Alliteration is an important part of "Tulips," adding to the musicality and meaning of key passages. For example, repeating, liquid /l/ and /w/ sounds create a soothing music in lines 3-4, reinforcing the "peacefulness" the speaker is describing:
I am learning peacefulness, lying by myself quietly
As the light lies on these white walls, this bed, these hands.
Later, line 21 bristles with /s/ alliteration (also known as sibilance) and /k/ consonance:
Their smiles catch onto my skin, little smiling hooks.
The combination of hissing sibilance and hard /k/ sounds turns this ordinary image of a "family photo" into something ominous and unpleasant. These consonants also slow down the poem's pace, as if digging "little [...] hooks" into the rhythm of the language.
A similar effect occurs in lines 53-54, as the speaker describes air currents "snag[ging]" on the bright red tulips:
Now the air snags and eddies round them the way a river
Snags and eddies round a sunken rust-red engine.
Here, sibilance and /r/ alliteration—along with other effects, such as /n/ consonance and the assonance of "sunken"/"rust"—slow the poem's rhythm considerably. Again, it's as if the language itself has hit a snag.
Alliteration is especially prominent at the very end of the poem, in lines 62-63:
The water I taste is warm and salt, like the sea,
And comes from a country far away as health.
Repeating /w/, /s/, and /k/ sounds help make the language musical and memorable, so that these last two lines function almost like a closing couplet.