Asymptotic Wake Research Articles

Mountain-Wave Turbulence in the Presence of Directional Wind Shear over the Rocky Mountains

Abstract Mountain-wave turbulence in the presence of directional wind shear over the Rocky Mountains in Colorado is investigated. Pilot reports (PIREPs) are used to select cases in which moderate or severe turbulence encounters were reported in combination with significant directional wind shear in the upstream sounding from Grand Junction, Colorado (GJT). For a selected case, semi-idealized numerical simulations are carried out using the WRF-ARW atmospheric model, initialized with the GJT atmospheric sounding and a realistic but truncated orography profile. To isolate the role of directional wind shear in causing wave breaking, sensitivity tests are performed to exclude the variation of the atmospheric stability with height, the speed shear, and the mountain amplitude as dominant wave breaking mechanisms. Significant downwind transport of instabilities is detected in horizontal flow cross sections, resulting in mountain-wave-induced turbulence occurring at large horizontal distances from the first wave breaking point (and from the orography that generates the waves). The existence of an asymptotic wake, as predicted by Shutts for directional shear flows, is hypothesized to be responsible for this downwind transport. Critical levels induced by directional wind shear are further studied by taking 2D power spectra of the magnitude of the horizontal velocity perturbation field. In these spectra, a rotation of the most energetic wave modes with the background wind, as well as perpendicularity between the background wind vector and the wavenumber vector of those modes at critical levels, can be found, which is consistent with the mechanism expected to lead to wave breaking in directional shear flows.

Study of the influence of imposed turbulence on the asymptotic wake deficit in a very long line of wind turbines

The influence of imposed turbulence on the development of the flow along a long row of wind turbines is studied, in search for an asymptotic wake deficit state. Calculations are performed using EllipSys3D, a CFD code that solves the Navier–Stokes equations in their incompressible form using a finite volume approach. In this code, the Large-Eddy Simulation technique is used for modeling turbulence, and the wind turbine rotors are represented as actuator discs whose loading is determined through the use of tabulated airfoil data by applying the blade-element method.Ten turbines are located along a row and separated from each other by seven rotor diameters, which is representative of the distance used in today's offshore wind farms. Turbulence is pre-generated with the Mann model, with imposed turbulence intensity levels of 4.5% and 8.9%. The aim with this study is to investigate features of the flow that depend solely on imposed turbulence and the presence of wind turbine rotors. For this reason, the turbines are isolated from their environment, and no effect from the presence of the atmospheric boundary layer is modeled, i.e., a non-sheared inflow is used. Analysis of the characteristics of the flow as a function of the position along the row of turbines is performed in terms of standard deviation of the velocity components, turbulence kinetic energy, mean velocity, and power spectra of the axial velocity fluctuations. The mean power production along the row of turbines is also used as an indicator.Calculations are performed below rated power, where a generator torque controller implemented in EllipSys3D renders it possible for the turbines to adapt to the flow conditions in which they operate.The results obtained for the standard deviation of the velocity components, turbulence kinetic energy, power and mean velocity as functions of downstream distance show that an asymptotic wake state seems close to be reached, in the conditions tested, near the end of the 10 turbine row. Significant changes towards this state are seen to happen faster when imposing turbulence in the domain. Power spectra of the axial velocity fluctuations are shown to provide interesting information about the turbulence in the flow, but are found not to be useful in determining if an asymptotic wake state is reached.

Turbulent wake solutions of the Prandtl alpha equations.

A derivation of the Navier-Stokes alpha equations for spatially dependent alpha is presented. It is shown that an extra term in the equation is necessary to ensure the conservation of momentum. The Prandtl form of these variable alpha equations are determined for both planar and axisymmetric pressure-gradient-driven boundary-layer flows correcting previous work on the subject. The Prandtl equations are then solved analytically for four different asymptotic wake flows: the classical planar wake, the classical axisymmetric wake, the planar dragless wake, and the axisymmetric dragless wake. Least-squares fits of the theoretical solutions with available turbulent mean-flow velocity data for classical planar and axisymmetric wakes are given. We point out that the dissipation coefficient does not have to be equal to the kinematic viscosity, and its numerical value may be estimated from experimental data.

Numerical simulations of orographic gravity waves in flows which back with height

AbstractNumerical simulations are carried out to determine the steady‐state gravity‐wave structure above a circularly‐symmetric, bell‐shaped hill when the wind backs with height. Two types of idealized basic state flow are considered; one representing the advection of a uniform baroclinic shear flow, and the other being a flow of constant speed but with a wind vector that rotates uniformly in height. of particular interest is the nature of the critical level processes that must appear at all heights due to the three‐dimensionality of the wave field and the rotation of the wind vector. As suggested by linear theory, a critical level or asymptotic wake is found downwind of the hill at any height, where ‘downwind’ refers to the wind at that level only. For small hills (where the nondimensional mountain height ≤ unity), the simulated vertical momentum flux profiles are in very good agreement with linear hydrostatic wave theory if it is assumed that wave‐component critical‐level absorption is total (i.e. no wave transmission or reflections are permitted). the vertical momentum fluxes generated by a 1 km high hill show nonlinear enhancement by a factor of about 1.5. an interesting feature of the gravity‐wave field generated by this 1 km mountain is an asymptotic wake that is dominated by a single intense shear layer in which the Richardson number is order unity.

