Showing papers on "Incompressible flow published in 1993"

Nearly incompressible fluids. II - Magnetohydrodynamics, turbulence, and waves

Abstract: The theory of nearly incompressible (NI) fluid dynamics developed previously for hydrodynamics is extended to magnetohydrodynamics (MHD). On the basis of a singular expansion technique, modified systems of fluid equations are derived for which the effects of compressibility are admitted only weakly in terms of the different possible incompressible solutions (thus ‘‘nearly incompressible MHD’’). NI MHD represents the interface between the compressible and incompressible magnetofluid descriptions in the subsonic regime. The theory developed here does not hold in the presence of very large thermal, gravitational, or field gradients. It is found that there exist three distinct NI descriptions corresponding to each of the three possible plasma beta (β ≡ the ratio of thermal to magnetic pressure) regimes (β≪1, β∼1, β≫1). In the β≫1 regime, the compressible MHD description converges in the low Mach number limit to the equations of classical incompressible three‐dimensional (3‐D) MHD. However, for the remaining plasma beta regimes, the imposition of a large dc magnetic field forces the equations of fully compressible 3‐D MHD to converge to the equations of 2‐D incompressible MHD in the low Mach number limit. The ‘‘collapse in dimensionality’’ corresponding to the different plasma beta regimes clarifies the distinction between the 3‐D and 2‐D incompressible MHD descriptions (and also that of 21/2‐D incompressible MHD). The collapse in dimensionality that occurs as a result of a decreased plasma beta can carry over to the weakly compressible corrections. For a β∼1 plasma, Alfven waves propagate parallel to the applied magnetic field (reminiscent of reduced MHD), while for a β≪1 magnetofluid, quasi‐1‐D long‐wavelength acoustic modes propagate parallel to the applied magnetic field. The detailed theory of weakly compressible corrections to the various incompressible MHD descriptions is presented and the implications for the solar wind emphasized.

A collocated finite volume method for predicting flows at all speeds

Abstract: An existing two-dimensional method for the prediction of steady-state incompressible flows in complex geometry is extended to treat also compressible flows at all speeds. The primary variables are the Cartesian velocity components, pressure and temperature. Density is linked to pressure via an equation of state. The influence of pressure on density in the case of compressible flows is implicitly incorporated into the extended SIMPLE algorithm, which in the limit of incompressible flow reduces to its well-known form. Special attention is paid to the numerical treatment of boundary conditions. The method is verified on a number of test cases (inviscid and viscous flows), and both the results and convergence properties compare favourably with other numerical results available in the literature.

Global linear stability analysis of weakly non-parallel shear flows

Abstract: The global linear stability of incompressible, two-dimensional shear flows is investigated under the assumptions that far-field pressure feedback between distant points in the flow field is negligible and that the basic flow is only weakly non-parallel, i.e. that its streamwise development is slow on the scale of a typical instability wavelength. This implies the general study of the temporal evolution of global modes, which are time-harmonic solutions of the linear disturbance equations, subject to homogeneous boundary conditions in all space directions. Flow domains of both doubly infinite and semi-infinite streamwise extent are considered and complete solutions are obtained within the framework of asymptotically matched WKBJ approximations. In both cases the global eigenfrequency is given, to leading order in the WKBJ parameter, by the absolute frequency ω0(Xt) at the dominant turning point Xt of the WKBJ approximation, while its quantization is provided by the connection of solutions across Xt. Within the context of the present analysis, global modes can therefore only become time-amplified or self-excited if the basic flow contains a region of absolute instability.

Mathematical Methods in Fluid Dynamics

Abstract: Quantities and equations describing the flow irrotational incompressible flow models of compressible and rotational inviscid flow mathematical theory and numerical solution of inviscid flow problems with trailing conditions stationary inviscid transonic flow the Euler quations and nonlinear hyperbolic systems mathematical theory and numerical solution of viscous flow.

Is the steady viscous incompressible two‐dimensional flow over a backward‐facing step at Re = 800 stable?

