k ‐ ϵ Model for the Atmospheric Boundary Layer Under Various Thermal Stratifications

2005 ◽  
Vol 127 (4) ◽  
pp. 438-443 ◽  
Author(s):  
Cédric Alinot ◽  
Christian Masson
Keyword(s):  
Boundary Layer ◽  
Atmospheric Boundary Layer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Similarity Theory ◽  
Turbulence Models ◽  
Boussinesq Approximation ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Production Term

This paper presents a numerical method for predicting the atmospheric boundary layer under stable, neutral, or unstable thermal stratifications. The flow field is described by the Reynolds’ averaged Navier-Stokes equations complemented by the k‐ϵ turbulence model. Density variations are introduced into the momentum equation using the Boussinesq approximation, and appropriate buoyancy terms are included in the k and ϵ equations. An original expression for the closure coefficient related to the buoyancy production term is proposed in order to improve the accuracy of the simulations. The resulting mathematical model has been implemented in FLUENT. The results presented in this paper include comparisons with respect to the Monin-Obukhov similarity theory, measurements, and earlier numerical solutions based on k‐ϵ turbulence models available in the literature. It is shown that the proposed version of the k‐ϵ model significantly improves the accuracy of the simulations for the stable atmospheric boundary layer. In neutral and unstable thermal stratifications, it is shown that the version of the k‐ϵ models available in the literature also produce accurate simulations.

Aerodynamic Simulations of Wind Turbines Operating in Atmospheric Boundary Layer With Various Thermal Stratifications

2002 ◽  
Author(s):  
Cedric Alinot ◽  
Christian Masson
Keyword(s):  
Boundary Layer ◽  
Atmospheric Boundary Layer ◽  
Wind Turbines ◽  
Power Output ◽  
Stokes Equations ◽  
Disk Surface ◽  
Boussinesq Approximation ◽  
Stable Stratification ◽  
Momentum Equation ◽  
Production Term

This paper presents a numerical method for performance predictions of wind turbines immersed into stable, neutral, or unstable atmospheric boundary layer. Tile flowfield around a turbine is described by the Reynolds’ averaged Navier-Stokes equations complemented by the k-ε turbulence model. The density variations are introduced into the momentum equation using the Boussinesq approximation and appropriate buoyancy terms are included into the k and ε equations. An original expression for the closure coefficient related to the buoyancy production term is proposed in order to improve the accuracy of the simulations. The turbine is idealized as actuator disk surface, on which external surficial forces exerted by the turbine blade on the flow are prescribed according to the blade element theory. The resulting mathematical model has been implemented in FLUENT. The results presented in the paper include the power output and wake development under various thermal stratifications of an isolated wind turbine. In stable stratification, the power output is 4% lower than in neutral condition, while in unstable situation, the power is 3% larger. The predicted wake velocity defects are qualitatively in agreement with experimental observations.

scholarly journals Validation of atmospheric boundary layer turbulence model by on-site measurements

2010 ◽  
Vol 14 (1) ◽  
pp. 199-207 ◽  
Author(s):  
Zarko Stevanovic ◽  
Nikola Mirkov ◽  
Zana Stevanovic ◽  
Andrijana Stojanovic
Keyword(s):  
Boundary Layer ◽  
Atmospheric Boundary Layer ◽  
Linear Models ◽  
Stokes Equations ◽  
Turbulence Models ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Turbulence ◽  
Askervein Hill ◽  
Neutral Conditions

Modeling atmosperic boundary layer with standard linear models does not sufficiently reproduce wind conditions in complex terrain, especially on leeward sides of terrain slopes. More complex models, based on Reynolds averaged Navier-Stokes equations and two-equation k-? turbulence models for neutral conditions in atmospheric boundary layer, written in general curvilinear non-orthogonal co-ordinate system, have been evaluated. In order to quantify the differences and level of accuracy of different turbulence models, investigation has been performed using standard k-? model without additional production terms and k-? turbulence models with modified set of model coefficients. The sets of full conservation equations are numerically solved by computational fluid dynamics technique. Numerical calculations of turbulence models are compared to the reference experimental data of Askervein hill measurements.

