Numerical discovery and experimental confirmation of vortex ring generation by microramp vortex generator

An implicitly implemented large eddy simulation (ILES), by using the modified fifth order WENO scheme, is applied to study the flow around the microramp vortex generator (MVG) at Mach 2.5 and Re_θ = 5760. A series of new discoveries on the flow around supersonic MVG have been made by the UTA LES team including source of the momentum deficit, inflection points (surface in 3-D), Kelvin–Helmholtz instability and vortex ring generation. Most of the new discoveries, which were made by the UTA LES team and presented in 2009, were confirmed by experiment conducted by the UTA experiment team in 2010. A new 5-pair-vortex-tube model near the MVG is given based on the ILES observation.     

LES and analyses on the vortex structure behind supersonic MVG with turbulent inflow

In this paper, an implicitly implemented high order large eddy simulation by using the fifth order bandwidth-optimized WENO scheme is applied to make comprehensive studies on ramp flows with and without control at Mach 2.5 and Re_θ = 5760. Flow control in the form of microramp vortex generators (MVG) is applied. The mechanism of vortex ring generation behind MVG has been studied in detail and shear layer instability has been studied and found as the mechanism of K–H vortex ring generation. A series of new observations on the flow around supersonic MVG have been made including inflection points (surface in 3-D), vorticity conservation, interaction of the primary vortex and new generated K–H vortex rings, and the K–H vortex ring structure. The numerical observations have been confirmed by the experimental work.     

Large eddy simulation of a sheet/cloud cavitation on a NACA0015 hydrofoil

A single fluid model of sheet/cloud cavitation is developed and applied to a NACA0015 hydrofoil. First, a cavity formation model is set up, based on a three-dimensional (3D) non-cavitation model of Navier–Stokes equations with a large eddy simulation (LES) scheme for weakly compressible flows. A fifth-order polynomial curve is adopted to describe the relationship between density coefficient ratio and pressure coefficient when cavitation occurs. The Navier–Stokes equations including cavitation bubble clusters are solved using the finite-volume approach with time-marching scheme, and MacCormack’s explicit-corrector scheme is adopted. Simulations are carried out in a 3D field acting on a hydrofoil NACA0015 at angles of attack 4°, 8° and 20°, with cavitation numbers σ = 1.0, 1.5 and 2.0, Re = 10⁶, and a 360 × 63 × 29 meshing system. We study time-dependent sheet/cloud cavitation structures, caused by the interaction of viscous objects, such as vortices, and cavitation bubbles. At small angles of attack (4°), the sheet cavity is relatively stable just by oscillating in size at the accumulation stage; at 8° it has a tendency to break away from the upper foil section, with the cloud cavitation structure becoming apparent; at 20°, the flow separates fully from the leading edge of the hydrofoil, and the vortex cavitation occurs. Comparisons with other studies, carried out mainly in the context of flow patterns on which prior experiments and simulations were done, demonstrate the power of our model. Overall, it can snapshot the collapse of cloud cavitation, and allow a study of flow patterns and their instabilities, such as “crescent-shaped regions.”     

An immersed boundary-lattice Boltzmann method combined with a robust lattice spring model for solving flow–structure interaction problems

An immersed boundary (IB)-lattice Boltzmann method (LBM) combined with a robust lattice spring model (LSM) was developed for modeling fluid–elastic body interactions. To include the effects of viscous flow forces on the deformation of a flexible body, rotational invariant springs were connected regularly inside the deformable body with square lattices. Fluid–solid interactions were due to an additional force density in the lattice Boltzmann equation enhanced by the split-forcing approach. To check the validity and accuracy of the numerical method, the flow over a rigid plate and the deformation of a cantilever beam were investigated. To demonstrate the capability of the new method, different test cases were examined. The deformation of a two-dimensional flexible vertical plate in a laminar cross-flow stream at different conditions was analyzed. The simulations were performed for different boundary conditions imposed on the elastic plate, namely, fixed-end corners and fixed middle point. Different flow conditions such as “steady flow regime”, “vortex shedding flow regime”, and the limit of “rigid body motion” were examined using the new IB-LBM-LSM approach. A general formulation for evaluating the deformation of the elastic body was also introduced, in which the position of the LSM nodes (inside the body) was updated implicitly at each time step. Two dimensionless groups, namely capillary number (Ca) and Reynolds number (Re), were used for parametric study of the behavior of the flow around the deformable plate. It was found that for low Reynolds numbers (Re < 50) and when the middle of the plate was fixed, decreasing the capillary number led to a decrease in the drag coefficient. The fluctuation of the plate during the vortex shedding flow regime was also explored. It was found that when the middle of the plate was fixed, the critical Reynolds number for the initiation of vortex shedding increased. For Re > 100, the Strouhal number was observed to increase with the decrease in capillary number.     

