From the Monster Manual, p. 325, the Giant Constrictor Snake can grapple a target. That grapple inflicts 2d8 + 4 damage. Up to here, no problem.
Now, let's assume the grapple works and the target fails his escaping skill check.
Will the 2d8 + 4 damage apply again on the next round without the snake having to re-roll its attack roll? After all, the constriction is certainly still going strong, right? Many other grappling effect seem to inflict damages on each round until the grapple is broken or the target dies... but in this case, it is not written in the description of that monster.