Docstoc

03Lect6BoundLayer

Document Sample
03Lect6BoundLayer Powered By Docstoc
					      Wind loading and structural response
         Lecture 6 Dr. J.D. Holmes



Atmospheric boundary layers and
         turbulence I
              Atmospheric boundary layers and turbulence



                                   153 metres       64 metres        12 metres

                          35
                          30
       Wind speed (m/s)




                          25
                          20
                          15
                          10
                          5
                          0
                               0         1      2           3    4               5
                                                Time (minutes)




Wind speeds from 3 different levels recorded from a synoptic gale
      Atmospheric boundary layers and turbulence


Features of the wind speed variation :

 • Increase in mean (average) speed with height

 • Turbulence (gustiness) at each height level

 • Broad range of frequencies in the fluctuations

 • Similarity in gust patterns at lower frequencies
           Atmospheric boundary layers and turbulence

• Mean wind speed profiles :
• Logarithmic law
                     dU
                        is a function of (z, ρ a τ 0 )
                     dz
          0 - surface shear stress        a - air density

                      dU             u
                          constant . 
                      dz              z
                     u = friction velocity =  (0/a)
 integrating w.r.t. z :
                    U  (1/ k ) . u log e z  constant
         Atmospheric boundary layers and turbulence



• Logarithmic law

                              u
                     U(z)       log e (z/z0 )
                              k

• k = von Karman’s constant (constant for all surfaces)

• zo = roughness length (constant for a given ground surface)
    logarithmic law - only valid for z >zo and z < about 100 m
        Atmospheric boundary layers and turbulence



• Modified logarithmic law for very rough surfaces
              (forests, urban)
                       u      z - zh 
                               z 
                 U(z)  log e         
                       k           o 


• zh= zero-plane displacement

 zh is about 0.75 times the average height of the roughness
       Atmospheric boundary layers and turbulence




• logarithmic law applied to two different heights

                    U(z 1 ) log e z1/z o 
                           
                    U(z 2 ) log e z 2 /z o 


• or with zero-plane displacement :

                  U(z 1 ) log e (z 1  z h )/z o 
                         
                  U(z 2 ) log e (z 2  z h )/z o 
         Atmospheric boundary layers and turbulence

• Surface drag coefficient :
  Non-dimensional surface shear stress :
                                                       0     2
                                                             u
                                                           2
                                                      U10 U10
                                                          2




                                      u        10 
   from logarithmic law :     U10       log e  
                                               z 
                                      k         o
                                                   2
                                              
                                              
                                            
                                       k
                                        10  
                                 log e   
                                        z 
                                
                                        o  
       Atmospheric boundary layers and turbulence

• Terrain types :


                   Terrain Type               Roughness       Surface Drag
                                              Length (m)       Coefficient

        Very flat terrain (snow, desert)      0.001 - 0.005   0.002 – 0.003

        Open terrain (grassland, few trees)   0.01 – 0.05     0.003 – 0.006

        Suburban terrain (buildings 3-5 m)      0.1 – 0.5     0.0075 – 0.02

        Dense urban (buildings 10-30 m)           1–5          0.03 – 0.3
       Atmospheric boundary layers and turbulence



• Power law
                                      
                                  z 
                   U ( z )  U10  
                                  10 


•  = changes with terrain roughness and height range

                           1      
                                
                    log ( z / z ) 
                       e   ref 0 



             zref = reference height
      Atmospheric boundary layers and turbulence

• Matching of power and logarithmic laws :

    zo = 0.02 m                          = 0.128                 zref = 50 metres


                                                Logarithmic law
                                                Power law

                            100

                             80
            Height, z (m)




                             60

                             40

                             20

                              0
                                  0.0     0.5         1.0           1.5
          Atmospheric boundary layers and turbulence

• Mean wind speed profiles over the ocean:
• Surface drag coefficient () and roughness length (zo) vary with mean
  wind speed
                                               2
                                 au 2* aκU10
                            zo                         (Charnock, 1955)
                                  g      g
         g - gravitational constant       a - empirical constant

        a lies between 0.01 and 0.02
                                      2
                                 
                                                                        2
                                                   a  kU10 
                 κ              
                          k
substituting :                                 zo                   
                           10                  g  log e 10/zo  
                    log e   
                           z 
                   
                           o  
  Implicit relationship between zo and U10
        Atmospheric boundary layers and turbulence

• Mean wind speed profiles over the ocean:
      Assume g = 9.81 m/s2 ; a = 0.0144 (Garratt) ; k =0.41

                 U10 (m/s)    Roughness Length (mm)

                     10                   0.21

                     15                   0.59

                     20                   1.22

                     25                   2.17

                     30                   3.51

 Applicable to non-hurricane conditions
           Atmospheric boundary layers and turbulence

• Relationship between upper level and surface winds :
• Geostrophic drag coefficient          Cg 
                                               u*
                                               Ug
                                  Ug
    Rossby Number :        Ro 
                                  fzo


   balloon measurements : Cg = 0.16 Ro-0.09                 (Lettau, 1959)

Can be used to determine wind speed near ground level over different terrains :

               Log law        Lettau       Lettau       Log law
  U10, terrain 1  u*,terrain 1  Ug  u*,terrain 2  U10, terrain 2
           Atmospheric boundary layers and turbulence

