Experiments in Fluids 24 (1998) 308—322 ( Springer-Verlag 1998

Topological structures near a heated rotating cylinder in cross flow M. Oesterle, M. Lauster, R. Waibel, V. Lippig, D. Straub

308 Abstract The heated rotating circular cylinder in cross flow is apt for a standard case in basic research of complex thermofluiddynamics. Respective investigations done at the Institute of Thermodynamics over the last decade were focused on the theoretical modeling and the experimental determination of various field properties, e.g. boundary layer separation points or the global Nusselt numbers, over a wide range of parameter combinations. This paper is concerned with the design and construction of a closed multi-purpose wind tunnel along with a flexible, fully computer-guided measurement equipment. The set-up allows the realization of an extended research program for a variety of flow configurations which are of interest in engineering practice, especially. The description and interpretation of the experimental results mainly concern some characteristic patterns of a channel flow along the wake of a heated and rotating cylinder. They include investigations on different fluids within the physically relevant range of molecular masses and of some factors such as the aspect ratio a and the ratio of peripheral speed of the cylinder to free stream velocity at various turbulence intensities of inflow. The Eckertnumber phenomenon is demonstrated for high values of a. A comparison of the theoretical predictions with the measured data will be published elsewhere.

1 Problems of exploring complexity Crossflows around a cylindrical body, normally arranged to the flow direction, belong to the most interesting flow configurations in practice and theory. Today’s technology provides

Received: 26 August 1996/Accepted: 22 August 1997 M. Oesterle, M. Lauster, R. Waibel, V. Lippig, D. Straub Institut fu¨ r Thermodynamik WE 10, Fakulta¨ t fu¨ r Luft-und Raumfahrttechnik, Universita¨ t der Bundeswehr Mu¨ nchen, D-85577 Neubiberg, Germany Correspondence to: D. Straub

a lot of examples for all kinds of fluids and flow regimes. For instance, intimate knowledge of the highly complex interactions between fluid and moving walls is necessary for calculating the so-called ‘side-load effects’ observed at spin-stabilized flying objects or during rocket launch. Due to turbomachinery flows, heat transfer rates from hot gases to rotating blades are decisively determined by the complex flow field topology around them. Hence, complicated and expensive measurement programs are needed. For such reasons simplified flow configurations like crossflows of gases around a surface-coated cylinder is of considerable interest for basic experiments, provided that the cylinder is vertically installed inside a straight channel and able to be rotated as well as heated. But also theoretical aspects concerning frictionless potential flows stimulate research work on this subject. Assuming, e.g., a rectangular cross-section of the channel, an additional parameter may be defined by the ratio of the projection area of the cylinder and just that cross-section. The resulting flow patterns, strongly influenced by blockage effects due to this aspect ratio, enables us to create optional features of complex matter. Ten years ago, such opportunities induced a team of some prominent experts to turn attention to this special field of research: ‘‘Two problems that would be useful to attack are the flow around a backward facing step and around a rotating cylinder, although the numerical details for the latter are difficult’’. (Nixon 1986, p. 25). Some contributions to the latter basic problem will be concisely dealt with in this paper concerning channel flows of a gas around a heated and rotating vertical cylinder. In case the flow to be investigated shows complex patterns with respect to space and time, it seems inevitable to characterize such a kind of complexity by some primitive indicators. As a rule, the latter cannot be immediately read off from the common field variables determined either experimentally or numerically. Yet, these variables may sometimes be transformed and summed up in such a way as to result in just this kind of simple characteristic which is particularly useful for a quantitative indication of that complexity and its theoretical description. To prepare this option two primary steps have to be done:

Dedicated to Herbert Wilhelmi on the occasion of his 65th birthday

z A classification of typical flow situations; z a listing of the relevant variables affecting the flow field.

The authors gratefully acknowledge the financial support of this project by the Deutsche Forschungsgemeinschaft (DFG). They also wish to thank Dipl.-Ing. Michael Gschwendtner, Mrs. Dagmar Feuerer, and Miss Carmen Webber for their help and technical assistance.

At a first glance to the references concerning the cylinder problem (cf. Oesterle 1996, pp. 1—8) it seems a likely supposition that a matrix-like classification scheme for the

309

Fig. 1. Classification scheme for characteristic flow situations

really occurring flow patterns may be useful: Horizontally, three groups, depending on the position of the separation point of the boundary layer for the non-rotating cylinder are distinguished in Fig. 1: z Subcritical flow with Reynolds numbers between about 300 and 105 related to the diameter of the cylinder and the upstream conditions of the gas; the separation point is located at an angle between 100° and 80° (angles measured from the front stagnation point). There is a laminar separation and a turbulent vortex street at the lee side of the cylinder. z Critical flow at Reynolds numbers between 105 and 3.5]105; the angles for the separation points steeply rise from angles 80° up to 140°. We have a transition from laminar to turbulent boundary layer flow with simultaneous separation and a significant drop of the drag. z Supercritical flow; Reynolds numbers range from 3.5]105 to 3.5]106. The transition from laminar to turbulent boundary flow takes place at the front part of the cylinder and the separation points remain nearly constant at an angle of 140°. Vertically, the influence of rotation is considered: Rotating clockwise, the cylinder walls change the position of front and rear stagnation points and the flow becomes asymmetric. The term ‘critical’ refers to the theory of potential flows where at

