Three-dimensional Turbulent Boundary Layer Research Articles

THEORETICAL STUDY OF A THREE-DIMENSIONAL TURBULENT BOUNDARY LAYER OVER A ROTATING CONE OR DISK

In this study, we investigated turbulent boundary layer flow due to a rotating cone or disk. A comprehensive theoretical analysis was performed and the results for several physical quantities of significant interest are presented. The problem was modeled such that both cone and disk cases could be achieved under the same formulation. In most situations, comparisons were made with the experimental results available in the literature. A primary objective of this study was to present reliable results of a rotating cone or disk with respect to several physical quantities of significant interest in order to facilitate future studies. The results were obtained at 45° and 60° angles in the cone case and at a 90° angle in the disk case. The Keller-box method was applied to numerically solve the Reynolds-averaged Navier-Stokes equations. The mixing-length turbulent model was implemented to incorporate the turbulence effects. The validity of this method was first confirmed by comparing the obtained solution for the disk case with the existing literature produced by different researchers. Circumferential and radial velocity distributions and polar plots in the cone and disk cases are presented for Reynolds numbers up to Re<i><sub>r</sub></i> = 1 × 10<sup>6</sup>. However, the circumferential and radial skin-friction coefficients, combined friction coefficient, moment coefficient, boundary layer thickness, displacement thickness, velocity shape factor, flow rate, and flow swirl angle are provided up to Re<i><sub>r</sub></i> = 1 × 10<sup>8</sup>. We also noted that the asymptotic nature of velocities can be achieved by setting the value of grid parameter <i>K</i>.

Strategies for Computational Fluid Dynamics Validation Experiments

Abstract The Benchmark Validation Experiment for Reynolds-averaged Navier–Stokes (RANS)/large eddy simulation (LES) Investigations (BeVERLI) aims to produce an experimental dataset of three-dimensional non-equilibrium turbulent boundary layers with various levels of separation that, for the first time, meets the most exacting requirements of computational fluid dynamics validation. The application of simulations and modeling in high-consequence engineering environments has become increasingly prominent in the past two decades, considerably raising the standards and demands of model validation and forcing a significant paradigm shift in the design of corresponding validation experiments. In this paper, based on the experiences of project BeVERLI, we present strategies for designing and executing validation experiments, hoping to ease the transition into this new era of fluid dynamics experimentation and help upcoming validation experiments succeed. We discuss the selection of a flow for validation, the synergistic use of simulations and experiments, cross-institutional collaborations, and tools, such as model scans, time-dependent measurements, and repeated and redundant measurements. The proposed strategies are shown to successfully mitigate risks and enable the methodical identification, measurement, uncertainty quantification, and characterization of critical flow features, boundary conditions, and corresponding sensitivities, promoting the highest levels of model validation experiment completeness per Oberkampf and Smith [1]. Furthermore, the applicability of these strategies to estimating critical and difficult-to-obtain bias error uncertainties of different measurement systems, e.g., the underprediction of high-order statistical moments from particle image velocimetry velocity field data due to spatial filtering effects, and to systematically assessing the quality of uncertainty estimates is shown.

Some Studies on Three Dimensional Turbulent Boundary Layer Flows

This paper describes some studies on flow separation and on the extended Cebeci-Smith turbulence and transition models, carried out using a three-dimensional turbulent boundary layer code developed at the Vikram Sarabhai Space Center (VSSC), Thiruvananthapuram, India. This code uses an orthogonal curvilinear surface fitted co-ordinate system. The governing equations are solved, by an implicit space marching numerical method called the Keller Box Scheme with suitable modifications for reverse crossflows. Initial data on two intersecting planes for starting the solution procedure, are obtained by solving the appropriate limiting form of the governing equations. Three transition models available in the Literature are studied. A few sample results are presented and compared with experimental measurements.

