Use of DNS Data for the Evaluation of Closure Models in Spanwise-Rotating Turbulent Channel Flow

Transcription

1 Universit of Colorado, Boulder CU Scholar Aerospace Engineering Sciences Graduate Theses & Dissertations Aerospace Engineering Sciences Spring --24 Use of DNS Data for the Evaluation of Closure Models in Spanwise-Rotating Turbulent Channel Flow Alan Sean-Ker Hsieh Universit of Colorado Boulder, Follow this and additional works at: Part of the Aerodnamics and Fluid Mechanics Commons, and the Mechanical Engineering Commons Recommended Citation Hsieh, Alan Sean-Ker, "Use of DNS Data for the Evaluation of Closure Models in Spanwise-Rotating Turbulent Channel Flow" (24). Aerospace Engineering Sciences Graduate Theses & Dissertations This Thesis is brought to ou for free and open access b Aerospace Engineering Sciences at CU Scholar. It has been accepted for inclusion in Aerospace Engineering Sciences Graduate Theses & Dissertations b an authorized administrator of CU Scholar. For more information, please contact cuscholaradmin@colorado.edu.

2 Use of DNS Data for the Evaluation of Closure Models in Spanwise-Rotating Turbulent Channel Flow b Alan Hsieh B.S., Universit of Southern California, 2 A thesis submitted to the Facult of the Graduate School of the Universit of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Aerospace Engineering Sciences 24

3 This thesis entitled: Use of DNS Data for the Evaluation of Closure Models in Spanwise-Rotating Turbulent Channel Flow written b Alan Hsieh has been approved for the Department of Aerospace Engineering Sciences Sedat Biringen Professor John Evans Professor Mahmoud Hussein Date The final cop of this thesis has been examined b the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarl work in the above mentioned discipline.

4 iii Hsieh, Alan (M.S., Aerospace Engineering Sciences) Use of DNS Data for the Evaluation of Closure Models in Spanwise-Rotating Turbulent Channel Flow Thesis directed b Professor Sedat Biringen A direct numerical simulation (DNS) of a spanwise-rotating turbulent channel flow was conducted for three rotation numbers: Ro c =, 5.2 and 26, at a Renolds number Re c = 8. The data base obtained from these simulations was used to evaluate several commonl used Renolds- Averaged Navier-Stokes (RANS-based) closure models for rotating turbulent channel flows. It was shown that the Renolds stresses predicted b the Speziale-Gatski (SG) model were the most consistent with the DNS results. A correction to the Girimaji turbulence model was proposed to remove a discontinuit in the non-rotational case. The pressure-strain functions of the explicit algebraic Renolds stress model (EARSM)-tpe SG and Girimaji models were examined and the modeled pressure-strain distributions of both turbulence models, especiall near the suction wall, were demonstrated to become more accurate with increasing rotation number. The accurac of the modeled pressure-strain was also shown to affect the accurac of the corresponding modeled Renolds stresses near the channel walls.

5 Dedication Dedicated to m famil, friends and mentors who have supported me throughout m path in education.

6 v Acknowledgements First and foremost, I would like to thank Professor Biringen for all of his great support and advice throughout m graduate stud. The qualit of m research and this thesis would not have been possible without the insightful input and great guidance given to me b m research advisor. I would also like to thank m colleague Alec Kucala for his ever-present availabiit to answer questions and his mentoring and tutelage especiall in the field of computation, and Scott Wagg for allowing me to use his atmospheric boundar laer code as the basis of m present direct numerical simulation code. I would also like to thank the National Institute of Computational Sciences and the Texas Advanced Computing Center for providing the necessar computational resources for m research. And finall, thanks to the Universit of Colorado s Department of Aerospace Engineering Sciences for providing me the necessar financial support over the course of m graduate studies.

7 Contents Chapter Introduction 2 Numerical Method and Case Descriptions 5 2. Time Integration Scheme Spatial Differences Case Parameters and Descriptions Present Direct Numerical Simulation Results Total shear stress, law of the wall, energ spectra and mean velocit Renolds stresses, turbulent kinetic energ and dissipation rate Velocit fluctuations RANS Turbulence Models Turbulence Model Overview v 2 f turbulence model Speziale-Gatski turbulence model Girimaji turbulence model RANS Model Testing Girimaji Model Correction Pressure-Strain Modeling in Explicit Algebraic Renolds Stress Models 4 4. Pressure-Strain Overview

8 4.2 Pressure-Strain Model Testing vii 5 Conclusion 46 Bibliograph 47 Appendix A Derivation of the v 2 f model 49 B Derivation of the Speziale-Gatski model 5 C Derivation of the Girimaji model 54 D Intercomponent energ transfer in the Renolds stress equation 57

9 viii Tables Table 2. Case descriptions and initial conditions Model coefficients for the v 2 f turbulence model Model coefficients for the Speziale-Gatski turbulence model Model coefficients for the Girimaji turbulence model

10 Figures Figure 2. Schematic diagram of rotating chanel flow simulation Present DNS turbulent Renolds number distribution over time (Case A) Present DNS turbulent Renolds number distribution over time (Case B) Present DNS turbulent Renolds number distribution over time (Case C) Present DNS distribution of shear stresses (Case A) Present DNS law of the wall (Case A) Present DNS one-dimensional energ spectra (Case A) Present DNS mean velocit distributions (Cases A-C) Present DNS Renolds stress distributions (Cases A-C) Present DNS turbulent kinetic energ distributions (Cases A-C) Present DNS turbulent dissipation rate distributions (Cases A-C) Present DNS instantaneous velocit fluctuation contour plots (Case A) Present DNS instantaneous velocit fluctuation contour plots (Case B) Present DNS instantaneous velocit fluctuation contour plots (Case C) Modeled Renolds stresses profiles (Case A) Modeled Renolds stresses profiles (Case B) Modeled Renolds stresses profiles (Case C) G coefficient groupings within the Girimaji-modeled Renolds shear stress (Case A) 32

