Must ablution be renewed for continuous, non-lust-induced discharge that appears before Fajr prayer, given the hardship due to cold weather and lack of hot water?

Vaginal discharge is pure (tahir), so there is no need to perform istinja' (ritual cleansing) from it, nor to purify clothes or the body if they are stained by it. However, it does nullify wudu' (ablution) according to the majority of scholars. A woman who is constantly afflicted by it and whose discharge does not stop during prayer time should perform wudu' for each prayer after its time has begun. With this wudu', she can perform both the obligatory (fard) and voluntary (nafl) prayers until the time for that prayer ends. Performing wudu' before the prayer time begins is not sufficient for her, and if she does so, her prayer is invalid according to the majority of scholars.