Transition of a temporally developing three-dimensional turbulent boundary layer

Transition of a temporally developing three-dimensional turbulent boundary layer

Wall-modelled large-eddy simulation of three-dimensional turbulent boundary layer in a bent square duct

We conduct wall-modelled large-eddy simulation (WMLES) of a pressure-driven three-dimensional turbulent boundary layer developing on the floor of a bent square duct to investigate the predictive capability of three widely used wall models, namely, a simple equilibrium stress model, an integral non-equilibrium model, and a partial differential equation (PDE) non-equilibrium model. The numerical results are compared with the experiment of Schwarz & Bradshaw (J. Fluid Mech., vol. 272, 1994, pp. 183–210). While the wall-stress magnitudes predicted by the three wall models are comparable, the PDE non-equilibrium wall model produces a substantially more accurate prediction of the wall-stress direction, followed by the integral non-equilibrium wall model. The wall-stress direction from the wall models is shown to have separable contributions from the equilibrium stress part and the integrated non-equilibrium effects, where how the latter is modelled differs among the wall models. The triangular plot of the wall-model solution reveals different capabilities of the wall models in representing variation of flow direction along the wall-normal direction. In contrast, the outer LES solution is unaffected by the type of wall model used, resulting in nearly identical predictions of the mean and turbulent statistics in the outer region for all the wall models. This is explained by the vorticity dynamics and the inviscid skewing mechanism of generating the mean three-dimensionality. Finally, the LES solution in the outer layer is used to study the anisotropy of turbulence. In contrast to the canonical two-dimensional wall turbulence, the Reynolds stress anisotropy exhibits strong non-monotonic behaviour with increasing wall distance.

Structural aspects of the attached turbulent boundary layer flow over a hill

Three-dimensional turbulent boundary layers under strong pressure gradients and curvature are the rule in real-world flow applications but are typically not well predicted by turbulence models due to isotropic eddy viscosity or equilibrium assumptions. Validation-quality data in complex 3D flows is necessary for continued efforts to improve simulation accuracy. The Benchmark Validation Experiments for RANS/LES Investigations (BeVERLI) Hill bump model, designed specifically for validation experiments, was tested in the Virginia Tech Stability Wind Tunnel to collect validation experiment data on the three-dimensional (3D) boundary layer flow over a 3D hill. Laser Doppler velocimetry measurements on the bump model were used to study the mean flow and turbulence structure and evaluate the impact of pressure gradient and curvature upon the total shear stress in the boundary layer and evaluate the impact of pressure gradient and curvature upon the total shear stress behavior in the near-wall region. From analysis of the BeVERLI Hill flow, including the boundary layer just upstream of the hill, and comparison with the 3D flow around a wing-body junction of Ölçmen et al. (Exp Fluids 31:219–228, 2001), it is shown that none of the stations studied exhibit a constant shear stress region over any significant region of the boundary layer.Graphical abstract

Stationary cross-flow breakdown in a high-speed swept-wing boundary layer

A new type-II secondary instability mode was recently identified in high-speed cross-flows using stability analysis, but its role in the transition process is not yet clear. Here, the breakdown of stationary cross-flow vortices at high speeds is examined using direct numerical simulation to determine differences from the low-speed case. The transition is achieved by disturbing stationary cross-flow vortices with unsteady blowing/suction in a swept-wing boundary layer with swept angle 45°, free-stream Mach number 6, and unit Reynolds number 8 ×106. The results reveal that, as in low-speed cases, the type-I secondary instability mode (with frequency ≈190 kHz) is crucial to the breakdown, but neither the traditional nor the new type-II secondary instability play a role. The vortical structure induced by the type-I secondary instability mode has two counter-rotating tubes stretched along the spanwise direction and a footprint aligned normal to the mean flow direction. The composite vortex structures are similar to rolls/braids in plane free-shear layers arising from Kelvin–Helmholtz instability and they evolve into hairpins in the late stage of the transition. Some preliminary statistics from a three-dimensional turbulent boundary layer are provided as a comparison to the two-dimensional ones. The fluctuating cross-flow velocity does not contribute to the momentum and heat transfer on average, probably due to the very weak mean cross-flow profile. Thus, the obtained three-dimensional turbulent boundary layer is the same as the two-dimensional one but inclined by a swept angle. To the authors’ knowledge, this is the first in-depth analysis of the high-speed cross-flow transition to full turbulence.

