In this subsection of a long essay on various yichud issues, Rav Eliezer Waldenberg deals with the question of whether or not there is a violation of yichud between a parent and their adopted child of the other gender. His initial reaction is to say that there should be a violation of yichud, even if the child does not know that he or she is adopted, since, at the end of the day, there is seclusion with a male and female who are not close biological relatives.
However, Rav Waldenberg then shifts gears and says that there are several angles by which such a situation can be permitted, particularly if the adoption took place before the child was old enough to be subject to a prohibtion of yichud. His first avenue for doing so is based on the Levush, who explains that the reason that a biological child can have yichud with a parent is because we are not worried about any sin occurring in such a situation. Similarly, rules Rav Waldenberg, in a case where the child has grown up relating to the adult as a parent, we assume that the feelings between them are those of a parent and child and that no untoward behavior will take place. He even extends this to allow them to hug and kiss as any other parent and child would.
On a related note, Rav Waldenberg borrows a logic from the case of a man with his menstruating wife. Even though she is classified as an ערוה, her husband can nevertheless be alone with her without a concern that they will have forbidden relations. The rationale given for this by the Chazon Ish is that they are used to each other (and thus presumably understand the boundaries and are capable of controlling themselves). So too in our case, says Rav Waldenberg, can we assume that the parent and adopted child are used to each other, specifically in the context of a parent-child relationship, and will not commit any illicit sexual act.
Rav Waldenberg then notes that the Yereim explains the permission to be secluded with one's menstruating wife as being a function of the rule that we do not make decrees on the community that people will not be able to uphold. Since it is not reasonable to expect spouses to live apart for 12 days each month, yichud with one's wife during her nidda period was not decreed. In our case as well, it would be well-nigh impossible for parents and children to have to find separate domiciles once the child reaches the relatively young age when yichud becomes an issue. Based on the Sefer Chikrei Lev, he notes that this would also be in violation of the spirit of דרכיה דרכי נועם- that the ways of the Torah are pleasant.
Rav Waldenberg concludes by adducing proofs from several stories of Sages who raised non-biological children from a young age and conducted themselves with them as they would with their own children. Finally, he notes that these leniencies are important in order to continue to encourage people to adopt those in need of it, rather than allow them to live unstable lives.