Vorticity Transport Equation Research Articles

Cavitation shedding dynamics around a hydrofoil simulated using a filter-based density corrected model

The unsteady turbulent cloud cavitation around a NACA66 hydrofoil was simulated using the filter-based density corrected model (FBDCM). The cloud cavitation was treated as a homogeneous liquid-vapor mixture and the effects of turbulent eddy viscosity were reduced in cavitation regions near the hydrofoil and in the wake. The numerical results (in terms of the vapor shedding structure and transient pressure pulsation due to cavitation evolution) agree well with the available experimental data, showing the validity of the FBDCM method. Furthermore, the interaction of vortex and cavitation was analyzed based on the vorticity transport equation, revealing that the cavitation evolution has a strong connection with vortex dynamics. A detailed analysis shows that the cavitation could promote the vortex production and flow unsteadiness by the dilatation and baroclinic torque terms in the vorticity transport equation.

Principal interval decomposition framework for POD reduced‐order modeling of convective Boussinesq flows

SummaryA principal interval decomposition (PID) approach is presented for the reduced‐order modeling of unsteady Boussinesq equations. The PID method optimizes the lengths of the time windows over which proper orthogonal decomposition (POD) is performed and can be highly effective in building reduced‐order models for convective problems. The performance of these POD models with and without using the PID approach is investigated by applying these methods to the unsteady lock‐exchange flow problem. This benchmark problem exhibits a strong shear flow induced by a temperature jump and results in the Kelvin–Helmholtz instability. This problem is considered a challenging benchmark problem for the development of reduced‐order models. The reference solutions are obtained by direct numerical simulations of the vorticity and temperature transport equations using a compact fourth‐order‐accurate scheme. We compare the accuracy of reduced‐order models developed with different numbers of POD basis functions and different numbers of principal intervals. A linear interpolation model is constructed to obtain basis functions when varying physical parameters. The predictive performance of our models is then analyzed over a wide range of Reynolds numbers. It is shown that the PID approach provides a significant improvement in accuracy over the standard Galerkin POD reduced‐order model. This numerical assessment of the PID shows that it may represent a reliable model reduction tool for convection‐dominated, unsteady‐flow problems. Copyright © 2015 John Wiley & Sons, Ltd.

Taylor’s hypothesis in turbulent channel flow considered using a transport equation analysis

Direct numerical simulations of turbulent channel flow at Reτ = 205 and 932 have been carried out to examine Taylor’s “frozen turbulence” hypothesis. The terms in Taylor’s hypothesis appear in the transport equation for instantaneous momentum (Navier-Stokes) in this flow. The additional terms, i.e., the additional convective acceleration term and the pressure gradient and viscous force terms, act to diminish the validity of Taylor’s hypothesis when they are relatively large compared to the Taylor’s hypothesis terms and are not in balance. A similar analysis has been applied to the transport equation for instantaneous vorticity. The additional terms in this equation, namely, the additional convective rates of change of vorticity terms, the stretching/compression/rotation of vorticity terms, and the viscous diffusion of vorticity terms, similarly act to diminish the validity of Taylor’s hypothesis when they are relatively large compared to the terms in the hypothesis and are not in balance. Where in the channel flow this diminishment occurs, and to what degree, and which of the non-Taylor’s hypothesis terms in the momentum and vorticity equations contribute most to this diminishment are unraveled here.

Numerical simulation and analysis of the internal flow in a Francis turbine with air admission

In case of hydro turbines operated at part-load condition, vortex ropes usually occur in the draft tube, and consequently generate violent pressure fluctuation. This unsteady flow phenomenon is believed harmful to hydropower stations. This paper mainly treats the internal flow simulation in the draft tube of a Francis turbine. In order to alleviate the pressure fluctuation induced by the vortex rope, air admission from the main shaft center is applied, and the water-air two phase flow in the entire flow passage of a model turbine is simulated based on a homogeneous flow assumption and SST k-ω turbulence model. It is noted that the numerical simulation reasonably predicts the pressure fluctuations in the draft tube, which agrees fairly well with experimental data. The analysis based on the vorticity transport equation shows that the vortex dilation plays a major role in the vortex evolution with air admission in the turbine draft tube, and there is large value of vortex dilation along the vortex rope. The results show that the aeration with suitable air volume fraction can depress the vortical flow, and alleviate the pressure fluctuation in the draft tube.