Calculation of the three-dimensional turbulent boundary layer in the vessel’s bow and midship area by the integral method

The quality of technical solutions applied in ship design and construction is substantially determined by the validity of viscous flow parameters, especially in the vicinity of thrusters. The influence of shapes on velocity, pressure and turbulence distribution can be assessed by physical experimentation. Along with undeniable advantages, physical modelling has a number of disadvantages: high labour input and cost, limited range of parameters variation (e.g. Reynolds numbers), difficulty in separating the influence of individual factors. Therefore, the development of mathematical models and numerical calculation schemes based on them is extremely relevant. Calculation of friction resistance and calculation of the velocity field in the vicinity of a vessel comes down to the calculation of its boundary layer and trace parameters. Current mathematical models with a planar representation, i.e. working with projections rather than real curves, do not give an accurate description. This has led to the need for a mathematical model that can address all of the required characteristics of a three-dimensional turbulent boundary layer. Both differential and integral methods are used to calculate three-dimensional boundary layers. The model is capable of calculating the characteristics of the spatial boundary layer in the stern and the viscous wake, in the bow and in the middle of the vessel. The model is quite versatile: it has been successfully used both for computing the boundary layer characteristics in the bow and midship area by the integral method based on the thick boundary layer concept, and for computing the turbulent flow in the stern and viscous wake by the differential method based on the partial parabolic concept using the k-ε turbulence model. In this paper we will elaborate on the integral method calculation.

Direct numerical simulation of a non-equilibrium three-dimensional turbulent boundary layer over a flat plate

Abstract

Experimental investigation of coherent structures of a three-dimensional separated turbulent boundary layer

Coherent structures of a three-dimensional (3D) separation due to an adverse pressure gradient are investigated experimentally. The flow set-up consists of a flat plate to develop a turbulent boundary layer upstream of an asymmetric two-dimensional diffuser with one diverging surface. The diffuser surface has an initial mild curvature followed by a flat section where flow separation occurs. The top and the two sidewalls of the diffuser are not equipped with any flow control mechanism to form a 3D separation. Planar particle image velocimetry (PIV) using four side-by-side cameras is applied to characterize the flow with high spatial resolution over a large streamwise-wall-normal field of view (FOV). Tomographic PIV (tomo-PIV) is also applied for volumetric measurement in a domain flush with the flat surface of the diffuser. The mean flow obtained from averaging instantaneous velocity fields of this intermittent unsteady flow appears as a vortex with an elliptical cross-section. The major axis of the ellipse is tilted with respect to the streamwise direction. As a result, the average velocity in the mid-span of the diffuser has an upstream forward flow and a downstream backward flow, separated by a point of zero wall shear stress. Sweep motions mainly carry out transport of turbulent kinetic energy upstream of this point, while ejections dominate at the downstream region. In the instantaneous flow fields, forward and backward flows have equivalent strength, and the separation front is extended in the spanwise direction. The conditional average of the separation instants forms a saddle-point structure with streamlines converging in the spanwise direction. Proper orthogonal decomposition (POD) of the tomo-PIV data demonstrates that about 42 % of the turbulent kinetic energy is present in the first pair of modes, with a strong spanwise component. The spatial modes of POD also show focus, node and saddle-point structures. The average of the coefficients of the dominant POD modes during the separation events is used to develop a reduced-order model (ROM). Based on the ROM, the instantaneous 3D separation over the diffuser is a saddle-point structure interacting with focus-type structures.

