Direct Numerical Simulation Of Turbulent Flow Over A Dimpled Flat Plate Using An Immersed Boundary Technique

Many methods of passive flow control rely on changes to surface morphology. Roughening surfaces to induce boundary layer transition to turbulence and in turn delay separation is a powerful approach to lowering drag on bluff bodies. While the influence in broad terms of how roughness and other means of passive flow control to delay separation on bluff bodies is known, basic mechanisms are not well understood. Of particular interest for the current work is understanding the role of surface dimpling on boundary layers. A computational approach is employed and the study has two main goals. The first is to understand and advance the numerical methodology utilized for the computations. The second is to shed some light on the details of how surface dimples distort boundary layers and cause transition to turbulence. Simulations are performed of the flow over a simplified configuration: the flow of a boundary layer over a dimpled flat plate. The flow is modeled using an immersed boundary as a representation of the dimpled surface along with direct numerical simulation of the Navier-Stokes equations. The dimple geometry used is fixed and is that of a spherical depression in the flat plate with a depth-to-diameter ratio of 0.1. The dimples are arranged in staggered rows separated by spacing of the center of the bottom of the dimples by one diameter in both the spanwise and streamwise dimensions. The simulations are conducted for both two and three staggered rows of dimples. Flow variables are normalized at the inlet by the dimple depth and the Reynolds number is specified as 4000 (based on freestream velocity and inlet boundary layer thickness). First and second order statistics show the turbulent boundary layers correlate well to channel flow and flow of a zero pressure gradient flat plate boundary layers in the viscous sublayer and the buffer layer, but deviates further away from the wall. The forcing of transition to turbulence by the dimples is unlike the transition caused by a naturally transitioning flow, a small perturbation such as trip tape in experimental flows, or noise in the inlet condition for computational flows.

Passive flow control achieved by surface dimpling can be an effective strategy for reducing drag around bluff bodies - an example of substantial popular interest being the flow around a golf ball. While the general effect of dimples causing a delay of boundary layer separation is well known, the mechanisms contributing to this phenomena are subtle and not thoroughly understood. Numerical models offer a powerful approach for studying drag reduction, however simulation strategies are challenged by complex geometries, and in applications the introduction of ad hoc turbulence models which introduce additional uncertainty. These and other factors provide much of the motivation for the current study, which focused on the numerical simulations of the flow over a simplified configuration consisting of a dimpled flat plate. The principal goals of the work are to understand the performance of the numerical methodology, and gain insight into the underlying physics of the flow. Direct numerical simulation of the incompressible Navier-Stokes equations using a fractional step method was employed, with the dimpled flat plate represented using an immersed boundary method. The dimple geometry utilizes a fixed dimple aspect ratio, with dimples arranged in a single spanwise row. The grid sizes considered ranged from approximately 3 to 99 million grid points. Reynolds numbers of 3000 and 4000 based on the inlet laminar boundary layer thickness were simulated. A turbulent boundary layer was induced downstream of the dimples for Reynolds numbers which did not transition for the flow over an undimpled flat plate. First and second order statistics of the boundary layer that develops agree reasonably well with those for turbulent channel flow and flat plate boundary layers in the sublayer and buffer layers, but differ in the outer layer. Inspection of flow visualizations suggest that early transition is promoted by thinning of the boundary layer, initiation of shear layer instabilities over the dimples, flow separation and reattachment, and tripping of the boundary layer at the trailing edge of the dimples.