Fig. 2. Basic configuration

the ‘critical’ speed of rotation front and rear stagnation point collapse to a free stagnation point. In the matrix scheme, the first row depicts the non-rotating cylinder while the second, third and fourth row show subcritical, critical and supercritical rotation, respectively. Figure 2 sketches the measuring segment of a wind tunnel where the experiments have to be performed. It is characterized by some geometrical and operational parameters as well as by a set of initial values (index R) concerning the fluid. The mass flow rate m5 of the test gas enters at cross section { with a certain velocity V , temperature T , density o , and = = =

310

degree of turbulence Tu flowing around the cylinder with diameter D, wall temperature T , and X as the ratio of speed w of rotation and fluid velocity. It leaves the tunnel — which is B units wide — at cross section r. The primary results of the experiments presented in this paper aim at a quantitative relationship between the heat transfer rates at the cylinder surface and these stationary flow variables for different parameter values prescribed. Dealing the problem in the classical theoretical manner with the conservative Navier—Stokes equations for compressible fluids and regarding the experimental and theoretical results of, e.g., Kestin et al. (1961) , Zukauskas and Ziugzda (1985) , Morgan (1975) , and Peller et al. (1984), we finally end up with a relation for the global Nusselt number which reads (for the details cf. Oesterle 1996, pp. 11—16)

Nu\f (Re , Pr , Ec , T* , k* , j* , o* , X, Tu, k , k* ). q w = = = w w w w

(1)

Here, Re , Pr , Ec , and Tu denote the Reynolds number, = = = the Prandtl number, the Eckert number, and the degree of turbulence for the fluid flow, respectively. The dimensionless values T* , k* , j* , and o* at the cylinder wall stand for the gas w w w w temperature, its viscosity, heat conductivity, and the mass density; they are all related to pertinent reference values. X is the ratio of speed of rotation and fluid velocity. Finally, k stands for the blockage factor calculated as the ratio of q diameter of the cylinder over channel width B, the parameter k characterizes the overall relative roughness of the cylinder w surface. As a rule, the function f usually is not known but subject to experimental investigations. Even if we combine the variables Pr , k* , j* , and o* to a new quantity called ‘MgasN’ and = w w w characterize it by a fixed set of data for the four variables, we still have to deal with a function having eight input quantities. Thus, these theoretical considerations imply the following requirements for the layout of an experimental device which is flexible enough to cover the wide range of flow situations while being capable of fast, highly accurate, and redundant measurements: z Variation of the Reynolds number over two to three orders of magnitude at least; z variation of X from 0 to values far beyond 2 depending on the Reynolds number; z flexible, but accurate heating for cylinder walls; z low degrees of turbulence for the main flow;

z variable blockage factors; z the possibility to use several sorts of gases. The respective measurements will be supplemented by an extended experimental program concerning local values of both velocity- and temperature fields behind the cylinder.

2 Design of the major mechanical parts of the wind tunnel 2.1 The closed loop flow system Figure 3 depicts a three-dimensional drawing of the complete set-up for the experiments. For a better overview the total assembly with its details is sketched once again in Fig. 4. The geometrical data refer to millimeter. More informations can be found in Oesterle’s report. According to the requirements set up in Sect. 1. the experimental device was designed as a closed wind tunnel allowing various gases to be used and, thus, the dependency of the heat transfer on the Prandtl number and the material functions k* , j* , and o* to be investigated. Additionw w w ally, the appropriate choice of test gases considerably extends the range of the Reynolds number as can be seen in Fig. 5. It goes without saying that special efforts have to be taken for the sealing when working with helium. Changing the gas makes it necessary to evacuate the tunnel; all mechanical parts are designed to the respective stability. Hence, the tunnel may be operated with excess pressure up to twice the atmospheric pressure as well. The whole equipment is built on massive footings isolated against the floor. Additional shock absorbers around the fan give an enhanced damping characteristics.

2.2 Fan system A radial fan with a wattage of 40 kW drives a dry air flow delivering a maximum volume rate of 10.800 m3/h at a speed of rotation of 2400 rpm. The total pressure loss of the whole device is about 100 mbar. With a thyristor direct current control equipment the speed of rotation may smoothly be adjusted from 60 rpm up to the maximum of 2400 rpm leading to flow speeds from 5 to 70 m/s. For other gases considerable differences arise. Thus, the maximum speeds shown in the following Table 1 strongly depend on the molecular mass at standard conditions. Thus, the range of the feasible Reynolds numbers, here e.g. related to a cylinder diameter of 116 mm,

Fig. 3. The test facility

311

Fig. 4. Experimental arrangement

Table 1. Properties of the various test gases Gas

Molecular mass [kg/kmol]

Mass density [kg/m3]

