According to the bible, what is the right way for a man to test if his wife is being an adulteress?

The test was part of the Law Covenant given to the Jews.

I'm not a Jew. The Law Covenant has passed away. Therefore the test is no longer valid.

Num 5:13 And a man lie with her carnally, and it be hid from the eyes of her husband, and be kept close, and she be defiled, and [there be] no witness against her, neither she be taken [with the manner];

Num 5:14 And the spirit of jealousy come upon him, and he be jealous of his wife, and she be defiled: or if the spirit of jealousy come upon him, and he be jealous of his wife, and she be not defiled:

Num 5:15 Then shall the man bring his wife unto the priest, and he shall bring her offering for her, the tenth [part] of an ephah of barley meal; he shall pour no oil upon it, nor put frankincense thereon; for it [is] an offering of jealousy, an offering of memorial, bringing iniquity to remembrance.

Num 5:16 ¶ And the priest shall bring her near, and set her before the LORD:

Num 5:17 And the priest shall take holy water in an earthen vessel; and of the dust that is in the floor of the tabernacle the priest shall take, and put [it] into the water:

Num 5:18 And the priest shall set the woman before the LORD, and uncover the woman's head, and put the offering of memorial in her hands, which [is] the jealousy offering: and the priest shall have in his hand the bitter water that causeth the curse:

Num 5:19 And the priest shall charge her by an oath, and say unto the woman, If no man have lain with thee, and if thou hast not gone aside to uncleanness [with another] instead of thy husband, be thou free from this bitter water that causeth the curse:

Num 5:20 But if thou hast gone aside [to another] instead of thy husband, and if thou be defiled, and some man have lain with thee beside thine husband:

Num 5:21 Then the priest shall charge the woman with an oath of cursing, and the priest shall say unto the woman, The LORD make thee a curse and an oath among thy people, when the LORD doth make thy thigh to rot, and thy belly to swell;

Num 5:22 And this water that causeth the curse shall go into thy bowels, to make [thy] belly to swell, and [thy] thigh to rot: And the woman shall say, Amen, amen.

Num 5:23 ¶ And the priest shall write these curses in a book, and he shall blot [them] out with the bitter water:

Num 5:24 And he shall cause the woman to drink the bitter water that causeth the curse: and the water that causeth the curse shall enter into her, [and become] bitter.

Num 5:25 Then the priest shall take the jealousy offering out of the woman's hand, and shall wave the offering before the LORD, and offer it upon the altar:

Num 5:26 And the priest shall take an handful of the offering, [even] the memorial thereof, and burn [it] upon the altar, and afterward shall cause the woman to drink the water.

Num 5:27 And when he hath made her to drink the water, then it shall come to pass, [that], if she be defiled, and have done trespass against her husband, that the water that causeth the curse shall enter into her, [and become] bitter, and her belly shall swell, and her thigh shall rot: and the woman shall be a curse among her people.

Num 5:28 And if the woman be not defiled, but be clean; then she shall be free, and shall conceive seed.

Doug's answer referencing Number chapter 5 is correct. One must remember that this test was given ONLY to the Covenant People and can be used ONLY while the Tabernacle or The Temple stand. It's very nature demonstrates that it can be used ONLY in Ancient Israel in the Temple era.

This command was never meant for Gentiles living at ANY time in history. This was part of the covenant made with the Hebrew people while they lived in their promised land and had the Ark of The Covenant among them. This is what it meant when it said the woman was to "stand before The LORD".

In the direct presence of God in the Temple, God would see to it that the correct result happened.

Old Test.... a concoction was put together to give to the woman.... if it caused her leg to rot and her belly to swell. I think it says "a bitter water to cause a curse" on the adulteress.

Get 2 lemons. Squeeze the lemons into a pitcher and add water and sugar. While the perfect solution is being put together to make lemonade--if she's too busy being banged by another guy to care--she's cheating on you.

The Bible has no test.

It assumes you're smarter than a rock in most respects.