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We learnt in our fluids course that boundary layer separation can occur even for laminar non-turbulent flow (for high viscosity fluid).

In this case, shown as ‘steady separation’ in the diagram below. Is the wake caused by the separated boundary layer actually non-turbulent?

enter image description here

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Yes, laminar wakes exist. There are beautiful examples in An Album of Fluid Motion. The plates of laminar wakes behind cylinders are 40-46. Plate 42, of flow past a cylinder with Re=26, is below. You'll see it looks very similar to the diagram in your question.

enter image description here

You might also be interested in plates 11-14, where the Reynolds numbers are even lower, Re<0.1. With such low Re, even the flow around a square and a plate perpendicular to the flow is laminar.

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