Egon Krause

A comparison of second- and sixth-order methods for large-eddy simulations

Abstract Large-eddy simulations of spatially developing planar turbulent jets are performed using a compact finite-difference scheme of sixth-order and an advective upstream splitting method-based method of second-order accuracy. The applicability of these solution schemes with different subgrid scale models and their performance for realistic turbulent flow problems are investigated. Solutions of the turbulent channel flow are used as an inflow condition for the turbulent jets. The results compare well with each other and with analytical and experimental data. For both solution schemes, however, the influence of the subgrid scale model on the time averaged turbulence statistics is small. This is known to be the case for upwind schemes with a dissipative truncation error, but here it is also observed for the high-order compact scheme. The reason is found to be the application of a compact high-frequency filter, which has to be used with strongly stretched computational grids to suppress high-frequency oscillations. The comparison of the results of the two schemes shows hardly any difference in the quality of the solutions. The second-order scheme, however, is computationally more efficient.

Numerical simulation of the blood flow in the human cardiovascular system

This paper describes a numerical model of the human cardiovascular system. The model is composed of 15 elements connected in series representing the main parts of the system. Each element is composed of a rigid connecting tube and an elastic reservoir. The blood flow is described by a one-dimensional time-dependent Bernoulli equation. The action of the ventricles is simulated with a Hills three-element model, adapted for the left and right heart. The closing of the four heart valves is simulated with the aid of time-dependent drag coefficients. Closing is achieved by letting the drag coefficient approach infinity. The resulting system of 32 non-linear ordinary differential equations is solved numerically with the Runge-Kutta method. The results of the simulation (pressure-time and volume-time dependence for the atria and ventricles and pressure forms in the aorta at a heart rate of 70 beats per minute) agree with the physiological data given in the literature. The models input aortic impedance is 31.5 dyn s cm-5 which agrees with literature data given for aortic input impedance in man 26-80 dyn s cm-5). Long-term stability of the system was achieved. The cardiovascular system presented here can also be simulated at higher and varying heart rates--up to 200 beats per minute. The results of calculations for some pathological changes (e.g. valvular abnormalities) are discussed.

The solution to the problem of vortex breakdown

A numerical solution of the Navier-Stokes equations is discussed for time-dependent, three-dimensional, incompressible flow. The solution is based on the artificial compressibility concept and a dual time stepping procedure. An implicit Roe-type flux-difference splitting is used for the discretization of the convective and central differencing for the viscous terms. The difference equations are written for a Cartesian grid. A new approximation for the downstream boundary conditions is derived. The solution was employed to describe the process of vortex breakdown in isolated slender vortices. It is shown, that stable breakdown can be achieved by iterating the side boundary conditons through a new integral technique. Comparison with experimental flow visualizations shows good qualitative agreement between the vortex structures observed in experiments and simulations.

Fourth order “mehrstellen”-integration for three-dimensional turbulent boundary layers

Abstract A new finite difference solution for Prandtls boundary-layer equations is described in detail for steady, incompressible luminar and turbulent flows. Only boundary sheets will be considered and curvature effects in the direction normal to the wall will be neglected. The governing equations are presented in form of a vector equation. Their numerical stability is discussed for an elementary finite difference molecule. Improved finite-difference approximations with a truncation error of fourth order are then introduced to enable either increased accuracy or shortened calculation times, in particular, for three-dimensional problems. Detailed studies of the behaviour of the overall error of the solution and several applications to real flow situations supplement the general considerations.

