Fundamental Principles & Equations

< 2.6. An application of the momentum equation > Drag of a 2-D body  Consider a two-dimensional body in a flow

Aerodynamics 2017 fall - 1 - Fundamental Principles & Equations

< 2.6. An application of the momentum equation > Drag of a 2-D body  Surface forces : two contributions

• The pressure distribution over the surface abhi   pnˆ dA abhi • The surface force on def created by the presence of the body

Aerodynamics 2017 fall - 2 - Fundamental Principles & Equations

< 2.6. An application of the momentum equation > Drag of a 2-D body  Resultant aerodynamic force : R'

 Because the body surface and volume surface have opposite normals n, this R' is precisely equal and opposite to all the def surface integrals for the control volume.

Aerodynamics 2017 fall - 3 - Fundamental Principles & Equations

< 2.6. An application of the momentum equation > Drag of a 2-D body  Considering the integral form of the momentum equation

d      Vdv   V nˆVdA    pnˆ dA  R dt abhi

 The right-hand side of this equation is physically the force on the fluid moving through the control volume

Aerodynamics 2017 fall - 4 - Fundamental Principles & Equations

< 2.6. An application of the momentum equation > Drag of a 2-D body  Assuming steady flow, above equation becomes    R   V nˆVdA  pnˆ dA abhi

Aerodynamics 2017 fall - 5 - Fundamental Principles & Equations

< 2.6. An application of the momentum equation > Drag of a 2-D body  The x component of R' is the aerodynamic drag D'  D   V nˆu dA   pnˆ iˆdA abhi  Because the boundaries of the control volume abhi are chosen far enough form the body, p is constant along these boundaries. So, we have

 pnˆ iˆdA  0 abhi

Aerodynamics 2017 fall - 6 - Fundamental Principles & Equations

< 2.6. An application of the momentum equation > Drag of a 2-D body  Finally, we obtain  D   V nˆu dA

Aerodynamics 2017 fall - 7 - Fundamental Principles & Equations

< 2.6. An application of the momentum equation > Drag of a 2-D body  The momentum-flux integral is zero on the top and bottom boundaries, since these are defined to be along streamlines, and hence have zero momentum flux. Only the momentum flux on the inflow and outflow planes remain.

 a b 2 2 (V nˆ)u dA   1u1 dy  2u2 dy  i h

( where dA=dy(1) )

Aerodynamics 2017 fall - 8 - Fundamental Principles & Equations

< 2.6. An application of the momentum equation > Drag of a 2-D body  Using continuity equation,

a b 1u1dy  2u2dy i h a b 2 1u1 dy  2u2u1dy i h So...  b b 2 V nˆu dA   2u2u1dy  2u2 dy  h h b   2u2 u1  u2 dy h

Aerodynamics 2017 fall - 9 - Fundamental Principles & Equations

< 2.6. An application of the momentum equation > Drag of a 2-D body  Therefore,

b D  2u2 u1 u2 dy h

 For incompressible flow, ρ=constant, equation becomes

b D   u2 u1 u2 dy h

Aerodynamics 2017 fall - 10 - Fundamental Principles & Equations

< 2.7. Energy equation > Energy conservation  Physical principle : Energy can be neither created nor destroyed; it can only change in form

 System and surroundings q w  de • δq : heat to be added to the system form the surroundings • δw : the work done on the system by the surroundings • de : the change of internal energy

Aerodynamics 2017 fall - 11 - Fundamental Principles & Equations

< 2.7. Energy equation > Energy conservation  The first law of thermodynamics

B1  B2  B3

• B1 : rate of heat added to fluid inside control volume form surroundings

• B2 : rate of work done on fluid inside control volume

• B3 : rate of change of energy of fluid as it flows through control volume

Aerodynamics 2017 fall - 12 - Fundamental Principles & Equations

< 2.7. Energy equation > Energy conservation  Rate of volumetric heating  q dv v  Heat addition to the control volume due to viscous effects  Qviscous

 Therefore,

B  q dv  Q 1  viscous v

Aerodynamics 2017 fall - 13 - Fundamental Principles & Equations

< 2.7. Energy equation > Energy conservation  Rate of work done by pressure force on S     p dS V S  Rate of work done by body forces    f dvV v  The total rate of work done on the fluid     B   pV dS   f V dv W 2     viscous S v

Aerodynamics 2017 fall - 14 - Fundamental Principles & Equations

< 2.7. Energy equation > Energy conservation  Net rate of flow of total energy across control surface    V 2   V dS e   S  2   Time rate of change of total energy inside v (control volume)   V 2   e  dv t v  2 

 In turn, B3 is the sum of above equations   V 2     V 2  B  e  dv  V dS e   3        t v  2  S  2 

Aerodynamics 2017 fall - 15 - Fundamental Principles & Equations

< 2.7. Energy equation > Energy conservation  Energy conservation equation     q dv  Q  pV dS   f V dv W  viscous     viscous v S v   V 2   V 2      e  dv   e  V dS t v  2  S  2 

Aerodynamics 2017 fall - 16 -