General Three-dimensional Flow Research Articles

The viscous force and torque on a closed irrotational surface

The force and torque on a solid body in a viscous potential flow are often taken to be independent of viscosity. Joseph et al. (Eur. J. Mech. B/Fluids, vol. 12, 1993, pp. 97–106; J. Fluid Mech., vol. 265, 1994, pp. 1–23) proved that this holds for (i) the force (not the torque) in any two-dimensional flow, and (ii) the drag force experienced by a purely translating three-dimensional body. The remaining components of the force and torque, along with general three-dimensional flows, were not considered. Importantly, the flow was assumed to be unbounded and irrotational everywhere. We eliminate this rarely satisfied assumption and consider the viscous force and torque experienced by any closed surface where the flow is irrotational locally; this can include a body's surface. Any vorticity distribution is permitted away from the closed irrotational surface. In so doing, we complete the analysis of Joseph et al. for all components of the viscous force and torque in two and three dimensions and enable application to real flows that inevitably contain regions of vorticity.

Modeling Anisotropic Electrical Conductivity of Blood: Translating Microscale Effects of Red Blood Cell Motion into a Macroscale Property of Blood.

Cardiovascular diseases are a leading global cause of mortality. The current standard diagnostic methods, such as imaging and invasive procedures, are relatively expensive and partly connected with risks to the patient. Bioimpedance measurements hold the promise to offer rapid, safe, and low-cost alternative diagnostic methods. In the realm of cardiovascular diseases, bioimpedance methods rely on the changing electrical conductivity of blood, which depends on the local hemodynamics. However, the exact dependence of blood conductivity on the hemodynamic parameters is not yet fully understood, and the existing models for this dependence are limited to rather academic flow fields in straight pipes or channels. In this work, we suggest two closely connected anisotropic electrical conductivity models for blood in general three-dimensional flows, which consider the orientation and alignment of red blood cells (RBCs) in shear flows. In shear flows, RBCs adopt preferred orientations through a rotation of their membrane known as tank-treading motion. The two models are built on two different assumptions as to which hemodynamic characteristic determines the preferred orientation. The models are evaluated in two example simulations of blood flow. In a straight rigid vessel, the models coincide and are in accordance with experimental observations. In a simplified aorta geometry, the models yield different results. These differences are analyzed quantitatively, but a validation of the models with experiments is yet outstanding.

Spike formation theory in three-dimensional flow separation

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in three-dimensional flows with arbitrary time dependence. Based on the exact evolution of the largest principal curvature on near-wall material surfaces, our theory identifies fixed and moving separation. Our approach is effective over short time intervals and admits an instantaneous limit. As a byproduct, we derive explicit formulas for the evolution of the Weingarten map and the principal curvatures of any surface advected by general three-dimensional flows. The material backbone we identify acts first as a precursor and later as the centrepiece of Lagrangian flow separation. We discover previously undetected spiking points and curves where the separation backbones connect to the boundary and provide wall-based analytical formulas for their locations. We illustrate our results on several steady and unsteady flows.

A weighted particle scheme for Enskog-Vlasov equation to simulate spherical nano-droplets/bubbles

The Enskog-Vlasov equation has been proven successful in capturing the complex behaviour of fluids undergoing a phase change. However, its numerical solution is computationally demanding, and this has restricted the studies to one- and two-dimensional planar flows. In this work, a weighted particle scheme is developed for the numerical solution of the Enskog–Vlasov equation in spherically symmetric geometry. It is shown how weighted schemes designed for the Boltzmann equation can be extended to cope with the non-local structure of the Enskog collision integral, and a compact expression of the mean force field is determined using the shell theorem. As an application, the growth rates of nano-droplets/bubbles in the bulk of a homogeneous metastable vapour/liquid are evaluated in a wide range of supersaturation ratios that was not possible until now due to the high computational cost required by alternate approaches. The proposed scheme significantly broadens the range of problems that can be investigated via the Enskog-Vlasov equation, and it is a stepping stone towards the simulation of general three-dimensional liquid-vapour flows.