Phenomenological modeling of nanosecond pulsed surface dielectric barrier discharge plasma actuation for flow control

One-zone inhomogeneous phenomenological nanosecond dielectric barrier discharge (NS-DBD) actuation model used for flow control simulation is established to investigate the flow control mechanisms, based on experiments and theoretical analysis. When the inhomogeneous phenomenological model is applied to a plate, the formation of spanwise vorticity is analyzed through the vorticity transport equation, and the spanwise vorticity is mainly engendered due to the baroclinicity of pressure gradient and density gradient, also due to the vorticity transfer by the flow convection in the vicinity of the actuation region. Agreement of the simulation with experiments on a column shows that the inhomogeneous phenomenological NS-DBD actuation model is reasonable. Separation control over NACA 0015 airfoil at high angle of attack indicates that the spanwise vortices induced by plasma actuation make the separated shear layer instable, promote interaction between shear layers, and downstream the separation point. Different excitation frequency has different effect on the lift, with the optimum reduced frequency F+ ≈ 6 in current simulation.

Pulsating instability and self-acceleration of fast turbulent flames

A series of three-dimensional numerical simulations is used to study the intrinsic stability of high-speed turbulent flames. Calculations model the interaction of a fully resolved premixed flame with a highly subsonic, statistically steady, homogeneous, isotropic turbulence. The computational domain is unconfined to prevent the onset of thermoacoustic instabilities. We consider a wide range of turbulent intensities and system sizes, corresponding to the Damköhler numbers Da = 0.1 − 6.0. These calculations show that turbulent flames in the regimes considered are intrinsically unstable. In particular, we find three effects. (1) Turbulent flame speed, ST, develops pulsations with the observed peak-to-peak amplitude STmax/STmin>10 and a characteristic time scale close to a large-scale eddy turnover time. Such variability is caused by the interplay between turbulence, which continuously creates the flame surface, and highly intermittent flame collisions, which consume the flame surface. (2) Unstable burning results in the periodic pressure build-up and the formation of pressure waves or shocks, when ST approaches or exceeds the speed of a Chapman-Jouguet deflagration. (3) Coupling of pressure gradients formed during pulsations with density gradients across the flame leads to the anisotropic amplification of turbulence inside the flame volume and flame acceleration. Such process, which is driven by the baroclinic term in the vorticity transport equation, is a reacting-flow analog of the mechanism underlying the Richtmyer-Meshkov instability. With the increase in turbulent intensity, the limit-cycle instability discussed here transitions to the regime described in our previous work, in which the growth of ST becomes unbounded and produces a detonation.

Dynamics and blocking of Rossby waves in quasi-two-dimensional shear flows

One of the most intriguing phenomena in the Earth’s atmosphere—blocking of weather anomalies—is often attributed to the appearance of stationary or blocked Rossby waves in zonal flows. The Obukhov-Charney equation, which is the quasigeostrophic form of the potential vorticity transport equation, is traditionally used to describe Rossby waves in quasi-two-dimensional flows. A class of exact solutions of this equation that describes Rossby waves in the zonal flow with a constant horizontal shear has been constructed in this work. It has been shown that the features of wave dynamics fundamentally depend on the ratio of the wavelength to the radius of Rossby deformation. If this ratio is quite large, there is a fairly long-term quasistationary stage of evolution in which the meridional wavenumber and total energy of a wave (close to the maximum value) hardly depend on time. This effect is implemented under the conditions of the prevailing contribution of the deformation of the free surface of the atmosphere to potential vorticity. It has been shown that this effect can lead to new scenarios of the phase and amplitude blocking of Rossby waves.

Near field vorticity distributions from a sharp-edged rectangular jet

Experimental results on the near field development of a free rectangular jet with aspect ratio 10 are presented. The jet issues from a sharp-edged orifice attached to a rectangular settling chamber at Reh∼23,000, based on slot width, h. Measurements on cross plane grids were obtained with a two-component hot wire anemometry probe, which provided information on the three dimensional characteristics of the flow field. Two key features of this type of jet are mean axial velocity profiles presenting two off axis peaks, commonly mentioned as saddleback profiles, and a predominant dumbbell shape as described by, for example, a contour of the axial mean velocity. The saddleback shape is found to be significantly influenced by the vorticity distribution in the transverse plane of the jet, while the dumbbell is traced to two terms in the axial mean vorticity transport equation that diffuse fluid from the centre of the jet towards its periphery. At the farthest location where measurements were taken, 30 slot widths from the jet exit, the flow field resembles that of an axisymmetric jet.

