A posteriori analysis of a positive streamwise invariant discretization of a convection-diffusion equation

We consider the finite element discretization of a convection-diffusion equation, where the convection term is handled via a fluctuation splitting algorithm. We prove a posteriori error estimates which allow us to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach.Nous considérons une discrétisation par éléments finis d’une équation de convection-diffusion, où un algorithme de décentrage est utilisé pour traiter le terme de convection. Nous prouvons des estimations d’erreur a posteriori qui permettent d’adapter le maillage pour optimiser la discrétisation de ces équations. Des résultats numériques confirment l’intérêt d’une telle approche.Junta de Andalucí

Large eddy simulation of mixed convection in a vertical slot and geometrical statistics of wall-bounded thermal flow

Buoyant flows are characterized with unsteady large-scale structures and thus time-dependent large eddy simulation (LES) is generally favored. In this dissertation, to further explore LES for buoyant flow, an LES code based on a collocated grid system is first developed. A multigrid solver using a control strategy is developed for the pressure Poisson equations. The control strategy significantly accelerated the convergence rate. A temperature solver using a fourth-order Runge-Kutta approach is also developed. The LES code is extensively tested before it is applied. Although the collocated grid system will introduce conservation errors, in tests of a steady lid-driven cavity flow and transient start-up flow, the effect of the non-conservation of the collocated grid system was not significant. In LES, the effect of SGS scales is represented by SGS models. A novel dynamic nonlinear model (DNM) for SGS stress is tested using isothermal channel flow at Reynolds number 395. The kinetic energy dissipation and geometrical characteristics of the resolved scale and SGS scale with respect to the DNM are investigated. In general, the DNM is reliable and has relatively realistic geometrical properties in comparison with the conventional dynamic model in the present study. In contrast to a pure advecting velocity field, a scalar (temperature) field displays very different characteristics. The modelling of SGS heat flux has not been as extensively studied as that of SGS stress partly due to the complexity of the scalar transport. In this dissertation, LES for a turbulent combined forced and natural convection is studied. The DNM model and a nonlinear dynamic tensor diffusivity model (DTDM-HF) are applied for the SGS stress and heat flux, respectively. The combined effect of the nonlinear models is compared to that of linear models. Notable differences between the nonlinear and linear SGS models are observed at the subgrid-scale level. At the resolved scale, the difference is smaller but relatively more distinguishable in terms of quantities related to the temperature field. Finally, the geometrical properties of the resolved velocity and temperature fields of the thermal flow are investigated based on the LES prediction. Some universal geometrical patterns have been reproduced, e.g. the positively skewed resolved enstrophy generation and the alignment between the vorticity and vortex stretching vectors. The present research demonstrates that LES is an effective tool for the study of the geometrical properties of a turbulent flow at the resolved-scales. The wall imposed anisotropy on the flow structures and orientation of the SGS heat flux vector are also specifically examined. In contrast to the dynamic eddy diffusivity model, the DTDM-HF successfully predicts the near-wall physics and demonstrates a non-alignment pattern between the SGS heat flux and temperature gradient vector

Numerical resolution of turbulent flows on complex geometries.