RETRACTED: Aqueous aluminium–copper hybrid nanofluid flow past a sinusoidal cylinder considering three-dimensional magnetic field and slip boundary condition

We analyzed the problem of the steady general three-dimensional magnetohydrodynamics stagnation-point boundary layer flow past an impermeable wavy circular cylinder considering aluminium–copper/water hybrid nanofluid as the working fluid and velocity slip as well as temperature jump boundary conditions. The induced magnetic field effect was also taken into account. The analytical procedure is based on the model that implements the nanoparticles and base fluid masses to formulate the equivalent volume fraction, equivalent density, and equivalent specific heat at constant pressure which is then substituted in the chosen single-phase thermophysical properties. Then, the foregoing relations were used in basic governing PDEs (partial differential equations), according to Tiwari–Das nanofluid scheme. It is worth mentioning that the bvp4c code from MATLAB software that is a famous finite-difference method has been exploited for solving the final similarity ODEs (ordinary differential equations). Results demonstrate that the developed mass-based model can be successfully employed with great confidence to study the flow and heat transfer of hybrid nanofluid in other similar problems. Moreover, it is proved that the nodal stagnation points possess higher values of skin friction coefficients and local Nusselt numbers relative to those for the saddle stagnation points. Besides, enhancing the second nanoparticle's mass leads to increase in all parameters of engineering interest including skin friction coefficients along x and y directions as well as local Nusselt number.

A robust approach for stability analysis of complex flows using high-order Navier-Stokes solvers

Global stability modes of flows provide significant insight into their dynamics. Direct methods to obtain these modes are restricted by the daunting sizes and complexity of Jacobians encountered in general three-dimensional flows. Jacobian-free iterative approaches such as those based on Arnoldi algorithm have greatly alleviated the required computational burden by considering a smaller subspace formed using Jacobian-vector products to obtain the desired subset of the eigenmodes. However, operations such as orthonormalization, as well as commonly used spectral transformations of matrices introduce computational and parameter-dependent costs that inhibit their routine application to general three-dimensional flowfields. The present work addresses some of these limitations by proposing and implementing a robust, generalizable high-order approach to extract the principal global modes; the method is shown to be straightforward to use with curvilinear coordinates and is exercised in a compressible framework. Linearized perturbation evolution snapshots, representing the desired Jacobian-vector products, are directly obtained from the same non-linear Navier-Stokes code as is used to generate the basic state by adding an easily derived body force, together with suitable initial conditions. This leverages the framework of the non-linear code, including existing advanced high-order spatio-temporal discretization methods, while obviating the need for a separate linearized solver. The subspace formed with the high-order accurate Jacobian-vector products, when subjected to dynamic mode decomposition (DMD), yield the desired physically meaningful principal stability modes without the need for an iterative procedure. The validity and versatility of the method are demonstrated with numerous examples encompassing essential elements expected in realistic flows, such as compressibility effects and complicated domains requiring general curvilinear meshes. Favorable quantitative comparisons are presented with an Arnoldi-based matrix-forming approach in extracting primary and secondary modes of interest. Considerations regarding the accuracy, linearity, convergence and overall efficiency of the method relative to other Jacobian-free methods are also presented. The proposed method is shown to offer an attractive alternative for stability analyses, particularly in flows where stretched, curvilinear meshes are warranted, or very low frequency modes are desired.

