Two-dimensional Viscous Flow Research Articles

Two-stage fourth-order gas-kinetic scheme for three-dimensional Euler and Navier-Stokes solutions

ABSTRACTFor the one-stage third-order gas-kinetic scheme (GKS), successful applications have been achieved for the three-dimensional compressible flows [Pan, L., K. Xu, Q. Li, and J. Li. 2016. “An Efficient and Accurate Two-stage Fourth-order Gas-kinetic Scheme for the Navier-Stokes Equations.” Journal of Computational Physics 326: 197–221]. The high-order accuracy of the scheme is obtained by integrating a multidimensional time-accurate gas distribution function over the cell interface within a time step without using Gaussian quadrature points and Runge-Kutta time-stepping technique. However, to the further increase of the order of the scheme, such as the fourth-order one, the one step formulation becomes very complicated for the multidimensional flow. Recently, a two-stage fourth-order GKS with high efficiency has been constructed for two-dimensional inviscid and viscous flow computations ([Li, J., and Z. Du. 2016. “A Two-stage Fourth Order Time-accurate Discretization for Lax-Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws.” SIAM Journal on Scientific Computing 38: 3046–3069]; Pan et al. 2016), and the scheme uses the time accurate flux function and its time derivatives. In this paper, a fourth-order GKS is developed for the three-dimensional flows under the two-stage framework. Based on the three-dimensional WENO reconstruction and flux evaluation at Gaussian quadrature points on a cell interface, the high-order accuracy in space is achieved first. Then, the two-stage time stepping method provides the high accuracy in time. In comparison with the formal third-order GKS [Pan, L., and K. Xu. 2015. “A Third-order Gas-kinetic Scheme for Three-dimensional Inviscid and Viscous Flow Computations.” Computers & Fluids 119: 250–260], the current fourth-order method not only improves the accuracy of the scheme, but also reduces the complexity of the gas-kinetic flux solver greatly. More importantly, the fourth-order GKS has the same robustness as the second-order shock capturing scheme [Xu, K. 2001. “A Gas-kinetic BGK Scheme for the Navier-Stokes Equations and its Connection with Artificial Dissipation and Godunov Method.” Journal of Computational Physics 171: 289–335]. Numerical results validate the outstanding reliability and applicability of the scheme for three-dimensional flows, such as the cases related to turbulent simulations.

Flowfield Reconstruction Method Using Artificial Neural Network

Multidimensional aerodynamic database development has become more and more important for the design, control, and guidance of modern aircraft. To relieve the curse of the dimensionality, a novel flowfield reconstruction method based on an artificial neural network is proposed. The idea is to design a simplified problem that is related to the target problem. Then, the map from the simplified problem to the target problem is built using an artificial neural network. Finally, the target problem can be predicted efficiently through solving the simplified problem instead. Examples of the efficiency of this approach include two-dimensional viscous nozzle flows, an inviscid M6 wing flow, a viscous hypersonic flow of a complex configuration, and an unsteady two-dimensional Riemann problem to evaluate the performance of the proposed method. A Gaussian process model is also incorporated for a comparative study. With an artificial neural network of moderate complexity, the solution of the target problem can be generated with good accuracy. Among other observations, it is found that shocks can be predicted well with sharp resolution for steady and unsteady cases. Overall, using a simplified problem that accounts for all the interested parameters as inputs tends to be more reliable than using input parameters.

Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations

In this paper, we consider the global strong solutions to the Cauchy problem of the compressible Navier-Stokes equations in two spatial dimensions with vacuum as far field density. It is proved that the strong solutions exist globally if the density is bounded above. Furthermore, we show that if the solutions of the two-dimensional (2D) viscous compressible flows blow up, then the mass of the compressible fluid will concentrate on some points in finite time.