11 x 3.5 Present DNS mean strain rate tensor S 2 (Case A) η term in the Girimaji turbulence model (Case A) G coefficient distribution for the Girimaji turbulence model (Case A) Modeled Renolds shear stress distribution for the Girimaji turbulence model (Case A) Girimaji-modeled Renolds stress distributions with correction (Case A) Girimaji-modeled Renolds stress distributions with correction for Schwartz s inequalit (Case A) Determinant of Girimaji-modeled Renolds stress tensor with correction (Case A) Modeled pressure-strain distributions (Case A) Modeled pressure-strain distributions (Case B) Modeled pressure-strain distributions (Case C)

12 Chapter Introduction When a turbulent channel flow is subject to rotation in the spanwise direction, asmmetr is observed across the channel and is distinguished b reduced turbulence levels near one wall and elevated turbulence levels near the opposite wall; these regions are known as the suction and pressure sides, respectivel (Grundestam, Wallin, and Johansson, 28). Subsequentl, the smmetric profiles of Renolds-stresses, mean velocit, and velocit fluctuations in a non-rotating channel become asmmetric with respect to the channel center-line as a result of rotation. The effects of spanwise rotation on the momentum characteristics of turbulent channel flow have been well documented in previous direct numerical simulation (DNS) studies b Grundestam et al. (28) and Wu and Kasagi (24). Turbulence is alread a complex scientific field due to the chaotic and unpredictable nature of turbulent flows and structures, and the rotation-induced additional bod forces (Coriolis, centrifugal) onl cause momentum transport mechanisms to become more complicated. Turbulent flows, not laminar flows, are predominantl found in nature, and with the prevalence of rotation-dependent machiner in engineering, a phsical understanding of turbulent flow in rotating sstems is prudent for engineering and scientific analses. Hence, research into this tpe of flow has relevance to practical engineering applications such as turbine blades and rotating turbomachiner design especiall with regards to surface heat transfer and skin friction. In order to assess the effects of spanwise rotation on the momentum balance in turbulent channel flow within this work, a direct numerical simulation was emploed to integrate the time-

13 2 dependent, three-dimensional Navier-Stokes equations. The resulting DNS data base was then used to test the accurac of three commonl-cited Renolds-Averaged Navier-Stokes (RANS) closure models, the v 2 f, Speziale-Gatski (SG) and Girimaji models, proposed b Reif, Durbin, and Ooi (999), Speziale and Gatski (993), and Girimaji (996), respectivel, in modeling Renolds stresses for spanwise-rotating turbulent channel flow. Computational modeling of all tpes of turbulent flows has been essential to the field of engineering since the advent of the computer, allowing for engineering analsis to extend beond theor and phsical experimentation. As computers have increased in complexit and computational power in the recent decades, three main forms of computational methods for turbulence modeling have arisen. Direct numerical simulation (DNS) is a computational simulation in which the three-dimensional, time-dependent Navier-Stokes equations are numericall solved along with the full range of spatial and temporal scales of turbulence, resulting in the most accurate computational solutions. However, direct numerical simulations are frequentl cost-prohibitive due to the high computational cost of resolving all scales, especiall in cases of complex geometr and flow configurations. Large edd simulation (LES) is another computational simulation that solves the three-dimensional, time-dependent Navier-Stokes equations similar to DNS, but onl for large-scale turbulence. For the smallest scales of turbulence such as the Kolmogorov microscales, model functions known as subgrid-scale models are used to approximate the contributions from these smaller scales into the rest of the LES (Renolds, 976). The flexibilit of LES in choosing the size of the filter separating the large and small scales allows for model adaptation to available computational resources, although LES has decreased accurac compared to DNS as some scales of turbulence are not full resolved, onl modeled. The third tpe of computational modeling is known as Renolds-Averaged Navier-Stokes (RANS) based modeling. In RANS modeling, there is no numerical solution of the Navier-Stokes equations such as in DNS and LES, instead an alternative form of the Navier-Stokes equations known the Renolds stress equation is derived and partial differential equations of various flow quantities within this equation are solved for (Renolds, 976). While RANS-based models are

14 3 the least accurate of the three main computational models, these models are also require far less computational power and therefore a RANS-based model that can model turbulence in flow fields with accurac similar to DNS or LES is ver desirable in the engineering field. There are man levels of RANS-based models: the simplest levels include zero-equation, one-equation, and twoequation models, which solve a pde related to the mean velocit field, a pde related to a turbulent velocit scale, and two pdes related to both a turbulent velocit and length scale, respectivel (Renolds, 976). However, these simpler RANS-based models are generall unable to properl model rotational flows and are subsequentl disregarded in this work. For flows involving rotation, nonlinear edd viscosit models are the most frequentl applied RANS-based models, which include the two model subtpes v 2 f models and explicit algebraic Renolds stress models, both of which are considered in this work. For a suitable data base to compare and test these RANSbased models, a more accurate computational solution was required; due to the relativel simple geometr of rotating turbulence channel flow, a direct numerical simulation was conducted for this stud instead of a large edd simulation. Other variations of computational modeling also exist. For example, detached edd simulation (DES) is a combination of RANS modeling and LES where subgrid-scale models are used to model flow regions where the turbulent length scale exceeds the grid dimensions and the RANS method is used to solve the flow regions where the turbulent length scale is less than the maximum grid dimension (generall in the near-wall region). Coherent vortex simulation is another computational approach in which the turbulent flow field is decomposed into coherent and noncoherent sections using wavelet filtering; the coherent section is then resolved in a similar method to LES. Unstead Renolds-Averaged Navier-Stokes (URANS) models, partiall integrated transport models (PITM), and hbrid LES-RANS (HLS) models are additional computational methods that are mixtures of LES and RANS to varing degrees. Of the three major flow simulation methods: DNS, LES and RANS modeling, RANS-based turbulence models are the most practical as the offer the lowest computational cost. The three aforementioned turbulence models, which are of the most common RANS-based model tpe used

