Experiments in Fluids (2005) 39: 771–783 DOI 10.1007/s00348-005-0010-z
R ES E AR C H A RT I C L E
O. O¨zcan Æ K. E. Meyer Æ P. S. Larsen
Measurement of mean rotation and strain-rate tensors by using stereoscopic PIV
Received: 21 June 2004 / Revised: 31 May 2005 / Accepted: 31 May 2005 / Published online: 12 August 2005 Springer-Verlag 2005
Abstract A technique is described for measuring the mean velocity gradient (rate-of-displacement) tensor by using a conventional stereoscopic particle image velocimetry (SPIV) system. Planar measurement of the mean vorticity vector, rate-of-rotation and rate-of-strain tensors and the production of turbulent kinetic energy can be accomplished. Parameters of the Q criterion and negative k2 techniques used for vortex identification can be evaluated in the mean flow field. Experimental data obtained for a circular turbulent jet issuing normal to a crossflow in a low speed wind tunnel for a jet-to-crossflow velocity ratio of 3.3 are presented to show the applicability of the proposed technique. The results reveal the presence of a secondary counter-rotating vortex pair (SCVP) which is located within the jet core and has a sense of rotation opposite to that of the primary one (PCVP). Consistency of the measurements is verified by the agreement of data obtained in two perpendicular planes. Accuracy of the data is discussed and algebraic relations for some measurement uncertainties are presented.
List of symbols D Jet (pipe) diameter G*ij Normalized mean velocity gradient tensor (GijD/ Uc) eijm Permutation symbol Normalized turbulent kinetic energy ðu0i u0j =2U2c Þ k* N Ensemble size O. O¨zcan (&) Department of Mechanical Engineering, Yildiz Technical University, 80626 Istanbul, Turkey E-mail:
[email protected] K. E. Meyer Æ P. S. Larsen Department of Mechanical Engineering, Technical University of Denmark, Building 403, 2800 Kgs. Lyngby, Denmark
P* q* Q* R*ij S*ij s*ij T*ij Uc Ui u¢i Urms u0i u0j W xi
Normalized production rate of turbulent kinetic energy (PD/U3c ) pffiffiffiffiffiffiffiffiffiffi Normalized speed ð Ui Ui =Uc Þ Normalized second invariant of the mean velocity gradient tensor (QD2/U2c ) Normalized mean rate-of-rotation tensor (RijD/ Uc) Normalized mean rate-of-strain tensor (SijD/Uc) Fluctuating rate-of-strain tensor Normalized truncation error in Gij, Sij and Rij (TijD/Uc) Crossflow velocity Mean velocity components Fluctuating velocity components Root-mean-square of velocity components Reynolds stress tensor (divided by density) Bulk velocity of jet (pipe) flow Spatial coordinates x,y,z
Greek letters d( ) Spacing between velocity vectors in the ( ) direction D( ) uncertainty in ( ) * Normalized dissipation rate of turbulent kinetic energy ( D/U3c ) ki Normalized eigenvalues (kiD2/U2c ) of [Sik Skj+Rik Rkj] m Kinematic viscosity x* Normalized magnitude of mean vorticity vector qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 R2zy þ R2xz þ R2yx D=Uc Þ Xi Normalized mean vorticity components (WiD/Uc)
1 Introduction The velocity gradient (rate-of-displacement) tensor is an important quantity in the analysis of problems in Fluid Mechanics. A knowledge of the decomposition into rotation- and strain-rate tensors is necessary for modeling turbulence, validating constitutive relations,
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identifying vortices and understanding physical structure of a flow. Measurement of mean rotation and strain-rate tensors is valuable especially in flows involving steady or quasi-steady reverse and secondary flow regions where vortices may be difficult to detect because of mean shear. However, measurement of the velocity gradient tensor is a challenging task which requires acquisition of all the three components of the velocity vector at a number of points that are slightly displaced along three mutually perpendicular directions. Wallace and Foss (1995) describe the difficulties in measuring the rate-of-rotation tensor in turbulent flows. The hot-wire anemometry and the Laser Doppler Anemometry (LDA) are, in principle, capable of measuring the velocity gradient tensor at a point. Andreopoulos and Honkan (1996) and Zhou et al. (2003) describe hotwire probes (consisting of 9 and 8 wires, respectively) capable of measuring the instantaneous velocity gradient tensor. However, long data