}+\mu(\nabla\mathbf{u}_{f}+\nabla\mathbf{u}_{f}^{T})/2$ is
the filtered fluid stress tensor, $\mathbf{u}_{p}$ is the Eulerian particle
velocity, and $\tau_{p}=\rho_{p}d_{p}^{2}/(18\mu_{f})$ is response time of
spherical particles with density $\rho_{p}$ and diameter $d_{p}$. The
subscript “$(\cdot)_{|p}$” indicates a quantity evaluated at the particle
locations. The first term in (3) is the stress exerted by the undisturbed
flow. The second term represents the drag force by the particle on the fluid,
which is represented using Stokes drag. The latter dominates the interphase
coupling (3) in a dusty gas since the particle-to-fluid density ratio is very
large ($\rho_{p}/\rho_{f}\gg 1$).

A scaling analysis of the drag terms shows that the mass loading
$M=\rho_{p}\phi_{p,0}/\rho_{f}$, where $\phi_{p,0}$ is the average particle
volume fraction, determines the strength of the coupling between the gas and
solid phases. Thus, for vanishingly small mass loadings, the merger of co-
rotating vortices follows the same dynamics as in a particle-free flow since
the solid phase has little effect on the gas phase in the limit $M\ll 1$.
Conversely, the merger dynamics are expected to deviate from the established
dynamics in particle-free flows when $M=O(1)$.

The particle phase is treated in the Lagrangian frame. The motion of the
$i$-th particle is given by (Maxey & Riley, 1983)

$\displaystyle\frac{d\mathbf{x}_{p}^{i}}{dt}(t)$ $\displaystyle=$
$\displaystyle{\mathbf{u}_{p}^{i}}(t),$ (4)
$\displaystyle\frac{d\mathbf{u}_{p}^{i}}{dt}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\rho_{p}}\nabla\cdot\bm{\tau}(\mathbf{x}_{p}^{i},t)+\frac{\mathbf{u}_{f}(\mathbf{x}_{p}^{i},t)-\mathbf{u}^{i}_{p}(t)}{\tau_{p}},$
(5)

where $\mathbf{x}_{p}^{i}$, and $\mathbf{u}_{p}^{i}$, are the Lagrangian
position and velocity of the “$i$”-th particle, respectively. Gravity is
ignored in this work to focus on inertial effects. Other interactions,
including particle collisions, are negligible due to the large density ratio
and low volume fraction in the semi-dilute regime. In the equations above, the
instantaneous particle volume fraction and Eulerian particle velocity field
are obtained from Lagrangian quantities using

$\displaystyle\phi_{p}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}V_{p}\mathcal{G}\left(\left|\left|\mathbf{x}-\mathbf{x}_{p}^{i}\right|\right|\right),$
(6) $\displaystyle\phi_{p}\mathbf{u}_{f|p}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\mathbf{u}_{f}(\mathbf{x}_{p}^{i}(t),t)V_{p}\mathcal{G}\left(\left|\left|\mathbf{x}-\mathbf{x}_{p}^{i}\right|\right|\right),$
(7) $\displaystyle\phi_{p}\mathbf{u}_{p}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\mathbf{u}_{p}^{i}(t)V_{p}\mathcal{G}\left(\left|\left|\mathbf{x}-\mathbf{x}_{p}^{i}\right|\right|\right),$
(8)

where $V_{p}=\pi d_{p}^{3}/6$ is the particle volume, $\mathcal{G}$ represents
a Gaussian filter kernel of size $\delta_{f}=3\Delta x$, where $\Delta x$ is
the grid spacing. Further details on the numerical aspects can be found in
(Capecelatro & Desjardins, 2013).

##  3 Particle-free vortex merger

Figure 1: The evolution of vortex separation $b/b_{0}$ as a function of time
in single-phase fluid at $\Rey_{\Gamma}=530$. Symbols: ●, Cerretelli &
Williamson (2003); ▫, present simulation. Snapshots represent normalized
vorticity.

Before addressing how inertial particles may modulate the merger dynamics, we
first describe the different stages of vortex pair merger in a particle-free
case that will be used in §4 to assess the effect of introducing inertial
particles.  To this end, we consider a simulation of particle-free vortex
merger at circulation Reynolds number
$\Rey_{\Gamma}=\rho_{f}\Gamma/\mu_{f}=530$, where $\Gamma$ is the vortex
circulation, that matches the experiments of Cerretelli & Williamson (2003).

We perform a “pseudo-2D” simulation in a periodic domain, subdivided into 4
anti-symmetric quadrants as previously done in (Shuai & Kasbaoui, 2022) and
(Shuai et al., 2022). Here, “pseudo-2D” refers to the fact that the axial
direction is considered periodic and discretized with only one grid point to
enable the inclusion of spherical particles at a later point. In each
quadrant, we initialize two co-rotating Lamb-Oseen vortices with equal initial
radii $a_{0}$ and separated by $b_{0}$, such that the ratio $a_{0}/b_{0}=0.17$
is initially below the merger threshold $(a/b)_{\mathrm{crit}}=0.29$. This
indicates that the two vortices will not merge immediately, but rather undergo
a first diffusive stage before the onset of a convective stage. Each quadrant
has a size $30a_{0}$-by-$30a_{0}$, which is sufficiently large to avoid
interactions between vortices in different quadrants for the length of the
simulations considered (Shuai & Kasbaoui, 2022). The simulation grid is
uniform with a high spatial resolution $a_{0}/\Delta x\approx 51$ to provide
good resolution of the vortex cores. Data collected from all four quadrants is
used in the computation of statistics after application of appropriate
symmetries.

