Scott Waggy

Direct Numerical Simulation of the Turbulent Ekman Layer: Turbulent Energy Budgets

Results from a direct numerical simulation of the Ekman layer at a Reynolds number of 400 were analyzed to compute energy spectra and energy budgets for this flow. Energy budgets showed that the majority of the turbulent kinetic energy in the flow is produced by the coupling between the streamwise primary Reynolds shear stress and the streamwise velocity gradient. It was also found that the coupling of the vertical velocity variance with the spanwise directions allows transfer of kinetic energy by means of the return-to-isotropy pressure strain terms due to the ‘splatting effect’.

Direct Numerical Simulation of the Turbulent Ekman Layer: Evaluation of Closure Models

AbstractA direct numerical simulation (DNS) at a Reynolds number of 1000 was performed for the neutral atmospheric boundary layer (ABL) using the Ekman layer approximation. The DNS results were used to evaluate several closure approximations that model the turbulent stresses in the Reynolds averaged momentum equations. Two first-order closure equations proposed by O’Brien and by Large, McWilliams, and Doney were tested; both models approximate the eddy diffusivity as a function of height using cubic polynomials. Of these two models, the O’Brien model, which requires data both at the surface layer and at the top of the boundary layer, proved superior. The higher-order k–e model also agreed well with DNS results and more accurately represented the eddy diffusivity in this rotational flow.

Direct Numerical Simulation of the Turbulent Ekman Layer: Instantaneous Flow Structures

A direct numerical simulation of the turbulent Ekman layer at a Reynolds number of 400 was performed. Twopoint velocity and pressure correlations were plotted to identify and estimate the average sizes and locations of instantaneous flow structures characteristic of turbulence. It was found that these structures are characterized by elongated eddies near the surface, which broaden away from the wall. The correlations roughly align with the mean sheardirectionnearthesurface;movingawayfromthewall,theyexhibittiltingandliftingofdownstreamsegments. The u 0 two-point correlation, in particular, showed significant tilting in the outer regions of the flow demonstrating that this is a significant deviation from typical nonrotating boundary-layer behavior.

PARALLEL IMPLEMENTATION OF A NAVIER–STOKES SOLVER: TURBULENT EKMAN LAYER DIRECT SIMULATION

A massively parallel direct numerical solution procedure for the turbulent Ekman layer is presented. The simulations study the dynamics of turbulence in this flow by solving the incompressible Navier–Stokes equations with Coriolis and buoyancy terms. The governing equations are integrated via a semi-implicit time advancement algorithm which is massively parallelized using the Portable, Extensible Toolkit for Scientific Computation (PETSc) libraries. Accuracy of the numerical scheme was validated by comparisons of simulation results with the hydrodynamic linear stability theory for Poiseuille flow. Two cases are presented to demonstrate the capabilities of the code: (a) a neutrally stable case of Reynolds number, Re = 400 and (b) an unstably stratified case at Re = 1,000 requiring very high resolution in all coordinate directions. Results indicate that the scalability is not limited by the overall size of the problem, but rather by the number of mesh points per processor. Strong scaling is demonstrated for both cases with as few as 10,000 unknowns per processor.

Wake effects on turbulent transport in the convective boundary layer

The effect of the downstream propagation of a wake on the transport of momentum, energy and scalars (such as humidity) in the convective boundary layer (CBL) is studied using a direct numerical simulation. The incompressible Navier–Stokes and energy equations are integrated under neutral and unstable thermal stratification conditions in a rotating coordinate frame with the Ekman layer approximation. Wake effects are introduced by modifying the mean velocity field as an initial condition on a converged turbulent Ekman layer flow. With this initial velocity distribution, the governing equations are integrated in time to determine how turbulent transport in the CBL is affected by the wake. Through the use of Taylor’s hypothesis, temporal evolution of the flow field in a doubly periodic computational domain is transformed into a spatial evolution. The results clearly indicate an increase in the scalar flux at the surface for the neutrally stratified case. An increase in wall scalar and heat flux is also noted for the CBL under unstable stratification, though the effects are diminished given the enhanced buoyant mixing associated with the hot wall.