The structure of two-dimensional separation

1990 ◽  
Vol 220 ◽  
pp. 397-411 ◽  
Author(s):  
Laura L. Pauley ◽  
Parviz Moin ◽  
William C. Reynolds
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Adverse Pressure Gradient ◽  
Navier Stokes ◽  
Stream Velocity ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Shedding Frequency

The separation of a two-dimensional laminar boundary layer under the influence of a suddenly imposed external adverse pressure gradient was studied by time-accurate numerical solutions of the Navier–Stokes equations. It was found that a strong adverse pressure gradient created periodic vortex shedding from the separation. The general features of the time-averaged results were similar to experimental results for laminar separation bubbles. Comparisons were made with the ‘steady’ separation experiments of Gaster (1966). It was found that his ‘bursting’ occurs under the same conditions as our periodic shedding, suggesting that bursting is actually periodic shedding which has been time-averaged. The Strouhal number based on the shedding frequency, local free-stream velocity, and boundary-layer momentum thickness at separation was independent of the Reynolds number and the pressure gradient. A criterion for onset of shedding was established. The shedding frequency was the same as that predicted for the most amplified linear inviscid instability of the separated shear layer.

scholarly journals The numerical solution of the asymptotic equations of trailing edge flow

1974 ◽  
Vol 340 (1620) ◽  
pp. 91-111 ◽  
Keyword(s):  
Boundary Layer ◽  
Outer Layer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Trailing Edge ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Asymptotic Equations ◽  
Displacement Thickness

According to Stewartson (1969, 1974) and to Messiter (1970), the flow near the trailing edge of a flat plate has a limit structure for Reynolds number Re →∞ consisting of three layers over a distance O (Re -3/8 ) from the trailing edge: the inner layer of thickness O ( Re -5/8 ) in which the usual boundary layer equations apply; an intermediate layer of thickness O ( Re -1/2 ) in which simplified inviscid equations hold, and the outer layer of thickness O ( Re -3/8 ) in which the full inviscid equations hold. These asymptotic equations have been solved numerically by means of a Cauchy-integral algorithm for the outer layer and a modified Crank-Nicholson boundary layer program for the displacement-thickness interaction between the layers. Results of the computation compare well with experimental data of Janour and with numerical solutions of the Navier-Stokes equations by Dennis & Chang (1969) and Dennis & Dunwoody (1966).

Navier–Stokes solutions of unsteady separation induced by a vortex

2002 ◽  
Vol 465 ◽  
pp. 99-130 ◽  
Author(s):  
A. V. OBABKO ◽  
K. W. CASSEL
Keyword(s):  
Boundary Layer ◽  
Large Scale ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Recirculation Region ◽  
Small Scale ◽  
Navier Stokes Equations ◽  
Scale Interaction

Numerical solutions of the unsteady Navier–Stokes equations are considered for the flow induced by a thick-core vortex convecting along a surface in a two-dimensional incompressible flow. The presence of the vortex induces an adverse streamwise pressure gradient along the surface that leads to the formation of a secondary recirculation region followed by a narrow eruption of near-wall fluid in solutions of the unsteady boundary-layer equations. The locally thickening boundary layer in the vicinity of the eruption provokes an interaction between the viscous boundary layer and the outer inviscid flow. Numerical solutions of the Navier–Stokes equations show that the interaction occurs on two distinct streamwise length scales depending upon which of three Reynolds-number regimes is being considered. At high Reynolds numbers, the spike leads to a small-scale interaction; at moderate Reynolds numbers, the flow experiences a large-scale interaction followed by the small-scale interaction due to the spike; at low Reynolds numbers, large-scale interaction occurs, but there is no spike or subsequent small-scale interaction. The large-scale interaction is found to play an essential role in determining the overall evolution of unsteady separation in the moderate-Reynolds-number regime; it accelerates the spike formation process and leads to formation of secondary recirculation regions, splitting of the primary recirculation region into multiple corotating eddies and ejections of near-wall vorticity. These eddies later merge prior to being lifted away from the surface and causing detachment of the thick-core vortex.

scholarly journals Natural convection in a shallow cavity with differentially heated end walls. Part 2. Numerical solutions

1974 ◽  
Vol 65 (2) ◽  
pp. 231-246 ◽  
Author(s):  
D. E. Cormack ◽  
L. G. Leal ◽  
J. H. Seinfeld
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Aspect Ratio ◽  
Excellent Agreement ◽  
Asymptotic Theory ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Shallow Cavity

Numerical solutions of the full Navier-Stokes equations are obtained for the problem of natural convection in closed cavities of small aspect ratio with differentially heated end walls. These solutions cover the parameter range Pr = 6·983, 10 ≤ Gr 2 × 104 and 0·05 [les ] A [les ] 1. A comparison with the asymptotic theory of part 1 shows excellent agreement between the analytical and numerical solutions provided that A [lsim ] 0·1 and Gr2A3Pr2 [lsim ] 105. In addition, the numerical solutions demonstrate the transition between the shallow-cavity limit of part 1 and the boundary-layer limit; A fixed, Gr → ∞.

