Bernoulli equation is often used to explain how rooftops are blown off during tornados: the usual explanation is that the high speed of the wind il related to a lower pressure outside the house, compared to the one inside; hence we have a net force lifting the roof.

I was trying how this description agrees with relativity. I tried to consider this simple example: consider two tunnels separated by a wall. In tunnel $1$ the air flows with a certain velocity $\vec{v}$ in one direction, while in tunnel $2$ the air is still. I wold expect a lower pressure in tunnel $1$, and hence a net force acting on the wall "from $2$ to $1$". Considering the reference frame moving with velocity $\vec{v}$, I would expect the opposite situation, i.e. a net force acting on the wall "from $2$ to $1$".

Where is the mistake in this naive reasoning? I've read something about Bernoulli equation in different reference frames, but the key point there was the work done by the surfaces, and I don't see how it could play a role here.