According to Cerretelli & Williamson (2003); Melander et al. (1988); Meunier
et al. (2002), the merger of two co-rotating vortices follows three stages
called the first diffusive stage, the convective stage, and the second
diffusive stage. All these stages are reproduced in our simulations as can be
seen in figure 1 which show the evolution of the normalized vortex separation
$b/b_{0}$, and axial vorticity $\omega_{z}$ normalized by the angular velocity
$\omega_{\Gamma}=\Gamma/(2\pi a_{0}^{2})$ from one of the four quadrants.
Initially, the two vortices rotate around one another, but their separation
remains constant as can be seen in figure 1. After a time period
$\mathrm{t_{D}}$, the cores grow sufficiently to reach the merger threshold
$(a/b)_{\mathrm{crit}}=0.29$, at which point, the convective merger is
initiated. During this stage, the separation decreases linearly. The vortices
are deformed significantly as shown at $(t-t_{D})\Gamma/b_{0}^{2}=0$ and 6 in
figure 1. The second diffusive stage occurs between
$6\lesssim(t-t_{D})\Gamma/b_{0}^{2}\lesssim 9$, after which the vortex pair
can be considered fully merged.

##  4 Particle-laden vortex merger

###  4.1 The role of disperse phase on vortex merger

The merger of vortices in the presence of a particulate phase involves two
additional processes – the dispersion of inertial particles by the background
flow and the feedback force from the disperse phase. The former depends on the
spatial distribution of strain-dominated regions over rotation-dominated
regions, while the latter depends on the local particle concentration and the
slip velocity between the two phases. These processes are coupled, and one
requires a full numerical simulation to understand completely how a dusty
vortex merger behaves differently from its particle-free counterpart. This
will be elaborated on in the next section. Here, we will attempt to identify
some expected behaviour in dusty vortex mergers based on dispersion and
feedback force in an idealized scenario of two liked-signed point vortices of
equal circulation.

(a) $\mathbf{u}$

(b) $-D\mathbf{u}/Dt$

Figure 2: Velocity field and negative of the acceleration field ($\propto$
feedback force) for a pair of identical point vortices in co-rotating frame.
The point vortices are located at $(\pm 1,0)$ and are marked as red circles.
In the velocity field (a) the stagnation points are at $(0,0)$ and
$(\pm\sqrt{5},0)$ and are labelled as $\mathbf{S_{0}}$, $\mathbf{S_{1}}$ and
$\mathbf{S_{2}}$ respectively. The elliptic fixed points are at
$(0,\pm\sqrt{3})$ and are labelled as $\mathbf{G_{1}}$ and $\mathbf{G_{2}}$
respectively.

The streamline pattern for a pair of point vortices in the co-rotating frame
has three stagnation points ($\mathbf{S_{0}}$, $\mathbf{S_{1}}$ and
$\mathbf{S_{2}}$) and two elliptic fixed points ($\mathbf{G_{1}}$ and
$\mathbf{G_{2}}$, the counterparts of the “ghost vortices” in the finite-sized
vortex merger processes) as shown in figure 2(a). Heavy inertial particles are
expected to centrifuge to infinity in open vortical flows. However, in the
flow field for a vortex pair, heavy inertial particles can get trapped in the
vicinity of the vortices (Angilella, 2010)! The location of fixed point for
inertial particles are different from tracers as the ghost vortices, elliptic
fixed points for tracers, now turn into stable spirals and migrate towards the
saddle points (Ravichandran et al., 2014). At a critical Stokes number, the
two fixed points merge and disappear. Beyond the critical Stokes number,
trapping is no longer possible, and all particles now fly off to infinity. The
rate at which particles are centrifuged from the vicinity of the vortices also
depends on the Stokes number; lower Stokes would lead to a slower depletion of
particles from the neighbourhood of the vortices.

(a) Low inertia merger

(b) High inertia merger - Initial stage

(c) High inertia merger - Later stage

Figure 3: Mechanism of vortex merger process in presence of a disperse phase.
Green arrows indicate the particle feedback force and the blue arrows indicate
the antisymmetric component of the vorticity.