Large Eddy Simulation and theoretical investigations of the transient cavitating vortical flow structure around a NACA66 hydrofoil

Compared to non-cavitating flow, cavitating flow is much complex owing to the numerical difficulties caused by cavity generation and collapse. In this paper, the cavitating flow around a NACA66 hydrofoil is studied numerically with particular emphasis on understanding the cavitation structures and the shedding dynamics. Large Eddy Simulation (LES) was coupled with a homogeneous cavitation model to calculate the pressure, velocity, vapor volume fraction and vorticity around the hydrofoil. The predicted cavitation shedding dynamics behavior, including the cavity growth, break-off and collapse downstream, agrees fairly well with experiment. Some fundamental issues such as the transition of a cavitating flow structure from 2D to 3D associated with cavitation–vortex interaction are discussed using the vorticity transport equation for variable density flow. A simplified one-dimensional model for the present configuration is adopted and calibrated against the LES results to better clarify the physical mechanism for the cavitation induced pressure fluctuations. The results verify the relationship between pressure fluctuations and the cavity shedding process (e.g. the variations of the flow rate and cavity volume) and demonstrate that the cavity volume acceleration is the main source of the pressure fluctuations around the cavitating hydrofoil. This research provides a better understanding of the mechanism driving the cavitation excited pressure pulsations, which will facilitate development of engineering designs to control these vibrations.

Numerical Simulation of Two Dimensional Laminar Wall Jet Flow over Solid Obstacle

A Computational Fluid Dynamics code was developed to study the flow characteristics of two dimensional laminar incompressible flow. Stream function-vorticity formulation was used for solving two dimensional continuity and momentum equations. The unsteady vorticity transport equation is solved by alternate direction implicit scheme. The stream function equation is solved by the successive over relaxation method. A computational code in c-language was developed to solve the tridiagonal system of algebraic equations. Two dimensional flow through a channel with rectangular block at the bottom wall was considered for the validation. The streamline patterns obtained for different Reynolds number shows good agreement with published results. The code was modified to simulate an incompressible laminar wall jet flow around a solid obstacle. Simulations were carried out for different Reynolds numbers. Contour plots of Stream line, u-velocity and v-velocity were obtained. The variations of flow patterns and the development of vortices were studied and reported.

A dynamic eddy-viscosity closure model for large eddy simulations of two-dimensional decaying turbulence

This paper puts forth a simplified dynamic eddy-viscosity subgrid-scale model for the vorticity transport equation which is employed in a large eddy simulation study of freely evolving isotropic two-dimensional turbulent flows. The dynamic parameter is averaged in space, thereby retrieving a spatially constant value which only varies in time. The proposed dynamic model is applied to a two-dimensional decaying turbulence problem in a square periodic box, which is a standard prototype of more realistic turbulent flows in the atmosphere and oceans, in order to eliminate any possible errors associated with the boundary conditions or mesh non-uniformities. Compared with high-resolution direct numerical simulations, the performance of the dynamic model is systematically investigated considering various filtering strategies by means of test filters. The effects of the computational resolution and the filtering ratio between the test and the grid filters are also studied by using a huge set of parameters.

Numerical simulation of three dimensional cavitation shedding dynamics with special emphasis on cavitation–vortex interaction

Recent experiments showed that there is an interaction between the fluid vortex formation and cavitation, but the mechanism is still an open problem. In the present paper, the structure of the cavitating flow around a twisted hydrofoil was investigated numerically using the mass transfer cavitation model and the modified RNG k-ε model with a local density correction for turbulent eddy viscosity. The predicted three dimensional cavity structures and the shedding frequency agree fairly well with experimental observations. Three types of flow behavior along the suction side of the twisted hydrofoil are discussed. Further analysis of the flow field reveals that cavitation promotes vortex production and increases the boundary layer thickness with local separation and the flow unsteadiness. Finally, the influence of cavitation on the vorticity distribution is illustrated using the vorticity transport equation in a variable density flow and is demonstrated by the contribution of vortex stretching, vortex dilatation and baroclinic torque terms.

A Lagrangian-Lagrangian Model for Two-Phase Bubbly Flow around Circular Cylinder

A Lagrangian-Lagrangian model is developed using an in-house code to simulate bubble trajectory in two-phase bubbly flow around circular cylinder. Random Vortex Method (RVM) which is a Lagrangian approach is used for solving the liquid phase. The significance of RVM relative to other RANS/LES methods is its capability in directly modelling the turbulence. In RVM, turbulence is modeled by solving the vorticity transport equation and there is no need to use turbulence closure models. Another advantage of RVM relative to other CFD approaches is its independence from mesh generation. For the bubbles trajectory, equation of motion of bubbles which takes into account effect of different forces are coupled with the RVM. Comparison of the results obtained from current model with the experimental data confirms the validity of the model. Effect of different parameters including flow Reynolds number, bubble diameter and injection point on the bubbles' trajectory are investigated. Results show that increase in the Reynolds number reduces the rising velocity of the bubbles. Similar behavior is observed for the bubbles when their diameter was decreased. According to the analysis carried out, present Lagrangian-Lagrangian model solves the issues of mesh generation and turbulence modelling which exist in common two phase flow modelling schemes.

