A pattern that I have had success with is as follows:
- Alternately swipe up ($\uparrow$) and left ($\leftarrow$) (or whichever two directions you choose) as quickly as you can.
- When you get stuck (or when you see an opportunity$^1$), swipe right ($\rightarrow$), then up ($\uparrow$) a few times$^2$, then continue with step 1.
- Keep this up until you get the 512 tile in the top-left corner.
Once you get to 512, you need to be more careful, because you can easily fill your entire board without the opportunity to collapse anything.
At this point, I usually adopt an alternating right($\rightarrow$)/up($\uparrow$) strategy, but making sure that I only swipe right ($\rightarrow$) when the top row is full. When you get two matching tiles in the top row, swipe left ($\leftarrow$) until you've collapsed all pairs, and then go back to the first strategy above (left($\leftarrow$)/up($\uparrow$)) for a bit, until the numbers on the top row get too big. Then continue with right($\rightarrow$)/up($\uparrow$), etc.
Of course, as with any strategy, you need to use your knowledge of the game and judgement to determine when to deviate from the "speed" tactic and pay closer attention to what you're doing, but as a general strategy, I have found this to work quite well.
With this method, I can fairly consistently get to 2048 in under 15 minutes. (Often under 10 minutes.)
1Look for situations like this:
128 & 64 & 16 & 4\\
64 & 16 & 4\\
32 & 16 & 2\\
4 & 2\\
Even though you can continue swiping up and left, it makes sense here to stop and swipe right, because the 2nd row matches up so nicely with the top row. You need to watch for these opportunities, because if you focus too heavily on the up/right strategy, you will clog up the whole board and leave yourself no moves.
2After you do the one swipe up, you may still have several large-ish numbers in rows 2 and 3. If you immediately continue with the up/left tactic, you will probably trap a small number on the left side of row 2, which will upset the entire strategy. Often, the numbers will line up quite nicely, and you'll be able to collapse many of your larger numbers into the top row. You have to use your judgement to determine how many up-swipes you need before you go back to the up/left tactic.