Flow rate measurement in a pipe flow by vortex shedding - Springer Link

Forsch Ingenieurwes (2010) 74: 77–86 DOI 10.1007/s10010-010-0117-0

O R I G I NA L A R B E I T E N / O R I G I NA L S

Flow rate measurement in a pipe flow by vortex shedding M. Reik · R. Höcker · C. Bruzzese · M. Hollmach · O. Koudal · T. Schenkel · H. Oertel

Received: 24 February 2010 / Published online: 9 March 2010 © Springer-Verlag 2010

Abstract The flow rate measurement of liquid, steam, and gas is one of the most important areas of application for today’s field instrumentation. Vortex meters are used in numerous branches of industry to measure the volumetric flow by exploiting the unsteady vortex flow behind a blunt body. The classical Kármán vortex street behind a cylinder shows a decrease in Strouhal number with decreasing Reynolds number. Considering the flow behind a vortex shedding device in a pipe the Strouhal-Reynolds number dependence shows a different behaviour for turbulent flows: a decrease in Reynolds number leads to an increase in Strouhal number. This phenomenon was found in the experimental investigations as well as in the numerical results and has been confirmed theoretically by a stability analysis. Volumenstrommessung in einer Rohrstroemung durch Wirbelabloesung Zusammenfassung Die Volumenströmung von Flüssigkeiten, Dampf und Gasen ist eine der wichtigsten Anwendungen im Bereich der Instrumentierung. WirbelstromMessgeräte werden in zahlreichen Bereichen der Industrie genutzt, um mit der instationären Wirbelablösung hinter stumpfen Körpern den Volumenstrom zu messen.

M. Reik · C. Bruzzese · T. Schenkel · H. Oertel () Institute for Fluid Mechanics, Karlsruher Institute of Technologie (KIT), 76128 Karlsruhe, Germany e-mail: [email protected] R. Höcker · M. Hollmach · O. Koudal Endress + Hauser Flowtec AG, Reinach, Switzerland R. Höcker e-mail: [email protected]

Die klassische Kármánsche Wirbelstrasse hinter einem Zylinder zeigt eine Verringerung der Strouhal-Zahl mit Abnehmen der Reynoldszahl. Bei der turbulenten Strömung hinter einem Wirbelgenerator zeigt die Strouhal-Zahl ein anderes Verhalten. Eine abnehmende Reynolds-Zahl führt zu einem Anstieg der Strouhal-Zahl. Dieses Verhalten wurde sowohl im Experiment als auch bei der numerischen Berechnung gefunden und wurde mit einer Stabilitätsanalyse theoretisch bestätigt. Nomenclature a [–] a Wave number [1/m] A Area [m2 ] b [–] D, d Pipe diameter, diameter of vortex shedding device [m] f Shedding frequency [1/s] K Calibration factor [m3 ] K Kinetic energy [J/kg] p Pressure [Pa] Red Reynolds number [–] Str Strouhal number [–] t Time [s] U Velocity [m/s] [m/s] ui u∞ Undisturbed flow velocity [m/s] u Averaged velocity [m/s] Fluid velocity [–] u¯ i [m/s] ui V Volume [m3 ] ˙ V Volume flow rate [m3 /s] Coordinate [m] xi ρ Fluid density [kg/m3 ] τ Viscous stress tensor [N/m2 ]

78

δi j τ¯ij μ ω

Forsch Ingenieurwes (2010) 74: 77–86

Boundary layer thickness [m] Shear stress [N/m2 ] Dissipation rate [J/(m3 s)] Viscosity [kg/(ms )] Angular velocity [1/s]

1 Introduction The industrial demand on flow rate measurement devices is 100 thousand pieces or 100 Mio. $ per year. The requirement for increasing performance and accuracy of these devices is very strong. Among the wide variety of measurement principles, the vortex shedding flow meter is well regarded to be reliable, robust and flexible (Alsalihi et al. [1]). The application range of the vortex shedding device reaches from liquids through saturated steam to pure gases. The measurement principle of vortex flow meters is based on the formation of vortices sched from an obstacle spread over the span of a pipe. Figure 1 shows the investigated CAD-geometry and picture of the triangular vortex shedding device in a pipe (d/D = 0.23, d diameter of vortex shedding device, D pipe diameter) as well as the results of numerical flow simulation. The upstream flow is a turbulent pipe flow. The flow structure reveals a horse shoe vortex at the connection of the shedding device with the wall. Downstream the bluff body the horse shoe vortex near the pipe wall interacts with a turbulent, three-dimensional, shedding wake flow. The corresponding shedding frequency allows to determine the flow rate. Abb. 1 Vortex shedding device and flow simulation in a pipe

