Wind Over Hills

1
Wind Over Hills

• Ryan Wallace
• MAE 741
• 4/26/06

2
Outline

• Flow Structure
• Momentum and Continuity Equations
• Scales of the Flow
• Turbulent Stress and its Model
• Momentum Scaling
• Interaction between Regions
• Vorticity Distortion and Turbulence Structure
• Separation
• Effect of Froude Number
• Spectra
• Conclusion
• References

3
Flow structure

• Upstream of the hill the wind is taken to be
  fully developed.
• The velocity profile of the atmospheric boundary
  layer is described in the equation to the right.
• Flow at the top of the hill is about 50 faster
  then the upstream velocity.
• Flow is divided up into three layers
• Upper
• Middle
• Inner
• Flow downstream is separated for hills with steep
  slopes

Figure 1 Flow geometry and asymptotic for flow
over a hill.
(Belcher Hunt, 1998)
4
Momentum and Continuity Equations
For hill with a small slope the perturbations are
small therefore the momentum and continuity
equations becomes
5
Scales

• Advection Time Scale
• Represents the distortion of the turbulence by
  the mean flow
• Lagrangian Time Scale
• Represents the decorrelation and relaxation time
  scale of the large energy-containing eddies
• The ratio of these two times gives rise to a
  suitable turbulence model
• Is the measure of distance that the turbulence is
  away from the local equilibrium
• Length scales are based on
• Height of the hill
• Length of the hill
• Thickness of the inner layer
• Surface roughness
• ? and a are constants

When H/L ltlt 1
Hill height Hill Length Inner layer
thickness Surface roughness
6
Turbulent Stress

• The ratio of the time scales is the same order of
  magnitude as dissipation over advection

• Reynolds shear stress for low slope hills
• Uses the mixing length model

(Belcher Hunt, 1998)
7
Momentum Scaling

• In the outer region, turbulence is distorted
  quickly
• Therefore the ratio of the turbulent stress to
  the mean flow is on the order of H/L
• The ratio to the perturbation stress gradient to
  mean flow advection in the outer region becomes
• In the inner region, turbulence can be estimated
  by the mixing model
• Therefore the perturbation velocity scales to
  ?u/L
• The ratio to the perturbation stress gradient to
  mean flow advection in the inner region becomes
• In middle region the velocity is governed be the
  Rayleigh equation

8
Interaction between Regions

• The displacement of a streamline at the top of
  the inner layer is due to
• Displacement over the hill itself
• Bernoulli displacement due to the pressure
  gradient in the upper layer
• Mean shear in the middle layer
• Frictional effects in the inner layer
• These mechanisms are all coupled
• The pressure gradient is related to the turbulent
  stress by the equation below
• This equation shows that the surface shear stress
  perturbation is in phase and proportional to the
  surface pressure perturbation
• The magnitude of the streamline displacement is
  equal to

9
Vorticity Distortion and Turbulence Structure

• The vorticity of the eddies in the rapid
  distorting regions are distorted mainly by
  anisotropic strain of the mean flow
• ?x is increased while ?z is decreased
• Turbulence intensities are also effected
• Normal stress changes are neglected in the inner
  region
• Small asymmetric intensity changes occur in the
  outer region due to an asymmetric displacement in
  the inner region
• A perturbation velocity interacts with the mean
  flow over a hill and produces an exponential
  growth of streamwise vorticity cause by rotation
  and stretching of the vertical vorticity by the
  mean shear
• Normal stress of the flow are larger over hills
  then over flat surfaces

10
Separation

• Separated flow downstream of a moderate to large
  sloped hill produces hydraulic jumps
• Downstream vorticities will persist in weak
  turbulent flows while the vorticities will
  quickly dissipate in strong turbulent flows
• Reattachment length scales are on the order of
  12H for a smooth hill to 3H for a highly
  turbulent flow and uneven hill
• A high Reynolds number flow will led to
  separation where the pressure gradient is
  greatest, i.e. the crest of the hill
• The wake region exhibits neutral flow and
  stratification

(Hunt Snyder, 1980)
11
Separation

• When the slope of the height of the hill over the
  length is greater then 30 the flow typically
  will separate
• 3-D hills require closure models and direct
  numerical simulation (DNS) needs to be utilized
  to solve for the flow
• Large eddy simulation will not give accurate
  results in the inner region but can lead to
  useful insights in the flow
• Lee wave are generated over the separated flow

(Hunt Snyder, 1980)
12
Effect of Froude Number

• Froude Number is the ratio of inertial force to
  gravitational force
• As the Froude number is increased from 0 to
  infinity the flow separation creeps up the back
  side of the hill
• At a low Froude number the lee waves are distinct
  and also have some separation from the flow and
  the hill
• As the Froude number increases the lee waves
  vanish
• Hydraulic jump moves away from the hill as the
  Froude number increases

(Hunt Snyder, 1980)
13
Spectra

• Integral time scale is approximately equal to the
  advection time scale at the top of the hill
• Horizontal components have more energy then the
  vertical components
• The spectrum the inertial subrange and with
  energy density has the following equation

(Tampieri, Mammarella, Maurizi, 2002)
14
Spectra

• Increasing shear decreases the integral scale
• Estimation for the integral turbulence scale
• a and b are obtain from experimental data
• Can use the estimation to find the reduction of lE

15
Conclusion

• Flow over hills is similar to flow around buff
  bodies
• Thermo-plumes was neglected in this summary of
  wind over hills but are important mechanisms
• Separated model are not very accurate yet
• There is still a lot to be discover for flows
  over hills

16
References

• S.E. Belcher, J.C. R. Hunt. 1998. Turbulent flow
  over hills and waves. Annu. Rev. Fluid Mech. 1998
  30507-38
• W. H. Snyder, R. S. Thompson, R. E. Eskridge, R.
  E. Lawson, I. P. Castro, J.C.R. Hunt, Y. Ogawa.
  1985. The structure of strongly stratified flow
  over hills dividing-streamline Concept. J. Fluid
  Mech. Vol. 152, pp. 249-288.
• F. T. Smith, P. W. M. Brighton, P. S. Jackson, J.
  C. R. Hunt. 1981 On boundary-layer flow past
  two-dimensional obstacles. J. Fluid Mech. Vol.
  113, pp. 123-152.
• I. P. Castro, W. H. Snyder, G. L. Marsh. 1983.
  Stratified flow over three-dimensional ridges. J.
  Fluid Mech. Vol. 135, pp. 261-282.
• F. Tampieri, I. Mammarella, A. Maurizi. 2003.
  Turbulence in complex terrain. Boundary-Layer
  Meterology. 109 85-97.
• J. C. R. Hunt, W. H. Snyder. 1980. Experiments on
  stably and neutrally stratified flow over a model
  three-dimensional hill. J. Fluid Mech. Vol. 96
  part 4, pp. 671-704.
• H. Tennekes, J. L. Lumley. 1972. A First Course
  in Turbulence. The MIT Press.