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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
€
[u,v,w]
€
[r,φ,z]
€
z
€
r
€
φ
1. Rotating Flow / No Boundary
Simple Vortex
€
0 = −∂p
∂r+ρV 2
r€
[0,V (r),0]
€
z
€
r
€
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
€
z
€
r
€
φ
€
z = 0 : u = v = w = 0
2. Rotating Flow/Frictional Boundary layer
€
0 = −∂p
∂r+ρV 2
r
€
0 = −∂p
∂r+ν∂ 2u
∂z2
For any , (radial inflow) near frictional boundary
€
z
€
r
€
φ
€
V (r)
€
u < 02. Rotating Flow/Frictional Boundary layer
Behavior of near frictional boundary depends on
€
(u,v,w)
€
V (r)
€
V ∝ rβ
Similarity solutions exist
Solid-body rotation
Potential vortex
€
β = 1
€
β > −1
€
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)
€
V =ωr
z
νω
3. Flow in Solid-Body Rotation Above a Stationary Disk
Solution exhibits “overshoot” and “pumping”
€
(vmax >V)
€
(w(∞) > 0)
z
νω
3. Flow in Solid-Body Rotation Above a Stationary Disk
€
u
€
v
€
0
€
0.5
€
−0.5
€
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
€
0 ≈ −V
r
2
+ν∂ 2u
∂z2
€
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
€
r /rmax
€
r /rmax
4. Potential Vortex Above a Stationary Disk: Experiment
€
V ∝ r−1
Phillips and Khoo (1987, Proc. Roy. Soc. London)
4. Potential Vortex Above a Stationary Disk: Experiment
€
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”( )
€
u < 0
€
v <V
decreasing radius
decreasing radius
€
−ru
€
rv
€
z
radial tangential
€
−ru
€
rv
€
102z
€
r = 0.14. Potential Vortex Above a Stationary Disk: Theory & Experiment
Phillips and Khoo (1987, Proc. Roy. Soc. London)
radial tangential
€
−ru
€
rv
€
102z
€
r = 0.14. Potential Vortex Above a Stationary Disk: Theory & Experiment
Phillips and Khoo (1987, Proc. Roy. Soc. London)
€
0 ≈ −V
r
2
+ν∂ 2u
∂z2
€
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
€
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
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