Aerodynamics is an important part of the vehicle development process. The following article is focused mainly on the cars as the industry has seen huge strides of development in technology to reduce drag, improve cooling, and thereby improving fuel economy.

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Aerodynamics is mainly concerned with the surface forces acting on the vehicle due to the fluid motion around it and hence becomes necessary to understand bits and pieces of fluid mechanics, thermodynamics, partial differential equations, and numerical methods to understand, quantify and improve them for a specific objective. The surface forces by the fluid on the vehicle can be projected along the road (or the surface in which the vehicle is moving) and perpendicular to the road which is termed as the drag and lift force respectively.

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First, let's discuss the parameters associated with Aerodynamics:

1. Coefficient of drag:

     It is the drag force normalized by the free stream dynamic pressure. It is important to note the coefficient of drag is not constant (ideally) and varies with the vehicle speed.

For example, in a coast down test, the drag coefficient is the average value of the minimum and maximum speed in the deceleration process. CFD engineers and wind tunnel experts tend to use a design test point at which the drag coefficient is calculated and iterated.

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There are various components to drag - form drag, friction drag, induced drag, wave drag, etc., The most important component is the form drag which arises due to the pressure difference.

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Q. How to analyze the aerodynamics of a body?

First, we consider the body to be fixed and the fluid to be moving at asymptotic velocity. Now, the reason is that, if the body were considered to be moving and the fluid particles are fixed, then the streamlines, pathlines, and streaklines aren't constant and is a function of time and position, even for the steady velocity of the body. But, since the body is fixed and assuming a steady flow of the fluid, the streamlines, pathlines, and streaklines become constant and is easy to analyze the problem.

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Since the fluid is composed of molecules and it is difficult to analyze every molecule, fluid particles are introduced. This is the continuum approach to analyze the fluid mechanics problem. The fluid particles are sufficiently big to contain large number of molecules and small enough to allow for macroscopic changes in the properties. 

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Now, is the fluid particle an open (or) closed (or) isolated thermodynamic system? If the fluid is inviscid there is no exchange of molecules within the particles and hence is a closed thermodynamic system. Additionally, if it is adiabatic, the fluid particles can be assumed as an isolated system. In general, the fluid particles are considered to be an open thermodynamic system. This allows for molecule exchange between the particles and hence the diffusion process (and) viscosity of the fluid can be represented in addition to the temperature. The temperature scales are smaller than the fluid dynamic scales and hence we can assume thermal equilibrium of fluid particles.

 

The motion of the fluid particle can be defined in terms of -isovolumic deformation, -rigid body rotation, -shear and isotropic expansion (or) contraction. There is no stress associated with rigid body rotation. The pressure force results in isovolumic expansion (or) contraction. The shear stress is proportional to the velocity gradient. Why? Because the stress is produced by a force (or) alternatively stress is equivalent to pressure ie. force per unit area. Now, the difference in momentum in the fluid particles creates shear stress. Shear stress can be perpendicular to the fluid particle but, is regarded with pressure as equivalent pressure.

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Using Cauchy's theorem, the surface forces can be defined as the sum of the pressure forces and shear stresses acting on the vehicle along the components vectors if we neglect the curvature of the body. So, the surface force tensor is a function of time, position, and orientation to the surface. Now, before discussing any further with the surface forces, it is to be elucidated that the shear stress at a point requires 9 component information in 3D space and hence is a rank 2 tensor (Tij).

 

Q. Why is this?

Because shear stress can't be defined at a point. A plane is needed to define the shear stress and hence an infinitely small cube is considered. Now, having known the 9 components of shear stress, the shear stress at any arbitrary plane can then be determined. Note, however, due to the conservation of angular momentum, the stress tensor is symmetric. If there are no body couples acting on the fluid element, then the stress tensor would require just 6 components to be defined.

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Since for a Newtonian fluid, the shear stress is proportional to the velocity gradient, to find the shear stress at a point, 81 constants are required to equate Tij with velocity gradients [Tij = Cijkl grad(u)]