Complex Turbulent Flow: Potentials of Second-Moment and Related Closures and Examples of Application

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1 Title /3 Complex Turbulent Flow: Potentials of Second-Moment and Related Closures and Examples of Application by K. Hanali Department of Multi-scale Physics, Delft University of Technology Delft, The Netherlands Aim and Content of the Lecture / Definition of complex turbulent flows ; Implications and Requirements on Turbulence models Numerical methods Boundary conditions Illustration and discussion of performances of basic Eddy Viscosity (EVM) and Differential Stress Model (DSM) in complex turbulent flows Some recent novelties in modelling and computing certain classes of complex turbulent flows Future prospects Conclusions and recommendations

2 Which Flows are Complex? 3 / Complexity, lie beauty, is in the eye of the beholder (H. McDonald, 980); Here the notion implies generally a departure from the thin shear layer approximations as well as any additional geometrical and physical complexities: Strong pressure gradient (acceleration and deceleration), pulsation Presence of extra strain rates (curvature, lateral divergence, bul dilatation, swirl..) Body forces (rotational, buoyancy, magnetic, ) Flow separation and recirculation Stagnation and streamlines reattachment Secondary motions (Sew-induced and Stress-induced) Laminar-to-turbulent and reverse transition (low Re-number effects) Three-dimensionality effects Compressibility effects and flow discontinuities (e.g. shoc waves) Strong property variations, chemical reactions, radiation; Multi-phases and multi-components Modelling & Computational Implications 4 / Listed phenomena influence directly the turbulent stresses, (mainly via the stress generation), but also the turbulence scales; These effects should appear in both the stress- and the scale-providing equations; Some cases require simply additional exact or modelled equations and terms in the model (e.g. body forces, rotation, compressibility, multiphase- and multi-component equations); In other cases the model is expected to account for physical complexities itself (separation and recirculation, secondary motions,turbulence hysteresis, etc.); Abilities to capture complexity effects is the basic criteria for udging the success and generality of a flow model; Most of the listed phenomena require modifications of basic numerical method (often designed to solve the Navier-Stoes type of equations)

3 Modifications of - Eqn. For Complex Flows 5/ Sensitizing the -equation to irrational strains (an extra term, Hanalic & Launder 980) improves the predictions flows with strong pressure gradients ~ (with C 4. 44) : C U U ~ ' i l lm ; In thin shear flow C ( u u ) x xm In some cases the term account also for the streamline curvature, e.g. bacward facing step - though spoils the curved shear layer predictions; A control of excessive length scale in wall region can be achieved by adding a term in -equation (where l 3 / / and C. 5) l U x Yap, 987: S l 0.83 Cl x n l C l xn Hanalic et al. 995: Sl max Cl l x n Cl l ~ A ;0 x n Realizability Constraints for - Models 6/ Recall that in EVM the turbulent stress is defined as U U i uu i ts where S 3 x x and t C / i If - equation is written as D C C PC ( ) Dt x xl note that the time scale can grow excessive (e.g. stagnation point), causing unrealistic diminishing of (detached from P?) and, consequently, excessive! Note also that in some cases some normal stress component may go negative! Durbin (995) proposed to limit by imposing realizability constraints u 0 on stress-strain equation. By noting that S is symmetric, with only diagonal components in principal-axis coordinates, its eigenvalues satisfy 3 SS S and 3 0 ( for const) It follows that: For principal axes, u t which for realizability constraint 3 leads to the time scale bounds or C 6C S 6 S Or, in a more convenient form: S /3 a min, with a C S u 0

Realizability Constraints for EVM Illustration: Heat Transfer on GT rotor blade 7/ Predictions of heat transfer coefficient with - and - EVM without and with time scale bound (Medic and Durbin, JT

4 Realizability Constraints for EVM Illustration: Heat Transfer on GT rotor blade 7/ Predictions of heat transfer coefficient with - and - EVM without and with time scale bound (Medic and Durbin, JT 00) Turbulence intensity: Standard - Standard - with bound Extra Strain-Rates 8 / This motion is associated with any extra rate of strain (and body force effect) in addition to the simple (basic) shear U / x

