PIV-based evaluation of pressure fluctuations within the turbulent boundary layer

Transcription

1 PIV-based evaluation of pressure fluctuations within the turbulent boundary layer Sina Ghaemi 1,*, Daniele Ragni 1, Fulvio Scarano 1 1: Department of Aerodynamics, Delft University of Technology, Kluyverweg 1, 2629 HS, Delft, The Netherlands * correspondent author: s.ghaemi@tudelft.nl Abstract The unsteady pressure field is obtained from time-resolved tomographic particle image velocimetry (Tomo-PIV) measurement within a fully developed turbulent boundary layer at free stream velocity of U = 9.3 m/s and Re θ = 24. The pressure field is evaluated from the velocity fields measured by Tomo-PIV at 1 khz invoking the momentum equation for unsteady incompressible flows. The spatial integration of the pressure gradient is conducted by solving the Poisson pressure equation with fixed boundary conditions at the outer edge of the boundary layer. The PIV-based evaluation of the pressure field is validated against simultaneous surface pressure measurement using calibrated condenser microphones mounted behind a pinhole orifice. The comparison shows agreement between the two pressure signals obtained from the Tomo-PIV and the microphones with a cross-correlation coefficient of.6 while their power spectral densities (PSD) overlap up to 3 khz. The use of the Tomo-PIV system with the application of three-dimensional momentum equation shows higher accuracy compared to the planar version of the technique. The combination of a correlation-sliding-average technique, the Lagrangian approach to the evaluation of the material derivative and the planar integration of the Poisson pressure equation results in the best agreement with the pressure measurement of the surface microphones. 1. Introduction The unsteady pressure field within the turbulent boundary layers is the source of engineering problems such as the cabin noise and structural vibration of aircrafts and high-speed trains (Willmarth 1975). These pressure fluctuations are also believed to result in acoustic scattering upon traveling over the trailing-edge of aircraft wing or high-lift devices (Ffowcs Williams & Hall 197). However, the characterization of the unsteady pressure field in a turbulent boundary layer has considerably suffered from the limitations of the measurement techniques especially their intrusive and point-wise nature. The intrusive nature of the available pressure measurement techniques has limited most of the investigations to the measurement of the wall pressure. To the authors knowledge, only Tsuji et al. (27) conducted high-frequency measurement of the pressure inside a turbulent boundary layer using a static pressure probe. In spite of that the investigation of the relation between the pressure field and turbulent flow structures was difficult due to the point-wise nature of the measurement. As a result, most of the previous investigations have been conducted using wall pressure measurement (Johansson et al. 1987) or numerical simulations (Chang et al. 1999, Kim et al. 22). Such simulations provide access to three-dimensional (3D) pressure and velocity field but extension to high Reynolds numbers and more complex geometries is still challenging. The investigation of passive and active methods for surface pressure control also benefits from the development of 2D or 3D non-intrusive pressure measurement techniques with appropriate spatio-temporal resolution. The evaluation of the pressure field from the application of the Navier-Stokes equation to the measured velocity field by PIV has already been considered in several studies. Gurka et al. (1999) solved the Poisson equation to determine the steady pressure field in a laminar pipe flow and a turbulent impinging jet. Baur and Köngeter (1999) tackled the more complicated situation of the unsteady pressure field generated by a wall-mounted obstacle and spatially integrated the unsteady Navier-Stokes. Hosokawa et al. (23) applied the technique to a two-phase flow and calculated the pressure distribution around bubbles within a laminar flow. Fujisawa et al. (24) evaluated - 1 -