The first theoretical description of vortex shedding dates back to Strouhal [23] and von Kármán [25] for the farfield of the wake. In 1923 Heisenberg [7] presented an analytical approach to the vortex schedding of a bluff body. Roshko [21] presented a first functional relationship of Strouhal num·d ) and Reynolds number (Red = ρ·uμ∞ ·d ), with ber (Str = fu∞ shedding frequency f , undisturbed flow velocity u∞ , diameter of bluff body d and viscosity μ: Str = a +

b , Red

with constants a and b. Oertel [15] and Monkewitz [10] further developed independently the mechanism of vortex schedding by introducing the concept of absolute instability of a wake flow, which has been applied for turbulent flows by Oertel [16]. In a region of absolute instability growing pertubations waves move upstream and downstream where as in a locally convective unstable region growing waves only move downstream. Most of the work on vortex schedding is related to the wake of a cylindrical body at velocities resulting in Reynolds numbers in the laminar range of Red = 50–150. In state of the art industrial flow meters rectangular of triangular shaped bluff bodies are employed, operating at Reynolds numbers of Red = 3000 and beyond with turbulent vortex shedding regime. In contrast to the majority of academic work on vortex schedding, the boundary conditions in industrial flow meters are significantly different. Von Kármán (and many of his successors) used long, thin rods in a flow


79

Fig. 2 Strouhal-Reynolds-number dependence of vortex shedding

with constant upstream velocity. In vortex flow meters, the turbulent flow is confined by the pipe wall which produces a flow with a significant velocity profile. At the junction between the bluff body and the pipe wall, a horse shoe vortex exists. Despite of these differences, the formation of vortices is to some extent similar to that of a von Kármán vortex street. The most common similarity is the fact that the frequency of vortex schedding is directly proportional to the flow velocity. This behaviour is employed in vortex flow meters. In the simplest case the vortex shedding frequency is counted and multiplied by a constant calibration factor yielding the volumetric flow rate. V˙ = K · f. This, however, implies that industrial flow meters are operated assuming a constant Strouhal number, indicated by a single calibration factor K associated with a flow meter. u·A=K ·f u ·A·d =K ⇔ f ·d ⇔

1 · A · d = K, Str

200 < Red < 1 · 105 . At Reynolds numbers Red ≥ 3 · 105 the laminar-turbulent transition in the cylinder boundary layer occurs, which corresponds to a characteristic increase of the Strouhal number. In context to this the vortex shedding flow in the pipe of Fig. 1 is turbulent in the whole regime. It is determined by a complex three dimensional interaction of horseshoe vortices near the pipe walls and the vortex shedding behind the vortex device. Nevertheless the concept of absolute instability (Oertel [15]) can be applied, which has been the first time derived in Oertel [16]. The question arising from the revelation that the Strouhal number varies with Reynolds number is, whether this behaviour is caused by the boundary conditions (confined flow, horse shoe vortex, velocity profile) or by an inherent property of the wake flow. To estimate the StrouhalReynolds-number dependence of Fig. 2 for the triangular vortex shedding device of Fig. 1 experiments as well as numerical RANS simulations have been performed. The decay of Strouhal-number with increasing Reynoldsnumber is theoretically confirmed by a stability analysis.

(1)

with the pipe area A. In fact, the Strouhal number is found to be a function of Reynolds number, which is the topic of this article. For the classical Kármán vortex street behind a bluff body in a two-dimensional infinite domain the Strouhal-Reynolds number dependence is well known (Fig. 2, Oertel et al. [18]). In the laminar regime of the Kármán vortex street a strong increase of the Strouhal number can be observed due to the decrease in the size of the absolute unstable region in the cylinder wake flow. The Strouhal number saturates to an approximately constant value of the Strouhal number Str = 0.21 in the turbulent wake Reynolds number regime of