5 Streamline Curvature 9 / We may distinguish between the local streamline curvature irrespective of the shape of flow boundaries (e.g. recirculating region behind a step), and the bul curvature (e.g. curved channels): Streamline curvature attenuates the turbulence when the mean flow angular momentum increases with curvature radius (e.g. over a convex surface - stabilizing curvature); It implies the turbulence in the opposite situation (e.g. over a concave surface - destabilizing curvature) Even a small extra strain rate can have a significant effect on stress production: e.g. in a thin shear flow with mild curvature U / x U / x, but u u hence both terms in P u U / x u U / x are of importance Modelling the Curvature Effects 0 / Basic EVMs fail to account for curvature effects in most turbulent flows and an ad hoc correction is usually applied by modifying the sin term in the -equation, (e.g. Launder 977, Durst & Rastogi 980): C (0. Rit) where Rit U ( RU ) R R R is the local radius of streamline curvature, whereas U mean velocity is the resultant Note: Rit (Richardson No.) changes its sign and magnitude in accord with the curvature sign (stabilizing or destabilizing effect); DSM captures these effects via exact production terms in the stress equations which show a selective sensitivity to the streamline curvature; Basic ASM performs marginally better than the basic EVM; With account of extra strain rate U / x (local curvature) or U (longitudinal curvature) ASM / R performs well (e.g. Cheng & Farouhi 99)

6 Local Curvature: Illustrations / DSM computation of a highly curved mixing layer (Gibson & Rodi 98): Longitudinal Curvature: Illustrations / ASM computations of U-bend and S-bend (Cheng & Farohi 99; see also Iacovides & Launder, 985): Mesh system for two-d S-bend duct geometry Static pressure coefficient distribution for a degree S-bend

7 Flow System Rotation 3 / Bul flow rotation affects both the mean flow and turbulence by action of the Coriolis force, influencing stress intensity and turbulence scale; Mean momentum equation: DU Dt i P U i uiu xi x x U Note: P P x x / and is the system angular velocity; Stress transport equation contains the exact rotational production term; R ( uium m u um im ) The Coriolis force U acts perpendicular to the velocity vector U and has no direct influence on the -budget (R ii =0), but redistributes the stress among components and modifies the turbulence production; Rotation causes a decrease of dissipation in isotropic turbulence (DNS, Bardina et al. 985) Effects of Rotation on Turbulence 4 / Effects of rotation on turbulence can be illustrated in a simple shear flow (e.g. a plane channel) where the angular velocity vector ( system vorticity ) (here 3 ) is aligned with the dual vector of the mean shear vorticity / ( U i / x U / xi ) (here / U / x ) If 3 and have the same direction, rotation attenuates the turbulence ( stabilizing effect ); if 3 and have the opposite directions, rotation will amplify the turbulence ( destabilizing effect ); This criterion is often expressed in term of Rossby number defined as the ratio of the mean shear vorticity to the system vorticity (here R o ( U ) or its / x ) / 3 reciprocal S = /R o

8 Modelling the Rotation Effects (DSM) 5 / In DSM, the effect of rotation should be accounted for in the model of the pressure-strain by replacing P in by the total stress generation P, R (or by P to ensure material frame-indifference, Thomas and Tahar,988); 0. 5 R -equation should, in principle, be modified to account for effects of rotation on turbulence scale; some proposals for extra terms: / Bardina et al. (987): C ( 0.5W W ) Shimomura (993): C ( ) where C 0.5 and W i is the intrinsic mean vorticity Successful DSM-computation of rotating channel flows, with no modification of equation, where reported by Launder et al., 987; ASM-computation of rotating mixing layer by Nilsen and Andersson shows also a negligible effect of the rotational source term in -equation System Rotation: Illustration 6 / DSM computation of a flow in a rotating plane channel (Launder et al. 987)

9 System Rotation: Illustration 7 / Mean velocity and shear stress in a rotating plane channel, Re m =590, Ro 0., Low-Re-DSM compared with DNS (Jairlic 997) Swirl 8 / Swirl provides another imposed fluid rotation with its axis usually aligned with the mean flow direction (Coriolis force = 0); Swirl enhances the turbulent mixing and often induces recirculation; these features are exploited in IC engines and gas-turbine combustors; EVM computations employ the swirl-independent coefficients in the modelled equations, but generally with a little success; Basic DSM did not improve much predictions of a swirling et; Simple IP model of performed better than LRR model of, performed better than LRR model, Recognizing that a maor deficiency lies in Fu et al. 987 proposed the inclusion of convection C in form to ensure material frame indifference, (negligible effects in non-swirling flows): C [( P R P 3 ) ( C, C 3, Still better predictions are expected with improved scale equation. )]