The feedback force of the particles on the fluid depends on the local particle
concentration and slip velocity (see equation (3)). In the limit of small
particle inertia, the slip velocity can be written as
$(\mathbf{u}_{p}-\mathbf{u}_{f|p})\approx-\tau_{p}D\mathbf{u}_{f|p}/Dt$,
proportional to the local fluid acceleration. The negative of fluid
acceleration is plotted in 2(b), revealing that each of the vortices
experiences a stretching force along the line joining the centers. The forces
are in balance when the particle inhomogeneity is minimal outside each
vortices. This would be a scenario for cases with low particle inertia, where
the merger dynamics would resemble that of particle-free cases. As we will
show later in section 4.3, the dynamics can be viewed as that of an effective
fluid with modified density. The merger dynamics will resemble that shown in
the schematic in figure 3(a) – the particle feedback forces are in balance,
and the outer antisymmetric vorticity components are responsible for bringing
the vortices together. When particle inertia is higher, the centroid of the
two vortex system ($\mathbf{S_{0}}$ in figure 2(a)) quickly gets devoid of any
particle. The particle depletion near the origin reduces the inward feedback
force compared to the outward feedback force, as shown in figure 3(b). Thus,
the vortex cores experience an increase in separation due to this imbalance in
the feedback force. Once the particles have centrifuged sufficiently far from
the vortices, the vortices reverse direction and start approaching each other
due to the induced velocity from the antisymmetric vorticity component. For
high inertia, the imbalance in the inward and outward feedback forces on each
vortex core can also have dramatic effects where they significantly stretch an
individual vortex to rip it into two cores as highlighted in figure 3(c).

Next, we will explore the role of the feedback force and antisymmetric
component of vorticity quantitatively using numerical simulations.

###  4.2 Simulation configuration and vortex center identification

Except for the introduction of randomly placed mono-disperse particles, the
simulation configuration remains the same as described in §3. The particles
have diameter $d_{p}$, density $\rho_{p}$, and are initialized with velocities
that match the fluid velocity at their locations.

We consider seven cases where the particle inertia and mass loading are
varied. Table 1 lists a summary of the non-dimensional parameters in each
case. Case A is the reference particle-free case. Cases B and C correspond to
the limit of very low particle inertia, characterized by a circulation Stokes
number $\mathrm{St}_{\Gamma}=\tau_{p}/\tau_{f}=0.01$.  Here, $\tau_{f}=2\pi
a_{0}^{2}/\Gamma$ is the characteristic fluid timescale associated with an
isolated vortex of radius $a_{0}$ and circulation $\Gamma$. In these two
cases, the mass loading is $\mathrm{M}=1.0$ and 0.5, respectively, obtained by
varying the average particle volume fraction $\phi_{p,0}$. In cases D-G, we
vary the Stokes number $\mathrm{St}_{\Gamma}$ by changing the particle
diameter as shown in Table 1. For all cases, the initial separation ratio is
$a_{0}/b_{0}=0.17$ and the density ratio $\rho_{p}/\rho_{f}$ is fixed at 2167.
In order to compute ensemble averages, we repeat the simulations several
times, each time with a different realization of the initial random
distribution of the particles. The total number of realizations $N_{r}$ for
each case is also shown in table 1. We chose $N_{r}$ such that
$N_{r}a/d_{p}\sim 8,000$, i.e., there are about 16,000 particles within the
core of a vortex across all realizations. This ensures that ensemble-averaged
quantities are not significantly impacted by discrete fluctuations due to
increasing particle size, particularly in cases D-G where $a/d_{p}<450$.

Case |  $\mathrm{St}_{\Gamma}$ |  $M$ |  $\phi_{p,0}$ |  $a/d_{p}$ |  Number of realizations $N_{r}$
---|---|---|---|---|---
A |  – |  – |  – |  – |  –
B |  0.01 |  1 |  $4.6\times 10^{-4}$ |  1008 |  8
C |  0.01 |  0.5 |  $2.3\times 10^{-4}$ |  1008 |  8
D |  0.05 |  1 |  $4.6\times 10^{-4}$ |  450 |  8
E |  0.1 |  1 |  $4.6\times 10^{-4}$ |  318 |  8
F |  0.2 |  1 |  $4.6\times 10^{-4}$ |  225 |  32
G |  0.3 |  1 |  $4.6\times 10^{-4}$ |  184 |  32
Table 1: Non-dimensional parameters considered in simulations. For all cases,
the initial separation ratio is $a_{0}/b_{0}=0.17$ and Reynolds number
$\Rey_{\Gamma}=\Gamma/\nu=530$ .

The vortex separation is a significant variable to investigate the vortex
merger process. Unlike in particle-free merger where smooth vorticity fields
allow easy detection of the centers, the feedback force from disperse
particles causes large vorticity fluctuations that make the detection of
vortex centers harder. The left picture in figure 4 shows an example of the
large vorticity fluctuations obtained in a particle-laden case. Due to this,
we have found it necessary to filter the data from Euler-Lagrangian
simulations to reliably detect the vortex centers.

To identify the vortex center, we first compute the filtered vorticity
$\overline{\omega}_{z}(\mathbf{x})$ from the instantaneous vorticity field
$\omega_{z}(\mathbf{x})$ by convolving with a triangle filter kernel $g$ with
support $\delta_{f}$ , as follows,

$\overline{\omega}_{z}(x,y,t)=\iint\omega_{z}(x^{\prime},y^{\prime})g(x-x^{\prime},y-y^{\prime})dx^{\prime}dy^{\prime}.$
(9)

To retain the vortex feature after filtering, $\delta_{f}$ is set to be half
of the initial vortex core radius, that is $a_{0}/2$. The right picture in
figure 4 shows the result of applying this filtering procedure to the field in
the left side of figure 4.  After filtering, we use the gradient descent
method to find the local extremums of the filtered vorticity field, thus
obtaining the locations of the two vortex centers. The distance between these
two centers gives the instantaneous vortex separation $b^{(i)}(t)$ for a
realization “$i$”. Lastly, we ensemble-average results across all realizations
to obtain the vortex separation in a dusty gas flow,
$b(t)=(1/N_{r})\sum_{i=1}^{N_{r}}b^{(i)}(t)$

Figure 4: To detect vortex centers, the vorticity field is filtered to remove
fluctuations induced by the Lagrangian forcing.

