A Generalization of Prefactored Compact Schemes for Advection Equations

A generalized prefactorization of compact schemes aimed at reducing the stencil and improving the computational efficiency is proposed here in the framework of transport equations. By the prefactorization introduced here, the computational load associated with inverting multi-diagonal matrices is avoided, while the order of accuracy is preserved. The prefactorization can be applied to any centered compact difference scheme with arbitrary order of accuracy (results for compact schemes of up to sixteenth order of accuracy are included in the study). One notable restriction is that the proposed schemes can be applied in a predictor-corrector type marching scheme framework. Two test cases, associated with linear and nonlinear advection equations, respectively, are included to show the preservation of the order of accuracy and the increase of the computational efficiency of the prefactored compact schemes

On deriving nonreflecting boundary conditions in generalized curvilinear coordinates

In this work, nonreflecting boundary conditions in generalized three-dimensional curvilinear coordinates are derived, relying on the original analysis that was done in Cartesian two-dimensional coordinates by Giles (AIAA Journal, 28.12, 2050-2058, 1990). A thorough Fourier analysis of the linearized Euler equation is performed to determine the eigenvalues and the eigenvectors that are then used to derive the appropriate inflow and outflow boundary conditions. The analysis lacks rigorous proof of the well-posedness in the general case, which is open to investigation (a weak assumption is introduced here to complete the boundary conditions). The boundary conditions derived here are not tested on specific applications.Comment: 44 page

Numerical Anisotropy in Finite Differencing

Numerical solutions to hyperbolic partial differential equations, involving wave propagations in one direction, are subject to several specific errors, such as numerical dispersion, dissipation or aliasing. In multi-dimensions, where the waves propagate in all directions, there is an additional specific error resulting from the discretization of spatial derivatives along grid lines. Specifically, waves or wave packets in multi-dimensions propagate at different phase or group velocities, respectively, along different directions. A commonly used term for the aforementioned multidimensional discretization error is the numerical anisotropy or isotropy error. In this review, the numerical anisotropy is briefly described in the context of the wave equation in multi-dimensions. Then, several important studies that were focused on optimizations of finite difference schemes with the objective of reducing the numerical anisotropy are discussed

Combined prefactored compact schemes for first- and second-order derivatives: conceptual derivation

The derivation of combined prefactored compact schemes for first and second order derivatives is described here, relying on the Fourier analysis of the original prefactored compact schemes. By this approach, the order of accuracy of the original schemes can be increased from sixth to eight, or from eight to tenth (depending on the order of the original scheme), while the number of grid points in the stencil is kept the same. Here, we only frame the conceptual derivation of the schemes, leading to a closed set of equations for the weights.Comment: arXiv admin note: text overlap with arXiv:1902.0442

Effect of Wall Transpiration and Heat Transfer on Nonlinear G\"{o}rtler Vortices in High-speed Boundary Layers

G\"{o}rtler vortices in boundary layer flows over concave surfaces are caused by the imbalance between centrifugal effects and radial pressure gradients. Depending on various geometrical and/or freestream flow conditions, vortex breakdown via secondary instabilities leads to early transition to turbulence. It is desirable, therefore, to reduce vortex energy in an attempt to delay the transition from laminar to turbulent flow, and thereby achieve a reduced frictional drag. To this end, we apply a proportional control algorithm aimed at reducing the wall shear stress and the energy of G\"{o}rtler vortices evolving in high-speed boundary layers. The active control scheme is based on wall transpiration with sensors placed either in the flow or at the wall. In addition, we evaluate the effect of wall cooling and heating on G\"{o}rtler vortices evolving in high-speed boundary layers, by reducing or increasing the upstream wall temperature. The numerical results are obtained by solving the full Navier-Stokes equations in generalized curvilinear coordinates, using a high-order numerical algorithm. Our results show that the active control based on wall transpiration reduces both the wall shear stress and the energy of the G\"{o}rtler vortices; the passive control based on wall cooling or heating reduces the wall shear stress, but slightly increases the energy of the vortices in both supersonic and hypersonic regimes.Comment: 23 page

Numerical anisotropy study of a class of compact schemes

We study the numerical anisotropy existent in compact difference schemes as applied to hyperbolic partial differential equations, and propose an approach to reduce this error and to improve the stability restrictions based on a previous analysis applied to explicit schemes. A prefactorization of compact schemes is applied to avoid the inversion of a large matrix when calculating the derivatives at the next time level, and a predictor-corrector time marching scheme is used to update the solution in time. A reduction of the isotropy error is attained for large wave numbers and, most notably, the stability restrictions associated with MacCormack time marching schemes are considerably improved. Compared to conventional compact schemes of similar order of accuracy, the multidimensional schemes employ larger stencils which would presumably demand more processing time, but we show that the new stability restrictions render the multidimensional schemes to be in fact more efficient, while maintaining the same dispersion and dissipation characteristics of the one dimensional schemes