This thesis aims at developing a numerical methodology suitable for the direct numerical simulation (DNS) and large-eddy simulation (LES) of turbulent flows in order to be used in complex flows, currently encountered in industrial application. At the same time, the study of such turbulent flows can be an opportunity for gaining insight into the complex physics associated with them. To accomplish these goals, the mathematical formulation, conservative spatial discretization on unstructured grids and time- integration scheme for solving the Navier-Stokes equations are presented. The spatial discretization proposed preserves the symmetry properties of the continuous differential operator and ensure both, stability and conservation of the global kinetic energy balance on any grid. Furthermore, the time-integration technique proposed is an efficient self-adaptive strategy, based on a one-parameter second-order-explicit scheme, which has been successfully tested on both Cartesian staggered and unstructured collocated codes, leading to CPU cost reductions of up to 2.9 and 4.3, respectively. After presenting the general methodology for computing flows in complex geometries with unstructured grids, different LES models and regularization models suitable for these kind of meshes are presented and assessed by means of the analysis of different flows. First, regularization models are tested by means of the simulation of different cases with different level of complexity of the mesh. From a structured grid to a very complex mesh, with zones composed of prism and tetrahedral control volumes. It has been shown, that regularization models are very dependent on the quality of the filtering process. Although good results can be obtained with structured or smooth unstructuredmeshes, their performance is affected under fully irregular unstructured grids. A possible remedy to circumvent this issue is also presented. The main idea is to formulate the C4 model within a LES template. Although preliminary results are promising, further testing is still required. After regularization model assessment, LES models are also tested in a natural convection flow. It is shown that, although first order statistics are well solved for most of the models tested (with the exception of the Smagorinsky model), QR- and dynamic-Smagorinsky models present a better prediction of the second-order statistics. However, if CPU time is considered, then QR model is the best alternative. The second part of the thesis is devoted to the study of turbulent flows past bluff bodies. The cases studied are: the flow past a sphere, the flow past a circular cylinder and the flow past a NACA 0012 airfoil. All these cases shares some characteristics encountered in turbulent flows with massive separations, i.e., flow separation, transition to turbulence in the separated shear-layers and turbulent wakes with periodic shedding of vortices. However there are intrinsic characteristics of the turbulence in each of them, which make them interesting for the studying of the turbulence. Furthermore, the results presented for the flow past a sphere at Re = 3700 and 10000, together with the flow past a NACA 0012 at Re=50000 and AoA = 8 are the first DNS results presented in the literature for both flows. Conclusions drawn from the good results obtained point out that the use of the conservative formulation presented in this thesis, is one of the keys for the success of the SGS models used. This formulation, together with the use of unstructured grids might be a step towards the use of LES models for solving industrial flows on complex geometries at high Reynolds numbers.La present tesi proposa una metodologia apte per a realitzar simulacions directes de la turbulència (DNS) i simulacions de les grans escales (LES) de fluxos turbulents en geometries complexes. Tanmateix també s'estudia detalladament els mecanismes bàsics de funcionament dels fluxos turbulents en diferents situacions d'interès industrial i acadèmic. Per acomplir aquest objectiu s'ha desenvolupat una innovadora formulació matemàtica que permet conservar discretament les propietats continues de les equacions governants en malles no estructurades. La formulació proposada preserva la simetries originals dels operadors diferencials, assegurant així l'estabilitat i la conservació de l'energia cinètica turbulent en qualsevol mallat. Posteriorment s'ha proposat una metodologia d'integració temporal basada en una formulació explicita de segon ordre. Aquesta nova tècnica ha demostrat ser entre 2.9 i 4.3 més rapida que les tècniques anteriorment utilitzades per la comunitat. Un cop presentada la formulació per a simular fluxos turbulents en geometries complexes, s'han validat diferents models LES adaptats a malles no estructurades. Els models s'han testejat usant diferents solucions de referencia de la literatura i simulacions d'alt nivell generades en el context de la present tesi. Finalment s'ha conclòs que la conjunció de la formulació bàsica proposada amb alguns del models LES sorgits en els darrers anys es molt efectiva per a simular fluxos turbulents en situacions complexes, essent el Variational Multiscale WALE i el model QR els més adequats per a simular situacions de interes industrial. La segona part de la tesi es dedicada a l'estudi aerodinàmic del flux turbulent al voltant de diferents perfils. El perfils seleccionats son: el flux al voltant d'una esfera, flux al voltant d'un cilindre i flux al voltant d'un perfil NACA 0012. Els tres casos comparteixen fenomenologies com ara separació massiva de capes límits, esteles turbulentes i desprendiment periòdic de remolins. Tot i així cadascun d'ells es comporta diferent a nivell turbulent així que es d'interès estudiar-los i entendre quins son les causes de les diferencies físiques que es troben. Cal recordar que la física estudiada es la que es pot trobar posteriorment en ales d'avió, perfils de turbines de vent, aerodinàmica de cotxes, etc. Finalment recalcar que els resultats DNS del flux al voltant de l'esfera a Re=3700 i Re=10000 conjuntament amb els DNS del flux al voltant del perfil NACA a Re=50000 i AoA =8 son els primers presentats en la literatura internacional en el seu àmbit. Finalment es pot concloure que la formulació conservativa presentada en la tesis juntament amb els diferents models LES d'última generació testejats en la tesis, han demostrat ser una eina eficaç tan per a resoldre fluxos turbulents d'interès acadèmic com per simular situacions d'interès industrial.Postprint (published version

An Asymptotic Self-Sustaining Process Theory for Uniform Momentum Zones and Internal Interfaces in Unbounded Couette Flow