2 Methods In the following chapters the experimental setup, the numerical model and the method of the stability analysis is presented. To be able to compare the results the numerical model is based on the CAD geometry for the experiment. The stability analysis is performed based on time averaged velocity profiles of the numerical flow simulations. 2.1 Experiments A vortex flow meter as commercially available as an E + H Prowirl 72 was investigated to produce experimental data directly comparable with numerical simulations. For the tests

80


Fig. 3 Geometry of experimental set up

a flow meter of nominal diameter DN150 was used. Figure 3 shows the geometry of the experimental and numerical set up, with the triangular vortex shedding device and the paddle, measuring the shedding frequency. The vortex sensing element is a paddle shaped device mounted in an opening of the bluff body close to the pipe wall. It senses the pressure fluctuations at the trailing edge of the bluff body caused by the alternating flow separation. The paddle is hold by a membrane sealing the pipe. At the opposite side (outside the flow) a cylinder is attached to the membrane. The cylinder performs the same movement as the paddle as it is moved by the pressure fluctuations. The cylinder is embedded in two semicircular shells but separated by a small gap, forming a capacitor. A charge amplifier amplifies the change of the capacity. The amplifiers output signal at a given (constant) flow rate was recorded as a time series with a sample rate of 20 kHz for several seconds totalising 8000 vortex shedding periods. The vortex shedding frequency was extracted from the time series by means of band pass filtering and zero crossing counting. The vortex flow meter was tested with water at room temperature (app. 24◦ C) and a pressure level was kept constant at 0.2 MPa downstream of the meter under test. The flow was provided by a pump upstream. A package of four orifice-plate flow-straighteners decoupled the test section from the feeding system. The test meter was mounted in a straight pipe 40 diameters downstream of the flowstraighteners. The outlet section had a length of 20 diameters. The vortex meter operating pressure was measured by a differential pressure transducer relative to the ambient pressure. The maximum flow rate given by the feeding system was 120 l/s equivalent to a Reynolds number of 290,000 based on the bluff body width. Test were performed at Reynolds number ranging from Red = 2800 to Red = 290,000, resulting in a dynamic range of 1:100. According to the feeding system, temperature and pressure

could not be varied. The reference flow rate was measured by means of Coriolis master flow meters. The entire reference system (incl. temperature measurement, pressure measurement, mineral content of the water and the conversion from mass flow rate to volumetric flow rate) is certified to be accurate within ± 0.08% of reading. The data was reduced by calculating the velocity of the flow inside the flow meter from the volumetric flow rate and the pipe area. Density and viscosity were calculated using correlations form IAPWS-IF97 and the measured temperature and pressure as input. The results are presented as Strouhal number vs. Reynolds number in Figs. 7 and 10. The confidence level of the frequency measurement is 99% including the 0.08% reference uncertainty. 2.2 Software and numerical model 2.2.1 Basic equations and physical models For the simulation of the turbulent unsteady flow of the vortex shedding device, the three-dimensional, incompressible and unsteady Continuity and Reynolds Averaged Navier Stokes (RANS) equations are used: ∂ρ ∂(ρ · u¯ i ) + = 0, ∂t ∂xi ∂ u¯ j u¯ i ∂ u¯ i ρ· + ∂t ∂xj = f¯i +

∂(ui uj ) ∂ τ¯ij ∂ p¯ + −ρ · . ∂xi ∂xj ∂xj

(2)

(3)

The left hand side of (3) represents the change in mean momentum of the fluid element due to the unsteadiness in the mean flow and the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress due to the mean pressure field, the viscous stresses and the stress


81

(−ρui uj ) due to the fluctuating velocity field, generally referred to as Reynolds stresses. The Reynolds stress tensor is a symmetric tensor. Due to these six additional terms the system of differential equations is no longer closed. A common approach to specify these stress terms is the Boussinesq assumption which bases on the assumption of isotropic turbulence. Hereby the Reynold stresses are described analog to the viscous stresses by an eddy viscosity μt . −ρ

· ui

· uj

2 ∂ u¯l μt · = μt · Sij + + K · ρ δij , 3 ∂xl

(4)

with Sij =

∂ u¯j ∂ u¯i . + ∂xj ∂xi

(5)