Pr/Pr A*3 p\const T\const

Ma/Ma A*3 T\const v\const

He Air Ar CO 2 SF 6 R114

4.003 28.95 39.948 44.010 146.056 170.922

0.17847 1.2923 1.7839 1.9769 6.602 7.670

0.9327 1 (0.72) 0.944 1.083 1.044 1.081

2.908 1 (333 [m/s]) 0.932 0.785 0.431 0.393

Fig. 5. Reynolds numbers for the various test gases related to the diameter D\116 mm of the cylinder

spans more than two orders of magnitude. After exiting the fan, the gas is decelerated by a diffuser. A special screen structure prevents separation at the divergent diffuser walls. To achieve stationary conditions the dissipation energy injected into the gas by the fan has to be removed via an especially designed cooling system allowing either to achieve a homogenous temperature in the gas or to apply a gradient lateral to the main stream. Additionally the honeycomb structure of the cooling fins strongly redirect the gas to parallel motion. A detailed description of the cooling system can be looked up elsewhere (Wurst et al. 1991). At the entry of the settling chamber, up to four turbulence screens with different meshes may be applied normal to the direction of the flow. The degree of turbulence for the influx is significantly influenced by the number of screens and the respective mesh numbers. Within the range of inlet flow

velocities from 20 up to 70 m/s the turbulence for dry air does not exceed values of about 1%. Only for small flow rates the achievable degrees of turbulence go beyond the 2%-limit.

2.3 The cylinder system The very core of the whole device, the heated rotating cylinder system, has been fully developed, produced, and tested at our institute. Within the frame of a tunnel with fixed walls the only possibility to vary the blockage factor is to use several cylinders with different diameters. Therefore, a special construction had been chosen to assure that the cylinders can conveniently be changed without removing major parts of the tunnel. A detailed overview of the total assembly is given by Fig. 6: the system is comprised of three assembly units, viz. the cylinder itself (sketched here with three different diameters used in our experiments), the motor unit, and the transmitter module. For the motor unit a modified motor spindle,

312

Fig. 6. Assembly units of the cylinder system

developed for high-speed machining of aluminum was chosen. The water-cooled two-pole stator is driven by a rotating field intermediate circuit reverser allowing speeds of rotation in the range of 60—30 000 min~1 exactly to be adjusted. The ball-bearings at the bottom and top of the motor unit are cooled and lubricated by highly pressurized cold hydraulic fluid. To fit the cylinder to the motor the upper end of the quill has a cut seating. In this carrier the cylinders are fixed with their slightly conical assembly stem, granting a self-centered slipless connexion. The transmitter module is screwed to a flange tightly mounted at the bottom of the quill; the centering is done by a horn-center with six alignment pins. The electro-mechanical system for all three cylinders consists of a hollow cylinder, the electrical heating, the assembly stem, a temperature reference brick, the temperature probes, the signal multiplexer and a 20-pole golden plug. Figure 6 shows the details for the largest cylinder. All cylinders are made of supertight aluminum (AlCuMg2) certified for aeronautical applications. The polished and hardened surface is smooth in the hydraulic sense with a resulting coefficient of roughness of approximately k +8.5]10~5. w Four electrical-heating elements with a wattage of 500 W each are vulcanized to the inner surface. Their temperature is controlled by two thermocouples located at the interface of the elements with the cylinder. Several other thermocouples are used to measure the wall temperature of the cylinder. The complete cylinder is screwed to the assembly stem with a press fit. In the core of the stem the reference brick made of copper and the wiring harness are installed; the multiplexer and the 20-pin golden plug are glued to the assembly stem. A lock nut secures the whole system to the quill of the motor. The coupling of electrical power and signals from the rotating parts to the non-rotating surroundings is done by the

Fig. 7. Transmitter module

transmitter module, sketched in Fig. 7. As well as the cylinder, the module carrier is made of supertight aluminum. On the shaft there is a 12-way slip ring unit with contacts made of hardened silver for the power supply of electronics and heating elements. The copper-graphite brushes have to be cooled with cold humid air to dissipate the heat from friction especially at high speeds of rotation; moreover the humidity lubricates brushes and contacts. In the core of the module carrier we find the optoelectronical transmitter housed in epoxy resin. A golden socket connector plays the counterpart to the 20-pin plug of the assembly stem. At the bottom of the module, exactly located in

313

Fig. 8. Control and diagnostic systems

the rotating axis, an infrared diode transmits the signals from the thermocouples to the non-rotating receiver. All rotating components were precisely balanced after manufacturing. Additionally, a mobile balancing device is used during the experiments to minimize balancing errors.

3 Control and diagnostic systems Much of the flexibility of the multipurpose wind tunnel is gained by the unique and sophisticated mechanical design but most of its ability to cover wide ranges of values for a high amount of parameters with reasonable effort of time and labor is provided by its fully-integrated computerized control and diagnostic systems. The block diagram in Fig. 8 shows (except for the speed control of the blower motor) the details of the electronic systems implemented at the wind tunnel. Note that control and diagnostic devices are complex and meshed in a network with two IBM-compatible computers (for the laser-Doppler anemometer and the hot-wire anemometer, respectively) running a fully self-developed software especially designed for vastly automated measurement processes.

3.1 The four major control systems In order to achieve stationary conditions, three parameters have to be stabilized to preset values with high accuracy. For this purpose respective controlling devices have been developed, tested, and calibrated.

3.1.1 The speed control for the main flow Due to the construction as a closed channel, the speed of the influx for the respective fluids in the measuring range is strictly depending on the speed of rotation of the blower. Therefore, a complex speed-control system for the fluid can be substituted by a comparably simple device controlling the speed of rotation of the separate-excitation direct-current blower motor. Setting the stator field to characteristic values, the speed of the motor may smoothly be adjusted from 0 to 3000 rpm. The current in the armature is delivered by a thyristor-based static converter in the range of 0—120 Am. The desired speed of rotation may be set either by a potentiometer or by software via the RS232 interface of the IBM computer.

