var search = 'something' // the search string toLowerCase();

db.ref('foo/'+uid)
    .orderByChild('searchIndex')
    .startAt(search)
    .endAt(search + "uf8ff")
    .once('value')
    .then(function(snap){ ... })
    .catch(function(error){ ... });

"foo": {
     ".read": "auth != null",
     "$uid": {
         ".write": "auth.uid == $uid)",
         ".indexOn": ["searchIndex"]
     }
 },