Predicting vortex-induced vibration from driven oscillation results

Two-dimensional simulations of flow past both an elastically-mounted cylinder and an externally-driven oscillating cylinder were performed at a Reynolds number of Re = 200. The results were compared to determine if the oscillations of the driven-oscillation model were consistent with the oscillations observed in the elastically-mounted system. It was found that while this is the case, there is considerable sensitivity to input forcing. This sensitivity could explain observed discrepancies between experimental results for the two systems.     

Numerical prediction of buoyancy driven micropolar fluid flow within uniformly heated eccentric annulus

In this paper, the problem of buoyancy driven micropolar fluid flow within an annulus formed between two circular concentric/eccentric tubes has been numerically investigated using Fourier spectral method. The annulus inner wall is uniformly heated and maintained at constant heat flux while the outer wall is cooled and kept at constant temperature. The full governing equations of linear momentum, angular momentum and energy have been solved to give the details of flow and thermal fields. The heat convection process in the annulus is mainly controlled by modified Rayleigh number Ra, Prandtl number Pr, radius ratio Rr, eccentricity, e and material parameters of Micropolar fluid. The material parameters are dimensionless spin gradient viscosity λ, dimensionless micro-inertia density B and dimensionless vortex viscosity D. The study considered a range of modified Ra up to 10⁵ and is carried out at three values of Pr, namely Pr = 0.1, 1.0 and 7.0, and at three values of parameter D, namely, D = 2, 4, 8 while the eccentricity is varied between −0.65 and +0.65. The radius ratio is fixed at 2.6 while the material parameters B and λ are assigned the value of 1. The effect of the controlling parameters on flow and thermal fields has been investigated with emphasis on the effect of these parameters on local and mean inner wall temperatures. The study has shown that for certain controlling parameters the steady mean temperature of inner wall of the annulus is maximum at a certain eccentricity. The study has also shown that as the parameter D increases the steady mean inner wall temperature increases. Moreover, the study has shown that as the Pr increases the mean inner wall temperature decreases.     

On the bin packing problem with a fixed number of object weights

We consider a bin packing problem where the number of different object weights is fixed to C. We analyze a simple approximate approach and show that it leads to an asymptotically exact polynomial algorithm with absolute error 0 if C = 2, at most 1 if 1 < C ? 6, and at most 1 + ⌊(C − 1)/3⌋ if C > 6. A consequence of our analysis is a new upper bound on the gap between the optimal value of the problem at hand and the round-up of the optimal value of the linear relaxation of its Gilmore–Gomory formulation.     

Computational study of decaying annular vortex flow using the Rε/k − ε turbulence model

In this article, we present three dimensional CFD study of turbulent vortex flow in an annular passage using OpenFOAM 1.6. The vortex flow is generated by introducing the flow through a tangential entry to the passage. For the analysis presented in this article, turbulence was modeled using the R_ε/k − ε model, in addition, a comparison between such model with the standard k − ε model was conducted and discussed. The main characteristics of the flow such as vortex structure and recirculation zone were investigated. It was found that flow is subjected to Rankine vortex structure with three forced vortex regimes and a free vortex region near to the outer wall. The phenomenon of vortex decay was investigated by depicting the swirl number trend along the axial direction of the flow domain. It was found that the vortex decay is subjected to an exponential decay behavior. New coefficients for the exponential decay correlation were derived based on local values of velocity components in different radial planes.     

