Aerodynamics - Laws of Conservation

Laws of Conservation

Aerodynamic problems are normally solved using conservation laws as applied to a fluid continuum. The conservation laws can be written in integral or differential form. In many basic problems, three conservation principles are used:

• Continuity: 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 in electric circuits. The differential form of the continuity equation is:

Above, 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 is then equal to zero. For flow through a tube with one inlet (state 1) and exit (state 2) as shown in the figure in this section, the continuity equation may be written and solved as:

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

• Conservation of Momentum: This equation applies Newton's second law of motion to a continuum, 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.

For the same figure, a control volume analysis yields:

Above, the force is placed on the left 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 which means it acts in the opposite direction as depicted in the figure. In aerodynamics, air is normally assumed to be a Newtonian fluid, which posits a linear relationship between the shear stress (the internal friction forces) 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.

Above, h is enthalpy, k is the thermal conductivity of the fluid, T is temperature, and is the viscous dissipation function. The viscous dissipation function governs the rate at which mechanical energy of the flow is converted to heat. The 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 side is a material derivative. Again using the figure, the energy equation in terms of the control volume may be written as:

Above, the shaft work and heat transfer are assumed to be acting on the flow. They may be positive (to the flow from the surroundings) or negative (to the surroundings from the flow) 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.