Error estimation and adaptive mesh refinement for aerodynamic flows

This lecture course covers the theory of so-called duality-based a posteriori error estimation of DG finite element methods. In particular, we formulate consistent and adjoint consistent DG methods for the numerical approximation of both the compressible Euler and Navier-Stokes equations; in the latter case, the viscous terms are discretized based on employing an interior penalty method. By exploiting a duality argument, adjoint-based a posteriori error indicators will be established. Moreover, application of these computable bounds within automatic adaptive finite element algorithms will be developed. Here, a variety of isotropic and anisotropic adaptive strategies, as well as $hp$-mesh refinement will be investigated.

A novel methodology for simulating contact-line behavior in capillary-driven flows

Despite the wide swath of applications where multiphase fluid contact lines exist, there is still no consensus on an accurate and general simulation methodology. Most prior numerical work has imposed one of the many dynamic contact-angle theories at solid walls. Such approaches are inherently limited by the theory accuracy. In fact, when inertial effects are important, the contact angle may be history dependent and, thus, any single mathematical function is inappropriate. Given these limitations, the present work has two primary goals: 1) create a numerical framework that allows the contact angle to evolve naturally with appropriate contact-line physics and 2) develop equations and numerical methods such that contact-line simulations may be performed on coarse computational meshes.

Fluid flows affected by contact lines are dominated by capillary stresses and require accurate curvature calculations. The level set method was chosen to track the fluid interfaces because it is easy to calculate interface curvature accurately. Unfortunately, the level set reinitialization suffers from an ill-posed mathematical problem at contact lines: a ``blind spot'' exists. Standard techniques to handle this deficiency are shown to introduce parasitic velocity currents that artificially deform freely floating (non-prescribed) contact angles. As an alternative, a new relaxation equation reinitialization is proposed to remove these spurious velocity currents and its concept is further explored with level-set extension velocities.

To capture contact-line physics, two classical boundary conditions, the Navier-slip velocity boundary condition and a fixed contact angle, are implemented in direct numerical simulations (DNS). DNS are found to converge only if the slip length is well resolved by the computational mesh. Unfortunately, since the slip length is often very small compared to fluid structures, these simulations are not computationally feasible for large systems. To address the second goal, a new methodology is proposed which relies on the volumetric-filtered Navier-Stokes equations. Two unclosed terms, an average curvature and a viscous shear VS, are proposed to represent the missing microscale physics on a coarse mesh.

All of these components are then combined into a single framework and tested for a water droplet impacting a partially-wetting substrate. Very good agreement is found for the evolution of the contact diameter in time between the experimental measurements and the numerical simulation. Such comparison would not be possible with prior methods, since the Reynolds number Re and capillary number Ca are large. Furthermore, the experimentally approximated slip length ratio is well outside of the range currently achievable by DNS. This framework is a promising first step towards simulating complex physics in capillary-dominated flows at a reasonable computational expense.

Low cost 3D global instability analysis and flow sensitivity based on dynamic mode decomposition and high-order numerical tools

We explore the recently developed snapshot-based dynamic mode decomposition (DMD) technique, a matrix-free Arnoldi type method, to predict 3D linear global flow instabilities. We apply the DMD technique to flows confined in an L-shaped cavity and compare the resulting modes to their counterparts issued from classic, matrix forming, linear instability analysis (i.e. BiGlobal approach) and direct numerical simulations. Results show that the DMD technique, which uses snapshots generated by a 3D non-linear incompressible discontinuous Galerkin Navier?Stokes solver, provides very similar results to classical linear instability analysis techniques. In addition, we compare DMD results issued from non-linear and linearised Navier?Stokes solvers, showing that linearisation is not necessary (i.e. base flow not required) to obtain linear modes, as long as the analysis is restricted to the exponential growth regime, that is, flow regime governed by the linearised Navier?Stokes equations, and showing the potential of this type of analysis based on snapshots to general purpose CFD codes, without need of modifications. Finally, this work shows that the DMD technique can provide three-dimensional direct and adjoint modes through snapshots provided by the linearised and adjoint linearised Navier?Stokes equations advanced in time. Subsequently, these modes are used to provide structural sensitivity maps and sensitivity to base flow modification information for 3D flows and complex geometries, at an affordable computational cost. The information provided by the sensitivity study is used to modify the L-shaped geometry and control the most unstable 3D mode.