Modeling high-order statistics in the turbulent Ekman layer

ABSTRACT Results from a direct numerical simulation (DNS) of the neutral and unstable turbulent Ekman layer at a Reynolds number of 1000 were used to evaluate turbulence closure models. For the neutrally stratified Ekman layer, the higher-order moments of velocity were examined and the accuracy of a kurtosis model was assessed. For the unstable Ekman layer, the analysis of higher-order moments was extended to temperature-velocity correlations. Model coefficients were optimised using DNS data and it was shown that the optimised models accurately captured the distributions of all fourth-order moments. These low-Reynolds number results can be extrapolated to higher Reynolds numbers to parameterise turbulence in other flow fields with rotational effects such as the atmospheric boundary layer.

Reply to ''Comments on 'Direct Numerical Simulation of the Turbulent Ekman Layer: Evaluation of Closure Models'''

In this note, we respond to J. C. Bergmann’s comments (Bergmann 2014) regarding our recent article assessing Ekman-layer turbulence closure models using direct numerical simulation (DNS) results. As Bergmann notes in his introduction, many of the comments that he provides are not specifically related to our paper. While many of these issues are of interest for understanding the dynamicsoftheatmosphericboundarylayer(ABL),wewill limit our responses primarily to those issues that directly relate to the objectives and the results of our paper. In the following reply, we first discuss of the applicabilityoftheneutral,turbulentEkmanlayerasamodelfor the ABL, and we then address some fundamental objections that Bergmann raises to the use of DNS. We also address Bergmann’s other concerns regarding turbulence models, momentum balance, and vertical exchange and eddy viscosity modeling assumptions.

Parallel Implementation of a Navier-Stokes Solver: Turbulent Ekman Layer Direct Numerical Simulation

A massively parallel direct numerical solution procedure for the turbulent Ekman layer is presented. The simulation studies the dynamics of turbulence for this flow by solving the incompressible Navier-Stokes equations with Coriolis and buoyancy terms. The governing equations are integrated via a semi-implicit time advancement algorithm which is massively parallelized using the Portable, Extensible Toolkit for Scientific Computation (PETSc) libraries. Accuracy of the numerical scheme was validated by comparisons of simulation results with the hydrodynamic linear stability theory for Poiseuille flow. Two cases are presented to demonstrate the capabilities of the code: a) a neutrally stable case of Reynolds number, Re = 400 and b) an unstably stratified case at Re = 1000 requiring very high resolution in all coordinate directions. Results indicate that the scalability is not limited by the overall size of the problem, but rather by the number of mesh points per processor. Strong scaling is demonstrated for both cases with as few as 10,000 unknowns per processor.

Analysis of signal propagation in an elastic-tube flow model

We combine linear and nonlinear signal analysis techniques to investigate the transmission of pressure signals along a one-dimensional model of fluid flow in an elastic tube. We derive a simple, generally applicable measure for the robustness of a simulated vessel against in vivo pressure fluctuations, based on quantifying the degree of synchronization between proximal and distal pressure pulses. The practical use of this measure will be in its application to simulated pulses generated in response to a stochastic forcing term mimicking biological variations of root pressure in arterial blood flow.

Instantaneous Turbulent Flow Structures of the Numerically Simulated Ekman Layer

Results from a direct numerical simulation of the Ekman layer at a Reynolds number of 400 have been evaluated to identify turbulent flow structures in the instantaneous velocity fields. Archetypal structures revealed by means of two dimensional two point velocity correlations show that the rotational boundary layer is many respects similar to other turbulent flowfields, with the near wall region populated by streaky, elongated eddies, while the flowfield away from the wall shows marked spanwise broadening of the characteristic flow structures. As the eddies move away from the wall, tilting and lifting of downstream segments are observed, similar to observations made of non-rotational boundary layers. Some structural differences are noted which may be a function of system rotation, especially in the spanwise sectional isocorrelation projections. Reversal of the observed tilting direction in the streamwise velocity two point correlations are attributed to vertical variations in the Reynolds stress and mean velocity gradient vectors.