acquisition times may render the point-based measurement techniques impractical. The holographic particle image velocimetry (PIV) technique (Meng and Hussain 1993, Tao 2000), which measures all three components of the velocity vector in a volume, readily produces the velocity gradient tensor. However, holographic PIV is a relatively new technique based on photographic recording, which requires long data processing times and large computer memory that severely limit the number of realizations used in data averaging. The scalar imaging velocimetry (Dahm et al. 1992) and the three-dimensional particle tracking velocimetry (Nishino et al. 1989) also produce all components of the instantaneous velocity gradient tensor in a volume. The latter technique generally provides poor spatial resolution whereas the former is applicable to high Schmidt number flows. The dual-plane stereo PIV (Hu et al. 2001, Mullin and Dahm 2004a, b) and highspeed scanning stereoscopic PIV (Hori and Sakakibara 2004) are probably the best techniques available today for measuring the instantaneous velocity gradient tensor. However, these systems have a high cost and additional alignment or calibration requirements. The present paper describes a simple technique for measuring the mean velocity gradient tensor in a plane by using a conventional stereoscopic particle image velocimetry (SPIV) system that records all three components of the instantaneous velocity vector. The technique involves taking PIV data in two or three closely spaced parallel planes at different times and is applicable to a broad range of three-dimensional flows (compressible, incompressible, steady, unsteady, laminar, turbulent). All components of the mean velocity gradient tensor are calculated by using finite differences. In general, the method requires large samples and good spatial resolution. Planar measurements of the mean vorticity vector, rotation- and strain-rate tensors and the production of turbulent kinetic energy can be readily accomplished. Invariants and eigenvalues of the mean velocity gradient tensor (such as the Q parameter which is useful for vortex identification, Hunt et al.
1988) can be evaluated. Dissipation rate of the turbulent kinetic energy can also be estimated if the interrogation area size is sufficiently small in a general, non-isotropic three-dimensional flow. Despite the fact that planar and non-intrusive measurements of the invariants of the velocity gradient tensor and the production rate of turbulent kinetic energy (using its exact definition) can also be accomplished by the more sophisticated techniques mentioned in the previous paragraph; such data are rarely reported in the literature and limited to intermediate and small scales of turbulence (Mullin and Dahm 2004a, b, van der Bos et al. 2002). The proposed technique has the potential of augmenting the usefulness and impact of stereoscopic PIV systems which are becoming increasingly available in research laboratories worldwide. An important limitation of the proposed technique is its inability to measure all components of the fluctuating velocity gradient tensor. The technique is applicable to three-dimensional flows only and does not provide more information than a conventional stereoscopic PIV system when the mean flow is homogeneous in one or two directions. Experimental data obtained for a jet in crossflow are presented to show the applicability of the proposed technique. The test flow, which is highly turbulent, vortical and three-dimensional, involves flow reversals in all three directions and can be best studied by a directionally sensitive non-intrusive technique. Measurement of the production and dissipation rates of turbulent kinetic energy and detection of vortices in the mean flow field are necessary for a better understanding of the turbulent flow structure. Data averaged over a thousand vector maps along the intersection of two perpendicular planes are compared with each other to assess the consistency of the technique. Some earlier results on the successful implementation of the technique was presented in Meyer et al. (2001) which reported that the deviator of the velocity gradient tensor is not aligned with the deviatoric Reynolds stress (i.e., the gradient-transport approximation is not valid) for a jet in crossflow.