There are numerous models to determine this eddy viscosity. At present, the most frequently used is the K − turbulence model, which consists of a transport equation for the turbulent kinetic energy K and for the dissipation rate . But the flow of the vortex shedding device is anisotropic. Non-linear models try to cater for this defect by adopting non-linear relationships between Reynolds stresses and the rate of strain. In this work a quadratic form of the K − turbulence model was used with the constitutive relations for the Reynolds stresses: −ρ · ui · uj K

=

2 μt ∂ u¯l μt · · Sij + ρ δij − 3 K ∂xl K μt 1 · Sil Slj − · δij Slm Slm + C1 · 3 μt · [il Slj + j l Sli ] + C2 · μt 1 · il j l − · δij lm lm (6) + C3 · 3

where ij is the mean vorticity tensor given by: ij =

∂ u¯j ∂ u¯i − ∂xj ∂xi

(7)

The coefficients C1 , C2 and C3 are empirical constants. In the K − turbulence model, the eddy viscosity μt is determined by μ t = ρ · f μ · Cμ ·

K2 .

(8)

Equations (9) and (10) show the transport equations for the turbulent kinetic energy K and for the dissipation rate .

∂K ∂K + ρ · u¯ j · ∂t ∂xj ∂ u¯ j ∂ u¯ i ∂ u¯ = μt · · + ∂xj ∂xj ∂xi ∂K μt ∂K ∂ μ· − ρ · ε, + · + ∂xj ∂xj σk ∂xj

ρ·

ρ·

(9)

∂ε ∂ε + ρ · u¯ j · ∂t ∂xj

∂ u¯ j ε ∂ u¯ i ∂ u¯ i · + · μt · K ∂xj ∂xj ∂xi ∂ ∂ε μt ∂ε ε2 + μ· − Cε2 · ρ · . − · ∂xj ∂xj σε ∂xj K

= Cε1 ·

(10)

2.2.2 Numerical methods Equations (2) and (3) are solved by a commercially available CFD solver (Star-CD© , Computational Dynamics Ltd. London). This software package is using the finite volume method (FVM) in the arbitrary Lagrange Euler formulation (ALE). The FVM starts from the conservation equation for mass and momentum in integral form. The ALE formulation is obtained from the standard Navier-Stokes equations by replacing the convective velocity by the relative velocity to the moving volume surface. For an arbitrary volume V with surface S moving with local surface velocity uiS the integral form is as follows ∂ ρdV + ρ(ui − uiS ) · n dS = 0 (11) ∂t V S ∂ ρui dV ∂t V + (ρui (ui − uiS ) + pI − τ ) · n dS = 0 (12) S

where ρ is the fluid density, ui the fluid velocity vector in the fixed coordinate system, uiS the velocity vector of the boundary S of the control volume V , n the normal outward unity vector of this boundary, p the pressure, I the unit tensor and τ the viscous stress tensor. V in (11) and (12) is an arbitrary volume with surface S. For the numerical solution of the ALE-Navier-Stokes equation set the fluid volume is subdivided into a computational grid consisting of elements or cells of finite volume (see Fig. 4). For each cell the integral equations (11) and (12) are approximated by algebraic differences in space and time. The approximation schemes used in this work are a modified linear spatial and a temporal time stepping scheme (MARS and Euler implicit). The resulting algebraic equation system is solved using the well known implicit PISO (Pressure Implicit with Splitting of Operators) algorithm.

82


global criterion, the local stability characteristics are combined to predict the shedding frequency. 2.3.1 Theoretical backround

Fig. 4 Computational grid with 2.3 · 106 cells

Figure 4 shows the numerical grid for the simulations. The grid is refined towards the bluff body and the walls. The grid at the walls incorporates a prism sub layer which was adapted for a hybrid wall function with 0.5 y + , z+ 25 with a growth ratio of 1.2 and a total number of prism cells of 3. The resulting grid has 2.3 · 106 grid points. The grid is designed and suited for unsteady RANS simulation of this problem. Depending on the inflow velocity a Large Eddy Simulation would require a wall resolution 10 to 40 times higher, resulting in a jump in grid size of about one order of magnitude. To minimise the influence of the boundary conditions the computational domain was elongated by extruding the grid upstream and downstream ten times the hydraulic diameter. At the inlet the analytical 1/7-power law turbulent profile is given as velocity boundary condition, while at the outlet an outflow gradient boundary condition is used. The time step size was chosen to fulfill the CFL criterion CFL < 1 for all simulations. The Reynolds number as well as the Strouhal number are defined with the average flow velocity u. ¯ Note that the width of the bluff body d represents the relevant characteristic length scale. Red =

u¯ · d , ν

Str =

f ·d u¯

(13)

