MAE 3241: AERODYNAMICS AND FLIGHT 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

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

CONSERVATION OF MASS Relative to CS Inertial 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

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

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

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