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On the sensitivity of jets and shearlayers
orthe solution of the Orr-Sommerfeld equation by the compound matrix
method (CMM)
György Paál, Péter Tamás Nagy
Motivation 1.
On the sensitivity of jets and shear layers
Determine the stability properties of self-similar flows
Jets Boundary layers
Motivation 2.
On the sensitivity of jets and shear layers
Applications
Modelling self-excited flowsEdge tone Cavity tone
Organ pipes, flueinstrumentsY branches inpipes
car door gapspantograph of trainsweapon bays onaircrafts
Analytical approximation of the jet velocity profile
On the sensitivity of jets and shear layers
𝑈 𝑦 = sech2 𝑦𝑛
𝑛=1: - the self-similar profile in the far field- non-dimensional Bickley profile
Plane jet
The perturbed NS equations
On the sensitivity of jets and shear layers
𝛻 ∙ 𝑢𝑡 = 0
The non-dimensional Navier-Stokes equations(conservation of mass and momentum):
𝜕𝑢𝑡
𝜕𝑡+ 𝑢𝑡 ∙ 𝛻𝑢𝑡 = −𝛻𝑝𝑡 +
1
𝑅𝑒∆ 𝑢𝑡
Apply perturbation , 𝑢𝑝, 𝑝𝑝
𝑢𝑡 = 𝑈 + 𝑢𝑝 𝑝𝑡 = 𝑃 + 𝑝𝑝
If U, P are known, time-independent variables, describe the base flow, fulfil the governing equations. up = (up, vp, wp)
If U(y) is independent of x (parallel flow, or self-similar flow), theperturbation can be assumed in a complex wave form:
𝑣𝑝 = 𝑣(𝑦) 𝑒𝑖(α𝑥−ω𝑡) α: wavenumber, ω: angular frequency
Temporal and spatial instability
𝜕𝑈
𝜕𝑥= 0, 𝑉 = 0
On the sensitivity of jets and shear layers
𝜔 = 5𝛼 = 5
𝑣𝑝 = 𝑣(𝑦) 𝑒𝑖(α𝑥−ω𝑡)
𝜔 = 5 + 0.3𝑖𝛼 = 5𝜇𝑡 = 𝐼𝑚 ω
ω = 5𝛼 = 5 − 0.3𝑖𝜇𝑥 = −𝐼𝑚 𝛼
Here, spatial stability investigation: 𝜇 = 𝜇𝑥 = −𝐼𝑚 𝛼
𝜇 : growth rate
The Orr-Sommerfeld equation
On the sensitivity of jets and shear layers
Substituting the perturbation into the NS equation, linearising and simplifying, the equation system can be expressed in terms of v, leading to a fourth order linearordinary differential equation:
𝑈 𝑦 ,𝑈′′(𝑦) describe the flow (known)𝑅𝑒 is a parameter (known)𝑣 𝑦 is the eigenfunction (unknown)α,ω are the eigenvalue pairs (one is a parameter, the otherone is unknown)
𝑣𝑖𝑣 − 2𝛼2𝑣′′ + 𝛼4𝑣 = 𝑖 𝑅𝑒 [ 𝛼𝑈 − 𝜔 𝑣′′ − 𝛼2𝑣 − 𝛼𝑈′′𝑣]
𝑣𝑝 = 𝑣(𝑦) 𝑒𝑖(α𝑥−ω𝑡)
The OS equation in the far field
On the sensitivity of jets and shear layers
Fal
The Orr-Sommerfeld equation:
𝑣𝑖𝑣 − 2𝛼2𝑣′′ + 𝛼4𝑣 = i 𝑅𝑒 [ 𝛼𝑈 − 𝜔 𝑣′′ − 𝛼2𝑣 − 𝛼𝑈′′𝑣]
In the far field (𝑦 → ∞) it becomes a constant coefficient linear
differential equation because 𝑈(𝑦) is a constant:
𝑈∞ = lim𝑦→∞
𝑈(𝑦) , lim𝑦→∞
𝑈′′ 𝑦 = 0
𝑣𝑖𝑣 − 2𝛼2 𝑣′′ + 𝛼4 𝑣 = 𝑖 𝑅𝑒 [ 𝛼𝑈∞ − 𝜔 𝑣′′ − 𝛼2 𝑣 ]
Analytical solution in the far-field:
𝑣 y =
𝑖=1
4
෨𝑘𝑖 𝑣𝑖𝑦 =
𝑖=1
4
෨𝑘𝑖 𝑒𝜆𝑖𝑦
𝜆 = − 𝛼,−𝑄, 𝛼, 𝑄 T
𝑄:= 𝛼2 − i𝑅𝑒(𝛼𝑈∞ − 𝜔)
The OS equation in the far field 2.
