Time-dependent Incompressible Flow Research Articles

Sound generation by flow over a two-dimensional cavity

Sound generated by flow over a cavity at a Mach number of 0.1 and a Reynolds number based on cavity length of 5000 is calculated. The computation utilizes a two part technique where the time-dependent incompressible flow is first obtained and then a second calculation is performed for the compressible aspects of the flow. This second calculation utilizes a grid and numerical scheme designed for resolution of acoustic waves. The cavity flowfield is observed to oscillate quite regularly at the Strouhal number of 0.58 which produces an acoustic source of the same frequency. Time histories, spectra, and directivity of the sound radiation are computed.

AN EXPLICIT FEM FOR 3-D VISCOUS INCOMPRESSIBLE FLOWS, WITH A EBE/PCG ITERATIVE ALGORITHM

SUMMARY A numerical procedure for the solution of the 3-D Navier-Stokes equations is developed and implemented for analyzing time-dependent incompressible viscous flows past arbitrary shapes. The equations are discretized using a finite element Galerkjn formulation with a streamwise upwinding option. The discretization in time is performed with fractional steps on the momentum equations to obtain the fractional step velocity field explicitly. The pressure field at each time level is obtained from an auxiliary potential function with the solution of a Poisson's equation, where an element-by-element (EBE) iteration procedure with preconditioned conjugate gradient (PCG) is employed. The method is used to study laminar flow past a circular cylinder with a swept bump. The flow field is solved for two different Reynolds numbers and the drag coefficient histories are obtained for the cylinder. The steady-state values of the drag coefficients show the 3-D relief effects on the cylinder. The computations are performed on an IBM 3090VF. The vectorization of the EBE algorithm allows a drastic reduction in terms of CPU time for each time step.

Numerical Study of Outflow Boundary Conditions for Time-Dependent Incompressible Flows.

The issue on outflow boundary conditions (OBCs) is very important in the numerical simulation of incompressible viscous flows. We consider a Sommerfeld radiation condition (SRC) from points of view : one is an analogy between the SRC and Taylor's hypothesis for large-scale vortex structures, and the other is consistency between the SRC and global mass conservation for primitive variables formulation. A new SRC is proposed, in which the convection velocity of the SRC is determined by an arithmetic mean of the maximum normal velocity and minimum normal velocity at the outflow boundary. Our new condition is tested in a numerical simulation using the finite difference method for the problem of Lamb dipole convection in a uniform flow, and its result is compared with the results obtained by adopting three SRCs. Our new condition automatically satisfies the global mass conservation and it gives a more accurate prediction than the old ones.

An acoustic/viscous splitting technique for computational aeroacoustics

A new technique for the numerical analysis of aerodynamic noise generation is developed. The approach involves first solving for the time-dependent incompressible flow for the given geometry. A “hydrodynamic” density correction to the constant incompressible density is then calculated from knowledge of the incompressible pressure fluctuations. The compressible flow solution is finally obtained by considering perturbations about the “corrected” incompressible flow. This fully nonlinear technique, which is tailored to extract the relevant acoustic fluctuations, appears to be an efficient approach to the numerical analysis of aerodynamic noise generation, particularly in viscous flows. Applications of this technique to some classical acoustic problems of interest, including some with moderately high subsonic Mach numbers, are presented to validate the approach. The technique is then applied to a fully viscous problem where sound is generated by the flow dynamics.

Passive scalars and three-dimensional Liouvillian maps

Global aspects of the motion of passive scalars in time-dependent incompressible fluid flows are well described by volume-preserving (Liouvillian) three-dimensional maps. In this paper the possible invariant structures in Liouvillian maps and the two most interesting nearly-integrible cases are investigated. In addition, the fundamental role of invariant lines in organizing the dynamics of this type of system is exposed. Bifurcations involving the destruction of some invariant lines and tubes and the creation of new ones are described in detail.

An inviscid flow with compact support in space-time

There exists a nonzero weak solution to the Euler equations of time-dependent incompressible fluid flow in the plane such that this solution has compact support in space-time.

