Two-dimensional Incompressible Fluid Research Articles

Starting Flow by Pitch Acceleration of Flat-Plate Foil

This paper presents the investigation of the starting flow in association with pitch acceleration using transient laminar-flow computation and an unsteady inviscid-flow theory. A thin flat-plate foil pitched at the leading edge is modeled in a two-dimensional incompressible fluid at chord Reynolds numbers of the order of . Several acceleration amplitudes () and acceleration durations () are maneuvered using a modified canonical kinematic equation at reduced pitch rates from 0.196 to 0.763. Comparisons between computational and theoretical data reveal that the formation of the starting vortex is determined by the acceleration amplitude scaled by the square of the convective time. The corresponding starting force coefficient that scales with the dimensionless acceleration amplitude is proportional to the acceleration angle , which decreases with increasing reduced pitch rate. The starting vortex, once formed at the trailing edge, is shed at the end of the acceleration period. The quasi-steady theory that includes only pitch motion as the strength of the dipolar sheet for apparent-mass flow can also yield an acceptable approximation to capture the apparent-mass force coefficient spike in acceleration.

Global well-posedness for the 2D chemotaxis-fluid system with logistic source

ABSTRACT In this paper, the two-dimensional incompressible chemotaxis fluid with logical source is studied as following: By taking advantage of a coupling structure of the equations and using a scale decomposition technique, the global existence and uniqueness of weak solutions to the above system for a large class of initial data is obtained.

Applying the immersed boundaries to the incompressible fluid flows using inverse source problem

A new method is suggested for imposing the non-regular no-slip/no-penetration conditions, on the two-dimensional incompressible viscous fluid flows. The method employs a new immersed boundary technique in the vorticity–stream function formulation of the Navier–Stokes equations. By semi-implicit time discretization, the vorticity–stream function equations are decomposed to the Helmholtz and Poisson equations at each time step. Then, these direct problems are converted to inverse ones and the solution domain is changed from a multiply connected domain to a simply connected one, where the inverse problems are solved. The method is flexible as it can be applied easily to the complex and moving geometries. Moreover, since the problem is solved on Cartesian grids, the fast Helmholtz and Poisson solvers are employed. In order to demonstrate the capability of the method, it is employed for analyses of a couple of stationary and moving boundary problems. First, the computational performance and the convergence rate of the solver are assessed using the method of manufactured solution. The results showed a first-order convergence rate with the computational efforts that scales by $$\left( {N\,\log \,N} \right)$$ . Then, as a classical non-regular fixed boundary problem, the flow around a stationary circular cylinder is studied. For the moving boundary problems, the method is applied to the problems of the oscillating cylinder in a quiescent fluid and two rotating cylinders placed in a channel. The results show good agreements with the available data in the literature.

External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid

We consider the motion of a rigid body due to the pressure of a surrounded two-dimensional irrotational perfect incompressible fluid, the whole system being confined in a bounded domain with an impermeable condition on a part of the external boundary. Thanks to an impulsive control strategy we prove that there exists an appropriate boundary condition on the remaining part of the external boundary (allowing some fluid going in and out the domain) such that the immersed rigid body is driven from some given initial position and velocity to some final position and velocity in a given positive time, without touching the external boundary. The controlled part of the external boundary is assumed to have a nonvoid interior and the final position is assumed to be in the same connected component of the set of possible positions as the initial position.

Affine Motion of 2d Incompressible Fluids Surrounded by Vacuum and Flows in $$\mathrm{SL}(2,\mathbb {R})$$

The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in $$\mathrm{SL}(2,\mathbb {R})$$. In the case of perfect fluids, the motion is given by geodesic flow in $$\mathrm{SL}(2,\mathbb {R})$$ with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in $$\mathrm{SL}(2,\mathbb {R})$$. A complete classification of the dynamics is given including rigid motions, rotating eddies with stable and unstable manifolds, and solutions with vanishing pressure. For perfect fluids, the displacement generically becomes unbounded, as $$t\rightarrow \pm \infty $$. For MHD, solutions are bounded and generically quasi-periodic and recurrent.

Control at a distance of the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid

We consider the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid with Navier slip-with-friction conditions at the solid boundary. The fluid-solid system occupies the whole plane. We prove the small-time global exact controllability of the position and velocity of the solid when the control takes the form of a distributed force supported in a compact subset (with nonvoid interior) of the fluid domain, away from the body. The strategy relies on the introduction of a small parameter: we consider fast and strong amplitude controls for which the “Navier-Stokes+rigid body” system behaves like a perturbation of the “Euler+rigid body” system. By the means of a multi-scale asymptotic expansion we construct a controlled solution to the “Navier-Stokes+rigid body” system thanks to some controlled solutions to “Euler+rigid body”-type systems and by using that the influence of the boundary layer on the solid motion turns out to be sufficiently small.

