BERNOULI'S EQUATION

Bernoulli equation

The equation above stated that, where

1)      Points 1 and 2 lie on the streamline

2)      The fluid has constant velocity

3)      The flow is steady

4)      There is no friction

Although these restriction sound severe, the Bernoulli equation is very useful, partly because it is very simple to use and partly because it can give great insight into the balance between pressure, velocity and elevation.

Pressure/velocity variation

Consider the steady, flow of the constant density fluid in a converging duct, without losses due to friction. The flow satisfies all the restrictions governing the use of Bernoulli’s equation. Upstream and downstream of the contraction we make the one dimensional assumption that the velocity is constant over the inlet and outlet areas and parallel.

When streamlines are parallel, the pressure is constant across them, except for hydrostatic head differences (if the pressure was higher in the middle of the duct). The gravity is ignored, then the pressures over the inlet and outlet areas are constant. Along the streamline on the centreline, the Bernoulli equation and the one dimensional continuity equation give respectively,

These two observation provide an intuitive guide for analysing fluid flows, even when the flow is not one dimensional. For example, when the fluid passes over a solid body, the streamlines get closer together, the flow density increases and the pressure will decrease. Airfoils are designed so that the flow over the top surface is faster than over the bottom surface. Therefore the average pressure over the top surface is less than the average pressure over the bottom surface and a resultant force due to this pressure difference is produced. This is the source of lift on an airfoil. Lift is defined as the force acting on an airfoil due to its motion, in a direction normal to the direction of motion. Likewise, the drag on an airfoil is defined as the force acting on an airfoil due to its motion, along the direction of motion.

Example 1

A table tennis ball placed in a vertical air jet becomes suspended in the jet, and it is very stable to small pertubations in any direction. Push the ball down, and its springs back to its equilibrium position: push it sideways and it rapidly returns to its original position in the center of the jet. In the vertical direction, the weight of the ball is balanced by a force due to pressure differences: the pressure over the rear half of the sphere is lower than over the front half because of losses that occur in the wake (large eddies form in the wake that dissipate a lot of flow energy). To understand the balance of forces in the horizontal direction, we need to know that the jet has its maximum velocity in the center and the velocity of the jet decrease towards its edge. The ball position is stable because if the ball moves sideways, its outer side move into a region of lower velocity and higher pressure whereas its inner side moves closer to the center where the velocity is higher and the pressure is lower. The differences in pressure tend to move the ball back towards the center.

Stagnation pressure and dynamic pressure

Bernoulli’s equation leads to some interesting conclusions regarding the variation of pressure along a streamline. Consider a steady flow on a perpendicular plate.

 There is one streamline that divides the flow goes under the plate. Along this dividing streamline, the fluid moves towards the plate. Since the flow cannot pass through the plate, the fluid must come to rest at the point where it meets the plate. In other words, it stagnates. The fluid along the dividing or stagnation streamline slows down and eventually comes to rest without deflection at the stagnation point. Bernoulli’s equation along the stagnation streamlines gives ,

Where the point e is far upstream and point 0 is at the stagnation point. Since the velocity at the stagnation point is zero,

The stagnation or total pressure, p₀ is the pressure measured at the point where the fluid comes to rest. It is the highest pressure found anywhere in the flowfield, and it occurs at the stagnation point. It is the sum of the static  pressure p₀ and the dynamic pressure measure far upstream. It is called the dynamic pressure because it arises from the motion of the fluid. We can also express the pressure anywhere in the flow in the form of a non-dimensional pressure coefficient, c_p where,

At the stagnation point c_p = 1 which is its maximum value. In the freestream, far from the plate, c_p = 0.

Pitot tube

One of the application of Bernoulli’s equation is in the measurement of velocity with a Pitot-tube. It is simply consist of a tube bent at the right angles.

By pointing the tube directly upstream into the flow and measuring the difference between the pressure sensed by the Pitot tube and pressure of the surrounding air flow, it can give a very accurate measure of the velocity. In fact, it is probably the most accurate method available for measuring flow velocity on a routine basis and accuracies better than 1% are easily possible. Bernoulli equation along the streamline that begins far upstream of the tube and comes to rest in the mouth of the pitit tube shows the pitot tube measures the stagnation pressure in the flow. Therefore to find the velocity, v_e, we need to know the density of air and the pressure difference (p₀ - p_e). the density can be found from standard tables if the temperature and the pressure are known. The pressure difference is usually found indirectly by using a static pressure tapping located on the wall of the wind tunnel or on the surface of the model.