3.1.2 The temperature of the influx The temperature T of the influx is controlled by the gas-cooler = system described above or (in detail) in Wurst et al. (1991). Independent of the operating state, the gas cooler is fed with cooling water at a constant mass flow rate and variable temperature in the flow pipe to achieve nearly constant heat-transfer coefficients at the walls of the cooler. Hence, the amount of heat absorbed only depends on the local temperature differences between gas and cooling water. For stationary conditions (i.e. constant velocity and temperature of the influx) the temperature of the gas is directly proportional to that one of the cooling water.

The temperature in the flow pipe is adjusted with a motorized valve cock. Depending on the desired gas temperature, hot water from the cooler reflux is admixed to the cold one from an cold-water reservoir. An electronic PID control circuit manipulates the position of the valve using the temperature information gathered by the Pt100 probe in the settling chamber (cf. Fig. 4). Thus, the gas temperature in the measuring range may be held constant over time at values from 15 °C to 60 °C.

314

3.1.3 The speed of rotation of the cylinder Due to the frictional contact between cylinder and quill (cf. Sect. 2.3), the speed of rotation of the cylinder is equal that one of the motor spindle. The quill is driven by a three-phase induction motor with a wattage of 15 (!) kW. At present, a digital control circuit with programmable PID characteristics adjusts the speed of rotation to values from 60—15 000 rpm with high accuracy; the information of the actual speed is given by an inductive probe reading the number of revolutions directly at the transmitter module. The set value may be fixed manually or automatically via the interface (IEEE488) of the PC (cf. Fig. 8).

3.1.4 The wall temperature of the cylinder A set of cylinders with different diameters each including the actual version of the measuring and control system for the wall temperature has been fully developed, produced, and tested at our institute (Wurst et al. 1991). Miniaturization and necessary stability could only be guaranteed with respect to the occurring centrifugal forces by use of modern Surface-Mounted-Device (SMD) components. Four heating panels are mounted axially at the inner walls. For each one of the panels, a separate control circuit has been installed thus assuring a uniform wall temperature. Panels 1 and 4 at top and bottom of the cylinder are used to compensate the heat transfer to the channel walls (protective heating concept). Thirteen thermocouples are used to measure the wall temperature of the cylinder. Their outputs are compared to a temperature independent reference voltage. The temperature of the copper reference brick is delivered by a Pt100 element and a Wheatstone-bridge circuit. The details for the cylinder heating are described elsewhere (Oesterle 1996).

3.2 The diagnostic system The diagnostic system of our multipurpose wind tunnel is comprised of three major subsystems serving for different purposes: (i) visualization of turbulent structures in the main flow, (ii) measurements of the velocities especially in the deadwater zone of the cylinder, and (iii) the exploration of the boundary layers at the cylinder wall.

3.2.1 Laser-light-sheet The idea of this method is to inoculate the flow with particles which follow the stream lines without slip and reflect the laser

Fig. 9. Principal sketch for the LLS method

light. To show the time dependent behavior the illuminated structures of the flow may then be photographed with either a reflex camera or a video camera through the optical crown window. (cf. Fig. 9). The aerosol generator located right behind the measuring range (cf. Fig. 4) which is needed for Laser-Doppler Anemometer (LDA) measurements is not applicable because of the insufficient life time of the aerosol particles. The optimum solution to the problem was a custom mist generator which is often found in discotheques. The best results with respect to diameter distributions and life times of the droplets are achieved provided that the mist is injected behind the third turbulence screen in the settling chamber with a special probe while the tunnel is operated at a weak vacuum. A remote control enables the user to adjust the particle density at any location of the measuring range. More details may be found in Oesterle (Oesterle 1996). Using a video camera to record the time dependent behavior one has to take into consideration that only fluctuations in the range of the video norm (one picture every 1/25 s) may be resolved. Therefore the LLS method is only conditionally suitable to observe transient effects. However, it is an appropriate tool to study large temporarily stable eddy structures in the wake of the cylinder.

3.2.2 The laser-doppler-anemometer (LDA) The LDA is a system for local, high-resolution, and extremely accurate velocity measurements in fluids. Coherent light, scattered at moving boundaries, is frequency-shifted and thus works as a vehicle for an information on the velocity. If the particles used as scattering medium are sufficiently small so that their dynamic behavior may be neglected then the measured particle velocity may directly be interpreted as the local velocity of the fluid. As will be described later in this paper, the flow around the cylinder is — at least in areas far away from the tunnel walls — planar with velocity vectors having only two (instead of three) significant components. With the two-components/two-colors LDA system these two orthogonal components may be measured simultaneously. Note from Fig. 8 that the optical transmitter/receiver unit has been separated from the large and heavy laser-light source. Both components are interconnected by a flexible fiber optics.