Finite-volume scheme for the solution of integral boundary layer equations

An unstructured-mesh finite-volume formulation for the solution of systems of steady conservation laws on embedded surfaces is presented. The formulation is invariant to the choice of local tangential coordinate systems and is stabilized by a novel up-winding scheme applicable also to mixed-hyperbolic systems. The formulation results in a system of non-linear equations which is solved by a quasi-Newton method. While the finite volume scheme is applicable to a range of conservation laws, it is here implemented for the solution of the integral boundary layer equations, as a first step in developing a fully coupled viscous-inviscid interaction method. For validation purposes, integral boundary layer quantities computed using a minimal set of three-dimensional turbulent integral boundary layer equations are compared to experimental data and an established computer code for two-dimensional problems. The validation shows that the proposed formulation is stable, yields a well-conditioned global Jacobian, is conservative on curved surfaces and invariant to rotation as well as convergent with regard to mesh refinement.

Modifications of the near-wall structures in thermally induced three-dimensional turbulent boundary layer

Statistical analysis is carried out in order to examine the modifications of near wall coherent structures in a thermally induced three-dimensional turbulent boundary layer (3DTBL). The buoyancy forces modify the counter rotating vortices in the near-wall region with different degrees in a similar way to that produced by the pressure and shear driven three-dimensional turbulent boundary layer flows. Evidence of such modifications that is highlighted in the present study may explain the reduction of Reynolds shear stress and turbulence activities in the near-wall region. It has been shown that the induced three-dimensionality weakens the strength of near-wall vortices in such a way that vortices of one sign are weakened to produce ejections and those of the other sign are weakened to produce sweeps. In contrast to the hypotheses explaining the reduction of turbulence activities in pressure and shear driven 3DTBL flows, vortices rotating in the same direction as the background mean spanwise vorticity in the near-wall region are weakened much more than those rotating in the opposite direction.

Assessment of Turbulence Model Predictions for an Aero-Engine Centrifugal Compressor

The accurate prediction of mean flow fields with high degrees of curvature, adverse pressure gradients, and three-dimensional turbulent boundary layers typically present in centrifugal compressor stages is a significant challenge when applying Reynolds averaged Navier–Stokes turbulence modeling techniques. The current study compares the steady-state mixing plane predictions using four turbulence models for a centrifugal compressor stage with a tandem impeller and a “fish-tail” style discrete passage diffuser. The models analyzed are the k-ε model (an industry standard for many years), the shear stress transport (SST) model, a proposed modification to the SST model denoted as the SST-reattachment modification (RM), and the Speziale, Sarkar, and Gatski Reynolds stress model (RSM-SSG). Comparisons with measured performance parameters—the stage total-to-static pressure and total-to-total temperature ratios—indicate more accurate performance predictions from the RSM-SSG and SST models as compared to the k-ε and SST-RM models. Details of the different predicted flow fields are presented. Estimates of blockage, aerodynamic slip factor, and impeller exit velocity profiles indicate significant physical differences in the predictions at the impeller-diffuser interface. Topological flow field differences are observed: the separated tip clearance flow is found to reattach with the SST, SST-RM, and RSM-SSG models, while it does not with the k-ε model, a larger shroud separation at the impeller exit seen with the SST and SST-RM models, and core flow differences are in the complex curved diffuser geometry. The results are discussed in terms of the production and dissipation of k predicted by the various models due to their intrinsic modeling assumptions. These comparisons will assist aerodynamic designers in choosing appropriate turbulence models, and may benefit future modeling research.