Simulação de fluxo de fluidos em meios porosos desordenados uma análise de efeito de escala na estimativa da permeabilidade e do coeficiente de arrasto

The present study provides a methodology that gives a predictive character the computer simulations based on detailed models of the geometry of a porous medium. We using the software FLUENT to investigate the flow of a viscous Newtonian fluid through a random fractal medium which simplifies a two-dimensional disordered porous medium representing a petroleum reservoir. This fractal model is formed by obstacles of various sizes, whose size distribution function follows a power law where exponent is defined as the fractal dimension of fractionation Dff of the model characterizing the process of fragmentation these obstacles. They are randomly disposed in a rectangular channel. The modeling process incorporates modern concepts, scaling laws, to analyze the influence of heterogeneity found in the fields of the porosity and of the permeability in such a way as to characterize the medium in terms of their fractal properties. This procedure allows numerically analyze the measurements of permeability k and the drag coefficient Cd proposed relationships, like power law, for these properties on various modeling schemes. The purpose of this research is to study the variability provided by these heterogeneities where the velocity field and other details of viscous fluid dynamics are obtained by solving numerically the continuity and Navier-Stokes equations at pore level and observe how the fractal dimension of fractionation of the model can affect their hydrodynamic properties. This study were considered two classes of models, models with constant porosity, MPC, and models with varying porosity, MPV. The results have allowed us to find numerical relationship between the permeability, drag coefficient and the fractal dimension of fractionation of the medium. Based on these numerical results we have proposed scaling relations and algebraic expressions involving the relevant parameters of the phenomenon. In this study analytical equations were determined for Dff depending on the geometrical parameters of the models. We also found a relation between the permeability and the drag coefficient which is inversely proportional to one another. As for the difference in behavior it is most striking in the classes of models MPV. That is, the fact that the porosity vary in these models is an additional factor that plays a significant role in flow analysis. Finally, the results proved satisfactory and consistent, which demonstrates the effectiveness of the referred methodology for all applications analyzed in this study.

Large Eddy Simulations applied to wind loading and pollutant dispersion

The application of Computational Fluid Dynamics based on the Reynolds-Averaged Navier-Stokes equations to the simulation of bluff body aerodynamics has been thoroughly investigated in the past. Although a satisfactory accuracy can be obtained for some urban physics problems their predictive capability is limited to the mean flow properties, while the ability to accurately predict turbulent fluctuations is recognized to be of fundamental importance when dealing with wind loading and pollution dispersion problems. The need to correctly take into account the flow dynamics when such problems are faced has led researchers to move towards scale-resolving turbulence models such as Large Eddy Simulations (LES). The development and assessment of LES as a tool for the analysis of these problems is nowadays an active research field and represents a demanding engineering challenge. This research work has two objectives. The first one is focused on wind loads assessment and aims to study the capabilities of LES in reproducing wind load effects in terms of internal forces on structural members. This differs from the majority of the existing research, where performance of LES is evaluated only in terms of surface pressures, and is done with a view of adopting LES as a complementary design tools alongside wind tunnel tests. The second objective is the study of LES capabilities in calculating pollutant dispersion in the built environment. The validation of LES in this field is considered to be of the utmost importance in order to conceive healthier and more sustainable cities. In order to validate the numerical setup adopted, a systematic comparison between numerical and experimental data is performed. The obtained results are intended to be used in the drafting of best practice guidelines for the application of LES in the urban physics field with a particular attention to wind load assessment and pollution dispersion problems.

Adaptive coatings based on polyaniline for direct 2D observation of diffusion processes in microfluidic systems

[EN] Therefore the understanding and proper evaluation of the flow and mixing behaviour at microscale becomes a very important issue. In this study, the diffusion behaviour of two reacting solutions of HCI and NaOH were directly observed in a glass/polydimethylsiloxane microfluidic device using adaptive coatings based on the conductive polymer polyaniline that are covalently attached to the microchannel walls. The two liquid streams were combined at the junction of a Y-shaped microchannel, and allowed to diffuse into each other and react. The results showed excellent correlation between optical observation of the diffusion process and the numerical results. A numerical model which is based on finite volume method (FVM) discretisation of steady Navier-Stokes (fluid flow) equations and mass transport equations without reactions was used to calculate the flow variables at discrete points in the finite volume mesh element. The high correlation between theory and practical data indicates the potential of such coatings to monitor diffusion processes and mixing behaviour inside microfluidic channels in a dye free environment.

