Transcript Chap.2 - 永達技術學院

Chap.2
Aerodynamics: Some
Fundamental Principles and
Equations
OUTLINE
 Review of vector relations
 Control volumes and fluid elements
 Continuity equation
 Momentum equation
 Pathlines and streamlines
 Angular velocity, vorticity and circulation
 Stream function and velocity potential
Review of vector relations
Vector algebra


Scalar product: A  B  A B cos
Vector product: A  B  ( A B sin )e  G
Orthogonal coordinate systems

Cartesian coordinate system
A  Ax i  Ay j  Az k
B  Bx i  B y j  Bz k
A  B  Ax Bx  Ay B y  Az Bz
i
A  B  Ax
j
Ay
k
Az
Bx
By
Bz

Cylindrical coordinate system
A  Ar e r  Ae   Az e z
B  Br e r  Be   Bz e z
A  B  Ar Br  A B  Az Bz
er
A  B  Ar
e
A
ez
Az
Br
B
Bz

Spherical coordinate
system
A  Ar e r  Ae   Ae 
B  Br e r  Be   Be 
A  B  Ar Br  A B  A B
er
A  B  Ar
Br
e
A
B
e
A
B
Gradient of a scalar field

Definition of gradient of a scalar p
 Its magnitude is the maximum rate of change of p per
unit length.
 Its direction is the maximum rate of change of p.



Isoline: a line of constant p values
Gradient line: a line along which p is tangent at
every point.
dp
Directional derivative: ds  p  n
where n is the unit vector in the s direction.

Expression for p in Cartesian coordinate system
p p
p
p 
i
j k
x
y
z
Divergence of a vector field


If V is the velocity of a flow, the divergence of V
will be the time rate of volume change per unit
volume.
Expression for divergence of V, V, in Cartesian
coordinate system
V  Vx i  Vy j  Vz k
Vx Vy Vz
V 


x
y
z
Curl of a vector field


The angular velocity  of a fluid element
translating along a streamline is equal to one-half
of the curl of V, denoted by V.
Expression for curl of V in Cartesian coordinate
system
V  Vx i  Vy j  Vz k
i

V 
x
Vx
j

y
Vy
k

z
Vz
Relations between line, surface and volume
integrals

Stokes’ theorem
 A  ds   (  A)  dS
c

s
Divergence theorem
 A  dS   (  A) dV
s

V
Gradient theorem
 pdS   pdV
s
V
Control volumes and fluid elements
Control volume
approach
Fluid element
approach
Continuity equation
Fixed control volume
Mass flow equation
  Vn A
m
Continuity equation
in a finite space

dV   V  dS  0

t V
S
Continuity equation
at a point

   ( V)  0
t
Momentum equation
Fixed control volume
Original form is Newton’s second law
Momentum equation in integral form

VdV   ( V  dS )V    pdS   fdV  Fviscous

t V
S
S
V
f is body force; Fviscous is viscous force on control surface
X-component of the momentum equation in
differential form (similar form for y- and zcomponent).
( u)
p
   ( uV)    f x  ( Fx )viscous
t
x
Navier-Stokes equations

The momentum equations for a viscous flow.
Euler equations
The momentum equations for a steady inviscid
flow.
p
  ( uV )  

x
p
  ( vV )  
y
p
  ( wV )  
z
Pathlines and streamlines
Pathline

Path of a fluid element.
Streamline

A curve whose tangent
at any point is in the
direction of the velocity
vector at that point.
For steady flow,
pathlines and
streamlines are
identical.
Streamline equation for steady flow

By definition, flow velocity V is parallel to
directed segment of the streamline ds, so
dsxV=0
i
j k
ds  V  dx dy dz  0
u
v
w
wdy  vdz  0
udz  wdx  0
vdx  udy  0

For two-dimensional flow
dy v

dx u
Angular velocity, vorticity and circulation
Angular velocity and vorticity


As a fluid element translate along a streamline, it
may rotate as well as shape distorted.
Angular velocity 
1  w v   u w   v u  
ω  
 i   
 j    k 
2  y z   z x   x y  



Vorticity  is defined to be 2, also equal to xV.
If xV≠0, the flow is rotational, and ≠0.
If xV=0, the flow is irrotational, and =0.
Circulation Γ



Definition
  C V  ds
Relation with lift: if an
airfoil is generating lift, the
circulation taken around a
closed curve enclosing the
airfoil will be finite.
By Stokes’ theorem
  C V  ds    (  V)  dS
S

If the flow is irrotational (xV=0) everywhere
with the contour of integration, then Γ= 0.
Stream function and velocity potential
Stream function

For two-dimensional steady flow, a streamline
equation is given by setting the stream function
equal to a contant.
 ( x, y )  c

For incompressible flow
  


u
y
,v  
x
Velocity potential


For an irrotational flow
  V  0    ( )  0
We can find a scalar function φ such that V is
given by the gradient of φ which is therefore
called velocity potential.
V  



u
,v 
,w 
x
y
z
Relation between  and φ

Equipotential lines (φ= constant) and streamlines
( = constant) are mutually prependicular.