This arrangements allows to install a highly accurate 4-axis traverser working without mirrors. For the generation of coherent light a standard 4 W Argon-ion laser is used. The most powerful lines of its spectrum, the green one at j \514.5 nm and the blue one at 1 j \488 nm are separated with a prism; at the optical outlet of 2 the color division box the horizontally polarized green and blue laser beams are available. The fiber link consists of two monomode fibers optimized for the respective wave length. To insure high efficiency a special coupling device is used to focus the laser spots to the fiber optics. At the outlet a collimator is used to realign the divergent laser beams. The optical components for the transmitter/receiver units are standard parts. For the blue and the green beam respectively, a Bragg cell dynamically shifts the frequency leading to an offset in the measured Doppler-shift. Thus, both value and direction for the two velocity components may accurately be determined. In the receiver unit a stereo filter system is installed to reduce reflections especially occurring near the walls of channel and cylinder. This decidedly improves the signal—noise ratio and leads to a better detection by the photomultipliers. The signals are then conditioned by two commercial counters; their binary signal may be read by the IBM computer via interface. As scattering medium, a DEHS aerosol1 is used; it is injected by an aerosol generator which produces tracer particles with an aerodynamical diameter of 3 lm. The best mixing characteristics for all test gases in all flow regimes are obtained by locating the aerosol generator in the reflux directly in front of the fan (cf. Fig. 4). Due to the symmetry, cylindrical coordinates are appropriate for the problem. Hence, a 4-axis traverser was designed and manufactured at our institute which uses two axis to locate the vertically mounted LDA probe in azimuthal and radial direction; another vertical and horizontal axis open the possibility to focus any location of the measuring range. The step motors driving the traverser are controlled by the PC with the RS232 interface.

315

Fig. 10. Block diagram for the Hot-Wire Anemometer

minimize aerodynamical perturbations. This ‘wing’ is fixed to the traverser at an adjustable angle giving an additional degree of freedom to position the probe tangentially to the cylinder wall. Measurements in the wake of the cylinder have shown that the eddy structures induce strong vibrations of the wing with high amplitudes. A special damping device works automatically provided that the plunging depth in y-direction of the wing exceeds a certain value: An electromagnet presses the wing to a Teflon slideway at the lower border of the slots decidedly reducing the vibrations. A two-axis traverser controlled by the PC for HWA measurements positions the probe in the x- and y-directions. Step motor driven ball screws move the sleds with increments of 1/100 mm. The traversing device is tightly fixed to the channel walls to reduce sensibility to vibrations of the whole test facility.

4 Experimental results 4.1 Accuracy of the measured data

3.2.3 The hot-wire anemometer For measurements of the local fluid temperature a Hot-Wire Anemometer (HWA) was used. In contrast to the determination of local flow velocities, it is well-known that the HWA-technique works with the Wheatstone bridge as a constant-current anemometer (CCA). Commonly, an electric current in the hot-wire probe of 1 mA is pre-set thus avoiding the disturbing effect of overheating the probe. Ideally, the latter is in thermal equilibrium with the surrounding fluid; hence its electrical resistance is a direct measure for the local fluid temperature. Figure 10 shows the block diagram for the HWA: Along the measuring range, narrow slots in one of the channel walls allow importation and positioning of the probe which is plugged to the probe adapter. The bearing for the adapter is a wing-shaped structure with a slender triangle profile to

1 DEHS is the commercial acronym for sebacic acid diethyl-hexyl

The exact analysis of measurement errors and the propagation of errors in complex experiments are problematic and most often dubious because of the unknown errors induced e.g. by the sensors or the interpretation of the gathered data. Here, we resign to carry out an exact theoretical analysis but periodically calibrate our data with redundant measuring systems depending on totally different principles. This sets up an empirical data base from which coefficients of accuracy are calculated as shown in Table 2. The details of the respective measuring procedures and the data processing are described elsewhere (Kestin et al. 1961, pp. 56—65).

4.2 Investigation of three-dimensional effects Instabilities of the stagnation flow at the front end of the cylinder (the so-called Go¨ rtler vortices) as well as special effects at the intersection of cylinder and channel walls (the so-called horseshoe vortices) cause three-dimensional flow structures to occur. Therefore experiments had to be done showing that at least in the regions where LDA and HWA

Table 2. Coefficients of accuracy for the measuring systems and derived values

316

Measured quantity

Procedure

Coefficient of accuracy

Temperature Temperature Temperature Temperature Difference pressure

Resistance thermometer Pt100 Thermocouple NiCr—Ni Infrared sensor Hot-wire probe Prandtl probe Betz manometer Prandtl probe MENSOR (quartz) PREMA 6001 PREMA 6001 LDA LDA Incremental pick-up

^0.5 K ^1.0 K ^0.5 K ^0.2 K ^2%

Stepper motor Heidenhain Derived value Derived value

^0.01 mm ^0.005 mm ^5% ^4%

Derived value Derived value

^8% ^7%

Difference pressure Electric current Voltage Velocity Turbulence Speed or rotation (cylinder) Distance (HWA) Distance (LDA) Reynolds number Electrical heating power Nusselt number Streamfunction

