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The ecological and evolutionary success of multicellular lineages is due in no small part to their increased size relative to unicellular ancestors. However, large size also poses biophysical challenges, especially regarding the transport of nutrients to all cells; these constraints are typically overcome through multicellular innovations (e.g., a circulatory system). Here we show that an emergent biophysical mechanism - spontaneous fluid flows arising from metabolically-generated density gradients - can alleviate constraints on nutrient transport, enabling exponential growth in nascent multicellular clusters of yeast lacking any multicellular adaptations for nutrient transport or fluid flow. Surprisingly, beyond a threshold size, the metabolic activity of experimentally-evolved snowflake yeast clusters drives large-scale fluid flows that transport nutrients throughout the cluster at speeds comparable to those generated by the cilia of extant multicellular organisms. These flows support exponential growth at macroscopic sizes that theory predicts should be diffusion limited. This work demonstrates how simple physical mechanisms can act as a 'biophysical scaffold' to support the evolution of multicellularity by opening up phenotypic possibilities prior to genetically-encoded innovations. More broadly, our findings highlight how co-option of conserved physical processes is a crucial but underappreciated facet of evolutionary innovation across scales.