Stagnation-point flow of an aqueous titania-copper hybrid nanofluid toward a wavy cylinder

PurposeThe purpose of this paper is to investigate analytically the steady general three-dimensional stagnation-point flow of an aqueous titania-copper hybrid nanofluid past a circular cylinder that has a sinusoidal radius variation.Design/methodology/approachFirst, the analytic modeling of hybrid nanofluid is presented, and using appropriate similarity variables, the governing equations are transformed into nonlinear ordinary differential equations in the dimensionless stream function, which is solved by the well-known function bvp4c from MATLAB.FindingsThe current solution demonstrates good agreement with those of the previously published studies in the special cases of regular fluid and nanofluids. Graphical results are presented to investigate the influences of the titania and copper nanoparticle volume fractions and also the nodal/saddle indicative parameter on flow and heat transfer characteristics. Here, the thermal characteristics of hybrid nanofluid are found to be higher in comparison to the base fluid and fluid containing single nanoparticles. An important point to note is that the developed model can be used with great confidence to study the flow and heat transfer of hybrid nanofluids.Originality/valueAnalytic modeling of hybrid nanofluid is the important originality of present study. Hybrid nanofluids are potential fluids that offer better heat transfer performance and thermophysical properties than convectional heat transfer fluids (oil, water and ethylene glycol) and nanofluids with single nanoparticles. In this investigation, titania (TiO2, 50 nm), copper (Cu, 20 nm) and the hybrid of these two are separately dispersed into the water as the base fluid and analyzed.

Performance of Partial and Cavity Type Squealer Tip of a HP Turbine Blade in a Linear Cascade

Three-dimensional highly complex flow structure in tip gap between blade tip and casing leads to inefficient turbine performance due to aerothermal loss. Interaction between leakage vortex and secondary flow structures is the substantial source of that loss. Different types of squealer tip geometries were tried in the past, in order to improve turbine efficiency. The current research deals with comparison of partial and cavity type squealer tip concepts for higher aerothermal performance. Effects of squealer tip have been examined comprehensively for an unshrouded HP turbine blade tip geometry in a linear cascade. In the present paper, flow structure through the tip gap was comprehensively investigated by computational fluid dynamic (CFD) methods. Numerical calculations were obtained by solving three-dimensional, incompressible, steady, and turbulent form of the Reynolds-averaged Navier-Stokes (RANS) equations using a general purpose and three-dimensional viscous flow solver. The two-equation turbulence model, shear stress transport (SST), has been used. The tip profile belonging to the Pennsylvania State University Axial Flow Turbine Research Facility (AFTRF) was used to create an extruded solid model of the axial turbine blade. For identifying optimal dimensions of squealer rim in terms of squealer height and squealer width, our previous studies about aerothermal investigation of cavity type squealer tip were utilized. In order to obtain the mesh, an effective parametric generation has been utilized using a multizone structured mesh. Numerical calculations indicate that partial and cavity squealer designs can be effective to reduce the aerodynamic loss and heat transfer to the blade tip. Future efforts will include novel squealer shapes for higher aerothermal performance.

Unsteady convective heat and mass transfer of a nanofluid in Howarth’s stagnation point by Buongiorno’s model

Purpose – The purpose of this paper is to do a comprehensive study on the unsteady general three-dimensional stagnation-point flow and heat transfer of a nanofluid by Buongiorno’s model. Design/methodology/approach – In this study, the convective transport equations include the effects of Brownian motion and thermophoresis. By introducing new similarity transformations for velocity, temperature and nanoparticle volume fraction, the basic equations governing the flow, heat and mass transfer are reduced into highly non-linear ordinary differential equations. The resulting non-linear system has been solved both analytically and numerically. Findings – The analysis shows that velocity, temperature and nanoparticle concentration profiles in the respective boundary layers depend on five parameters, namely unsteadiness parameter A, Brownian motion parameter Nb, thermophoresis parameter Nt, Prandtl number Pr and Lewis number Le. It is found that the thermal boundary layer thickens with a rise in both of the Brownian motion and the thermophoresis effects. Therefore, similar to the earlier reported results, the Nusselt number decreases as the Brownian motion and thermophoresis effects become stronger. A correlation for the Nusselt number has been developed based on a regression analysis of the data. This correlation predicts the numerical results with a maximum error of 9 percent for a usual domain of the physical parameters. Originality/value – The stagnation point flow toward a wavy cylinder (with nodal and saddle stagnation points) that a little attention has been given to it up to now. The examination of unsteadiness effect on the general three-dimensional stagnation-point flow. The application of an interesting and global model (Boungiorno’s model) for the nanofluid that incorporates the effects of Brownian motion and thermophoresis. The study of the effects of Brownian motion and thermophoresis on the nanofluid flow, heat and mass transfer characteristics. The prediction of correlation for the Nusselt number based on a regression analysis of the data. General speaking, we can tell the problem with this geometry, characteristics, the applied model, and comprehensive results, was Not studied and analyzed in literature up to now.

