The "Union Army" was the US Army - all of it. Its overall commander stayed in Washington DC until Grant was put in that position, and Grant stayed in the field with his field armies. The Union Army had several field armies out conducting operations throughout the war.
The US Army at the start of the war was under General in Chief Winfield Scott, who was 75 years old and had been in the Army since 1808 - 53 years. There was no system of retirement pay then, and officers served until they died. "Old Fuss and Feathers" had been a hero of the War of 1812, but was now enormously fat - over 400 pounds. He was very tall too, 6' 8", and when on those rare occasions when he decided to try to ride his poor, long suffering horse, he had to be hoisted up on it with a block and tackle. He tended to fall asleep in the middle of conversations.
Scott finally retired in 1862 and his place was taken by "The Young Napoleon", George B. McClellan. At the time McClellan was also commanding a field army, the Army of the Potomac. He assured Lincoln he could do both jobs. Lincoln could never seem to get McClellan to do anything, especially not fight battles, and eventually replaced him as General in Chief with "Old Brains", Henry W. Halleck, but for the time left McClellan in command of his field army. Halleck was a disappointment too, and was replaced by Grant in 1864.
But I suspect what you are really asking is how many commanders did the most famous Union field army, the Army of the Potomac, go through in the first two years of the war. The answer is seven. Irvin McDowell, McClellan, John Pope, McClellan again, Ambrose Burnside, Joe Hooker, and George Meade. Meade actually remained in command of the Army of the Potomac for the rest of the war, but beginning in 1864, when Grant became General in Chief, he made his HQ with Meade's Army and moved with it, so people tend to forget Meade was there and that Grant did not directly command the that field army.
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The commanders were: General Rosencrans for the Union and General Braxton Bragg for the Confederacy.
The Union
the union was winning the civil war
The first shot in the civil war was fired by the south at the Union Fort Sumter in South Carolina
Where was the first battle civil war fought at
The commanders were: General Rosencrans for the Union and General Braxton Bragg for the Confederacy.
Nebraska was the first state admitted into the union after the civil war. Nebraska was admitted into the Union on March 1, 1867 becoming the 37th state to join the Union.
National labor union
Winfield Scott: July 5, 1841 - November 1, 1861
Tennessee
General McDowen
The Union
the union was winning the civil war
Mississippi was first settled in 1699 and joined the Union in 1817. After the Civil War, Mississippi was readmitted to the Union in February of 1870.
Union: James S. Negley Confederacy: Edmund Kirby Smith
Nebraska- In 1867 it was the 37th state to join the Union.
Tennessee