A contribution to the problem of vortex breakdown

Abstract Associated with the breakdown process is the formation of a stagnation point on the axis of the vortex. This requires the deceleration of the axial velocity component, which must be enforced by a positive axial pressure gradient. The analysis presented here shows, how the pressure gradient along the axis of the vortex is influenced by the radial and azimuthal velocity components. An explicit expression for ∂p/∂x (x, 0) can be obtained by integration of the momentum equation for the radial velocity component with respect to the radial and subsequent differentiation of the integral with respect to the axial direction. In an order of magnitude analysis it is then demonstrated that for large Reynolds numbers one component of the frictional force in the azimuthal direction cannot be neglected. In order to obtain an estimate for the pressure gradient rigid body rotation is assumed for the vortex core, and a distribution similar to that of a potential vortex w = kr −n , for the outer portion. The estimate shows that a positive axial pressure gradient can exist only, if the radial velocity component is positive and if the exponent n is less than unity. It is also verified that a potential vortex cannot support an axial pressure gradient, that the pressure gradient in magnitude is directly proportional to the square of the maximum of the azimuthal velocity, referenced to the freestream velocity.

Three-dimensional computation of unsteady flows around a square cylinder

Tetsuro Tamura ORI, Shimizu Corporation, 2-2-2 Uchisaiwai-cho, Chiyoda-ku, Tokyo 100, Japan Egon Krause Aerodynarnisches Institut, RWTH Aachen, WOllnerstraBe zw, 5. u. 7, Aachen, West Germany Susumu Shirayarna and Katsuya Ishii Institute of Computational Fluid Dynamics, 1-22-3 Haramachi, Meguro-ku, Tokyo 152, Japan Kunio Kuwahara The Institute of Space and Astronautical Science, 3-I-1 Yoshinodai, Sagamihara-shi, Kanagawa 229, Japan

Computational fluid dynamics: Its present status and future direction

Abstract Developments and advances in numerical fluid dynamics are being reviewed with emphasis on physical aspects in preference to methodical questions. The governing equations of fluid dynamics, describing the conservation of mass, momentum, and energy are discussed first. Recent work on predictions of inviscid, and of boundary-layer flows is then described in the following two sections. Thereafter, computations of fully viscous flows by numerical solutions of the Navier-Stokes equations are elucidated with several examples of the recent literature.

Simulation of incompressible and compressible flows on vector-parallel computers

Explicit and implicit schemes for compressible and incompressible unsteady three-dimensional flows implemented on high-performance vector-parallel computers are presented with selected applications. The explicit scheme for compressible flows is based on a multi-stage Runge-Kutta scheme with multigrid acceleration. For incompressible flows an explicit Adams-Bashforth method and an implicit dual-time stepping scheme with a conjugate gradient method and a local ILU preconditioning is applied. The essential details of the solution schemes are presented are, their implementation on parallel computer architectures, and their performance for different flow problems on different hardware platforms.

Computation of transition to turbulence in the compression stage of a reciprocating engine

The transition to turbulent flow in the compression stage of a reciprocating engine is studied by obtaining the finite-difference numerical solutions to the governing Navier-Stokes equations without using explicit turbulence models. A computational method is developed under the assumption that the flow is in a low-subsonic regime with strong compression. The numerical method is a simple extension of the well known MAC method. Computations were performed for three different chamber geometries at the engine speed of 1400 rpm. The results of the computations clearly demonstrate the transient process in which large tumbling vortices break down into smaller ones near the end of the compression process. The transition process is also caught experimentally by using Mach-Zehnder interferometry.

Boundary-layer investigations on a model of the ELAC 1 configuration at high Reynolds numbers in the DNW

Abstract Results of boundary-layer investigations on the leeward side of a 1: 12 scale model of the ELAC 1 configuration of a space transportation system are presented. The configuration has the shape of a thick delta wing with a rounded leading edge. The model length is 6 m; the experiments were carried out in the 8×6 m 2 low-speed German-Dutch-Windtunnel at Reynolds numbers up to Re=40·10 6 . In a first series of experiments mean velocity profiles were determined in the turbulent boundary layer on the leeward side of the model, with a single hot-wire probe in the plane of symmetry at four positions. Comparison calculations with a numerical solution of the boundary-layer equations showed good agreement up to angles of attack α =10° . In a second series of tests the laminar-turbulent transition of the flow and its separation near the rounded leading edge were investigated at three positions with multi-sensor hot-film arrays with 40, 56, and 96 elements. These measurements demonstrated that the flow near the rounded leading edge is markedly influenced by the nose radius.