Lattice Boltzmann Simulation of Blood Flow Properties and Vessel Geometry in an Open and Closed Vessel: A Numerical Study

In the present article, Lattice Boltzmann method is utilized to simulate two-dimensional incompressible viscous flow in an open and closed microchannel (vessel). The main focus of the present research is to study physical parameters of blood flow in a vessel. To find the effect of oscillatory flow inside the vessel, we take account of the Reynolds number from 0.05 to 1.5 for numerical computation in the present manuscript in an open straight vessel. In addition, the accuracy of Poiseuille Law is investigated for blood flow in open vessel too. For this purpose, the effect of the vessel diameter and blood viscosity on the blood flow is studied numerically. As extra results, the effect of blood injection to a coronary bifurcation with two closed ends are studied. The blood pressure drop is high at the beginning of the vessel (pressure variation is high between the adjacent points along the vessel), but after the path along the vessel, the speed of dropping pressure decreases and the pressure difference between the adjoining points decreases along the vessel. Finally, the present results have been compared with the available experimental and numerical results that show good agreements.

Finite element analysis of magnetohydrodynamic flow over flat surface moving in parallel free stream with viscous dissipation and Joule heating

PurposeThe purpose of this paper is to investigate two-dimensional viscous incompressible magnetohydrodynamic boundary layer flow and heat transfer of an electrically conducting fluid over a continuous moving flat surface considering the viscous dissipation and Joule heating.Design/methodology/approachSuitable similarity variables are introduced to reduce the governing nonlinear boundary layer partial differential equations to ordinary differential equations. A numerical solution of the resulting two-point boundary value problem is carried out by using the finite element method with the help of Gauss elimination technique.FindingsA comparison of obtained results is made with the previous work under the limiting cases. Behavior of flow and thermal fields against various governing parameters like mass transfer parameter, moving flat surface parameter, magnetic parameter, Prandtl number and Eckert number are analyzed and demonstrated graphically. Moreover, shear stress and heat flux at the moving surface for various values of the physical parameters are presented numerically in tabular form and discussed in detail.Originality/valueThe work is relatively original, as very little work has been reported on magnetohydrodynamic flow and heat transfer over a continuous moving flat surface. Viscous dissipation and Joule heating are neglected in most of the previous studies. The numerical method applied to solve governing equations is finite element method which is new and efficient.

Unsteady flow and heat transfer over a permeable stretching/shrinking sheet with generalized slip velocity

PurposeThis study aims to investigate the unsteady two-dimensional viscous flow and heat transfer over an unsteady permeable stretching/shrinking sheet (surface) with generalized slip velocity condition.Design/methodology/approachSimilarity transformation is used to reduce the system of partial differential equations into a system of nonlinear ordinary differential equations. The resulting equations are then solved numerically using “bvp4c” function in MATLAB software.FindingsDual solutions are found for a certain range of the unsteady, suction and stretching/shrinking parameters. Stability analysis is performed, and it is revealed that the first (upper branch) solution is stable and physically realizable, whereas the second (lower branch) solution is unstable.Practical implicationsThe results obtained can be used to explain the characteristics and applications of the generalized slip in boundary layer flow. Such condition is applied for particulate fluids such as foams, emulsions, polymer solutions and suspensions. Furthermore, the phenomenon of stretching/shrinking sheet can be found on the manufacturing of polymer sheets, rising and shrinking balloon or moving and shrinking polymer film.Originality/valueThe present numerical results are original and new for the study of unsteady flow and heat transfer over a permeable stretching/shrinking sheet with generalized slip velocity.

Immersed boundary-simplified lattice Boltzmann method for incompressible viscous flows

An immersed boundary-simplified lattice Boltzmann method is developed in this paper for simulations of two-dimensional incompressible viscous flows with immersed objects. Assisted by the fractional step technique, the problem is resolved in a predictor-corrector scheme. The predictor step solves the flow field without considering immersed objects, and the corrector step imposes the effect of immersed boundaries on the velocity field. Different from the previous immersed boundary-lattice Boltzmann method which adopts the standard lattice Boltzmann method (LBM) as the flow solver in the predictor step, a recently developed simplified lattice Boltzmann method (SLBM) is applied in the present method to evaluate intermediate flow variables. Compared to the standard LBM, SLBM requires lower virtual memories, facilitates the implementation of physical boundary conditions, and shows better numerical stability. The boundary condition-enforced immersed boundary method, which accurately ensures no-slip boundary conditions, is implemented as the boundary solver in the corrector step. Four typical numerical examples are presented to demonstrate the stability, the flexibility, and the accuracy of the present method.