Modeling and Simulation of Fiber Orientation in Injection Molding of Polymer Composites

We review the fundamental modeling and numerical simulation for a prediction of fiber orientation during injection molding process of polymer composite. In general, the simulation of fiber orientation involves coupled analysis of flow, temperature, moving free surface, and fiber kinematics. For the governing equation of the flow, Hele‐Shaw flow model along with the generalized Newtonian constitutive model has been widely used. The kinematics of a group of fibers is described in terms of the second‐order fiber orientation tensor. Folgar‐Tucker model and recent fiber kinematics models such as a slow orientation model are discussed. Also various closure approximations are reviewed. Therefore, the coupled numerical methods are needed due to the above complex problems. We review several well‐established methods such as a finite‐element/finite‐different hybrid scheme for Hele‐Shaw flow model and a finite element method for a general three‐dimensional flow model.

Transport of spherical gyrotactic organisms in general three-dimensional flow fields

In this paper we consider spherical organisms whose orientations are determined by the combination of viscous torque with a self-righting torque (either due to bottom-heaviness, shape, or through preferential swimming due to active sensing): a phenomenon known as gyrotaxis. We restate the stable equilibrium orientation of such an organism in a homogeneous shear flow, which exists for all shear values if the two torques are not parallel, and only when the nondimensional shear is below a critical value if the torques are parallel. We calculate the long-term trajectories of such an organism in a flow, which consists of linear motion in the direction of the vorticity and a conic section in the plane of shear, giving, for instance, helical motion in the case of pure rotation. We compute the orientation probability density function when the reorientation is coupled with rotational diffusion and show that including rotational diffusion reduces the mean swimming speed and rotates the mean equilibrium orientation toward the vertical. Numerical simulation of stochastic gyrotactic cells in synthetic turbulence demonstrates the interaction between swimming and more general flow fields. The calculation of the mean drift and diffusion demonstrates that the expected transition between swimming-dominated transport (for low kinetic energy dissipation rate ϵ) and turbulence-dominated transport (for large ϵ) occurs over dissipation rates such motile organisms experience in nature (ϵ∼10−10–10−3 m2 s−3).

A family of vortex rings and a variational application to potential flows around three-dimensional bodies

A variational formulation and solution of general three-dimensional potential flows gave rise to the construction of a special family of ‘trial functions’. This family is composed by circular-sector vortex rings, here named α-rings, i.e., rings that are positioned on the border of a circular sector with aperture angle α. An explicit formula for the velocity potential describing the α-rings family is here derived. A particular case is the well-known circular vortex-ring. The formula is given in terms of a uniformly valid series involving trigonometric and Hypergeometric functions. Results concerning the complete circular ring are compared to the well-known solution given, in closed form, in terms of Bessel functions, validating the present formula. Convergence is discussed. Graphical examples are shown for various rings of different sector angles. As an elementary application, the steady potential flow around three-dimensional bodies in unbounded fluid is formulated and solved under the variational approach. The variational method is fully validated through the sphere problem and for a family of spheroids. Examples concerning either translatory or rotatory motion around a transversal axis are presented for the spheroid family.