On the sensitivity of jets and shear layers
Analytical solution in the far field:
If 𝑅𝑒 ≫ 1 then 𝑄 ≫ 𝛼 the differential equation becomes stiff.
Special method: the Compound Matrix Method
𝑣 y =
𝑖=1
4
෨𝑘𝑖 𝑣𝑖𝑦 =
𝑖=1
4
෨𝑘𝑖 𝑒𝜆𝑖𝑦
𝜆 = − 𝛼,−𝑄, 𝛼, 𝑄 T
𝑄:= 𝛼2 − i𝑅𝑒(𝛼𝑈∞ − 𝜔)
Transformation of variables
On the sensitivity of jets and shear layers
A compound matrix method alkalmazása az Orr-Sommerfeld egyenletre
In the far field the perturbation vanishes 𝑣 𝑦 → ∞ = 𝑣′ 𝑦 → ∞ = 0
The „true” solution reduces to
In the far field: 𝑣𝑖𝑦 → ∞ = 𝑣
𝑖𝑦
Let us transcribe the 4th order diff. eq. into a first order diff. eq.
system:
𝜙 = [𝑣, 𝑣′, 𝑣′′, 𝑣′′′]
𝜆 = − 𝛼,−𝑄, 𝛼, 𝑄 T ෨𝑘3 = ෨𝑘4 = 0𝑣 y =
𝑖=1
4
෨𝑘𝑖 𝑒𝜆𝑖𝑦
𝑣 𝑦 =
𝑖=1
2
𝑘𝑖 𝑣𝑖𝑦
Transformation of variables 2.
On the sensitivity of jets and shear layers
A compound matrix method alkalmazása az Orr-Sommerfeld egyenletre
Use the so-called compound matrix variables:
𝜂1 = det𝜙1(1)
𝜙2(1)
𝜙1(2)
𝜙2(2)
= 𝜙1(1)𝜙2(2)
− 𝜙1(2)
𝜙2(1)
= 𝑣1𝑣′
2− 𝑣
2𝑣′
1…
𝜂2 = det𝜙1(1)
𝜙3(1)
𝜙1(2)
𝜙3(2)
, 𝜂3 = det𝜙1(1)
𝜙4(1)
𝜙1(2)
𝜙4(2)
, 𝜂4 = det𝜙2(1)
𝜙3(1)
𝜙2(2)
𝜙3(2)
, …, 𝜂6
In the far field:
The problem of stiffness is solved! Furthermore, by introducing the 𝜂 =
𝜂
− 𝑄+𝛼 𝑒− 𝛼+𝑄 𝑦 variable, non-homogenous bc.s can be prescribed at „infinity”
𝑣1
𝑦 ~𝑒−𝛼𝑦
𝑣2
𝑦 ~𝑒−𝑄𝑦
𝜂1 = 𝑣1𝑣′
2− 𝑣
2𝑣′
1~ −(𝑄 + 𝛼) 𝑒− 𝛼+𝑄 𝑦
𝜂2 ~ − (𝑄2 − 𝛼2)𝑒− 𝛼+𝑄 𝑦, …, 𝜂6 ~ 𝑒− 𝛼+𝑄 𝑦
𝜙 = 𝑣, 𝑣′, 𝑣′′, 𝑣′′′ = [𝜙1, 𝜙2, 𝜙3, 𝜙4]
Boundary conditions in the near field
On the sensitivity of jets and shear layers
The far field boundary conditions (𝑣 𝑦 → ∞ = 𝑣′ 𝑦 → ∞ = 0) are
fulfilled after transformation. Two other boundary conditions are
necessary. Near field boundary conditions
Symmetric profile 𝑣′(0) = 𝑣′′′(0) = 0 → 𝜂5 0 = 0
Antisymmetric profile 𝑣(0) = 𝑣′′(0) = 0 → 𝜂2 0 = 0
Wall 𝑣(0) = 𝑣′(0) = 0 → 𝜂1 0 = 0
Jet
Shooting method
On the sensitivity of jets and shear layers
𝑦 = 0𝑦 = 𝐿
Integration (�ු�, 𝜔) Error
Desired BC
(𝛼, 𝜔)
The boundary value problem should be solved as an initial value
problem. The solution can be initialised at a point (𝑦0 = L), where the
velocity is approximately constant 𝑈 𝑦 → ∞ ≈ 𝑈(𝐿)
The integration should start from there towards 𝑦 = 0.
