AERO 325 Kinematics
What is kinematics?
• Study of flows without any forces present
• Used to describe how a fluid element moves and deforms without forces acting
on the element
• Each fluid element with volume 𝛿𝑉 is sized such that 𝜆3 ≪ 𝛿𝑉 ≪ 𝐿3 where 𝜆
is the mean free path
• 2 primary frames of reference to analyze the motion of a fluid element
• Lagrangian Frame: frame of reference moves together with the fluid element
• Eulerian Frame: frame of reference is stationary w.r.t observer of the fluid flow
Relation between Lagrangian and Eulerian frames
• Relate Lagrangian and Eulerian frames through material derivatives
(also known as total or substantial derivatives)
• Rate of change of a property measured by a fluid element
• Material derivative = Eulerian derivative + convective derivative
• Eulerian derivative: rate of change measured at fixed position 𝑥Ԧ →
𝜕
𝜕𝑡
• Convective derivative: change seen by fluid element due to motion → 𝑣Ԧ ∙ ∇ =
𝜕
𝜕
𝜕
𝑢 +𝑣 +𝑤
𝜕𝑥
𝜕𝑦
𝜕𝑧
Material Derivatives
𝐷
𝜕
• Combined together: =
+ 𝑣Ԧ ∙ ∇
𝐷𝑡
𝜕𝑡
• Applied to some field quantity T:
•
𝐷𝑇
𝜕𝑇
=
+ 𝑣Ԧ ∙ ∇T
𝐷𝑡
𝜕𝑡
• As a summary:
𝐷𝑇
•
= rate of change of T measured by a moving fluid element
𝐷𝑡
𝜕𝑇
•
= rate of change seen by an observer in Eulerian frame at fixed 𝑥Ԧ
𝜕𝑡
• 𝑣Ԧ ∙ ∇T = change in T seen by fluid element due to its motion
• Material derivatives can be calculated for vector fields as well, but
then some of the terms are tensors
Flow Lines
• Used to help visualize the flow. There are 4 basic curves:
• Pathline: trace of the trajectory of a single individual fluid element (intrinsic to
Lagrangian view)
• Streamline: curve that is tangent everywhere to the velocity vector that is
drawn for one “snapshot” of the flow (intrinsic to Eulerian view)
• Streakline: curve generated by continuous release of markers from a single
point (easy to produce experimentally)
• Timeline: curve generated by markers released at the same time along a
continuous curve in space (easy to produce experimentally)
Pathlines and Streamlines
• Shape of the pathline depends on:
• Starting location of the fluid element marker
• Time of the release (if unsteady)
• Equation for the pathline 𝑥(𝑡)can
Ԧ
be obtained by
solving a differential equation using the initial
𝑑 𝑥Ԧ
condition and the given velocity field: = 𝑣(
Ԧ 𝑥,
Ԧ 𝑡)
Pathline in unsteady flow
𝑑𝑡
• In unsteady flow, streamline depends on the time
the snapshot is taken
• In steady flow, streamlines coincide with pathlines
• Equation for streamline can be solved by solving
𝑑 𝑥Ԧ
= 𝑣(
Ԧ 𝑥,
Ԧ 𝑡0 ) where s is “pseudo time” variable
𝑑𝑠
that parametrizes the positions along the streamline
that 𝑡0 < 𝑠 < 𝑡
Streamline
Streakline and Timelines
• Both experimentally generated by release of
markers (ex. Smoke, dyes, or bubbles)
• Streaklines: continuous release of markers at
different positions
• Timelines: short burst release of markers at one
position at the same time
Streakline in an
unsteady flow
• In steady flow, streaklines, streamlines, and
pathlines all coincide
• Timelines generally do not coincide with other
lines
Timeline
• https://forms.gle/psrfC3bDPfPcYDUg8
Vorticity
• Measure of how rotational a flow is
• 𝑉𝑜𝑟𝑡𝑖𝑐𝑖𝑡𝑦 = 𝜉Ԧ = ∇ × 𝑣Ԧ (curl of velocity)
• 𝜉Ԧ = 0 -> flow is irrotational
• 𝜉Ԧ != 0 -> flow is rotational