Derivation of a sphere theorem for the Stokes flow following Imai's procedure

A sphere theorem for general three-dimensional Stokes flow is shown to be derived by the use of Imai's procedure for solving the Stokes equation for spherical boundary in terms of one vector harmonic function and Kelvin's inversion theorem.

High order level contour reconstruction method

Complex interfacial physics arising from geometric curvature associated with surface tension as well as phase transformation make it a formidable task to design an accurate, reliable, and yet simple method for direct computation of multiphase flows. Hybrid methods mixing conventional, Volume-of-Fluid, Level Set, Phase Field, and Front Tracking methods have recently become popular in an attempt to overcome the shortcomings of each method alone. We developed the Level Contour Reconstruction Method (LCRM) as part of a hybrid method for treating the complex interface geometry associated with general three-dimensional multiphase flows. The main idea in that work focused on a simple and robust algorithm especially suited for dynamic interfaces in the three-dimensional case by combining characteristics of both Front Tracking and Level Set methods. In this article we describe a modification to the LCRM which introduces a high order interpolation kernel during the course of the interface reconstruction along with a new hybrid surface tension formulation. With this we can essentially eliminate any mass redistribution between regions of differing curvature and reconstruct the interface accurately and smoothly. The improvement with high order reconstruction is also noticeable vis a vis spurious currents which are further decreased by two orders of magnitude over the previous linear reconstruction method. Moreover, there is no disturbance concurrent with reconstruction and the solution fidelity is not influenced by the reconstruction time step. This High Order Level Contour Reconstruction Method retains the simplicity of the original LCRM and avoids complicated interface smoothing procedures.

Techniques for deriving explicit algebraic Reynolds stress models based on incomplete sets of basis tensors and predictions of fully developed rotating pipe flow

Different techniques for deriving explicit algebraic Reynolds stress models (EARSMs) using incomplete sets of basis tensors are discussed. The first is the Galerkin method which has been used by several authors. The second alternative technique, proposed here, is based on the least-squares method. The idea behind the latter method is to minimize the error induced in the implicit relation, i.e., the algebraic Reynolds stress model (ARSM) equation, due to the use of incomplete sets of basis tensors. It is argued that since the system matrix of the ARSM equation is not symmetric and positive definite, the Galerkin method does not give EARSMs that are optimal in the strict classical sense. The possible singular behavior depending on the choice of the basis tensors has also been investigated. It is demonstrated that many of the EARSMs based on incomplete tensor bases, expressed in general three-dimensional mean flows, have singularity problems in some flows, such as general two-dimensional (2D) mean flows or more specifically, strain- and/or rotation-free 2D mean flows. The different EARSMs emanating from the two derivation methods are investigated by computing fully developed rotating pipe flow. The results indicate that the EARSMs derived with the least-squares method capture the behavior of the complete EARSMs significantly better than those derived with the Galerkin method. Furthermore, by using mean flow data from the complete EARSMs to evaluate the square error of the incomplete EARSMs it is demonstrated that the least-squares based EARSMs have square errors significantly smaller than the Galerkin EARSMs, very close to minimum.

An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications

An eddy-viscosity model is proposed and applied in large-eddy simulation of turbulent shear flows with quite satisfactory results. The model is essentially not more complicated than the Smagorinsky model, but is constructed in such a way that its dissipation is relatively small in transitional and near-wall regions. The model is expressed in first-order derivatives, does not involve explicit filtering, averaging, or clipping procedures, and is rotationally invariant for isotropic filter widths. Because of these highly desirable properties the model seems to be well suited for engineering applications. In order to provide a foundation of the model, an algebraic framework for general three-dimensional flows is introduced. Within this framework several types of flows are proven to have zero energy transfer to subgrid scales. The eddy viscosity is zero in the same cases; the theoretical subgrid dissipation and the eddy viscosity have the same algebraic structure. In addition, the model is based on a fundamental realizability inequality for the theoretical subgrid dissipation. Results are shown for a transitional and turbulent mixing layer at high Reynolds number and a turbulent channel flow. In both cases the present model is found to be more accurate than the Smagorinsky model and as good as the standard dynamic model. Unlike the Smagorinsky model, the present model is able to adequately handle not only turbulent but also transitional flow.

