A body having the general shape of a half-body is placed in...

A body having the general shape of a half-body is placed in a stream of fluid. At a great distance upstream the velocity is $U$ as shown in Fig. $\mathrm{P} 6.67 .$ Show how a measurement of the differential pressure between the stagnation point and point $A$ can be used to predict the free-stream velocity, $U .$ Express the pressure differential in terms of $U$ and fluid density. Neglect body forces and assume that the fluid is nonviscous and incompressible.

Key Concepts

Bernoulli's Principle

Bernoulli's principle states that for an incompressible, non-viscous fluid undergoing steady flow, the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline. This principle is fundamental in relating the pressure differences observed in the flow to the fluid velocity.

Stagnation Pressure

Stagnation pressure is the local pressure experienced by a fluid when it is brought to a complete stop isentropically. It represents the maximum pressure in a flow and is used, along with Bernoulli's principle, to connect the pressure at a stagnation point with the free-stream velocity.

Incompressible and Non-viscous Flow Assumptions

Assuming the fluid is incompressible and non-viscous simplifies the application of Bernoulli's equation by ensuring that the density remains constant and viscous effects can be neglected. This makes it easier to establish a direct relation between pressure differences and velocity changes in the fluid system.

Free-stream Velocity

Free-stream velocity refers to the velocity of a fluid far from any disturbances such as bodies or boundaries, where the flow is undisturbed. In many practical problems, the free-stream velocity serves as an important reference point for analyzing how the fluid interacts with objects, and it is often derived from measurable pressure differentials in the flow.

Differential Pressure Measurement

Differential pressure measurement involves determining the difference in pressure between two points in a fluid system. In the context of flow analysis, measuring the pressure difference between a stagnation point and another point in the flow allows for quantitative determination of local and free-stream velocities, as this difference is directly related to the kinetic energy per unit volume of the fluid.

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