An advanced laser-Doppler velocimeter for full-vector particle position and velocity measurements

An advanced laser-Doppler measurement technique has been developed for fully resolved three-component particle position and velocity vector measurements in turbulent flows. The instrument deemed the ‘comprehensive’ laser-Doppler velocimeter employs a novel optical arrangement to measure multiple-component sub-measurement-volume-scale seed particle positions simultaneously with the velocity vector measurements of conventional laser-Doppler velocimetry (LDV). In the current paper, the effectiveness of the position resolution capabilities is considered, which allows for velocity statistics measurements at multiple locations within the measurement volume. Design estimates for the position vector uncertainties for a particle passing the measurement volume are about 3 µm root mean square in any direction, although in situ estimates indicate an uncertainty value closer to 14 µm with the possibility of further refinement through optimized alignment. To validate the operation of the instrument, measurements are presented in turbulent boundary layers previously examined with high-resolution conventional LDV. The flat-plate turbulent boundary layer is studied at two Reynolds numbers up to Reθ = 7500. Measurements are also presented in a pressure-driven three-dimensional turbulent boundary layer created beside a wing/body junction. These measurements illustrate the effectiveness of the technique for obtaining highly resolved velocity profiles within the measurement volume and give the highest spatial resolution velocity statistics published for Reynolds numbers of the magnitude studied.

Turbulence structural measurements using a comprehensive laser–Doppler velocimeter in two- and three-dimensional turbulent boundary layers

The advanced ‘comprehensive’ laser–Doppler velocimeter is used to acquire spatially and temporally resolved turbulence structural measurements in high Reynolds number two- and three-dimensional turbulent boundary layers. The new instrument directly measures three-dimensional particle trajectories at high repetitions. These trajectories are analyzed in post-processing to obtain fluctuating velocity gradient tensor fields, which lead to direct measurements of turbulent viscous dissipation rates. Such data acquired in two- and three-dimensional boundary layers with an approach flow momentum thickness Reynolds number, Re θ = 7500 are presented. Results indicate that anisotropy of the dissipation rate of Reynolds stresses persists to similar heights in viscous wall units as obtained with direct numerical simulations at lower Reynolds numbers. Measurements in a three-dimensional turbulent boundary layer in the vicinity of a wing/body junction also indicate that a reduction in the value of the velocity/pressure-gradient correlations for the Reynolds normal stresses reduces the turbulent energy redistribution and contributes to reduced shear stress magnitudes, as observed previously through DNS (Moin, P., Shih, T.-H., Driver, D., Mansour, N.N., 1990. Direct numerical simulation of a three-dimensional turbulent boundary layer, Phys. Fluids A 2 (10), 1846–1853).

Experimental transport-rate budgets in complex 3-D turbulent flow near a wing/body junction

Experimental flow measurements are presented for a wing–body junction flow obtained using laser-Doppler velocimetry. Mean velocity and Reynolds stress data are used to calculate the complete transport-rate budgets of Reynolds stresses and turbulent kinetic energy. The measurements were carried out in the Virginia Tech Boundary Layer Tunnel at a nominal air speed of 27.5 m/s around a NACA 0020 tail and 3:2 elliptical nose wing shape. Data are presented for a two-dimensional turbulent boundary layer (2DTBL), a strongly skewed three-dimensional turbulent boundary layer (3DTBL), a location in the vicinity of a 3-D separation line, and around the center of the vortex in the horse-shoe vortex that forms around the wing. Terms in the transport-rate equations were calculated also using the measured triple order fluctuating velocity products. Results show that the pressure-diffusion approximated by Lumley [Lumley, J.L., 1978. Computation modeling of turbulent flows. Adv. Appl. Mech. 18, 124–176] is an important term in the balance of v 2 ¯ , uv ¯ , and vw ¯ stress budgets; there were distinct differences between the two-dimensional and three-dimensional turbulent boundary layer budgets. Qualitative comparisons of experimental stress-transport-rate budgets to previous DNS results show a better agreement using the anisotropic dissipation rate of Hallbäck et al. [Hallbäck, M., Groth, J., Johansson, A.V., 1990. An algebraic model for nonisotropic turbulent dissipation rate in Reynolds stress closure, october. Phys. Fluids A 2 (10), 1859–1866].