4.3 Global Nusselt numbers

^1.25% ^2% ^0.5% ^1% ^5% ^0.01%

measurements take place the problem could be dealt as a planar one, due to additional symmetry requirements for the flow field. Investigations with the LLS method especially in the boundary regions where cylinder and channel walls intersect showed that only 20% of the total cylinder height at bottom and top are perturbed by three-dimensional effects. In other words: the fluid moves around the cylinder in a planar flow without horseshoe vortices over 60% of the total cylinder height. It is symmetrical to the middle plane of the channel where the LDA and HWA measurements are done. No indicators for the existence of stable Go¨ rtler vortices in the whole range of the Reynolds number have been found. Additionally to these more or less qualitative results, a quantitative statement was derived from LDA data measured in different planes of the channel: the largest deviation in value and phase of the velocities were found to be about 5%. As is well-known from theoretical fluiddynamics, the structure of a two-dimensional stationary flow is given by its stream function t. The partial derivatives of t with respect to (Cartesian) coordinates x and y deliver the respective components v and u of the velocity vector v\(u, v)T if compressibility of the fluid is considered to be sufficiently small:

Lt Lt \u, \[v. Ly Lx

(2)

Hence, numerical integration of the velocity components u and v with respect to y and x, respectively, should result in two functions t and t which are identical up to small deviations 1 2 induced by errors in measurement or the hypothesis of incompressibility:

t \:uLy; t \[:vLx; t +t . 1 2 1 2

For test purpose, we calculated the streamfunctions t and 1 t for the nonrotating cylinder with a diameter of 85 mm at 2 a Reynolds number of Re \105. The two functions deviate less = than 10% in the whole region; qualitative and quantitative results thus support the hypothesis of a planar flow.

(3)

4.3.1 Calculation of Nusselt numbers One of the primary results of our experimental campaigns were the relations between the global Nusselt numbers and the Reynolds numbers for the rotating and non-rotating cylinders. The global Nusselt number is the dimensionless characteristic describing the heat transfer between cylinder and fluid. It is defined as usual by

Nu:\

a)D j f

(4)

where a, D and j are the global heat transfer coefficient, the f diameter of the cylinder and the heat conductivity of the fluid near the cylinder wall, respectively. The global heat transfer coefficient a is derived as the arithmetic mean of the coefficients a and a for the two heating panels in the middle of 2 3 the cylinder. They can easily be calculated from the electrical power used to maintain constant temperature at stationary flow conditions:

P el,i ; i\2, 3. a :\ i T [T w,i =

(5)

P stands for the electrical power for the ith heating panel, T el,i w,i and T are the temperatures of the wall (measured separately = by thermocouples) and the fluid, respectively. As was shown in Sect. 3.1, the electrical power inputs for panels 1 and 4 are used for the protective heating concept to compensate the heat transfer to the channel walls. The Reynolds number for the flow

v )D Re :\ = = l

(6)

and the Reynolds number for the cylinder rotation

v )D ReX :\ u l

(7)

are calculated from the kinematic viscosity of the fluid l and the speed of the flow v or the speed of the cylinder wall v at = u a number of revolutions u. As usual, the data are recorded in log—log diagrams.

4.3.2 The non-rotating cylinder Figure 11 depicts the ‘original’ data (i.e. without any corrections for the blockage factor) for the three cylinders and various test gases. Comparisons with results from former experiments by other authors reveal that our measurements extend the knowledge of the relations Nu(Re ) in all regimes from subcritical to = supercritical flow. Moreover, excellent accordance is achieved

317

Fig. 11. Nusselt numbers as a function of the Reynolds number for various test gases

Fig. 12. Static pressure along the measuring range for the three cylinders Fig. 13. Nusselt diagram for air with X'0

after ‘blockage correction’ indicating the correct choice of the experimental set-up and the measuring procedures. Especially, the uncorrected data measured for the smallest cylinder highly correlate with Zukauskas’s ‘ideal case’ (cf. Zukauskas et al. 1985). This accordance of the experimental data for the smallest cylinder with the ideal case is explained by measurements of the static pressure along the measuring range: In Fig. 12 the pressure coefficient c is drawn for ten locations; the p data were gathered by Prandtl probes installed in the channel walls. Obviously, the smallest cylinder is least affected by the walls, thus approximating the ideal case of a cylinder in free stream. On the other hand, from these results for the cylinders with diameters D\85 mm and D\116 mm, respectively, strong influences of the channel walls on the wake have to be expected.

4.3.3 The rotating cylinder For speed ratios XP0.5 the results correspond to those from Peller (Peller et al. 1984). In contrast to Peller’s data no significant rise is detected in the global Nusselt number at the onset of rotation for ratios below 0.5 (cf. Oesterle 1996, pp. 104—107). Up to values X+2 the heat transfer is mainly influenced by the parameters of the influx, above this limit the rotation is the dominant effect. Geropp (Geropp 1969) cites a relation for the turbulent heat transfer which is drawn into the diagrams as a line being approached by the experimental results for XA2. The influence of very high speeds of rotation on the Eckert number will be dealt with separately in section 4.6. Figs. 13 and 14 show

318

Fig. 14. Nusselt diagram for He

the experimental data for the various test gases and different Reynolds numbers of the influx. For comparison, results from other authors are shown if available. In all test gases for the cylinder with a diameter of D\116 mm, a very interesting phenomenon is observed for 1\X\2 and the critical Reynolds number of the flow: First, the heat transfer decreases before it then approaches Geropp’s limiting line. The smaller cylinders do not show this behavior. Two conclusions may be drawn: z At values X[2 the blockage factor loses influence on the heat transfer. z The transition to a heat transfer dominated by the rotation is shifted to values significantly less than X\2 for critical influx (group 2, Fig. 11).