10 Swirling Flows: Illustration 9 / Confined swirling coaxial ets (EVM, DSM, Hogg & Leschziner, 989) RSM Swirling Flows: Illustration 0/ Axial and tangential mean velocities in a swirling flow in a long pipe Low- Re-DSM compared with experiments (Jairlic 997)

11 Secondary Currents / This notion implies usually secondary motion with longitudinal, streamwise vorticity superimposed on the mean flow in x -direction; Sew-induced (pressure-driven) (Prandtl s first ind of secondary flows) is essentially an inviscid process, generated by the deflection of existing mean vorticity; viscous and turbulent stresses cause to diffuse Turbulent-stress-induced (Prandtl s nd ind of secondary flow), is generated by the turbulent stresses; Secondary motion can appear in form of cross flow lie in 3-D thin shear flows ( 0.5 U 3 / x ), or in form of recirculating cross currents lie in non-circular ducts ( 0.5 ( U 3 / x U / x3 ), Sew-induced secondary velocity can be high, while the stress-induced currents are wea, though still very important for turbulent transport. Modelling Secondary Flows / Mean vorticity dynamics is defined by ( for component): D Dt x x U x U U 3 x x3 " sewinduced" uu3 uu3 u u3 x x3 xx3 " stressinduced" generation of Sew-induced secondary motion, driven essentially by pressure does not require a complex turbulence model; Stress-induced motion can not be handled with EVM and requires a model which can compute individual turbulent stress components, (DSM or ASM)

12 3-Dimensionality 3 / Even mild 3-dimensionality of the mean flow produces significant changes in turbulence structure; Current turbulence models have been developed on the basis of our nowledge of -D flows; Plausible extensions to third dimension does not give always satisfactory results; Experimental investigations of 3-D flows have been intensified recently broadening the basis for refinement of turbulence models and numerical methods for computation of complex 3-D flows; Complex flows require turbulence models of higher order than EVM, employment of non-orthogonal body-fitted coordinates and integration through the viscous sublayer to the wall (low-re-number models); This is still a formidable tas; current achievements in the computation of complex 3-D flows rest at the most on ASM with separate treatment of viscous wall layer with - or even -mixing length models Computations of 3-D Boundary Layers 4 / Even in simple 3-D boundary layers eddy viscosity is not isotropic, i.e: uu u3u U / x U / x 3 (x is the coordinate normal to the wall); Hence, basic EVMs can not reproduce any 3-dimensionalities; Shima 99 computed three cases of 3-D boundary layer with a low-re-number DSM; Predictions of fully developed pseudo Eman layer are good, but BL s on infinite swept wing and rotating cylinder are less satisfactory.

13 Pressure Dominated Flows 5 / Strong pressure gradients modify the mean rate of strain and depending on the sign amplify or attenuate turbulence; Extreme cases are the laminarization of an originally turbulent flow, when dp / dx 0 (severe acceleration) and the flow separation when dp / dx 0 (strong deceleration); Both extreme cases represent a challenge to turbulence modelling; Turbulent flows subected to periodic variations of pressure gradient or other external conditions (pulsating and oscillating flows) fall into the same category with an additional feature: hysteresis of turbulence field lagging in phase behind the mean flow perturbations; Basic EVM can not capture these features; DSM performs generally better, though additional modifications (mainly in scale-equation) are needed; Wall functions are inapplicable for specifying boundary conditions Wall B.C. and Near-Wall Treatment 6 / Wall function treatment becomes inapplicable in most complex flows: - Extra strain-rates and strong p disturb local equilibrium in near-wall region (reattachment, impingement, recirculation, rotation..); - Laminar-to-turbulent and reverse transition (low Re-numbers), high acceleration, rotation, buoyancy or magnetic force, property variation; More exact approach is to employ a turbulence model with incorporated lo-re-number and wall proximity modification; Near-wall (Low-Re-number) models require fine numerical mesh within the viscosity affected wall sub-layer; Computations become demanding on computer resources and impractical for complex flows, but can not be avoided if e.g. transitional phenomena are in focus; Possible compromises: Two-layer approach, -, or even ML in the viscous sublayer, DSM in the rest of flow (Frane & Rodi 99, Lien 99), or Elliptic Relaxation (Durbin 99, 993)