###  4.3 Weakly inertial particles

Figure 5: Successive snapshots of normalized particle volume fraction
$\phi_{p}$ and filtered vorticity $\overline{\omega}_{z}$ iso-contours for
cases B, E, and G. See supplementary materials for animations.

(a)

(b)

(c)

(d)

Figure 6: Evolution of the normalized separation $b/b_{0}$ for (a) the weakly
inertia cases B-C, (b) the moderately inertial cases D and E, and the highly
inertial cases (c) F, and (d) G.

Figure 5 shows the isocontours of normalized particle volume fraction and
normalized filtered vorticity from $t\Gamma/b_{0}^{2}=0$ to 14 for
$\mathrm{St}_{\Gamma}=0.01$, 0.1 and 0.3 with mass loading $M=1$. Videos of
these cases are provided in supplementary materials. From the snapshots in
figure 5, it is clear that semi-dilute inertial particles alter the merger
dynamics even at very small Stokes numbers. For the case at
$\mathrm{St}_{\Gamma}=0.01$, shown on the first and second rows of figure 5,
the dynamics of the vorticity field remain qualitatively similar to those of
the particle-free vortex pair, however, the merger takes significantly longer
and the vortices are further stretched than in the single-phase case. During
this process, the particles are gradually ejected from the two vortex cores
under the effect of preferential concentration, similar to the dynamics
observed for an isolated dusty vortex in (Shuai & Kasbaoui, 2022). By
$t\Gamma/b_{0}^{2}=7$, this results in the formation of two distinct void
fraction bubbles. These structures get progressively larger and more
stretched, as can be seen at $t\Gamma/b_{0}^{2}=10.5$ and 14. As the vortices
approach one another, a line of particles can be seen dividing the two
vortices at $t\Gamma/b_{0}^{2}=14$. This line of particles forms because the
region between the two vortices is dominated by straining, which draws in
particles originally located outside the vortices, and those that have been
ejected from the cores.

Figure 6(a) shows the evolution of the vortex-pair separation for the particle
free case A, and cases B $(\mathrm{St}_{\Gamma}=0.01,M=1.0)$ and C
$(\mathrm{St}_{\Gamma}=0.01,M=0.5)$. While the particle-free vortices merge
around $t\Gamma/b_{0}^{2}\approx 16$, merger occurs much later, around
$t\Gamma/b_{0}^{2}\approx 28$, with mass loadings $\mathrm{M}=1.0$.

The observation in figures 5 and 6(a) suggest that, at first approximation,
the merger of dusty vortex pair at $\mathrm{St}_{\Gamma}\ll 1$ is similar to
the merger of particle-free vortices in an effective fluid with density
$\rho_{\mathrm{eff}}=(1+M)\rho_{f}$. To justify this reasoning, consider a
Two-Fluid model of the particle phase where the particle velocity field is
expanded in the limit of small inertia as done in (Kasbaoui et al., 2015;
Shuai et al., 2022)

$\displaystyle\frac{\partial\phi}{\partial t}+\bm{u}_{p}\cdot\nabla\phi$
$\displaystyle=$ $\displaystyle-\phi\nabla\cdot\bm{u}_{p},$ (10)
$\displaystyle\bm{u}_{p}$ $\displaystyle=$
$\displaystyle\bm{u}_{f}-\tau_{p}\left(\frac{\partial\bm{u}_{f}}{\partial
t}+\bm{u}_{f}\cdot\nabla\bm{u}_{f}\right)+O(\tau_{p}^{2}),$ (11)

Combining these equations with the fluid conservation equations,

$\displaystyle\nabla\cdot\bm{u}_{f}=0,$ (12)
$\displaystyle\rho_{f}\left(\frac{\partial\bm{u}_{f}}{\partial
t}+\bm{u}_{f}\cdot\nabla\bm{u}_{f}\right)$ $\displaystyle=$
$\displaystyle-\nabla
p+\mu\nabla^{2}\bm{u}_{f}+\frac{\rho_{p}\phi}{\tau_{p}}(\bm{u}_{p}-\bm{u}_{f}),$
(13)

yields the following mixture equations

$\displaystyle\frac{\partial\rho_{\mathrm{eff}}}{\partial
t}+\bm{u}_{f}\cdot\nabla\rho_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\tau_{p}\left\\{\left(\rho_{\mathrm{eff}}-\rho_{f}\right)\nabla\bm{u}_{f}:\nabla\bm{u}_{f}+\frac{\nabla\rho_{\mathrm{eff}}}{\rho_{\mathrm{eff}}}\cdot\left(-\nabla
p+\mu\nabla^{2}\bm{u}_{f}\right)\right\\},$ (14)
$\displaystyle\rho_{\mathrm{eff}}\left(\frac{\partial\bm{u}_{f}}{\partial
t}+\bm{u}_{f}\cdot\nabla\bm{u}_{f}\right)$ $\displaystyle=$
$\displaystyle-\nabla p+\mu\nabla^{2}\bm{u}_{f},$ (15)

where
$\rho_{\mathrm{eff}}=\left(1+M\frac{\phi_{p}}{\phi_{p,0}}\right)\rho_{f}$ is
the local effective density, the first term on the right-hand side of (14)
represents preferential concentration and the second term is due to the slip
between the two phases. Thus, in the limit of negligible inertia, i.e.,
$\tau_{p}\rightarrow 0$, or equivalently, $\mathrm{St}_{\Gamma}\rightarrow 0$,
the inertial effects due to preferential concentration and slip vanish making
equations (14) and (15) identical to those of a single-phase fluid with
effective density $\rho_{\mathrm{eff}}$.

