Introduction to Turbulence Modeling

Transcription

1 Introducton to Turbulence Modelng Professor Ismal B. Celk West Vrgna nversty CFD Lab. - West Vrgna nversty I-1

2 Introducton to Turbulence CFD Lab. - West Vrgna nversty I-2

3 Introducton (contnued) What s turbulence? Flud flow occurs prmarly n two regmes: lamnar and turbulent flow regmes. Lamnar flow: smooth, orderly flow usually restrcted to low values of key parameters such as Reynolds number Turbulent flow: fluctuatng, dsorderly (random) moton of fluds CFD Lab. - West Vrgna nversty I-3

4 Introducton (contnued) Steady and nsteady Lamnar and Turbulent Flow CFD Lab. - West Vrgna nversty I-4

5 Introducton (contnued) Characterzaton of Turbulence: Irregular (dsorderly or random ) Transent (always unsteady) Three-dmensonal (spatally varyng n 3D) Dffusve: enhances mng and entranment Dsspates knetc energy nto heat Occur at large Reynolds numbers CFD Lab. - West Vrgna nversty I-5

6 Introducton ( contnued) What s turbulence (contnued)? Beyond the crtcal values of some dmensonless parameters (e.g. Reynolds number, Grashof number, Froud number, Rchardson number, etc.) the lamnar flow becomes unstable and transtons tself nto a more stable but chaotc mode called turbulence characterzed by unsteady, and spatally varyng (three-dmensonal) random fluctuatons whch enhance mng, dffuson, entranment, and dsspaton. CFD Lab. - West Vrgna nversty I-6

7 Lamnar Flow Eamples (From Woods et al., 1988) (From Van dyke, 1982) Ppe Flow Re Flow past a crcular cylnder Re 41.0 CFD Lab. - West Vrgna nversty I-7

8 Turbulent Flow Eamples (From Van Dyke, 1982) (From Van Dyke, 1982) Homogeneous turbulence behnd a grd Turbulent water et CFD Lab. - West Vrgna nversty I-8

9 Turbulence Scales Root mean square (rms) Velocty fluctuatons: u Length (eddy sze): Tme, /u Turbulence Reynolds number Re t u /ν Large eddes n a turbulent boundary layer (From Tennekes and Lumley, 1992): ~ L t boundary layer thckness Turbulent knetc energy: k 1/2(u 2 v 2 w 2 ) 3u 2 /2 Dsspaton rate: ε ~ u 3 / Kolmogorov scales K (ν/ε) 1/2 K (ν 3 /ε) 1/4 u K (νε) 1/4 CFD Lab. - West Vrgna nversty I-9

10 Why turbulence modelng? Memory and tme requrements for Drect Numercal Smulatons (DNS) s prohbtve Industral applcatons nvolvng comle geometres, multphases, and reactons are mpossble to smulate Remedy: Solve tme averaged equatons and model the turbulence statstcs. Grd ponts and tme step requrements for DNS of channel flow (After Wlco, 1993) Re H Re N DNS No. of Grd Nodes Number of Tmesteps 12, ,00 30, ,00 61,600 1, ,00 Re 10 Re t ; N ~ (l/η) 3 ~ Re t ,000 4, ,00 CFD Lab. - West Vrgna nversty I-10

11 Transport Equatons: Conserved scalar J w Control volume balance: {tme rate of change} {net flu through the surfaces} {net source} Δ y ( ρ φ Δ Δ y Δ z ) t ( ρφ ) J n ( ρ φδvol.) Δ CFD Lab. - West Vrgna nversty J s t conventon Tme Rate of Change ( J J ) ( J J ) ( J J ) source snk e w J e J ( ρuφ ) ( Γ φ ) n e dffuson s J w top ( J ) Δ Smlarly for J n, J s, J top, J bot m J m φ ρa; bot w AΓ φ Convecton Dffuson Sφ net source (Flu) mass flow rate I-11

12 Naver Stokes Equatons Vector Notaton ( ) ρ t ( ) ρ P θ ~ ~ ~ ~ Tensor Notaton ( ρ ) ( ρ ) t P Cartesan Coordnates, (,y,z); (,V,W) -Momentum Equaton ( ρ ) ( ρ ) ( ρv ) ( ρw ) t I II y II y z II z P III IV ITme rate of Change IIIPressure Gradent IIConvecton IV Vscous dffuson y y z z CFD Lab. - West Vrgna nversty I-12

13 ( ρ ) ( ρ h) h t dh C p dt; q Energy Equaton: (q T k ) P Φ t P (Heat flu n - drecton); Fourer s Law Φ Vscous dsspaton functon, h enthalpy, T Temperature CFD Lab. - West Vrgna nversty I-13

14 Newtonan Fluds Stress Tensor [ ] y z Normal Stresses yy zz 2μ V 2μ W 2μ y yy zy 2 μ. 3 2 μ. 3 2 μ. 3 z yz zz yz z : Symmetrc 2 nd Order Tensor y (t t, e.g. t y t y ) z zy Shear Stresses y μ y V. V W μ. z y ~ ~ W μ. z ~ ~ ~ ~ CFD Lab. - West Vrgna nversty I-14