2 Velocity gradient tensor The mean velocity gradient tensor Gij is defined by Gij ¼
@Ui @xj
ð1Þ
where Ui are the mean velocity components and xj are the space variables in a Cartesian coordinate system, i and j (=1,2,3) being free indices. Gij can be decomposed into the summation of the symmetrical rate-of-strain (deformation) tensor Sij and the skew-symmetrical rateof-rotation (spin) tensor Rij which are given by (Tennekes and Lumley 1972) 1 @Ui @Uj Sij ¼ ð2Þ þ 2 @xj @xi
Rij ¼
1 @Ui @Uj 2 @xj @xi
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ð3Þ
The mean vorticity vector Wi, which is twice the angular velocity vector, is related to the rate-of-rotation tensor by Xi ¼ eijm Rmj
ð4Þ
where eijm is the permutation symbol. The strain-rate tensor Sij can be written as the summation of deviatoric and isotropic tensors which are measures of the rate-ofdistortion and the rate-of-dilatation (volumetric expansion), respectively. The rates of production and viscous dissipation of turbulent kinetic energy (P and ), which are important parameters in most models of turbulence, are related to the mean and fluctuating rate-of-strain tensors Sij and s¢ij, respectively by (Tennekes and Lumley 1972) P ¼ u0i u0j Sij
ð5Þ
¼ 2ms0ij s0ij
ð6Þ
where overbar denotes ensemble or time averaging, m is the kinematic viscosity, u¢i are the fluctuating velocity components and u0i u0j is the Reynolds stress tensor (divided by density). The repeated indices i and j imply summation in Eqs. 5 and 6. The fluctuating rate-ofstrain tensor s¢ij is defined by an equation similar to Eq. 2 where Ui is replaced by u¢i. The Q criterion technique used in vortex identification evaluates the following scalar (Hunt et al. 1988) Q¼
Rij Rij Sij Sij 2
ð7Þ
which is the second invariant of the mean velocity gradient tensor. Jeong and Hussain (1995) report that the second largest eigenvalue of (SikSkj+RikRkj), which is named k2, is generally a better parameter than Q in identifying a vortex. High positive values of Q and large negative values of k2 identify vortical flow regions where the rotation rate dominates the strain rate in the mean flow field. In other words, the vortex detection criterion can resolve whether a certain vorticity is due to a vortex or shear. A conventional stereoscopic PIV system measures all three components of the instantaneous velocity vector in a plane. Averaging of PIV vector maps produces all three components of the mean velocity and six components of the Reynolds stress tensor. In-plane gradients of all velocity components can be calculated by using the simple central difference scheme which is accurate to second order. An important shortcoming of a conventional PIV system is its inability to measure the mean velocity gradients in the out-of-plane direction. This limitation can be overcome by acquiring stereoscopic PIV data in two or three closely-spaced parallel planes at different times. The gradients of all mean velocity components in the out-of-plane direction can be evaluated
by using the forward difference scheme which is accurate to first order when PIV data are obtained in only two closely-spaced parallel planes. Accuracy of the velocity gradients in the out-of-plane direction can be improved by taking data in three closely-spaced parallel planes and employing the central difference scheme. Once Gij is known, Sij, Rij, Wi, P and Q can be calculated from Eqs. 2, 3, 4, 5 and 7, respectively. Calculation of the dissipation rate of turbulent kinetic energy is not as straightforward as the parameters just mentioned and will be described in the next paragraph. All three components of the fluctuating velocity vector are calculated as the difference between the instantaneous and mean velocity vectors. The simple central difference scheme is employed to calculate the components of the fluctuating rate-of-strain tensor involving in-plane gradients of the fluctuating velocity components. However, since data in parallel planes are not obtained simultaneously, the three out-of-plane derivatives of the fluctuating velocity components cannot be measured. Yet, one of these can be derived from a knowledge of two in-plane gradients of the fluctuating velocity field which is divergence-free for incompressible flow. By neglecting the remaining two out-of-plane gradients in Eq. 6, one can obtain an estimate of the turbulent dissipation rate for a general (non-isotropic) three-dimensional flow. (The estimate could potentially be improved by simply using 9/7 times the seven recorded contributions on account of the approach to isotropy of small scales.) The value of is computed from Eq. 6 as a summation performed over the PIV vector maps (processed to yield the fluctuating rate-ofstrain tensor) of the ensemble. An accurate measurement of the dissipation rate of the turbulent kinetic energy requires a rather small interrogation area size and small separation distance between the parallel laser planes (both smaller than a few Kolmogorov viscous length scales) as will be discussed later.