Preconditioned upwind methods to solve incompressible Navier-Stokes equations

Introduction T O solve the three-dimensional incompressible NavierStokes equations, the methods of solution can generally be divided into three concepts. These concepts include methods employing pressure iteration to satisfy the continuity equation, methods using the vorticity formulation, and methods based on preconditioned incompressible equations that can be reduced to the equations using artificial compressibility. By adding an artificial time derivative of pressure (artificial compressibility) to the continuity equation, the elliptic-parabolic nature of the unsteady incompressible Navier-Stokes equations is changed to a hyperbolic-parabolic one. Therefore, well-developed compressible flow algorithms can be applied readily. To accelerate convergence to a steady state, one arbitrary parameter must be determined. This concept was first proposed by Chorin. To generalize Chorin's approach, Turkel suggested a preconditioned method for solving the incompressible flow equations. This method introduces artificial time derivatives not only in the continuity equation but also in the momentum equation. Now, the resultant time-dependent incompressible flow equations contain two arbitrary parameters to be determined for faster convergence to a steady state. The idea is to choose these two parameters such that the disparity in wave speeds will be minimized during the unphysical transient state. In the present study, a preconditioned upwind method is developed. Constant and adaptive preconditioning parameters are then exercised to assess the numerical convergence for a fully developed flow in a straight square duct. Several related issues have been addressed in Ref. 3.

Computational study of unsteady viscous flow around a transversely and longitudinally oscillating circular cylinder in a uniform flow at high reynolls numbers

A finite difference simulation method for the time dependent viscous incompressible flow around a transversely and longitudinally oscillating circular cylinder at Reynolds numbers of Re=4×103 and 4×104 is presented. The Navier-Stokes equations in finite difference form are solved on a moving grid system, based on a time dependent coordinate transformation. Solution of the vortex street development behind the cylinder is obtained when the cylinder remains stationary and also when it is oscillating. Time eholution of the flow configuration is studied by means of stream lines, pressure contours and vorticity contours. The computer results predict the lock-in phenomenon which occurs when the oscillation frequency is close to the vortex shedding frequency in the transverse mode or around double the vortex shedding frequency in the longitudinal mode. The time dependent lift and drag coefficients are obtained by the integration of the pressure and shear forces around the body. The drag, lift and the displacement relations are also discussed.

Numerical simulation of unsteady viscous free surface flow

Abstract A finite element method is presented for the numerical simulation of time-dependent incompressible viscous free surface flows. The time-dependent primitive equations are solved sequentially using explicit time marching procedure. The method is based on a velocity correction approach to the time integration of the Navier-Stokes equations in which only the incompressibility condition is treated implicitly. A special arbitrary mixed Lagrangian-Eulerian description has been used to avoid the typical problems encountered in a purely Lagrangian description. The method appears applicable for small computers; problems requiring several thousand nodes can be solved on personal computers. Numerical experiments have been performed that show that this approach is reasonably efficient and robust for a range of complicated highly nonlinear problems.

Time-dependent analysis of incompressible viscous flow by the finite-element method using the variable transformation.

A storage requirement of the finite-element method (FEM), which is a result of using un-structured meshes, can be considerably reduced by using iterative solvers. However, the acceleration of the convergence is still an important and urgent problem in spite of currently available vector processors. In this paper, the convergence acceleration for solving the time-dependent incompressible viscous flow by using the iterative solver is given. The variable transformation is applied to the finite-element discretized equations in order to obtain the diagonal-dominant equation system. The usual framework of the primitive variables (velocity and pressure) is transformed into that of newly introduced variables : rotational velocity and velocity potential. The effectiveness of the present technique is demonstrated through the lid-driven cavity flow and the flow past a cylinder.

Numerical 3D-simulation of pulsatile wall shear stress in an arterial T-bifurcation model

The structure of pulsatile blood flow and wall shear stress in a 90° T-bifurcation model is analysed numerically. The nonlinear Navier-Stokes equations for time-dependent incompressible Newtonian fluid flow are approximated using a newly developed pressure correction, finite element method. The wall shear stress is calculated from the finite element velocity field. The investigation shows viscous flow phenomena such as flow separation and stagnation and the distribution of high and low wall shear stress during the pulse cycle. Furthermore, the effect of a sharp corner the bifurcation edge on the wall shear stress is analysed. Detailed local flow investigation is required to examine fluid dynamic contribution to the development of arterial diseases such as atherosclerosis and thrombosis.

Development of the mask method for incompressible unsteady flows

A new method is developed to solve 2-dimensional time dependent incompressible viscous flows in an arbitrary geometry. The Navier-Stokes equations are solved in primitive variables using a pseudospectral formulation. The geometry is introduced by directly forcing the velocity field near the boundaries. The method extends a similar computational technique introduced by Basdevant and Sadourny [1]. Some examples are shown to evaluate its efficiency and accuracy.

A numerical method for the time-dependent Stefan problem in Czochralski crystal growth

Based on a MAC (marker and cell) scheme for time-dependent incompressible viscous flow a finite difference algorithm has been developed which includes the dynamics of the solid-liquid interface and a flat melt free surface in the Czochralski crystal growth process.