Some geometrical properties of free boundaries in the Hele-Shaw flows

In this paper, we are concerned with certain geometric properties of the moving boundary in the case of two-dimensional viscous fluid flows in Hele-Shaw cells under injection. We study the invariance in time of free boundary for such a bounded flow domain under the assumption of zero surface tension. By applying various results in the theory of univalent functions, we consider the invariance in time of starlikeness of a complex order, almost starlikeness of order α ∈ [0, 1), and almost spirallikeness of type γ∈(−π/2,π/2) and order α ∈ (0, cos γ). This work complements recent work on planar Hele-Shaw flow problems in the case of zero surface tension.

High Reynolds Viscous Flow Simulation Past the Elliptical Airfoil by Random Vortex Blob

In this paper, numerical simulation for a two-dimensional viscous and incompressible flow past the elliptical airfoil is presented by Random Vortex Blob (RVB). RVB is a numerical technique to solve the incompressible, two-dimensional and unsteady Navier-Stocks equations by converting them to rotational non-primitive formulations. In this method, the velocity vector at a certain point can be calculated without considering any grid around it, so the RVB method can be treated as a meshless method. Accordingly, the turbulent flow past a cylinder as well as an elliptical airfoil is investigated. In both cases, the obtained mean time velocities are compared with available numerical and experimental results where an acceptable agreement is observed. Having known the velocity field, by employing momentum balance, the drag and lift coefficients caused by flow past the elliptical airfoil with different diameter ratios and Re=105 are calculated and compared with experimental data where a good consistency is achieved.

Lift and drag in three-dimensional steady viscous and compressible flow

In a recent paper, Liu, Zhu, and Wu [“Lift and drag in two-dimensional steady viscous and compressible flow,” J. Fluid Mech. 784, 304–341 (2015)] present a force theory for a body in a two-dimensional, viscous, compressible, and steady flow. In this companion paper, we do the same for three-dimensional flows. Using the fundamental solution of the linearized Navier-Stokes equations, we improve the force formula for incompressible flows originally derived by Goldstein in 1931 and summarized by Milne-Thomson in 1968, both being far from complete, to its perfect final form, which is further proved to be universally true from subsonic to supersonic flows. We call this result the unified force theorem, which states that the forces are always determined by the vector circulation Γϕ of longitudinal velocity and the scalar inflow Qψ of transverse velocity. Since this theorem is not directly observable either experimentally or computationally, a testable version is also derived, which, however, holds only in the linear far field. We name this version the testable unified force formula. After that, a general principle to increase the lift-drag ratio is proposed.

Numerical Study of Elastic Red Blood Cell Motion and Deformation Using Improved Lattice Boltzmann-Immersed Boundary Method

In the present article, a new approach is proposed to enhance the hybrid Lattice Boltzmann-Immersed Boundary Method (IB-LBM) for simulation of two-dimensional incompressible viscous flows involving flexible immersed boundaries. The efficiency and accuracy of the proposed approach is justified through the simulation of the motion and deformation of two-dimensional red blood cell in a microchannel. In the case of conventional IB-LBM, the equation of solid immersed boundary has a certain mathematical formulation and it is not changed during the calculations. As a result, there are difficulties in the deformation of solid particle during its transmission, especially in the sharp points. For this reason, an innovation is applied in this research which solves this problem. Besides, after updating the Lagrangian points, the nodes can become too close to (or too far away from) each other, which may contribute and cause numerical instabilities. So, the boundary needs to be rediscretized. In order to represent the trace of the immersed boundary accurately in the absence of any sharp discontinuities, we regrid the updated Lagrangian points at each time step using spline interpolation. The presented computational results agree reasonably well with the available data in the literature.

Stability of the Kolmogorov flow and its modifications

Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for general two-dimensional viscous incompressible shear flows. It is shown that the eigenvalues of the linear eigenvalue problem are odd functions of the wave number, while the critical values of viscosity are even functions. If the velocity averaged over the long period is nonzero, then the loss of stability is oscillatory. If the averaged velocity is zero, then the loss of stability can be monotone or oscillatory. If the deviation of the velocity from its period-average value is an odd function of spatial variable about some x 0, then the expansion coefficients of the velocity perturbations are even functions about x 0 for even powers of the wave number and odd functions about for x 0 odd powers of the wave number, while the expansion coefficients of the pressure perturbations have an opposite property. In this case, the eigenvalues can be found precisely. As a result, the monotone loss of stability in the Kolmogorov flow can be substantiated by a method other than those available in the literature.

