I always see logs got stuck on the middle pier (I believe it’s at the middle of the river) of the bridge. The logs pile up on that pier during the rising water, and are revealed when the water level falls. Is it the secondary flow (like in the tea leaf paradox) that causes the logs to flow in the middle of the river?
Do you mean a bridge like this?
The length of the longest pieces of drift determines the maximum width of the common types of drift accumulation. Long logs hold together large accumulations and support them against lateral forces. The width of the channel influences the length of drift delivered to the bridge, and thereby helps to determine accumulation potential and characteristics.
For this bridge the side supports cannot jam, because at the side ones, the logs hitting them will rotate and will continue with the flow.
Is it the secondary flow (like in the tea leaf paradox) that causes the logs to flow in the middle of the river?
That's key. From Potential Drift Accumulation at Bridges, Publication No. FHWA-RD-97-028,
In most streams, secondary circulation currents converge at the surface, causing floating material to be transported along a relatively narrow drift path within the channel. Piers located in the drift path are the most common sites of drift accumulation.
Where that narrow drift path exists depends on the nature of the stream. From the same reference, the drift path tends to be in the middle in straight sections of the stream, but toward the outside of the bend where the river curves. The bridge in question is probably over a straight section of a stream.