Evolution of vortex-surface fields in viscous Taylor-Green and Kida-Pelz flows

In order to investigate continuous vortex dynamics based on a Lagrangian-like formulation, we develop a theoretical framework and a numerical method for computation of the evolution of a vortex-surface field (VSF) in viscous incompressible flows with simple topology and geometry. Equations describing the continuous, timewise evolution of a VSF from an existing VSF at an initial time are first reviewed. Non-uniqueness in this formulation is resolved by the introduction of a pseudo-time and a corresponding pseudo-evolution in which the evolved field is ‘advected’ by frozen vorticity onto a VSF. A weighted essentially non-oscillatory (WENO) method is used to solve the pseudo-evolution equations in pseudo-time, providing a dissipative-like regularization. Vortex surfaces are then extracted as iso-surfaces of the VSFs at different real physical times. The method is applied to two viscous flows with Taylor–Green and Kida–Pelz initial conditions respectively. Results show the collapse of vortex surfaces, vortex reconnection, the formation and roll-up of vortex tubes, vorticity intensification between anti-parallel vortex tubes, and vortex stretching and twisting. A possible scenario for understanding the transition from a smooth laminar flow to turbulent flow in terms of topology of vortex surfaces is discussed

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions

For a strictly inviscid barotropic flow with conservative body forces, the Helmholtz vorticity theorem shows that material or Lagrangian surfaces which are vortex surfaces at time t = 0 remain so for t > 0. In this study, a systematic methodology is developed for constructing smooth scalar fields φ(x, y, z, t = 0) for Taylor–Green and Kida–Pelz velocity fields, which, at t = 0, satisfy ω·∇φ = 0. We refer to such fields as vortex-surface fields. Then, for some constant C, iso-surfaces φ = C define vortex surfaces. It is shown that, given the vorticity, our definition of a vortex-surface field admits non-uniqueness, and this is presently resolved numerically using an optimization approach. Additionally, relations between vortex-surface fields and the classical Clebsch representation are discussed for flows with zero helicity. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Both uniqueness and the distinction separating the evolution of vortex-surface fields and Lagrangian fields are discussed. By tracking φ as a Lagrangian field in slightly viscous flows, we show that the well-defined evolution of Lagrangian surfaces that are initially vortex surfaces can be a good approximation to vortex surfaces at later times prior to vortex reconnection. In the evolution of such Lagrangian fields, we observe that initially blob-like vortex surfaces are progressively stretched to sheet-like shapes so that neighbouring portions approach each other, with subsequent rolling up of structures near the interface, which reveals more information on dynamics than the iso-surfaces of vorticity magnitude. The non-local geometry in the evolution is quantified by two differential geometry properties. Rolled-up local shapes are found in the Lagrangian structures that were initially vortex surfaces close to the time of vortex reconnection. It is hypothesized that this is related to the formation of the very high vorticity regions

More efficient time integration for Fourier pseudo-spectral DNS of incompressible turbulence

Time integration of Fourier pseudo-spectral DNS is usually performed using the classical fourth-order accurate Runge--Kutta method, or other methods of second or third order, with a fixed step size. We investigate the use of higher-order Runge-Kutta pairs and automatic step size control based on local error estimation. We find that the fifth-order accurate Runge--Kutta pair of Bogacki \& Shampine gives much greater accuracy at a significantly reduced computational cost. Specifically, we demonstrate speedups of 2x-10x for the same accuracy. Numerical tests (including the Taylor-Green vortex, Rayleigh-Taylor instability, and homogeneous isotropic turbulence) confirm the reliability and efficiency of the method. We also show that adaptive time stepping provides a significant computational advantage for some problems (like the development of a Rayleigh-Taylor instability) without compromising accuracy

Direct numerical simulation of 'short' laminar separation bubbles with turbulent reattachment