2.3 Stability analysis To confirm theoretically the numerical results of the Strouhal-number dependence versus Reynolds-number, absolutely and convectively unstable wake flow is investigated in two steps. Firstly, a local linear stability analysis of several time averaged velocity profiles at different distances from the trailing edge of the bluff body is performed under locally parallel flow assumption. Secondly, by means of a

In order to perform a local linear stability analysis, a basic state has to be defined. Here, the basic state is obtained by time averaging several periods of the saturated unsteady wake flow resulting from the numerical flow simulations. A more natural choice of the basic state is the state of the wake at the beginning of the linear growth process of disturbances. This is called the quasi-steady state. It consists of a pair of vortices attached to the bluff body (Hannemann et al. [6]). Since experimental data of the quasi-steady state is currently not available, it is computed by a transient simulation. However, our stability investigations based on numerically determined quasi-steady states yield results that can’t be used to apply an appropriate global criterion to predict the shedding frequency. It is concluded that due to numerical dissipation the quasi-steady states obtained by numerical simulation with Star CD© do not represent sufficiently accurate physical states. Indeed, the quasi steady states which develop during simulation are very short living and sensitive. diffusion has to be avoided when simulating separated flows. The analysed basic state consists of a velocity profile u0 (z) in the x, z-plane in the middle of the bluff body (x stands for the main flow direction and z for the direction perpendicular to the longitudinal axis of the bluff body). Thereby u0 (z) is the mean flow (averaged over several periods) in the saturation state obtained by RANS simulations. So the basic state is a time averaged (of turbulent fluctuations) turbulent state. The application of the stability analysis to turbulent basic states is presented in Oertel [16]. The disturbances superimposed on the basic state are described by a complex perturbation stream function (x, z, t) which is decomposed in normal modes ˆ (x, z, t) = (z) exp (i(ax − ωt)),

(14)

ˆ with complex amplitude function (z), wave number a and frequency ω. This leads (for a detailed derivation, see Schlichting and Gersten [22]) to an eigenvalue problem given by the Orr-Sommerfeld equation

ˆ d 2 d 2 u0 2 ˆ ˆ − a · −a· · 2 dz dz2 4 2ˆ ˆ i d 2 d 4 ˆ + · − 2 · a · + a · = 0, Red dz4 dz2

(a · u0 − ω) ·

(15)

together with boundary conditions at the wall of the pipe: ˆ wall = 0,

ˆ wall d = 0. dz

(16)


83

Fig. 5 Impulse response of a one-dimensional perturbation

This eigenvalue problem defines an implicit relation between the wave number a and the frequency ω which is called the dispersion relation D(a, ω) = 0.

(17)

To determine whether a flow is absolutely or convectively unstable, a theoretical framework from plasma physics (Briggs [3]) is used. This concept is applied to hydrodynamic stability analysis in Hannemann et al. [6], Koch [11], Monkewitz [13], Monkewitz [14], Triantafyllou et al. [24], Yang [26], Oertel et al. [19], Oertel [17] and is reviewed in Oertel [15]. The main idea is to analyse the asymptotic behaviour of the impulse response of the flow for large times. An ideal impulse excites modes of all frequencies and all wave numbers and the flow amplifies those modes or a band of modes that are unstable. The results of an analysis of the time-asymptotic impulse response according to Monkewitz [13], Hannemann et al. [6] and Oertel [15] can be summarised as follows: On a ray xt = const. in the (x, t)-plane (Fig. 5), the mode with wave number a ∗ , given by the group velocity dω ∗ x da (a ) = t , is amplified with the temporal growth rate σi = ωi (a ∗ ) − ai∗

dω ∗ (a ). da

(18)

Subscripts i and r denote the imaginary and the real part of complex quantities. After long times (time-asymptotic behaviour), the wave packet generated by an impulse at (x, t) = (0, 0) lies within a wedge formed by the two rays of vanishing amplification σi . Inside the wedge disturbances grow (σi > 0) and outside they decay (σi < 0) respectively. Growth and decay are exponentially in time. The two different impulse responses shown in Fig. 5 differ in the position of the ray x x 0 t = 0. The wave number a along the ray t = 0, with group dω 0 velocity da (a ) = 0, is amplified with the growth rate σi0