Stationary gravity‐wave structure in flows with directional wind shear

AbstractMost theoretical studies of flow over mountains assume that the wind direction is constant. For two‐dimensional orographic ridges this is not an issue, since only the component of the wind across the ridge forces internal gravity waves. A special case of rectilinear flow is when the wind component changes sign (i.e. the wind direction changes discontinuously through 180 degrees): such a point is a critical level for all stationary gravity waves. For isolated three‐dimensional mountains the gravity‐wave response comprises waves of all azimuthal orientations, and each of these may have a different critical‐level height‐depending on whether the wind backs or veers with height.Using a combination of ray‐tracing methods and stationary‐phase solutions, the structure of the stationary, hydrostatic gravity‐wave field generated by an isolated mountain is studied for a simple wind field which backs with height. In contrast to the behaviour for unidirectional flow, wave action does not accumulate indefinitely beneath the critical level but is continuously advected downwind by the component of the wind parallel to the phase lines. Instead of the wave energy becoming unbounded following a ray as in the 2D case, the wave energy decays to zero along a trailing wake zone downwind of the mountain along phase lines. This is just an extension of the ‘asymptotic wake’ found in the linear solutions of Smith for constant flow over an isolated mountain. The work described here underlines the desirability of including the effects of selective critical‐level absorption in the gravity‐wave drag parametrization schemes of numerical weather‐prediction models.

Energy balance for turbulent flow around a surface mounted cube placed in a channel

Results of an experimental investigation of the inhomogeneous, three-dimensional flow around a surface mounted cube in a channel are presented. LDA measurements of single-point velocity correlations are used to determine the production, convection and transport of the turbulence kinetic energy, k, in the obstacle wake. The turbulence dissipation rate is obtained as a closing term to the balance of the k-transport equation. The results provide some insight to the evolution of the turbulence dissipation rate from the near field recirculation zone to the asymptotic wake. Also presented is a comparison between measured and modeled transport terms.

Laminar flow at the trailing edge of a flat plate

Numerical solutions of the Navier-Stokes equations for laminar flow around and beyond the trailing edge of a flat plate at a Reynolds number of 10E5 are presented. The size of the computational domain has been varied over a wide range to show that previous calculations of this type have employed domains which were too small to obtain a domain-independent solution. The present large-domain solutions are compared with those obtained with the triple-deck and interactive boundary-layer theories. Satisfactory agreement is found in regions of validity of these theories. However, the present results are more complete insofar as they extend well into the asymptotic wake.

The symmetric turbulent wake of a flat plate

Detailed measurements of mean flow and turbulence in the developing symmetric wake of a smooth, flat plate are presented. The results are discussed in the light of previous data and theories for near and far wakes. It is shown that evolution of the upstream boundary layers into the classical asymptotic wake occurs in three quite distinct stages and takes about 350 wake momentum thicknesses.

Fluid drift caused by a rising bubble

The drift (or translation) of liquid by a rising bubble with a closed wake is shown by the motion of tracers suspended in the liquid. At low Reynolds numbers, Re, the tracers move upward continuously, while at higher Re the tracers have a looping motion. The drift is largest on the axis of rise and decays rapidly with radial distance from the axis. There are presently no models which accurately predict the drift, but the asymptotic wake expression of Batchelor provides a lower limit on the volumetric flow rate of drifted fluid.

In-line interaction of a pair of bubbles in a viscous liquid

The axial velocity behind a single bubble rising in a viscous liquid has been measured. It is lower than the asymptotic wake velocity even 30 diameters behind the bubble. The inline interaction of two bubbles can be described by a superposition approximation in which the trailing bubble rises at a velocity equal to its terminal velocity plus the wake velocity behind the leading bubble and the leading bubble is unaffected by the trailing bubble. A wake velocity correlation and the critical conditions for coalescence are provided for 10<Re<100 and M > 4 × 10 −3.

Similar Solutions of the Mean Velocity, Turbulent Energy and Length Scale Equation

A semiempineal approximation of the equations for second order correlations and moments was quantitatively investigated. The resulting partial differential equations for mean velocity, turbulent kinetic energy and length scale are supposed to be valid for plane and axisymmetric free as well as bounded turbulent flows. They were converted into ordinary differential equations by means of similarity transformations for six different flow cases (plane free jet, plane asymptotic wake, free mixing layer, plane channel flow, axisymmetric free jet, axisymmetric pipe flow) and then solved numerically. The values of the coefficients introduced for the closure of the equations were determined from investigations of the dependence of the solutions on them and by comparison with experimental results. A single set of coefficients was obtained, which reproduces the measurements of mean velocity for all six treated cases and the decay of homogeneous turbulence with satisfactory agreement.

Bubble coalescence in viscous liquids

When a gas is bubbled into a viscous liquid, coalescence of the bubbles occurs. This in turn influences the bubble-induced liquid flow and the mass transfer rate, matters of considerable practical importance. Experiments have been conducted on the relative motion of vertically aligned bubble pairs, each having volumes from 10 to 40 cm 3, in a 67 per cent wt solution of sucrose in water. The Reynolds numbers encoutered were 40–90. It was demonstrated that bubbles up to 70 cm apart initially, coalesced, even when the difference in the infinite fluid rise velocities of single bubbles would suggest that coalescence is impossible. Although more detailed information upon wake structure and the motion of bubbles in such wakes is required before a full explanation is possible, a theoretical treatment relying on an asymptotic wake velocity profile provides a general explanation of the phenomenon.