2 the steady pressure field and investigated the interaction of two parallel plane jets. Liu and Katz (26) proposed the use of a four-exposure system to evaluate the material derivative of the velocity. Charonko et al. (21) assessed the effect of different approaches such as integration method, grid resolution, sampling rate. The pressure field of a cavity flow and the subsequent acoustic emissions have been evaluated by Koschatzky et al. (211) using time-resolved 2C-PIV. De Kat and van Oudheusden (211) also evaluated the pressure field behind a square cylinder using Stereoscopic PIV and Tomo-PIV and discussed the effect of measurement parameters on the resulting accuracy. As this brief summary demonstrates, the application of PIV-based pressure evaluation to the turbulent boundary layers remains unexplored and most of the previous works were concentrated on 2D fluctuations of large vortical structures from periodical or quasi-periodical vortex shedding at low frequencies. A turbulent boundary layer is a more extreme situation due to small 3D velocity fluctuations and the wide range of length and time scales. The present work aims at obtaining the unsteady pressure field within a turbulent boundary layer from time-resolved 3D velocity fields measured by Tomo-PIV in a thin-volume configuration. Emphasis is placed on the validation and characterization of the PIV-based pressure field obtained from the integration of the Poisson equation in comparison to the surface pressure fluctuations measured by the condenser microphones. 2. Experimental apparatus Flow setup The experiments were performed in an open test-section vertical wind tunnel with a circular crosssection of.6 m diameter. A flat plate of 2 m length with an elliptical leading-edge and a sharp symmetric trailing-edge is used to obtain a fully developed turbulent boundary layer. The plate spanned the entire test section and was installed at zero angle-of-attack. Spanwise uniform laminar-to-turbulent transition of the boundary layer is forced at 15 mm downstream of the leading edge by applying a 12 mm long strip of sparsely spread carborundum particles of.8 mm grain size on both sides of the plate. The flat plate is made of chipboard except the aluminum replaceable part where the microphones are installed. The measurement region is located 1.5 m downstream of the leading-edge where the boundary layer is fully developed. The turbulent boundary layer is characterized by ensemble averaging the cross-correlation maps of 2C-PIV images (Meinhart et al. 2). The parameters are summarized in Table 1. Parameter Units U m/s 9.3 δ 99 (δ) mm 3.9 θ mm 3.8 Re δ 19,6 Re θ 2,4 Re τ 77 u τ m/s.365 λ µm 4 H 1.39 Table 1. Boundary layer parameters at the measurement location. Wall pressure measurement The pressure fluctuations at the wall have been measured using two electret condenser Sonion 81T microphones. This instrument can measure from 1 to 2, Hz while it offers a flat frequency response from 25 Hz to 75 Hz. The microphone has a sensitivity of db at 1 khz that is equivalent to 21 mv/pa and the maximum noise level is 28 db which is equivalent to.5 mpa rms pressure fluctuation in the range of Hz. The analog signal is conditioned - 2 -

3 by a low-pass filter starting at 1.6 khz followed by an amplifier. The signals are sampled at 5 khz using a National Instrument NI-9215 data acquisition system with 16 bits resolution. The data acquisition is synchronized with the PIV image recording. The microphones are placed behind a pinhole orifice of d =.2 mm (5λ) diameter. The pinhole orifice prevents attenuation of the high frequency fluctuations and also shields the microphones from heating by the laser light. The resonant frequency of the considered cavity (14 khz) is higher than the cut-off frequency of the analog filter (1.6 khz) to minimize the effect on the microphone response. Further assessment of the installed microphones and the pinhole orifice relative to a reference microphone was conducted confirming that the influence of the Helmholtz resonance is negligible. Amplitude calibration is conducted due to the applied signal amplification measuring conversion factor of the overall chain to be and V/Pa for microphones 1 and 2, respectively. Planar PIV A 2C-PIV system has been applied to characterize the turbulent boundary layer and also evaluate the pressure field. The illumination is provided by a diode pumped dual cavity Nd:YLF laser (Litron Lasers, LDY33HE). Each cavity delivers a pulse energy of 22.5 mj/pulse at 1 khz operation frequency. A laser sheet of approximately 1 mm thickness and 6 mm width was formed in the streamwise direction (parallel to the wall). A Photron Fast CAM SA1 camera with a 12-bit CMOS sensor of pixels (pixel pitch 2µm) equipped with a Nikon objective of 2 mm focal length set to aperture of f / 4 was used to record the light scattered by the illuminated tracers. The test section was seeded with 1µm SAFEX droplets. An ensembles of 5, double-frame recordings at 2 Hz with 7 µs pulse separation was acquired and analyzed using DaVis 8. (LaVision) with correlation ensemble averaging for the mean velocity distribution. The time-resolved measurements are acquired at 1, Hz in continuous mode and analyzed using the iterative window deformation technique (WIDIM) algorithm (Scarano and Riethmuller 2). Any velocity vector at or below the wall is imposed to be zero, which improves the stability of the iterative cross-correlation in the vicinity of the wall (Theunissen et al., 28). Because the measurements are performed at relatively high temporal rate (1, Hz), a short-time-averaging technique (sliding-average correlation) becomes possible. Such an averaging process is applied to the cross-correlation maps (Scarano et al. 21) and the sliding-averaged cross-correlation at time t is obtained by averaging over r (r = m+k+1) successive single-pair cross-correlations (R) following,, 1 = ++1,, + (1) where t is time between successive correlation maps expressed in the domain of spatial shifts x and y. The system parameters are shown in Table 2. Time-resolved Tomo-PIV The Tomo-PIV system (Elsinga et al. 26) consists of the same equipment (laser, camera and particle generator) as the detailed 2C-PIV system. However, the laser sheet is expanded in the z direction to a thickness of 3.5 mm and the imaging system consists of four Photron Fast CAM cameras subtending a solid angle around the normal to the illuminated plane as illustrated in Figure 1. The cameras are equipped with Scheimpflug adapters and 15mm Nikon objectives with aperture setting of f / 5.6. The illuminated volume has been seeded with 1µm droplets at a concentration of approximately 1 particles/mm 3, resulting in a particle image number density of approximately.8 particles per pixel (ppp). The mapping function of the tomographic system is - 3 -