15 4 to model rotational flows, are all nonlinear edd viscosit models. These RANS-based models produce model closure for the Renolds stress equation b solving partial differential equations of important flow quantities, such as the turbulent kinetic energ k and the turbulent energ dissipation rate ε. Both the SG and Girimaji models, explicit algebraic Renolds stress models (EARSM), additionall construct an implicit model for the anisotropic tensor b ij from the partial differential equations of the energ budget terms in the Renolds stress equation, and then derive an explicit form for b ij analticall. The effect of the anisotropic tensor is seen in the pressure-strain function proposed b Speziale, Sarkar, and Gatski (99) (SSG) that is present in both of these models, and the modeled pressure-strain values of these two models are directl compared to the pressurestrain values extracted from the DNS. Previous research has shown that the pressure-strain correlation term exerts a powerful influence on man turbulent quantities such as the turbulent kinetic energ (Launder, Reece, and Rodi, 975), and hence we also investigate these two EARSM-tpe models in regards to correlating the accurac of the modeled pressure-strain with the accurac of the modeled Renolds-stresses for the SG and Girimaji turbulence models. This paper is organized as follows: the second section will cover the numerical method, parameters and specifications used in the direct numerical simulation and the DNS validation. The third section will provide an overview of the three RANS-based turbulence models considered in this work and a comparison of the modeled Renolds stresses with the DNS; a correction to the Girimaji model for the non-rotational case is also proposed in this section. The final section will analze the accurac of the SG and Girimaji pressure-strain functions and draw correlations between the modeled pressure-strain and modeled Renolds stresses for the two turbulence models.

16 Chapter 2 Numerical Method and Case Descriptions The time-dependent, three-dimensional incompressible Navier-Stokes equations were integrated in a doubl periodic (in x and z-directions) turbulent channel flow using a fractional step method (Wagg, Biringen, and Sullivan, 23). With all spatial coordinates non-dimensionalized b the channel half-height δ and velocities b the centerline velocit U c, the Navier-Stokes equations read (in conservative form) u i x i = (2.) u i t + u iu j = p + 2 u i Ω j Ro c ε x j x i Re c x i x ijk j Ω u k (2.2) where Re c = U c δ/ν, ν is the kinematic viscosit, and the vector u= u,v,w is composed of three velocit components in the x (streamwise), (wall-normal), and z (spanwise) directions, respectivel. Also, p represents non-dimensionalized pressure and t represents non-dimensionalized time. The rotation number, or Rossb number, is defined as Ro c = 2Ωδ/U c where Ω is the spanwise angular rotation vector. The flow geometr is shown in figure 2., and the flow field is assumed to be statisticall homogeneous in the streamwise (x) and spanwise (z) directions. Hence, periodic boundar conditions are applied in both of those directions. Due to turbulent channel flow being a wall-bounded flow, velocit boundar conditions at the walls (=, 2) were assumed to be no-slip.

17 Figure 2.: Diagram of turbulent channel flow with rotation in the spanwise direction. 6

18 7 2. Time Integration Scheme All of the numerical methods and schemes presented in this work were performed using a Navier-Stokes solver developed b Wagg (29) for the simulation of the turbulent Ekman laer. This code was further modified for plane channel flow (Kucala and Biringen, in press.) and periodic channel flow; rotational effects were added and implemented smoothl in the present work. The time integration scheme used is the semi-implicit Adams-Bashforth/Crank-Nicolson (ABCN) method, which makes the numerical procedure second order accurate in time (Wagg et al., 23). The Crank-Nicholson method uses an implicit scheme for the diffusion terms in equation 2.2, which is written as û i u n i t 2 û i 2Re x ui n 2Re x3 2 +O ( t 2) (2.3) where the predicted velocit û i is solved implicitl using the linear sstem of equations shown in equation 2.4. ( t 2Re 2 x 2 3 ) û i u n i + t 2 Mn i (2.4) with M n i = Re 2 u n i x 2 3 (2.5) The remaining terms in equation 2.2 are solved using the explicit Adams-Bashforth scheme as shown in equation 2.6, û i u n i t 3 2 Ln i 2 Ln i +O ( t 2) (2.6) where ( Li n = un i un j x j Ro un j ε ji3 P + 2 ui n x i Re x u n i x 2 2 ) (2.7) Combining both expressions from the Adams-Bashforth and Crank-Nicholson schemes ields the expression for the predicted velocit in equation 2.8. ( t 2Re 2 x 2 3 ) ( û i = ui n + t 2 Mn i Ln i ) 2 Ln i (2.8)

