ChapterII.PartI

# ChapterII.PartI - II Governing Equations In aerodynamics or...

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II. Governing Equations In aerodynamics, or fluid mechanics, there are six properties of the flow an engineer is usually interested in – pressure p, density ρ , the three velocity components (u,v,w), and the temperature T. For gases and mixtures of gases (e.g. air) the equation of state links p, ρ and T by: Equation of State: p = ρ RT (2.1) We need to come up five additional equations linking the 6 properties. These five equations are PDEs and turn out to be: a) Conservation of Mass or Continuity b) Conservation of u- momentum c) Conservation of v-momentum d) Conservation of w-momentum e) Conservation of energy These equations may be derived using a Lagrangean approach, or an Eulerian approach. In the Lagrangean approach, we follow a fixed set of fluid particles (e.g. a cloud, a tornado, tip vortices from an aircraft) and write down equations governing their motion. This is somewhat like tracking satellites and missiles in space, using equations to describe their position in space and the forces acting on them. In the Eulerian approach, we look at a (usually) fixed or (sometimes) moving volume in space surrounded by permeable boundaries. We develop equations describing what happens to the fluid inside the control volume as new fluid enters and old fluid particles leave. Eulerian approach is the preferred approach in most fluid dynamics applications. This is what we will follow in our derivations. Conservation of Mass (Continuity): The conservation of mass stems from the principle that mass can not be created or destroyed inside the control volume. Obviously, we are situations (e.g. nuclear reactions) involving the conversion of mass into energy. Let V be a control volume, a balloon like shape in space. We will assume that it, and its surface S remain fixed in space. The surface is permeable so that fluid can freely enter in and leave. The continuity equation says The time rate of change of Mass within the control volume V = Rate at which mass enters V through the boundary S

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ChapterII.PartI - II Governing Equations In aerodynamics or...

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