2013/5/22 ( ) ( ) ( ) 10.1 Crocco s equation for entropy. 10 Multi-dimensional compressible flows. Consdier inviscid flows with no body force.
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1 lti-dimesiol ompressible flows Assme: irrottiol d ivisid. roo s etio for etropy. Goverig etios for flid motio.3 Smll pertrbtio theory (for stedy d slightly ompressible flow.3. Prdtl-Glert rle (for sbsoi flow.3. Akert s theory (for spersoi flow.3.3 Jze-Ryleigh epsio (slightly ompressible. Prdtl-eyer flow. roo s etio for etropy osdier ivisid flows with o body fore. Eler etio: ( P vetor idetity : P P Rell : dh d e Pd Tds d ( ( P ( ω ( h T s ( h ω T s dp Tds. roo s etio for etropy T s ( h ω h ω h h stgtio ethlpy Disssios: for stedy flows (i log stremlie: T( s h ω h ( s if h ostt log stremlie T (ii h ostt everywhere i the flow T s ω s (isetroi flow if d oly if ω (irrottiol flow. Goverig etios for ivisid d irrottiol flows w/ot body fore (i irrottiol φ, φ veloity potetil (ii ivisid: Eler etio ( ( P φ ( P φ dp ostt φ dp φ φ ostt
2 (iii isetropi flows P P dp d P dp d d d d Rell: Berolli etio φ dp φ φ ostt φ ( φ φ dp φ ( φ φ dp dp d d d t (iv ss oservtio ( or dp P d Rell Eler etio: { ( P } ( ( ( i i j i j i i i ( i i j i j i i i ( ( φ ( φ φ φ ( φ φ φ φ φ φ ( φ φ φ φ φ t s (iompressible flow φ φ φ ( φ φ.3 Smll pertrbtio theory for stedy d slightly ompressible flow φ φ t φ φ φ ( φ φ φ φ φ φ j j j i i j φ φ φ φ j j j i i j φ φ φ φ j j j i i j
3 .3 Smll pertrbtio theory γrt Rell T P ~ for stedy sbsoi d spersoi flows γ γ ~ for reltively sleder bodies h U γ γ U h mber fr from flyig body U the freestrem veloity (the flyig veloity of the body the sod speed fr from body Assme: dibti flow d idel gs h h ostt everywhere h T T T U P P P U γ γ ( γ ( U U γ γ ( γ ( U γ φ φ.3 Smll pertrbtio theory ~ for stedy sbsoi d spersoi flows ~ for reltively sleder bodies Assme: the body ses smll pertrbtio to the free strem. veloity potetil: φ U φ where φ << U ( γ ( eergy oservtio: U U υ w U U γ U.3 Smll pertrbtio theory φ U φ γ U φ φ φ φ φ φ φ j j j i i j j i i j ji j i ( U ( U υ w ( U y z ( U φ υ ( U υ w υ z w ( U υ w w z 3
4 ( U ( U υ w y z υ υ υ ( U w y z w w w w ( U υ w z φ υ υ ( U U ( U U U ( γ υ U γ U w w wu wu υ φ υ U υ U γ U φ U U φ φ φ φ φ z φ φ φ z ( ~ vlid for reltively sleder bodies ~ ellipti for sbsoi flow ~ hyperboli for spersoi flow Pressre oeffiiet (pressre lift d drg P P P U isetropi reltio: γp γp P P P P Rell γ U P / γ γ U P P U ( γ P / γ / γ / γ P P P P P P γ γ / γ P U ( γ P γ U U ( γ γ γ P U γ P U γ P P P P U U γu P U γ P U EXAPLE: D flow over wve-shped wll etio of the wvy srfe: y ( Bodry oditio: π η εsi λ ελ<< υ dy π π ε os d λ λ
5 EXAPLE: D flow over wve-shped wll Bodry oditio: υ υ π π ε os U U λ λ ε π (, η πu os (, λ λ y Remrk: φ φ goverig etio: ( ε π Bs: (, πu os λ λ fiite s y EXAPLE: D flow over wve-shped wll (i < φ φ goverig etio: ( ξ dξ ξ d ξ ξ ξ dξ ξ d φ φ ξ ξ Seprtio of vribles: φ ( ξ, y F( ξ G( y y φ φ ξ d F α F d ξ dg α G dy d F d G dξ dy G ( y F d F d G Fdξ Gdy α ~ trigoometri i ξ d epoetil i y os si αy αy F ξ A αξ B αξ G y e De B: fiite s y, α> y (, y F G( y De α { Aos Bsi} φ ξ ξ αξ αξ y (, y e α { A os B si} φ ξ αξ αξ αy α α φ ( y, e A os B si ε π Bs: (, πu os λ λ αy α α ε π α e A os B si πu os λ λ y ε α π α A πu d B d λ λ 5
6 A B Uε π α λ Uε π π φ (, y os ep y λ λ Uε π π π si ep y λ λ λ π ε for <<, we eed << U λ P symmetri o drg!, π ε π si U λ λ EXAPLE: D flow over wve-shped wll (i > φ φ goverig etio: ( φ φ ( ~ D wve etio (, y f ( y g y φ f : pertrbtios geerted by the wll trvel dowstrem g : pertrbtios geerted by the wll trvel pstrem (physil impossible Veloity ompoets d pressre re ostt log the lies ostt y dy t θ d or si θ ~ lled h lies (, y f ( y f φ ξ ε π Bs: (, πu os λ λ π dξ λ λ y df ε π U os df ε π πu os d λ λ df ( Uε π π os d λ λ Uε π f ( si λ Uε π φ (, y si ( y λ 6