19 Once the predicted velocit û i is computed, the velocit at the next time step (n+) is found b appling a corrector step as shown in equation 2.9, with φ representing the pseudo-pressure. u n+ i = û i t φ x i (2.9) The pseudo-pressure is computed b taking the divergence of equation 2.9 and enforcing equation 2. (continuit) at the next time step, ielding the expression shown in equation Spatial Differences 2 φ t = û i (2.) x i x i x i 8 The spatial derivatives are computed using a finite-difference approximation in all coordinate directions with fourth order central differences b means of high-order Lagrangian polnomials. The streamwise (u) and spanwise (w) velocit components are discretized on a centered vertical mesh, while the wall-normal (v) velocit component is discretized on a staggered vertical mesh to enhance coupling between the vertical velocit and the pressure (Wagg et al., 23). In order to dissipate excess energ that remains unsolved b the grid when appling finite difference methods, an artificial viscosit in phsical space must be implemented to ensure numerical smoothing and account for dealiasing. In this numerical integration scheme, this is applied b adding a high-order dissipation term to the finite difference equations as shown in equation 2.; the dissipation term β was set to a value of.5. Li n = + β x 4 4 ui n j x 4 j (2.) 2.3 Case Parameters and Descriptions The Renolds number based on the laminar centerline velocit Re c = 8, was kept constant for the present simulations; the friction Renolds number Re τ was allowed to var. The grid resolution used was in the streamwise (x), wall-normal (), and spanwise (z) directions, respectivel. The domain lengths were selected as L x = 4πδ, L = 2δ, and L z = 2πδ for

20 9 Table 2.: Case Descriptions and Initial Conditions Case Re m Re τ Ro m Ro τ u c u τ A B C all simulations to adequatel fit the largest scales of motion. A summar of the different cases is found in table 2.. Quantities denoted b subscripts of m, c, and τ correspond to scaling using the mean bulk velocit, mean channel center-line velocit, and global friction velocit, respectivel. For all cases, the Navier-Stokes equations were integrated until a quasi-periodic stead-state was reached and each successive case with increasing rotation number was initiated using the steadstate data base attained b the previous case. The original data base for the non-rotating case (A) was initialized from an non-converged data field of non-rotating turbulent channel flow given to this work s author b colleague Alec Kucala. The numerical procedure outlined in the previous section was programmed using the Portable, Extensible Toolkit for Scientific Computation (PETSc). As PETSc routines are designed for solving massive sstems of equations, implementation of the PETSc libraries enabled the simulation code to run optimall using as few processors as possible. As linear solvers are necessar to compute the pseudo-pressure and predicted velocit, all solutions were obtained using Krlov subspace methods found within the routines of the PETSc librar; the convergence tolerance of the linear sstems was e 7. Simulations were run on the Kraken Cra XT5 sstem located at the National Institute for Computational Sciences for a total run time of over nondimensional time units. The value of the time step was.2 for all simulations and was chosen to optimize the flow field convergence time while ensuring numerical stabilit; the total number of iterations exceeded half a million. The quasi-periodic stead-state for all simulation cases was characterized b two requirements: a convergence of the normal and shear Renolds stress profiles and a sufficient leveling

21 of the turbulent Renolds number over a nondimensional time period of T = 2. This second requirement was established due to a consistent decrease of the turbulent Renolds number over time after an initial increase in value from the added rotation, and to ensure a large enough sample size so that a satisfactor data base for time-averaging could be collected. The distribution of the friction Renolds number Re τ over nondimensional time T for simulation cases A, B and C are shown in figures 2.2, 2.3 and 2.4, respectivel; these figures also displa the time spans over which the time-averaged data used for the present DNS profile distributions was collected. Figure 2.2: Re τ (Ro m = ) : Distribution over time. : Time span used for time-averaged data Reτ T

22 Figure 2.3: Re τ (Ro m =.) : Distribution over time. : Time span used for time-averaged data Reτ T

23 2 Figure 2.4: Re τ (Ro m =.5) : Distribution over time. : Time span used for time-averaged data Reτ T As the streamwise and spanwise directions are assumed periodic and therefore infinite, all profiles were averaged along those directions. For all cases, the momentum statistics were timeaveraged and denoted b an overbar () for a dimensionless time period of 2 with a hundred evenl-spaced samples. These quantities were then scaled b the global friction velocit u τ. 2.4 Present Direct Numerical Simulation Results 2.4. Total shear stress, law of the wall, energ spectra and mean velocit For the nonrotating case (A), the total kinematic shear stress is shown along with the primar Renolds shear stress u v in figure 2.5. The profile of the total shear stress is linear in the case of full developed channel flow (Kim et al., 987), and the computed result shows agreement. Additionall, the law of the wall is plotted for the non-rotating case (A) along with the accepted approximations for the viscous sublaer and log-law regions in figure 2.6. The Von Karman con-

24 3 stant κ and constant C have the values.4 and 5.8, respectivel, and the normalized velocit distribution displas the appropriate profiles in the viscous sublaer and log-law regions. To show the adequac of the grid resolutions and computational domain for high wavenumbers, the energ spectra in the streamwise (x) and spanwise (z) directions are shown in figure 2.7 at selected points near the center-line and channel wall. The results demonstrate that the energ densit corresponding to high wavenumbers is several orders of magnitude smaller than the energ densit associated with low wavenumbers and that there is no evidence of aliasing at high wavenumbers at this grid resolution. Figure 2.5: Distribution of the shear stresses. : u v : u v + Re τ du/d u v

25 4 Figure 2.6: Law of the wall. : DNS velocit distribution; - - -: U + = + ; - : U + = κ ln(+ )+C U

26 Figure 2.7: One-dimensional energ spectra: (a) Streamwise ( δ =.3); (b) Spanwise ( δ =.3); (c) Streamwise ( δ =.85); (d) (b) Spanwise ( δ =.85); 5 (a) (b) Eii Eii 2 2 k x (c) 2 2 k z (d) Eii Eii 2 2 k x 2 2 k z

27 Figure 2.8: Mean velocit profiles. : Case A; - - -: Case B; : Case C, - : 2Ω lines C U + 5 B A The expected asmmetric stratification of the mean velocit profiles subjected to spanwise rotation (cases B C) is shown in figure 2.8. As observed experimentall and computationall b Johnston, Halleen, and Lezius (972) and Kristofferson and Andersson (993), respectivel, a region in the channel where the mean velocit varies linearl across the channel walls is created as a result of the spanwise rotation and is characterized b a linear slope of 2Ω in the mean velocit profile. This aspect is reflected in both rotating simulation cases in figure 2.8. The width of this linear region also grows larger with increasing rotation number, which correlates well with Kristofferson and Andersson (993). As case A does not include rotational effects, the corresponding mean velocit profile is expectedl smmetric about the channel centerline ( = ) and does not include a linear region.