= ωi0

with ω = ω(a ) 0

0

(19)

called the absolute growth rate. In general, those points a 0 and ω0 where dω da = 0 are branch point singularities of the

dispersion relation. The location of these branch points in the complex ω-plane determines the absolute or convective instability of the analysed velocity profile. If ω0 lies in the upper half of the complex ω-plane (ωi0 > 0) an unstable velocity profile is absolutely unstable, whereas it is convectively unstable if ω0 lies in the lower half plane (ωi0 < 0) (for further details about the relevancy of the found branch point singularities for absolute and convective instabilities, see Monkewitz [13]). In the complex a-plane, the branch point singularities are saddle points. After having determined the absolute or convective character of instability of several velocity profiles in the wake of the bluff body, a global criterion is applied to these local stability results in order to predict the shedding frequency. According to the classification in Monkewitz [14], Oertel [15], the bluff body has a wake of type AB (for absolutebounded). The wake contains a region of absolute instability bounded by a solid boundary on one side (corresponding to the trailing edge of the bluff body) and a convective unstable region on the other side. An appropriate criterion for flows of type AB is Koch’s mode selection criterion (Koch [11]). Koch suggests that the shedding frequency in the saturation state (even for supercritical Reynolds numbers, see Hannemann et al. [6] and Oertel [15]) is determined by a local resonance occurring at the point xk which separates the absolutely and the convectively unstable regions. According to Koch’s criterion, the vortex shedding frequency of the wake flow is the real part of the branch point ω0 at the transition point xk from locally absolutely unstable to locally convectively unstable. The dimensionless shedding frequency, the Strouhal number Str, is given by the relation Str =

ωr0 (xk ) . 2π

(20)

2.3.2 Numerical analysis In order to investigate the absolute and convective instability of a velocity profile, the dispersion relation (17) has to be solved for complex wave numbers and complex frequencies. Following the procedure in Yang [26], the saddle point a 0 can be identified by first solving the dispersion relation

84


are defined as the negative sum of the off-diagonal elements of the corresponding row. This ensures that the numerical differentiation of a constant function yields the null vector. The numerical analysis (including analytical transformations) is performed with MATLAB© and MAPLE© . The implemented codes are verified in two steps. Firstly, eigenvalues of the plane Poiseuille flow are computed and correspond with results in Orszag [20] and Herbert [8]. Finally, the existence of absolutely unstable velocity profiles of a cylinder wake is shown, in agreement with results in Yang [24].

3 Results Fig. 6 Eigenvalues of the symmetric mode (Kármán mode) for a velocity profile at a distance of x/d ≈ 0.4 (with d = 3.5 · 10−2 m) from the bluff body, Red = 1916

for wave numbers a lying on an orthogonal grid in the complex a-plane. Then lines of constant ωr and ωi of the corresponding complex eigenvalues ω of the symmetric modes (Kármán modes with maximal ωi ) are plotted. One of these plots for a velocity profile at a distance of x/d ≈ 0.4 (with d = 3.5 · 10−2 m) from the bluff body and a Reynolds number Red = 1916 is shown in Fig. 6. At the branch point singularity, ωi ≈ 0.3. Since ωi is positive, the analysed velocity profile is absolutely unstable. The numerical solution of the dispersion relation is performed by the Chebyshev collocation method introduced in Gottlieb et al. [5]. For the computation, the Gauss-Lobatto collocation points kπ , k = 0, . . . , N, −1 ≤ ζk ≤ 1 (21) ζk = cos N are used. The interval of computation [−1, 1] has to be transformed into the interval [− D2 , D2 ], where D is the local diameter of the pipe. The computational accuracy of high derivatives (e.g. fourth derivative in Orr-Sommerfeld equation) is improved by a procedure presented in Don [4], where a grid mapping (derived in Kosloff [12]) that reallocates the collocation points is applied. Thus, the complete transformation of the collocation points is D arcsin (αζk ) · , 2 arcsin (α) D D ζk ∈ [−1, 1], zk ∈ − , , α = cos (0.5). 2 2

zk =

(22)

The value of the mapping parameter α is proposed in Hesthaven et al. [9]. An additional improvement in the accuracy of high derivatives is obtained by using a computation method for the diagonal elements of the differentiation matrices suggested in Bayliss et al. [2]. The diagonal elements

