Quantitative Characterization of Pressure-related Turbulence Transport Terms using Simultaneous Nonintrusive Pressure and Velocity Measurement*

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Qantitative Characterization of Pressre-related Trblence Transport Terms sing Simltaneos Nonintrsive Pressre and Velocity Measrement* Xiaofeng Li and Joseph Katz Department of Aerospace Engineering

1 Qantitative Characterization of Pressre-related Trblence Transport Terms sing Simltaneos Nonintrsive Pressre and Velocity Measrement* Xiaofeng Li and Joseph Katz Department of Aerospace Engineering San Diego State University Department of Mechanical Engineering Johns Hopkins University UMich/NASA Symposim on Advances in Trblence Modeling Ann Arbor, Michigan Jly -, 07 *Sponsors: ONR, NSF and SDSU

2 Otline of Presentation Backgrond and Motivation: Overview of the Notre Dame Wake Stdy Pressre Reconstrction Methods Eperimental Setp Measrement Reslts on Pressre-Related Trblence Pressre-Velocity Correlation Pressre Diffsion Distribtion and Comparison with Trblence Diffsion and Total Prodction Terms Comparisons of Velocity-Pressre-Gradient, Pressre-Strain and Pressre Diffsion Terms Conclsions

3 ND Wake Stdy*: High-Lift Wake Flow in Pressre Graidents Two major featres associated with the wake flow generated by pstream elements in a high lift system : The wake development invariably occrs in a strong pressre gradient environment. The wake profile is highly asymmetric. 4 0 Mean Velocity Profile of the Initial Slat Wake (From A.M.O. Smith, 975) Umean/Uma (From Thomas, Nelson and Li, 998) *Sponsor: NASA Langley Research Center (NASA NAGI-987) (Thanks to Ben Anders, Christopher Rmsey and John Carlson)

ND Wake Stdy: Eperimental Facility Schematic of Notre Dame Sbsonic Wind Tnnel for the Wake Stdy Contraction ratio of inlet 0.5: Speed of wind tnnel ~ 0 m/s Reynolds nmber (based on c) Re.

4 ND Wake Stdy: Eperimental Facility Schematic of Notre Dame Sbsonic Wind Tnnel for the Wake Stdy Contraction ratio of inlet 0.5: Speed of wind tnnel ~ 0 m/s Reynolds nmber (based on c) Re Dimension of test section ft.(width) ft.(height) ft. (length) Chord of the splitter plate c = 48 in.m Instrmentation for Flow Srvey X-wire and LDV Reynolds nmber based on the initial wake momentm thickness Re =

5 Wake Strctre Nomenclatre

6 ND Wake Stdy: Constant Pressre Gradient Environment One Uniqe Featre: The imposed pressre gradient is constant in the flow field. Common Zero Pressre Gradient Zone Station for TKE bdget measrement / = for symmetric wake; 5 for asymmetric wake

7 Epansion of the Trblent Kinetic Energy Eqation For steady, -D in the mean, incompressible, homogeneos trblence flow, we have 0 t, 0 U and 0. Also we have, from continity eqation, U U. Ths, the trblent kinetic energy eqation can be simplified as follows: q U q U Convection p p Pressre Diffsion Trblence Diffsion U U U q q Prodction Viscos Diffsion () Dissipation No device is capable of measring this term to date. Assme This term is negligible. This term reqires miniatre probes to captre the finest eddy of motion

Parallel Probe Aspe type AHWG-00, tngsten wire, diameter 5m, length 0.9mm, spacing between dal sensors 0.mm. Twin X-wire Configration Two identical X-wire probes of Aspe type AHWX-00.

8 Sensors for the TKE Bdget Measrements Constant Temperatre Hot-wire Anemometer IFA-00 anemometer. X-wire Probe (Adapted from Dentec Catalog) Parallel Probe (Adapted from Dentec Catalog) X-wire Probe Aspe type AHWX-00, tngsten wire, diameter 5m, length.mm. Parallel Probe Aspe type AHWG-00, tngsten wire, diameter 5m, length 0.9mm, spacing between dal sensors 0.mm. Twin X-wire Configration Two identical X-wire probes of Aspe type AHWX-00. Ct-off Freqency of Low Pass Filter 0 khz Sampling Freqency 40 khz ( Nyqist Freqency = 0 khz) Total record length per sample. sec. Twin X-wire Configration Kolmogorov microscales of wake flow L K = ( ) /4 0. mm; T K = () / 0.4 ms.