To verify this hypothesis, we conducted additional simulations of single-phase
merger where the fluid density equals $\rho_{\mathrm{eff}}=(1+M)\rho_{f}$ for
$M=0.5$ and $M=1.0$. Comparison of the vortex-pair separation from these
simulations with the separation measured in the particle-laden cases B and C
(see figure 6(a)) shows excellent agreement during most of the merger.
Deviations that can be seen for $t\Gamma/b_{0}^{2}\gtrsim 18$ are likely due
to inertial effects which, as suggested by the growth of the void bubbles,
become significant as time progresses, despite the low Stokes number
$\mathrm{St}_{\Gamma}=0.01$.

Although the effective fluid analogy captures well the merger dynamics in the
limit $\mathrm{St}_{\Gamma}\ll 1$, it is worth further investigating the
precise mechanisms activated by the particles causing the slowing down of the
merger and which may play a larger role with increasing
$\mathrm{St}_{\Gamma}$. For this reason, we investigate the dynamics of the
ensemble-averaged axial vorticity $\langle\overline{\omega}_{z}\rangle$ and
the particle feedback $\langle\overline{\bm{F}}\rangle$ as the interplay
between these two fields controls the merger. Following Cerretelli &
Williamson (2003), we investigate these quantities in a rotated reference
frame $(x^{\prime},y^{\prime})$ such that the $x^{\prime}$-direction connects
the two vortex centers. The $y^{\prime}$-direction is orthogonal to the latter
and represents a plane of symmetry before the vortex cores start deforming.
Further, we decompose these quantities into symmetric and antisymmetric parts.
For the axial vorticity, this decomposition reads

$\displaystyle\langle\overline{\omega}_{z}\rangle(x^{\prime},y^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\langle\overline{\omega}_{z}\rangle(x^{\prime},y^{\prime})+\langle\overline{\omega}_{z}\rangle(-x^{\prime},y^{\prime})\right]+\frac{1}{2}\left[\langle\overline{\omega}_{z}\rangle(x^{\prime},y^{\prime})-\langle\overline{\omega}_{z}\rangle(-x^{\prime},y^{\prime})\right]$
(16) $\displaystyle=$
$\displaystyle\langle\overline{\omega}_{z}^{S}\rangle+\langle\overline{\omega}_{z}^{A}\rangle,$
(17)

where $\langle\overline{\omega}_{z}^{S}\rangle$ and
$\langle\overline{\omega}_{z}^{A}\rangle$ denote the symmetric and
antisymmetric vorticity, respectively. For illustration, figure 7 shows the
filtered ensemble-averaged vorticity field
$\langle\overline{\omega}_{z}\rangle$ in the laboratory reference frame, in
the rotated reference frame, its symmetric part
$\langle\overline{\omega}_{z}^{S}\rangle$, and antisymmetric part
$\langle\overline{\omega}_{z}^{A}\rangle$ at an arbitrary time during the
merger.

Figure 7: Iso-contours of ensemble-averaged and filtered (a) vorticity
$\langle\overline{\omega}_{z}\rangle(x,y)$ in the laboratory reference frame,
(b) vorticity $\langle\overline{\omega}_{z}\rangle(x^{\prime},y^{\prime})$ in
the rotated reference frame. (c) symmetric vorticity
$\langle\overline{\omega}^{S}_{z}\rangle(x^{\prime},y^{\prime})$, and (d)
antisymmetric vorticity
$\langle\overline{\omega}^{A}_{z}\rangle(x^{\prime},y^{\prime})$.

As argued by Cerretelli & Williamson (2003), it is only the antisymmetric
vorticity field $\langle\overline{\omega}_{z}^{A}\rangle$ that contributes to
the change of separation in particle-free cases. Depending on the symmetries
of $\langle\overline{\omega}_{z}^{A}\rangle$, the induced velocity field may
either pull together or push apart the vortex cores. When inertial particles
are dispersed in the flow, the antisymmetric part of the component of the
particle feedback force in the direction parallel to the line connecting the
two vortex centers $\bm{c}_{1}$ and $\bm{c}_{2}$, i.e.,

$\langle\overline{F}^{A}_{||}\rangle=\langle\overline{\bm{F}}^{A}\rangle\cdot\frac{\bm{c}_{2}-\bm{c}_{1}}{||\bm{c}_{2}-\bm{c}_{1}||}$
(18)

also affects the pair separation. Note that, the dynamics of the symmetric
part $\langle\overline{F}_{||}^{S}\rangle$ in the parallel direction control
the translational drift of the whole vortex-pair but do not impact the
separation. The dynamics of the symmetric and antisymmetric parts of particle
feedback force in the normal direction, i.e.,
$\langle\overline{F}_{\perp}^{S}\rangle$ and
$\langle\overline{F}_{\perp}^{A}\rangle$, influence the normal stretching and
rotation-rate of the vortex pair, respectively. Since our primary concern is
the rate at which a vortex-pair merges in a dusty flow, we focus on analyzing
$\langle\overline{\omega}_{z}^{A}\rangle$ and
$\langle\overline{F}_{||}^{A}\rangle$ as these are the only two fields
impacting the vortex separation.