Phase velocity effects of the wave interaction with exponentially sheared current

The nonlinear interaction between the unidirectional bichromatic wave-train and exponentially sheared current in water of an infinite depth is investigated. The model is based on the vorticity transport equation and the exact free surface conditions, without any assumptions for the existence of small physical parameters. Earlier works of the wave–current interaction were mainly restricted to either current acted on the monochromatic wave or irregular waves limited to irrotational current. Different from these previous works, no constraint is made in our model for amplitudes of the primary wave, and the current owns an exponential type profile along the vertical line. To ensure that the effect of vorticity on the phase velocity is consistent with earlier derivation, the case of a small amplitude wave traveling on the exponentially sheared current is examined firstly. Then the effect of nonlinearity on the phase velocity of primary waves in a bichromatic wave-train is considered. Accurate high-order approximations of the phase velocity are obtained under consideration of both the nonlinear wave self–self and mutual interactions. Finally, the combined effect of vorticity and nonlinearity on the phase velocity is investigated through the case of a bichromatic wave-train propagating on an exponentially sheared current. It is found that the characteristic current slope determines the effect of vorticity on the phase velocity caused by nonlinear wave self–self and mutual interactions, and the surface current strength may amplify/reduce this effect.

Navier –Stokes Second Exact Transformation

In this article second Navier - Stokes (NSE) exact transformation to the simpler equations is covered. This transformation is executed by classical methods of Mathematical Analysis. It is shown that 3-D NSE can be conversed to a traditional vector form which looks like 2-D Vorticity Transport Equations. Second result, as well as first one, is very important for the solution of Navier-Stokes existence and smoothness one of seven Millennium Prize Problems that were stated by the Clay Mathematics Institute. The proof of solution' existence of such equations is simpler than traditional NSE. New equations will simplify the solutions of many other problems of Applied Mathematics in engineering, aeronautics, etc.

Dynamic analysis of electro- and magneto-rheological fluid dampers using duct flow models

Magneto-rheological (MR) and electro-rheological (ER) fluid dampers provide a semi-active control mechanism for suppressing vibration responses of a structure. MR and ER fluids change their viscosity under the influence of magnetic and electrical fields, respectively, which facilitates automatic control when these fluids are used in damping devices. The existing models, namely the phenomenological models for simulating the behavior of MR and ER dampers, rely on various parameters determined experimentally by the manufacturers for each damper configuration. It is of interest to develop mechanistic models of these dampers which can be applied to various configurations so that their fundamental characteristics can be studied to develop flexible design solutions for smart structures. This paper presents a formulation for dynamic analysis of electro-rheological (ER) and magneto-rheological (MR) fluid dampers in flow and mix mode configurations under harmonic and random excitations. The procedure employs the vorticity transport equation and the regularization function to deal with the unsteady flow and nonlinear behavior of ER/MR fluid in general motion. The finite difference method has been used to solve the governing differential equations. Using the developed approach, the damping force of ER/MR dampers can be calculated under any type of excitation.

Three-dimensional vortex wake structure of flapping wings in hovering flight

Flapping wings continuously create and send vortices into their wake, while imparting downward momentum into the surrounding fluid. However, experimental studies concerning the details of the three-dimensional vorticity distribution and evolution in the far wake are limited. In this study, the three-dimensional vortex wake structure in both the near and far field of a dynamically scaled flapping wing was investigated experimentally, using volumetric three-component velocimetry. A single wing, with shape and kinematics similar to those of a fruitfly, was examined. The overall result of the wing action is to create an integrated vortex structure consisting of a tip vortex (TV), trailing-edge shear layer (TESL) and leading-edge vortex. The TESL rolls up into a root vortex (RV) as it is shed from the wing, and together with the TV, contracts radially and stretches tangentially in the downstream wake. The downwash is distributed in an arc-shaped region enclosed by the stretched tangential vorticity of the TVs and the RVs. A closed vortex ring structure is not observed in the current study owing to the lack of well-established starting and stopping vortex structures that smoothly connect the TV and RV. An evaluation of the vorticity transport equation shows that both the TV and the RV undergo vortex stretching while convecting downwards: a three-dimensional phenomenon in rotating flows. It also confirms that convection and secondary tilting and stretching effects dominate the evolution of vorticity.