4.4 Velocities and temperatures Even in stationary flow the field of the velocities, especially in the wake of the cylinder is a function of time. Hence, to get reproducible results, a method is chosen which was proposed by Geropp and Leder (Geropp et al. 1985): At many positions of the flow field a certain number of measurements is taken and averaged.2 This mean is regarded as the valid velocity of the flow at the respective spot. Thus, the instationary wake of the cylinder with its well-known von Ka` rman vortex street is mapped to two large stationary eddies behind the cylinder; their length is a direct function of the Reynolds number for the influx (cf. Leder 1992). Hence, at stationary conditions the flow behind a cylinder may be divided into two major parts: A combination of cylinder and eddy structure on the one hand and a potential flow around this ‘replacement body’ on the other. Figure 15 depicts the velocity vectors and — derived with a special numerical integration algorithm — the stream lines of the time averaged flow for the small cylinder with diameter

2 Actually, 500 measurements are taken for each of the two components of the veocity vector.

D\50 mm, Reynolds number of the influx of Re \7]104 = and X varying from 0 to 2. Here only clockwise rotation is considered.3 For the nonrotating cylinder, the time averaged flow field is symmetrical. With increasing number of revolutions for clockwise rotation the lower eddy is compressed until it vanishes completely; simultaneously the forward and aft stagnation points are moved out of the plane of symmetry. For values XP2 the lower eddy is shrunk to a comparatively small separated structure indicating the coincidence of the upper and lower separation points. The cylinder no longer possesses a separated boundary layer and the figure resembles the flow around a rotating cylinder in a uniform gas at rest. This perfectly agrees with the heat transfer behavior: At XP2 the heat transfer is dominated by the rotation. The temperature distribution is determined by HWA measurements (for details cf. Oesterle 1996, pp. 64—65): For the large cylinder (diameter D\116 mm) the distributions were investigated at Reynolds number Re \105 and wall = temperatures T \60 °C and T \100 °C. The rotation was w w varied from X\0 (no rotation), X\2 to X\3. Figure 16 shows the results in the form of temperature iso-lines; it is noteworthy that the excellent precision of the HWA system allows a resolution of 0.5 K from line to line! Note as general outline that at this Reynolds number the structure of the temperature distribution is not changed significantly by an increase in the wall temperature from 60 °C to 100 °C. Just like the streamlines, the temperature iso-lines are deformed to asymmetric shape due to rotation.

4.5 Boundary layers 4.5.1 Velocities The LDA trials for the velocity profiles of the boundary layer primarily aimed at the investigation of the complex transition and separation behavior of the flow near the wall. It is well-known from literature that the LDA systems have limited efficiency near walls due to an adverse signal—noise ratio and reduced density of the tracer particles. To achieve reliable data, the number of measurements had to be increased notably.4 Additionally, only the tangential component was investigated; this is admissible because even near the separation points this component and the total velocity vector differ less than 5% (cf. Peller 1986). Figure 17 shows the development of the boundary layer of the large cylinder for a Reynolds number Re \7.4]104 and X varying from 0 to 2.4. Obviously, the = boundary layer behavior shows the same properties as could be seen from the global flow situation.

3 Prior to our main campaigns comparisons had been done with clockwise and counterclockwise rotation but no significant differences could be detected. Hence, the symmetry of the set-up was assumed. 4 The investigation of the profiles at 12 positions of the circumference of the cylinder with 20 points each and 1000 measurements at every point (which gives a total amount of 2.4]105 outcomes!) took about 48 h with an optimally adjusted optics.

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Fig. 15. Velocity vectors and stream lines for the small cylinder (D\50 mm) at Re \7]104 and X\0, X\1, and X\2, respectively =

Fig. 16. Measured temperature distribution in the wake of the cylinder (D\116 mm) at Re \105, non-rotating and at X\2 and X\3, = respectively

At X\0 the separation points are located symmetrically at an angle of approximately ^110° measured from the front stagnation point. With increasing number of revolutions, the boundary layer becomes asymmetric due to wall friction. At last, at values X[2, no separation points occur any more and the boundary layer is like that one of a rotating cylinder in a gas at rest.

4.5.2 Temperatures Measurements of the temperature profiles near the wall require a lot of care due to the influences of the probe on the boundary

layer. Here, simultaneous investigations with the hot-wire probe and the LDA were done, indicating that at high Reynolds numbers Re [105 (which produce comparably thin boundary = layers) no reliable data could be derived. Former measurements, done by Peller and Straub (Peller et al. 1988) with a Reynolds number of Re \5.6]104 are in = good accordance with our results. Furthermore, they could be extended up to Re \105 as shown in Fig. 18. The relative = temperature h, defined as

h:\

T[T = T [T w =

(8)

Fig. 17. Velocity profiles of the boundary layer for the large cylinder (D\116 mm) at Re \7.4]104 and X\0 and = X\2.4, respectively

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heat transfer from fluid to cylinder or vice versa breaks down. Geropp gave a systematical explanation for this phenomenon (Geropp 1969): He introduced a special Eckert number EcX with

EcX :\

Fig. 18. Measured profiles of the temperature boundary layer at Re \105 and a wall temperature of T \60 °C = w

was measured for the large cylinder (D\116 mm) with a constant wall temperature of 60 °C at numerous positions. Only the non-rotating cylinder was considered at present, no data are available for rotation. The profiles are drawn for various angles (front stagnation point 0°, rear stagnation point 180°).

