Ülgen Gülçat

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 Poissons 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 perfo...

3-D Viscous Flow Solutions Over Bodies in Relative Motion

A space-time finite element method based on the Arbitrary Lagrangian-Eulerian description of the incompressible Navier-Stokes Equations is developed. The developed method is used for predicting the flows past bodies in relative motion. The governing equations are expressed in a fixed frame of reference, wherein the terms related to grid motion are included. Superparametric space-time elements are used in the discretization of the domain in which, finite elements are both allowed to move and deform in (i) simple and (ii) automatic manner. The equations for the rigid body motion are integrated to calculate the trajectory of the moving object under aerodynamic and gravitational forces. The code developed here is first tested on the flow about a drifting and falling sphere which starts to move at Reynolds number of 104 from a steady state. To study the flows about the bodies in relative motion, store-separation from a wing problem is investigated as a turbulent flow for the Reynolds number of 2.5 x 106 based on the length of the store.

3D FLOW CALCULATIONS WITH AN A‐L‐E DESCRIPTION USING SPACE–TIME FINITE ELEMENTS

A space-time finite element method based on an arbitrary Lagrangian-Eulerian description is developed and implemented for the solution of Navier-Stokes equations for predicting the unsteady incompressible flows past arbitrary geometries. The governing equations are expressed in the fixed frame of reference wherein the terms related to grid motion are included. Superparametric space-time elements are used in discretization of the domain in which the finite elements are both allowed to move and deform. The code developed here is calibrated and tested on the flow about a drifting sphere. First, the unidirectionally drifting sphere is set to drift from a steady state at an initial Reynolds number of 1000. In addition, laminar flow about a drifting and falling sphere is studied, starting from the steady state at a Reynolds number of 10,000.

Incompressible Flow About Thin Wings

Thin wing theory is an efficient tool for the study of the spanwise variation of aerodynamic characteristics which has effect on the total lift and moment coefficient of a finite wing. This variation is considerably slow except at the tip region of the high aspect ratio wings

Aerodynamics: The Outlook for the Future

The outlook and future of the aerodynamics are discussed. The present advances imply that aerodynamics in the future is heading towards the analysis of unmanned very small and very slow vehicles, and manned or unmanned very fast and large vehicles. For the former Nature is closely observed. For the latter, however, several countries are collaborating, since the financial requirements are enormous for such projects. It seems some such aerospace projects have been canceled already because of financial burdens. Before the turn of the twentieth century, some scientific views were presented for the forecast of the development in aerospace projects during the twenty-first century. However, after 9/11 many military, and civilian, projects and studies were devoted to the national security. Inevitably, the discipline of aerodynamics will be influenced with this change and the research needed for increasing the wind and flowing water energy densities one order of magnitude.

Subsonic and Supersonic Flows

Compressible flow past thin wings and slender bodies are studied. First, subsonic flow past thin wings are analyzed by means of potential flow theory. The kernel function method is introduced for arbitrary planforms undergoing simple harmonic oscillations. The spanwise polynomial approximation and the chordwise trigonometric function approximations which automatically satisfy the Kutta condition and inherit the leading edge singularity are used. The Doublet-Lattice method as a more general numerical approach is given for the analysis of the flow past additional surfaces like tail or store surfaces, which are not necessarily in plane with the wing surface or surfaces having spanwise deflections. Airfoil response to the arbitrary unsteady motion is also given for subsonic flows. A brief review of shock waves and Mach waves in a supersonic flow is given. Afterwards, unsteady supersonic potential flow is studied for a simple harmonically oscillating point source. First, unsteady flow past an airfoil is considered. Then, supersonic flow about thin wings are analyzed using the Mach box technique. Introduction to supersonic kernel function method is briefly provided. Arbitrary unsteady motion of an airfoil in a supersonic flow is presented. Slender body theory is introduced to analyze the cross flow past bodies of considerable fineness where the cross flow is shown to be approximately incompressible under certain conditions. In connection with the slender body approximation, the Munk’s airship theory is utilized to predict the stability derivatives of missile like bodies.

