Directory-specific naming of grep buffer as *grep-dir*

[Nordlöw]

The following code enables me to perform several concurrent
compilations in separate directories:

(setq compilation-buffer-name-function
      '(lambda (mode)
         (concat "*" (downcase mode) "-" compilation-directory "*"))
      )

This because a compilation in directory DIR will produce output in
buffer: *compile-DIR*.

The logic also applies to grep-mode but the compilation-directory is
not affected by the grep command and there is no variable grep-
directory to use instead. How can I solve this problem?

Thanks in advance,
Nordlöw