Direct numerical simulation of the incompressible Navier–Stokes equations is used to study flows where laminar boundary-layer separation is followed by turbulent reattachment forming a closed region known as a laminar separation bubble. In the simulations a laminar boundary layer is forced to separate by the action of a suction profile applied as the upper boundary condition. The separated shear layer undergoes transition via oblique modes and [Lambda]-vortex-induced breakdown and reattaches as turbulent flow, slowly recovering to an equilibrium turbulent boundary layer. Compared with classical experiments the computed bubbles may be classified as ‘short’, as the external potential flow is only affected in the immediate vicinity of the bubble. Near reattachment budgets of turbulence kinetic energy are dominated by turbulence events away from the wall. Characteristics of near-wall turbulence only develop several bubble lengths downstream of reattachment. Comparisons are made with two-dimensional simulations which fail to capture many of the detailed features of the full three-dimensional simulations. Stability characteristics of mean flow profiles are computed in the separated flow region for a family of velocity profiles generated using simulation data. Absolute instability is shown to require reverse flows of the order of 15–20%. The three-dimensional bubbles with turbulent reattachment have maximum reverse flows of less than 8% and it is concluded that for these bubbles the basic instability is convective in nature

Active flow control methods for aerodynamic applications

The cylinder in cross flow has been the subject of many numerical and experimental studies since it provides a deep insight of the physical phenomena occurring in a wide range of flow regimes. Despite a number of investigations at Reynolds number (Re = 3900), there has been a constant debate on the important aspects of the flow such as spanwise resolutions, lateral domain extent, convergence of turbulent statistics in the near wake, the so called U-V streamwise velocity profiles at x = 1D, where D is the cylinder diameter, and the critical Re for the onset of shear layer instability together with its characterization. In this thesis, an attempt has been made to address some of these issues and report new results through Direct numerical simulations (DNS) by employing spanwise domain extents i.e. Lz = 1.5D; 2D; 2.5D; pD at the moderate flow regime i.e. Re = 2000, where boundary layer is still laminar while the near wake has gone fully turbulent. Intermittent bursts of shear layer instability have been spotted at this Re indicating the signs of the incipient laminar to turbulent transition in the separating shear layer. It is further confirmed that the secondary instability develops in the regions between the opposite sign large scale spanwise vortices and features a phase lag of 135 degree. Pseudo-Floquet analysis gives a good prediction of fastest growing mode consistent with the reported numerical and experimental measurements. In the second part of the thesis, active flow control (AFC) past circular cylinder has been thoroughly investigated with the aid of parametric analysis at the same Re. We applied spanwise-dependent fluidic actuation, both steady and time-dependent, on the flow past a circular cylinder at Re = 2000. The actuation takes place in two configurations: in-phase blowing and suction from the slits located at ±90 degree (top and bottom) with respect to the upstream stagnation point for both steady and time periodic actuation, and blowing and suction from the top and bottom slits traveling oppositely with respect to each other in the spanwise direction. Optimal forcing amplitude and wavelength are obtained by sweeping across the parametric space. Spanwise-dependent time-independent forcing with wavelength ¿z = 2D has been found the optimal one in terms of drag reduction and attenuation in lift fluctuations. The time-dependent forcing with sinusoids travelling oppositely with respect to each other along the span produced significant reduction in drag force and lift fluctuations, however, the in-phase time periodic actuation with forcing frequency four times the natural vortex shedding frequency resulted in significant increased drag and lift fluctuation, signalling to a potential candidate for the energy harvesting applications. Finally, in the last part of the thesis, time-dependence of flow inside novel laminar-fluidic-oscillator has been analyzed using