AAE 3710
Fundamentals of Aerodynamics
Lecture 9 Finite control volume
analysis + Differential analysis
02/13/2006
Applying first law of thermodynamics to control
volume
Governing eqns based on
system method
(Normally differential form)
Reynolds Transport Theorem
Control volume formulas
(Normally integral form)
Physical laws that will be considered:
1. Conservation of Mass
1.1 fixed, nondeforming C.V.
1.2 moving, nondeforming C.V.
1.3 deforming C.V
2. Newton’s 2nd law
linear momentum equation
3. First law of thermodynamics
Applying first law of thermodynamics to control
volume
The first law of thermodynamics for a system in symbols
or
Sign convention:
Energy in +
Energy out Reynolds transport theorem
Fixed, nondeforming C.V.
Applying first law of thermodynamics to control volume
Energy exchanged between the control volume contents and surroundings because of a
temperature difference
Adiabatic flow
Shaft work
Normal stress work
Normally negligibly small
Shear stress work
steady
Uniform across control surfaces
One dimensional energy equation for steady-in-the-mean flow
Zero elevation
Internal energy rise and
Pressure rist
Could be found by flow rate
Zero: adiabatic
Energy equation in relation to Bernoulli Equation
If no shaft work
One dimensional
flow of a single
stream between two
sections
Bernoulli Equation
0
For steady incomressible inviscid flows, as requried by
Bernoulli equation.
For steady incomressible viscous flows
Useful or available energy
Chapter 6 Differential Analysis of Fluid Flow
Finite control volume
Analysis(chap. 5)
Differential analysis(chap. 6)
Not focused on details of fluid;
Consider a finite C.V. and
control surfaces
Integral equations
Fluid Element Kinematics
Focused on details of fluid;
Consider a point or an
Infinitesimal C.V.
Differential equations
Differential Analysis of Fluid Flow
Translation:
Every point in the element have the same vleocity
Linear deformation:
The rate of change of the volume per unit volume
Volumetric dilation rate
Causing linear deformation
Cross derivative causing angular deformation
Differential Analysis of Fluid Flow
Angular Motion
Angular velocity of line OA
z
Angular velocity of line OB
z
-
+
Differential Analysis of Fluid Flow
Vorticity
A vector that twice the rotation vector
Irrotational flow field
Differential Analysis of Fluid Flow
Angular deformation:
Rate of shearing strain (rate of angular deformation)
Differential representation of Mass Conservation Law
Applied onto an infinitesimal C.V.
Differential Analysis of Fluid Flow
Mass conservation
For differential
volume,
Only density
variation is
Considered.
Mass flow rate in and
out of the differential
volume
Continuity equation
Differential Analysis of Fluid Flow
Continuity equation
Steady flow
Incompressible flow
Differential Analysis of Fluid Flow
Continuity equation in cylindrical coordinate system
Stream function
Consider a 2 dimensional, steady, incompressible plane flow, its continuity equation is
Stream function(x,y) is defined such that It is always the solution to the above PDE
Lines along which the stream function is constant are streamlines
Change in the value of is related to the volume flow rate between two adjacent
streamlines
Differential Analysis of Fluid Flow
Stream function for cylindrical coordinates