Circulation
• Contour integral of velocity
• 𝐶𝑖𝑟𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = Γ = − ‫𝑣 𝐶׬‬Ԧ ∙ 𝑑 𝑙Ԧ
• C is come closed contour where the integral is evaluated CCW
• Applying Stoke’s theorem, vorticity and circulation are related:
• Γ = − ‫𝜉 𝑆׬‬Ԧ ∙ 𝑑 𝑆Ԧ
• Circulation for a closed contour C is equal to minus the integral of
vorticity over the enclosed surface
Conservation of Mass
• Antoine Lavoisier’s discovery of mass conservation: mass
can neither be created nor destroyed
• Consider a control volume with volume V and surface S
• For this CV, mass flowrate out of the CV = time rate of
decrease of the mass inside the CV
• Mass flux through the S of the CV = 𝜌𝑣Ԧ ∙ 𝑛
• Integrate mass flux over S to obtain mass flow in and out of
the CV
• 𝑚𝑎𝑠𝑠 𝑓𝑙𝑜𝑤 𝑜𝑢𝑡 𝑜𝑓 𝑉 = ‫𝑣𝜌 𝑆׬‬Ԧ ∙ 𝑛 𝑑𝑆
𝑑
• Rate of decrease of mass in CV = ‫𝑉׬‬
𝑑𝑡
𝜌 𝑑𝑉
Conservation of Mass (a.k.a continuity equation)
𝑑
• ‫𝑉׬‬
𝑑𝑡
𝜌 𝑑𝑉 + ‫𝑣𝜌 𝑆׬‬Ԧ ∙ 𝑛 𝑑𝑆 = 0 (integral form)
• Apply divergence theorem to the surface integral
• ‫𝑣𝜌 𝑆׬‬Ԧ ∙ 𝑛 𝑑𝑆 = ‫𝑣𝜌 ∙ ∇ 𝑉׬‬Ԧ 𝑑𝑉
• Assuming CV does not change with time,
• ‫𝑉׬‬
𝜕𝜌
+ ∇ ∙ (𝜌𝑣)
Ԧ
𝜕𝑡
𝑑𝑉 = 0
𝜕𝜌
• + ∇ ∙ (𝜌𝑣)=0
Ԧ
(differential form)
𝜕𝑡
𝐷𝜌
•
+ 𝜌∇ ∙ 𝑣=0
Ԧ (total derivative form)
𝐷𝑡
Velocity Potential and Stream Function
• For special cases, the velocity field can be fully described by a scalar field
when the flow is either
1. Irrotational
2. 2D incompressible
1. Irrotational: ∇ × 𝑣Ԧ = 0
• Curl of a gradient is always 0. Thus, if 𝑣Ԧ = ∇𝜙, ∇ × 𝑣Ԧ = ∇ × ∇𝜙 = 0
• 𝜙 is the velocity potential
2. Incompressible 2D flow: density is constant
𝜕𝜌
• Continuity equation + ∇ ∙ (𝜌𝑣)
Ԧ reduces down
𝜕𝑡
• ∇ ∙ 𝑣Ԧ = 0
• Divergence of curl of a vector field is always 0. Thus, if 𝑣Ԧ = ∇ ×
𝜓𝑘෠ , where 𝜓 𝑥, 𝑦 some scalar field, ∇ ∙ 𝑣Ԧ = ∇ ∙ ∇ × 𝜓𝑘෠ = 0
• 𝜓 is the stream function
Velocity Potential and Stream Function
• In cartesian coordinates,
• 𝑢=
𝜕𝜓
𝜕𝜓
,𝑣 = −
𝜕𝑦
𝜕𝑥
• In cylindrical coordinates
• 𝑣𝑟 =
1 𝜕𝜓
𝜕𝜓
, 𝑣𝜃 = −
𝑟 𝜕𝜃
𝜕𝑟
Rotation and Deformation of Fluid Elements
• Understanding fluid element
deformation is crucial to
deriving stress-strain
relationship in dynamics
• 2 types of strain
• Normal strain: along a single axis
• Shear strain: angle change
between the pair of axes
Vorticity and strain tensors
• Velocity gradient tensor: ∇𝑣Ԧ =
𝜕𝑢
𝜕𝑥
𝜕𝑢
𝜕𝑦
𝜕𝑣
𝜕𝑥
𝜕𝑣
𝜕𝑦
(2D)
• Rotation arises from antisymmetric part of ∇𝑣Ԧ
• Deformation arises from symmetric part of ∇𝑣Ԧ
1
1
𝑇
• ∇𝑣Ԧ = ∇𝑣Ԧ − ∇𝑣Ԧ + ∇𝑣Ԧ + ∇𝑣Ԧ 𝑇
2
2
1
1
𝑇
• 𝛀 = ∇𝑣Ԧ − ∇𝑣Ԧ = Ω𝑖𝑗 = (𝜕𝑖 𝑣𝑗 − 𝜕𝑗 𝑣𝑖 ) Vorticity tensor
2
2
1
1
𝑇
• 𝝐 = ∇𝑣Ԧ + ∇𝑣Ԧ = 𝜖𝑖𝑗 = (𝜕𝑖 𝑣𝑗 + 𝜕𝑗 𝑣𝑖 ) rate-of-strain tensor
2
2
Rotation of a fluid element
1
• Angular velocity 𝜔 = ∇ × 𝑣Ԧ = 𝜔𝑖
2
0
1
• 𝛀 = 𝜕2 𝑣1 − 𝜕1 𝑣2
2
𝜕3 𝑣1 − 𝜕1 𝑣3
𝜕1 𝑣2 − 𝜕2 𝑣1
0
𝜕3 𝑣2 − 𝜕2 𝑣3
𝜕1 𝑣3 − 𝜕3 𝑣1
0
𝜕2 𝑣3 − 𝜕3 𝑣2 = −𝜔3
𝜔2
0
𝜔3
0
−𝜔1
−𝜔2
𝜔1
0
• In rigid body motion, relative velocity of 2 points Δ𝑣Ԧ separated by Δ𝑟,
Ԧ which is Δ𝑣Ԧ =
𝜔 × Δ𝑟Ԧ = Δ𝑟Ԧ ∙ 𝛀
Deformation
• Proof that 𝝐 related to deformation in the notes. Take a look at it
• In summary, for 2-D,
𝜖𝑥𝑥 𝜖𝑥𝑦
• 𝝐= 𝜖
𝑦𝑥 𝜖𝑦𝑦
1 𝑑Δx
• 𝜖𝑥𝑥 =
Δ𝑥 dt
1 𝑑Δ𝑦
• 𝜖𝑦𝑦 =
Δ𝑦 dt
• 𝜖𝑥𝑦 = 𝜖𝑦𝑥 = −
Δy
k
1 𝑑k
2 dt
Δx
• There exists principal direction of strain where shear rates are 0. In the coordinate
system aligned with this direction,
𝜖𝐼 0
0
• 𝜖 = 0 𝜖𝐼𝐼 0
0 0 𝜖𝐼𝐼𝐼
Element in pure
shear