Dendrite growth is a complex liquid-solid phase transition process involving multiple physical factors. A phase-field lattice-Boltzmann method was developed to simulate the two- and three-dimension dendrite growth of Al-Cu alloy. The effect of fully coupled thermal-solute-convection interaction on the dendrite growth was investigated by incorporating a parallel-adaptive mesh refinement algorithm into the numerical model. By accurately reproducing the latent heat release, solute diffusion and convective transport behaviors at the liquid solid interface, the interaction mechanism among thermal-solute-convection transport as well as their coupling effects on the dendrite growth dynamics were discussed. The simulation results show that the release of latent heat slows down the dendrite growth rate, and both natural and forced convection disrupt the symmetrical growth of dendrites. Their combination makes the growth of dendrites more complex, capturing important physical aspects such as recalescence, dendrite tip splitting, dendrite tilting, dendrite remelting, and solute plume in the simulation case. Based on the robustness and powerful ability of the numerical model, the formation mechanisms of these physical aspects were revealed.