METHODS TO ANALYSE INSTABILITIES IN FLUID DYNAMICS

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1 Scientific Bulletin of the Politehnica University of Timisoara Transactions on Mechanics Special issue The 6 th International Conference on Hydraulic Machinery and Hydrodynamics Timisoara, Romania, October 21-22, 2004 METHODS TO ANALYSE INSTABILITIES IN FLUID DYNAMICS Ildiko Tulbure*, University of Petrosani / Technical University of Clausthal * Str. Universitatii 20, , Petrosani, Romania Tel.: (+40) , Fax: (+40) , ildiko.tulbure@tu-clausthal.de ABSTRACT In the present paper some of the mathematical methods for analysing unstable flows will be presented and described. Stability calculations for the Navier-Stokes equations have been carried out using the Orr-Sommerfeld equations. From this analysis conclusions can be drawn regarding the transformation conditions of a laminar in a turbulent flow. The Bénard-convection shows another specific example of instabilities in fluid dynamics, which will be discussed in the present paper. The Lorenz equations, as a further development, used to describe meteorological processes in the atmosphere, demonstrates that instabilities can lead to chaotic behaviour of the system. Such kind of unpredictable behaviour is called deterministic chaos. These processes are very important in other fields as well, for instance when describing air pollution phenomena. Following a discussion of the existing mathematical tools to describe these processes an overview of newer mathematical possibilities will be given. For instance the fractal geometry is nowadays considered a promising method to understand and to describe instabilities not only in fluid dynamics, but also in other scientific fields where dynamic processes occur. KEYWORDS Instabilities, Turbulence, Deterministic chaos, Attractors, Fractal dimension NOMENCLATURE v [-] velocity vector k c p Pr [-] Prandtl-number q w L N u T [-] Nusselt-number 3 g L ( TW T ) Ra [-] 2 Rayleigh-number D [-] fractal dimension p [N/m 2 ] pressure T [K] temperature Subscripts and Superscripts k, j cartesian coordinates 1. STABILITY ANALYSIS FOR A BOUNDARY-LAYER FLOW BY USING THE DISTURBANCES THEORY Fluid flows are very often characterised by instabilities. Laminar and stable flows are rather exceptions in fluid mechanics. One of the most interesting problems in fluid dynamics is the problem of stability of a laminar flow and its transition to turbulence. Already more than 100 years ago Reynolds understood from his researches that the turbulence issue represents a stability problem. It took a little bit time until the year 1929 when Tollmien carried out the first stability calculation for the boundary-layer flow along a flat plate, i.e. for the Blasius-solution [3]. He considered the basic stationary flow (with the velocities v i and pressure p) perturbed by a two-dimensional small disturbance (v i and p ). The disturbed flow respects

2 again the continuity equations, the Navier-Stokes equations as well as the boundary conditions. The resulting disturbances differential equation is called Orr-Sommerfeld equation of hydrodynamic stability [6]. This equation can be linearized and in this way the stability of the flow can be analyzed. As a result of the Tollmien calculations it came out that there is a minimum critical Reynolds number of about 420, below which the flow is stable [3]. Later, in 1943, Schubauer and Skramstad were the first ones to verify experimentally the occurrence of the oscillations, so-called Tollmien-Schlichting oscillations, predicted by the Orr-Sommerfeld equations in boundary-layer flow over a flat plate [11]. There are some other stability calculations using the disturbances theory for other flow types as well. All these calculations have actually not properly solved the general problem of mathematical modelling and understanding of turbulence flow phenomena. 2. BÉNARD-CONVECTION The heat transfer by convection is described by using the three equations for mass, impulse and energy conservation. With the condition that the density is constant ( = m = const.), with the exception of only one term and that the other material parameters are constant as well (this is the so-called Boussinesq-approximation) following system of equations is obtained: (T 1 ) and cooled on the upper side (T 2 < T 1 ) (Figure 1). In these conditions a vertical gradient of density results in the horizontally stratified fluid, because the lower liquid Figure 1. Bénard-cells and representation of the surface of hot coffee [10]. layers are lighter than the upper layers (exception is water!). For small T = T 1 - T 2 the phenomenon is characterised only by heat conduction. If T goes over a critical value, the phenomenon of convection appears. In this transition from order to disorder the first mode of instability consists in creating dynamic order states, that become clearly by observable stable structures (Figure 2). Under these conditions in the case of twodimensional flow