28 Renolds stresses, turbulent kinetic energ and dissipation rate Figure 2.9: Renolds-stress profiles: (a) u u ; (b) v v ; (c) w w ; (d) u v. : Case A; - - -: Case B; : Case C. (a) (b) 4 3 u u 5 v v (c) (d) 2 w w 3 2 u v Renolds-stresses profiles are shown in figure 2.9 for all rotation numbers. For cases involving rotation (B C), the strength of the turbulence shifts towards the pressure side in figures 2.9(b d) as shown b the higher amplitudes of turbulent stresses in the region near the pressure wall ( = 2). This correlates with increased turbulence levels that would be expected in the pressure region due to the appearance of Talor-Gortler vortices (roll cells) as described b Grundestam et al. (28). There is also a subsequent decrease in the amplitudes of the turbulent normal stresses near the suction wall at =, which correlate with the expected suppressed turbulence levels in the suction region. For Ro m =., there is a slight increase in the peak value of u u on the pressure side of the channel, but this peak amplitude subsequentl decreases at the high rotation number Ro m =.5. In the profiles for v v, an increasing rotation number results in an increase in

29 8 the peak value at =.5. Similarl, when subject to rotation, the w w profiles displa a consistent peak value at =.8 that increases with the rotation number. The results of these simulations correspond well with Kristofferson and Andersson (993), who found that at a rotation number of approximatel Ro m =.2, the levels of u u saturate near the pressure wall, and instead energ is distributed to the other Renolds normal stresses components. In the DNS profiles for the shear stress u v, the introduction of spanwise rotation in case B produces a constant positive offset from case A throughout the entire domain with exception of the near-wall region, with a slight positive amplitude increase as the rotation number increases further. These Renolds stress profiles were consistent with the DNS results of Kristofferson and Andersson (993) for all rotation numbers. Figure 2.: Total volume-averaged turbulent kinetic energ. : Case A; - - -: Case B; : Case C k

30 Figure 2.: Turbulent dissipation rate ε. : Case A; - - -: Case B; : Case C ε The distributions of the total volume-averaged turbulent kinetic energ k are shown in figure 2. for all rotation numbers. Similar to the present DNS distributions of the Renolds-stresses for varing rotation numbers, the distribution of the turbulent kinetic energ across the channel is smmetric for non-rotating channel flow (case A); when rotation is introduced, the turbulent kinetic energ distribution becomes asmmetric, and the levels of turbulence are demonstrated to increase with increasing rotation number near the pressure wall ( = 2). The distributions of the turbulent dissipation rate ε are also displaed in figure 2., which similarl displa asmmetr due to added rotation Velocit fluctuations Various instantaneous streamwise and wall-normal velocit fluctuation contour plots at different rotation numbers are shown to assist visualization of the pressure and suction regions of the channel; colors within the contour plots represent the magnitudes of the respective velocit fluctu-

31 2 ations. Figure 2.2 shows the (x,)-planes of instantaneous streamwise and wall-normal velocit at the spanwise coordinate z/δ=π for simulation case A. It is observed that turbulence is evenl distributed throughout the channel for both fluctuation velocities. Figures 2.3 and 2.4 show the same instantaneous velocit contour plots for simulation cases B and C, respectivel, at the same spanwise coordinate. For low rotation number Ro m =., figure 2.3(a) demonstrates that wallnormal velocit fluctuations are suppressed in the region nearest to the suction wall ( = ) and high turbulence interactions are becoming restricted to the pressure region of the channel. The streamwise fluctuation contour plot shows a similar trend, but the suction region still contains long elongated structures. For the high rotation number Ro m =.5, the wall-normal fluctuations become more concentrated into a series of small quasi-periodic structures on the pressure side of the channel, and turbulence is highl suppressed elsewhere. This same tpe of trend is also observable in the contour plots for the streamwise fluctuations. These figures collectivel indicate that turbulence in spanwise-rotating turbulent channel flow is stratified in the wall-normal direction and the level of stratification increases as the rotation number grows larger. The Coriolis force increases with the rotation number, which forces fluid from the suction region into the pressure region, thus enhancing mixing and turbulence in pressure region while laminarizing the suction region where no mixing is occuring. Hence the enhanced and suppressed aspects of turbulence in the pressure and suction regions of the channel, respectivel, are clearl seen in contour plots of streamwise and wall-normal instantaneous velocit.