4 obtained by a calibration procedure with a two-layered target. The residual system pointing accuracy is monitored with the Volume-Self-Calibration technique (Wieneke, 28). Minimum intensity subtraction was followed by the subtraction of a local neighborhood minimum for background light elimination. The image intensity was normalized by the average over a kernel of 51 pixel. The three-dimensional object reconstruction is obtained with the iterative application of the multiplicative algebraic reconstruction technique (MART, Herman & Lent 1976). The calculation is performed on a 48-cores PC with the LaVision software Davis 8.. The crosscorrelation is performed with multi-grid iterative volume deformation (VODIM, Scarano & Poelma 29). Also in this case the correlation sliding-average technique is applied, similarly to equation 1 to obtain the instantaneous velocity fields. The case of single-pair, two and three pairs (r =1, 2, 3) are considered here for comparison. The no-slip condition is applied at the wall to stabilize the calculation of the velocity closest to the surface (Theunissen et al., 28). The system parameters associated with the Tomo-PIV are summarized in Table 3. Figure 1. The experimental setup of the Tomo-PIV system measuring the turbulent boundary layer on a vertically mounted flat-plated. The volume is illuminated from downstream, parallel to the flat plate to reduce light scattering from the wall. The light passes through a knife-edge filter. Repetition rate 2 Hz 1, Hz FOV (x y) px mm δ 1.54δ px mm δ.9δ Digital resolution (S) 21.5 px/mm 21.3 px/mm M Number of samples 5, 1, Method ensemble correlation sliding-average correlation IA 4 4 px mm IA overlap 75 % 75 % Vectors per field px mm Table 2. System parameters of the 2C-PIV at recording rates of 2 and 1, Hz

5 Repetition rate Reconstructed volume (x y z) 1, Hz voxels mm 3 1.6δ 1.59δ.11δ Digital resolution (S) 18.3 voxels/mm M.37 Number of samples 1, Method IV sliding-average correlation voxels mm IV overlap 75 % Vectors per field Table 3. System parameters of the time-resolved Tomo-PIV. PIV-based pressure evaluation The evaluation of the instantaneous pressure field from the 3D velocity field is based on the use of the Navier-Stokes equation, assuming incompressibility and constant viscosity as given by = +, (2) where P represents the instantaneous pressure, U is the velocity vector, ρ density and µ is the dynamic viscosity. At the right-hand side the symbol D/Dt indicates the material derivative of the velocity field. The present work follows the approach proposed by Gurka et al. (1999) and de Kat et al. (211) whereby the problem is formulated in terms of the Poisson pressure equation obtained by applying the divergence operator to equation (2). The viscous effects are negligible in comparison to the inertia terms resulting in =.. (3) The right hand side of equation (3) requires calculation of the material derivative from the PIV vector fields. The main scheme for calculating the material derivative in this work is the Lagrangian method formulated as " # ' " ( $)* +(' " ( $,* +( $ -.. (4) In this equation, U p indicates the velocity of the fluid parcel travelling on a trajectory passing through the grid node at location x p at time t. The location of the fluid parcel at t +n t and t -n t is estimated by following the 3D velocity vector U(t ) forward and backward in time, respectively. After the right hand side of the equation is determined, the Poisson equation is solved with a - 5 -

