1. MAE 5360: Hypersonic Airbreathing Engines
Integral Forms of Mass and Momentum Equations
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 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

4. Integral 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

7. Application to Rocket Engines
Chemical
Energy F
Rocket Propulsion (class of jet propulsion) that produces thrust by ejecting stored matter
Propellants are combined in a combustion chamber where chemically react to form high T&P gases
Gases accelerated and ejected at high velocity through nozzle, imparting momentum to engine
Thrust force of rocket motor is reaction experienced by structure due to ejection of high velocity matter
Same phenomenon which pushes a garden hose backward as water flows from nozzle, gun recoil
Thermal
Energy
Kinetic
Energy

8. Application to Airbreathing Engines
Chemical
Energy
Thermal
Energy
Kinetic
Energy
Flow through engine is conventionally called THRUST
Composed of net change in momentum of inlet and exit air
Fluid that passes around engine is conventionally called DRAG