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A fractional step method for solving the steady-state, incompressible, Navier-Stokes equations is presented. The proposed iterative method uses an estimate for the optimal under-relaxation coefficient at each iteration. This estimate uses prior information that is already available in the iterative method so the calculation has negligible cost. Numerical tests show that the convergence rate of the fractional step method using this optimal relaxation parameter is nearly independent of the mesh size, resulting in significant cost savings for problems with fine meshes. The ideas demonstrated in this paper are general and can be used to improve many existing iterative methods for steady flows, as well as methods for the unsteady Navier-Stokes equations and methods for other saddle-point problems.

The Oriented-Eddy Collision (OEC) model treats turbulent flow as a non-Newtonian fluid where the average behavior of turbulence is modeled as a collection of interacting fluid particles which have inherent orientation. The model is derived from the two-point velocity correlation transport equation, and has the form of a collection of Reynolds-stress transport equations, with one set of transport equations for each representative eddy direction. The addition of eddy orientation information adds important physics to the model and allows the model to represent structural (two-point) information about the turbulence. This structural information is sufficient to allow the model to capture the effect of external forces and imposed mean strains (such as rapid distortion theory) exactly. The only physical effects that must be empirically modeled are those that are due to turbulence-turbulence interactions, referred to as eddy collisions. The performance of the model in a number of canonical flow situations is presented.

Prior work has demonstrated the effectiveness of using two-equation closures as the basis for universal, self-adapting turbulence models that are effective at any mesh resolution (Perot and Gadebusch in Phys. Fluids 19:115105, 2007). In order to demonstrate the broad applicability of the fundamental approach, the same behavior is now demonstrated for a second-moment closure (SMC). The SMC has the advantage over the earlier two-equation universal closure of being more accurate in the coarse mesh limit and of having a natural mechanism for backscattering energy from the modeled to the resolved turbulent fluctuations. The mathematical explanation for why Reynolds averaged (RANS) transport equation closures are applicable at any mesh resolution, including the large eddy simulation (LES) regime, is reviewed. It is demonstrated that for the problem of isotropic decaying turbulence, the SMC model produces good predictions at any mesh resolution and with arbitrary initial conditions. In addition, it is shown that the proposed model automatically adapts to the mesh resolution provided. The self-adaptive nature of the method is clearly observed when different initial conditions are used. It is shown that classic RANS models (often thought to produce steady and smooth solutions) can produce three-dimensional, unsteady, and chaotic solutions when generalized correctly and when provided with sufficient mesh resolution. The implications of these observations on the fundamental theories of RANS and LES turbulence modeling are discussed.

Direct numerical simulations (DNSs) are used to investigate the drag-reducing performance of superhydrophobic surfaces (SHSs) in turbulent channel flow. SHSs combine surface roughness with hydrophobicity and can, in some cases, support a shear-free air–water interface. Slip velocities, wall shear stresses and Reynolds stresses are considered for a variety of SHS microfeature geometry configurations at a friction Reynolds number of Reτ ≈ 180. For the largest microfeature spacing studied, an average slip velocity over 75% of the bulk velocity is obtained, and the wall shear stress reduction is found to be nearly 40%. The simulation results suggest that the mean velocity profile near the superhydrophobic wall continues to scale with the wall shear stress but is offset by a slip velocity that increases with increasing microfeature spacing.

The turbulent potential model is a Reynolds-averaged (RANS) turbulence model that is theoretically capable of capturing nonequilibrium turbulent flows at a computational cost and complexity comparable to two-equation models. The ability of the turbulent potential model to predict nonequilibrium turbulent flows accurately is evaluated in this work. The flow in a spanwise-driven channel flow and over a swept bump are used to evaluate the turbulent potential model's ability to predict complex three-dimensional boundary layers. Results of turbulent vortex shedding behind a triangular and a square cylinder are also presented in order to evaluate the model's ability to predict unsteady flows. Early indications suggest that models of this type may be capable of significantly enhancing current numerical predictions of turbomachinery components at little extra computational cost or additional code complexity.

