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.

Conservation Laws

Aerodynamic problems are normally solved using conservation of mass, momentum, and energy,referred to as continuity, momentum, and energy equations. The conservation laws can be written in 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`

where 𝜌 is the fluid density, u is a velocity vector, and t is time. Physically the equation also shows that mass is neither created nor destroyed in the control volume. For a steady-state process, the rate at which mass enters the volume is equal to the rate at which it leaves the volume.Consequently, the first term on the left-hand side is then equal to zero. For flow through a tube with one inlet (state 1) and exit (state 2) as shown in Figure 1, the continuity equation may be written and solved 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.

Figure 1

Conservation of Momentum

The momentum equation applies Newton’s second law of motion to a control volume in a flow field, whereby force is equal to the time derivative of momentum. Both surface and body forces are accounted for in this equation. For instance, F could be expanded into an expression for 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`

where the force F is placed on the left-hand side of the equation, assuming it acts with the flow moving in a left-to-right direction. Depending on the other properties of the flow, the resulting force could be negative that means it acts in the opposite direction as depicted in Figure 1. In aerodynamics, air is normally assumed to be a Newtonian fluid, which posits a linear relationship between the shear stress and the rate of strain of the fluid.

The equation above is a vector equation: in a three-dimensional flow, it can be expressed as three scalar equations.The conservation of momentum equations are often called the Navier–Stokes equations, while others use the term for the system that includes conversation of mass, conservation of momentum, and conservation of 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:

The design of aircraft wings is significantly impacted by conservation laws. Engineers use the principles of conserving mass and momentum to efficiently create lift. Wing shape, size, and airfoil profiles are precisely calculated to balance air pressure and momentum for ideal lift production.

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.