An Euler Code Calculation of Blade-Vortex Interaction Noise

An Euler code has been developed for calculation of noise radiation due to the interaction of a distributed vortex with a Joukowski airfoil. The time-dependent incompressible flow field is first determined and then integrated to yield the resulting sound production through use of the elegant low-frequency Green’s function approach. This code has several interesting numerical features involved in the vortex motion and in continuous satisfaction of the Kutta condition. In addition, it removes the limitations on Reynolds number and is much more efficient than an earlier Navier-Stokes code. Results indicate that the noise production is due to the deceleration and subsequent acceleration of the vortex as it approaches and passes the airfoil. Predicted acoustic levels and frequencies agree with measured data although a precise comparison would require the strength, size, and position of the incoming vortex to be known.

Application of a fractional-step method to incompressible Navier-Stokes equations

A numerical method for computing three-dimensional, time-dependent incompressible flows is presented. The method is based on a fractional-step, or time-splitting, scheme in conjunction with the approximate-factorization technique. It is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions. Appropriate boundary conditions for the intermediate velocity field are derived and tested. Numerical solutions for flows inside a driven cavity and over a backward-facing step are presented and compared with experimental data and other numerical results.

A study on the behavior of separated flow around elliptic airfoils and the unsteady aerodynamic forces under dynamic stall. (1st report, Numerical analysis. Part 1)

A numerical scheme is developed for use on two-dimensional time-dependent incompressible turbulent flows of thigh Reynolds number, and applied to the flow over an impulsively started 15% thickness elliptic airfoil at an angle of attack of 30° at a Reynolds number of 5×105. The scheme divides the flow field around an airfoil into two regions, the boundary-layer region and its outer flow region. The Reynolds equations with vorticity and stream function as variables are solved in the outer flow domain. The equation of integrated form for the vorticity is solved using a combination of finite differences and grid-free point vortices, while that for the stream function is solved by a similar technique to "Cloud In Cell" method. Results for the problem which involved massive flow separation were compared with experimental data and flow fields and aerodynamic forces acting on the airfoil compared well with mesurements.

Finite element solution of the unsteady Navier-Stokes equations by a fractional step method

A finite element method is presented for the numerical simulation of time-dependent incompressible viscous flows. The method is based on a fractional step approach to the time integration of the Navier-Stokes equations in which only the incompressibility condition is treated implicitly. This leads to a computational scheme of extremely simple algorithmic structure that is particularly attractive for cost-effective solutions of large-scale problems. Numerical results indicate the versatility and effectiveness of the proposed method.

Finite-element method for time-dependent incompressible free surface flow

We present a finite-element method for time-dependent incompressible free surface fluid flow problems described by the Navier-Stokes equations. The elements chosen have dimensions in both space and time, and the resulting system of equations is block-tridiagonal and lends itself to solution by standard techniques. In the present article we restrict our attention to two-dimensional problems although three-dimensional problems may be solved by a straightforward generalization. The method is essentially an implicit time stepping technique and therefore stable even for relatively large time steps. With this choice of elements, the method is completely adaptive to the changing nature of the solution. An iterative procedure is used to find the position of the free surface; this procedure is found to be rapidly convergent determining accurately the shape of the free surface within a few iterations. Numerical results are given for the problem of entrainment of fluid by a vertically moving plate, which has applications to the chemical engineering problems of the free coating of metals. We also consider the problem of circulation flow in a rectangular channel.

Dynamic Analysis of Incompressible Viscous Liquids

A Lagrangian finite-element analysis technique is presented for two-dimensional plane strain problems involving time-dependent incompressible flow of viscous liquids. The incompressibility feature is obtained with an iterative procedure which adjusts the nodal velocities until the anticipated volumetric strains are within specified limits. By eliminating the compressibility, it is possible to determine the integration time increment from the rate of distortion, rather than the sound speed transit time, as is required with various numerical methods involving wave propagation in compressible materials. This paper includes the formulation of the numerical method and illustrative examples.

Time-dependent laminar incompressible flow through a spherical cavity

A finite-difference numerical method for the solution of the unsteady flow of a viscous incompressible fluid through axisymmetric circular ducts of variable axial geometry is developed and applied to the flow in a spherical-cavity geometry approximating the human aortic valve. The presence and motion of the valve leaflets are considered only as long as they can be assumed to present negligible impedance to the flow. The numerical solution is based on the vorticity/streamfunction approach, and is carried out for the systolic acceleration phase of the heart beat. A hybrid-mesh design consisting of a fine cell structure in the region close to the solid walls and a coarser grid in the core region is used. An experimental flow-visualization study in an acrylic model of the spherical cavity shows good agreement with the numerical simulation. An early separation of flow occurs at the entrance to the cavity, and an annular eddy grows in the wake until it occupies most of the cavity. The use of the hybrid mesh also makes possible the simulation of fine secondary-flow features in the cavity under peak-flow conditions.