By using the lattice Boltzmann method

the Kelvin-Helmholtz instability of two dimensional immiscible and incompressible fluids was numerically simulated. Taking the rolling height H as a reference value

the effects of density ratio

surface tension and shear stress on the appearance of Rayleigh-Taylor (R-T) instability in Kelvin Helmholtz(K-H) instability was studied. The results show that the density ratio plays a decisive role in the coupling effect of the two instabilities. When the density ratio is close to 1

the R-T instability does not occur in the K-H instability. When the density ratio increases

the R-T instability begins to occur in the K-H instability. An increase in the surface tension coefficient has no effect either on the K-H instability

or on the coupling height variation of the two instabilities

but it has an inhibitory effect on the inward movements of the fluids

and the thickness of the rolled fluid is significantly increased. The shear stress has an inhibitory effect on the coupling of the K-H instability and R-T instability.