32 2 Figure 2.2: Instantaneous velocit fluctuation contour plots (Ro m = ). (a) Streamwise (b) Wallnormal /δ (a) x/δ. -. /δ (b) x/δ Figure 2.3: Instantaneous velocit fluctuation contour plots (Ro m =.). (a) Streamwise (b) Wallnormal /δ (a) x/δ. -. /δ (b) x/δ.5 -.5

33 22 Figure 2.4: Instantaneous velocit fluctuation contour plots (Ro m =.5). (a) Streamwise (b) Wallnormal /δ (a) x/δ. -. /δ (b) x/δ

34 Chapter 3 RANS Turbulence Models 3. Turbulence Model Overview Renolds-Averaged Navier-Stokes (RANS) based turbulence models are based upon analzing flows in two parts, an average (or mean) component and a fluctuating component. This concept of averaging is applied to the equations of mass and momentum conservation, producing terms known as the Renolds stresses. The modeling of these Renolds stresses is the primar objective of RANS-based models, which must solve the turbulence closure problem associated with these equations. One of the most common methods within these RANS-based models to obtain closure is to relate the Renolds stresses linearl to the mean strain rate tensor S ij through the definition of an edd viscosit. However, these models, known as linear edd viscosit models, have been known to fail significantl in complex flows, including rotational flows. Nonlinear edd viscosit models have also been developed, which use a nonlinear relationship between the Renolds stresses, mean strain rate and rotation tensors in order to more accuratel model complex flows; unfortunatel, a constitutive relationship between the Renolds stress and mean velocit gradient has been demonstrated to not exist in turbulence (Lumle, 97). This work considers three nonlinear edd viscosit models: the v 2 f, SG and Girimaji turbulence models, proposed b Reif, Durbin, and Ooi (999), Speziale and Gatski (993), and Girimaji (996), respectivel. In this section, three RANS models are being tested due to their practical computational costs and popularit in the engineering communit. However, the deficiencies of RANS models must be considered especiall the information that is lost from the results of RANS modeling

35 24 compared to that of a direct numerical simulation. As mentioned above, RANS models appl the notion of averaging to the Navier-Stokes equations, which eliminates phase information and retains onl amplitude information from the governing equations of the flow. Phase information has a plethora of uses in fluid mechanics research as it includes data such as the pressure signals at the channel walls, which are crucial information in research areas such as flow control as man methods of flow control design are centered about a certain frequenc range. Therefore the results obtained from RANS modeling would be useless to researchers in need of this information even if the results obtained from the RANS model were ver accurate relative to DNS data. 3.. v 2 f turbulence model The v 2 f turbulence model is similar to man linear edd viscosit models in which partial differential equations for the turbulent kinetic energ k and turbulent dissipation energ rate ε, shown in equations 3. and 3.2, are solved. This model also incorporates the two additional partial differential equations in equations 3.3 and 3.4, for v 2 and f, which correspond to a scalar analogous to the wall-normal Renolds stress and an elliptic relaxation function, respectivel. These equations (eq ) are also a function of the production term (P), kinematic viscosit (ν), edd viscosit (ν t ), turbulent length scale (L), and turbulent time scale (T); the model coefficients are shown in table 3.. The model expressions for the turbulent length and time scales are found in equation 3.5. Dk Dt = P ε+ x j ((ν+ν t ) k x j ) (3.) Dε Dt = C ǫ P C ǫ2ε + ((ν+ ν t ) ε ) (3.2) T x j σ ǫ x j Dv 2 Dt = k f v2 k + x j ((ν+ν t ) v2 x j ) (3.3) f L 2 2 f =(C ) 2/3 v2 /k k + C 2 P k (3.4) L=C L max( k3/2 ; C η ( ν3 ε ε )/4 ), T = max( k ε ; 6( ν ε )/2 ) (3.5)

36 25 Table 3.: Coefficients for the v 2 f Model (Reif et al., 999) Cǫ C ǫ2 σ ǫ C L C η f.4 ( +.45 ) v2 B solving equations , the v 2 f model produces an explicit expression for the Renolds stresses shown in equation 3.6, where Cµ is a function of the strain rate (S ij ) and rotation tensors (W ij ). More information on the v 2 f model s formulation and derivation is found in the appendices. u i u j = 2 3 kδ ij 2C µv 2 TS ij (3.6) 3..2 Speziale-Gatski turbulence model The Speziale-Gatski model, or SG model, is an explicit algebraic Renolds stress model (EARSM), a common subtpe of nonlinear edd viscosit models. In EARSM-tpe models, an implicit model for the anisotropic tensor b ij is constructed from the Renolds stress equation and then an explicit form for b ij is derived analticall. Using the relationship between the anistropic tensor and Renolds stress tensor shown in equation 3.7 (Pope, 975), the SG model proposes the following explicit expression for the Renolds stresses in equation 3.8. The model coefficients are found in table 3.2, and it is important to note the non-absolute nature of several coefficients. The terms η and ζ are equivalent to invariants of the scaled strain rate and rotation tensors, or S ij S ij and W ij W ij, respectivel; similarl, Π b represents the second invariant of the anisotropic tensor, b ij b ij. Model expressions for α, S ij, and W ij are found in equations 3.9, 3., and 3., respectivel, with S ij and w ij representing the dimensional strain rate and rotation tensors, respectivel. Additionall, the SG model applies the partial differential equations used in the v 2 f model and shown in equations 3. and 3.2 to compute the turbulent kinetic energ k and dissipation rate ε. For more information on the SG turbulence model and its formulation and derivation, readers are