scholarly journals Laminar and weakly turbulent oceanic gravity currents performing inertial oscillations

2011 ◽  
Vol 8 (5) ◽  
pp. 2001-2045
Author(s):  
A. Wirth
Keyword(s):  
Mathematical Models ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Gravity Current ◽  
Boussinesq Approximation ◽  
Turbulent Fluxes ◽  
Navier Stokes ◽  
Small Scale ◽  
Mean Flow ◽  
Inertial Oscillations

Abstract. The small scale dynamics of a weakly turbulent oceanic gravity current is determined. The gravity current considered is initially at rest and adjusts by performing inertial oscillations to a geostrophic mean flow. The dynamics is explored with a hierarchy of mathematical models. The most involved are the fully 3-D Navier-Stokes equations subject to the Boussinesq approximation. A 1-D and 0-D mathematical model of the same gravity current dynamics are systematically derived. Using this hierarchy and the numerical solutions of the mathematical models, the turbulent dynamics at the bottom and the interface is explored and their interaction investigated. Three different regimes of the small scale dynamics of the gravity current are identified, they are characterised by laminar flow, coherent roll vortices and turbulent dynamics with coherent streaks and bursts. The problem of the rectification of the turbulent fluxes, that is how to average out the fluctuations and calculate their average influence on the flow is considered. It is shown that two different regimes of friction are superposed, an Ekman friction applies to the average geostrophic flow and a linear friction, not influenced by rotation, to the inertial oscillations. The combination of the two makes the bulk friction non-local in time for the 0-D model. The implications of the results for parametrisations of the Ekman dynamics and the small scale turbulent fluxes in the planetary boundary layer are discussed.

Self-induced flow in a rotating tube

1991 ◽  
Vol 230 ◽  
pp. 505-524 ◽  
Author(s):  
S. Gilham ◽  
P. C. Ivey ◽  
J. M. Owen ◽  
J. R. Pincombe
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Laser Doppler Anemometry ◽  
Glass Tube ◽  
The Self ◽  
Navier Stokes ◽  
Ekman Number ◽  
Induced Flow ◽  
Short Tube

When a tube, sealed at one end and open to a quiescent environment at the other, is rotated about its axis, fluid flows from the open end along the axis towards the sealed end and returns in an annular boundary layer on the cylindrical wall. This paper describes the first known study to be made of this self-induced flow. Numerical solutions of the Navier–Stokes equations are shown to be in mainly good agreement with experimental results obtained using flow visualization and laser–Doppler anemometry in a rotating glass tube.The self-induced flow in the tube can be described in terms of the length-to-radius ratio, G, and the Ekman number, E. However, for large values of G (G [ges ] 20), the flow outside the boundary layer on the endwall of the tube can be characterized by a single, modified, Ekman number, E*, where E* = GE. Although most of the fluid entering the open end of the tube is entrained into the annular (Stewartson-type) boundary layer, for small values of E* (E* < 0.2) some flow reaches the sealed end. For this so-called 'short-tube case’, the flow in the boundary layer on the endwall is shown to be similar to that associated with a disk rotating in a quiescent environment: the free disk. The self-induced flow for the short-tube case is believed to be responsible for the ’ hot-poker effect’ used, on some jet engines, to provide ice protection for the nose bullet.

Validation of Reynolds-Averaged Navier-Stokes Simulations for International America’s Cup Class Spinnaker Force Coefficients in an Atmospheric Boundary Layer

2007 ◽  
Vol 51 (01) ◽  
pp. 22-38
Author(s):  
William C. Lasher ◽  
Peter J. Richards
Keyword(s):  
Boundary Layer ◽  
Atmospheric Boundary Layer ◽  
Turbulence Models ◽  
Reynolds Stress Model ◽  
Design Tool ◽  
Navier Stokes ◽  
Force Coefficients ◽  
America’S Cup ◽  
Reynolds Averaged Navier Stokes ◽  
Better Than

Three semirigid models for International America's Cup Class spinnakers were tested in a wind tunnel with a simulated atmospheric boundary layer. These experiments were also simulated using a commercial Reynolds-averaged Navier-Stokes (RANS) solver with three different turbulence models. A comparison between the experimental and numerical force coefficients shows very good agreement. The experimentally measured differences in the driving force coefficients among the three sails were predicted well by all three turbulence models. The realizable k-e model produced the best results, and the standard k-e model produced the worst. The Reynolds stress model did not perform significantly better than the standard k-e model. The results suggest that RANS can be used as a design tool for optimizing spinnaker shape.

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