Similar to this answer, I will say that quantum computers that are capable of doing calculations that are industry-relevant, do not yet exist (and probably won't exist for at least a couple more decades).
Until then, the answer to the question below is "no", or the word "real" needs to be changed to "theoretical":
"Are there any engineering problems that quantum computing can solve/simulate and demonstrate a real advantage over conventional computing?"
In theory, if a quantum computer could be built with a low enough error-rate, and with enough qubits, and a suitable enough gate-set, then they could be used for calculating energies (and other properties) of molecules, materials and other types of matter (i.e. matter modeling), including the battery design suggestion that was made in one of the comments.
A real engineering example
A real example of a difficult problem for classical computers, is the calculation of the energies of the iron-molybdenum complex (FeMoco), which is the active site of the nitrogenase enzyme, which is able to make ammonia from N2 and H2. Currently this reaction is accomplished by the Haber-Bosch process, which consumes 1-2% of the world's energy supply because it has to be done at 300–500°C and 60-180 atm pressure, and because it produces the fertilizers that are used by farmers that feed something like 80% of the world's population. The nitrogenase enzyme exists in nitrogen-fixing bacteria that accomplish the nitrogen-fixing task at basically room temperature and ambient pressure, so if we could figure out how that enzyme works, then we may be able to get it to work on a larger industrial scale (i.e. chemical engineering). It has been estimated that a quantum computer would need millions of qubits to solve just one small part of this problem after including the qubits used in error correcting codes, the best attempts using classical computers have used trillions of classical bits and haven't made much progress either.
For a quantum computer to have a "real" advantage over classical computers for a problem like this (or probably for any problem associated with simulating molecules or materials), not only would we need millions of qubits, but the error rates would have to go down by a lot. It's not going to happen any time soon, and by soon I mean decades. Another related thread on this site is this one: Is quantum computing just pie in the sky?