9 ND Wake Stdy: TKE Bdget for the Symmetric Wake in ZPG Prodction ~±% Convection Trblent Diffsion Prodction 0.05 Dissipation 0.0 Gain Trblent Diffsion Pressre Diffsion ~±9% Convection ~±% Pressre Diffsion Loss Dissipation y/delta (0 = Convection + Prodction + Trblent Diffsion + Pressre Diffsion + Dissipation) ~±5%

10 ND Wake Stdy: Comparison of TKE Bdget for the Symmetric Wake in ZPG with DNS Reslt (Moser, Rogers & Ewing, 998) b Term * U 0.0 DNS d Pressre Diffsion_DNS Pressre Diffsion_ZPG*Ue/U b Term * U 0.04 DNS d Trblent Diffsion_DNS Trblent Diffsion_ZPG*Ue/U y b DNS y b DNS Note the difference between the DNS and the eperimental data: DNS reslt is based on data obtained in the similarity region; Or eperimental reslt is in the near wake region with / = 4 (The DNS Data Cortesy of Michael M. Rogers)

11 ND Wake Stdy: Comparison of TKE Bdget for the Symmetric Wake in ZPG with DNS Reslt (Moser, Rogers & Ewing, 998) b Term * U 0.0 DNS d Dissipation_DNS Dissipation_ZPG*Ue/U b Term * U 0. DNS d Prodction_DNS Prodction_ZPG*Ue/U y b DNS 0.0 Time Derivative_DNS Convection_ZPG*Ue/U y b DNS 0.0 b Term * U DNS d The DNS Data Cortesy of Michael M. Rogers. Reslts pblished in Eperiments in Flids (Li and Thomas, 004) y b DNS

12 D Dt velocity- pressre - gradient Reynolds Stress Transport Eqation ij k i j Pij i k k prodction ij ij ij Material derivative of Reynolds stress (nsteady term convection ) where P ij i k trblence diffsion U k j j k U viscos diffsion k i i ij k j k dissipatio n ij p p i j j i j p i p i j D ( p ij velocity -pressre -gradient ) pressre diffsion, p j i j i pressre -strain, R ij i.e., ij D ( p) ij R ij Note: Pressre diffsion is also called the gradient of the Reynolds stress fl de to flctating pressre.

13 Importance of Measrement of Pressre-Strain Terms Importance of intercomponent energy transfer R R R 0 0 Becase for incompressible flow p Pressre-strain terms are responsible for redistribtion of energy among components of the trblence normal stresses, i.e., intercomponent energy transfer among flctating components. Ths pressre-strain terms serve as the primary mechanism for the retrn-to-isotropy process (Pope, 000) of the anisotropic trblence. Pressre-strain terms play a major role in defining trblence development. However, two eqation models like k-epsilon make no attempt to differentiate between the three flctating velocity components. Qantifying the intercomponent energy transfer is a key to nderstanding the physics of trblent shear flow.

14 The State-of-the-Art of the Non-intrsive Pressre Measrement Techniqes The instantaneos spatial pressre distribtion in an incompressible trblent flow field can be measred non-intrsively by integration of the measred material acceleration, which is the dominant contribtor to pressre gradient for flow at high Reynolds nmber: p DU Dt Dominant term Representative work for the direct line integration approach incldes Li and Katz, (omni-directional integration, 006, Ep. Flids; 008, Phys. Flids; 0 JFM); Joshi, Li and Katz (04 JFM); Li et al. (AIAA Paper ) etc. In addition, the pressre distribtion can also be obtained by solving the Poisson eqation, as shown in Violato et al. (008, Ep. Flids), and de Kat and van Odhesden (0, Ep. Flids), etc. The robstness of the omni-directional integration method has been confirmed by Charonko et al. (00). Least Sqare Reconstrction (e.g., Jeon et al., 05), or Direct Matri Inversion (Li and Katz 006), POD-based Irrotation Correction (Wang et al., 06). Poisson eqation math eqivalent Least Sqare Reconstrction (Wang et al., 07). U Negligible for high Re flow and in regions away from the wall

Integration over the entire flow field to

distribtion: Material Acceleration Vector

Integration Instantaneos Pressre Distribtion

word: Pressre PIV Li and Katz 0, Jornal of

Li and Katz 008, Physics of Flids, 0, 0470.