• Mean wind profiles in hurricanes :
• Aircraft flights down to 200 metres


• Drop-sonde (probe dropped from aircraft - tracked by satellite) : recently started


• Sonic radar (SODAR) measurements in Okinawa


 •   Tower measurements
      •   not enough
      •   usually in outer radius of hurricane and/or higher latitudes
          Atmospheric boundary layers and turbulence

• Mean wind profiles in hurricanes :
                      North
                                           US Navy
                      West Cape            antennas

                                  Exmouth


                                    EXMOUTH
                                    GULF




                                   100 km

 • Northern coastline of Western Australia
 • Profiles from 390 m mast in late nineteen-seventies
                        Atmospheric boundary layers and turbulence

• Mean wind profiles in hurricanes :
• In region of maximum winds : steep logarithmic profile to 60-200 m

• Nearly constant mean wind speed at greater heights

                        1000

                                                                     log e ( z / 0.3)
                                                        U z  U10
        Height z, (m)




                                                                    log e (10 / 0.3)    for z < 100 m
                         100


                                                         Uz =U100 for z  100 m

                         10
                               0.0      1.0       2.0

                                     U(z)/U(10)
           Atmospheric boundary layers and turbulence

• Mean wind profiles in thunderstorms (downbursts) :
• Doppler radar

• Some tower measurements (not enough)

• Horizontal wind profile shows peak at 50-100 m

• Model of Oseguera and Bowles (stationary downburst):
                                                λR 2 
                                             U
                                                2r   1 e r/R 2
                                                                       e 
                                                                          z/z 
                                                                                  e z/ε   
                                                     
   r - radial coordinate
   R - characteristic radius
   z* - characteristic height out of the boundary layer
    - characteristic height in the boundary layer
    - scaling factor
         Atmospheric boundary layers and turbulence

• Mean wind profiles in thunderstorms (downbursts) :
 Model of Oseguera and Bowles (stationary downburst) :



                                                            r/R = 1.121
    R = 1000 m                             600

    r/R = 1.121
                                           400
    z* = 200 metres           Height (m)

     = 30 metres                          200

     = 0.25 (1/sec)                        0
                                                 0     20         40      60
                                                     Wind speed (m/s)
         Atmospheric boundary layers and turbulence

• Mean wind profiles in thunderstorms (downbursts) :
 Add component constant with height (moving downburst) :



    R = 1000 m                              600


    r/R = 1.121
                                            400


                               Height (m)
    z* = 60 metres
     = 50 metres                           200



     = 1.3 (1/sec)                          0
                                                  0   20   40   60   80   100
                                                       Wind speed (m/s)

    Uconst = 35 m/s
            Atmospheric boundary layers and turbulence


                                 153 metres       64 metres        12 metres

                        35
                        30
     Wind speed (m/s)




                        25
                        20
                        15
                        10
                        5
                        0
                             0         1      2           3    4               5
                                              Time (minutes)


Turbulence represents the fluctuations (gusts) in the wind speed


It can usually be represented as a stationary random process
        Atmospheric boundary layers and turbulence


Components of turbulence :

 • u(t) - longitudinal - parallel to mean wind direction
        - parallel to ground (usually horizontal)

 • v(t) - parallel to ground - right angles to u(t)

 • w(t) - right angles to ground (usually vertical)

                  w(t)     v(t)
                                   U+u(t)


                                              ground
       Atmospheric boundary layers and turbulence


Turbulence intensities :
                                                        2

                                   u  {  U (t )  U  dt} 2
                                           T
                                         1                   1
• standard deviation of u(t) :
                                         T 0

  Iu = u /U (longitudinal turbulence intensity) (non dimensional)


  Iv = v /U (lateral turbulence intensity)


  Iw = w /U (vertical turbulence intensity)
       Atmospheric boundary layers and turbulence

Turbulence intensities :

near the ground, u  2.5u*
                            2.5u                   1
   Iu = u /U                              
                    u  /0.4 log e z/z 0  log e z/z 0 
                     from logarithmic law

                                0.88
   v  2.2u*         Iv 
                             log e z/z 0 


                              0.55
   w  1.37u*        Iw 
                           log e z/z 0 
         Atmospheric boundary layers and turbulence

Turbulence intensities :

rural terrain, zo = 0.04 m :

                     Height, z (m)    Iu
                          2          0.26

                          5          0.21

                          10         0.18

                          20         0.16

                          50         0.14

                         100         0.13
          Atmospheric boundary layers and turbulence

 Probability density :

• The components of turbulence (constantU) can generally be
represented quite well by the Gaussian, or normal, p.d.f. :

                                             1  u  U 2 
                       f u u  
                                     1
   for u(t) :                            exp  σ   
                                  σ u 2π     2 u  
                                                          
                                              1  v 2 
                        f v v  
                                      1
   for v(t) :                             exp   
                                                  
                                   σ v 2π     2  σv  
                                                       
                                              1  w 2 
                        f w w  
                                      1
   for w(t) :                             exp   
                                                  
                                   σ w 2π     2  σw  
                                                       
  End of Lecture 6

       John Holmes
225-405-3789 JHolmes@lsu.edu

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:0
posted:5/6/2013
language:English
pages:27
gegouzhen12 gegouzhen12
About