Abstract: A detailed case study is made of one particular solution of the 2D incompressible Navier-Stokes equations. Careful mesh refinement studies were made using four different methods (and computer codes): (1) a high-order finite-element method solving the unsteady equations by time-marching; (2) a high-order finite-element method solving both the steady equations and the associated linear- stability problem; (3) a second-order finite difference method solving the unsteady equations in streamfunction form by time-marching; and (4) a spectral-element method solving the unsteady equations by time-marching. The unanimous conclusion is that the correct solution for flow over the backward-facing step at Re=800 is steady - and it is stable, to both small and large perturbations.

Performance of compressible flow codes at low Mach numbers

Abstract: The accuracy and the performance of three two-dimensional compressible flow codes at freestream Mach numbers as low as 0.001 are examined. Two of the codes employ a finite volume discretization scheme along with a multistage time-stepping algorithm to solve the Euler equations. The two codes differ in their respective use of cell-centered and node-centered differencing schemes. The third code uses an implicit finite difference procedure to solve the unsteady Navier-Stokes equations. Computational test cases are the inviscid steady flow over a circular cylinder and the impulsively started viscous flow over a cylinder

Skin Friction and Velocity Profile Family for Compressible Turbulent Boundary Layers

Abstract: The paper presents a general approach to constructing mean velocity profiles for compressible turbulent boundary layers with isothermal or adiabatic walls. The theory is based on a density-weighted transformation that allows the extension of the incompressible similarity laws of the wall to the compressible regions. The velocity profile family is compared to a range of experimental data, and excellent agreement is obtained. A self-consistent skin friction law, which satisfies the proposed velocity profile family, is derived and compared with the well-known Van Driest II theory for boundary layers in zero pressure gradient. The results are found to be at least as good as those obtained by using the Van Driest II transformation.

Application of scattering chaos to particle transport in a hydrodynamical flow.

Abstract: The dynamics of a passive particle in a hydrodynamical flow behind a cylinder is investigated. The velocity field has been determined both by a numerical simulation of the Navier–Stokes flow and by an analytically defined model flow. To analyze the Lagrangian dynamics, we apply methods coming from chaotic scattering: periodic orbits, time delay function, decay statistics. The asymptotic delay time statistics are dominated by the influence of the boundary conditions on the wall and exhibit algebraic decay. The short time behavior is exponential and represents hyperbolic effects.

An experimental study of a turbulent wing-body junction and wake flow

Abstract: Extensive measurements were conducted in an incompressible turbulent flow around the wing-body junction formed by a 3∶2 semi-elliptic nose/NACA 0020 tail section and a flat plate. Mean and fluctuating velocity measurements were performed adjacent to the wing and up to 11.56 chord lengths downstream. The appendage far wake region was subjected to an adverse pressure gradient. The authors' results show that the characteristic horseshoe vortex flow structure is elliptically shaped, with ∂ (W)/∂Y forming the primary component of the streamwise vorticity. The streamwise development of the flow distortions and vorticity distributions is highly dependent on the geometry-induced pressure gradients and resulting flow skewing directions. The primary goal of this research was to determine the effects of the approach boundary layer characteristics on the junction flow. To accomplish this goal, the authors' results were compared to several other junction flow data sets obtained using the same body shape. The trailing vortex leg flow structure was found to scale on T. A parameter known as the momentum deficit factor (MDF = (Re T)2 (θ/T)) was found to correlate the observed trends in mean flow distortion magnitudes and vorticity distribution. Changes in δ/T were seen to affect the distribution of u′, with lower ratios producing well defined local turbulence maxima. Increased thinning of the boundary layer near the appendage was also observed for small values of δ/T.