15 Circlar Virtal Bondary Omni-Directional Integration (Li and Katz, 006, 008, 0) Circlar Virtal Bondary Omni-Directional Integration over the entire flow field to obtain the instantaneos spatial pressre distribtion: Material Acceleration Vector Map Virtal Bondary Omni- Directional Integration Instantaneos Pressre Distribtion Integration to obtain pressre Google key word: Pressre PIV Li and Katz 0, Jornal of Flid Mechanics, vol. 78, pp Li and Katz 008, Physics of Flids, 0, Li and Katz 006, Eperiments in Flids, 4, 7-40.

16 Rotating Parallel Ray Omni-Directional Integration Old Algorithm (007, 008) Circlar virtal bondary omnidirectional integration algorithm. Virtal bondary New Algorithm (Li et al. AIAA Paper ) Rotating parallel ray omni-directional integration algorithm. Real image bondary =.0% = 0.% An inherent defect in the old algorithm: Location dependence of integration weight. Other than the points near the geometric center, points at other places do not see a niform weight of contribtion from all directions. New algorithm: Eqal Weights of integration involvement can be generated, ths eliminates the defect in location dependence.

17 Time-Resolved PIV Measrements High Speed Camera PCO.dima, bit Resoltion: fps High Repetition Rate Laser Photonics DM60-57 Nd:YLF Maimm plse rate - 0 khz Plse energy: 60mJ at KHz Tracer Particles -- Hollow glass spheres, 8-m Dimensions of Cavity Cavity width (L): 8.mm Cavity depth (H): 0.0mm Flow Direction Leading Edge Cavity Wall Field of View (55 mm) Trailing Edge H=0.0 mm L=8. mm Free Stream Speed in Eperiment.m/s Reynolds Nmber Re=40,000 (based on cavity width) Re =6, bondary layer trblent Image size: 55 mm Vector Spacing: 0. mm Interrogation window size: mm

(m/s) -0.0007 0.55 0.0.8 0.0075 0.077-0. 0.5 0.7.0 0.0075 0.077-0.6 0.

18 Incoming Trblent Bondary Layer Profile (b) /U /L = -0.6 Flow L=8. mm H=0.0 mm Location /L Displacement thickness * (mm) Momentm thickness (mm) Shape factor H (= */ ) Skin friction coefficient C f Friction velocity (m/s) Re = 6. Recall: for Blasis profile, H.6 for Klebanoff profile, H. Conclsion: The incoming bondary layer is trblent.

19 U Unit: m/s Flow Direction Leading Edge Trailing Edge Cavity Wall Note: the step of the scale below 0 m/s is 0.5m/s; above 0m/s, the scale is not linear

20 Instantaneos Pressre Map: Sample Ue=0m/s Re=5,000 (based on cavity width)

21 Instantaneos Pressre Map: Sample Streamlines: U - Ue/ Ue=0m/s Re=5,000 (based on cavity width)

Sample Raw Data: Characteristic Flow Phenomena Vorte Shearing and Appearance/ Disappearance of Low Pressre at Corner Swirling Strength ci ci, the imaginary part of the comple eigenvale of the

22 Sample Raw Data: Characteristic Flow Phenomena Vorte Shearing and Appearance/ Disappearance of Low Pressre at Corner Swirling Strength ci ci, the imaginary part of the comple eigenvale of the local velocity gradient tensor, represents the strength of local swirling motion and can be sed to identify vortices. (Zho, et al 996). Pressre Coefficient C p Streamline: c, assminging c as 0.5U

23 Flow Field of View H=0.0 mm Mean Velocity and Pressre (Ue=. m/s) Based on an average of 0,000 realizations /Ue v /Ue L=8. mm U V Cp mean Cp rms

24 Reynolds Stress Distribtions v v Flow Field of View H=0.0 mm L=8. mm Other trblence statistics obtained from measrements (to be shown later on): Pressre-velocity correlations; Pressre-strain correlations. Reslts of spectrm analysis (Li and Katz, 0) are in agreement with theory and or cavitation visalization for this same flow (Li and Katz, 008). Negative prodction de to large negative vales of V