Figure 8: Iso-contours of ensemble-averaged and filtered antisymmetric
particle-feedback force along the two vortex centers
$\langle\overline{F}_{||}^{S}\rangle$, antisymmetric axial vorticity
$\langle\overline{\omega}_{z}^{A}\rangle$, particle volume fraction
$\langle\overline{\phi}_{p}\rangle$, and total axial vorticity
$\langle\overline{\omega}_{z}\rangle$ at representative times during the
merger. Data corresponds to case B with $\mathrm{St}_{\Gamma}=0.01$,
$\mathrm{M}=1$, and $\Rey_{\Gamma}=530$.

Figure 8 shows the normalized, filtered, and ensemble-averaged particle
feedback force in the parallel direction $\langle\overline{F}_{||}^{A}\rangle$
and antisymmetric vorticity $\langle\overline{\omega}_{z}^{A}\rangle$ in the
rotated reference frame for case B ($\mathrm{St}_{\Gamma}=0.01$ and $M=1$). To
facilitate comparison, the ensemble-averaged particle volume fraction
$\langle\overline{\phi}_{p}\rangle$, fluid vorticity
$\langle\overline{\omega}_{z}\rangle$ in the rotated reference frame are also
included in this figure. The antisymmetric vorticity field
$\langle\overline{\omega}_{z}^{A}\rangle$ displays features that are similar
to those observed by Cerretelli & Williamson (2003) in particle-free vortex
merger. Before the convective merger is initiated, at about $\Gamma
t/b_{0}^{2}\sim 14$, each vortex center is surrounded by two inner and two
outer regions where the antisymmetric vorticity is large. The induced velocity
by the inner antisymmetric vorticity pushes the two cores apart, while the
outer antisymmetric vorticity has the opposite effect of pulling the two cores
together. In particle-free merger, there is a balance between these two
effects during the first diffusive stage resulting in no change of the
separation $b$. With the initiation of the convective merger, the balance
between inner and outer antisymmetric vorticity is broken, with the former
dominating and causing the separation $b$ to decrease. In the dusty flow case
B, figure 8 shows that the disperse particles exert a force in the inner
region of each core that pulls the two vortices together, and a force on the
outer regions that pushes the vortices apart. Later towards $\Gamma
t/b_{0}^{2}=14$, the inner attractive force vanishes leaving only the outer
force with net repulsive effect. Thus, the particles oppose the attractive
pull generated by the antisymmetric vorticity leading to slower merger as
observed in figure 6(a).

###  4.4 Moderately inertial particles

While the merger dynamics of a dusty flow with weakly inertial particles
($\mathrm{St}_{\Gamma}\ll 1$) are qualitatively similar to those of a
particle-free flow, new dynamics emerge with increasing particle inertia. The
most notable change noted in our simulations with
$0.05\leq\mathrm{St}_{\Gamma}\leq 0.1$ is that the eventual merger of the
vortex pair starts first by the two vortices pushing apart.

The void bubbles in case E ($\mathrm{St}_{\Gamma}=0.1$, $M=1.0$), shown in
figure 5, grow significantly faster than in the low inertia case B, as the
effects of preferential concentration intensify with increasing particle
inertia (Shuai & Kasbaoui, 2022). Further, the deformation of vortex cores and
the void bubbles starts earlier suggesting that this process is related to
particle inertia. Due to the faster depletion of the cores, the particle line
separating the two vortices appears earlier, at around $t\Gamma/b_{0}^{2}=7$,
and becomes thinner as the merger progresses. During this early transient
$t\Gamma/b_{0}^{2}\lesssim 10.5$, the two cores push apart leading to an
increase in separation compared to the initial state. The cores start
approaching one another only once the line of particles separating the cores
becomes sufficiently thin, which eventually ruptures.

Figure 6(b) shows the evolution of the normalized separation $b/b_{0}$ for
cases D and E, alongside the data for the particle-free case A. While
$t\Gamma/b_{0}^{2}\lesssim 7$, the separation remains approximately constant.
During this stage, the two vortices are mostly independent from one another
and evolve according to dynamics similar to those reported in (Shuai &
Kasbaoui, 2022). Unlike in single-phase merger where the growth of the vortex
cores is exclusively driven by viscosity, the growth of the cores and void
bubbles are interlinked as the feedback force from the particles exiting the
cores causes greater spreading of the vorticity field.  Time
$t\Gamma/b_{0}^{2}\simeq 7$ marks the start of a new stage, that we call
_repulsion stage_ , and which ends by $t\Gamma/b_{0}^{2}\simeq 12$ in case D
($\mathrm{St}_{\Gamma}=0.05$) and $t\Gamma/b_{0}^{2}\simeq 10$ in case E
($\mathrm{St}_{\Gamma}=0.1$). During this stage, the vortex pair separation
increases monotonically, up to a saturation limit $b/b_{0}\ \sim 1.35$ and 1.5
in cases D and E, respectively. Then, the merger is initiated. This stage
resembles the convective stage in particle-free merger, during which the
vortex pair separation drops rapidly and lasts until $t\Gamma/b_{0}^{2}\simeq
18$ in case D and $t\Gamma/b_{0}^{2}\simeq 16$ in case E. At the end of this
stage, the two void regions have merged, resulting in a large particle-free
region containing the two vortices. The dynamics from here and onward follow
those of single-phase merger as the particles have been ejected from the
central region.