37 26 Table 3.2: Coefficients for the SG Model (Speziale and Gatski, 993) C C 2 C 3 C 4 g P/ε Π b referred to the appendices. u i u j = 2 3 kδ ij b ij = u i u j 2k 3 δ ij (3.7) 6(+η 2 )α k 3+η 2 + 6ζ 2 η 2 + 6ζ 2(S ij + S ij W kj W ik S kj ) 2(S ik S kj 3 S mns mn δ ij ) (3.8) α = C 2 4/3 C 3 2 (3.9) S ij = 2 g k ε (2 C 3)S ij (3.) W ij = 2 g k ε (2 C 4)[w ij +( C 4 4 C 4 2 )e mjiω m ] (3.) 3..3 Girimaji turbulence model The Girimaji turbulence model is also an EARSM-tpe model and its basic formulation is ver similar to that of the SG model. The primar difference from the SG model occurs once the implicit model for the anistropic tensor is constructed consisting of nonlinear algebraic equations derived from the Renolds stress equation. With the amount of unknowns exceeding the amount of equations in the Renolds stress equation causing the well-known turbulence closure problem, these two models resolve this closure problem using different methods. The SG model uses mathematical functions known as Pade approximants to reduce the number of unknowns and form a regularized, or approximated, model that relates the anisotropic, strain rate and rotation tensors. Alternativel, the Girimaji model full solves the nonlinear equations under an imposed weak equilibrium assumption which presumes a constant ratio between the production and dissipative energ rate budget terms. More information regarding these models solutions to

38 27 Table 3.3: Coefficients for the Girimaji Mode (Girimaji, 996) L L L 2 L 3 L the closure problem is found in the appendices. The explicit expression for the Girimaji-modeled Renolds stresses is shown in equation 3.2. The coefficients of the Girimaji model are found in table 3.3, and it is noted that all of the coefficients are absolute. Similar to the SG model, the terms η and η 2 represent invariants of the normalized strain rate and rotation tensors: S ij S ij and W ij W ij, respectivel, and in this case these tensors are scaled b the edd turnover time k/ε. The expressions for the G coefficients are shown in equations 3.3 and 3.4; the various dependencies of these terms, such as model parameters D and b, are found in the appendices. For more detail regarding the formulation and derivation of the Girimaji turbulence model, readers are referred to the appendices. u i u j = 2k(G S ij + G 2 (S ik W kj + W ik S kj )+G 3 (S ik S kj 3 S mns mn δ ij )) 2 3 kδ ij (3.2) L L 2 L 2 +2η 2L 2 4 when η = p 3 G = +( b 2 + D) /3 +( b 2 D) /3 when D> p a 3 cos( θ 3 ) when D<and b< (3.3) p a 3 cos( θ 3 + 2π 3 ) when D<and b> L G 2 = 4 G L 2 η L G L G 3 = 4 G L η 2L 3 + η L G (3.4)

39 RANS Model Testing As noted in the previous section, the three RANS-based models in this work are all nonlinear edd viscosit models, and so the approximations for the Renolds stresses in these models are various functions of the mean strain rate tensor, rotation tensor, turbulent kinetic energ and dissipation rate. When these turbulence models are used for predictive purposes, partial differential equations of the velocit, turbulent kinetic energ, and dissipation rate are numericall solved to ield these quantities. However, since the purpose of this work is to test the modeling capabilities of these three turbulence models, these implemented quantities will be directl extracted from the DNS data base, thus ielding more accuratel modeled Renolds stresses for comparison with the Renolds stresses directl extracted from the DNS. The normal Renolds stresses (u i u i ) and shear Renolds stress (u v ) calculated from the DNS data using the three aforementioned turbulence models are compared directl to the results of the DNS for the three rotation numbers used in the present DNS stud. For a case of nonrotation (A), the results are shown in figure 3.. The approximations of the SG and Girimaji models are similar, and more accuratel model the Renolds normal stresses compared to the v 2 f model; the v 2 f model has the most accurate distribution of the Renolds shear stress, however. It is important to note that there are slight irregularities in the Girimaji-modeled Renolds normal stresses and a significant discontinuit in the Girimaji-modeled Renolds shear stress at the channel center-line.

40 Figure 3.: Modeled Renolds stresses profiles (Ro m = ): (a) u u ; (b) v v ; (c) w w ; (d) u v. : DNS; : v 2 f ; - -: SG; - - -: Girimaji. 29 (a) (b) 8 3 u u v v (c) (d) 2 w w 2 u v For low rotation number (case B), the modeled Renolds stresses and DNS results are plotted in figure 3.2, and for high rotation number (case C), the results are shown in figure 3.3. It is observed that once rotational effects are considered in the Girimaji turbulence model, the discontinuit and irregularities disappear from its modeled Renolds stresses. For the cases involving rotation, the SG model generall produces the most accurate modeled Renolds stresses compared to the distributions directl obtained from the DNS data.

41 Figure 3.2: Modeled Renolds stresses profiles (Ro m =.): (a) u u ; (b) v v ; (c) w w ; (d) u v. : DNS; : v 2 f ; - -: SG; - - -: Girimaji. 3 (a) (b) 4 3 u u 5 v v (c) (d) 2 w w 3 2 u v

42 Figure 3.3: Modeled Renolds stresses profiles (Ro m =.5): (a) u u ; (b) v v ; (c) w w ; (d) u v. : DNS; : v 2 f ; - -: SG; - - -: Girimaji. 3 (a) (b) u u 4 v v (c) (d).5 w w 2 u v Girimaji Model Correction In this section, the aforementioned discontinuit in the Girimaji-modeled Renolds shear stress profile in the non-rotational case (A) is removed b adding a corrective function to the turbulence model. The explicit expression for the Renolds stresses in equation 3.2 is examined for the dominant influences on the modeled Renolds shear stress term. The various groupings on the right hand-side (RHS) of equation 3.2 are individuall plotted for each G coefficient in figure 3.4, and the grouping composed the G coefficient, turbulent kinetic energ k and mean strain rate tensor S 2 is found to be the primar contributor to the modeled Renolds shear stress. No anomalies are present in the distributions of k in figure 2. and S 2 in figure 3.5, so it is concluded that the G coefficient is the primar cause of the discontinuit. The G coefficient has a strong dependenc on η, whose distribution is plotted in figure 3.6 and shown to approach zero at the

