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Every bow that I have seen used to play an orchestral stringed instrument, regardless of the style of music being played, has the same concave design. By concave I mean that there is a slight curve of the wood towards the bow hair (which is pronounced by loosening the bow).
As best I can tell, bows of the 17th century were convex. Why are convex bows virtually never seen in use now?
It's a fun experience to play with a convex Baroque bow, but if you've ever tried to use it for anything romantic or later, you'll quickly want to get back to the concave Tourte design that everybody has nowadays.
The thing with convex bows is that they bounce around like mad. This can work quite well for the elegant-rhythmic dance feel of Baroque and early Classical music. But it works completely against you if you need biting attack and/or dramatic long tenuto notes. With such a playing style, a baroque bow feels like immersing styrofoam in water: the string seems to outright repel the bow. Whereas a Tourte bow can pretty easily be forced to “sink into he string” for straight sustained notes.
Some “historically informed” performers do use convex bows for Baroque music, but it's not like a Tourte bow doesn't still have some bounce and can convey that gracious character, especially when held in “Baroque position” (i.e. closer to the center of gravity, rather than right at the frog).
If you trace the development of the orchestra, you'll see that there has been a shift toward larger groups in larger spaces between the baroque period and now. A natural consequence of this is that instruments had to adapt to project more sound to fill those spaces up. The modern bow is just one of those adaptations. The concave design can hold significantly more tension in the bow hair than a convex design. This allows for more pressure to be applied to the strings when playing which increases the volume of the sound. It's not the only adaptation to string instruments over that time either. The angle of the neck and fingerboard used to be much flatter than it is today and the bridge is higher. All of this allows more tension on the strings to make the instruments project more into larger performance spaces.
Late in the day, but: The concave bow allows a lighter bow to be used for the same level of tension in the hair, because it reduces the torque over the majority of the stick's length (and particularly at the centre where the effect of a bend is greatest). Greater tension means that, for a given change in force, you have to move the bow less far, which is useful in loud-and-fast passages – less-weight-and-more-tension is a double-whammy for high-speed changes of bow pressure.
There are costs, of course: for example, you cannot afford to reduce the tension to play multiple strings simultaneously (the technique of tightening the bow with thumb or forefinger is no longer relevant).
BTW, I don't think the reduction of rotational inertia around the length of the stick is an advantage – as you really don't want to rotate the stick in that direction in the first instance.
N.B.1: that the improved stiffness was already well understood and implemented before Tourte. Indeed, late baroque and very early classical bows included sticks where, before tightening, the stick was almost touching the hair in the upper half if the bow. I believe that the advantage of the Tourte is supposed to be increased flexibility when playing at the tip, but I could be entirely wrong on this.
N.B.2: I personally prefer a slightly short, stiff concave snakewood bow for most 17th-century and earlier music. Indeed, even for late romantic and modern works, I find it preferable to some brazilwood bows supposedly built to Tourte's design, but I'm probably somewhat atypical in this.
