View
3
Download
0
Category
Preview:
Citation preview
MECH 5810 Module 4: Differential FluidDeformation & Conservation of Mass
D.J. Willis
Department of Mechanical EngineeringUniversity of Massachusetts, Lowell
MECH 5810 Advanced Fluid DynamicsFall 2017
Outline
1 Kinematics and Mathematics
Operator Basics
Aside: Gauss’s Theorem
Differential Conservation of Mass
Aside: Stokes Theorem (Optional in module 4)
Kinematics
We would like to examine differential elements of fluid:
How can we mathematically describe the motion for the fluid?How does a fluid element deform due to the velocity field?Can we simplify the deformation into consistent sub-deformations?
Kinematics: Linear Strain Rate – ε
Similar idea to solid mechanics - normalized change in length
The difference is strain rate – continues to deform
Linear strain rate: represents the normalized rate of change in
length of a fluid element with respect to time:
Strain Rate =1lo
D(l)Dt
=∂u∂x
So, how do we get ∂u∂x ?
Kinematics : Linear Strain Rate
Strain Rate =
(1lo
D(l)Dt
)'(
1lo
∆l∆t
)=
(1lo
(u + ∂u
∂x lo)
∆t − (u) ∆t∆t
)
Strain Rate = εxx =
(1
��lo
(�u + ∂u
∂x��lo)��∆t − (�u)��∆t
��∆t
)=
(∂u∂x
)
εxx, εyy, εzz: Linear strains in x, y, z
Kinematics : Volumetric/Bulk Strain Rate
Volumetric strain rate = Rate of change of the volume per unit
volume
The infinitesimal volume δV = δxδyδz.
Volumetric Strain Rate =1δV
DDt
(δV) =1
δxδyδzDDt
(δxδyδz)
=1δx
DDt
(δx) +1δy
DDt
(δy) +1δz
DDt
(δz)
Simplifying, we get:
Volumetric Strain Rate =∂u∂x
+∂v∂y
+∂w∂z
=∂ui
∂xi︸︷︷︸indicial notation
Which can be represented using the divergence operator:
Volumetric Strain Rate = ∇ ·~u =∂u∂x
+∂v∂y
+∂w∂z
εxx, εyy, εzz: Linear strains in x, y, z
Vector Calculus : Gradient – grad(p) = ∇p
The gradient operator provides the path of steepest descent
Sidenote: Fluids will tend to follow minimum energy paths
grad(p) = ∇(p) =
∂p∂x∂p∂y∂p∂z
www.mathworks.com
Grad-dot-u: volumetric strain rate→incompressible conservation of mass
Outline
1 Kinematics and Mathematics
Operator Basics
Aside: Gauss’s Theorem
Differential Conservation of Mass
Aside: Stokes Theorem (Optional in module 4)
Aside 1: Gauss’s Theorem
Gauss’s Theorem or Divergence Theorem∫∫∫V∇ ·~udV =
∫∫S~u · n̂dS
Or, more generally:∫∫∫V∇ · ~QdV =
∫∫S
~Q · n̂dS
States that the volume integral of the divergence of a quantity (~Q)
is equal to the surface flux of that quantity.
This can be loosely translated as: The divergence of a quantity is
similar to the rate of change of a volume quantity with respect to
time.
Concept/Illustration of Divergence Theorem
Illustration of Divergence Theorem
A conceptual view of the divergence theorem:
Consider a volume V to be made up of many elemental volumes
dV
The divergence is, by definition, the normalized rate of change of
a volume
Hence, if the divergence is positive (volume is increasing), and the
boundary of our volume is fixed, then, there must be outflux of
fluid equal to the rate of change of the volume.