6 second-order central difference scheme, yielding the pressure field. The required boundary conditions are of Dirichlet type for a known pressure outside of the boundary layer or of Neumann type at the wall, where no value on the pressure can be imposed, following equation (2). Having the right hand side of equation (6) from the time-resolved Tomo-PIV data, the integration can be conducted on a plane that involves integration of 3. Results /, = 2 2 = /, 3 4 4/ 56 /, , (5) Statistical properties of the TBL The 2C-PIV mean velocity profile that is obtained from the average of the correlation maps (detailed in the first column of Table 2) is shown in Figure 2(a). This high spatial resolution enables measurement within the inner layer and makes it possible to calculate the friction velocity (u τ ) by fitting a line to sublayer region of this profile. The obtained inner layer variables (u τ and λ) are used to scale the profiles of Figure 2(a) as y + = y/λ and U + = U/u τ. The slight discrepancy (about 1%) between the 2C-PIV and Tomo-PIV velocity in the outer layer is ascribed to a small difference of free stream velocity during the two experiments. The turbulent fluctuations of Figure 6(b) show the typical trend of <u 2 >, <v 2 > and <uv> within a zero pressure gradient turbulent boundary layer and a rather good agreement is observed between the two measurement systems. Figure4 (a) (b) U C-PIV Tomo-PIV <u i u j >/u τ <u 2 >, 2C <v 2 >, 2C <uv>, 2C <u 2 >, Tomo <v 2 >, Tomo <w 2 >, Tomo <uv>, Tomo y y + Figure 2. (a) turbulent boundary layer profile in semi-log scale measured using 2C-PIV and Tomo-PIV systems. The dashed lines show the law of the wall and log law. (b) Profiles of normal and Reynolds shear stress fluctuations. The pre-multiplied power spectral density (PSD) of the streamwise velocity fluctuations (u) at y + = 5 is shown in Figure 3. The streamwise velocity fluctuations (u) is obtained from the timeresolved 2C-PIV and Tomo-PIV data acquired at 1 khz and processed using the correlation sliding-average method by r = 1, 2 and 3 pairs of successive PIV correlations. The PSDs obtained from 2C-PIV and Tomo-PIV show slight difference at the low frequency range, which may be due to poor convergence at low frequencies. The difference decreases with increase of frequency and eventually the distribution reaches a minimum (f > 3kHz) ascribed to measurement noise. The amplitude of the noise level corresponding to the minimum in the pre-multiplied PSD is slightly higher for the 2C-PIV in comparison to the Tomo-PIV considering the same number of image pairs - 6 -

