Modern Compressible Flow Chapter 2 Notes

==Integral Forms of the Conservation Equations for Inviscid Flows==
:"The ultimate objective of analyzing fluid flow is to predict the forces on objects immersed in the flow" - Anderson
:The fluid flow obeys fundamental relationships that are derived from Newton's laws of mechanics, the first 3 laws.
:Approach - Expressing the conservation equations of fluid motion.
:*Conservation of mass
:*Conservation of momentum
:*Conservation of energy
:*Appropriate fluid equation of state
:*Appropriate boundary conditions
===Finite Control Volume (Application of the Conservation Laws)===
:Choices for analysis
:*Stationary control volume
:*Moving control volume
:In figure 2.3, pg. 44, 3rd edition, there are illustrations of stationary and moving control volumes.
:Control volume is defined as the region of space and its boundary surfaces. Control volume may be macroscopic or differential (i.e. infinitesimal in size). Even an infinitesimal control volume is large enough such that fluid flow around and through it continuum flow (at least for continuum fluid mechanics).
:The integral statements of the conservation of mass and momentum are powerful analysis tools for fluid flow. For compressible flow, the integral statement of the conservation of energy is also needed.
====Continuity Equation====
:Mass cannot be created or destroyed (which ignores nuclear effects).
:Control Volume Application:
:insert integral continuity equation here
:Stationary or fixed control volume
:mdot through ds is:
:\dot m = \rho V dS
====Momentum Equation====
:Control Volume Application:
:\oint\limits_S\left(\rho\boldsymbol{V}\cdot d\boldsymbol{S}\right)\boldsymbol{V} + 
\oint\limits_V\frac{\partial (\rho\boldsymbol{V})}{\partial t} dV = 
\oint\limits_V\rho\boldsymbol{f} dV -
\oint\limits_S p d \boldsymbol{S}
:Define the various terms
\oint\limits_V\frac{\partial (\rho\boldsymbol{V})}{\partial t} dV 	The momentum stored in the control volume
\oint\limits_S\left(\rho\boldsymbol{V}\cdot d\boldsymbol{S}\right)\boldsymbol{V}	The net momentum flow into or out of the control volume
\oint\limits_V\rho\boldsymbol{f} dV 	The net body force on the control volume
\oint\limits_S p d \boldsymbol{S} 	The integral of the pressure force on the control volume
====Energy Equation====
:Anderson provides a description of the formulation of the first law statement for an open system or control volume:
:\dot Q_S + \dot W_{shaft} + \dot W_{viscous} + \dot W_{electric} -
\oint\limits_S p\boldsymbol{V}\cdot d\boldsymbol{S} +
\oint\limits_V \rho\left(\boldsymbol{f}\cdot\boldsymbol{V}\right)dV = 
::\oint\limits_V \frac{\partial}{\partial t}\left[\rho\left(e+\frac{V^2}{2}\right)\right]dV + 
\oint\limits_S \rho\left(e+\frac{V^2}{2}\right) \boldsymbol{V} \cdot d\boldsymbol{S}
:Define the various terms
\dot Q_S 	Integral of surface heat flux
\dot W_{shaft} 	Shaft work on control volume
\dot W_{viscous} 	Viscous force work on control volume
\dot W_{electric} 	Electric force work on control volume
\oint\limits_S p\boldsymbol{V}\cdot d\boldsymbol{S} 	Rate of work done on the fluid inside the control volume due to pressure forces on the surface.
\oint\limits_V \rho\left(\boldsymbol{f}\cdot\boldsymbol{V}\right)dV 	Rate of work done on the fluid inside the control volume due to body forces.
\oint\limits_V \frac{\partial}{\partial t}\left[\rho\left(e+\frac{V^2}{2}\right)\right]dV	Time rate of change of energy inside the control volume due to transient variations of the flowfield variables.
\oint\limits_S \rho\left(e+\frac{V^2}{2}\right) \boldsymbol{V} \cdot d\boldsymbol{S} 	Net rate of flow of energy across the control surface.
:Ordinarily in aeronautical applications, the viscous work is neglected, except in boundary layers at high speeds. Also, shaft work is neglected, unless the CV encompasses a compressor or turbine. Further, most problems in practice are steady-state and with a CV having fixed boundaries, or fixed geometry. Therefore the surface integrals are non-zero over specific parts of the boundary. Thus, the steady flow, steady-state energy conservation law for control volumes is:
:go into chapter 3 for this equation. (3.9ish)
==Application of the Momentum Equation to a Jet Propulsion Engine==
:We summarize the momentum equation results for a jet engine.
:see Figure 2.7 with streamtubes for inlet instead
:insert thrust equation derivations...
:The external applied force balances the thrust produced by the engine. The thrust force generated by the engine is given by:
:T=-\left(\dot m_e V_e + \left(p_e-p_0\right)A_e\right)\hat i + \left(\dot m_1 V_1 + \left(p_1-p_0\right)A_1\right)\hat i
:Thus, thrust is usually in the direction of motion. The resisting force, F, is then:
:F = -T
:In free, steady flight, the drag force opposes the thrust
:F = D = -T
:In practice, most people just calculate the scalar value of T:
:T=-\left(\dot m_e V_e + \left(p_e-p_0\right)A_e\right) + \left(\dot m_1 V_1 + \left(p_1-p_0\right)A_1\right)
:Note that the drawing of the engine showed a stream tube different from P0. If P1 ≠ P0, then the flow into the inlet must either diffuse or slow down, P1 > P0, or accelerate, P1 < P0.
:The stream tubes shapes are shown in the sketches below:
:(insert stream tube pictures)
===Gross Thrust of a Bare Engine===
:Compare gross thrust uninstalled to installed.
:The additive drag is a force equal to the integrated effects of the pressure field created by the engine and part of the wing that disturbs the flow stream. The disturbance flow field in front of the engine may also, in part, be attributed to the pylon supporting the engine (and the wing), although some of these disturbances are lumped into vehicle drag called "interference drag."