Transcript Governing Equations - Florida Institute of Technology

MAE 5130: VISCOUS FLOWS
Momentum Equation: The Navier-Stokes Equations, Part 1
September 7, 2010
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
1
DEVELOPMENT OF N/S EQUATIONS: ACCELERATION
 
ma  F
•
•

•


 F  
a   f  f body  f surface  f external
V

 DV
a
Dt



DV 

 f body  f surface  f external
Dt


f body  g

 
DV

 g  f surface
Dt
•
•
•
•
Momentum equation, Newton’s second law
System is fluid particle so convenient to divide by
volume, V, of particle so work with density, 
Concerned with:
– Body forces
• Gravity
• Applied electromagnetic potential
– Surface forces
• Friction (shear, drag)
• Pressure
– External forces
Eulerian description of acceleration
Substitution in to momentum
– Forces are per unit volume
Recall that body forces apply to entire mass of fluid
element
– Gravitational body force, g
Now ready to develop detailed expressions for surface
forces (and how they related to strain, which are
related to velocity derivatives)
2
t ij
Stress on a face normal to i axis
Stress acting in j direction
SURFACE FORCES
•
•
•
Surface forces are those applied by external stresses on the side of the element
Quantity tij is a tensor (just as strain rate eij)
Pay attention to sign convention for stress components
3
FORCES ON FRONT FACES
dFx  t xx dydz  t yx dxdz  t zx dxdy
dFy  t xy dydz  t yy dxdz  t xy dxdy
dFz  t xz dydz  t yz dxdz  t zz dxdy
t xx t xy t xz 


t ij  t yx t yy t yz 
t

t
t
zy
zz 
 zx
t ij  t ji
 e xx

e ij   e yx
e
 zx
e xy e xz 

e yy e yz 
e zy e zz 
• Force in x-direction due to stress
• Force in y-direction due to stress
• Force in z-direction due to stress
• Stress can also be written as symmetric
tensor
• Written in this way to keep analogy with
strain rate tensor, eij, because stress tensor is
also symmetric
• Symmetry is required to satisfy equilibrium
of moments about three axes of element
• Rows of tensor correspond to applied force
in each coordinate direction
4
SYMMETRIC STRESS TENSOR
• Stress tensor
– Viscous Flows, 3rd Edition,
by F. White
• Stress tensor
– Fluid Mechanics, 3rd
Edition, by F. White
• Recall tij
– i: Stress on a face normal to
i axis
– j: Stress acting in j
direction
5
EXPRESSION OF STRESS FORCES
dFx  t xx dydz  t yx dxdz  t zx dxdy
•
•
If element in equilibrium, this forces balanced by equal and
opposite force on back face of element
If accelerating, front and back face stresses will be different
by differential amounts
 
t yx 

 

t xx 
t zx 


t

dx
dydz

t
dydz

t

dy
dxdz

t
dxdz

t

dz
dxdy

t
dxdy






xx
yx
zx
 xx x
  yx y
 zx z





 


 
 t yx 
 t

 t

dFx ,net   xx dx dydz  
dy dxdz   zx dz dxdy
 x

 z

 y

•
fx 
t xx t yx t zx


x
y
z

t ij
f surface   t ij 
x j


DV

 g   t ij
Dt
•
•
•
•
•
•
Net force in the x-direction
– Compare this with conservation of mass derivation
Put force on per unit volume basis (divide by dxdydz)
Force per unit volume in x-direction is equivalent to taking
the divergence of the vector (txx, txy, txz), which is the upper
row of the stress tensor (shown in previous slide)
Total vector surface force
Divergence of a tensor is a vector
Newton’s second law
All that remains is to express tij in terms of velocity
– Assume viscous deformation-rate law between tij and eij
6
FLUID AT REST: HYDROSTATICS


DV

 g   t ij
Dt

V 0
t ij  0 for i  j
t xx  t yy  t zz   p
• Newton’s second law of motion
• Fluid at rest
– Velocity = 0
– Viscous shear stresses = 0
• Normal stresses become equal to the
hydrostatic pressure

 p  g
7
HYDROSTATICS EXAMPLE
• Depths to which submarines can dive are limited by the strengths of their hulls
• Collapse depth, popularly called crush depth, is submerged depth at which a
submarine's hull will collapse due to surrounding water pressure
• Seawolf class submarines estimated to have a collapse depth of 2400 feet (732 m),
what is pressure at this depth?
• P = gh = (1025 kg/m3)(9.81 m/s2)(732 m) = 7.36x106 Pa = 73 atmospheres
• HY-100 a a yield stress of 100,000 pounds per square inch
8
TENSOR COMMENT
 T11 T12 T13 


Tij   T21 T22 T23 
T T

 31 32 T33 
• Tensors are often displayed as a matrix
• The transpose of a tensor is obtained by
interchanging the two indicies, so the
transpose of Tij is Tji
 T11 T21 T31 


T ji   T12 T22 T32 
T T

 13 23 T33 
Qij  Q ji
Rij   R ji
Tij 
1
Tij  T ji   1 Tij  T ji 
2
2

 1
 1 2 3 

 42
4
0
5


 2 1 3  2
23



 2
24
2
0
1 5
2
3 2  
  1
2  
5 1   4  2

2   2
  23
3  
  2
24
2
0
1 5
2
3 2

2 
5 1 
2 

3 

• Tensor Qij is symmetric if Qij = Qji
• Tensor is antisymmetric if it is equal to
the negative of its transpose, Rij = -Rji
• Any arbitrary tensor Tij may be
decomposed into sum of a symmetric
tensor and antisymmetric tensor
 1 0.5 
 1 2 3   1 3 2.5   0

 
 

0
2 
 4 0 5   3 0 3    1
 2 1 3   2.5 3 3    0.5  2 0 

 
 

9
INDEX NOTATION RULES AND COORDINATE ROTATION
•
•
•
Key to classifying scalars, vectors, or tensors is how their components change if the
coordinate axes are rotated to point in new directions
A scalar (temperature, density, etc.) is unchanged by rotation – it has the same value in any
coordinate system, which is a defining characteristic of a scalar
Vectors and tensors change with rotating coordinate system
10
INDEX NOTATION, VECTORS, AND TENSORS
• Index notation
– A free index occurs once and only once in each and every term in an equation
– A dummy or summation index occurs twice in a term
• Vector has a magnitude and direction that is measured with respect to a chosen
coordinate system
– Alternative description is to give three scalar components
– Not every set of 3 scalar components is a vector
• Essential extra property of a vector is its transformation properties as coordinate
system is rotated
• 3 scalar quantities vi(i=1,2,3) are scalar components of a vector if they transform
according to:
• A tensor (2nd rank) is defined as a collection of 9 scalar components that change
under rotation of axes according to:
11