Colloquium in honour of Geneviève Comte-Bellot. Similarities and Differences Rainer Friedrich, TU MünchenM

Transcription

1 Colloquium in honour of Geneviève Comte-Bellot Compressible Turbulent Channel & Pipe Flo: Similarities and Differences Rainer Friedrich, TU MünchenM

2 Compressible turbulent flos Motivation Compressible turbulent flos are an important element of high-speed flight G. Comte-Bellot Coll. 009

3 Stud of compressible turbulent flos Names from the earl das: Yaglom (948): eqs. for -point correlations van Driest (95): transformation Kovaszna (953): ep., modes of compressible turbulence Morkovin (96): hpothesis, SRA Still man open questions: Measurement difficult... DNS: important contributions since the 80s Literature: Revie articles: Bradsha (977), Lele (994) Agardographs: Fernholz & Finle (977) Books: Smits & Dussauge (006), Chassaing et al. (00), Gatski & Bonnet (009), Garnier, Sagaut, Adams (009) G. Comte-Bellot Coll

4 DNS of compressible channel and pipe flo Wh spending time ith so simple flos? Understanding the phsical mechanisms that eplain similarities & differences beteen plane & aismmetric flos forms a first step toards improved turbulence modelling Isolating such mechanisms needs simplifications such as e.g. full-developed flo G. Comte-Bellot Coll

5 Contents DNS of supersonic channel and pipe flo Some computational details Compressibilit effects Comparison of mean flo/renolds stresses Comparison of Renolds stress budgets Analsis of pressure fields G. Comte-Bellot Coll

6 Some computational details h R z L Cartesian/clindrical coordinates Treatment of ais singularit as in Mohseni & Colonius (JCP, 000) Compact 5th order LD upind schemes (Adams-Shariff, JCP, 996) Compact 6th order central schemes (Lele, JCP, 99) 3rd order lo-storage RK scheme L G. Comte-Bellot Coll

7 Some computational details h R L L z Flo Re τ M τ Re m M m L H B Grid channel πh h 4πh/ pipe R R Rπ Reτ = ρuτl μ, Mτ = uτ γrt, Rem = ρmuml μ, l = h,r, ρmum = 0 ρud ( h) G. Comte-Bellot Coll

8 Previous ork h R z L Coleman et al. (JFM 305, 995) Ghosh et al. (IJHFF 9, 008) Huang et al. (JFM 305, 995) Morinishi et al. (JFM 50, 004) Fosi et al. (JFM 509, 004) L G. Comte-Bellot Coll

9 Previous ork h R z L Similarities and differences beteen incompressible channel & pipe flo: Schlichting (968): Similarit beteen velocities not perfect Wosnik et al. (JFM 000): Theor for vel. & skin friction, Re effects Nieustadt & Bradsha (997): Similarit fails beond nd order moments L G. Comte-Bellot Coll

10 Compressibilit effects Supersonic flo: isothermal alls Sharp all-normal gradients of mean densit and temperature: mean propert variations Van Driest transformation is successful. SRA needs modification Mean dilatation effects negligible Pressure-dilatation and compressible dissipation rate (intrinsic compressibilit effects) are unimportant Densit variations responsible for change in pressurestrain correlation and Renolds stress anisotrop G. Comte-Bellot Coll

11 Comparison of mean flo variables Viscous and Renolds shear stresses (normalized ith inner variables) μ μ du d ρu v τ = l = a p Channel Pipe l = h, R, a a =, =, p p = = h R,, p = ρ μ u τ τ dp d, τ = ρ u τ. R : Kármán number l G. Comte-Bellot Coll. 009

12 Comparison of mean flo variables Mean streamise velocit Talor series epansion for viscous sublaer: u ρu v ρu v μ μ σ μ 3 3 = d d d..., σ = μ μ l 6 μ τ A B. Pressure-gradient term looses importance at high Re τ u u μ = d μ 0 AB Incompressible u =... l isothermal flo Differencesin thecoreregion Channel Pipe G. Comte-Bellot Coll. 009

13 Comparison of mean flo variables Mean pressure Integrated all-normal momentum balance: p = p ρv v 0 d ( ρv v ρ ) (pipe), p = p ρv v (channel) ( ) R p p Channel Pipe Differences due to transverse curvature. Wall-normal pressure gradients are small compared to densit & temperature gradients l G. Comte-Bellot Coll

