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Wind Over Hills

Wind Over Hills. Ryan Wallace MAE 741 4/26/06. 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

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Wind Over Hills

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  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 • κ and α are constants • When H/L << 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 Δu/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 • ωxis increased while ωz 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 30:507-38 • 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.

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