What does it indicate if a professor from other departments is included in applied math department? Currently I am preparing for postgraduate application. My research interest is fluid mechanics. There are several professors whose research interests are similar to mine and all of them are included in both applied math department and some non-math departments, such as geophysics department. Most of their papers are published in Journal of Fluid Mechanics, which is one of the leading journals in that field. However, it seems that those papers do not involve advanced math knowledge, such as abstract algebra or analysis. Even epsilon-delta terminology seldom appears in those papers.

This slightly contradicts to my previous understanding. In the math department where I finished my undergraduate study, real analysis is a basic requirement for every postgraduate student. Does this mean the field of applied fluid mechanism does not require students with solid math background? What does it imply if a professor is included in applied math department? Does this indicate he/she will supervise a student with only math background? Or does this mean he/she has an intention to publish papers in math journals?
Affiliate faculty are common in many university departments. This indicates faculty who have a departmental home (faculty line, voting rights, tenure and promotion line, office space) but for a variety of reasons are given courtesy appointments (sometimes called in the USA as 0 FTE appointments) in another department. This may be because they teach a crosslisted course or occasionally sit on doctoral committees of the other department, or occasionally just for the academic lulz (ie, they requested an affiliation and there was no reason to deny it).

Affiliate faculty rarely occasionally (depending on discipline and university) serve as chairs of doctoral committees and similarly are rarely/occasionally (depending on discipline) given students in that department to supervise. Their ability though to sit as members on committees is useful if you were to want to do a project in geophysics (in this case) but otherwise you can ignore them.

Note that this is different from faculty with dual appointments where their faculty line is split (0.5 FTE in one department, 0.5 in another). They have voting and advisory rights in both departments.

FTE = Full-Time Equivalent, the essential counting unit of faculty lines. 
A department with two full-time faculty and two joint-appointments would 
have 2*1 FTE + 2 * 0.5 FTE = 3 FTE


TL;DR: Don't worry about affiliate faculty. Focus on the core faculty and their strengths when making decisions.