Flow separation is encountered in many engineering devices, e.g., turbines, diffusers, wings and aftbodies of aircrafts. The physical mechanisms of separated turbulent boundary layers over curved walls are not yet well understood. The main objectives of the present study are to: (i) develop an efficient numerical methodology to perform direct numerical simulations (DNS) of spatially-developing turbulent boundary layers (SDTBLs) over curved walls, and (ii) enhance our knowledge on the dynamics of turbulence in SDTBLs separating over curved walls. To achieve these objectives, we have developed a new pressure-correction method, called FastRK3, for simulating incompressible flows over curved walls. FastRK3 solves the incompressible Navier-Stokes (NS) equations written in orthogonal curvilinear coordinates. The orthogonal formulation of the NS equations substantially reduces the computational cost of the flow solver and the numerical stencils of its second-order finite difference discretization mirror that of the Cartesian formulation. This property allows us to develop an FFT-based Poisson solver for pressure, called FastPoc, for those cases where the components of the metric tensor are independent of one spatial direction: surfaces of linear translation (e.g., curved ramps and bumps) and surfaces of revolution (e.g., axisymmetric shapes). Our results show that the new FFT-based Poisson solver, FastPoc, is thirty to sixty times faster than the multigrid-based linear solver, and the new flow solver, FastRK3, is overall four to seven times faster when using FastPoc rather than multigrid. FastRK3 is an explicit, three-stage, third-order Runge-Kutta based projection-method which requires solving the Poisson equation for pressure only once per time step. We show theoretically and numerically that (i) FastRK3 has the same temporal order of accuracy for pressure and velocity as the standard RK3 method for both free-shear and wall-bounded flows when the RK3 coefficients and the pressure extrapolation scheme satisfy specific conditions herein theoretically derived, (ii) FastRK3 is third-order accurate in time for velocity and second-order accurate in time for pressure for free-shear flows, and (iii) FastRK3 is second-order accurate in time for velocity and pressure for `stiff' wall-bounded flows. In summary, given that the computational mesh satisfies the property of orthogonality, FastRK3 simulates flows over curved walls with second-order accuracy in both space and time. Using FastRK3, we perform DNS of a SDTBL separating over a curved wall. We validate FastRK3 by comparing our numerical results with published experiments. For the first time, we derive the budget equations of the turbulence kinetic energy and of the Reynolds stresses in orthogonal coordinates, and report the results from our DNS. We study the dynamics of turbulence of the separated flow over the curved wall by analyzing these budget equations. Our analysis shows that, in the separated region over the curved ramp, the TKE production occurs through the production of (u2) as well as (v2) in contrast to a ZPG SDTBL where the TKE production is mostly through the production of (u2). In the curved ramp region, the viscous diffusion and dissipation of (v2) and (uv) are not zero at the wall, unlike that for both a ZPG SDTBL over a flat-plate as well as a pressure-gradient induced turbulent flow separation over a flat plate. And, the curved ramp region of the flow is characterized by enhanced transport of the Reynolds stresses compared to those of the upstream ZPG SDTBL due to the mixing layer created in the flow by the flow separation. Finally, our results have shown, for the first time, that the Reynolds stress profiles and budgets in the orthogonal curvilinear coordinates are very similar to those in the APG region of the 'pressure-gradient induced flow separation' in a flat-plate turbulent boundary layer. Such a comparison is only possible because (i) we employ a structured orthogonal grid over the curved ramp in our simulations, and (ii) FastRK3 solves the governing equations written in orthogonal curvilinear coordinates.

Direct Numerical Simulation of Turbulence with Scalar Transfer around Complex Geometries Using the Immersed Boundary Method and Fully Conservative Higher-Order Finite-Difference Schemes.

A three-dimensional, turbulent flow in a channel with a sudden expansion was studied by direct numerical simulation of the incompressible Navier-Stokes equations. The objective of this study was to provide statistical data of backwardfacing step flow for turbulence modelling. Additionally, analysis of the statistical and dynamical properties of the flow is performed. The Reynolds number of the main simulation was Reh = 9000, based on the step height and mean inlet velocity, with the expansion ratio ER = 2:0. The discretisation is performed using the spectral/hp element method with stiffly-stable velocity correction scheme for time integration. The inlet boundary condition is a fully turbulent velocity and pressure field regenerated from a plane downstream of the inlet. A constant flowrate was ensured by applying Stokes flow correction in the inlet regeneration area. Time and spanwise averaged results revealed, apart from the primary recirculation bubble, secondary and tertiary corner eddies. Streamlines show an additional small eddy at the downstream tip of the secondary corner eddy, with the same circulation direction as the secondary vortex. The analysis of the 3D, timeonly average shows the wavy spanwise structure of both primary and secondary recirculation bubble, that results in spanwise variations of the mean reattachment location. The visualisation of spanwise averaged pressure uctuations and streamwise velocity showed that the interaction of vortices with the recirculation bubble is responsible for the apping of the reattachment position. The characteristic frequency St = 0:078 was found. The analysis of small-scale energy transfer was performed to reveal large backscatter regions in strong Reynolds stress areas in the mixing layer. High correlation of small-scale transfer with non-linear interaction of large-scale velocity and small-scale vorticity was found. The data of the flow fields was archived. It contains the averages for velocities, pressure and Reynolds stress tensor, as well as 3D instantaneous pressure and velocity history.