Meinhart \& Adrian (Phys. Fluids, vol. 7, 1995, p 694) were the first investigators to document that the wall-normal ($y$) structure of the instantaneous streamwise velocity in the turbulent boundary layer exhibits a staircase-like profile: regions of quasi-uniform momentum are separated by internal shear layers across which the streamwise velocity jumps by an O(1) amount when scaled by the friction velocity $u_\tau$. This sharply-varying instantaneous profile differs dramatically from the well-known long-time mean profile, which is logarithmic over much of the boundary layer, and prompted Klewicki (Proc. IUTAM, vol. 9, 2013, p. 69--78) to propose that the turbulent boundary layer is singular in two distinct ways. Firstly, spanwise vorticity and mean viscous forces are concentrated in a near-wall region of thickness $\mathit{O}(h/\sqrt{Re_\tau})$, where $Re_\tau$ is the friction Reynolds number and $h$ is the boundary-layer height. Secondly, in a turbulent boundary layer, spanwise vorticity and viscous forces are also significant away from the wall (outboard of the peak in the Reynolds stress), but \emph{only} in spatially-localized regions, i.e. within the internal shear layers. This interpretation accords with Klewicki\u27s multiscale similarity analysis of the mean momentum balance for turbulent wall flows (J. Fluid Mech., vol. 522, 2005, pp. 303--327). The objective of the present investigation is to probe the governing Navier--Stokes equations in the limit of large $Re_\tau$ in search of a mechanistic self-sustaining process (SSP) that (i) can account for the emergent staircase-like profile of streamwise velocity in the inertial region and (ii) is compatible with the singular nature of turbulent wall flows. Plausible explanations for the formation and persistence of sharply-varying instantaneous streamwise velocity profiles all implicate quasi-coherent turbulent flow structures including streamwise roll motions that induce a cellular flow in the transverse (i.e. spanwise/wall-normal) plane. One proposal is that the large-scale structures result from the spontaneous concatenation of smaller--scale structures, particularly hairpin and cane vortices and vortex packets. A competing possibility, explored here, is that these large--scale motions may be \emph{directly} sustained via an inertial--layer SSP that is broadly similar to the near-wall SSP. The SSP theory derived in this investigation is related to the SSP framework developed by Waleffe (Stud. Appl. Math, vol. 95, 1995, p. 319) and, especially, to the closely-related vortex-wave interaction (VWI) theory derived by Hall \& Smith (J. Fluid Mech., vol. 227, 1991, pp. 641--666) and Hall \& Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178--205) in that a rational asymptotic analysis of the instantaneous Navier--Stokes equations is performed. Nevertheless, in this dissertation, it is argued that these theories cannot account for organized motions in the inertial domain, essentially because the roll motions are predicted to be viscously dominated even at large $Re_\tau$. The target of the present investigation is an inherently multiscale SSP, in which inviscid streamwise rolls differentially homogenize an imposed background shear flow, thereby generating uniform momentum zones and an embedded internal shear layer (or interface), and are sustained by Rayleigh instability modes having asymptotically smaller streamwise and spanwise length scales. The Rayleigh mode is supported by the inflectional wall-normal profile of the streamwise--averaged streamwise velocity. Because the thickness of the internal shear layer varies comparably slowly in the spanwise direction, the Rayleigh mode is refracted and rendered fully three--dimensional. This three--dimensional mode is singular, necessitating the introduction of a critical layer inside the shear layer within which the mode is viscously regularized. As in VWI theory, a jump in the spanwise Reynolds stress is induced across the critical layer, which ultimately drives the roll motions. This multiscale and three--region asymptotic structure is efficiently captured using a complement of matched asymptotic and WKBJ analysis. The resulting reduced equations require the numerical solution of both ordinary differential eigenvalue and partial differential boundary-value problems, for which pseudospectral and spectral collocation methods are employed. Crucially, in contrast to Waleffe\u27s SSP and to VWI theory, the rolls are sufficiently strong to differentially homogenize the background shear flow, thereby providing a plausible mechanistic explanation for the formation and maintenance of both UMZs and interlaced internal shear layers

Research in Applied Mathematics, Fluid Mechanics and Computer Science

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period October 1, 1998 through March 31, 1999