Statistical analysis of the effects of helicity in inhomogeneous turbulence

Abstract: Effects of helicity in three‐dimensional incompressible inhomogeneous turbulence are examined with the aid of a two‐scale direct‐interaction approximation (DIA). The turbulent helicity gives a measure of the reflectional asymmetry in a turbulent flow and its inhomogeneity contributes to the sustainment of large‐scale vorticity field in a three‐dimensional mean flow. The importance of helicity effects is discussed in the context of flows in a rotating system and swirling flows in a pipe. A three‐equation model with the turbulent helicity incorporated is proposed using the theoretical results. The validity of the model is confirmed quantitatively through the application to a decaying swirling flow in a pipe.

Fluid Dynamics of Two Miscible Liquids with Diffusion and Gradient Stresses

Abstract: This chapter is based on papers by Joseph [1990b], Galdi, Joseph, Preziosi and Rionero [1991], Joseph anu [1991] and Hu and Joseph [1992]. The density of incompressible fluids can vary with concentration and temperature, but not with pressure. The velocity field u of such incompressible fluids is not in general solenoidal: div u ≠ 0. We require that the mass per unit total volume of one of the liquids in a material volume is conserved in the absence of diffusion. This yields the diffusion equation for the mass fraction ψ. Alternatively, if we obtain an equation for the volume fraction φ, then the left hand side of the diffusion equation differs from the usual substantial derivative of φ by the addition of φ div u.

Progress in high-lift aerodynamic calculations

Abstract: The current work presents progress in the effort to numerically simulate the flow over high-lift aerodynamic components, namely, multi-element airfoils and wings in either a take-off or a landing configuration. The computational approach utilizes an incompressible flow solver and an overlaid chimera grid approach. A detailed grid resolution study is presented for flow over a three-element airfoil. Two turbulence models, a one-equation Baldwin-Barth model and a two equation k-omega model are compared. Excellent agreement with experiment is obtained for the lift coefficient at all angles of attack, including the prediction of maximum lift when using the two-equation model. Results for two other flap riggings are shown. Three-dimensional results are presented for a wing with a square wing-tip as a validation case. Grid generation and topology is discussed for computing the flow over a T-39 Sabreliner wing with flap deployed and the initial calculations for this geometry are presented.

Preconditioning and the Limit to the Incompressible Flow Equations

Abstract: The use of preconditioning methods to accelerate the convergence to a steady state for both the incompressible and compressible fluid dynamic equations are considered. The relation between them for both the continuous problem and the finite difference approximation is also considered. The analysis relies on the inviscid equations. The preconditioning consists of a matrix multiplying the time derivatives. Hence, the steady state of the preconditioned system is the same as the steady state of the original system. For finite difference methods the preconditioning can change and improve the steady state solutions. An application to flow around an airfoil is presented.

A cell-centered ICE method for multiphase flow simulations

Abstract: The Implicit Continuous-fluid Eulerian (ICE) method is a finite-volume scheme that is stable for any value of the Courant number based on the sound speed. In the incompressible limit, the ICE method becomes essentially identical to the Marker and Cell (MAC) method, so the two schemes are closely related. In this article, the classical ICE method is extended to multiple interpenetrating phases, and employed with a single control volume (nonstaggered) mesh framework. The incompressible limit is preserved, so that problems involving equations of state, or those exhibiting constant material densities, can be addressed with the same computer code. The scheme reduces properly to a single-fluid method, enabling benchmarking using well-known test cases. Thus, the numerical issues focus only on those aspects unique to problems having multiple density, velocity and temperature fields. The discussion begins with a derivation of the exact, ensemble-averaged equations. Examples of the most basic closures axe given, and the well-posedness of the equations is demonstrated. The numerical method is described in operator notation, and the discretization is sketched. The flow patterns in a bubble column are computed as an incompressible flow example. For a compressible flow example, the expansion and compression of a bubble formed by high-explosivemore » gases under water is shown. In each case, comparison to experimental data is made.« less

Development of the wake behind a circular cylinder impulsively started into rotatory and rectilinear motion