7 for the sliding-average correlation (same r). Therefore the random noise is mostly associated to the out-of-plane motion of particles in the planar measurement. The single-pair sliding-average (r = 1) shows energy distribution varying over approximately three decades up to a frequency of approximately 3 khz where the minimum is reached. The use of two and three pairs decreases the level of such a minimum by about one order of magnitude. However, this also results in an earlier drop of the spectral energy at lower frequency, which may indicate a reduction of the temporal resolution. Any further increase of the correlation-averaging kernel (number of pairs) will produce a visible reduction of the temporal resolution and the case of more than 3 pairs (r > 3) is not considered in the present work. The discrepancy between the 2C-PIV and Tomo-PIV is also observed to increase by using a higher number of correlation pairs. The lowest background noise corresponds to Tomo-PIV data processed by r = 3 which is about (m/s) 2 equivalent to a measurement noise of.2 m/s (.4 voxels, relative error of.2%) observed at 3.5 khz. This Tomo-PIV velocity field (r = 3) is used as the main data through the rest of the analysis for pressure evaluation. 1-1 f E uu, (m/s) C-PIV, r =1 2C-PIV, r =2 2C-PIV, r =3 Tomo-PIV, r =1 Tomo-PIV, r =2 Tomo-PIV, r = f, Hz Figure 3. The pre-multiplied PSD of streamwise velocity fluctuation measured using 2C-PIV and Tomo-PIV at y + =5 and processed by 1, 2 and 3 correlation pairs of the sliding-average algorithm. Validation of PIV-based pressure The pressure fluctuations (p) obtained from the Tomo-PIV system and the simultaneous surface pressure measurement using the electret microphone is shown over a 2 ms time span in Figure 4(a). Both of the signals are band-pass filtered between 25 to 35 Hz in which the lower limit of 25 Hz is applied due to the non-linear sensitivity of the microphones and the upper cut-off is due to the noise dominated region of the Tomo-PIV system with the 3 pair sliding-average correlation (r = 3) as observed in Figure 3. The material derivative of velocity is evaluated using the 3D Lagrangian method with steps of n = 3. A close agreement of the Tomo-PIV pressure and the microphone surface pressure measurement is observed in Figure 4(a). The comparison is further quantified by temporal cross-correlation (R pp' ) between the two pressure signals defined as :: ; = : =>? D =>? (7) where p PIV and p mic are the pressure measured simultaneously by the Tomo-PIV and the surface microphone at the same wall location (pinhole location) and < > denotes the ensemble average operator. The rms of pressure measured by the PIV and the microphones are σ PIV =.49 and σ mic = - 7 -

8 .57 Pa band-pass filtered between Hz. A cross-correlation coefficient of R pp' =.6 is observed in Figure 8(b) along with a signal to noise ratio of about 6 defined as the ratio of the correlation peak to the rms of the cross-correlation values from 5 to 1 ms. (a) (b) p, Pa R pp T omo-piv microphone t, m s t, ms Figure 4. Comparison of the pressure signal measured by the surface microphone and evaluated from time-resolved Tomo-PIV in (a) time-domain and (b) quantified using cross-correlation coefficient. The PSD of the pressure fluctuations measured by the surface microphone, 2C-PIV and by the Tomo-PIV system is shown in Figure 5(a). The PSD is obtained using a method similar to Figure 3 applied to unfiltered signals with f res =1 Hz. The comparison shows close match between the PSD of microphone and Tomo-PIV pressure from 3 Hz till about 3 khz, with the signal energy decreasing over approximately 2.5 decades. Beyond 3 khz (or below two and half decades from the peak of the PSD) the spectrum given by Tomo-PIV measurements is flattened, which is ascribed to noise. The PSD of the 2C-PIV pressure signal shows close agreement to the microphone and Tomo-PIV for frequencies less than 1 khz. Beyond this limit the 2C-PIV deviates and at about 2 khz starts to flatten due to the background noise. This trend follows the higher noise level of the 2C-PIV system observed in the velocity PSD of Figure 3. In order to quantify the agreement of the microphone and Tomo-PIV pressure spectra the coherence function (Cpp') between the pressure obtained from the microphone (p) and the Tomo-PIV systems (p') defined as E :: ; = FG HH;F IG HH G H;H;. (8) is illustrated in Figure 5(b). In this equation Epp' is the absolute value of the cross spectra between the two signals and Epp and Ep'p' are the PSD of the microphone and the Tomo-PIV signals, respectively. The results shows the highest spectral agreement of the two signals at about 5 Hz and values higher than.5 within the range of 25 to 1 Hz. The low value of the coherence number observed at frequencies smaller than 2 Hz is ascribed to the non-linear sensitivity of the electret microphone at this range. While the reduction in the coherence number at frequencies higher than 1 Hz is currently ascribed to the noise of the Tomo-PIV system. The loss of correlation between the two signals is partly associated to the unresolved minute turbulent structures at the wall vicinity especially around the buffer layer (y + 2) where strong pressure sources exist (Chang et al. 1999)

