The Importance of Conservation Law in Aerodynamics

Introduction

The study of how air interacts with solid objects, aerodynamics, is crucial in aircraft design, performance, and safety. Conservation laws, based on physics, underlie aerodynamic principles and help engineers create efficient aircraft. This blog will discuss the significance of conservation laws in aerodynamics and their influence on the aviation sector.

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Conservation Laws

Engineers normally solve aerodynamic problems by applying the conservation of mass, momentum, and energy—known as the continuity, momentum, and energy equations. They express these conservation laws in either integral or differential form.

Conservation of Mass

If a certain mass of fluid enters a volume, it must either exit the volume or change the mass inside the volume. In fluid dynamics the continuity equation is analogous to Kirchhoff’s current law (that is, ‘the sum of the currents flowing into a point in a circuit is equal to the sum of the currents flowing out of that same point’) in electric circuits. The differential form of the continuity equation is

`frac{partialrho}{partial t}+nablacdotleft(rho uright)=0`

We denote the fluid density by $ρ\rho$ , the velocity vector by $u\mathbf{u}$ , and time by $t$ . Physically, the continuity equation states that mass remains conserved within the control volume—no creation or destruction of mass occurs. In a steady-state process, mass enters and leaves the control volume at the same rate. This condition causes the first term on the left-hand side of the continuity equation to become zero. For flow through a tube with one inlet (state 1) and one exit (state 2), as shown in Figure 1, we can write and solve the continuity equation as:

`rho_1u_1A_1=rho_2u_2A_2`

where A is the variable cross-sectional area of the tube at the inlet and exit. For incompressible flows, the density remains constant.

Diagram showing a pipe with a narrowed middle section. An arrow labeled "u" indicates fluid flow from left to right, marked as 1 to 2.

Figure 1

Conservation of Momentum

Engineers apply Newton’s second law of motion to a control volume in a flow field to derive the momentum equation, where force equals the time rate of change of momentum. This equation includes both surface forces and body forces. For example, one can expand the force term $F$ to express the frictional force acting on an internal flow.

`frac{Du}{Dt}=F-frac{nabla p}rho`

For the pipe flow in Figure 1, control volume analysis gives

`rho_1A_1+rho_1left(u_1right)^2A_1+F=rho_2left(u_2right)^2A_2+rho_2A_2`

Place the force $F$ on the left-hand side of the equation, assuming it acts in the same direction as the flow, typically from left to right. Depending on the flow properties, the resulting force may turn out negative, indicating it acts in the opposite direction, as shown in Figure 1. In aerodynamics, we generally treat air as a Newtonian fluid, meaning the shear stress varies linearly with the fluid’s rate of strain.

The equation above represents a vector equation. In a three-dimensional flow, you can express it as three separate scalar equations. Engineers and scientists often refer to the conservation of momentum equations as the Navier–Stokes equations. Some also use the term to describe the complete system that includes conservation of mass, momentum, and energy.

Conservation of Energy

Although energy can be converted from one form to another, the total energy in a given closed system remains constant:

`rhofrac{Dh}{Dt}=frac{Dp}{Dt}+nablacdotleft(knabla Tright)+phi`

where h is enthalpy, k is the thermal conductivity of the fluid, T is temperature, and Φ is the viscous dissipation function. The viscous dissipation function governs the rate at which mechanical energy of the flow is converted to heat. This term is always positive since, according to the second law of thermodynamics, viscosity cannot add energy to the control volume.The expression on the left-hand side is a material derivative. Again using the pipe flow in Figure 1, the energy equation in terms of the control volume may be written as

`rho_1u_1A_1left(h_1+frac{u_1^2}2right)+W+Q=rho_2u_2A_2left(h_2+frac{u_2^2}2right)`

where the shaft work Ẇ and heat transfer rate Q̇ are assumed to be acting on the flow. They may be positive or negative depending on the problem.

The ideal gas law or another equation of state is often used in conjunction with these equations to form a determined system to solve for the unknown variables.

Applications of Conservation Law in Aircraft Design

1. Wing Design and Lift Generation:

Conservation laws significantly impact how engineers design aircraft wings. They apply the principles of mass and momentum conservation to generate lift efficiently. Engineers precisely calculate wing shape, size, and airfoil profiles to balance air pressure and momentum, producing optimal lift.

2. Propulsion Systems:

The key to aviation lies in efficient propulsion systems, with conservation laws playing a vital role in guiding the design of engines and propellers. Engineers can create propulsion systems that achieve maximum fuel efficiency and performance by utilizing their knowledge of energy conservation.

3. Flight Control Surfaces

The principles of conservation laws are utilized in creating flight control surfaces like ailerons, elevators, and rudders. These surfaces manage airflow to steer the aircraft while following the conservation of momentum, leading to stable and maneuverable control surfaces.

Conclusion

In short, conservation laws are the foundation of aerodynamics, influencing aircraft design and performance. With the evolution of the aviation sector, a thorough comprehension of these laws is vital. By applying the principles of mass, momentum, and energy conservation, engineers drive the aviation field forward, developing airplanes that advance efficiency, safety, and innovation in the air.