Discrete linear stability and sensitivity analysis of fluid flows

Este trabajo presenta un método discreto para el cálculo de estabilidad hidrodinámica y análisis de sensibilidad a perturbaciones externas para ecuaciones diferenciales y en particular para las ecuaciones de Navier-Stokes compressible. Se utiliza una aproximación con variable compleja para obtener una precisión analítica en la evaluación de la matriz Jacobiana. Además, mapas de sensibilidad para la sensibilidad a las modificaciones del flujo de base y a una fuerza constante permiten identificar las regiones del campo fluido donde una modificacin (ej. fuerza puntual) tiene un efecto estabilizador del flujo. Se presentan cuatro casos de prueba: (1) un caso analítico para comprobar la derivación discreta, (2) una cavidad cerrada a bajo Reynolds para mostrar la mayor precisión en el cálculo de los valores propios con la aproximación de paso complejo, (3) flujo 2D en un cilindro circular para validar la metodología, y (4) flujo en un cavidad abierta, presentado para validar el método en casos de inestabilidades convectivamente inestables. Los tres últimos casos mencionados (2-4) se resolvieron con las ecuaciones de Navier-Stokes compresibles, utilizando un método Discontinuous Galerkin Spectral Element Method. Se obtuvo una buena concordancia para el caso de validación (3), cuando se comparó el nuevo método con resultados de la literatura. Además, este trabajo muestra que para el cálculo de los modos propios directos y adjuntos, así como para los mapas de sensibilidad, el uso de variables complejas es de suprema importancia para obtener una predicción precisa. El método descrito es aplicado al análisis para la estabilización de la estela generada por un disco actuador, que representa un modelo sencillo para hélices, rotores de helicópteros o turbinas eólicas. Se explora la primera bifurcación del flujo para un disco actuador, y se sugiere que está asociada a una inestabilidad de tipo Kelvin-Helmholtz, cuya estabilidad se controla con en el número de Reynolds y en la resistencia del disco actuador (o fuerza resistente). En primer lugar, se verifica que la disminución de la resistencia del disco tiene un efecto estabilizador parecido a una disminución del Reynolds. En segundo lugar, el análisis hidrodinmico discreto identifica dos regiones para la colocación de una fuerza puntual que controle las inestabilidades, una cerca del disco y otra en una zona aguas abajo. En tercer lugar, se muestra que la inclusión de un forzamiento localizado cerca del actuador produce una estabilización más eficiente que al forzar aguas abajo. El análisis de los campos de flujo controlados confirma que modificando el gradiente de velocidad cerca del actuador es más eficiente para estabilizar la estela. Estos resultados podrían proporcionar nuevas directrices para la estabilización de la estela de turbinas de viento o de marea cuando estén instaladas en un parque eólico y minimizar las interacciones no estacionarias entre turbinas. ABSTRACT A discrete framework for computing the global stability and sensitivity analysis to external perturbations for any set of partial differential equations is presented. In particular, a complex-step approximation is used to achieve near analytical accuracy for the evaluation of the Jacobian matrix. Sensitivity maps for the sensitivity to base flow modifications and to a steady force are computed to identify regions of the flow field where an input could have a stabilising effect. Four test cases are presented: (1) an analytical test case to prove the theory of the discrete framework, (2) a lid-driven cavity at low Reynolds case to show the improved accuracy in the calculation of the eigenvalues when using the complex-step approximation, (3) the 2D flow past a circular cylinder at just below the critical Reynolds number is used to validate the methodology, and finally, (4) the flow past an open cavity is presented to give an example of the discrete method applied to a convectively unstable case. The latter three (2–4) of the aforementioned cases were solved with the 2D compressible Navier–Stokes equations using a Discontinuous Galerkin Spectral Element Method. Good agreement was obtained for the validation test case, (3), with appropriate results in the literature. Furthermore, it is shown that for the calculation of the direct and adjoint eigenmodes and their sensitivity maps to external perturbations, the use of complex variables is paramount for obtaining an accurate prediction. An analysis for stabilising the wake past an actuator disc, which represents a simple model for propellers, helicopter rotors or wind turbines is also presented. We explore the first flow bifurcation for an actuator disc and it suggests that it is associated to a Kelvin- Helmholtz type instability whose stability relies on the Reynolds number and the flow resistance applied through the disc (or actuator forcing). First, we report that decreasing the disc resistance has a similar stabilising effect to an decrease in the Reynolds number. Second, a discrete sensitivity analysis identifies two regions for suitable placement of flow control forcing, one close to the disc and one far downstream where the instability originates. Third, we show that adding a localised forcing close to the actuator provides more stabilisation that forcing far downstream. The analysis of the controlled flow fields, confirms that modifying the velocity gradient close to the actuator is more efficient to stabilise the wake than controlling the sheared flow far downstream. An interesting application of these results is to provide guidelines for stabilising the wake of wind or tidal turbines when placed in an energy farm to minimise unsteady interactions.