3.1 Model validation To validate the numerical model, the results of numerical simulations were compared with the results of measurements. For this purpose an experimental rig was build. The validation measurements were performed for a geometrically similar (d/D = 0.27) but smaller flow meter due to restrictions in laboratory space. Experiment fluid was air at ambient pressure and temperature. The measurements were done in a Reynolds number range of 10,000 < Red < 28,000. The necessary pressure was provided by a radial flow pump. The mass flow through the pipe was measured using an impeller flow meter. The shedding frequency was evaluated from the raw data of the vortex shedding flow meter. The CAD geometry provided a basis for the numerical model. Grid, boundary conditions and numerical setup for the validation case are the same as shown in Sect. 2.2.2. The frequency of the vortex shedding out of the numerical simulations was appraised and compared with the results of the measurements. Figure 7 shows the comparison between measured and simulated Strouhal number over Reynolds number. There is very good agreement in the perfectly linear Str–Re relation over the measurement range with a slight offset which is within the experimental uncertainty. The increase in Strouhal number for low Reynolds numbers could not be observed in the validation experiments since the signal to noise ratio was too high for Reynolds numbers smaller than 10,000. 3.2 Flow simulation and experiments 3.2.1 Unsteady flow configuration The numerical simulations reveal a complex flow structure around and behind the bluff body (see Fig. 8). At the junction of the bluff body with the pipe wall a horse shoe vortex


85

Fig. 9 Strouhal numbers for different turbulent intensities at the inlet boundary Fig. 7 Results of the validation

Fig. 10 Comparison between the Strouhal numbers determined by the numerical flow simulation and by the stability analysis

Fig. 8 Three dimensional flow in the vortex shedding device at 4 time steps of one period T0

can be detected. The size of it depends on the Reynolds number. In the center plane of the bluff body a periodic vortex shedding similar to the flow structure of a Kármán vortex street occurs. Towards the upper and lower pipe walls the influence of the horse shoe vortex and the smaller distance to the side wall change the flow characteristics and creates a complex periodic three dimensional vortex shedding structure. Figure 8 shows the periodic three dimensional flow structure by visualising the isovolume of the vorticity. The simulations with varying Reynolds numbers were done with a constant turbulent intensity (Tu = 5%) at the inlet. To determine the influence of the turbulent intensity simulations at Reynolds number Red = 13,000 for Tu = 1%, 3%, 5% and 10% were performed (see Fig. 9).

The results of this study show a minor influence of turbulent intensity on the frequency of the vortex shedding. Furthermore the influence of different geometrical setups upstream of the vortex shedding device like an elbow pipe in different distances in front of the device or different muff connections were investigated. In all the simulations, the same flow structure was found. 3.2.2 Strouhal number dependence The results of the numerical simulation show an increase of the Strouhal number with decreasing Reynolds number, starting at a Reynolds number of Red = 4800. Figure 10 shows the Strouhal-Reynolds number dependence determined by the numerical flow simulation in comparison with the results of the stability analysis. Additionally the experimental results are displayed in the same illustration. Figure 10 shows an increase of the Strouhal number for small Reynolds numbers in the simulations. The same tendency can be observed in the experimental data. The abso-

86

lute discrepancy arises from the tolerances in the geometry of the bluff body in the experimental setup. The increase of the Strouhal number for small Reynolds numbers is also confirmed by the results of the stability analysis. Differences in the absolute values of the Strouhal numbers for higher Reynolds numbers are due to the constraints of the local stability analysis.

4 Discussion and conclusions The results show that it is possible to predict the overall vortex shedding behaviour in its complexity using a simple RANS representation of turbulence without the need of the additional numerical overhead that comes with the higher order turbulence modelling like e.g. Large Eddy Simulation (LES) or Detached Eddy Simulation (DES). The results of the measurements as well as the results of the numerical simulations show an increase of the Strouhal number for small Reynolds numbers. This result is supported by the results of the stability analysis. At first glance the vortex shedding device is based on the effect of a Kármán vortex street which shows a decrease of Strouhal number for small Reynolds numbers. So it is astonishing that the vortex shedding device shows a different behaviour. But in a Kármán vortex street there is always a laminar upstream flow. In the presented case the upstream pipe flow is in a turbulent state. Besides, the observed flow in the vortex shedding device exhibits a completely three dimensional behaviour. So the results of the study show that the flow field in the vortex shedding device is not comparable with the wake flow of a Kármán vortex street. The three dimensional, turbulent vortex shedding behind a bluff body in a pipe is a new phenomenon.