A Control Forced Concurrent Precursor Method for LES Inflow

With the increased application of large eddy simulation techniques, the generation of realistic turbulence at inflow boundaries is crucial for the accuracy of a simulation. The Control Forced Concurrent Precursor Method (CFCPM) proposed in this work combines an existing concurrent precursor method and a mean flow forcing method with a new extension of the controlled forcing method to impose turbulent inflow boundary conditions primarily, although not exclusively, for domains that require periodic boundary conditions. Turbulent inflow boundary conditions are imposed through a region of body forces added to the momentum equations of the main simulation that transfers the precursor simulation into the main domain. Controlled forcing planes, which come into play as body forces added to the momentum equations on planes perpendicular to the flow, located in the precursor simulation, allow for specific Reynolds stress tensors and mean velocities to be imposed. The mean flow controlled forcing method only modifies the mean velocity profiles, leaving the fluctuating velocity field untouched. The proposed fluctuating flow controlled forcing methods extends the application of the original controlled forcing method to multiple fluctuating velocity components and couples their calculation in order to amplify the existing fluctuations present in the precursor flow field so that prescribed anisotropic Reynolds stress tensors can be reproduced. The new method was tested on high Reynolds number turbulent boundary layer flow over a wall-mounted cube and low Reynolds number turbulent boundary layer flow over a backward-facing step. It was found that the new extension of the controlled forcing method reduced the development time for both test cases considered here when compared to not using controlled forcing and only using the original controlled forcing method.Comment: 18 pages, 11 figure, under consideration for publication in 'Flow, Turbulence and Combustion

Wakes in stratified atmospheric boundary layer flows: an LES investigation

Large eddy simulations and three-dimensional proper orthogonal decomposition were used to study the interaction between a large stationary and moving bluff body and a high Reynolds number stably-stratified turbulent boundary layer. An immersed boundary method is utilized to take into account the effect of the bluff body on the boundary layer flow and turbulent inflow conditions upstream of the bluff body are imposed by employing a concurrent precursor simulation method both because a pseudo-spectral method is utilized in the horizontal directions. The dominant POD mode type was observed to occur at higher energy levels in the more turbulent wakes. Varying the thermal stratification showed only a slight effect on the turbulent statistics, but a significant effect on the number of dominant POD modes at the highest energy levels.Comment: 22 page

Assessment of the impact of two-dimensional wall deformations' shape on high-speed boundary layer disturbances

Previous experimental and numerical studies showed that two-dimensional roughness elements can stabilize disturbances inside a hypersonic boundary layer, and eventually delay the transition onset. The objective of this paper is to evaluate the response of disturbances propagating inside a high-speed boundary layer to various two-dimensional surface deformations of different shapes. We perform an assessment of the impact of various two-dimensional surface non-uniformities, such as backward or forward steps, combinations of backward and forward steps, wavy surfaces, surface dips, and surface humps. Disturbances inside a Mach 5.92 flat-plate boundary layer are excited using periodic wall blowing and suction at an upstream location. The numerical tools consist of a high-accurate numerical algorithm solving for the unsteady, compressible form of the Navier-Stokes equations in curvilinear coordinates. Results show that all types of surface non-uniformities are able to reduce the amplitude of boundary layer disturbances to a certain degree. The amount of disturbance energy reduction is related to the type of pressure gradients that are posed by the deformation (adverse or favorable). A possible cause (among others) of the disturbance energy reduction inside the boundary layer is presumed to be the result of a partial deviation of the kinetic energy to the external flow, along the discontinuity that is generated by the wall deformation

Large eddy simulation study of the humidity variation in the shadow of a large wind farm

Numerous studies have shown that wind turbine wakes within a large wind farm bring about changes to both the dynamics and thermodynamics of the atmospheric boundary layers (ABL). Previously, we investigated the relative humidity budget within a wind farm via field measurements in the near-wake region and large eddy simulations (LES). The effect of the compounding wakes within a large wind farm on the relative humidity was also investigated by LES. In this study, we investigate how the areas of relative humidity variation, that was observed in the near-wake, develop downstream in the shadow region of a large wind farm. To this end, LES of a wind farm consisting of 8x6 wind turbines with periodic boundary condition in the lateral direction (inferring an infinitely wide farm) interacting with a stable ABL is carried out. Two wind farm layouts, aligned and staggered, are considered in the analysis and the results from both configurations are compared to each other. It is observed that a decrease of relative humidity underneath the hub height and an increase above the hub height build up within the wind farm, and are maintained in the downstream of the farm for long distances. The staggered farm layout is more effective in keeping a more elongated region of low relative humidity underneath the hub, when compared to the aligned layout.Comment: 12 page