Numerical and Experimental Investigations of the Cavitating Behavior of an Inducer

A robust CFD model is described, suitable for general three‐dimensional flows with extensive cavitation at large density ratios. The model utilizes a multiphase approach, based on volume‐scalar‐equations, a truncated Rayleigh‐Plesset equation for bubble dynamics, and specific numerical modifications (in a finite‐volume solution approach) to promote robust solutions when cavitation is present. The model is implemented in the CFD software CFX TASCflow 2.12. The validation of the model was done on an inducer designed and tested at LEMFI. First, The physical model and the numerical aspects are described. Then, the experimental and numerical methodologies, at cavitating regime, are presented. Finally, for several flow rates, the comparisons between experimental and simulated results on the overall performances, head drop and cavitation figures, are discussed. For a range of flow rates, good agreement between experiment and prediction was found.

Vorticity decomposition and its application to sectional flow characterization

Great insights have been gained by applying the concept of vorticity to the study of the natural flow of rocks. Geologists have used various schemes of vorticity decomposition, depending on the purpose of their studies. The fundamental decomposition divides the overall vorticity into that of the material with respect to the instantaneous stretching axes and the spin of these axes in an external reference frame in which the flow is defined. This represents a new contribution to continuum mechanics, material science and structural geology. Its implications for the kinematic analysis of geological structures and fabrics are more significant than have been realized. In virtually every geological situation the flow has a spin component, and therefore vorticity decomposition needs to be considered. The decomposition of vorticity is applied to the characterization of the two-dimensional flow on a material plane (sectional flow) in a general three-dimensional flow field. The analysis of the sectional flow and sectional non-coaxiality is a precondition for a determination of the initial orientation of structural elements (such as the orientation of a fold developed in a competent layer that is embedded in a high-strain zone) and their temporal evolution. Theoretically, it may also be possible to derive the symmetry of a natural deformation path by examining fabric symmetries on differently oriented sections from deformed rock samples.

On the Instability of Certain Motions of an Ideal Incompressible Fluid

Short-wave instabilities in the general three-dimensional time-dependent flows of an ideal incompressible fluid are considered. A new geometrical optics equation for the perturbed vorticity amplitude is derived. It is shown that under certain conditions this equation can be solved explicitly. Using this observation a number of known instability conditions are obtained and some new ones are derived.

Microstructure suspended in three-dimensional flows

The dynamical behaviour of stretchable, orientable microstructure suspended in a general three-dimensional fluid flow is investigated. Model equations given by Olbricht, Rallison & Leal (1982) are examined in the case of microstructure travelling through arbitrarily complicated flows of the carrier fluid. As in the two-dimensional analysis of Szeri, Wiggins & Leal (1991), one must first treat the orientation dynamics problem; only then can the equation for stretch of the microstructure be analyzed rationally. In three-dimensional flows that are steady in the Lagrangian frame, attractors for the orientation dynamics are shown to be equilibria or limit cycles; this asymptotic behaviour was first deduced by Bretherton (1962). In three-dimensional flows that are time periodic in the Lagrangian frame (e.g. recirculating flows), the orientation dynamics may be characterized by periodic or quasi-periodic attractors. Thus, robust (generic) behaviour in these cases is always characterized by a single global attractor; there is no asymptotic dependence of orientation dynamics on the initial orientation. The type of asymptotic orientation dynamics – steady, periodic, or quasi-periodic - is signified by a simple criterion. Details of the relevant bifurcations, as well as history-dependent strong flow criteria are developed. Examples which illustrate the various types of behaviour are given.