Numerical prediction of turbulent boundary layer subjected to a sudden transverse strain at the wall

This study is about the behaviour of a two dimensional turbulent boundary layer on a flat plate subjected to a transverse motion of the bounding wall, producing a sudden shear force. The resulted flow is three-dimensional. Numerical prediction is achieved using the k-e model at high Reynolds number with the assumption of an isotropic turbulent viscosity. Close wall treatment constitutes the main difficulty within the three-dimensional turbulent boundary layer modelling. Therefore, we propose a law of wall along the transverse direction. Whilst the standard logarithmic function suggested in the literature for the two-dimensional case is used for the longitudinal component.

Similarity solution of mean streamwise flow along an external corner bisector

Turbulent boundary layer development, with zero pressure gradient, over an orthogonal external corner was investigated both experimentally and numerically. The results suggest that within the experimental and numerical limitations of the study that a similarity solution for the mean streamwise velocity profile exists along an external corner bisector. However, to confirm the self-similar solution, further independent measurement of the wall shear stress would be required. For the numerical simulation the Reynolds stress model was used. The experimental and numerical results were all consistent with a logarithmic self-similar solution. However the constants appearing in the self-similar solution were found to differ between the experimental and numerical results and reasons for these discrepancies are discussed. The results also provide scope for benchmarking of commercial software packages against a generic three-dimensional turbulent boundary layer that occurs in many engineering applications.

Three-dimensional Turbulent Boundary Layer in a Shrouded Rotating System

A thre-dimensional direct numerical simulation is combined with a laboratory study to describe the turbulent flow in an enclosed annular rotor-stator cavity characterized by a large aspect ratio G = (b − a)/h = 18.32 and a small radius ratio a/b = 0.152, where a and b are the inner and outer radii of the rotating disk and h is the interdisk spacing. The rotation rate Ω considered is equivalent to the rotational Reynolds number Re = Ωb2/ν= 9 .5 × 104 (ν the kinematic viscosity of water). This corresponds to a value at which experiment has revealed that the stator boundary layer is turbulent, whereas the rotor boundary layer is still laminar. Comparisons of the computed solution with velocity measurements have given good agreement for the mean and turbulent fields. The results enhance evidence of weak turbulence by comparing the turbulence properties with available data in the literature (Lygren and Andersson, J Fluid Mech 426:297–326, 2001). An approximately self-similar boundary layer behavior is observed along the stator. The wall-normal variations of the structural parameter and of characteristic angles confirm that this boundary layer is three-dimensional. A quadrant analysis (Kang et al., Phys Fluids 10:2315–2322, 1998) of conditionally averaged velocities shows that the asymmetries obtained are dominated by Reynolds stress-producing events in the stator boundary layer. Moreover, Case 1 vortices (with a positive wall induced velocity) are found to be the major source of generation of special strong events, in agreement with the conclusions of Lygren and Andersson (J Fluid Mech 426:297–326, 2001).

Characterization of the three-dimensional turbulent boundary layer in a concentric annulus with a rotating inner cylinder

Direct numerical simulations of fluid flow through a concentric annulus with a rotating inner wall were performed at ReDh=8900. To elucidate the modifications of the near-wall turbulent structure induced by rotation of the inner wall, we compared data obtained at rotation rates of N=0.0 and 0.429 for a system with a radius ratio (R*) of 0.5. Conditional quadrant/octant analysis and probability density functions of the velocity fluctuations revealed distinctive features of the three-dimensional turbulent boundary layer (3DTBL) in the concentric annulus with a rotating inner wall. Coherent structures near the inner wall were identified by a λ2-based eduction scheme to give the detailed information on the activated near-wall turbulent structures. The ensemble-averaging of the educed coherent vortices showed that enhanced ejections near the vortices were primarily responsible for the augmented turbulent structures. The alteration of the turbulent structures was attributed to the centrifugal force arising from rotation of the inner wall. The assumption of Littell and Eaton on the cause of the altered turbulent structures in 3DTBLs was invalid in the present study. Taken together, the present results showed that the 3DTBL in a rotating concentric annulus has features different from those observed in other types of 3DTBL due to the transverse curvature.