ENTROPY GENERATION AND IRREVERSIBILITY ANALYSIS ON FREE CONVECTIVE UNSTEADY MHD CASSON FLUID FLOW OVER A STRETCHING SHEET WITH SORET/DUFOUR IN POROUS MEDIA

The two-dimensional unsteady incompressible Casson fluid flow over a nonlinearly stretching sheet is studied here under the influence of an external magnetic field. The nonlinear thermal radiation heat transfer in the porous media is considered. The novelty of the present study is in the entropy generation and irreversibility analysis together with the Soret and Dufour effects for Casson fluid. The fluid velocity, temperature, concentration, friction factor, entropy generation rate, and Nusselt, Sherwood, and Bejan numbers are studied through solving the governing nonlinear equations in a transformed/converted system by a shooting method. It is found that the energy transport may be maximized by varying the magnetic field and unsteadiness factor. Moreover, an increase in the porosity parameter raises the temperature and concentration profiles, while it reduces the fluid's velocity. The Bejan number is augmented due to an increase in radiation, while it is reduced due to the increases in magnetic field and Casson fluid parameter. Finally, the irreversibility ratio is augmented as the magnetic field increases.

Analysis of Transport and Mixing Phenomenon to Invariant Manifolds Using LCS and KAM Theory Approach in Unsteady Dynamical Systems

The Lagrangian approach for the two-dimensional incompressible fluid flows has been studied with the help of dynamical systems techniques: Kolmogorov-Arnold-Moser (KAM) theory, stable, unstable manifold structures, and Lagrangian coherent structures (LCSs). For the time-dependent perturbation problems, we analyze in detail the development of transport barriers that play an important role in the transport process. Firstly, the analytical study of KAM theory is adopted to explain the transport and mixing phenomenon in measured and simulated airfoil flow. Then, the flow topology of unsteady flow behind an airfoil is investigated for the low Reynolds number problem. Simulations are carried out based on a particular Finite-Time Lyapunov Exponent (FTLE) technique for the detection of invariant manifolds of the hyperbolic trajectories. Besides, the Characteristic Base Split (CBS) scheme combined with a dual time stepping technique is utilized to simulate such transient flow problems. Thus, in the course of the current research, the role of the velocity phase plot during vortex formation is explored that is highly periodic and resulted in the formation of a stable pattern of manifolds and invariant tori. Hence, the proposed study encouraged the new picture of the vortex shedding and flow separation process. As a conclusion, our results give a better understanding of invariant tori control transport phenomena that will lead to a new heuristic for unsteady flows.

Oblate to prolate transition of a vesicle in shear flow.

Vesicles are micrometric soft particles whose membrane is a two-dimensional incompressible fluid governed by bending resistance leading to a zoology of shapes. The dynamics of deflated vesicles in shear flow with a bottom wall, a first minimal configuration to consider confined vesicles, is investigated using numerical simulations. Coexistence under flow of oblate (metastable) and prolate (stable) shapes is studied in details. In particular, we discuss the boundaries of the region of coexistence in the (v, Ca -plane where v is the reduced volume of the vesicle and Ca the Capillary number. We characterize the transition from oblate to prolate and analyse the divergence of the transition time near the critical capillary number. We then analyse the lift dynamics of an oblate vesicle in the weak flow regime.

A Galerkin boundary cover method for viscous fluid flows

Abstract A Galerkin boundary cover method (GBCM) is developed for modeling of two-dimensional incompressible viscous fluid flows. In this approach, a combination of the finite cover system employed in the numerical manifold method with the Galerkin approximation of the boundary integral equations is presented. It is applicable to both interior and exterior domains due to its naturally variational formulations. In contrast to the domain-type approach, only the boundary data is required in the newly explained method, makes it suitable for the exterior problems. To increase the solution accuracy without refining the local mesh, different cover functions can be implemented on different covers; thereby, the p-adaptive computations can be carried out conveniently. Further, the boundary conditions are imposed accurately and the system matrices are both symmetric and positive definite. The given numerical examples demonstrate the high accuracy and convergence rates of the proposed methodology by using a few covers.

Dynamics of rigid bodies in a two dimensional incompressible perfect fluid

We consider the motion of several rigid bodies immersed in a two-dimensional incompressible perfect fluid, the whole system being bounded by an external impermeable fixed boundary. The fluid motion is described by the incompressible Euler equations and the motion of the rigid bodies is given by Newton's laws with forces due to the fluid pressure. We prove that, for smooth solutions, Newton's equations can be recast as a second-order ODE for the degrees of freedom of the rigid bodies with coefficients depending on the fluid vorticity and on the circulations around the bodies, but not anymore on the fluid pressure. This reformulation highlights geodesic aspects linked to the added mass effect, gyroscopic features generalizing the Kutta-Joukowski-type lift force, including body-body interactions through the potential flows induced by the bodies' motions, body-body interactions through the irrotational flows induced by the bodies' circulations, and interactions between the bodies and the fluid vorticity.

