Conditions can often be difficult at the final Grand Slam tournament of the season, which takes place in the New York summer and regularly has matches played in sweltering conditions.
This was one of those occasions. Temperatures hit 35C at Flushing Meadows as Dart faced Kostyuk in the maelstrom of the mid-afternoon heat.
As a result, both players struggled to stay composed in an erratic match which lacked quality.
Double faults were thrown around regularly from each server, including one by Kostyuk on a set point in the tie-break, with another from Dart also handing over the first break of the second set.
Wild returning was another feature of a match which, particularly in the first set, was a difficult watch.
Dart will become the British number two after the US Open, climbing above Emma Raducanu, when she rises into the world’s top 70 for the first time on the back of her first-round victory over France’s Chloe Paquet.
Beating Kostyuk would have moved Dart into the top 60 and secured one of the biggest wins of her career by ranking.
She made a sharp start but was left to rue not being clinical enough in a scrappy first set and punishing her flustered Kostyuk.
A suspect second serve has been something which Dart’s opponents have pounced on and here she punished herself with the volume of double faults.
Dart looked close to tears as she returned to the court and continued to gesticulate towards her team, which was led by mum Susie who has been coaching her in New York.
Reassuring words could not help Dart settle back into her rhythm and the fight was sucked out of her as Kostyuk wrapped up a victory which ended up being more comfortable than it should have been.