Flow topology and small-scale dynamics in turbulent Rayleigh-Bénard convection

Without fluid turbulence, life might have rather different look. The atmosphere and oceans could nearly maintain a much larger temperature differences resulting in ultimate heating or cooling to the earth surface. The water and air flow could rather run much faster at rates of the speed of sound. Turbulence is a highly active nature of chaotic, random and three-dimensionality of swirling fluid. Its nonlinear convective property transports the momentum and energy in a helical mechanism leading eventually to an enriched fluid mixing and generating of small scale motions. These scales chiefly rule the hairpin vorticity dynamics, the strain production and the cascade of kinetic energy mechanisms. Hence, the key feature in turbulence is around disclosing the small scale motions. Studying the fine-scale dynamics gives us fundamental perspectives of flow topology and thus, improves our knowledge of turbulence physics. The turbulence dynamo becomes more complex when the active thermal gradient constitutes into the pure generator of turbulence. This particularly happens in the so-called buoyancy-driven Rayleigh-Bénard convection (RBC), when an infinite/bounded lying fluid is heated from below and cooled from above in the field of gravity. The main goal of this thesis is investigating the flow topology and small-scale dynamics in turbulent RBC, in order to better understanding its thermal turbulence mechanism and improve/validate the turbulence modeling for the foreseeable Computational-Fluid-Dynamics future. To do so, a complete direct numerical simulation (DNS) of turbulent RBC in a rectangular air-filled cavity of aspect ratio unity and pi spanwise open-ended distance, has been presented at Rayleigh numbers Ra={1e8, 1e10}, in chapter 1. A global kinetic energy conservation is inherited using a fourth-order symmetry-preserving scheme for the spatial discretization, and the flow dynamics is explored by analysis of kinetic and thermal energy power spectra, probability density function (PDF) of viscous and thermal dissipation rates, and identification of the wind in RBC. In chapter 2, the DNS dataset is used to investigate several universal small-scale features observed in various turbulent flows and recaptured here in turbulent RBC through the bulk. For instance, the inclined "teardrop" shape of joint PDF velocity gradient tensor invariants (Q,R), the preferential alignment of vorticity with the intermediate eigenstrain vector, and the spiraling degenerated behavior of the average rates invariants (,). It is found that a self-amplification of viscous straining -Qs results at Ra=1e10, helps in contracting the vorticity worms and enhances slightly the linear contributions of the vortex stretching mechanism. On the other hand, the evolution of relevant small-scale thermals has been addressed by investigating the average rate of invariants pertained to the traceless part of velocity-times-temperature gradient tensor i.e., (,). The new invariants are shown to follow correctly the evolution and lifetime of thermal plumes in RBC and hence disclose interactions of buoyant production and viscous dissipation. In chapter 3, the DNS dataset is employed to understand the underlying physics of the subgrid-scale (SGS) motions in turbulent RBC in the spirit of Large-eddy simulation (LES) turbulence modeling. To do so, the key ingredients of eddy-viscosity, eddy-diffusivity and turbulent Prandtl number, are calculated a priori and investigated in a topological point-of-view. As a result, it has been suggested the restricted application of the hypothesis of a constant turbulent Prandtl number only in the large-scale strain-dominated areas. More arguments have been attained through a priori investigation of the alignment trends imposed by existing parameterizations for the SGS heat flux. Finally, a new tensorial approach of modeling the SGS of thermal turbulence is sought, that opens new research trends in the future.Sin turbulencia, la vida tendría un aspecto bastante diferente. La atmósfera y los océanos podrían mantener una diferencia de temperatura mucho mayor que causaría un gran calentamiento o enfriamiento de la superficie de la tierra. Las corrientes de agua y aire podrían llegar a la velocidad del sonido. La turbulencia es un fenómeno caótico, aleatorio y tridimensional del flujo vortical. Su propiedad convectiva no lineal transporta moméntum y energía con un mecanismo helicoidal que conduce finalmente a una mezcla efectiva del fluido y una generación de escalas pequeñas de movimiento. Estas escalas dominan la dinámica de pequeña vorticidad, la disipación y la cascada de energía cinética. Por lo tanto, la clave de la turbulencia está en entender las escalas pequeñas. El estudio de la dinámica de estas escalas nos aportará una perspectiva de la topología del flujo y así mejorará nuestro conocimiento de la física de la turbulencia. El mecanismo turbulento se hace más complejo cuando el gradiente térmico constituye el productor principal de turbulencia. Esto sucede particularmente en la convección natural de Rayleigh-Bénard (RBC), donde una capa de fluido se calienta desde abajo y se enfría desde arriba. El objetivo principal de la tesis es investigar la topología del flujo y la dinámica de las escalas pequeñas en flujos turbulentos RBC con el fin de comprender mejor su mecanismo de turbulencia térmica y mejorar/validar su modelización en el futuro. En el capítulo 1 se presenta una completa Simulación Numérica Directa (DNS) de un flujo de aire turbulento RBC en una cavidad rectangular con sección cuadrada y longitud igual a pi, para números de Ra={1e8, 1e10}. Se utiliza un esquema de cuarto orden para la discretización espacial que garantiza la conservación la energía cinética global. La dinámica del flujo se explora con un análisis de: el espectro de energía térmica y cinética, la función densidad de probabilidad (PDF) de la disipación cinética viscosa y térmica, y la identificación del viento, en RBC. En el capítulo 2, los datos del DNS se utilizan para investigar las características universales de escalas pequeñas observadas en otros flujos turbulentos. Por ejemplo, el aspecto inclinado de "gota" del PDF conjunto de los invariantes del tensor gradiente de velocidad (Q,R), la alineación preferente de la vorticidad con el vector propio intermedio de la deformación y el comportamiento espiral deteriorado del ratio promediado de los invariantes (,). Se observa una amplificación de la deformación viscosa en el caso de Ra=1e10 que ayuda a contraer los tubos de vorticidad y mejora ligeramente la contribución lineal del mecanismo de estiramiento-vórtice. Por otro lado, se aborda la evolución de las escalas pequeñas térmicas con el estudio del ratio promediado de los invariantes del tensor gradiente de velocidad-por-temperatura (,). Los nuevos invariantes demuestran seguir correctamente la evolución de las plumas térmicas en RBC, revelando las interacciones de la producción flotante y la disipación viscosa. En el capítulo 3, los datos del DNS se emplean para comprender la física subyacente de los movimientos de la escala subrejilla (SGS) en turbulencia RBC con el espíritu de la modelización tipo simulación de grandes remolinos (LES). Para ello, se calculan a priori los componentes clave de viscosidad y difusividad de remolino y número de Prandtl turbulento, y se investigan desde un punto de vista topológico. Como resultado, se propone la aplicación de la hipótesis del número de Prandtl turbulento constante sólo en las áreas dominadas por la deformación a gran escala. Además, se alcanzan más argumentos con la investigación a priori de las tendencias de alineación geométrica impuestas por las parametrizaciones existentes del flujo de calor turbulento SGS. Finalmente, se estudia un nuevo enfoque tensorial para modelar la turbulencia térmica SGS, que abrirá nuevas líneas de investigación para el futuro