A combination of implicit and adaptative upwind tools for the numerical solution of incompressible free surface flows

This paper is concerned with the numerical solutions of time dependent two-dimensional incompressible flows. By using the primitive variables of velocity and pressure, the Navier-Stokes and mass conservation equations are solved by a semi-implicit finite difference projection method. A new bounded higher order upwind convection scheme is employed to deal with the non-linear (advective) terms. The procedure is an adaptation of the GENSMAC (J. Comput. Phys. 1994; 110: 171-186) methodology for calculating confined and free surface fluid flows at both low and high Reynolds numbers. The calculations were performed by using the 2D version of the Freeflow simulation system (J. Comp. Visual. Science 2000; 2:199-210). In order to demonstrate the capabilities of the numerical method, various test cases are presented. These are the fully developed flow in a channel, the flow over a backward facing step, the die-swell problem, the broken dam flow, and an impinging jet onto a flat plate. The numerical results compare favourably with the experimental data and the analytical solutions. Copyright (c) 2006 John Wiley & Sons, Ltd.

Three-dimensional finite element analysis of MHD duct flow by the penalty function formulation

A numerical scheme based on the Finite Element Method (FEM) is presented to calculate the full solution of a three-dimensional steady magnetohydrodynamic (MHD) flow with moderately high Hartmann numbers and interaction parameters. An incompressible, viscous and electrically conducting liquid-metal is considered. Assuming a low magnetic Reynolds number, the solution method solves the coupled Navier-Stokes and Maxwell's equations through the use of a penalty function method. Results are presented for Hartmann numbers in the range 10(2)-10(3).

Reduced order modeling of three dimensional external aerodynamic flows

A method is presented to construct computationally efficient reduced-order models (ROMs) of three-dimensional aerodynamic flows around commercial aircraft components. The method is based on the proper orthogonal decomposition (POD) of a set of steady snapshots, which are calculated using an industrial solver based on some Reynolds averaged Navier-Stokes (RANS) equations. The POD-mode amplitudes are calculated by minimizing a residual defined from the Euler equations, even though the snapshots themselves are calculated from viscous equations. This makes the ROM independent of the peculiarities of the solver used to calculate the snapshots. Also, both the POD modes and the residual are calculated using points in the computational mesh that are concentrated in a close vicinity of the aircraft, which constitute a much smaller number than the total number of mesh points. Despite these simplifications, the method provides quite good approximations of the flow variables distributions in the whole computational domain, including the boundary layer attached to the aircraft surface and the wake. Thus, the method is both robust and computationally efficient, which is checked considering the aerodynamic flow around a horizontal tail plane, in the transonic range 0.4?Mach number?0.8, ?3°?angle of attack?3°.

A 2D computational parametric analysis of the sheltering effect of fences on a railway vehicle standing on a bridge and experiencing crosswinds

In a crosswind scenario, the risk of high-speed trains overturning increases when they run on viaducts since the aerodynamic loads are higher than on the ground. In order to increase safety, vehicles are sheltered by fences that are installed on the viaduct to reduce the loads experienced by the train. Windbreaks can be designed to have different heights, and with or without eaves on the top. In this paper, a parametric study with a total of 12 fence designs was carried out using a two-dimensional model of a train standing on a viaduct. To asses the relative effectiveness of sheltering devices, tests were done in a wind tunnel with a scaled model at a Reynolds number of 1 × 105, and the train’s aerodynamic coefficients were measured. Experimental results were compared with those predicted by Unsteady Reynolds-averaged Navier-Stokes (URANS) simulations of flow, showing that a computational model is able to satisfactorily predict the trend of the aerodynamic coefficients. In a second set of tests, the Reynolds number was increased to 12 × 106 (at a free flow air velocity of 30 m/s) in order to simulate strong wind conditions. The aerodynamic coefficients showed a similar trend for both Reynolds numbers; however, their numerical value changed enough to indicate that simulations at the lower Reynolds number do not provide all required information. Furthermore, the variation of coefficients in the simulations allowed an explanation of how fences modified the flow around the vehicle to be proposed. This made it clear why increasing fence height reduced all the coefficients but adding an eave had an effect mainly on the lift force coefficient. Finally, by analysing the time signals it was possible to clarify the influence of the Reynolds number on the peak-to-peak amplitude, the time period and the Strouhal number.