Large Eddy Simulation of turbulent vortex-cavitation interactions in transient sheet/cloud cavitating flows

The objectives of this study are to: (1) quantify the influence of sheet/cloud cavitation on the hydrodynamic coefficients and surrounding flow turbulent structures, (2) provide a better insight in the physical mechanisms that govern the dynamics and structure of a sheet/cloud cavity, (3) improve the understanding of the interaction between unsteady cavitating flow, vortex dynamics and hydrodynamic performance. Results are presented for a 3D Clark-Y hydrofoil fixed at an angle of attack of α=8 degrees at a moderate Reynolds number, Re=7×105, for both subcavitating (σ=2.00) and sheet/cloud cavitating conditions (σ=0.80). The experimental studies were conducted in a cavitation tunnel at Beijing Institute of Technology, China. The numerical simulations are performed via the commercial code CFX using a transport equation-based cavitation model, the turbulence model utilizes the Large Eddy Simulation (LES) approach with the Wall-Adapting Local Eddy-viscosity model. The results show that numerical predictions are capable of capturing the initiation of the cavity, growth toward the trailing edge, and subsequent shedding, in accordance with the quantitative features observed in the experiment. The detailed analysis of the vorticity transport equation shows strong correlation between the cavity and vorticity structure, the transient development of sheet/cloud cavitation has significantly changed the interaction between the leading edge and trailing edge vortices, and hence the magnitude as well as the frequency of the hydrodynamic load fluctuations. Compared to the subcavitating case, the sheet/cloud cavitation leads to much higher turbulent boundary layer thickness and substantial increase in velocity fluctuation.

VORTEX METHOD SIMULATION OF BLASIUS' FLAT-PLATE BOUNDARY LAYER

The vortex method has been extensively used to simulate external flows around bluff and streamlined bodies. It relies on the discretization of the vorticity field into a cloud of vortex blobs to simulate the convective−diffusive transport of vorticity. Vortex blobs are generated in the neighborhood of the solid wall in order to satisfy the no-slip and the no-penetration boundary conditions, and they move in a Lagrangian manner to solve the vorticity transport equation. Despite the tremendous development that this powerful mesh-free technique has recently achieved, the numerical implementation of the wall boundary conditions is currently under intense investigation, since it is intimately connected to the vorticity generation process in the vicinity of the body surface. In this paper we describe an efficient two-dimensional vortex method algorithm, with emphasis on a new model for the vortex creation near the surface that increases the accuracy of the simultaneous implementation of the wall boundary conditions. We employ the adaptive fast multipole method to calculate the induced velocities and the corrected core-spreading method to simulate the vorticity diffusion in the boundary layer and wake. The method is second-order accurate in space when Gaussian vortex blobs are used and second-order accurate in time when the Adams-Bashforth scheme is used to march the integration process in time. The algorithm is tested against the well-known two-dimensional, incompressible boundary-layer flow over a flat plate. The agreement between the numerical results and the exact Blasius solution indicates that the algorithm provides an excellent representation of the vorticity field.

Simulation of a high Reynolds number reactive transverse jet and the formation of a triple flame

This paper briefly describes a hybrid Eulerian–Lagrangian approach for the numerical simulation of turbulent combustion and its application to the study of transverse reactive jets. Because of their interesting mixing properties, transverse jets are important to a variety of industrial applications such as film cooling, primary or dilution jets in gas turbines, and flame stabilization in high speed combustion. To capture the jet complex structure and the associated reaction dynamics, we developed a fast, multiscale and parallel 3D code using a Lagrangian particle method to solve the vorticity transport equation and an Eulerian adaptive grid-based method to solve the reactive transport equations.Flame anchoring computations were done for a pure methane jet in a crossflow of air at a Reynolds number Re=1000 and at a velocity ratio of 5 between the jet and the crossflow velocity. The results show that the reactive jet structure strongly resembles that of the non-reactive case, although the reactive jet is slower to bend into the cross flow and it is wider in the spanwise direction. The flame is anchored downstream of the nozzle in a relatively low velocity region and shows a triple flame structure. This triple flame seats in a region of relatively low scalar dissipation rate. We describe the three dimensional topology of the triple flame, as well as the impact of the jet counter-rotating vortex pair (CVP) on the flame front.