Correction to: Scientific Reports https://doi.org/10.1038/s41598-017-02143-y, published online 18 May 2017

This Article contains typographical errors in the Supplementary Information file.

In Equation S1,

\(\frac{D\rho }{Dt}+\nabla \cdot (\rho {\bf{u}})=0,\)

should read:

\(\frac{D\rho }{Dt}+\rho \nabla \cdot {\bf{u}}=0,\)

In Equation S8,

\(\frac{{D}_{0}{\bf{u}}^{\prime} }{{D}_{0}t}+({\bf{u}}^{\prime} \cdot \nabla ){{\bf{u}}}_{0}+\frac{\rho ^{\prime} }{{\rho }_{0}^{2}}\nabla {p}_{0}+\frac{1}{{\rho }_{0}}\nabla p^{\prime} =0,\)

should read:

\(\frac{{D}_{0}{\bf{u}}^{\prime} }{{D}_{0}t}+({\bf{u}}^{\prime} \cdot \nabla ){{\bf{u}}}_{0}-\frac{\rho ^{\prime} }{{\rho }_{0}^{2}}\nabla {p}_{0}+\frac{1}{{\rho }_{0}}\nabla p^{\prime} =0,\)

In Equation S13,

\(\frac{{D}_{0}{\bf{u}}^{\prime} }{{D}_{0}t}+\frac{1}{{\rho }_{0}}\nabla p^{\prime} =-({\bf{u}}^{\prime} \cdot \nabla ){{\bf{u}}}_{0}-\frac{\rho ^{\prime} }{{\rho }_{0}^{2}}\nabla {p}_{0}.\)

should read:

\(\frac{{D}_{0}{\bf{u}}^{\prime} }{{D}_{0}t}+\frac{1}{{\rho }_{0}}\nabla p^{\prime} =-({\bf{u}}^{\prime} \cdot \nabla ){{\bf{u}}}_{0}+\frac{\rho ^{\prime} }{{\rho }_{0}^{2}}\nabla {p}_{0}.\)

And finally,

‘By using equation (S5), \(\nabla {p}_{0}/{\rho }_{0}\) in the second term on RHS of equation (S13) is replaced by \(({{\bf{u}}}_{0}\cdot \nabla ){{\bf{u}}}_{0}\).’

Should read:

‘By using equation (S5), \(\nabla {p}_{0}/{\rho }_{0}\) in the second term on RHS of equation (S13) is replaced by \(-({{\bf{u}}}_{0}\cdot \nabla ){{\bf{u}}}_{0}\).’