9 (a) (b) f E pp, Pa C pp microphone Tomo-PIV 2C-PIV f, Hz f, Hz Figure 5. (a) The pre-multiplied PSD of the wall-pressure in the turbulent boundary layer obtained from the electret microphone, Tomo-PIV and 2C-PIV systems. (b) The coherence number for the pressure fluctuations measured by the microphone and the Tomo-PIV. Pressure field analysis within the TBL The pressure field obtained from the time-resolved Tomo-PIV system provides an indirect planar/volumetric measurement of pressure within the turbulent boundary layer. To the authors knowledge this is the first experiment where the planar pressure field is measured in a turbulent boundary layer since previous measurements where conducted using point-wise measurement techniques. The pressure field in a turbulent boundary layer consists of minute pressure fluctuations. In the present experiment, their maximum peak amplitude is on the order of 2σ p as shown in the instantaneous pressure field of Figure 6(a). The pressure field shows small blobs of relatively strong fluctuations adjacent to the wall while larger blobs of much weaker amplitude travel further away from the wall. It is also observed that the strongest pressure fluctuation (adjacent blobs of low and high pressure) exist till about y + = 2 (the inner and the overlap layers) and beyond that within the outer layer the fluctuations become even smaller, with amplitude of about.5σ p. Vortex identification using three dimensional formulation of the Q criterion (Hunt et al. 1988) shows the overlap of the vortex cores and the regions of minimum local pressure. Three vortex cores identified as A, B, and C in Figure 6(a) belong to the heads of three hairpin vortices traveling together within a hairpin packet (Adrian et al. 2). This is inferred from the magnified view of Figure 6(b) in which after subtracting the convection velocity of.78u the center of the circular streamlines (tangent to the velocity vectors) locate within the region of high Q confirming that the vortex cores are moving with the same advection velocity. Further evidence on the dynamics of this hairpin packet is provided in movie 1 at t = 1.5 till 2.5 ms. It is clearly observed in Figure 6(a) and (b) that the vortex heads coincide with low pressure regions. The positive pressure blob in between the vortex cores A and B is due to the stagnation region caused by the opposite inductions of the sweep event of vortex A and the ejection event of vortex B as it is observed in the vector field of Figure 6(b). This high pressure region is along the shear layer between the two hairpin vortices (Johansson et al. 1987)

10 (a) 2 (b) y y A stagnation point B C 2 A B C x x + P/σ p Figure 6. (a) A snapshot of pressure fluctuations along with contours of Q = 3e+4 1/s 2 (solid-thin) and Q = 3e+5 1/s 2 (solid-thick) obtained from Tomo-PIV. The pressure different Δ = P P ref is calculated relative to a reference pressure of P ref = Pa. (b) magnified view of the rectangular box containing the vortex heads A, B, and C. The velocity vectors are shown relative to U =.78U. 4. Conclusion The pressure field within a fully developed turbulent boundary layer has been evaluated from timeresolved Tomo-PIV measurements conducted at an acquisition frequency of 1 khz. The obtained pressure field is compared with surface pressure measured using electret condenser microphones. The comparison shows agreement of the two signals quantified by a cross-correlation coefficient of.6 and an agreement of the spectral properties up to 3 khz with the coherence number peaking at approximately 5 Hz. The accuracy of pressure evaluation based on the 3D velocity field is higher than that available from the surrogate measurements with the 2C-PIV technique. The PIV image analysis by the sliding-average correlation algorithm is observed to reduce the background noise level of both velocity and pressure spectra. Acknowledgement This work was conducted as a part of the FLOVIST project (Flow Visualization Inspired Aeroacoustics with Time Resolved Tomographic Particle Image Velocimetry), funded by the European Research Council (ERC), grant no References Adrian RJ, Meinhart C D, Tomkins CD (2) Vortex organization in the outer region of the turbulent boundary layer. J Fluid Mech 422:1 54. Baur T, Köngeter J (1999) PIV with high temporal resolution for the determination of local pressure reductions from coherent turbulent phenomena. Proceedings of the 3rd International Workshop on Particle Image Velocimetry, Santa Barbara, USA

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