КВАЗИ-ЛАГРАНЖЕВОЕ ИНТЕГРИРОВАНИЕ ДВУМЕРНЫХ УРАВНЕНИЙ ПРАНДТЛЯ И БУССИНЕСКА В НЕВЯЗКОМ ПРЕДЕЛЕ

For the hydrodynamic type systems locally describing incompressible twodimensional fluid flows without dissipation we suggest quasi-Lagrangian approach for their integration. This method is based on application of incomplete Legendre transformation when the independent variables become a Lagrangian invariant (i.e. the constant quantity along the fluid particle trajectory), instead of one of the spatial coordinate, and the rest ones which are another spatial coordinate and time. Thus, this method is based on the inverse transform of one of the spatial coordinate and by this reason differs from the the complete Legendre transformation. The classical example of the complete Legendre transformation is the Hodograph transformation applying to solve the equations for one-dimensional isoentropycal gas flows. In this paper it is shown that equation for the Lagrangian invariant after applying the incomplete Legendre transformation and introducing the stream function transforms into the linear eqation can be resolved by means of the generating function introduction. This method is turned to be effective for solving the inviscid twodimensional Prandtl equation (this equation describes the boundary layer behavior) that allows one to integrate this equation completely. In the case of the constant pressure along the boundary the parallel velocity component represents the Lagrangian invariant. The obtained solution is written through the initial data and satisfies the non-penetrate boundary condition. Analysis of this solution shows the formation of the singularity for the velocity gradient on the wall. This singularity appears as the result of breaking. At the breaking point the velocity gradient tends to infinity according to the power ~(t0–t)-1 where t0 is the singular time. This solution describes the appearance of the folding type singularity. It is shown also that the Prandtl equation admits complete integration for arbitrary dependence of pressure on the longitudinal coordinate. The simplest solution is written for the case of the constant pressure gradient. For the Boussinesq system the equation for density can be resolved by this method that reduces to one equation for the generating function. This work was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».

Dynamics of a rigid body in a two-dimensional incompressible perfect fluid and the zero-radius limit

In this survey we report some recent results on the dynamics of a rigid body immersed in a two-dimensional incompressible perfect fluid, with an emphasis on the zero-radius limit.

A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion.

A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation include how the pressure boundary condition is treated, the introduction of a stream function into the Euler–Poincaré variations, the derivation of free surface variations and how the equations for the vessel path in the Euclidean group, coupled to the fluid motion, are generated automatically.

Prediction of wake structure and aerodynamic characteristics of flow around square cylinders at different arrangements

Two-dimensional incompressible fluid flows around square cylinders at different arrangements have been numerically analyzed in the present work. The calculations are carried out for several values of Reynolds number (Re) ranging from 20 to 180. The results are presented in the form of vorticity contours and temporal histories of drag and lift coefficients. Besides, the physical parameters, namely, the average drag and lift coefficients and Strouhal number, are evaluated as a function of Re. Two different states of flow are predicted in the current investigation by systematically varying Re for steady and unsteady regimes. Vortex shedding is studied at different arrangements of the square cylinders allowing the investigation of three possible configurations. Special attention is paid to compute the drag and lift forces acting on the different obstacles, which allowed determining the optimal configuration in terms of both drags and lifts. The unsteady periodic wake is characterized by the Strouhal number, which varies with the Reynolds number and the obstacle geometry. The values of vortex shedding frequencies are consequently calculated in this study.

TRAJECTORY SIMULATION OF SELF-PROPELLED ELASTIC RODS IN FLUID

The research for flexible structures coupling with fluid simulates and promotes the development of soft robotics. A fast and accurate numerical method is significant for a real-time simulation of robots. This research provides theoretical and critical information for experiments to reduce the possibility of failure, by anticipating the path of the underwater soft robots and the possible requiring parameters for materials. This paper researches for bio-swimming elastic rods coupling with two-dimensional incompressible Newton fluid. First, we discretize elastic rods to extensive springs and rotational springs and stablish the kinetic equations based on energy reflecting the influence of internal force on swimming rods, and solve the governing equation by leap-frog algorithm. Second, semi-Lagrangian method is used to establish a fluid solver. Finally, the simplified coupling algorithm based on immersed-boundary method is raised, calling immersed-boundary method for momentum equations. These equations update the velocity of grid near the coupling interface directly to replace the function source force play in immersed-boundary method. Combining the fluid solver, rods solver and the coupling algorithm, a integral program solving the problem of the dynamics for immersed rods numerically. Simulating the rods swimming with a referenced pose of sine curvature. Comparing the simulating result to existed experiments of filament swimming and to the swimming trajectory of soft-body fish, we find the numerical result conform to experiment result, leading to the conclusion that this algorithm and dynamic model can simulate the trajectory of discrete elastic rods swimming underwater smoothly under the influence of both internal elastic force and coupling soft body-fluid interaction. Using this program, we test several critical parameters relating to swimming performance of rods, including iteration numbers, frequency and initial phases, finding that changing the initial phase of rods will alter onward direction of elastic rods. These results prove the feasibility and possibility of this algorithm and program being used for guiding development of real soft swimmers.