References 1. Alsalihi Z, Debatin K, Maul J, Ohle F, Oertel H Jr (1998) Numerical computations of internal flows over bluff bodies. Z Angew Math Mech 79:661–662 2. Bayliss A, Class A, Matkowsky BJ (1994) Roundoff error in computing derivatives using the Chebyshev differentiation matrix. J Comput Phys 116:380–383 3. Briggs RJ (1964) Electron-stream interactions in plasmas. MIT Press, Cambridge, Massachusetts 4. Don WS, Solomonoff A (1997) Accuracy enhancement for higher derivatives using Chebyshev collocation and a mapping technique. SIAM J Sci Comput 18(4):1040–1055

Forsch Ingenieurwes (2010) 74: 77–86 5. Gottlieb D, Hussaini MY, Orszag SA (1984) Theory and applications of spectral methods. In: Voigt RG et al (eds) Spectral methods for partial differential equations. SIAM, Philadelphia 6. Hannemann K, Oertel H Jr (1989) Numerical simulation of the absolutely and convectively unstable wake. J Fluid Mech 199:55– 88 7. Heisenberg W (1922) Absolute dimensions of Karman vortex motion. Phys Z 23:363–366 8. Herbert T (1977) Die neutrale Fläche der ebenen PoiseuilleStrömung. Habilitationsschrift, Fakultät für Luft- und Raumfahrttechnik, Universität Stuttgart 9. Hesthaven JS, Dinesen PG, Lynov JP (1999) Spectral collocation time-domain modelling of diffractive optical elements. J Comput Phys 155:287–306 10. Huerre P, Monkewitz PA (1990) Local and global instabilities in spatially developing flows. Annu Rev Fluid Mech 22:473–534 11. Koch W (1985) Local instability characteristics and frequency determination of self-excited wake flows. J Sound Vib 99(1):53–83 12. Kosloff D, Tal-Ezer H (1993) A modified Chebyshev pseudospectral method with an O(N −1 ) time step restriction. J Comput Phys 104:457–469 13. Monkewitz PA, Nguyen LN (1987) Absolute instability in the near-wake of two-dimensional bluff bodies. J Fluids Struct 1:165– 184 14. Monkewitz PA, Sohn KD (1986) Absolute instability in hot jets and their control. AIAA Paper Nr. 86-1882 15. Oertel H Jr (1990) Wakes behind blunt bodies. Annu Rev Fluid Mech 22:539–564 16. Oertel H Jr (1995) Bereiche der reibungsbehafteten Strömung. Z Flugwiss Weltraumforsch 19:119–128 (37. Ludwig Prandtl Gedächtnis Vorlesung, GAMM Tagung, Braunschweig) 17. Oertel H Jr (2009) Fluidmechanical instabilities. In: Prandtl Essentials of Fluid Mechanics, 3rd edn. Springer, New York 18. Oertel H Jr, Böhle M, Dohrmann U (2008) Strömungsmechanik, 5th edn. Vieweg+Teubner, Wiesbaden 19. Oertel H Jr, Delfs J (2005) Strömungsmechanische Instabilitäten. Universitätsverlag, Karlsruhe 20. Orszag SA (1971) Accurate solution of the Orr-Sommerfeld stability equation. J Fluid Mech 50(4):689–703 21. Roshko A (1954) On the development of turbulent wakes from vortex streets. Technical report, NACA report 22. Schlichting H, Gersten K (2006) Grenzschicht-Theorie, 10th edn. Springer, Berlin 23. Strouhal V (1878) Über eine besondere Art der Tonerregung. Annalen der Physik und Chemie 24. Triantafyllou GS, Triantafyllou MS, Chryssostomidis C (1986) On the formation of vortex streets behind stationary cylinders. J Fluid Mech 170:461–477 25. v. Kármán T (1912) Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfährt. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Math Phys Klasse, Berlin 26. Yang X, Zebib A (1989) Absolute and convective instability of a cylinder wake. Phys Fluids A 1(4):689–696