DNS. Again, pseudoFloquet stability analysis has been utilized to predict the fastest growing Fourier modes along the homogeneous direction. Supplementary three-dimensional numerical study has also been conducted for the suitable cases at various Re. It has been found that steady flow inside fluidic oscillator’s cavity bifurcates from steady state to time-periodic state through supercritical Hopf bifurcation. The secondary transition inside fluidic oscillator’s cavity occurs through the breaking of flow symmetry about the cavity axis by pitchfork supercritical bifurcation.El cilindro en flujo cruzado ha sido objeto de muchos estudios numéricos y experimentales, ya que proporciona una visión profunda de los fenómenos físicos que ocurren en una amplia gama de regímenes. A pesar de una serie de investigaciones en el número de Reynolds (Re = 3900), ha habido un debate constante sobre los aspectos importantes del flujo, como las resoluciones en el span, la extensión del dominio lateral, la convergencia de estadísticas turbulentas en la estela cercana, el tipo de perfil (U o V) en la estela a x = 1D, donde D es el diámetro del cilindro, y el Re crítico para el inicio de la inestabilidad de la capa de cizalla y su caracterización. En esta tesis, se han intentado abordar algunos de estos problemas e informar nuevos resultados a través de simulaciones numéricas directas (DNS) mediante el uso de extensiones de dominio spanwise de Lz = 1.5D; 2D; 2.5D; pD en un régimen de flujo transicional a Re = 2000, donde la capa límite todavía es laminar mientras que la estela cercana se ha vuelto completamente turbulenta. A este Re h sido detectada inestabilidad intermitente, lo que indicando una transición incipiente laminar-turbulenta de la capa de cizalla. Se confirma además que la inestabilidad secundaria se desarrolla en las regiones entre los vórtices a gran escala del signo opuesto y presenta un desfase de 135 grados. El análisis de pseudo-Floquet da una buena predicción del modo de crecimiento más rápido consistente con las mediciones numéricas y experimentales reportadas. En la segunda parte de la tesis, el control de flujo activo (AFC) sobre el cilindro circular se ha investigado a fondo con la ayuda de análisis paramétrico al mismo Re. Aplicamos una actuación fluídica dependiente de la envergadura, tanto constante como dependiente del tiempo, en el flujo alrededor de un cilindro circular a Re = 2000. La actuación se realiza en dos configuraciones: soplado y succión en fase desde las ranuras ubicadas a ± 90 grados (arriba y abajo) con respecto al punto de estancamiento aguas arriba (tanto para la actuación periódica constante como dependiente del tiempo), y para el soplado y la succión con dependencia temporal tal que viajan en sentido opuesto a lo largo de las ranuras superior e inferior. La amplitud y la longitud de onda de forzado óptimas se obtienen barriendo el espacio paramétrico. Se ha encontrado que el forzado independiente del tiempo pero de amplitud variable en la envergadura con longitud de onda ¿z = 2D es el óptimo en términos de reducción de la resistencia y atenuación en las fluctuaciones de sustentación. El forzado dependiente del tiempo con sinusoides que viajan en sentido opuesto entre sí a lo largo del tramo produce una reducción significativa en la fuerza de resistencia aerodinámica y la fluctuación de la sustentación, sin embargo, la actuación periódica en el tiempo en fase con una frecuencia de forzado cuatro veces mayor que la frecuencia natura de desprendimiento de vórtices resultó en un aumento significativo de la resistencia y fluctuaciones de sustentación, lo cual lo coloca como potencial candidato para aplicaciones de recolección de energía. Finalmente, en la última parte de la tesis, la dependencia temporal del flujo dentro de un nuevo oscilador fluídico laminar se ha analizado utilizando DNS. Nuevamente, el análisis de estabilidad pseudoFloquet se ha utilizado para predecir los modos de Fourier de más rápido crecimiento en la dirección homogénea. También se ha realizado un estudio numérico tridimensional suplementario para varios de los Re considerados. Se ha encontrado que el flujo constante dentro de la cavidad del oscilador fluídico bifurca del estado estacionario al estado periódico en el tiempo mediante una bifurcación supercrítica de Hopf. La transición secundaria dentro de la cavidad del oscilador fluídico ocurre a través de la ruptura de la simetría del flujo en relación al eje de simetría de la cavidad por bifurcación supercrítica pitchfork