Numerical exploration of new features on three-dimensional transition of a cylinder wake

The three-dimensional transition of the wake flow behind a circular cylinder is studied in detail by direct numerical simulations using 3D incompressible N-S equations for Reynolds number ranging from 200 to 300. New features and vortex dynamics of the 3D transition of the wake are found and investigated. At Re = 200, the flow pattern is characterized by mode A instability. However, the spanwise characteristic length of the cylinder determines the transition features. Particularly for the specific spanwise characteristic length linear stable mode may dominate the wake in place of mode A and determine the spanwise phase difference of the primary vortices shedding. At Re = 250 and 300 it is found that the streamwise vortices evolve into a new type of mode’“dual vortex pair mode” downstream. The streamwise vortex structures switch among mode A, mode B and dual vortex pair mode from near wake to downstream wake. At Re = 250, an independent low frequency f _m in addition to the vortex shedding frequency f _s is identified. Frequency coupling between f _m and f _s occurs. These result in the irregularity of the temporal signals and become a key feature in the transition of the wake. Based on the formation analysis of the streamwise vorticity in the vicinity of cylinder, it is suggested that mode A is caused by the emergence of the spanwise velocity due to three dimensionality of the incoming flow past the cylinder. Energy distribution on various wave numbers and the frequency variation in the wake are also described.     

Bounding the gap between extremal Laplacian eigenvalues of graphs

Let G be a graph whose Laplacian eigenvalues are 0 = λ₁ ? λ₂ ? ? ? λ_n. We investigate the gap (expressed either as a difference or as a ratio) between the extremal non-trivial Laplacian eigenvalues of a connected graph (that is λ_n and λ₂). This gap is closely related to the average density of cuts in a graph. We focus here on the problem of bounding the gap from below.     

Convergence of rank-type equations

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k = 1, and the median of an odd number p of equations, for k = (p + 1)/2. In the non-hyperbolic case, conjectures are stated about the eventual periodicity of the equations, generalizing long-standing conjectures of G. Ladas.     

A numerical simulation of flow between two rotating coaxial frustum cones

The behavior of flow between two coaxial frustum cones, with the inner one rotating and the outer stationary, is studied in this paper. It is found that the fluid at the outlet does not flow out directly, but flows up till a certain height. This reflux generates a vortex area with a quite large velocity and pressure magnitude. This reflux area, between Z/H = 0.05 and 0.30, has the trend to move up with increasing Reynolds number Re. The velocity magnitude is linear in the radial direction if the Re is small. This linear relation converts to quasi-quadratic function as the Re increasing. If the frustum cone inclination is small, the flow will tend to be unstable with a quite large velocity and pressure magnitude. Finally, a comparison is made with Taylor–Couette flow.     

Prospective teachers’ reactions to Right-or-Wrong tasks: The case of derivatives of absolute value functions

This paper illustrates the role of a Thinking-about-Derivatives task in identifying learners’ derivative conceptions and for promoting their critical thinking about derivatives of absolute value functions. The task included three parts: Define the derivative of a function f(x) at x = x_0,Solve-if-Possible the derivative of f(x) = |x| at x = 2 and at x = 0, and evaluate the correctness of suggested solutions in a Right-or-Wrong part. Three prospective teachers, Noa, Anat and Daniel were individually interviewed when solving the task. We found that while the participants correctly solved the Define part, they exhibited some erroneous images in the Solve-if-Possible part, and their work on the Right-or-Wrong part contributed to their critical thinking about functions and derivatives. All three participants expressed their appreciation of their work on the Right-or-Wrong part of the task.     

Co-cliques and star complements in extremal strongly regular graphs

Suppose that the positive integer μ is the eigenvalue of largest multiplicity in an extremal strongly regular graph G. By interlacing, the independence number of G is at most 4μ² + 4μ − 2. Star complements are used to show that if this bound is attained then either (a) μ = 1 and G is the Schläfli graph or (b) μ = 2 and G is the McLaughlin graph.     