4.6 The Eckert-number phenomenon Presumably, Yildiz’s dissertation (Yildiz 1964) first allowed to find out that, at a high speed of rotation of the cylinder, the

1 ) v2 2 u c (T [T ) p w =

(9)

relating the kinetic energy of the rotation as a measure for the dissipation to the specific enthalpy of the heat flux. This connects up the rotation and the temperature gradient which is the predominant variable for the heat transfer. Negative gradients will result in a heating of the cylinder walls from the fluid, positive ones will cool the walls. Now, assuming high speeds of rotation of the cylinder one can easily imagine that, due to friction, the temperatures of the fluid in the near-wall region and the cylinder walls equalize thus interrupting the heat transfer. This effect is predicted to occur at Eckert numbers EcX+1. To achieve high rotational Reynolds numbers ReX , all investigations for the Eckert number phenomenon were done with SF as test gas. 6 Sophisticated and careful calibration of the measurement systems granted a maximum deviation in the temperatures of 0.1 K. To prevent the gas from being heated locally in the cylinder area, a main flow with v \5 m/s was chosen (the very = low speed of the main flow does not influence the results because of the high values for XA2; cf. Sect. 4.4). The principle idea was to work with a temperature of the cylinder wall slightly higher than that of the gas and look for the behavior of the heating control system (cf. Sect. 3.1) when EcX+1 is reached: If indeed the heat transfer breaks down when reaching critical Eckert numbers, then the heating control system has to reduce the electrical power for the heating panels in order to hold the wall temperatures constant. The results shown in Fig. 19 were derived with the test parameters:

T \25 °C at 5 m/s, T \27 °C = w held constant by the control system, and ReX\0.15]105—3]106. The relative electrical power5 needed to maintain a constant temperature of the cylinder walls at 27 °C is drawn as a function of the rotational Reynolds number. After

5 P0 is the electrical power needed at ReX\0.15]105. el

speed of rotation of the cylinder may be adjusted from 60—15 000 rpm. Its wall temperature is held constant at preset levels by a complex electric heating system. For the measurements, three major systems are available: z The Laser-light sheet (LLS) for photographs and other visual methods; z the Laser-Doppler anemometer (LDA) for a precise analysis of the velocity field, and z the Hot-Wire Anemometer (HWA) for the acquisition of accurate temperature data.

Fig. 19. Relative electrical power as a function of the rotational Reynolds number

monotonically increasing to a value of approximately 4.5, at ReX+106 the electrical power impressively decreases6 until it reaches P0 again at ReX+2.6]106, thus indicating a drastic el breakdown in heat transfer. Even though the results show some deviation, Geropp’s curve (the bold face line) may easily be identified.7 As far as we know, this is the first time that the Eckert number phenomenon could be shown experimentally in a reproducible manner.

5 Summary The closed wind tunnel with its exchangeable cylinder system fully meets the requirements to study complex non-equilibrium phenomena in thermofluiddynamics. This set-up provides a basis for efficient, fully-computerized wide-range experiments to evaluate a flow situation which experts consider to be one of the two scientific key problems of modern fluid dynamics. The carefully designed fan system secures stable and repeatable conditions for the influx with adjustable degrees of turbulence. Special choice of test gases allows the variation of the Reynolds number over more than two orders of magnitude. The sophisticated rotating cylinder system opens the possibility to reach values of X far beyond 2; the influence of heated walls on the boundary layers and the flow regime downstream the cylinder may easily be studied. In our experiments three sample cylinders were available with a blockage factor from 12.5%, 20%, and 30%, respectively. The sophisticated mechanical design and the fully computerized electronic control and diagnostic systems of the closed wind tunnel offer a platform for fast and highly efficient measurements of the complex effects occurring near a rotating cylinder and along its wake. Reproducible conditions of the influx are set by control units for the speed of the main flow and the temperature. The

6 For the large cylinder with D\116 mm, a rotational Reynolds number of ReX\106 corresponds to a speed of rotation of 12 000 rpm. 7 In his original work (Geropp 1969) Geropp gave the Nusselt number as a function of the Reynolds number.

After having shown that the flow in the plane of measurements may be considered to be two-dimensional, the primary investigations aimed at the connection between the global Nusselt number and the rotational Reynolds number. The results well fit and extend and data known from literature; Geropp’s limiting line is confirmed for the various test gases. The measurements of the velocity and temperature fields as well as the boundary layer investigations revealed that the term ‘complexity’ needs a precise meaning. Following Leder (1992), the problem is discussed by Straub (1997). This is in particular true for the complex influences of rectilinear and rotational motion on the topological structure of the flow field and the heat transfer. From the streamlines of the time averaged flow the drag coefficient and the Magnus forces acting on the rotating cylinder may be calculated. For the first time systematic and reproducible experiments showed that Geropp’s assumption on the so-called Eckert-number phenomenon, stating that the heat transfer breaks down at a critical value of EcX+1 holds at least for SF as test gas 6 and certain preset values. At present, further experimental campaigns are in progress.

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