Figure 9: Iso-contours of ensemble-averaged and filtered antisymmetric
particle-feedback force along the two vortex centers
$\langle\overline{F}_{||}^{S}\rangle$, antisymmetric axial vorticity
$\langle\overline{\omega}_{z}^{A}\rangle$, particle volume fraction
$\langle\overline{\phi}_{p}\rangle$, and total axial vorticity
$\langle\overline{\omega}_{z}\rangle$ at representative times during the
merger. Data corresponds to case D with $\mathrm{St}_{\Gamma}=0.05$,
$\mathrm{M}=1$, and $\Rey_{\Gamma}=530$.

To elucidate the mechanism driving the repulsion stage, we report in figure 9
isocontours of the fields $\langle\overline{\omega}_{z}^{A}\rangle$,
$\langle\overline{F}_{||}^{A}\rangle$, $\langle\overline{\phi}_{p}\rangle$ and
$\langle\overline{\omega}_{z}\rangle$ for case D ($\mathrm{St}_{\Gamma}=0.05$)
at representative times in the rotated reference frame. As previously
discussed in §4.3, it is the symmetries of
$\langle\overline{\omega}_{z}^{A}\rangle$ and
$\langle\overline{F}_{||}^{A}\rangle$, and their interplay, that dictate the
evolution of the vortex pair separation. Up until $t\Gamma/b_{0}^{2}\simeq
10.5$, there is a relative balance between the inner and outer antisymmetric
vorticity $t\Gamma/b_{0}^{2}=10.5$. This suggests that the vorticity dynamics
do not have a significant effect on the vortex pair separation during this
time. In contrast, a gradual imbalance develops between the inner and outer
parts of the parallel antisymmetric particle feedback force
$\langle\overline{F}_{||}^{A}\rangle$. This imbalance favors the outer regions
which have a net repulsive effect on the vortex pair. It is caused by the drag
force exerted by inertial particles ejected away from the vortex pair. The
weakening of the particle feedback force in the inner regions results from the
growth of the void fraction bubbles and their gradual merger. This imbalance
leads to a gradual increase of the vortex pair separation. This effect
accelerates significantly at about $t\Gamma/b_{0}^{2}=10.5$, which represents
the time around which the inner regions become fully depleted from particles
and no longer exert any pull on the vortex pair. Later, the antisymmetric
vorticity develops an imbalance between inner and outer regions in turn, which
can be seen in the isocontours at $t\Gamma/b_{0}^{2}=14$. The induced velocity
by the breaking of this balance causes the vortex cores to pull together. When
this attractive effect of the antisymmetric vorticity overcomes the repulsive
effect of the disperse particles, merger is initiated and the vortex pair
separation decreases rapidly.

###  4.5 Highly inertial particles

Figure 10: Iso-contours of ensemble-averaged and filtered antisymmetric
particle-feedback force along the two vortex centers
$\langle\overline{F}_{||}^{S}\rangle$, antisymmetric axial vorticity
$\langle\overline{\omega}_{z}^{A}\rangle$, particle volume fraction
$\langle\overline{\phi}_{p}\rangle$, and total axial vorticity
$\langle\overline{\omega}_{z}\rangle$ at representative times during the
merger. Data corresponds to case F with $\mathrm{St}_{\Gamma}=0.2$,
$\mathrm{M}=1$, and $\Rey_{\Gamma}=530$.

With increasing Stokes number, the feedback force from the particles
increasingly distorts the vortical structures making the merger more complex.
This is illustrated in case G ($\mathrm{St}_{\Gamma}=0.3$, $M=1$), in figure
5, where the vortices appear highly stretched at a time as early as
$t\Gamma/b_{0}^{2}=3.5$. This extreme distortion causes each vortex to split
into two smaller vortices, an inner one and an outer one, as can be clearly
seen at $t\Gamma/b_{0}^{2}=7$ in the instantaneous fields in figure 5 and the
ensemble-averaged fields in figure 10. From figure 6(c) and 6(d), the inner
vortices start merging around $t\Gamma/b_{0}^{2}=18$ for case F
($\mathrm{St}_{\Gamma}=0.2$) and $t\Gamma/b_{0}^{2}=14$ for case G
($\mathrm{St}_{\Gamma}=0.3$), with no repulsion stage. Meanwhile, the outer
vortices start with a repulsive stage for $3\lesssim t\Gamma/b_{0}^{2}\lesssim
9$, during which the centers push apart to a maximum distance $b/b_{0}\simeq
1.9$. This stage is followed by a convective stage and a second diffusive
stage between $9\lesssim t\Gamma/b_{0}^{2}\simeq 16$. At the end, a single
distorted vortex is left, enclosed inside a larger void fraction bubble.

