058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 1 Chapter 6: Viscous Flow in Ducts 6.2 Stability and Transition Stability: can a physical state withstand a disturbance and still return to its original state. In fluid mechanics, there are two problems of particular interest: change in flow conditions resulting in (1) transition from one to another laminar flow; and (2) transition from laminar to turbulent flow. (1) Transition from one to another laminar flow (a) Thermal instability: Bernard Problem A layer of fluid heated from below is top heavy, but only undergoes convective “cellular” motion for Raleigh #: 4 2 / cr g d g d Ra Ra wd k α α υ ν Γ Γ = = > force viscous force bouyancy α = coefficient of thermal expansion = P T ∂ ∂ − ρ ρ 1 / dT T d dz Γ=∆ =− ( ) T ∆ − = α ρ ρ 1 0 d = depth of layer k, ν =thermal, viscous diffusivities
Transcript
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 1
Chapter 6: Viscous Flow in Ducts 6.2 Stability and Transition Stability: can a physical state withstand a disturbance and still return to its original state. In fluid mechanics, there are two problems of particular interest: change in flow conditions resulting in (1) transition from one to another laminar flow; and (2) transition from laminar to turbulent flow. (1) Transition from one to another laminar flow
(a) Thermal instability: Bernard Problem
A layer of fluid heated from below is top heavy, but only undergoes convective “cellular” motion for
Raleigh #: 4
2/ crg d g dRa Raw d kα α
υ νΓ Γ
= = > forceviscousforcebouyancy
α = coefficient of thermal expansion =PT
∂∂
−ρ
ρ1
/ dTT d dzΓ = ∆ = − ( )T∆−= αρρ 10 d = depth of layer k, ν =thermal, viscous diffusivities
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 2
w=velocity scale: convection (wΓ) = diffusion (kΓ/d) from energy equation, i.e., w=k/d
Solution for two rigid plates: Racr = 1708 for progressive wave disturbance αcr/d = 3.12 ^ ( ) [cos( ) sin( )]
c ti x ct iw we e x ct x ctα −= = − + − λcr = 2π/α = 2d ^ ( )i x ctT T e α −= α = αr c=cr + ici
αr = 2π/λ=wavenumber cr = wave speed ci : > 0 unstable = 0 neutral < 0 stable Thumb curve: stable for low Ra < 1708 and very long or short λ.
For temporal stability
Ra > 5 x 104 transition to turbulent flow
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(b) finger/oscillatory instability: hot/salty over
cold/fresh water and vise versa.
(Rs – Ra)cr = 657 ( )
)1(/
0
4
STkdzdsdgRs S
∆+∆−==
βαρρνβ
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 4
(c) Centrifugal instability: Taylor Problem
Bernard Instability: buoyant force > viscous force Taylor Instability: Couette flow between two rotating cylinders where centrifugal force (outward
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 5
from center opposed to centripetal force) > viscous force.
Square counter rotating vortex pairs with helix streamlines
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 6
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 7
(d) Gortler Vortices
Longitudinal vortices in concave curved wall boundary layer induced by centrifugal force and related to swirling flow in curved pipe or channel induced by radial pressure gradient and discussed later with regard to minor losses.
For δ/R > .02~.1 and Reδ = Uδ/υ > 5
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(e) Kelvin-Helmholtz instability
Instability at interface between two horizontal parallel streams of different density and velocity with heavier fluid on bottom, or more generally ρ=constant and U = continuous (i.e. shear layer instability e.g. as per flow separation). Former case, viscous force overcomes stabilizing density stratification.
( ) ( )22 2
2 1 1 2 1 2 0ig U U cρ ρ αρ ρ− < − → > (unstable)
21 UU ≠ large α i.e. short λ always unstable Vortex Sheet 1 2 1 2
1 ( )2ic U Uρ ρ= → = + > 0
Therefore always unstable
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 9
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 10
(2) Transition from laminar to turbulent flow Not all laminar flows have different equilibrium states, but all laminar flows for sufficiently large Re become unstable and undergo transition to turbulence. Transition: change over space and time and Re range of laminar flow into a turbulent flow.
υδU
cr =Re ~ 1000 δ = transverse viscous thickness
Retrans > Recr with xtrans ~ 10-20 xcr Small-disturbance (linear) stability theory can predict Recr with some success for parallel viscous flow such as plane Couette flow, plane or pipe Poiseuille flow, boundary layers without or with pressure gradient, and free shear flows (jets, wakes, and mixing layers). Note: No theory for transition, but recent DNS helpful.
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 11
Outline linearized stability theory for parallel viscous flows: select basic solution of interest; add disturbance; derive disturbance equation; linearize and simplify; solve for eigenvalues; interpret stability conditions and draw thumb curves.
2
2
^ ^ ^ ^ ^ ^ ^1
^ ^ ^ ^ ^ ^ ^1
^ ^0
t x x y y x
t x x y y x
xx
u u u u u vu v u p u
v u v u v v v v v p u
u v
υρ
υρ
+ + + + = − + ∇
+ + + + = − + ∇
+ =
Linear PDE for ^u ,
^v ,
^p for (u ,v , p ) known.
Assume disturbance is sinusoidal waves propagating in x direction at speed c: Tollmien-Schlicting waves.
^ ( )( , , ) ( )^^ ( )
^^ ( )
i x ctx y t y e
i x ctu ey
i x ctv i ex
αφ
ψ αφ
ααφ
−Ψ =
∂ −′= =∂
∂Ψ −= − = −∂
^
^
^
u u u
v v v
p p p
= +
= +
= +
=vu, mean flow, which is solution steady NS
=^^
,vu small 2D oscillating in time disturbance is solution unsteady NS
Stream function y =distance across shear layer
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 12
0^^=+ yx vu Identically!
ir iααα += = wave number = λ
π2
ir iccc += = wave speed = αω
Where λ = wave length and ω =wave frequency Temporal stability:
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 13
Inserting ^u ,
^v into small disturbance equations and
eliminating ^p results in Orr-Sommerfeld equation:
2 2 4( )( '' ) '' ( 2 '' )Re
IV
inviscid Raleigh equationiu c uφ α φ φ φ α φ α φ
α− − − = − − +
Uuu /= Re=UL/υ y=y/L 4th order linear homogeneous equation with homogenous boundary conditions (not discussed here) i.e. eigen-value problem, which can be solved albeit not easily for specified geometry and (u ,v , p) solution to steady NS.
058:0160 Chapter 6- part2 Professor Fred Stern Fall 2014 14
Although difficult, methods are now available for the solution of the O-S equation. Typical results as follows
(1) Flat Plate BL: 520Re ==
+
υδU
crit
(2) αδ* = 0.35 λmin = 18 δ* = 6 δ (smallest unstable λ) ∴ unstable T-S waves are quite large (3) ci = constant represent constant rates of damping
(ci < 0) or amplification (ci > 0). ci max = .0196 is small compared with inviscid rates indicating a gradual evolution of transition.
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(4) (cr/U0)max = 0.4 unstable wave travel at average velocity.
(5) Reδ*crit = 520 Rex crit ~ 91,000
Exp: Rex crit ~ 2.8 x 106 (Reδ*crit = 2,400) if care taken, i.e., low free stream turbulence
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