Fundamentals of Aerodynamics: Lecture 7 - Fluid Kinematics and Reynolds Transport Theorem, Study notes of Aerodynamics

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AAE 3710

Fundamentals of Aerodynamics

Lecture 7 Example problems for

fluid kinematics

Reynolds Transport Theorem^ It shifts from the governing laws expressed using Acceleration Vector system concepts (consider a given mass of the fluid) to those expressed using control volume concepts (consider a given volume) Time derivative term Convective terms Because of convective terms, even the fluid is steady it still possibly has accelerations.

Streamlinedefinition

Apply conservation of mass to

control volume

-Extensive property of the system: B sys (= M mass) -Intensive property of the system: b (=1; M=M*b) -B sys

Mass conservation law

time rate of change of the system mass = 0 D

M

sys/

D

t = 0 This is applied to a system of mass. We are interested not in a specific fluid but how fluid behaves in a field or control volume.^ How to proceed? Resort to Reynolds Transport Theorem

Apply conservation of mass to control volume

time rate of change ofthe mass of the con-tents of the controlvolume net rate of mass flow through the control surface zero Sign convention Outflow: “+” Inflow: “-” Continuity eqn with afixed, nondeformingcontrol volume Mass flow rate calculation : --If density and velocity are both uniform through a section of control surface (most common expression) Normal component of velocity For general nounifo replaced by average value

Apply conservation of mass to control volume Continuity equation There are 3 control surface, i.e. inlet ,^ outlet , and the pipe wall

. Only first two are valid contributions to the integral while the pipe wall is parallel to the flowing direction so there isno flow penetrating it.

Apply conservation of mass to control volume Different forms of continuity equations: Steady flows Steady, incompressible flows Multiple inlets and outlets