The splitting of each vortex results from the extreme distortion caused by the
particles. Figure 10 shows that the initial vortex pairs stretches under the
influence of the disperse particles. By $t\Gamma/b_{0}^{2}=3.5$, the vortices
assume elliptical shapes with similarly shaped void-fraction bubbles. Due to
the fast depletion of the inner region between the two cores, the particle
feedback force has a net repulsive effect on the vortex pair. However, since
this force is largest at the opposite ends of the vortex pair, it causes
significant stretching of the cores and ultimately causes the appearance of
two vorticity extrema for each initial vortex core.

The inner vortices start merging when they become decoupled from the disperse
particles. This occurs at around $t\Gamma/b_{0}^{2}=7.0$ as the inner region
is devoid of particles at this point. Figure 11 shows vector fields, where
vectors are scaled by magnitude, of the ensemble-averaged particle forcing
$\langle\overline{\bm{F}}\rangle$ in the laboratory reference frame at times
$t\Gamma/b_{0}^{2}=7.0$ and $t\Gamma/b_{0}^{2}=8.5$. Whereas the outer
vortices are dragged outward by the centrifuging particles, the inner vortices
are not subject to any forcing due to the absence of particles locally.
Meanwhile, the induced velocity by the inner vortices pulls them together,
following similar dynamics to those acting during the convective merger of
particle-free vortices, as can be seen from the antisymmetric vorticity in
figure 10 at times $t\Gamma/b_{0}^{2}=7.0$ and $t\Gamma/b_{0}^{2}=8.5$.
Ultimately, this causes the inner vortices to merge before the outer ones.

The merger of the outer vortices starts around $t\Gamma/b_{0}^{2}\sim 9$ and
concludes around $t\Gamma/b_{0}^{2}\sim 18$ and $t\Gamma/b_{0}^{2}\sim 16$ for
cases F and G, respectively. The outer vortices merge when the antisymmetric
vorticity exerts a greater pull than the repulsion caused by the particles,
following dynamics similar to those noted in §4.4.

(a)

(b)

Figure 11: Vector plot of $\langle\overline{\bm{F}}\rangle$ overlayed on
isocontours of axial vorticity $\langle\omega_{z}\rangle$ from case F
($\mathrm{St}_{\Gamma}=0.2$, $\mathrm{M}=1$) at non-dimensional times (a)
$t\Gamma/b_{0}^{2}=7$ and (b) $t\Gamma/b_{0}^{2}=7$. The vectors are scaled by
magnitude.

##  5 Conclusion

Eulerian-Lagrangian simulations of the merger of co-rotating vortices laden
with inertial particles reveal new mechanics specific to dusty flows. The
present simulations were carried out in the semi-dilute regime, specifically
for average particle volume fractions $\phi_{p,0}=2.3-4.6\times 10^{-4}$ and
mass loading $M=0.5-1.0$. Despite the low particle concentration, dusty flows
in this regime have strong momentum coupling between the carrier and disperse
phase since the mass loading is order unity. To investigate the effect of
particle inertia, we varied the Stokes number $\mathrm{St}_{\Gamma}$ in the
range 0.01 to 0.3. We found that these particles can be classified into three
main categories. Particles that have a Stokes number $\mathrm{St}_{\Gamma}\leq
0.01$ are considered weakly inertial. With such particles, the merger of dusty
vortices is delayed compared to the merger of particle-free vortices. However,
the merger dynamics are not much different from those of the particle-free
case, if one considers the particle-fluid mixture as an effective fluid with
density $\rho_{\mathrm{eff}}=(1+M)\rho_{f}$. Particles with Stokes number in
the range $\sim 0.05$ to $\sim 0.1$ are considered as moderately inertial. In
this case, the merger of a dusty vortex pair exhibits an additional stage
characterized by a temporary repulsion of the vortex cores, before undergoing
successive convective merger and second diffusive stages. Analyzing the
antisymmetric vorticity field and the antisymmetric particle feedback force in
a corotating reference frame, we find that the vortex separation increase is
caused by a repulsive force generated by the ejection of particles from the
vortex cores. Once all particles have been ejected from the inner region
separating the two cores, the attractive effect of the antisymmetric vorticity
field dominates, which triggers the merger of the vortex pair. For highly
inertial particles ($\mathrm{St}_{\Gamma}\gtrsim 0.2$), the particle feedback
force causes each core to stretch to such extent that it rips it into two
cores. In this case, the merger of the inner and outer vortices takes place in
sequence. The inner vortices initiate the merger as soon as the inner region
becomes devoid of particles. The outer vortices push apart temporarily and
then initiate the merger once the inertial particles are ejected sufficiently
away. In all cases, the final outcome of the merger is a single vortex with a
core that is depleted from all particles and a surrounding hallo of high
particle concentration.

Acknowledgments. The authors acknowledge support from the US National Science
Foundation (award #2148710, CBET-PMP), and from IIT Madras for the
“Geophysical Flows Lab” research initiative under the Institute of Eminence
framework.

Declaration of Interests. The authors report no conflict of interest.

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