15 I-15 CFD Lab. - West Vrgna nversty Equatons for Incompressble Fluds/Flows Equaton of Contnuty Momentum equatons Thermal energy equaton c p, c v constant; dh c p dt ( ) ( ) T h p p T k T c. T c t Φ ρ ρ 2 1 S 0 0; ~ ( ) ( ) ( ) g S 2 P t ρ μ ρ ρ ; stran rate tensor

16 Averagng Technques: Reynolds Averagng u u u' ( t) u ' ( t) < > Statonary Turbulence nstatonary Turbulence CFD Lab. - West Vrgna nversty I-16

17 Averagng Technques: Reynolds Averagng <> u; Notaton u u fluctuatng component of (,t) Tme average: < Ensemble average: Phase Averagng: < < > > Δt wndow wdth 1 N 1 Δt N 1 1 Δ t t 0 Δt t Δ t 0 (t)dt ;Lmt (, t ); N Large 2 (, t ) > (, t Δ t 2 ) d as Δt CFD Lab. - West Vrgna nversty I-17

18 Averagng Rules: Averagng < V > <> <V>; < <> > <>; <><V>> <><V> <d/dt> d(<>)/dt; <d(v)/d> d (<V>)/d average of a dervatve dervatve of the average <u> 0; average of the fluctuatons s zero <V> <><V> <uv> ; <uv> 0. (non-lnear terms!) Comment: Average of lnear terms s the same wth the averaged quanttes substtuted, Non-lnear terms, e.g. d(v)/d, lead to etra terms that need to be calculated separately. CFD Lab. - West Vrgna nversty I-18

19 I-19 CFD Lab. - West Vrgna nversty Reynolds Averaged Equatons: ncompressble fluds wth constant propertes 0 u 0; Note: T<T>ϕ; Drop < > when not necessary ( ) ( ) ( ) ( ) Stresses Reynolds ; ; u u S P t ρ μ ρ ρ ( ) ( ) ( ) ϕ ρ ρ ρ p t T p p u c T k T c T c t ( ) flues Turbulent u q S u ; conserved scalar; ; φ φ ρ ρ Φ Φ Γ Φ Φ t u ρu

20 Typcal shear flows: Mean flow Fully developed lamnar and turbulent flow n a channel (From Whte, 1991) Epermental turbulent-boundary layer velocty profles for varous pressure gradents (From Whte, 1991) CFD Lab. - West Vrgna nversty I-20

21 Typcal shear flows: Mean flow Structure of turbulent flow n a ppe (a) Shear stress (b) Average velocty (From Munson, et al., 1994) CFD Lab. - West Vrgna nversty I-21

22 Typcal shear flows: Mean flow nversal plot of turbulent velocty profles n zero pressure gradent After Hoffmann and Perry[11] (From Cebec and Bradshaw, 1988) nversal plot of turbulent temperature profles n zero pressure gradent. After Hoffmann and Perry [11] CFD Lab. - West Vrgna nversty I-22

23 Typcal shear flows: Mean flow 1 u ln( y ) B κ Comparson of Spaldng s nner-law epresson wth the ppe-flow data of Indgren (1965) CFD Lab. - West Vrgna nversty (From Cebec and Bradshaw, 1988) Mean-temperature dstrbuton across the layer as a functon of molecular Prandtl number I-23

24 Typcal shear flows: Mean flow Epermental rough-ppe velocty profles, showng the downward shft ΔB of the logarthmc overlap layer Composte plot of the profle-shft parameter ΔB(k) for varous roughness geometres, as compled by Clauser (1956) Boundary layer velocty profles for rough walls Notaton v* * 1 ln y B Δ B(k s ) ; κ / * ; y * y * /ν (From Whte, 1991) 1 1 y ΔB ln(1 0.3k s ) for ks > 60; ln( ) 8.5 κ κ k CFD Lab. - West Vrgna nversty s I-24

25 Fluctuatng Veloctes n a boundary Layer (From Whte, 1991) (From Frsch, 1995) CFD Lab. - West Vrgna nversty I-25

26 Transton and Intermttency Fg. 9. Predcton of bypass transton n zero-pressure-gradent flat plate boundary layer at 3 percent free-stream turbulence (taken from Henkes et. al. [49]) (from Leschzner, 2000) CFD Lab. - West Vrgna nversty I-26

27 Transton and Intermttency Flow regmes and drag coeffcent for varous obects (From Munson et. al, 1994) CFD Lab. - West Vrgna nversty I-27

28 Summary Part-I Flud flow ehbt two dstnct flow regons Lamnar flow : regular, smooth, low Re Turbulent flow: random, agtatve, hgh Re Transton from lamnar to turbulent flow regme s a functon of Reynolds number Inlet turbulence ntensty and length scale Wall roughness, other factors Important dmensonless parameters: Rchardson number, Grashof number, Froude number, Mach number etc. Turbulent flows ehbt a wde separaton of length and tme scales Kolmogorov scales are the smallest scales Complete resoluton of turbulent flows (.e. all scales) s very epensve and tme consumng and n many cases t s mpossble Remedy: se tme averaged (Reynolds) equatons and model the apparent turbulent stresses and turbulent flues. CFD Lab. - West Vrgna nversty I-28