An efficient segregated algorithm for two-dimensional incompressible fluid flow and heat transfer problems with unstructured grids

This paper introduces a newly developed segregated algorithm (named SIMPLERR algorithm) to overcome two major drawbacks in the SIMPLE-family algorithm. The full coupling between velocity and pressure is ensured in an easy segregated manner. Comprehensive comparisons among the SIMPLER algorithm, the IDEAL algorithm and the SIMPLERR algorithm are carried out via four benchmark incompressible fluid flow and heat transfer problems analyzed with unstructured grids. The results demonstrate that the IDEAL algorithm and the SIMPLERR algorithm have substantial advantages over the SIMPLER algorithm in both convergence rate and robustness, moreover, compared with the IDEAL algorithm, the SIMPLERR algorithm can offer a higher computational efficiency for its simplicity, especially for high-Re/Ra and fine-mesh cases; the computational results given by the present algorithm agree very well with benchmark data.

Solution of incompressible fluid flow problems with heat transfer by means of an efficient RBF-FD meshless approach

The localized radial basis function collocation meshless method (LRBFCMM), also known as radial basis function generated finite differences (RBF-FD) meshless method, is employed to solve time-dependent, two-dimensional (2D) incompressible fluid flow problems with heat transfer using multiquadric RBFs. A projection approach is employed to decouple the continuity and momentum equations for which a fully implicit scheme is adopted for the time integration. The node distributions are characterized by non-Cartesian node arrangements and large sizes, i.e., in the order of 105 nodes, while nodal refinement is employed where large gradients are expected, i.e., near the walls. Particular attention is given to the accurate and efficient solution of unsteady flows at high Reynolds or Rayleigh numbers, in order to assess the capability of this specific meshless approach to deal with practical problems. Three benchmark test cases are considered: a lid-driven cavity, a differentially heated cavity and a flow past a circular cylinder between parallel walls. The obtained numerical results compare very favorably with literature references for each of the considered cases. It is concluded that the presented numerical approach can be employed for the efficient simulation of fluid-flow problems of engineering relevance over complex-shaped domains.

Odd surface waves in two-dimensional incompressible fluids

We consider free surface dynamics of a two-dimensional incompressible fluid with odd viscosity. The odd viscosity is a peculiar part of the viscosity tensor which does not result in dissipation and is allowed when parity symmetry is broken. For the case of incompressible fluids, the odd viscosity manifests itself through the free surface (no stress) boundary conditions. We first find the free surface wave solutions of hydrodynamics in the linear approximation and study the dispersion of such waves. As expected, the surface waves are chiral and even exist in the absence of gravity and vanishing shear viscosity. In this limit, we derive effective nonlinear Hamiltonian equations for the surface dynamics, generalizing the linear solutions to the weakly nonlinear case. Within the small surface angle approximation, the equation of motion leads to a new class of non-linear chiral dynamics governed by what we dub the chiral Burgers equation. The chiral Burgers equation is identical to the complex Burgers equation with imaginary viscosity and an additional analyticity requirement that enforces chirality. We present several exact solutions of the chiral Burgers equation. For generic multiple pole initial conditions, the system evolves to the formation of singularities in a finite time similar to the case of an ideal fluid without odd viscosity. We also obtain a periodic solution to the chiral Burgers corresponding to the non-linear generalization of small amplitude linear waves.

Strong solutions to the Cauchy problem of two-dimensional incompressible fluid models of Korteweg type

This paper studies the local existence of strong solutions to the Cauchy problem of the incompressible fluid models of Korteweg type with vacuum as far field density. The corresponding 3D problem has been solved by Tan and Wang (2010) [21]. Notice that the technique used by Tan and Wang fails treating the 2D case, because the Lp-norm (p>2) of the velocity u cannot be controlled in terms only of ρu and ∇u here. In the present paper, we will use the framework of weighted approximation estimates introduced by Liang (2015) [14] for Navier–Stokes equations to obtain the local existence of strong solutions provided the initial density does not decay very slowly at infinity, with the compact support case included.