3 Experimental method and set-up The velocity gradient tensor was measured in the flow field of a turbulent non-buoyant jet in crossflow in a low-speed wind tunnel with test-section dimensions of 300 by 600 mm at the Technical University of Denmark. Figure 1 gives a schematic description of the experimental set-up. The jet issued 1,350 mm downstream of the leading edge of a flat plate insert which had a length of 1,950 mm and a width of 598 mm through a circular pipe of diameter D=24 mm with a bulk velocity of W=4.95 m/s. The crossflow velocity along the flat plate was Uc=1.50 m/s, producing a jet-to-crossflow velocity ratio W/Uc of 3.3. The distance between the flat plate and the opposite sidewall was 264 mm. Shop air was used to produce the jet, issuing at the end of a 2.5-mlong Perspex pipe of the same diameter which was fed through a 2-m-long hose-pipe. The thickness of the
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Fig. 1 Experimental set-up (schematic)
turbulent boundary-layer on the flat plate approaching the jet exit was 70 mm. The Reynolds number based on the jet diameter D and the crossflow velocity Uc was 2400 nominally. Special consideration was given to establish fully developed and self-preserved incoming turbulent flows in the pipe and on the flat plate, respectively. The results are discussed in reference to the (x,y,z) Cartesian coordinate system whose origin is at the center of the pipe at the jet exit as shown in Fig. 1. More information on the flow conditions can be found in Meyer et al. (2001), Meyer et al. (2002) and O¨zcan and Larsen (2003) who reported some early findings. Pedersen (2003) studied the large scale structures in the flow field by using the Proper Orthogonal Decomposition (POD) analysis. The PIV system shown in Fig. 1 consisted of two Kodak Megaplus ES 1.0 cameras with 60 mm Nikon lenses mounted in Scheimpflug condition (angle between cameras was approximately 80). A double cavity NdYAG laser delivering 100 mJ light pulses was employed to create a light sheet which was 1.5-mm-thick. Cameras and light sheet optics were mounted on the same traverse mechanism in order to accurately displace the measurement plane. Separate atomizers were used to seed the jet and the crossflow with 2–3 lm droplets of glycerol containing 15% water. The size of the seed particles were measured by an APS TSI 3320 time-of-flight spectrometer. The PIV images were monitored as the seeding rates were adjusted to produce approximately the same particle seeding level (density) in the jet and crossflow. The system was controlled by a Dantec PIV2100 processor and the data were processed with Dantec Flowmanager version 3.4 using adaptive velocity correlation which is suitable for flow fields having large velocity gradients. The images were processed by using refinement steps from an initial resolution of 64·64 pixels to a final resolution of 32·32 pixels per interrogation area. Between each refinement step, the vector maps were filtered to remove spurious vectors by replacing them with a weighted-average in the neighborhood of 3·3 vectors. An overlap of 25% was used
between interrogation areas. A calibration target aligned with the light sheet plane was used to obtain the geometrical information required for the reconstruction of the velocity vectors. The calibration images were recorded for five planes separated by displacements of 0.5 mm and the reconstruction was performed by using a linear transformation. Image maps were recorded with an acquisition rate of 0.5 Hz to yield N=1,000 instantaneous vector maps used to calculate moments. A low data acquisition rate was chosen to ensure the statistical independence of the moments. Two different configurations of the cameras and the light sheet were used to obtain data in constant y (as shown in Fig. 1) and constant z planes, having fields of view of 108 by 86 mm and 65 by 53 mm (optical magnifications of 0.107 and 0.174), respectively. Measurements in a constant z plane revealed a cross-section of the jet that had a limited region of variation compared to the data obtained in a constant y plane. Therefore, a larger magnification was chosen for the constant z plane data without any loss of information. For both planes, the velocity vector maps contained 33 by 37 vectors. Therefore, the linear dimensions of the interrogation areas varied between 1.5 and 3.4 mm. In order to minimize typical PIV errors, the design rules of Keane and Adrian (1990) were followed, i.e., the particle image density was larger than 10, the maximal value of the inplane displacement was generally less than 25% of the interrogation area size and the out-of-plane displacements were sufficiently small. The mean flow field has significant variations in a region defined by 0