43 32 channel center-line. An inspection of the Girimaji model parameters shows an inverse relationship between η and G, which would cause the values of G to be elevated significantl near the channel center-line where η approaches zero. This combination of elevated magnitudes of the G coefficient and a change of sign in the S 2 tensor produces the discontinuit at the channel centerline for modeled Renolds shear stress. Therefore, this discontinuit is attributed to a deficienc in the G coefficient expressions as η. Figure 3.4: G coefficient groupings within the Girimaji-modeled Renolds shear stress (Ro m = ). : G ; - - -: G 2 ; - -: G u v

44 Figure 3.5: Mean strain rate tensor distribution (Ro m = ) S Figure 3.6: η distribution in the Girimaji model (Ro m = ). 2 5 η

45 34 The Girimaji model has an alternate expression for G at η = as shown in equation 3.3. However, this equation is shown in figure 3.7 to have similar values to the regular G coefficient used, including in the region of the discontinuit. Consequentl, the substitution of this expression into the model would not alter the value of the modeled Renolds shear stress enough to reduce or eliminate the discontinuit. Figure 3.7 also displas the expected large magnitudes of the G coefficient near the channel center-line relative to the rest of the channel. A correction to the Girimaji model is made b introducing a new limiting value for the G coefficient as η approaches zero. Through analsis and comparison of the profiles for the G coefficient and η, an adequate value of G was determined for a certain span of values of η that evened out the discontinuit. A corrective value for G of -.5 corresponding to values of η 2. was assigned to the Girimaji model as shown in equation 3.5, which produced the following u v profile shown in figure 3.8 and the G coefficient profile in figure 3.7. G =.5 for η 2. (3.5)

46 35 Figure 3.7: G coefficient in the Girimaji model (Ro m = ). : Original; - - -: With corrective function; : Girimaji model expression for η = G

47 36 Figure 3.8: Girimaji-modeled Renolds shear stress profile (Ro m = ). : DNS; - -: Prediction with correction; : Prediction without correction..5.5 u v As demonstrated in figure 3.8, the discontinuit is evened out in the modeled shear Renolds stress profile and a more accurate slope in relation to the DNS data is displaed compared to the original Girimaji-modeled u v profile. This correction was unable to eliminate the slight irregularities present in the modeled Renolds normal stresses in the Girimaji model. However, as shown in equation 3.2, the modeled Renolds normal stresses have a greater dependenc on the coefficients G 2 and G 3, therefore a similar analsis can be applied for those coefficients to eliminate the irregularities at the channel center-line in those profiles. Realizabilit is a set of mathematical and phsical principles that need to be satisfied to prevent the turbulence model from generating non-phsical results. The three realizabilit conditions demonstrated b Schumann (977) are shown in equations u i u i (3.6) u i u j 2 u i u i u j u j (3.7)

48 det(u i u j ) (3.8) 37 Realizabilit testing of the Girimaji model with the proposed correction was conducted and distributions corresponding to the conditions in equations 3.6, 3.7 and 3.8 are shown in figures 3.9, 3. and 3., respectivel. Figures 3.9(a) - (c) demonstrate that the modeled Renolds normal stress distributions are non-negative which satisfies equation 3.6. Figure 3. displas the inequalit in equation 3.7, also known as Schwartz s inequalit, and the expression is shown to hold true. Finall, figure 3. demonstrates that the determinant of the modeled Renolds stress tensor is non-negative, which satisfies the condition in equation 3.8; consequentl the ad-hoc correction to the Girimaji model was established to satisf all three realizabilit conditions. Figure 3.9: Girimaji-modeled Renolds-stress distributions with correction (Ro m = ). (a) u u ; (b) v v ; (c) w w ; (d) u v. (a) (b) u u 2 v v (c) (d) 2 w w 2 u v

49 38 Figure 3.: Girimaji-modeled Renolds stress distributions for equation 3.7 [Schwartz s inequalit] (Ro m = ). : u u 2 2 ; - - -: u u u 2 u u i u j

50 Figure 3.: Determinant of Girimaji-modeled Renolds stress tensor (Ro m = ) det(u i u j )

51 Chapter 4 Pressure-Strain Modeling in Explicit Algebraic Renolds Stress Models 4. Pressure-Strain Overview As the SG and Girimaji turbulence models have been demonstrated to have usefulness in modeling the Renolds stresses in spanwise-rotating turbulent channel flows, a closer examination of these models can ield useful information on the influences of various model terms on the accurac of the modeled Renolds stresses. The SG and Girimaji models are both EARSM-tpe models, which utilize the Renolds stress equation (eq. 4.) b modeling the higher order terms (budget terms) that appear in that equation. The budget term variables listed in equation 4. include the pressure-strain term (Φ ij ), dissipation term (ε ij ) and pressure diffusion term (Dij T); R ij is equivalent to u i u j. DR ij Dt = R ik u j x k R jk u i x k + Φ ij ε ij 2Ω m (e mkj R ik + e mki R jk )+D T ij + ν 2 R ij (4.) Φ ij = p( u i x j + u j x i ) (4.2) Π ij = C ǫb ij + C 2 ks ij + C 3 k(b ik S kj + S ik b kj 2 3 b mns mn δ ij ) C 4 k(b ik W kj W ik b kj ) (4.3) In these models, the pressure-strain term (eq. 4.2) is of paramount importance as the correlations between pressure and strain fluctuations pla a dominant role in intercomponent energ transfer (Launder, Reece, and Rodi, 975). Consequentl, the distributions of the pressure-strain function for each of these models are compared to the pressure-strain budget values directl extracted from the DNS simulations. Both of these models appl the linear Speziale, Sarkar, and