25 The TKE Prodction Shear Prodction Dilatational Prodction Total In-plane TKE prodction U V v L y 0.4 U v y PL / U e V L v v y P Conclsion: Prodction is shear dominated in the shear layer. Dilatational prodction becomes significant near the corner. The total In-plane TKE prodction is defined as: Shear prodction v v y Dilatational prodction

Pressre-Velocity Correlation Ue =. m/s; L=8.

p vp In shear layer: Streamwise acceleration increases, p decreases (+), p (-) p (-); Streamwise deceleration

Impinging on wall: Above the wall: Streamwise acceleration increases, p decreases (+), p (+) p (+); Streamwise

26 Pressre-Velocity Correlation Ue =. m/s; L=8.mm; Re=40,000 (based on cavity width); Re = 6; Ensemble size: 80,000 data points for 8 seconds. p vp In shear layer: Streamwise acceleration increases, p decreases (+), p (-) p (-); Streamwise deceleration decreases, p increases (-), p (+) p (-). Impinging on wall: Above the wall: Streamwise acceleration increases, p decreases (+), p (+) p (+); Streamwise deceleration decreases, p increases (-), p (-) p (+). High favorable pressre gradient (+), v (+), p (-) p (-), v p (-); Low favorable pressre gradient (-), v (-), p (+) p (-), v p (-). In shear layer, -p correlation negative; in front of trailing corner, -p-correlation positive.

acceleration increase, p decrease (+), p (-) p (-); Streamwise deceleration decrease, p increase (-), p (+) p

27 Pressre-Velocity Correlation Ue = 0.0 m/s; L=8.mm; Re=5,000 (based on cavity width); Re = 40, Ensemble size:,000 data points for 500 seconds p vp Streamwise acceleration increase, p decrease (+), p (-) p (-); Streamwise deceleration decrease, p increase (-), p (+) p (-); Hooper and Msgrove (997) also reported strong negative correlation between flctating pressre and streamwise velocity component in a developed pipe flow sing a cobra (4-hole) probe.

The -component Diffsions and the TKE Prodction Trblence diffsion of Pressre diffsion of Total In-plane TKE prodction v L pl y 0.9 PL / U e -0. -0.4-0.

28 The -component Diffsions and the TKE Prodction Trblence diffsion of Pressre diffsion of Total In-plane TKE prodction v L pl y 0.9 PL / U e Note: The magnitde of pressre diffsion is of the same order as the trblence diffsion. The magnitde of the pressre diffsion is abot the same as the total trblence kinetic energy (TKE) prodction. Conclsion: Trblence diffsion of dominates in the shear layer. Close to the corner, pressre diffsion is significant (ths cannot be neglected in RANS simlation there); away from the corner, pressre diffsion is negligible. Pressre diffsion term (at least for the -component) cannot be modeled after the trblence diffsion since their shape of distribtion is different. Pressre diffsion has similar shape bt opposite sign with the prodction near corner. A hint for possible modeling relationship.

The v-component Diffsions and the TKE Prodction Trblence diffsion of v Pressre diffsion of v Total In-plane TKE prodction v v L y vp L y PL 0.5 / U e -0. -0.5-0.

29 The v-component Diffsions and the TKE Prodction Trblence diffsion of v Pressre diffsion of v Total In-plane TKE prodction v v L y vp L y PL 0.5 / U e Again, the magnitde of pressre diffsion is of the same order as the trblence diffsion. The magnitde of the pressre diffsion is at least comparable with the total trblence kinetic energy prodction. Conclsion: Trblence diffsion of dominates in the shear layer. Close to the corner, pressre diffsion is significant; away from the corner, pressre diffsion is negligible.

30 Remarks on Diffsion Term Modeling In the poplar eddy viscosity models of RANS (Reynolds-Averaged Navier-Stokes) simlation approach, a common practice is to combine the transport terms and model them as (Chen and Jaw 998; Pope 000; Lmley 978; F 99; Schwarz and Bradshaw 994): p i i j j s k Knowing the patterns of pressre and trblence diffsions are fndamentally different, it seems that collectively modelling the diffsion terms all together as shown above may not be jstifiable for this trblent -D open cavity shear layer flow. Qestion : Based on the eperimental evidence, can we model the diffsion terms in terms of prodction in stead of the gradient of k? p i p i P ij j ij p i P (?) ij _ dillational s (?) j ij j P ij ij T (?) k

4 R D ( p) Velocity-pressre-gradient tensor p v L y R D ( p)

31 Velocity-pressre-gradient tensor p L Pressre-rate-of-strain tensor R p L = pl U e Pressre diffsion of R D ( p) Velocity-pressre-gradient tensor p v L y R D ( p) Pressre-rate-of-strain tensor R = + Ue =.5 m/s; Re=40,000 Pressre diffsion of v p L y vpl U e 0.5 y v

Reynolds Stress Distribtions v 0.06 0.0 v Flow Field of View H=0.0 mm L=8.