7 Uε π (, y si ( y φ λ ( Uε π π os y λ λ P, ε π π os U λ λ D trsoi theory ivisid lier theory π y η ( εsi λ ~ 9º ot of phse with the wve shpe of the wll wve drg! oliervisos effet.3. Prdtl-Glert rle for sbsoi flow ~ trsformtio tht redes ll sosoi-flow problems to eivlet iompressible-flow problems Sbsoi-flow problems: φ φ ( trsformtio: φ φ η (, φ η φ η y φ Bodry oditios: ( df (, U d fiite s y where y f is the body srfe. trsformtio: (, φ η φ η y η φ df ( (, U η d fiite s y Potetil flow theory 7
8 .3. Akert s theory for spersoi flow > y t : mimm thikess α : gle of ttk h : mimm mber pper srfe : y η lower srfe : y η t ( ( h α (hord pper soltio : (, ( φ y f y lower soltio : (, y g( y φ both trvellig dowstrem.3. Akert s theory for spersoi flow bodry oditios: ( pper solt io : φ, y f y dη pper srfe : (, U d η df d U d d ( df veloity: w (, d pressre oeffiiet: P ( w df U U d ηd d (.3. Akert s theory for spersoi flow bodry oditios: pper srfe :, ( y g( y lower soltio : φ, ( dη U d ( η dg U d d d dg veloity: w (, d pressre oeffiiet: P w dg U U d ηd d ( Lift fore: ( (, (, L P P d U ( ( P, P(, d U d U η η αu η dη d d d { ( ( } ( ( η η η η siα α 8
9 > y Lift oeffiiet: t h α ( ( (hord η η η η siα α L L α U ~ idepedet of mber h d thikess t Drg: α ( (, (, D P P dy dη dη dy d η dη U P (, P(, d d d dη d d dη U d Drg oeffiet: D D U dη d η d d d o the pper srfe o the lower srfe > Defie thikess prmeter: δ t mber prmeter: hlf-thikess ftio: ε h ( lol hlf thikess t lol hlf thikess ( τ δ mber ftio: τ( γ ( lol mber h lol mber ( ε y γ ( (hord > τ δτ( ( τ ( t γ ( γ ( h εγ( α y si α α ( ( ( ( η α εγ δτ η α εγ δτ dη dη d d { ( ( } { ( ( } α εγ δτ α εγ δτ α εγ δτ ( ( ( ( { ( ( ( } α α εγ ε γ δ τ dη dη d d d α αε γ γ ε γ d δ τ d 9
10 D d η d η d d d ( d ( d α ε γ δ τ α ε δ γ ( d τ d ~ Spersoi irfoils shold be s stright s possible d s thi s possible.. Goverig etios for ivisid d irrottiol flows w/ot body fore φ γ φ φ φ φ φ φ j j j i i j ~ Shrp orers re preferble to roded orers..3.3 Jze-Ryleigh Epsio for stedy d slightly ompressible flow Look for soltio for smll <.5 i the followig form: (i Sbstitte it ito φ, yz, U φ ( yz,, U (pertrbtio soltio γ φ φ γ (,, (,, y z y z φ φ ( γ ( yz,, ( yz,, φ φ φ( yz,, φ( yz,, { ( } { ( O O } φ φ φ φ φ φ φ φ φ φ O γ φ( yz,, φ( yz,, ( γ { O( } φ φ φ φ ( γ φ φ ( O(
11 ( γ φ φ ( γ ( O( φ φ ( γ ( O( φ φ ( O( (ii Sbstitte it ito φ, yz, U φ ( yz,, φ φ φ φ j j j i i j j j j i i j φ U φ φ φ U φ φ φ j i i j φ φ φ O( j j j φ φ O j j j j ( γ ( ( φ φ O( φ φ φ φ O( O( j j i i φ φ O i j i j ( thig oeffiiets of like powers: φ :: φ j j φ φ φ φ :: j j j i i j ~ OK for <.5 ~ OK for y shpe of body ( γ φ φ φ φ φ i i i j i j φ φ φ φ φ φ φ φ φ φ i j i j :: j j i j i j i j i j
12 . Prdtl-eyer flows > Prdtl-eyer f very wek, otios epsio wves ~ look for et soltio ~ pressre d veloity ompoets re ostt log h lie ~ ledig h lie β si AB d si ( β θ β θ βθ A d os( β θ dθ dθ d dθ si d B ABsi β θ dθ d β θ dθ A B A B d dθ os β θ d d si β θ si ( β θ d ot β θ d from eergy oservtio: γ γ ( : sod speed t the stgtio stte ( γ ( γ ( γ ( γ ( γ d d γ ( γ d d γ o the other hd, d d d d d ( γ { γ } d ( γ d d d γ { } d {( γ }
13 d dθ d {( γ } ξ dξ dθ ( γ ξ ( γ ( ξ I: θ s (ledig h lie dθξ ξ ξ d ξdξ d ξdξ ξdξ ξ ξdξ {( γ( ξ } ( ξ ( γ dξ γ ξ γ ( ξ dξ ( γ ( ξ ξ ( γ γ γ t γ γ θ ξ ξ t ξ γ γ θ t t γ γ I: θ s (ledig h lie γ γ t t f γ γ Prdtl-eyer ftio θ f ( f ( θ f ( f ( ( i s θ flow elertes s it deflets. π γ ( ii θ m lim θ ( 3 for γ. γ 3
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