Basics of Rotating Boundary-Layer Flow

NCAR is funded by the National Science Foundation

Richard RotunnoNCAR, Boulder CO

http://www.mmm.ucar.edu/people/rotunno/

Outline

1. Rotating Flow / No Boundary

2. Rotating Flow / Frictional Boundary Layer

3. Boundary Layer of a Solid-Body-Rotation Vortex

4. Boundary Layer of a Potential Vortex

5. Boundary Layer of a Rankine-Type Vortex

6. Summary

Cylindrical Polar Coordinates

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[u,v,w]

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[r,φ,z]

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z

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r

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φ

1. Rotating Flow / No Boundary

Simple Vortex

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0 = −∂p

∂r+ρV 2

r€

[0,V (r),0]

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z

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r

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φ

1. Rotating Flow / No Boundary

http://web.mit.edu/hml/ncfmf.html

2. Rotating Flow/Frictional Boundary layer

Dye Injected atWater Surface(earlier)

Drain

“Secondary Flow”

2. Rotating Flow/Frictional Boundary layer

Bathtub Vortex

Dye Injected atWater Surface(later)

Drain

Bathtub Vortex

“Secondary Flow”

2. Rotating Flow/Frictional Boundary layer

Dye Injected nearBottom (earlier)

Drain

“Secondary Flow”

Bathtub Vortex

2. Rotating Flow/Frictional Boundary layer

Dye Injected nearBottom (later)

Drain

“Secondary Flow”

Bathtub Vortex

2. Rotating Flow/Frictional Boundary layer

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z

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r

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φ

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z = 0 : u = v = w = 0

2. Rotating Flow/Frictional Boundary layer

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0 = −∂p

∂r+ρV 2

r

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0 = −∂p

∂r+ν∂ 2u

∂z2

For any , (radial inflow) near frictional boundary

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z

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r

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φ

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V (r)

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u < 02. Rotating Flow/Frictional Boundary layer

Behavior of near frictional boundary depends on

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(u,v,w)

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V (r)

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V ∝ rβ

Similarity solutions exist

Solid-body rotation

Potential vortex

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β = 1

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β > −1

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β =−1

Rott and Lewellen (1966 Prog Aero Sci)

2. Rotating Flow/Frictional Boundary layer

3. Flow in Solid-Body Rotation Above a Stationary Disk*

* Bödewadt (1940) (Schlichting 1968 Boundary Layer Theory)

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V =ωr

z

νω

3. Flow in Solid-Body Rotation Above a Stationary Disk

Solution exhibits “overshoot” and “pumping”

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(vmax >V)

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(w(∞) > 0)

z

νω

3. Flow in Solid-Body Rotation Above a Stationary Disk

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u

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v

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0

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0.5

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−0.5

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1.0

Bödewadt

Ekman

hodograph

Ekman is linearized version of Bödewadt

3. Flow in Solid-Body Rotation Above a Stationary Disk

0 1 0 1 10-1 -10

z

νω

8

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0 ≈ −V

r

2

+ν∂ 2u

∂z2

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du

dt≈v 2 −V

r

2

> 0Inertial layer:

Friction layer:

Rotunno and Bryan (Submitted to JAS)

tangential

radial

3. Flow in Solid-Body Rotation Above a Stationary Disk

LaboratoryExperiment

“Secondary Flow”

3. Flow in Solid-Body Rotation Above a Stationary Disk

Hurricane Inner-Core Flow

Zhang, Rogers, Nolan & Marks (2011, Monthly Weather Review)

radial tangential

3. Flow in Solid-Body Rotation Above a Stationary Disk

Composite Dropsonde Analysis of Hurricane Inner-Core Flow

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r /rmax

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r /rmax

4. Potential Vortex Above a Stationary Disk: Experiment

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V ∝ r−1

Phillips and Khoo (1987, Proc. Roy. Soc. London)

4. Potential Vortex Above a Stationary Disk: Experiment

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V ∝ r−1

Phillips and Khoo (1987, Proc. Roy. Soc. London)

4. Potential Vortex Above a Stationary Disk: Theory

radial tangential

Burggraf, Stewartson and Belcher (1971, Phys. Fluids)

Radial inflow ( ), but no “overshoot”( )

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u < 0

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v <V

decreasing radius

decreasing radius

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−ru

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rv

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z

radial tangential

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−ru

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rv

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102z

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r = 0.14. Potential Vortex Above a Stationary Disk: Theory & Experiment

Phillips and Khoo (1987, Proc. Roy. Soc. London)

radial tangential

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−ru

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rv

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102z

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r = 0.14. Potential Vortex Above a Stationary Disk: Theory & Experiment

Phillips and Khoo (1987, Proc. Roy. Soc. London)

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0 ≈ −V

r

2

+ν∂ 2u

∂z2

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du

dt≈v 2 −V

r

2

< 0

Inertial layer:

Friction layer:

Wilson and Rotunno (1986, Phys. Fluids)

4. Potential Vortex Above a Stationary Disk: Experiment

Phillips (1985, Proc. Roy. Soc. London)

End-Wall Vortex

Vortex Breakdown

April 22, 2010 Texas Panhandle (H. Bluestein)

4. Potential Vortex Above a Stationary Disk: Tornado?

Kuo (1971, JAS)

5. Boundary layer of a Rankine-Type Vortex

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V (r)

Summary Rotating Flow / Frictional Boundary Layer Radial Inflow in BL, if Convergent, Transport from BL to InteriorBoundary Layer of Solid-Body-Rotation VortexApplication: HurricanesBoundary Layer of Potential VortexApplication: Certain Types of TornadoesBoundary Layer of Rankine-Type VortexApplication: Hurricanes& Certain Types of Tornadoes