Effect of aligned magnetic field on the steady viscous flow past a circular cylinder

The steady viscous incompressible and slightly conducting fluid flow around a circular cylinder with an aligned magnetic field is simulated for the range of Reynolds numbers 100 ? Re ? 500 using the Hartmann number, M. The multigrid method with defect correction technique is used to achieve the second order accurate solution of complete non-linear Navier–Stokes equations. The magnetic Reynolds number is assumed to be small. It is observed that volume of the separation bubble decreases and drag coefficient increases as M is increased. We noticed that the upstream base pressure increases slightly with increase of M whereas downstream base pressure decreases with increase of M. The effect of the magnetic field on the flow is discussed with contours of streamlines, vorticity, plots of surface pressure and surface vorticity.     

Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α ≈ 1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.     

Inventory models for perishable items with inventory level dependent demand rate

This paper presents inventory models for perishable items with inventory level dependent demand rate. The models with and without backlogging are studied. In the backlogging model, it is assumed that the backlogging rate is dependent on the waiting time and the amount of products already backlogged simultaneously. Two cases that holding inventory is profitable or not are studied, respectively. The smallest shelf space to ensure shortage not occur when holding inventory is not profitable is obtained. In the model without backlogging, it is assumed that the remaining stock at the end of the inventory cycle is disposed of with salvage value. The necessary and sufficient conditions for the existence and uniqueness of the optimal solution of these models are investigated. At last, some numerical examples are presented to illustrate the effectiveness of the proposed model. The model in this paper is generalization of present ones. In particularly, the model is reduced to Padmanabhan and Vrat’s when δ₁ = 0, and Dye and Ouyang’s when δ₂ = 0. If S = s and δ₂ = 0, it is Chang, Goyal and Teng’s model.     

Minimal Positive Solutions to Some Singular Second-Order Differential Equations

The existence of a minimal C¹[0, 1] positive solution is established for some second-order singular boundary value and initial value problems by new schemes, which are related to x′. Our nonlinearity may be singular at t = 0, 1, x = 0, or x′ = 0.     

An analytical comparison of the LP relaxations of integer models for the k-club problem

Given an undirected graph G = (V, E), a k-club is a subset of nodes that induces a subgraph with diameter at most k. The k-club problem is to find a maximum cardinality k-club. In this study, we use a linear programming relaxation standpoint to compare integer formulations for the k-club problem. The comparisons involve formulations known from the literature and new formulations, built in different variable spaces. For the case k = 3, we propose two enhanced compact formulations. From the LP relaxation standpoint these formulations dominate all other compact formulations in the literature and are equivalent to a formulation with a non-polynomial number of constraints. Also for k = 3, we compare the relative strength of LP relaxations for all formulations examined in the study (new and known from the literature). Based on insights obtained from the comparative study, we devise a strengthened version of a recursive compact formulation in the literature for the k-club problem (k > 1) and show how to modify one of the new formulations for the case k = 3 in order to accommodate additional constraints recently proposed in the literature.     

Fractal oscillations of self-adjoint and damped linear differential equations of second-order

For a prescribed real number s ∈ [1, 2), we give some sufficient conditions on the coefficients p(x) and q(x) such that every solution y = y(x), y ∈ C²((0, T]) of the linear differential equation (p(x)y′)′ + q(x)y = 0 on (0, T], is bounded and fractal oscillatory near x = 0 with the fractal dimension equal to s. This means that y oscillates near x = 0 and the fractal (box-counting) dimension of the graph Γ(y) of y is equal to s as well as the s dimensional upper Minkowski content (generalized length) of Γ(y) is finite and strictly positive. It verifies that y admits similar kind of the fractal geometric asymptotic behaviour near x = 0 like the chirp function y_ch(x) = a(x)S(φ(x)), which often occurs in the time-frequency analysis and its various applications. Furthermore, this kind of oscillations is established for the Bessel, chirp and other types of damped linear differential equations given in the form y″ + (μ/x)y′ + g(x)y = 0, x ∈ (0, T]. In order to prove the main results, we state a new criterion for fractal oscillations near x = 0 of real continuous functions which essentially improves related one presented in [1].