Question

Air flows over the top of an airplane wing, area \(A\), with speed \(v_{1}\) and past the underside of the wing with speed \(v_{\mathrm{u}} .\) Show that Bemoulli's equation predicts that the upward lift force \(L\) on the wing will be $$ L=\frac{1}{2} \rho A\left(v_{\mathrm{t}}^{2}-v_{\mathrm{u}}^{2}\right) $$ where \(\rho\) is the density of the air. (Hint: Apply Bernoulli's equation to a streamline passing just over the upper wing surface and to a streamline passing just beneath the lower wing surface. Can you justify setting the constants for the two streamlines equal?)

Short answer

Bernoulli's equation predicts that the upward lift force \(L\) on the wing will be, \(L=\frac{1}{2} \rho A\left(v_{t}^{2}-v_{u}^{2}\right). This result is derived by applying Bernoulli's equation to the different flow regions above and below the wing and the varying speeds.

Step-by-step solution

Identify Relevant Variables

The variables provided in the problem are: The area of the wing \(A\), the speed of the air over the top of the wing \(v_{1}\), the speed of the air on the underside of the wing \(v_{u}\) and the density of air \(\rho\). The lift force \(L\) is what we're looking to calculate.

Define Bernoulli's Equation

Bernoulli's equation can be written as \(P_1 + \frac{1}{2}\rho v_{1}^{2} = P_2 + \frac{1}{2}\rho v_{2}^{2}\), where \(P_1\) and \(P_2\) are pressures at the points 1 and 2 respectively, with corresponding speeds \(v_1\) and \(v_2\). It states that total energy in a steadily flowing fluid system is conserved at all points along a streamline.

Apply Bernoulli's Equation

Apply Bernoulli's equation to the air flowing above and below the wing. Assume air pressure farther from the wing is the same for top and underside, and so these two pressures cancel each other. The different pressure on the top and underside of the wing is then given by \(\Delta P = P_{u} - P_{t} = \frac{1}{2}\rho v_{t}^{2} - \frac{1}{2}\rho v_{u}^{2} = \frac{1}{2}\rho (v_{t}^{2} - v_{u}^{2})\). Where \(P_{u}\) is the pressure beneath the wing and \(P_{t}\) is the pressure above the wing.

Express Lift Force

The lift force \(L\) is the result of the pressure difference across the surface area of the wing. Therefore, \(L = \Delta P \cdot A = \frac{1}{2}\rho A(v_{t}^{2} - v_{u}^{2})\).

Key concepts

Aerodynamics

Aerodynamics is the study of how air flows around objects, a crucial field when it comes to understanding how airplanes fly. Specifically, it involves the behavior of air as it interacts with solid objects like airplane wings. It's not just about resistance or 'drag' that air provides; it's also about 'lift', which is an upward force that keeps the plane aloft. Aerodynamics is grounded in the principles of fluid dynamics, as air behaves similarly to a fluid.

In the case of an airplane wing, the shape is key. A wing is designed to force air to move faster over its top than beneath it, creating a difference in pressure that results in lift according to Bernoulli's principle, a cornerstone of aerodynamics. The efficiency of this lift is determined by factors like the wing's shape, angle, and the air's velocity and density.

Fluid Dynamics

Fluid dynamics is a sub-discipline of physics within fluid mechanics that deals with the flow of fluids—liquids and gases. It is essential for understanding how air (a gas) moves around objects, such as airplane wings, because air follows the same basic laws as any other fluid. The fluid's behavior is described by various equations and principles, with Bernoulli's equation being one of the significant breakthroughs in the field.

Fluid dynamics analyzes the relationships between velocity, pressure, and density of a fluid in motion, and this analysis helps us predict lift force on airplane wings. The exercise outlined shows a direct application of fluid dynamics principles to explain the lift generated on an airplane wing, by comparing the fluid's, in this case, air's, speed and pressure at different points.

Airplane Wing Lift

The lift force on an airplane wing is what enables an aircraft to rise into the air. This force is a result of the difference in pressure created between the top and bottom surfaces of the wing, as governed by the principles of aerodynamics. The wing's unique shape causes air to flow more rapidly over the top surface than the bottom, decreasing the pressure above the wing according to Bernoulli's principle.

The greater pressure from the slower-moving air underneath then pushes the wing upwards, creating lift. The magnitude of this lift force can be calculated with Bernoulli's equation, as demonstrated in the textbook exercise, which relies on the known variables such as air density, the speed of air at different points, and the wing's surface area.

Streamline Flow

Streamline flow, also known as laminar flow, describes the situation where a fluid flows in parallel layers with no disruption between them. This type of flow is crucial in the application of Bernoulli's equation to determine lift on an airplane wing. The assumption of streamline flow means that we can apply the principles of fluid dynamics and Bernoulli's equation to predict pressures and velocities along a single path.

When dealing with an airplane wing, it is the streamline flow of air over and under the wing that allows for the distinct pressure differences leading to lift. In the exercise, we use Bernoulli's equation to find the pressure change between two streamlines running just above and below the wing's surface. The streamline assumption simplifies the complex real-world scenario, making it possible to calculate the lift force with the given parameters.