Finite-Volume Filtering in Large-Eddy Simulations Using a Minimum-Dissipation Model

Large-eddy simulation (LES) seeks to predict the dynamics of the larger eddies in turbulent flow by applying a spatial filter to the Navier-Stokes equations and by modeling the unclosed terms resulting from the convective non-linearity. Thus the (explicit) calculation of all small-scale turbulence can be avoided. This paper is about LES-models that truncate the small scales of motion for which numerical resolution is not available by making sure that they do not get energy from the larger, resolved, eddies. To identify the resolved eddies, we apply Schumann’s filter to the (incompressible) Navier-Stokes equations, that is the turbulent velocity field is filtered as in a finite-volume method. The spatial discretization effectively act as a filter; hence we define the resolved eddies for a finite-volume discretization. The interpolation rule for approximating the convective flux through the faces of the finite volumes determines the smallest resolved length scale δ. The resolved length δ is twice as large as the grid spacing h for an usual interpolation rule. Thus, the resolved scales are defined with the help of box filter having diameter δ= 2 h. The closure model is to be chosen such that the solution of the resulting LES-equations is confined to length scales that have at least the size δ. This condition is worked out with the help of Poincarés inequality to determine the amount of dissipation that is to be generated by the closure model in order to counterbalance the nonlinear production of too small, unresolved scales. The procedure is applied to an eddy-viscosity model using a uniform mesh

Investigation of low-dissipation monotonicity-preserving scheme for direct numerical simulation of compressible turbulent flows

© 2014 Elsevier Ltd. The influence of numerical dissipation on direct numerical simulation (DNS) of decaying isotropic turbulence and turbulent channel flow is investigated respectively by using the seventh-order low-dissipation monotonicity-preserving (MP7-LD) scheme with different levels of bandwidth dissipation. It is found that for both benchmark test cases, small-scale turbulence fluctuations can be largely suppressed by high level of scheme dissipation, while the appearance of numerical errors in terms of high-frequency oscillations could destabilize the computation if the dissipation is reduced to a very low level. Numerical studies show that reducing the bandwidth dissipation to 70% of the conventional seventh-order upwind scheme can maximize the efficiency of the MP7-LD scheme in resolving small-scale turbulence fluctuations and, in the meantime preventing the accumulation of non-physical numerical errors. By using the optimized MP7-LD scheme, DNS of an impinging oblique shock-wave interacting with a spatially-developing turbulent boundary layer is conducted at an incoming free-stream Mach number of 2.25 and the shock angle of 33.2°. Simulation results of mean velocity profiles, wall surface pressure, skin friction and Reynolds stresses are validated against available experimental data and other DNS predictions in both the undisturbed equilibrium boundary layer region and the interaction zone, and good agreements are achieved. The turbulence kinetic energy transport equation is also analyzed and the balance of the equation is well preserved in the interaction region. This study demonstrates the capability of the optimized MP7-LD scheme for DNS of complex flow problems of wall-bounded turbulent flow interacting with shock-waves