A complex-valued first integral of Navier-Stokes equations: Unsteady Couette flow in a corrugated channel system

For a two-dimensional incompressible viscous flow, a first integral of the governing equations of motion is constructed based on a reformulation of the unsteady Navier-Stokes equations in terms of complex variables and the subsequent introduction of a complex potential field; complementary solid and free surface boundary conditions are formulated. The methodology is used to solve the challenging problem of unsteady Couette flow between two sinusoidally varying corrugated rigid surfaces utilising two modelling approaches to highlight the versatility of the first integral. In the Stokes flow limit, the results obtained in the case of steady flow are found to be in excellent agreement with corresponding investigations in the open literature. Similarly, for unsteady flow, the results are in accord with related investigations, exploring material transfer between trapped eddies and the associated bulk flow, and vice versa. It is shown how the work relates to the classical complex variable method for solving the biharmonic problem and perspectives are provided as to how the first integral may be further utilised to investigate other fluid flow features.

Numerical simulation of motion and deformation of healthy and sick red blood cell through a constricted vessel using hybrid lattice Boltzmann-immersed boundary method

In the present article, hybrid lattice Boltzmann-immersed boundary method is utilized to simulate two-dimensional incompressible viscous flow involving flexible immersed red blood cell (RBC) in a microchannel. The main focus of the present research is to study motion and deformation of both healthy and sick RBCs in a vessel with different sizes of stenosis. The presented computational results consent reasonably well with the available data in the literature. Two different channels i.e. a simple and a constricted channel are investigated in the present manuscript. The results show that the RBC transfer and deform without any lift force and rotation induced when it is located on the symmetry axis of the microchannel. However, when the RBC is located off the symmetry axis, the pressure difference produced in the flow around the RBC would apply lift forces on them and expel them towards the center of the channel. The healthy RBC always shows more deformation related to the sick one along the channel. Another important result of the present research is that for the ratio of a sick RBC cannot pass the stenosis, and it reasons serious difficulties for body. The present results have been compared with the available experimental and numerical results which show good agreements.

The stability of a flexible cantilever in viscous channel flow

Most studies of the flow-induced flutter instability of a flexible cantilever have assumed inviscid flow because of the high flow speeds and the large scale of the structures encountered in the wide range of applications of this fluid–structure interaction (FSI) system. However, for instance, in the fields of energy harvesting and biomechanics, low flow speeds and small- and micro-scale systems can give relatively low Reynolds numbers so that fluid viscosity needs to be explicitly accounted for to provide reliable predictions of channel-immersed-cantilever stability. In this study, we employ a numerical model coupling the Navier–Stokes equations and a one-dimensional elastic beam model. We conduct a parametric investigation to determine the conditions leading to flutter instability of a slender flexible cantilever immersed in two-dimensional viscous channel flow for Reynolds numbers lower than 1000. The large set of numerical simulations carried out allows predictions of the influence of decreasing Reynolds numbers and of the cantilever confinement on the single-mode neutral stability of the FSI system and on the pre- and post-critical cantilever motion. This model's predictions are also compared to those of a FSI model containing a two-dimensional solid model in order to assess, primarily, the effect of the cantilever slenderness in the simulations. Results show that an increasing contribution of viscosity to the hydrodynamic forces significantly alters the instability boundaries. In general, a decrease in Reynolds number is predicted to produce a stabilisation of the FSI system, which is more pronounced for high fluid-to-solid mass ratios. For particular fluid-to-solid mass ratios, viscous effects can lower the critical velocity and lead to a change in the first unstable structural mode. However, at constant Reynolds number, the effects of viscosity on the system stability are diminished by the confinement of the cantilever, which strengthens the importance of flow inertia.