14 Failure of Wall Functions 7 / Velocity distribution in the recovery region after reattachment behind a step (Low-Re-DSM, Hanalic & Jairlic, 997) RSM for Near-wall &Low-Re Number Flows 8 / RSM have been extended to account for Low-Re-number and near-wall effects (e.g. Hanalic & Launder 976, Launder & Shima 989, Launder & Tselepidais 99, Hanalic & Jairlic 993, and others); Modifications involve: - Inclusion of viscous diffusion in all equations; uu i - In uu i eq'n : [ fs ( fs) ]; fs fs(re t ) 3 - Additional term is added to the -equation (to model P 3 ) - Coefficient C,C,C..) are replaced by functions of turbulence Re number, Re /( ) and/or A t Satisfactory results have been obtained for a range of -D and some 3-D low- and high Re-number wall flows including cases of acceleration (laminarizing) and oscillating turbulent flows, by-pass and separationinduced transition, airfoil, and 3-D boundary layer,..

15 Elliptic Relaxation Second-moment Closure 9 / Instead of local damping functions, Durbin (99,993) proposed to solve the Elliptic Relaxation (ER) equation for f which modifies near a wall accounting for inhomogeneity and wall blocing*: L f f *Note that instead of, Durbin uses h f ( 3 uiu ) ( ), where p u i x u p x i Any model for can be used for h (usually Rotta+IP model suffice) Durbin s ER mimics better the physics, but requires solution of tensor elliptic equation for f with exact (complex!) wall boundary conditions. With lower scale bounds to account for viscosity integration to the wall 3/ 3 / 4 max, C ; L CL max, C / 4 DSM with Elliptic Blending (EB-DSM) 30 / Durbin s EVM v f model performs significantly better than -, but still can not match performance of DSM (stress anisotropy?) A proposition: elliptic blending DSM model (DSM-EB), (DSM with a scalar elliptic relaxation function, Manceau and Hanalic 00): uu * w h ( ) ( ) i where: h C a C( P P ) or SSG, or... 3 w 5 [ uu i nn uunn i uun l nl( nn i )] D CPC ( l Cuu l ) and Dt x xl The geometrical effects are reproduced from an ER equation for : L and n where 0, Kolmogorov scales used as low bounds as in the original Durbin model w

03( ) uu i nn i h and the coefficients in are chosen from the adopted model (IP, SSG,.

16 Illustration: EBM of Plane Channel Flow 3 / Comparison of computations for a plane channel flow at Re =590: Durbin s uiu f model; EB-DSM; Symbols: DNS Coefficients in EMB: C 0.8 C.85 C 6.0 C L 0.6 C 80 C.40.03( ) uu i nn i h and the coefficients in are chosen from the adopted model (IP, SSG,..) Illustration: EBM of Multiple Impinging Jets 3 / Heat transfer in multiple-impinging ets (Thielen, Hanalic and Joner, 004) ; symbols: experiments (Geers 004) SGDH: t ui T T x i GGDH: u C i u u i T x AFM: T U i u i C c u iu c u c 3 x x a u

17 Elliptic relaxation EVM: -v -f model 33 / Maor role of the damping function f in near-wall EVMs is to account for inviscid wall blocing effect that : suppress primarily the wall-normal fluctuations Durbin (99) proposed an EVM in which the eddy viscosity is defined as t C where is a scalar variables wall-normal stress close to a solid wall for which a separate equation is solved: D t f n Dt x x Here f is an elliptic relaxation function analogue to f in ER-DSM, defined by an elliptic equation f C P f L C ( n) x 3 B.C for n=: 0 f B.C. for n=6: f w =0 (numerically stiff) w ; 4 y (convenient but inferior ) Lower scale bounds (with realizability constraints): / max 3/ 3/ / min,, C max min, L, C 6 C S 6 C S Coefficients: C =0. C =6; C L =0.5; C =85; C =.4; C =0.3; v =.0 Elliptic relaxation EVM: --f model 34 / A robust version of ER EVM: instead of, an equation for is solved (Hanalic & Popovac 004): D with t f P X Dt x x t c where X is extra term (from direct transformation of and equations (X= 0). The elliptic relaxation equation (using SSG model for h ) is now: f P 3 P f L C C ' C x Advantages: - instead of in the source of the equation, we have P in the equation (easier to reproduce accurately near a wall) 4 In contrast to y, y what maes the wall boundary conditions less sensitive to grid clustering: w 0 f w w y y A variant with zero wall B.C. for f: Solve elliptic equation for f (the same as above) but with f and insert in equation w 0 / f f xn Coefficients modification: C L =0.35; C =0.85; =.; C =.4(+0.0/) /