Abstract: The temporal development of a 2D viscous incompressible flow generated by a circular cylinder started impulsively into steady rotatory and rectilinear motion is studied by integration of a velocity/vorticity formulation of the governing equations, using an explicit finite-difference/pseudo-spectral technique and an implementation of the Biot-Savart law. Results are presented for a Reynolds number of 200 (based on the cylinder diameter 2a and the magnitude U of the rectilinear velocity) for several values of the angular/rectilinear speed ratio alpha = omega(a)/U (where omega is the angular speed) up to 3.25. Several aspects of the kinematics and dynamics of the flow not considered earlier are discussed. For higher values of alpha, the results indicate that for Re = 200, vortex shedding does indeed occur for alpha = 3.25. However, consecutive vortices shed by the body can be shed from the same side and be of the same sense, in contrast to the nonrotating case, in which mirror-image vortices of opposite sense are shed alternately on opposite sides of the body. The implications of the results are discussed in relation to the possibility of suppressing vortex shedding by open or closed-loop control of the rotation rate.

Prediction of Pressure Drop for Incompressible Flow Through Screens

Abstract: A new pressure loss correlation predicts flow through screens for the wire Reynolds number range of 10 -4 to 10 4 using the conventional orthogonal porosity and a function of wire Reynolds number. The correlation is extended by the conventional cosine law to include flow that is not perpendicular to the screen. The importance of careful specification of wire diameter for accurate predictions of porosity is examined. The effective porosity is influenced by the shape of the woven wires, by any local damage, and by screen tension

Incompressible Computational Fluid Dynamics: On Some Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow

Abstract: In this article we discuss the solution of the Navier-Stokes equations modelling unsteady incompressible viscous flow, by numerical methods combining operator splitting for the time discretization and finite elements for the space discretization. The discussion includes the description of conjugate gradient algorithms which are used to solve the advection-diffusion and Stokes type problems produced at each time step by the operator splitting methods. Introduction and Synopsis The main goal of this article is to review several issues associated to the numerical solution of the Navier-Stokes equations modelling incompressible viscous flow. The methodology to be discussed relies systematically on variational priciples and is definitely oriented to Galerkin approximations. Also, we shall take advantage of time discretizations by operator splitting to decouple the two main difficulties occuring in the Navier-Stokes model, namely the incompressibility condition Δ u=0 and the advection term (u.Δ)u, u being here the velocity field. The space approximation will be based on finite element methods and we shall discuss with some details the compatibility conditions existing between the velocity and pressure spaces; the practical implementation of these finite element methods will also be addressed. This article relies heavily on [1]-[7] and does not have the pretention to cover the full field of finite element methods for the Navier-Stokes equations; concentrating on books only, pertinent references in this direction are [8]-[14] (see also the references therein). This article is organized in sections whose list is given just below. The Navier-Stokes equations for incompressible viscous flow Operator splitting methods for initial value problems. Application to the Navier- Stokes equations Iterative solution of the advection-diffusion sub-problems […]

Benchmark solutions for the incompressible Navier–Stokes equations in general co‐ordinates on staggered grids

Abstract: Benchmark problems are solved with the steady incompressible Navier-Stokes equations discretized with a finite volume method in general curvilinear co-ordinates on a staggered grid. The problems solved are skewed driven cavity problems, recently proposed as non-orthogonal grid benchmark problems. The system of discretized equations is solved efficiently with a non-linear multigrid algorithm, in which a robust line smoother is implemented. Furthermore, another benchmark problem is introduced and solved in which a 90 o change in grid line direction occurs

On solving the compressible Navier-Stokes equations for unsteady flows at very low Mach numbers

Abstract: The properties of a preconditioned, coupled, strongly implicit finite difference scheme for solving the compressible Navier-Stokes equations in primitive variables are investigated for two unsteady flows at low speeds, namely the impulsively started driven cavity and the startup of pipe flow. For the shear-driven cavity flow, the computational effort was observed to be nearly independent of Mach number, especially at the low end of the range considered. This Mach number independence was also observed for steady pipe flow calculations; however, rather different conclusions were drawn for the unsteady calculations. In the pressure-driven pipe startup problem, the compressibility of the fluid began to significantly influence the physics of the flow development at quite low Mach numbers. The present scheme was observed to produce the expected characteristics of completely incompressible flow when the Mach number was set at very low values. Good agreement with incompressible results available in the literature was observed.