32 Reynolds Stress Distribtions v v Flow Field of View H=0.0 mm L=8. mm Other trblence statistics obtained from measrements (to be shown later on): Pressre-velocity correlations; Pressre-strain correlations. Reslts of spectrm analysis (Li and Katz, 0) are in agreement with theory and or cavitation visalization for this same flow (Li and Katz, 008). Negative prodction de to large negative vales of V

33 Observations Pressre and velocity correlation: p and are negatively correlated in most of the shear layer. However, close to the cavity trailing corner, the p- correlation gradally decreases in magnitde, and eventally changes its sign, creating a positive peak jst pstream of the trailing edge, followed by a negative correlation again above the trailing corner. p and v are positively correlated in most of the shear layer, and become negatively correlated in the area srronding the cavity trailing corner. Diffsion: In the shear layer, the -component trblence diffsion dominates. Close to the corner, pressre diffsion is significant, and its magnitde is on the same order as those of the trblence diffsion and the total in-plane trblence prodction. The distribtion patterns of the trblence diffsion and the pressre diffsion are considerably different.

34 Observations (contined) Diffsion (contined): The - and the v-component pressre diffsion terms have opposite signs at corresponding locations srronding the trailing corner of the cavity, where the peaks of the v-component pressre diffsion are smaller in magnitde than those of the -component conterparts. Velocity-pressre-gradient tensor and pressre-strain: In the shear layer, the -component velocity-pressre-gradient tensor and the pressre-strain R have dominant vales in comparison with their v- component conterparts. The pressre-strain term R keeps a strong negative vale throghot the shear layer, peaking at the impingement point on the trailing wall of the cavity. The pressre-strain term R has a weak positive vale in the shear layer. The intercomponent flctation energy transfer completely changes its scheme on top of the trailing corner, where R takes a positive vale and R a negative one. Considering the negative trblence prodction there, the pressre-strain intercomponent energy transfer is the major mechanism that is responsible for the high -component flctation energy (and also the low v-component flctation energy) occrring on top of the trailing corner of the open cavity.

35 Conclding Remarks Pressre diffsion and trblence diffsion follow different patterns. Collectively modelling the pressre diffsion terms with other diffsion terms may not be jstifiable for the trblent -D open cavity shear layer flow. Pressre gradient plays a critical role in defining the pictre of trblence transport. Both the pressre diffsion and the pressre strain terms are intensified at places (e.g., impingement or re-attachment regions) of high pressre gradients. The change in pressre gradient reslts in the change in both the pressre diffsion and pressre strain terms. The pressre-related terms have sbstantial impact on the dynamics of trblence transport throghot the shear layer. Lack of data, especially high qality, right type of data, hinders the development of trblence modeling in the past 0 years. The relationships between the pressre-related terms and other transport terms are far from being ehasted. Ths more eplorations are needed and legitimate. The complicated intercomponent energy transfer process clearly shows the challenges (and perhaps opportnities) that trblence modeling faces in trblent cavity shear layer flow in particlar, and perhaps separation and reattachment flow in general.

36 Acknowledgments The research was sponsored in part by the Office of Naval Research (Ki-Han Kim and Debbie Nalchajian are the Program Officers), and in part by National Science Fondation grant CBET-480 (R. Joslin is the Program Director). The financial spport from the UGP program of the San Diego State University is also grateflly acknowledged. Most of the content of this talk can be fond in Li, X. and Katz, J., Pressre rate of strain, pressre diffsion and velocity pressre gradient tensor measrements in a cavity shear layer flow, 55th AIAA Aerospace Sciences Meeting, AIAA SciTech Form, (AIAA ), Also sbmitted to AIAA Jornal for review. Other information: Google keyword: Pressre PIV. SDSU Lab website: Xiaofeng.Li@mail.sds.ed

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