Incompressible Flow About an Airfoil

Two dimensional incompressible flows past thin lifting surfaces are studied. The lifting surface is modeled with a vortex sheet. The Biot–Savart law is used to establish the relation between the vortex sheet strength and the downwash w(x,t). The strength of the vortex sheet is calculated via the inversion of a singular integral equation given in terms of the downwash at the surface as a boundary condition. First, for an airfoil steady state solution after the impulsive start in a uniform stream is considered. Carleman’s formula is used to invert the integral relation which explicitly gives the vortex sheet strength satisfying the well known Kutta condition. The lifting pressure distribution and the sectional lift and the moment coefficients are found with integration of lifting pressure along the chord. Furthermore, the concept of center of pressure and aerodynamic center are introduced. The locations of both centers on the chord are then calculated. Unsteady flow case is studied with distribution of a vortex sheet in the wake as well. The wake vorticity and the bound vorticity are tied together with the unsteady Kutta condition which is nothing but imposing zero lifting pressure at the wake region. The Laplace transform technique is employed to establish the relation between the total bound circulation and the downwash. Then the transformed form of the bound vortex sheet strength and the downwash relation is used to obtain the expression for the lifting surface pressure in the Laplace domain. Since the inverse Laplace transform of the pressure expression is quite complex, the inversion is performed for the simple harmonic pressure variation. The lifting surface pressure has three terms each signifying different aspects of aerodynamic phenomena. The first term is the quasi steady term which is identical with the steady pressure term, the second term accounts for the contribution of the wake vorticity and, finally the third term is the ‘apparent mass’ term which is responsible for the non circulatory lifting pressure without the presence of free stream. Depending on the type of aerodynamics we use, the relevant terms are kept in the expression given for the pressure. Accordingly, (i) if all three terms are retained then the approach is called ‘unsteady aerodynamics’ and it is used for the problems having oscillations of order of 40 Hz, (ii) for the case of ‘quasi unsteady aerodynamics’ we neglect the apparent mass term where the approach is applicable for the 5–15 Hz range, and finally, (iii) ‘quasi steady aerodynamics’ requires retaining the quasi steady term only where the approach is good for the frequencies of 1 Hz or less. As a special type of an unsteady flow, Simple Harmonic Motion of an airfoil in pitch and/or in vertical translation is considered. The Theodorsen function here is indicative of the effect of the wake vorticity on the circulatory term in the lift expression. The effect of the wake vorticity on the profile shows itself as reduction of the magnitude of the lift coefficient, and the lagging of the response of the airfoil with the angle of attack change in time. In order to demonstrate this several basic examples of SHM are provided, wherein the hysteresis under the lift versus angle of attack curve is accounted for. In addition, returning wake problem of Loewy is studied with the help of a new function which depends on the wake spacing, reduced frequency and the rotational frequency of the blade, replace the Theodorsen function. Arbitrary unsteady motions are analyzed with the concept of the ‘indicial admittance’ function applicable to linear systems. First, the Wagner function as the indicial admittance to the arbitrary angle of attack change of an airfoil in a free stream is considered. Then, the Kussner function of the sharp edged gust impingement is obtained as the indicial admittance. In analyzing the response of an airfoil to an arbitrary motion the integral Fourier Transform Technique is utilized. The sinusoidal gust problem is studied by establishing the Sears function as the indicial admittance. For the analysis of the moving gust problem the concept of the Miles function is introduced. The Miles function is generally utilized in rotor aerodynamics. However, when the moving gust velocity becomes zero the Miles function transforms itself into the Kussner function. Finally, as an application of Wagner function an airfoil immersed in a sinusoidally varying free stream velocity is considered. This problem can be utilized in obtaining the estimate of the total lift coefficient of a blade in a forward flight from two dimensional considerations only.

Separate treatment of momentum and heat flows in parallel environment

In this study, already developed two separate solvers, one for the velocity field and one for the temperature field are run concurrently on two different machines. Each machine here also has multiprocessors which solve each equation in parallel using the domain decomposition technique. The parallel processing is performed via PVM on SGI Origin 2000 and Origin 3000 connected through 10 Mbit Ethernet. Since the machines are not dedicated the communication times may vary for different runs. The runs are first performed for a single domain on each machine handling the mass and the heat flow separately. Afterwards, two overlapping domains are used on each machine. Overall elapsed computational times indicate satisfactory speed up.