stationary, stable convection rolls (Taylor-instability) and in the case of threedimensional flow regular cells build up. (1) wit h This system of differential equations is basically resolvable when the initial and boundary conditions are known. The searched entities are depending on space and time: In the following a more simple solution for the Bénard-convection will be discussed, for details see [10]. The Bénard-convection appears when a horizontal liquid layer is heated on the lower side Figure 2. Building up of convection rolls under the influence of a temperature gradient [9]. For a known geometry (b, l, h) the problem is described by the two characteristic numbers: Prandtl-number (Pr) and Rayleigh-number (Ra). The characterising number for the heat transferred is the Nusselt-number. The dependence Nu = f (Ra, Pr) is to be obtained from measurements or from analytically solving the equations system (1). An analysis of the stability of the system can be carried out similar to the stability analysis of the boundary-layer flow, i. e. by linearisation of the conservation equations [3, 6]. The results show that

3 there is a minimum critical Rayleigh-number, below which the flow is stable. This is Ra krit = 657,5... for free surfaces and Ra krit = 1707,6... for walls [6]. In the nature there are a lot of examples for rolls and Bénard-cells, but in most of the cases the cells or rolls are invisible. In the meteorology a certain type of clouds (Cumuli) on the heaven shows unstable air layers. Moist air condenses in its upwards motion in Bénard-cells, which become visible as clouds. The same situation is valid for bands of clouds. The phenomenon is to be found also in the upper layers of sees and oceans, as well as earth boundary surface. 3. DETERMINISTIC CHAOS IN THE CASE OF THE LORENZ-SYSTEM To describe the Bénard-experiment theoretically, Lorenz truncated the complicated differential equations (1). He started with the study of convection processes in order to describe meteorological phenomena in the atmosphere and ended by developing a new non-linear theory. The conclusions he got can be enlarged for various types of dynamic problems. Taking the equations system (1) and making some mathematical substitutions the Lorenz equations system (2) can be obtained [9]: (2) where x is proportional to intensity of convection motion, y to temperature gradient between fluid layers and z to disturbance of the vertical temperature distribution. The parameters, r are specific hydrodynamic/thermal parameters and b is a specific geometric parameter, see [7, 9]. The Lorenz equations have been obtained from a meteorological calculation for the earth climate system (Figure 3). There are actually modelling a flat liquid layer, on which a temperature gradient is acting. The model couples the convection with heat conduction and is applied in meteorology and climate predictions [3, 9]. Lorenz tried 1963 to solve numerically the system of equations (2). He repeated one of calculations and did an error of typing a number with three numbers after comma instead of six numbers, as the computer calculated before. Although the difference between the two numbers were almost neglectable, with the new number Lorenz got totally different results Figure 3: A simple scheme on how Climate" is produced, from [1]. compared with the results he got before. This means that for some types of equation systems by using slightly different initial values totally different results can be obtained. The behaviour of such systems can not be surely predicted. For meteorology this result meant that because of nonlinearities there is not possible to predict weather changes for long term [9]. The Lorenz typing error example is known as "butterfly-effect", in the sense that very little causes can have big effects. This is the case for nonlinear systems, in which there are positive couplings among variables. As a result from these studies can be stated that linear systems are stable regarding uncrisp start values. Nonlinear systems can be very sensitive regarding uncrisp initial values. In the scientific world the notion of "deterministic chaos" is handled for such situations. Stability analysis for such nonlinear systems are very important to understand the behaviour of the system. One method to analyse stability is by using Ljapunov-exponents. Calculating for instance the Ljapunov-exponents as eigenvalues of the determinant of the Jacobian matrix it is possible to get information about the type of fixed points and/or attractors of the system [9]. Stable fixed points of a dynamic system are those points to whom all curves in the phase diagram in the neighbourhood of the fixed points are attracted. Unstable fixed points are those points from whom the curves in the phase diagram in the neighbourhood of the fixed points move away. An attractor of a nonlinear dynamic system is a curve or a surface having the property that the curves in the phase diagram in its neighbourhood are moving to that attractor.

