I have an OpenMP program (thousands of lines, impossible to reproduce here) that works as follows:
It consists of worker threads along with a task queue.
A task cons
These sort of situations can be quite hard to figure out. One key is to look at memory locality. Without seeing your code, it's impossible to say EXACTLY what is going wrong, but we can discuss some of the things that amke "multithreading less good":
In all NUMA systems, when the memory is located with processor X and the code running on processor Y (where X & Y aren't the same processor), every memory access will be bad for performance. So, allocating memory on the right NUMA node will certainly help. (This may require some special code, such as setting affinity masks and at least hinting to the OS/Runtime Systems that you want Numa-aware allocations). At the very least, ensure that you don't simply work on one large array that is allocated by the "first thread, then start lots more threads".
Another thing that is even worse is sharing or false sharing of memory - so if two or more processors are using the same cache-line, you will get a ping-pong match between those two processors, where each processor will do "I want memory at address A", get hold of the memory content, update it, and then the next processor will do the same thing.
The fact that results gets bad just at 12 threads seem to indicate that it's to do with "sockets" - either you are sharing data, or the data is located "on the wrong node". At 12 threads, it's likely that you start using the second socket (more), which will make these sort of problems more apparent.
For best performance, you need memory to be allocated on the local node, no sharing, and no locking. Your first set of results also look like they are not "ideal". I have some (absolutely non-sharing) code that gives exactly n-times better for number of processors, until I run out of processors (unfortunately, my machine only has 4 cores, so it's not very much better, but it's still 4x better than 1 core, and if I ever got my hands on a 48 or 64-core machine, it would produce 48 or 64 better results in calculating "weird numbers").
Edit:
The "Socket issue" is two things:
Memory locality: Basically, memory is attached to each socket, so if the memory is allocated from the region belonging to the "previous" socket, then you get extra latency reading the memory.
Cache/sharing: Within a processor, there are "fast" links to share data (and often a "bottom level shared cache", e.g. L3 cache), which allows for the cores within a socket to share data more efficiently than with those in a different socket.
All this amounts to something like working on servicing cars, but you don't have your own toolbox, so every time you need a tool, you have to ask your colleague next to you for a screwdriver, 15mm spanner, or whatever you need. And then give the tools back when your work area gets a bit full. It's not a very efficient way of working... It would be much better if you had tools of your own (at least the most common one - one of those special spanners that you only use once a month isn't a big issue, but your common 10, 12 and 15mm spanners and a few screwdrivers, for sure). And of course, it would get even worse if there are four mechanics, all sharing the same toolbox. This is the case where you have "all memory allocated on one node" in a four socket system.
Now imagine that you have a "box of spanners", and only one of the mechanics can use the box of spanners, so if you need a 12mm spanner, you have to wait for the guy next to you to finish using the 15mm spanner. This is what happens if you have "false cache-sharing" - the processor isn't really using the same value, but because there are more than one "thing" in the cacheline, the processors are sharing the cacheline (box of spanners).