Question

What is the difference between the upstream velocity and the free-stream velocity? For what types of flow are these two velocities equal to each other?

Short answer

Answer: Upstream velocity refers to the velocity of a fluid in a pipe or channel before it encounters an obstacle or a change in flow conditions, whereas free-stream velocity is the velocity of fluid far away from a solid boundary, where it is not influenced by the presence of the solid object. They can be equal in uniform, unidirectional flow where fluid velocity is constant and not changed by any disturbances, or in low-Reynolds-number flow where the viscous effects dominate and the effect of an object on the surrounding fluid can be negligible. However, in many practical situations, upstream and free-stream velocities will generally differ.

Step-by-step solution

Upstream Velocity

Upstream velocity refers to the velocity of a fluid in a pipe or channel before it encounters an obstacle or a change in flow conditions. It is essentially the fluid's velocity in its initial, undisturbed state.

Free-Stream Velocity

Free-stream velocity is the velocity of fluid far away from a solid boundary, such as an airfoil in an airstream or a submerged object in a flow. At this distance, the fluid is not influenced by the presence of the solid object, and its velocity is considered uniform and unaffected by the object.

Comparing Upstream and Free-Stream Velocities

While both upstream and free-stream velocities describe fluid velocities in undisturbed states, the key difference lies in their application. Upstream velocity usually pertains to flow in a pipe or channel, where the fluid encounters an obstacle or change in flow conditions downstream. Conversely, free-stream velocity relates to flows around solid objects, like air around an airplane wing or water around a submarine.

Equality Between Upstream and Free-Stream Velocities

In some types of flow, upstream and free-stream velocities can be equal. For example, in uniform, unidirectional flow (flow with constant velocity and direction), upstream and free-stream velocities are the same because the fluid velocity in the pipe or channel is not changed by any downstream disturbances or obstacles. Another case where these velocities may coincide is in a low-Reynolds-number flow, where the viscous effects dominate. In such a situation, it might be possible for the effect of an object on the surrounding fluid to be negligible, causing upstream and free-stream velocities to be equal. However, it is important to note that in many practical situations, particularly when there is a significant disturbance to the flow like an object in the flow or a change in flow conditions, upstream and free-stream velocities will generally differ.