The effect of droplet oscillation on internal heat transfer was investigated using a transient, three-dimensional Navier-Stokes solver for free-surface flows. The code solves conservation of momentum and energy on a three-dimensional deforming domain. The investigation sought to explore the effect that oscillations might have on temperature non-uniformity within droplets. Biot numbers of 0.1 and 0.25 were investigated for three different degrees of distortion: no perturbation, 13% and 35%. The droplets were given an initial perturbation and allowed to relax back to a spherical shape. The results show that the effect of the oscillation is minimal. As expected, the effect was greatest at large distortion and large Biot number. However, it appears that distortion does not contribute much to heat transfer within droplets.Copyright © 2002 by ASME

The effect of axisymmetric contraction and expansion on isotropic turbulence and its subsequent decay are evaluated by way of direct numerical simulation. Low, moderate, and high rates of contraction and expansion are evaluated at a moderate Reynolds number using two different sets of initial conditions of isotropic turbulence. The initial turbulence is generated via mechanical mixing so that the large scales of the turbulence develop naturally. Length scales, decay rates, anisotropy, and two-point correlations are investigated both during and after the straining process, with the goal of quantifying how anisotropic turbulence behaves during return-to-isotropy.

Direct numerical simulation is used to evaluate the effect of plane strain on isotropic homogeneous turbulence. The subsequent return to isotropy after the removal of the strain is also investigated. Large, moderate, and small strain rates are computed at moderate turbulence Reynolds numbers. The initial turbulence is generated via mechanical mixing so that the large scale turbulence develops relatively naturally. Turbulence length scales, Reynolds numbers, decay rates, and anisotropy are computed over the range of the simulations, with the goal of quantifying how anisotropic decay behaves. The simulations indicate that large scale anisotropy may not decay to zero at very large times. In agreement with experimental data, the presence of a recovery region is discerned before the return process is observed. Trajectory crossing is observed on the anisotropy invariant map indicating that anisotropy itself is not sufficient to determine its time evolution. Model constants for classic return-to-isotropy models are determined from the data and shown to vary with time. The oriented-eddy collision model [M. B. Martell and J. B. Perot, “The oriented-eddy collision turbulence model,” Flow, Turbul. Combust. 89(3), 335 (2012)], which includes turbulent structure information, is shown to predict the salient structure of the straining and return process.

Direct numerical simulation is used to investigate the decay exponent of isotropic homogeneous turbulence over a range of Reynolds numbers sufficient to display both high and low Re number decay behavior. The initial turbulence is generated by the stirring action of the flow past many small randomly placed cubes. Stirring occurs at 1/30th of the simulation domain size so that the low-wavenumber and large scale behavior of the turbulent spectrum is generated by the fluid and is not imposed. It is shown that the decay exponent in the resulting turbulence matches the theoretical predictions for a k2 low-wavenumber spectrum at both high and low Reynolds numbers. The transition from high Reynolds number behavior to low Reynolds number behavior occurs relatively abruptly at a turbulent Reynolds number of around 250 ( Re λ≈41).

These surfaces have been shown to provide drag reduction in laminar and turbulent flows. In this work, direct numerical simulation is used to investigate the drag reducing performance of superhydrophobic surfaces in turbulent channel flow. Slip velocities, wall shear stresses, and Reynolds stresses are determined for a variety of superhydrophobic surface microfeature geometry configurations at friction Reynolds numbers of Re180, Re395, and Re590. This work provides evidence that superhydrophobic surfaces are capable of reducing drag in turbulent flow situations by manipulating the laminar sublayer. For the largest microfeature spacing, an average slip velocity over 80% of the bulk velocity is obtained, and the wall shear stress reduction is found to be greater than 50%. The simulation results suggest that the mean velocity profile near the superhydrophobic wall continues to scale with the wall shear stress and the log layer is still present, but both are offset by a slip velocity that is primarily dependent on the microfeature spacing. © 2010 American Institute of Physics. doi:10.1063/1.3432514

A two-equation transport model is used to model turbulence at any mesh resolution, from Reynolds-averaged Navier–Stokes (RANS), to large eddy simulations (LES), to direct numerical simulations. The two-equation model used is a slight modification of the standard k∕e model that allows the backscatter of energy to resolved scales. A mathematical explanation is provided for why RANS models (such as this two-equation model) are applicable to LES. The model automatically adapts to the mesh resolution provided and no interaction from the user is necessary. This approach is tested on the problem of moderately high Reynolds number isotropic decaying turbulence and gives good predictions at any mesh resolution and with different initial conditions. A detailed analysis shows that at LES resolutions the solution remains fully unsteady and three-dimensional and does not approach a RANS-like solution.

From simulation data it is shown that the difference between the true rate of dissipation of turbulent kinetic energy and the ‘‘isotropic dissipation’’ used by Reynolds‐averaged modelers is less than 2% in a conventional viscous wall region, and negligible elsewhere. The difference is a contribution to viscous diffusion which is a small fraction of the total.