4 There are also so-called "strange attractors" as in the case of the Lorenz model (Figure 4). One observes that the trajectory is attracted to a bounded region in phase space, the motion is erratic, there is a sensitive dependence of the trajectory on the initial conditions, i.e. if instead of (0, 0.01, 0) an adjacent initial condition is taken, the new solution soon deviates from the old, and the number of loops is different. This actually means that non-predictable chaotic motions are a result of nonlinearities and positive couplings among the variables of the system and that such systems can have not only stable fixed points, but also unstable ones and even attractors [2, 9]. Figure 4: Results of numerical integration of Lorenz equations in the phase diagram. 4. FRACTALS AND FRACTAL DIMENSION Mandelbrot found out more than 20 years ago that there are some geometric figures, which contains self-similar geometric structures. For instance the so-called Koch s curve is one of them, being a line of infinite length that encloses a finite area [4] (Figure 5). Figure 5: Building up the Koch s curve. Each segment is divided into four smaller segments respecting the same rules. It is understood that the ramifications continue ad infinitum [4]. Mandelbrot demonstrated that such geometric constructions have a geometric dimension, which is not anymore an integer number. That is why this dimension is called fractal dimension. There are several calculation definitions for the fractal dimension of such curves, the most used one is the Hausdorff dimension (D) [9]. The Hausdorff dimension can be calculated with the formula log( number of divisions) D (3) log( number of segments) With this formula, the fractal dimension D for the Koch s curve will be: log 4 D 1,26 log 3 For more complicated geometric forms, like for instance attractors of trajectories of different dynamic models, there are more complicated formulas to calculate the fractal dimension, where the Ljapunov exponents play an important role [2, 9]. For the Lorenz system the attractor has a Hausdorff dimension which is noninteger and lies between two and three. For the Lorenz attractor the fractal dimension is D=2,06. This actually means that systems presenting nonpredictable chaotic motions are in some cases characterised by strange attractors with fractal dimensions. 5. BIFURCATIONS THEORY AND INSTABI- LITIES WITH VERHULST DYNAMICS In the following will be shown that not only nonlinear systems of equations can present chaotic behaviour, but also quite simple equations. A very famous example is the nonlinear recursive equation which is called logistic map. This equation has been written for the first time by Verhulst in 1845, as he was studying the dynamics of populations in biology [9]. (4) where: x t is the population for time t, x t+1 is the population for time (t+1) and r is the growth rate of the population in a certain time interval. For the sake of avoiding most complicated calculations the population number will be normalized :. Verhulst understood that the change of population number depends in a nonlinear way on the population number one year before. The growth rate r plays the role of a control parameter, because

5 it is influencing significantly the population dynamics. Equation (4) also describes the angles x n of a strongly damped kicked rotator. First of all the fixed points of the system will be calculated as the intersection between the curves and The fixed point is x P 1 1 Because x 0 it is r 1. r For different values of r the values of the fixed point can be calculated, getting following values: for r = 1 ; x P = 0 for r = 2 ; x P = 0,5 for r = 3 ; x P = 0,666 for r = 4 ; x P = 0,75 For r = 3 something very strange is happening with the dynamics of the system. A bifurcation appear, the variable x t oscillates between two values around the fixed point (Figure 6) [9]. Figure 7: Bifurcation diagram of the logistic map. Chaotic behaviour emerges for values of r > 3,5. The logistic map is the simplest example of equation with which chaos can be produced. A lot of other equations show similar behaviour when values of control parameters are lower or higher than the critical values. Figure 6: Bifurcation of the logistic map for the growth rate r = 3. Starting with r = 3,44865 there are four fixed points, with the value r = 3,54413 there are eight fixed points. For the value r = 3,56445 there will be 16 fixed points and so on. For bigger values of r the number of the fixed points increases very much. This phenomenon is called bifurcation cascade. For r = 3,56994 it is the end of the order state of the system and the chaotic behaviour emerges. There are a lot of fixed points and the curves cover all the domain between 0 and 1 (Fígure 7). 