18 --f model: illustrations 35 / Axisymmetric impinging et flow (Re=3000, Exp: Baughn & Shimizu): (Comp.: Popovac & Hanalic 004): Nusselt number Velocity profiles A hybrid EVM-DSM Model 36 / For complex flow, an interesting combination of DSM and EVM has been proposed by Basara and Jairlic (003): Solve momentum equation decoupled from uu equation, i.e. with eddy viscosity i t C / where C const but a function of uu i and S uu Solve i equation with U i from momentum equation and from its equation (one-way coupling) Evaluate C from the least square error for all stress components (in fact imposing exact inetic energy production in the - model): uu i S U U i C, where S S ; Si S S / x x i Disadvantage (c.w. EVM): need to solve equations for all stress components Advantages: Numerically robust (comparable to standard - model) Performance in predicting complex flows closer to full DSM than to EVMs

19 Illustration of Low-Re number DSMs: Pressure-induced Separation 37 / Separation bubble on a flat plate (DSM, Hadzic & Hanalic, 998), DNS: Spalart & Coleman, 997 DNS Low-Re-number DSM: Illustration 38 / Oscillating channel flow (DSM Hadzic & Hanalic, 99, Experiment Jensen et al. 989)

20 Illustration of DSM: Transition 39 / Separation-induced transition on a round-leading edge flat plate (Hadzic & Hanalic, 996) Illustration:Flat-wall Separation Bubble 40 / Transition in a separation bubble on a flat wall: comparison with DNS of Spalart and Strelets (997) (Hadzic and Hanalic 999)

21 3-D Boundary Layers: Illustration 4 / Response of turbulent stress components to a suddenly imposed transverse shear (rotating aft-cylinder), (DSM, Hanalic, Jairlic & Durst, 994, Experiment Bissonet & Mellor, 974) Vortex brea down in a Rotating Flows 4 / Streamlines and axial velocity in a spin-down RCM: Low-Re-DSM compared DNS experiments (Jairlic et al. 999)

22 Illustration of DSM: Swirl + Compression 43 / Velocity vector and swirl velocity in a RCM, Low-Re-DSM vs DNS (Jairlic et al. 999) Illustration of DSM: Sharp-edge Separation 44 / Flow behind a bacward-facing step, streamlines, friction and pressure coefficients, (DSM Hanalic & Jairlic, 997; Le & Moin, 994; Experiment: Jovic & Driver, 995)

23 Illustration of DSM: Smooth Flow Separation 45 / NACA 44 Airfoil, Re=.5 x 0 6, Low-Re-DSM (Hanalic & Jairlic, 999) Illustration of DSM: 3-D Separation 46 / Separation at car rear window, DSM (SSG) (Basara et al. 996)

Illustration of DSM: 3-D Separation 47 / Ahmed car model (35

00) Pressure field in the central plane (True LES: ~ 5-0 mill.

) Illustration of DSM: Flow in an Engine 48 / Temperature and

24 Illustration of DSM: 3-D Separation 47 / Ahmed car model (35 deg slant): U-RANS, U SSG +WF (500 cells) (Ouhlous and Hanali,, 00) Pressure field in the central plane (True LES: ~ 5-0 mill. cells!) Illustration of DSM: Flow in an Engine 48 / Temperature and velocity field in a direct-inection stratified charge engine (DISC) at top dead center (TDC): EVM and DSM (Yang et al. 998)

25 Illustration of DSM:Car Cabin 49 / Velocity and turbulence inetic energy in a simplified car interior ventilation system: EVM and DSM (Basara et al., 997) Conclusions 50 / A close loo at various phenomena in complex flows show that they can be captured only with DSMs! Extensive comparisons have confirmed this: even basic DSM s is more reliable than EVMs; In some cases ASM (NEV) with ad hoc remedies may give acceptable accuracy, if applied nowledgeably The integration up to the wall can not be avoided if transition phenomena and wall friction and heat transfer are to be solved No grid cells log Re, but the thicness of the viscous layer decreases with Re 7/8 : hence integration to the wall at very high Re 3-D flows is still difficult: the unstructured or bloc-structured grid with local refinement may ease the problem A minimum of both, the modelling and computing expertise is a prerequisite for trustworthy results.

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