mech 5810 module 5: conservation of linear...
TRANSCRIPT
MECH 5810 Module 5: Conservation of LinearMomentum
D.J. Willis
Department of Mechanical EngineeringUniversity of Massachusetts, Lowell
MECH 5810 Advanced Fluid DynamicsFall 2017
Outline
1 Differential Conservation of Momentum
Summation of forces = Mass × acceleration or Change in
momentum
Forces Per Unit volume
Euler Equation in s− n Coordinates
Examples
Outline
1 Differential Conservation of Momentum
Summation of forces = Mass × acceleration or Change in
momentum
Forces Per Unit volume
Euler Equation in s− n Coordinates
Examples
Equation of motion
Newton’s Law applied to a point mass/constant mass differential
element of fluid, expressed in per unit volume form:
m~aVol
= ρ~a =∑(
~FVol
)
~a is acceleration in a Lagrangian description:
~a =∂~ua
∂t
Which means, in an Eulerian Description we must use the
Material or Total Derivative to determine the acceleration:
ρ~a =D~uDt
=∂~u∂t
+ (~u · ∇)~u =∑(
~FVol
)
Outline
1 Differential Conservation of Momentum
Summation of forces = Mass × acceleration or Change in
momentum
Forces Per Unit volume
Euler Equation in s− n Coordinates
Examples
Pressure force per unit volume acting on a fluidelement
Free body diagram of a differential element
Fxpressure
Vol=
Net x−pressure force︷ ︸︸ ︷− (P2 − P1) · A
Vol
=− (P2 − P1) · dy · dz
dx · dy · dz
=− (P2 − P1)
dx
→ −∂p∂x
Fxpressure
Vol= −∂p
∂x;
Fypressure
Vol= −∂p
∂y;
Fzpressure
Vol= −∂p
∂z
Pressure force per unit volume acting on a fluidelement
What really matters is the gradient of the pressure:
~Fpressure
Vol= −∇p
Body force per unit volume acting on a fluidelement
Let’s look at gravity forces as an example:
~Fgravity = m~g = mg~iz
But we express everything per unit volume:
~Fgravity
Vol= ρg~iz
We follow similar procedures for other body forces.
Viscous forces per unit volume acting on a fluidelement
For now, we will simply write ~FviscVol
Later in the course we will return to this
General equations of motion
Equate change in momentum per unit volume with the forces per
unit volume:
ρ~a = ρD~uDt
= −∇p + ρ~g +~Fvisc
Vol
Expanded in three dimensions:
x− direction : ρ
(∂u∂t
+ u∂u∂x
+ v∂u∂y
+ w∂u∂z
)= −∂p
∂x+ ρgx +
Fxvisc
Vol
y− direction : ρ
(∂v∂t
+ u∂v∂x
+ v∂v∂y
+ w∂v∂z
)= −∂p
∂y+ ρgy +
Fyvisc
Vol
z− direction : ρ
(∂w∂t
+ u∂w∂x
+ v∂w∂y
+ w∂w∂z
)= −∂p
∂z+ ρgz +
Fzvisc
Vol
IF ~FviscVol = 0, then the above equations are the Euler Equations for an
incompressible fluid.
Outline
1 Differential Conservation of Momentum
Summation of forces = Mass × acceleration or Change in
momentum
Forces Per Unit volume
Euler Equation in s− n Coordinates
Examples
Euler Equation in s− n Coordinates
ρ~a = ρD~uDt
= −∇p + ρ~g����
+~Fvisc
Vol
Assumptions:
Steady flow (∂~u∂t = 0)Inviscid flow (~Fvisc
Vol = 0)Conservative bodyforces only(~g = −∇V(r))
(~u · ∇)~u =−∇pρ−∇V
Euler Equation in s− n Coordinates
Let’s start with the term:
(~u · ∇)~u
We know ~u is, by definition, tangent to the streamline:
~u = |u|is
(~u · ∇)~u = us∂
∂s
(|u|is
)→ Chainrule
(~u · ∇)~u = us
∂u∂s· is + u · ∂ is
∂s︸︷︷︸=?
But, we need to know what ∂∂s
(is)
is.
Euler Equation in s− n Coordinates: ∂∂s
(is)
We use the following picture to determine what ∂∂s
(is)
is:
dis = −indθ
= −indSR
∂ is∂s
=−inR
Euler Equation in s− n Coordinates
Knowing ∂ is∂s = −in
R we can continue to apply the chain rule to the
material derivative:
(~u · ∇)~u = v∂
∂s
(uis)
= u
∂u∂s· is + u · ∂ is
∂s︸︷︷︸−in
R
= u
∂u∂s
is +[−u2
Rin
](~u · ∇)~u = u
∂u∂s
is −u2
Rin
Euler Equation in s− n Coordinates: s-Direction
Each direction (is, in, il) can be examined separately:
In the is-direction: u∂u∂s︸︷︷︸
∂∂s
(u22
)is =
−1ρ
∂p∂s− ∂V
∂s︸︷︷︸V=gz
∂
∂s
[u2
2+ gz
]=−1ρ
∂p∂s
If we assume ρ = const along streamlines (not as rigorous asρ = const. everywhere), then:
∂
∂s
[v2
2+
pρ+ gz
]= 0 or
v2
2+
pρ+ gz = const.
Euler Equation in s− n Coordinates: n-direction
In the in-direction−u2
R=−1ρ
∂p∂n− ∂V
∂n︸︷︷︸V=gz
If ρ = const, then, the expression for the n-direction is:
∂
∂n(p + ρgz) =
ρu2
R
If we have gz = 0 then the expression states that the pressure
gradient in the direction normal to the streamline is inversely
proportional to the radius of curvature (R) of that streamline and
proportional to the square of the velocity. This is a very useful
result (as we shall see in shortly).
Euler Equation in s− n Coordinates: l-direction
In the il-direction
0 =−1ρ
∂p∂l− ∂V
∂l︸︷︷︸∂V∂S =gil
If ρ = const, then, in the l-direction:
∂
∂l(p + ρgz) = 0
This result simply says that any pressure gradient in the l-direction
is directly related to the body forces, and that no fluid acceleration
effects are present.
Outline
1 Differential Conservation of Momentum
Summation of forces = Mass × acceleration or Change in
momentum
Forces Per Unit volume
Euler Equation in s− n Coordinates
Examples
Bernoulli Equation & Streamline curvatureexamples
Assumptions:
Steady flow (∂~u∂t = 0)Inviscid flow (~Fvisc
Vol = 0)Conservative body forces only (~g = −∇V(r))
In the s-direction
∂
∂s
[u2
2+
pρ+ gz
]= 0 or
u2
2+
pρ+ gz = const.
In the n-direction ∂
∂n(p + ρgz) =
ρu2
R
In the l-direction ∂
∂l(p + ρgz) = 0