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Air New Zealand flight NZ675 left Auckland after midday yesterday and passengers had no issues until the plane prepared to land.
A passenger said the aircraft began to shake back and forth as if it were being "shunted around" on descent.
The plane failed to land twice, sharply accelerating upwards again.
Passengers burst into applause as the plane momentarily landed on its third descent, before it immediately took to the skies again.
The passenger said a flight attendant remarked, "well that was a tease," before the pilot announced a mixture of head and tail winds was causing the problems.
Another passenger sitting in the back of the plane said there were unaccompanied minors "throwing up all around him".
By the fourth descent, one passenger was sweating with her head in her hands, as kids vomited into sick bags.
Despite the passengers' reactions, the pilot remained calm, the Herald was told.
The plane finally landed on its fourth attempt - but passengers waited until the landing music was playing before applauding again.
Due to arrive at 2.15pm, the plane arrived in Dunedin nearly 30 minutes late.
An Air New Zealand spokeswoman said the flight performed a common "go around" procedure this afternoon due to "windshear and turbulence".
Pilots will often make another landing approach when they are satisfied landing conditions have improved, she said.
This is the second time in as many days our national carrier has made the news, after a flight bound for Sydney yesterday had to return to Christchurch after circling the South Island city for two hours.