A parallel implicit incompressible flow solver using unstructured meshes

Abstract: : An incompressible flow solver based on unstructured grids is implemented on a parallel distributed memory computer architecture An important phase in the flow solver is the solution of the elliptic equations for the velocities and pressure This elliptic solver is parallelized and incorporated into both the explicit and implicit versions of the incompressible flow solver Performance and scalability studies are carried out on both Intel iPSC 860 and the Intel Delta prototype, and show that the cod scalable A parallelizable load balancing algorithm is developed to be used in conjunction with the incompressible flow solver Steady and unsteady flows over a tri-element airfoil and NACA0012 airfoil are computed using the parallel incompressible flow solver

On L1-vorticity for 2-D incompressible flow

Abstract: We prove the existence of a classical weak solution for the 2-D incompressible Euler equations with initial vorticity ω0=ω0′ + ω0″, where ω0′ is inL1(R2)⌢H−1(R2), compactly supported, and ω0″ is a compactly supported positive Radon measure inH−1(R2).

Predicting equilibrium states with Reynolds stress closures in channel flow and homogeneous shear flow

Abstract: Turbulent channel flow and homogeneous shear flow have served as basic building block flows for the testing and calibration of Reynolds stress models. A direct theoretical connection is made between homogeneous shear flow in equilibrium and the log-layer of fully-developed turbulent channel flow. It is shown that if a second-order closure model is calibrated to yield good equilibrium values for homogeneous shear flow it will also yield good results for the log-layer of channel flow provided that the Rotta coefficient is not too far removed from one. Most of the commonly used second-order closure models introduce an ad hoc wall reflection term in order to mask deficient predictions for the log-layer of channel flow that arise either from an inaccurate calibration of homogeneous shear flow or from the use of a Rotta coefficient that is too large. Illustrative model calculations are presented to demonstrate this point which has important implications for turbulence modeling.

A least‐squares finite element method for time‐dependent incompressible flows with thermal convection

Abstract: SUMMARY The time-dependent Navier-Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity-pressure-vorticity-temperature-heat-flux (u-P-w- T-q) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the 12-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to lo6, lid-driven cavity flow at Reynolds numbers up to lo4 and flow over a square obstacle at Reynolds number 200, are presented to validate the method. The past decade has witnessed a great deal of progress in the area of computational fluid dynamics. Numerous flow problems have been successfully solved by finite difference, finite volume and finite element methods. Most finite element methods are based on the Galerkin method, 9 ' the Taylor-Galerkin method and the Petrov-Galerkin method. 3-5 Mixed-order interpolation and penalty approach are commonly used in these methods. It is well known that these methods often lead to large, sparse, unsymmetric linear systems which are difficult to solve numerically. This explains why finite element analysis for three-dimensional fluid flow problems is not a common practice. To overcome this difficulty, we propose and develop a Least-Squares Finite Element Method (LSFEM) for time-dependent incompressible flow problems. The linear systems resulting from the discretization of LSFEM are always symmetrical and positive-definite. Therefore, they can be solved more easily and efficiently. This is the main reason for investigating the least-squares finite element approach. Least-squares finite element methods have already been applied with some success to com- pressible Euler and hyperbolic equations. Jiang and Carey6- ' and Jiang and Povinelli' used an implicit method for compressible flows. To further the capabilities of the method, Lefebvre et al.' applied unstructured triangular meshes to compressible flow problems. For transient advection problems, Donea and Quartapelle lo classified four different LSFEM approaches: characteristic LSFEM proposed by Li," LSFEM by Carey and Jiang," Taylor LSFEM by Park and Liggett l3 and space-time LSFEM by Nguyen and Reynen.I4