If you were "profitable but no cash flow" then you must have non-cash additions to your profit, not deductions.
A classic example of 'profit but no cashflow' might be where you made a profit but spent a lot of money on stock that you haven't sold yet. Or you made a lot of sales that you are yet to be paid for.
In the PE world it is just as likely that you made a profit before interest and tax, but you paid it all in interest. You would then have an operating profit but no cashflow due to a cash item. It could still make it a good business to own, if you didn't need the debt, or wanted to have the interest paid to you.
Maybe you made a profit but paid it all in dividends to a holding company. Then you have a profit but no cash flow due to cash items that don't affect the p&l.
You're right about non-cash additions. I was confusing this with an enterprise showing a loss (especially for tax purposes) despite a positive cash flow. The classic example would be residential real estate, where depreciation can cause a net loss despite the landlord receiving enough rent to pay mortgage/property tax/maintenance. This is why in the U.S. there are rules that limit current deductions on the tax return for passive losses.
So I would think the "other way" from profitable/no cash flow is loss/with cash flow.