Stokes’ and Lamb's viscous drag laws

Since Galileo used his pulse to measure the time period of a swinging chandelier in the 17th century, pendulums have fascinated scientists. It was not until Stokes' (1851 Camb. Phil. Soc. 9 8–106) (whose interest was spurred by the pendulur time pieces of the mid 19th century) treatise on viscous flow that a theoretical framework for the drag on a sphere at low Reynolds number was laid down. Stokes' famous drag law has been used to determine two fundamental physical constants—the charge on an electron and Avogadro's constant—and has been used in theories which have won three Nobel prizes. Considering its illustrious history it is then not surprising that the flow past a sphere and its two-dimensional analog, the flow past a cylinder, form the starting point of teaching flow past a rigid body in undergraduate level fluid mechanics courses. Usually starting with the two-dimensional potential flow past a cylinder, students progress to the three-dimensional potential flow past a sphere. However, when the viscous flow past rigid bodies is taught, the three-dimensional example of a sphere is first introduced, and followed by (but not often), the two-dimensional viscous flow past a cylinder. The reason why viscous flow past a cylinder is generally not taught is because it is usually explained from an asymptotic analysis perspective. In fact, this added mathematical complexity is why the drag on a cylinder was only solved in 1911, 60 years after the drag on a sphere. In this note, we show that the viscous flow past a cylinder can be explained without the need to introduce any asymptotic analysis while still capturing all the physical insight of this classic fluid mechanics problem.

An adaptive-gridding lattice Boltzmann method with linked-list data structure for two-dimensional viscous flows

An adaptive-gridding lattice Boltzmann method with linked-list data structure for two-dimensional viscous flows

SPH simulation of turbulent flow past a high-frequency in-line oscillating cylinder near free-surface

This paper studies a two-dimensional incompressible viscous flow past a circular cylinder with in-line oscillation close to a free-surface. The sub-particle scale (SPS) turbulence model of a Lagrangian particle-based smoothed-particle hydrodynamics (SPH) method has been used to solve the full Navier–Stokes equations together with the continuity equation. The accuracy of numerical code has been verified using two cases consisting of an oscillating cylinder placed in the stationary fluid, and flow over a fixed cylinder close to a free-surface. Simulations are conducted for the Froude number of 0.3, the Reynolds numbers of 40 and 80, various gap ratios for fully-submerged and half-submerge cylinders. The dimensionless frequency and amplitude of oscillating have been chosen as 0.5, 0.8 and 10, 15, respectively. The selection of such a high oscillating frequency causes the flow regime to become turbulent. It is seen that the gap ratio defined as the ratio of cylinder distance from free-surface and its diameter, strongly affects the flow pattern and the magnitude of the drag and lift coefficients. The jet-like flow (the region above the cylinder and beneath the free-surface) creation is discussed in detail and showed that the strength of this jet-like flow is weakened when the gap ratio shrinks. It is seen that by decreasing the gap ratio, the lift and drag coefficients increase and decrease, respectively. It is found that the Reynolds number has an inverse effect on the drag and lift coefficients. Also, it is concluded that by increasing the amplitude of oscillation the drag coefficient increases.

INSTABILITY OF THE RAYLEIGH-BENARD CONVECTION FOR INCLINED LOWER WALL WITH TEMPERATURE VARIATION

This paper deals with an analysis of a two-dimensional viscous fluid flow between the two parallel plates inclined with respect to the horizontal plane, where the lower plate is heated and the upper one is cooled. The temperature difference between the plates is gradually increased during a certain time period after which it is temporarily constant. The temperature distribution on the lower plate is not constant in x-direction, there is a longitudinal sinusoidal temperature variation imposed on the mean temperature. We have investigated the wave number and amplitude influence of this variation on the subcritical stability and the onset of the Rayleigh-Bénard convective cells, by direct numerical simulation of 2D Navier-Stokes and energy equation.

Analysis of UAS-S4 Éhecatl aerodynamic performance improvement using several configurations of a morphing wing technology

ABSTRACTThe paper presents the results of the aerodynamic optimisation of an Unmanned Aerial System's wing using a morphing approach. The shape deformation of the wing is achieved by placing actuator lines at several positions along its span. For each flight condition, the optimal displacements are found by using a combination of the new Artificial Bee Colony algorithm and a classical gradient-based search routine. The wing aerodynamic characteristics are calculated with an efficient nonlinear lifting line method coupled with a two-dimensional viscous flow solver. The optimisations are performed at angles of attack below the maximum lift angle, with the aim of improving the Hydra Technologies UAS-S4 wing lift-to-drag ratio. Several configurations of the morphing wing are proposed, each with a different number of actuation lines, and the improvements obtained by these configurations are analysed and compared.