14 Comparison of mean flo variables Mean densit and temperature Integrated mean energ balance (pipe): R ( T T ) d v T γ λ ρ Pr d u T λ ρ τ mol. heat flu turb. heat flu γ Bq { all heat fl. γ R μ du μ d ( γ ) M ε d h.o.t. t 0 direct & turb. dissipation T T Mean mol. heat flu Direct dissipation Channel Pipe Turb. dissipation ρ ρ Turb. heat flu G. Comte-Bellot Coll. 009 l Differences due to transverse curvature... 4

15 Comparison of Renolds stresses ρ v v τ ρ u u τ Channel Pipe = ρ ρ μ μ Similarit close to the all. Wall-normal stress interacts ith the mean pressure. G. Comte-Bellot Coll

16 Streamise Renolds stress balance Terms normalized ith semi-local values τ /μ(t) MF TT PR VD DS PS Similarit close to the all, ecept for DS, VD (different curvature of u rms in -direction) Reduced energ redistribution (PS) & TT in the channel core G. Comte-Bellot Coll

17 RMS vorticit fluctuations Terms normalized ith τ /μ Channel Pipe Different curvature of u rms in -direction close to the all is also reflected in ω = v u z G. Comte-Bellot Coll

18 RMS pressure fluctuations p rms τ Channel Pipe l channel/pipe: compressible, Re τ =45 Wu & Moin (JFM 608, 008) Re τ =80, 4, incompressible pipe Subtle differences beteen channel and pipe flo. Compressibilit reduces pressure fluctuations in the all laer. G. Comte-Bellot Coll

19 Analsis of pressure fluctuations Laplacian of pressure fluctuations in channel & pipe u ~ u p = ρ A R u ~ C u ~ u r p = ρ r A R ρ ( u u u u ) ( u u u u ) ρ i j i j i j A S i j i j i j j D ρ τ Dt i j ij ( ρ u ) ( ρ u u ρ u u ) ρf. u u ru u u u r ϕ ρ B r r r ϕ C C 3 B j V j ρ u u ( u u u u ) ru ru ϕ u ϕu ϕ ( u ϕu ϕ u ϕu ϕ ) r r r r r ρ r r r r ϕ r r r ϕ A S B A S C C -C 3 terms are small in supersonic channel and pipe flo C D ρ τ Dt C 3 V B ρf. G. Comte-Bellot Coll

20 Analsis of pressure fluctuations Variable densit ansatz neglecting ave-propagation effects (Poisson equation for pressure fluctuations in the pipe) FT in homogeneous directions (, φ): d dr pˆ r dpˆ dr k r k ϕ pˆ = ρ fˆ ith b.c. dpˆ dr r = = 4 3 μ d ûr dr dμ dr dû dr r r =. Replace r.h.s b δ(r-r 0 ) and compute the Green function G to obtain the solution of the Poisson equation pˆ ( k, r, k ) ρ( r ) Ĝ ( k,k,r, r ) fˆ ( k,k, r ) r dr Bˆ ( k,k,r) = ϕ 0 ϕ 0 ϕ ϕ The Green function Ĝ and the boundar condition Bˆ depend on modified Bessel functions. G. Comte-Bellot Coll

21 Green function for channel and pipe flo The Green function Ĝ for a point source at /l=0.4 and sets of avenumbers decas faster in the pipe: k l=5, k z l=9 channel k l=5, k z l=5 G. Comte-Bellot Coll. 009

22 Green function solution for Π X-component of pressure-strain correlation for pipe flo: Π u ( ) = p ( r) = ρ ( r ) G f (, r, ϕ;r ) u r 0 0 r0dr0 0 u B Channel Pipe Lines: DNS Contributions of rapid and slo terms all terms Differences are due to different mean densities! G. Comte-Bellot Coll. 009

23 Summarizing remarks Supersonic turbulent channel and pipe flos ere compared at equal friction Renolds and Mach numbers DNS data reveal more differences than similarities beteen both flos Differences in mean propert variations are due to transverse curvature and loose importance as the Renumber increases Differences in mean densit directl affect pressurestrain correlations. Challenge for turbulence modelling! G. Comte-Bellot Coll