Outline
1 Kinematics and Mathematics
Operator Basics
Aside: Gauss’s Theorem
Differential Conservation of Mass
Aside: Stokes Theorem (Optional in module 4)
Differential conservation of mass
Consider a small fluid element dV = dxdydz fixed in space
Differential conservation of mass
Apply the integral conservation of mass equation to this small
volume:
0 =ddt
∫∫∫CV(t)
ρdV +
∫∫CS(t)
ρurndA
Applying divergence theorem to the second integral
0 =ddt
∫∫∫CV(t)
ρdV +
∫∫∫CV(t)
∇ · (ρ~u)dV︸ ︷︷ ︸∫∫CS(t) ρ~urndA
Differential conservation of mass
Since the differential control volume is fixed in space, we can bring
the ddt term inside the integral:
0 =
∫∫∫CV(t)
[∂
∂tρ+∇ · (ρ~u)
]dV
We also know that conservation of mass must apply at all points in
the fluid, as a result, we know that the integrand must be zero
everywhere (not just at select points):
0 =∂ (ρ)
∂t+∇ · (ρ~u)
This is the differential form of the conservation of mass statement
and it must hold true at all locations in a flowfield.
Differential conservation of mass
Means that (1) Local change in density w.r.t. time must be
balanced by the (2) divergence of the product of the density and
velocity:∂ρ
∂t︸︷︷︸1
+∇ · (ρ~u)︸ ︷︷ ︸2
= 0
Differential conservation of mass
For a constant density flow, ρ = const., this conservation of mass
statement becomes:
∇ · (~u) = 0
Does this make sense? The divergence of a constant density fluid
is zero.
Differential conservation of mass
Let’s try some examples...
Outline
1 Kinematics and Mathematics
Operator Basics
Aside: Gauss’s Theorem
Differential Conservation of Mass
Aside: Stokes Theorem (Optional in module 4)
Kinematics : Shear Strain Rate
Shear strain rate: Rate of decrease of the angle formed by two
mutually perpendicular lines on the element
The shear strain rate value depends on the orientation of the line
pair
εij =dα+ dβ
dt
Kinematics : Shear Strain RateLet’s look at dα first:
dα = atan(OA
) ' OA
' ∆llo
'∂u∂y · lo · dt
lo
' ∂u∂y
dt
Kinematics : Shear Strain Rate
So:dαdt
=∂u∂y
Similarly, it can be shown that:
dβdt
=∂v∂x
εxy =dα+ dβ
dt=∂u∂y
+∂v∂x
Kinematics : Strain Rate Tensor
εij =12
(∂ui
∂xj+∂uj
∂xi
)
∂u∂x
12
(∂u∂y + ∂v
∂x
)12
(∂u∂z + ∂w
∂x
)12
(∂v∂x + ∂u
∂y
)∂v∂y
12
(∂v∂z + ∂w
∂y
)12
(∂w∂x + ∂u
∂z
) 12
(∂w∂y + ∂v
∂z
)∂w∂z
The diagonal is the normal strain rate terms (εii)
The off-diagonals are half the shear strain rate terms
εij = εji
Kinematics : Vorticity and Curl - curl(~u) = ∇× ~u
Rotation or angular velocity is an important component of
kinematics as we shall see
We define vorticity to be twice the angular velocity of the fluid
element
Kinematics : Vorticity and Curl - curl(~u) = ∇× ~u
The curl of a vector indicates whether there is rotationality in the vector
field.
curl(~u) = ∇×~u =
∂∂x∂∂y∂∂z
×
u
v
w
=
∂w∂y −
∂v∂z
∂u∂z −
∂w∂x
∂v∂x −
∂u∂y
=
ωx
ωy
ωz
= ~ω
Pure rotation of an element:
Kinematics : Summary of Possible Fluid Motions
Vorticity, Rotation Rate
Shear Strain Rate
Linear Strain Rate
Volumetric Strain Rate
Aside 2: Stoke’s Theorem
∫∫A∇×~udA =
∮l~u · ~dl
If we put to use our latest kinematics knowledge, ω = ∇×~u, then:∫∫Vω︸︷︷︸∇×~u
dS =
∮l~u · ~dl = Γ︸︷︷︸
Circulation
Essentially a statement of the circulation theorem
The ~dl is an infinitesimal length of the boundary
The∮
represents the path integral
Recommended