6. NEW APPROACHES FOR DESCRIBING TURBULENCE PHENOMENA In the last 100 years successful researches have been carried out in order to develop turbulent theories, with which engineers and scientists have been able to work in solving practical turbulence problems. The most important step in the modelling of turbulent flows has been made by Prandtl in 1945 with his k model and later by Launder et. a. with the further developments to the k - model and derivative models [3]. In the last years several scientific works carried out in the field of modelling turbulent flows have tried to approach turbulent flows by using new mathematic instruments. There are basically two directions in the newer developments. One direction consists in modelling turbulent flows with the understanding that turbulent flows are examples of chaotic behaviour of the system under certain conditions. For certain conditions, exactly as for the logistic map, bifurcations appear and this means that the system has a lot of fixed points, among them the variables are oscillating. Newhouse, Ruelle and Takens demonstrated by

6 using the Hopf-bifurcation that a strange attractor is possible for certain cases in a turbulent motion [5]. It is in fact one of the aims of the study of deterministic chaos in fluiddynamic systems to understand mechanisms for fully developed turbulence, which implies irregular behaviour in time and space. The most fascinating and difficult question is how the onset of fluid turbulence in time (if we do not consider the distribution of spatial inhomogenities) is related to the emergence of a strange attractor. The second newest direction in analysing turbulent flows is oriented mostly on the geometry of a turbulent flow. The idea behind this is that exactly like self-similar geometric figures, a turbulent motion could have a fractal dimension. In the nature there are a lot of structures or surfaces, which are of fractal nature. Our skin, the lines which perfectly follow all the details of the outlines of clouds, of trees or of landscapes can all be interpreted as fractals because some self-similar structures can be recognised. If it is possible to demonstrate that in turbulent flows strange attractors emerge, this would mean that a fractal dimension could be calculated. This would lead to a new understanding of the geometry of the turbulent flow with the help of the fractal geometry. 7. CONCLUSIONS Treating the problem of turbulence like a problem of chaotic motion in spatially coupled nonlinear systems, like the Navier-Stokes equations actually are, some promising results have been obtained in the last decades. By generalising the behaviour of nonlinear dynamic systems from different disciplines with the help of the systems theory it is possible to use the results from one certain field into another one [12]. For instance if the behaviour of the logistic equation, which comes from biology, is known, this means that the same behaviour can be found in a technical system as well, which is described by the same equation. This is very important for analysing the phenomenon of turbulence by using newer approaches like deterministic chaos and fractal geometry. Experiences already collected with other nonlinear dynamic systems can be used to study nonlinearities connected with turbulent flows. The major conclusion from this paper is that since the nature is nonlinear, one has always to reckon with deterministic chaos and fractal structures. And this is valid for turbulent flows as well. REFERENCES 1. Canty, M., J. (1995): Chaos und Systeme, Vieweg, Braunschweig 2. Herrmann, D. (1994): Algorithmen für Chaos und Fraktale. Addison-Wesley, Bonn 3. Jischa, M., F. (1982): Konvektiver Impuls-, Wärme- und Stoffaustausch. Vieweg, Braunschweig 4. Mandelbrot, B., B. (1982): The Fractal Geometry of Nature, Freeman, San Francisco 5. Newhouse, S., Ruelle, D., Takens, F. (1978): Occurrence of Strange Attractors near Quasiperiodic Flow on Tm, m 3, Commun. Math. Phys., 64:35 6. Oertel, H., jr., Delfs, J. (1996): Strömungsmechanische Instabilitäten. Springer. Berlin 7. Peitgen, H.-O., Jürgens, H., Saupe, D. (1994): Chaos, Bausteine der Ordnung, Springer/Klett-Cotta, Berlin 8. Platten, J., K., Legros, J., C. (1984): Convection in Liquids, Springer, Berlin 9. Schuster, H., G. (1984): Deterministic Chaos. Physik.Velag, Weinheim 10. Steeb, W.-H., Kunick, A. (1989): Chaos in dynamischen Systemen, BI Wissenschaftsverlag, Mannheim 11. Streeter, V., L. (ed.) (1961): Handbook of Fluid Dynamics. McGraw-Hill Book Company, New York 12. Tulbure, I. (2003): Integrative Modellierung zur Beschreibung von Transformationsprozessen. VDI-Verlag, Düsseldorf, Reihe 16, Nr. 154