MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

1. MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS Views of Motion: Eulerian and Lagrangian Conservation Equation Summary January 19, 2011 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. KINEMATIC PROPERTIES: TWO ‘VIEWS’ OF MOTION • Lagrangian Description • Follow individual particle trajectories • Choice in solid mechanics • Control mass analyses • Mass, momentum, and energy usually formulated for particles or systems of fixed identity • ex., F=d/dt(mV) is Lagrangian in nature • Eulerian Description • Study field as a function of position and time; not follow any specific particle paths • Usually choice in fluid mechanics • Control volume analyses • Eulerian velocity vector field: • Knowing scalars u, v, w as f(x,y,z,t) is a solution

3. CONSERVATION OF MASS • This is a single scalar equation • Velocity doted with normal unit vector results in a scalar • 1st Term: Rate of change of mass inside CV • If steady d/dt( ) = 0 • Velocity, density, etc. at any point in space do not change with time, but may vary from point to point • 2nd Term: Rate of convection of mass into and out of CV through bounding surface, S • 3rd Term (=0): Production or source terms Relative to CS Inertial

4. INTEGRAL FORM VS. DIFFERENTIAL FORM • Integral form of mass conservation • Apply Divergence (Gauss’) Theorem • Transform both terms to volume integrals • Results in continuity equation in the form of a partial differential equation • Applies to a fixed point in the flow • Only assumption is that fluid is a continuum • Steady vs. unsteady • Viscous vs. inviscid • Compressible vs. incompressible

5. SUMMARY: INCOMPRESSIBLE VS. CONSTANT DENSITY • Two equivalent statements of conservation of mass in differential form • In an incompressible flow • Says particles are constant volume, but not necessarily constant shape • Density of a fluid particle does not change as it moves through the flow field • INCOMPRESSIBLE: Density may change within the flow field but may not change along a particle path • CONSTANT DENSITY: Density is the same everywhere in the flow field

6. MOMENTUM EQUATION: NEWTONS 2nd LAW Inertial Relative to CS • This is a vector equation in 3 directions • 1st Term: Rate of change of momentum inside CV or Total (vector sum) of the momentum of all parts of the CV at any one instant of time • If steady d/dt( ) = 0 • Velocity, density, etc. at any point in space do not change with time, but may vary from point to point • 2nd Term: Rate of convection of momentum into and out of CV through bounding surface, S or Net rate of flow of momentum out of the control surface (outflow minus inflow) • 3rd Term: • Notice that sign on pressure, pressure always acts inward • Shear stress tensor, t, drag • Body forces, gravity, are volumetric phenomena • External forces, for example reaction force on an engine test stand • Application of a set of forces to a control volume has two possible consequences • Changing the total momentum instantaneously contained within the control volume, and/or • Changing the net flow rate of momentum leaving the control volume