Comparison of Lagrangian Coherent Structures and Relative Dispersion in a Mixing Layer

Thesis (Master's)--University of Washington, 2016-06

Studies of turbulent flows in continuous casting of steel with and without magnetic field

This thesis develops and tests various transient and steady-state computational models such as direct numerical simulation (DNS), large eddy simulation (LES), filtered unsteady Reynolds-averaged Navier-Stokes (URANS) and steady Reynolds-averaged Navier-Stokes (RANS) with and without magnetic field to investigate turbulent flows in canonical as well as in the nozzle and mold geometries of the continuous casting process. The direct numerical simulations are first performed in channel, square and 2:1 aspect rectangular ducts to investigate the effect of magnetic field on turbulent flows. The rectangular duct is a more practical geometry for continuous casting nozzle and mold and has the option of applying magnetic field either perpendicular to broader side or shorter side. This work forms the part of a graphic processing unit (GPU) based CFD code (CU-FLOW) development for magnetohydrodynamic (MHD) turbulent flows. The DNS results revealed interesting effects of the magnetic field and its orientation on primary, secondary flows (instantaneous and mean), Reynolds stresses, turbulent kinetic energy (TKE) budgets, momentum budgets and frictional losses, besides providing DNS database for two-wall bounded square and rectangular duct MHD turbulent flows. Further, the low- and high-Reynolds number RANS models (k-ε and Reynolds stress models) are developed and tested with DNS databases for channel and square duct flows with and without magnetic field. The MHD sink terms in k- and ε-equations are implemented as proposed by Kenjereš and Hanjalić using a user defined function (UDF) in FLUENT. This work revealed varying accuracies of different RANS models at different levels. This work is useful for industry to understand the accuracies of these models, including continuous casting. After realizing the accuracy and computational cost of RANS models, the steady-state k-ε model is then combined with the particle image velocimetry (PIV) and impeller probe velocity measurements in a 1/3rd scale water model to study the flow quality coming out of the well- and mountain-bottom nozzles and the effect of stopper-rod misalignment on fluid flow. The mountain-bottom nozzle was found more prone to the longtime asymmetries and higher surface velocities. The left misalignment of stopper gave higher surface velocity on the right leading to significantly large number of vortices forming behind the nozzle on the left. Later, the transient and steady-state models such as LES, filtered URANS and steady RANS models are combined with ultrasonic Doppler velocimetry (UDV) measurements in a GaInSn model of typical continuous casting process. LES-CU-LOW is the fastest and the most accurate model owing to much finer mesh and a smaller timestep. This work provided a good understanding on the performance of these models. The behavior of instantaneous flows, Reynolds stresses and proper orthogonal decomposition (POD) analysis quantified the nozzle bottom swirl and its importance on the turbulent flow in the mold. Afterwards, the aforementioned work in GaInSn model is extended with electromagnetic braking (EMBr) to help optimize a ruler-type brake and its location for the continuous casting process. The magnetic field suppressed turbulence and promoted vortical structures with their axis aligned with the magnetic field suggesting tendency towards 2-d turbulence. The stronger magnetic field at the nozzle well and around the jet region created large scale and lower frequency flow behavior by suppressing nozzle bottom swirl and its front-back alternation. Based on this work, it is advised to avoid stronger magnetic field around jet and nozzle bottom to get more stable and less defect prone flow.

Numerical investigations of flow and energy fields near a thermoacoustic couple

The flow field and the energy transport near thermoacoustic couples are simulated using a 2D full Navier-Stokes solver. The thermoacoustic couple plate is maintained at a constant temperature; plate lengths, which are short and long compared with the particle displacement lengths of the acoustic standing waves, are tested. Also investigated are the effects of plate spacing and the amplitude of the standing wave. Results are examined in the form of energy vectors, particle paths, and overall entropy generation rates. These show that a net heat-pumping effect appears only near the edges of thermoacoustic couple plates, within about a particle displacement distance from the ends. A heat-pumping effect can be seen even on the shortest plates tested when the plate spacing exceeds the thermal penetration depth. It is observed that energy dissipation near the plate increases quadratically as the plate spacing is reduced. The results also indicate that there may be a larger scale vortical motion outside the plates which disappears as the plate spacing is reduced. (C) 2002 Acoustical Society of America.

A new wavelet-based adaptive method for solving population balance equations

A new wavelet-based adaptive framework for solving population balance equations (PBEs) is proposed in this work. The technique is general, powerful and efficient without the need for prior assumptions about the characteristics of the processes. Because there are steeply varying number densities across a size range, a new strategy is developed to select the optimal order of resolution and the collocation points based on an interpolating wavelet transform (IWT). The proposed technique has been tested for size-independent agglomeration, agglomeration with a linear summation kernel and agglomeration with a nonlinear kernel. In all cases, the predicted and analytical particle size distributions (PSDs) are in excellent agreement. Further work on the solution of the general population balance equations with nucleation, growth and agglomeration and the solution of steady-state population balance equations will be presented in this framework. (C) 2002 Elsevier Science B.V. All rights reserved.