The 𝛼,𝜔 parameters should be tuned to fulfil the boundary conditions
at y=0 and find the correct missing initial condition at y = L.
Thumb curve
On the sensitivity of jets and shear layers
𝜇 = 𝐼𝑚(𝛼)
The curves show constant αi contours. The solid curveseparates the stable and the unstable regions. The criticalReynolds number can be read from the curve.
Recrit
Jet: thumb curves
On the sensitivity of jets and shear layers
𝑈 𝑦 = sech2 𝑦𝑛
Bickley profile: 𝑛 = 1
Symmetric: Antisymmetric:
Results: sim.–antisym.
On the sensitivity of jets and shear layers
Thomas, F. O., Goldschmidt, V. W., 1985. The
possibility of a resonance mechanism in the
developing two-dimensional jet. Phys. Fluids 28
SYMMETRIC
ANTISYMETRIC
SYMMETRIC
Mode
SYMMETRIC
Mode
ANTISYM.
Mode
Numerical velocity profiles I.
On the sensitivity of jets and shear layers
The analytical profiles do not describe
• The planar jet with parabolic velocity profile at the orifice
• The effect of the nozzle wall thickness
• The transition of the velocity profiles. The parameter 𝑛 must be integer.
Calculate the base flow with CFD software. (Computational
Fluid Dynamics = numerical flow simulation)
Numerical velocity profiles II.
On the sensitivity of jets and shear layers
Prescribed
velocity profile
Long, parallel nozzle:
parabolic
Short, convergent
nozzle:
„top hat” profile
Wall/opening
Opening
𝑈 𝑦 Matlab
Numerical velocity profiles III.
On the sensitivity of planar jets
One result
On the sensitivity of jets and shear layers
|𝑢𝑡| = |𝑈 + 𝑢𝑝|
𝑈
|𝑢𝑝|
Growth rates for various velocity profiles
On the sensitivity of jets and shear layers
Parabolic „Top hat”
Wall BC, and symmetric perturbation
Symmetric-antisymmetric
On the sensitivity of jets and shear layers
Opening Wall
𝑥/𝛿 = 0.1
Boundary condition around the orifice
”Strange” mode
On the sensitivity of jets and shear layers
𝜇
ω = 0.1
On the sensitivity of jets and shear layers
Why is the jet more sensitive at the exit?
There are three reasons for the increased sensitivity of laminar,
plane jets close to the orifice.
1., the steeper velocity profile amplifies the disturbances at a
higher growth rate.
2., the local length scale is smaller close to the orifice, meaning
that the dimensional growth rate of disturbances is larger than
that in downstream region.
3., the disturbances excited upstream have more space to grow
exponentially.
Application for boundary layers: motivation
On the sensitivity of jets and shear layers
Aim: drag reduction
Examples in nature:
Laminar-turbulent transition I.
On the sensitivity of jets and shear layers
μ < 0 μ > 0
Laminar-turbulent transition II.
On the sensitivity of jets and shear layers
Forrás: YouTube
|𝑢𝑡|
𝑈
|𝑢𝑝|
Simple controller I.
On the sensitivity of jets and shear layers
𝑢(𝑡) = ǁ𝑐𝑝τ(𝑡 − Δ𝑡) 𝑢𝑝
|𝑢𝑡|
𝑈
|𝑢𝑝|
𝑐𝑝 = 0.5, Δ𝑡 = 0
Simple controller II.
On the sensitivity of jets and shear layers
𝑢(𝑡) = ǁ𝑐𝑝τ(𝑡 − Δ𝑡)
Orr-Sommerfeld equation
Reynolds-Orr equation
Stable Unstable
Stable
Unstable
Unstable
Passive coating
On the sensitivity of jets and shear layers
Rigid wall
Coated wall
Material: silicone rubber
𝑙1 = 25.5 𝜇𝑚
𝑙2 = 8653 𝜇𝑚
ℎ1 = 168 𝜇𝑚
ℎ2 = 14.5 𝜇𝑚
𝑤1 = 10 𝜇𝑚
𝑤2 = 33 𝜇𝑚
Thank you for your attention!
On the sensitivity of jets and shear layers